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abstract: 'We study gauge theories obtained by adding finite mass terms to Yang-Mills theory. The Maldacena dual is nonsingular: in each of the many vacua, there is an extended brane source, arising from Myers’ dielectric effect. The source consists of one or more $(p,q)$ 5-branes. In particular, the confining vacuum contains an NS5-brane; the confining flux tube is a fundamental string bound to the 5-brane. The system admits a simple quantitative description as a perturbation of a state on the Coulomb branch. Various nonperturbative phenomena, including flux tubes, baryon vertices, domain walls, condensates and instantons, have new, quantitatively precise, dual descriptions. We also briefly consider two QCD-like theories. Our method extends to the nonsupersymmetric case. As expected, the matter cannot be decoupled within the supergravity regime.'
author:
- 'Joseph Polchinski [^1] and Matthew J. Strassler [^2]'
title: ' The String Dual of a Confining Four-Dimensional Gauge Theory'
---
epsf \#1\#2\#3[[Nucl. Phys.]{} [**B\#1**]{} (\#2) \#3]{} \#1\#2\#3[[Phys. Rev.]{} [**D\#1**]{} (\#2) \#3]{} \#1\#2\#3[[Phys. Lett.]{} [**\#1B**]{} (\#2) \#3]{}
Introduction
============
The proposal of ’t Hooft [@tHooftlargeN], that large-$N$ non-abelian gauge theory can be recast as a string theory, has taken an interesting turn with the work of Maldacena [@maldacon]. The principal Maldacena duality applies not to confining theories but to conformal gauge theories, which are dual to IIB string theory on $AdS_5
\times S^5$. Starting with this duality one can perturb by the addition of mass terms preserving a smaller supersymmetry, or none at all, and in this way obtain a confining gauge theory. The problem is that the perturbation of the dual string theory appears to produce a spacetime with a naked singularity [@gppz]. As a consequence, even basic quantities such a condensates are incalculable.
In this paper we show that the situation is actually much better. There is no naked singularity, but rather an expanded brane source, and all physical quantities are calculable. We believe that this is the first example of a dual supergravity description of a four-dimensional confining gauge theory. It is also gives new insight into the resolution of naked singularities in string theory.
We focus on perturbations that preserve supersymmetry, though in fact our solutions are stable under the addition of small additional masses that break the supersymmetry completely. The vector multiplet contains an vector multiplet and three chiral multiplets. We will add finite supersymmetry-preserving masses to the three chiral multiplets. For brevity we will refer to this theory as ‘’. This theory has been studied by many authors [@cvew; @rdew; @mjsnotes; @doreya; @doreykumar]. It is known to have a rich phase structure [@cvew; @rdew; @lattice], which includes confining phases that are in the same universality class as those of pure Yang-Mills theory. We will show that the rich structure of this theory is reflected in supergravity in remarkable ways.
To study pure or Yang-Mills theories would require working at small ’t Hooft coupling and taking the masses of the extra multiplets to infinity. This is not tractable without an understanding of classical string theory in Ramond-Ramond backgrounds at large curvature. At large ’t Hooft coupling, where supergravity is valid, the masses of the extra multiplets must be kept finite. However, we emphasize these multiplets are four-dimensional and the ultraviolet theory is conformal. An alternative approach to obtaining a string dual of confining theories is via high-temperature five-dimensional supersymmetric field theories [@highTa; @highTb; @highTc; @grossooguri], whose low-energy limit is four-dimensional strongly-coupled non-supersymmetric Yang-Mills theory. The dual spacetime is non-singular, and the infrared cutoff provided by the temperature does indeed lead to confinement of electric flux tubes. In this case, however, there is a full set of massive five-dimensional states that do not decouple.
Our work was motivated by the observation of Myers [@myers], that D-branes in a transverse Ramond-Ramond (RR) potential can develop a multipole moment under fields that normally couple to a higher-dimensional brane. This ‘dielectric’ property is analogous to the induced dipole moment of a neutral atom in an electric field. For example, a collection of $N$ D0-branes in an electric RR 4-form flux develops a dipole moment under the corresponding 3-form potential. One can think of them as blowing up into a spherical D2-brane, and in a strong field the latter is the effective description. This happens because the D0-brane coordinates become noncommutative. The original D0-brane charge $N$, which of course is conserved in this process, shows up as a nonzero world-volume field strength on the D2-branes. Even earlier, Kabat and Taylor [@ktsphere] had observed that $N$ D0-branes with noncommuting position matrices could be used to build a spherical D2-brane in matrix theory, generalizing the flat membranes of matrix theory [@bfss]. For finite $N$ the sphere is ‘fuzzy’; or better, perhaps, it is somewhat granular. The equations describing this sphere bear a marked similarity to those which appear in the theory, which were first analyzed in [@cvew].
It is then natural to guess that Myers’ mechanism is at work in this theory. The mass perturbation corresponds to a magnetic RR 3-form flux, which is dual to an electric RR 7-form flux. The latter couples to the D3-brane in the same fashion as the electric 4-form flux does to the D0-brane, and so the D3-branes polarize into D5-branes with world-volume $\RR^4\times S^2$. One difference is that Myers considers D-branes in flat spacetime ($gN$ small), whereas for the gauge/gravity duality the background is $AdS_5 \times S^5$. In Myers’ case a small field produced a small D2-sphere, but in the conformal field theory there is no invariant notion of a small mass perturbation, and on the supergravity side there is no such thing as a small transverse two-sphere. Rather, the D5-spheres, which are dynamically (though not topologically) stable, wrap an equator of the $S^5$. We will show that there exist supergravity solutions in which the only ‘singularity’ is that due to the D5-brane source on the $S^5$.
However, this is far from the whole story. First, the classical theory has many isolated vacua [@cvew]. For each partition of $N$ into integers $n_i$, there must be a separate solution involving multiple D5-branes with D3-branes charges $n_i$, each wrapped on an equator of $S^5$ but at different $AdS$ radii $r_i$ proportional to $n_i$. We will study these vacua, and their properties, in our discussion below. Second, the quantum theory has even more vacua, which are permuted under the duality the field theory inherits from [@rdew]. In particular, the transformation $\tau \to -{1\over\tau}$, which takes the maximally Higgsed vacuum into the confining vacuum, will replace the D5-brane sphere with an NS5-brane sphere: [*this is the effective string description of the confining vacuum.*]{} The confining flux tubes are bound states of a fundamental string to the NS5-brane, or equivalently, instantons of the 5-brane world-volume noncommutative gauge theory. Meanwhile, the leading nonperturbative condensate corresponds to the three-form field generated by the NS5-brane’s magnetic dipole moment.
Our removal of the singularity resembles phenomena that occur on the Coulomb branch [@coul1; @coul2; @coul3; @coul4] and with the repulson singularity that arises in supergravity duals [@jpp]. There are certainly connections which need to be developed further, but the detailed mechanism is different. In particular, the appearance of NS-branes is new. Our result also gives insight into perturbations of the Randall-Sundrum compactification [@RS], and into recent proposals for the solution to the cosmological constant problem [@stanford].
We begin in section II with a review of the classical and quantum field theory vacua, and a discussion of the corresponding brane configurations. In fact, there are more brane configurations than vacua, but later we will argue that only one configuration is applicable for any given value of the parameters. In section III we review perturbations of the $AdS$/CFT duality, with attention to the issue of the naked singularity. We show that there is a small parameter: the system can be regarded as a perturbation of one that has only D3-brane charges. This enables us to obtain a quantitative description even for the rather asymmetric and nonlinear supergravity configuration that results from the expansion of the branes. In section IV we study a simplified calculation, in which $n \ll N$ probe D3-branes are introduced into a fixed background. We find that their potential has minima where they form a D5-brane or NS5-brane, or more generally one or more $(c,d)$ 5-branes, wrapped on an equator of the $S^5$. In section V we consider the case that all $N$ D3-branes expand into 5-branes. Although this substantially deforms the geometry, serendipitous cancellations allow us to find the effective potential in a simple form: it is the same as in the probe case. We discuss the stability of the solution, arguing that it survives even when supersymmetry is broken completely. In section VI we use the dual description to discuss the physics of the gauge theory, including flux tubes and confinement, baryons, domain walls, condensates, instantons, and glueballs. In section VII we briefly discuss extensions, including the case and orbifolds, and in section VIII we discuss implications and future directions.
Ground States
==============
Field Theory Background
-----------------------
In the language of four-dimensional supersymmetry, the theory consists of a vector multiplet $V$ and three chiral multiplets $\Phi_i$, $i=1,2,3$, all in the adjoint representation of the gauge group. In addition to the usual gauge-invariant kinetic terms for these fields, the theory has additional interactions summarized in the superpotential[^3]
$$W = \frac{2\sqrt 2}{g_{\rm YM}^2} \tr ([\Phi_1,\Phi_2]\Phi_3) \ .$$
The theory has an $SO(6)$ $R$-symmetry which is partially hidden by the notation; only the $U(1)$ $R$-symmetry of the supersymmetry and the $SU(3)$ that rotates the $\Phi_i$ are visible. However, if we write the lowest component of $\Phi_i$ as $$\phi_i = \frac{A_{i+3} + i
A_{i+6}}{\sqrt{2}}$$ (the reason for this notation will become evident later), then the potential energy for the scalar fields $A_m$, $m=4,\dots,9$, is explicitly $SO(6)$ invariant: $$V(A_m) \propto
\sum_{m,n=4}^9\tr\left([A_m,A_n][A_m,A_n]\right)\ .
\label{supmass}$$ The theory is conformally invariant, and consists of a continuous set of theories indexed by a marginal coupling $\tau = {\theta\over 2\pi} + i
{4\pi\over g_{\rm YM}^2}$, where $\theta $ and $g_{\rm YM}$ are the theta angle and gauge coupling of the theory.
We can partially break the supersymmetry by adding arbitrary terms to the superpotential. Consider the addition of mass terms $$\label{Wmass}
\Delta W = \frac{1}{g_{\rm YM}^2} (m_1\, \tr\,\Phi_1^2 + m_2\, \tr\,
\Phi_2^2 + m_3
\, \tr\, \Phi_3^2)\ .$$ If $m_1=m_2$ and $m_3=0$ the theory has supersymmetry; otherwise it has . If $m_1=m_2=0$ and $m_3\neq 0$ then the theory flows to a conformal fixed point with a smooth moduli space and duality [@twoadj; @karch; @coul3; @ALIS]. With two nonzero masses, the theory has a moduli space containing special subspaces where charged particles are massless and the Kähler metric is singular. However, in , where all three masses are non-zero, there is no moduli space; the theory has a number of isolated vacua. In the limit i ,m\_i , \^3 = m\_1m\_2m\_3 e\^[2i/N]{} [fixed]{} the theory becomes pure Yang-Mills theory. For gauge group $SU(N)$ the pure theory has $N$ vacua related by a spontaneously broken discrete $R$-symmetry. Note that this $R$-symmetry is not present in ; it is an accidental symmetry present only in the limit [Eq. [(\[nonelimit\])]{}]{}.
The classical vacua were described by Vafa and Witten [@cvew]. Assuming all masses are nonzero, we may rescale the fields $\Phi_i$ so as to make all the masses equal; having computed the vacua in this case one may undo this rescaling. In this case the $F$-term equations for a supersymmetric vacuum read $$[\Phi_i,\Phi_j] = -\frac{m}{\sqrt 2}\epsilon_{ijk} \Phi_k \ .
\label{fterm}$$ Consider the case of $SU(N)$. Recalling that the $\Phi_i$ are $N\times N$ traceless matrices, it is evident that the solutions to these equations are given by $N$-dimensional, generally reducible, representations of the Lie algebra $SU(2)$. The irreducible spin $(N-1)/2$ representation is one solution; $N$ copies of the trivial representation give another ($\Phi_i=0$). Since for every positive integer $d$ there is one irreducible $SU(2)$ representation of dimension $d$, each vacuum corresponds to a partition of $N$ into positive integers: $$\label{partN}
\{k_d \in \ZZ\geq 0\}\ {\rm such\ that\ } \sum_{d=1}^N\ dk_d = N\ ,$$ where $k_d$ is the number of times the dimension $d$ representation appears. The number of classical vacua of the theory is given by the number of such partitions.
Generally, for a given partition, the unbroken gauge group is $\left[\otimes_d U(k_d)\right]/ U(1)$. For example, if $k_d=1$ and $k_{N-d}=1$, then the $\Phi_i$ are block diagonal with blocks of dimension $d$ and ${N-d}$; the diagonal traceless matrix which is ${\bf 1}$ in each block generates an unbroken $U(1)$ gauge symmetry. Clearly we obtain $U(1)^{k-1}$ if there are $k$ such blocks. However, if $k_d=2$, then the two blocks of size $d$ can be rotated into each other by additional generators, giving altogether an $SU(2)$ instead of a $U(1)$. More generally we obtain $SU(k_d)$. Among these vacua there is a unique one which we will call the ‘Higgs’ vacuum, in which the $SU(N)$ gauge group is completely broken. This is the only ‘massive vacuum’ (meaning that it has a mass gap) at the classical level. For each divisor $d<N$ of $N$ we may take $k_d = N/d$ with all others zero, giving a vacuum with a simple unbroken gauge group $SU(N/d)$. All other vacua have one or more $U(1)$ factors; these are ‘Coulomb vacua.’
Quantum mechanically, the story is even richer. Donagi and Witten [@rdew] found an integrable system which permitted them to write the holomorphic curve and Seiberg-Witten form describing the quantum mechanical moduli space of the theory with $m_1=m_2$ and $m_3=0$.[^4] They considered the effect of breaking the supersymmetry to through nonzero $m_3\ll m_1,m_2$, and showed that the theory has a number of remarkable properties. Each classical vacuum which has unbroken gauge symmetry $SU(k)$ splits into $k$ vacua, all of which have a mass gap. (Coulomb vacua with non-abelian group factors split as well, although a complete accounting of these vacua was not given in [@rdew]; since the photons remain massless, such vacua do not have mass gaps.) The vacuum with $SU(N)$ unbroken ($k_1=N$, $\Phi_i=0$) splits into $N$ massive vacua, exactly the number which would be needed in the Yang-Mills theory obtained in the limit [Eq. [(\[nonelimit\])]{}]{}. The massive quantum vacua are those without $U(1)$ factors, and as noted above are associated with the divisors of $N$. Their total number is obviously given by the sum of the divisors of $N$; it therefore depends in an interesting way, one which does not have a large-$N$ limit, on the prime factors of $N$. The number of Coulomb vacua is exponential in $\sqrt{N}$.
Donagi and Witten showed the massive vacua were in a beautiful one-to-one correspondence with the phases of gauge theories classified by ’t Hooft. Let us review this classification [@tHclass]. $SU(N)$ gauge theories with only adjoint matter can be probed by sources which carry electric charges in the $\ZZ_N$ center of $SU(N)$ and magnetic charges in the $\ZZ_N = \pi_1[SU(N)/\ZZ_N]$ which characterizes possible Dirac strings. We may think of these charges as lying in an $N\times N$ lattice, a $\ZZ_N\times \ZZ_N$ group $L$. ’t Hooft showed that the possible massive phases of $SU(N)$ gauge theories are associated to the dimension-$N$ subgroups $P$ of $L$. In each phase, the charges corresponding to the $N$ elements of $P$ are screened, and all others are confined; the flux tubes which do the confining are represented by the elements of $L/P$. For example, if the ordinary Higgs mechanism creates a mass gap, all sources with magnetic charge are confined; the only unconfined elements of $L$ are the $(m,0)$, $m=0,\dots,N-1$. Thus $P$ is generated by the single element $(1,0)$. Every magnetic flux tube carries a $\ZZ_N$ charge $n=0,\dots, N-1$ and confines the sources with charge $(m,n)$ for any $m$. In an ordinary confining vacuum, the roles of $m$ and $n$ are reversed, but otherwise the story is the same. Vacua with oblique confinement are given by groups $P$ generated by $(m,1)$, where $m=0,\dots,N-1$.
More generally, however, the vacua are more complex. As mentioned earlier, each classical vacuum with unbroken $SU(k)$ symmetry splits into $k$ vacua. These vacua correspond to subgroups $P$ generated by $(k,0)$ and $(s,d)$, where $dk=N$ and $s=0,1,\dots, k-1$. This map of vacua to subgroups is one-to-one and onto. Note the Higgs vacuum is the case $d=N$, while the $N$ vacua which survive in the pure Yang-Mills theory are the cases $d=1$ for $ s=0,1,\dots ,N-1$, with $s=0$ being the confining vacuum.
The action of on the massive vacua is then straightforward [@rdew]. The $T$ transformation $\tau\to\tau + 1$ shifts each element $(m,n)$ of the group $L$ to $(m+n\mod N,n)$; all electric charges shift by their magnetic charge, through the Witten effect [@ewtheta]. The $S$ transformation $\tau\to -{1\over \tau}$ reverses electric and magnetic charges [@om]: $(m,n)\to (-n\mod
N,m)$. Thus $S$ and $T$ map $L$ to itself, but act nontrivially on its subgroups $P$. This action then corresponds to a permutation of the massive vacua. In particular, note that the Higgs and confining vacua are exchanged by $S$, while $T$ rotates the confining and oblique confining vacua into each other while leaving the Higgs vacuum unchanged. $S$ and $T$ then generate the entire group and its action on the vacua. The Coulomb vacua have not been fully classified, and the action of on them has not yet been understood.[^5]
We close the discussion of field theory by noting that this theory is very different from Yang-Mills theory in certain respects. (Recently, many of these qualitative points were emphasized in [@doreykumar].) Although it is a four-dimensional theory, it still has massive degrees of freedom (three Weyl fermions and six real scalars in the adjoint representation) with masses of order $m$. These massive states ensure that far above the scale $m$ (actually, as we will see, above $mg_{{\rm YM}}^2N$ in the confining phase) the theory becomes conformal, with gauge coupling $\tau$. The important $\ZZ_{2N}$ non-anomalous $R$-symmetry of the pure Yang-Mills theory, a $\ZZ_2$ of which is unbroken and a $\ZZ_N$ of which permutes the $N$ vacua of the theory, is broken explicitly by the presence of the massive fields. Consequently the confining and oblique confining vacua, although still permuted by $\tau\to\tau + n$ with $n$ an integer, are not related by a discrete $R$-symmetry and are not isomorphic. In particular their superpotentials have different magnitudes and the domain walls between them have a variety of tensions [@doreykumar]. In the limit of [Eq. [(\[nonelimit\])]{}]{}, for fixed $N$, the strong coupling scale and the corresponding gluino condensate, domain wall tension, and string tension are all much below the scale $m$ of the masses, and so the strong dynamics is not affected by the massive fields. However, we want to study the gravity dual of this theory, which requires large $g_{\rm YM}^2 N$. In this limit $\Lambda
= m \exp({-8\pi^2/ g_{\rm YM}^2 N})$ is of order $m$, and so all of the physics of the theory takes place near the scale $m$. We will not find the exponentially large hierarchy expected from dimensional transmutation; this can only be seen at small $g_{\rm YM}^2 N$, outside the supergravity regime.
Brane Representations
---------------------
Consider the Higgs phase, in which $$A_7 = - m L_1\ ,\quad
A_8 = - m L_2\ ,\quad
A_9 = - m L_3\ ,$$ where $L_i$ is the $N$-dimensional irreducible representation of $SU(2)$. The scalars $A_m$ are the collective coordinates of the D3-branes, normalized $x^m = 2\pi\alpha'A_m$ [@dbranes]. These are therefore noncommutative, but lie on a sphere of radius $r = \pi \alpha' m N$ $$x^m x^m = (2\pi \alpha' m)^2 L_i L_i \approx \pi^2 \alpha'^2 m^2 N^2\ .$$ The nonzero commutator of the collective coordinates corresponds to higher-dimensional brane charge, a fact familiar from matrix theory. Specifically [@ktsphere] the D3-branes can be equivalently represented as a single D5-brane of topology $\RR^4\times S^2$, the two-sphere having radius $r$, with $N$ units of world-volume magnetic field on the two-sphere. The Higgs vacuum of the four-dimensional theory is represented by this D5-brane.
Similarly, a vacuum corresponding to the reducible representation $\{
k_d \}$, defined as in [Eq. [(\[partN\])]{}]{}, corresponds to concentric D5-branes, where $k_d$ have radius $\pi\alpha' md$ for each $d$. Consider the case of two spheres, with $k_d = k_{N-d}=1$. If $d \neq
N-d$ then the spheres have different radii; the gauge group of the field theory is $U(1)$. However, if $d = N-d$, the two spheres coincide and the field theory has gauge group $SU(2)$. For $N-2d$ small, the $SU(2)$ is broken at a low scale and its W-bosons have mass proportional to $N-2d$. More generally, $k$ coincident D5-branes correspond to a classical vacuum with $SU(k)$ symmetry. Just as in the case of flat branes with sixteen supercharges, the curved D5-branes with four supercharges and only four-dimensional Lorentz invariance show enhanced gauge symmetry when they coincide, and when separated have W-bosons with masses of order the separation distance.[^6] Each classical vacuum of the theory is given by a set of D5 branes of radius $n_i$, with $\sum_i n_i=N$.
Quantum mechanically the situation is much more complicated. The $S$ transformation $\tau\to -{1\over \tau}$ should exchange the Higgs and confining vacua; therefore by Type IIB duality the confining vacuum is a single NS5-brane. A $T^n$ transformation ($\tau\to\tau+n$) leaves D5-branes unchanged and shifts an NS5-brane to a $(1,n)$ 5-brane. It follows that the $n^{\rm th}$ oblique confining vacuum is given by a $(1,n)$ 5-brane. The $S$-duality implies also that there should be vacua with multiple NS5-branes, or generally $(1,n)$ 5-branes, possibly coincident. In fact we may expect there to be vacua in which different types of 5-brane coexist. For example, suppose we partition $N$ using $k_s=1$ and $k_1= N-s$, so that the lower $(N-s)\times(
N-s)$ block of the fields $\Phi_i$ is zero, leaving $SU(N-s)$ unbroken. In this case we would expect a D5-brane of radius $s$ representing the broken part of the gauge group, and an NS5-brane (or a $(1,q)$ 5-brane) representing an (oblique) confining phase of the unbroken $SU(N-s)$ subgroup.
We will show that all of these brane configurations do indeed appear in the dual of the theory. This is a puzzle, however: the number of brane configurations is much larger than the number of phases. For $N=pq$, for example, the vacuum with $k_p = q$ is described by $q$ D5-branes of radii $\pi \alpha' m p$. In supergravity, this is clearly $S$-dual to the vacuum with $k_q = p$, which is therefore described by $q$ NS5-branes. However, from investigation of the field theory [@rdew], this vacuum also has a description in terms of $p$ D5-branes. We will see, in this and other examples, that our solutions exist only in limited ranges of parameter space, such that only one of the descriptions is valid at a time. Ideally, however, a more complete understanding of how the theory resolves this puzzle would be desirable.
Perturbations on $A{\lowercase{ d}}S_5 \times S^5$
==================================================
In this section we first review deformations of the $AdS$/CFT duality with attention to the issue of singularities, introduce the small parameter that makes the problem tractable, and discuss the field theory perturbation and its supergravity dual. We then give the IIB field equations, develop the necessary tensor spherical harmonics, and solve the field equations to first order in an expansion around $AdS_5 \times S^5$.
$AdS$/CFT and its Deformations
------------------------------
The $d=4$, Yang-Mills theory is dual to IIB string theory on $AdS_5 \times S^5$ [@maldacon]. The Yang-Mills coupling is related to the string coupling by $g_{\rm YM}^2 = 4\pi g$, and the common radius of the two factors of spacetime is $R = (4\pi gN \alpha'^2)^{1/4}$. To each local operator ${\cal O}_i$ of dimension $\Delta_i$ in the CFT corresponds two solutions of the linearized field equations [@witads; @GPK], a nonnormalizable solution which scales as $r^{\Delta_i - 4}$ with $r$ the $AdS$ radius, and a normalizable solution which scales as $r^{-\Delta_i}$. A supergravity solution which behaves at large $r$ as $$a_i r^{\Delta - 4} + b_i r^{-\Delta} \label{linear}$$ is dual to a field theory with Hamiltonian $$H = H_{\rm CFT} + a_i {\cal O}_i\ ,$$ and where the vacuum expectation value (vev) is [@bala] $$\langle 0 | {\cal O}_i | 0 \rangle = b_i\ .$$
We will be interested in relevant perturbations, those with $\Delta <
4$. In the field theory these are unimportant in the UV, while in the IR they become large and take the theory to a new fixed point or produce a mass gap. Correspondingly the perturbation [(\[linear\])]{} is small at large $r$, but at small $r$ it becomes large and nonlinear effects become important.
For a theory with a unique (or at least isolated) vacuum, the dynamics should determine the vev once the Hamiltonian is specified. This is in accord with the general experience with second order differential equations, where some condition of nonsingularity at small $r$ would give one relation for each pair $a_i$ and $b_i$.
Now let us summarize what is known, with attention first to two special cases that make sense:
[**1.**]{} In the theory, $a_i = 0$, it is actually possible to vary the particular $b$ that corresponds to $\cal O$ being a scalar bilinear. The point is that the theory does not have an isolated vacuum, and varying $b$ gives a state on the Coulomb branch. It is important to note that the supergravity solution is still singular, but that the singularity is physically acceptable, corresponding to an extended D3-brane source [@coul1; @coul2; @coul3; @coul4].
[**2.**]{} Certain perturbations give a nontrivial fixed point in the IR. These correspond to supergravity solutions with $AdS$ behavior at large and small $r$, with a domain wall interpolating [@ir1; @ir2; @twoadj; @karch; @ir3]. The vacua do have moduli, but most or all analyses have imposed symmetries which determine a unique vacuum and restrict to a single pair $(a,b)$. In these cases the differential equation does indeed determine $b$. The condition of $AdS$ behavior in the IR gives a boundary condition, which takes the form of an initial condition for damped potential motion.
[**3.**]{} More generally, for perturbations that produce a mass gap and destroy the moduli space, the known solutions are singular for all values of $b_i$ [@gppz1; @gppz]; for a recent discussion see [@gubs]. It does not make sense, however, that such singularities can all be understood as physically acceptable brane or other sources, because that would mean that the vevs are undetermined even though the vacua are isolated. This is another example of the important observation made by Horowitz and Myers in the context of negative mass Schwarzschild [@garyrob]: string theory does not repair all singularities; many singular spacetimes do not correspond to any state in string theory.
We will show that the perturbations corresponding to the masses [(\[Wmass\])]{} actually produce spacetimes with extended brane sources. The spacetime geometry is singular, but in a way that is fixed by the source, and so in particular the values of $b_i$ are determined.
This resembles the case [**1**]{} in that there are extended branes, and could in principle be analyzed by supergravity means as in that case: for some subset of the supergravity solutions the singularity will have an acceptable physical interpretation as a brane source. There has in fact been a search for just such solutions [@priv1; @gubs]; it has thus far been unsuccessful, but some features of our solution have been anticipated. This approach is extremely difficult, and has generally been restricted to special solutions with constant dilaton. In fact, the branes in our solution couple to the dilaton, which is therefore position-dependent.
We are able to treat these rather asymmetric geometries without facing the full nonlinearity of supergravity because of the existence of a small parameter. Consider the case of a single D5-brane with D3-brane charge $N$, wrapped on an equator of the $S^5$. The area of the two-sphere is of order $R^2$, so the density of D3-branes is $$\frac{N}{R^2} \sim \frac{N^{1/2}}{g^{1/2} \alpha'}\ . \label{densest}$$ Under the rather weak condition $N/g \gg 1$, this is large in string units and the effect of the D3-brane charge dominates that of the D5-charge charge.[^7] The system is therefore well approximated by a Coulomb branch configuration of the parent theory, where the general solution is given by linear superposition in the harmonic function. Thus we can work by treating the D5-brane charge, and the 3-form field strengths that are generated by it, as perturbations. It is less obvious, but will be seen in section IV.A, that the full 3-form field strength is effectively proportional to the same small parameter.
For the NS5 solution the corresponding condition is given by $g \to
1/g$ and so $Ng \gg 1$. This is precisely the condition for the gauge theory to be strongly coupled. We then recognize the earlier condition $N/g \gg 1$ as the condition for the dual gauge theory to be strongly coupled. When both of these conditions are satisfied the supergravity description is valid, so the D5 and NS5 solutions are both valid in the entire supergravity regime.
We will begin with a simpler problem, where we place a probe D5-brane of D3-brane charge $n$ into the linearized perturbation of the $AdS_5\times S^5$ background. In this case the condition for the D5-brane solution to be valid is similarly $$\frac{n^2}{gN} \gg 1\ .\label{condit}$$ We will use this condition at several points. In section V.B we will infer that this condition is not just a convenience but in fact a necessity in order for the solution to exist.
Field Equations and Background
------------------------------
The IIB field equations can be derived from the Einstein frame action [@iibact] $$\begin{aligned}
&&\frac{1}{2\kappa^2} \int d^{10}x (-G)^{1/2} R - \frac{1}{4\kappa^2}
\int \biggl( d\Phi \wedge *d\Phi + e^{2\Phi} dC \wedge *dC +
\nonumber\\
&&\qquad g e^{-\Phi} H_{\it 3} \wedge * H_{\it 3} +
g e^{\Phi} \tilde F_{\it 3} \wedge * \tilde F_{\it 3}
+ \frac{g^2}{2} \tilde F_{\it 5} \wedge * \tilde F_{\it 5}
+ g^2 C_{\it 4} \wedge H_{\it 3} \wedge F_{\it 3}\biggr) \ ,\qquad\end{aligned}$$ supplemented by the self-duality condition $$* \tilde F_{\it 5} = \tilde F_{\it 5}\ .$$ Here $$\begin{aligned}
\tilde F_{\it 3}{ &=& }F_{\it 3} - C H_{\it 3}\ ,\quad F_{\it 3}
= d C_{\it 2}\ ,
\nonumber\\
\tilde F_{\it 5}{ &=& }F_{\it 5} - C_{\it 2} \wedge H_{\it 3}\ ,\quad F_{\it 5}
= d C_{\it 4}\ .
\label{fstrens}\end{aligned}$$ We define the Einstein metric by $(G_{\mu\nu})_{\rm Einstein} = g^{1/2}e^{-\Phi/2}
(G_{\mu\nu})_{\rm string}$, so that it is equal to the string metric in this constant background. As a result $g$ appears in the action, explicitly and also through $2\kappa^2 = (2\pi)^7 \alpha'^4 g^2$.
The field equations are [@schwarz] $$\begin{aligned}
\nabla^2 \Phi { &=& }e^{2\Phi} \partial_M C \partial^M C
-\frac{ge^{-\Phi}}{12} H_{MNP} H^{MNP} +\frac{ge^{\Phi} }{12}
\tilde F_{MNP} \tilde F^{MNP} \ ,
\nonumber\\
\nabla^M( e^{2\Phi} \partial_M C) { &=& }- \frac{ge^{\Phi} }{6}
H_{MNP} \tilde F^{MNP} \ ,
\nonumber\\
d{*}(e^{\Phi} \tilde F_{\it 3}) { &=& }gF_{\it 5} \wedge H_{\it 3} \ ,
\nonumber\\
d{*}(e^{-\Phi} H_{\it 3} - C e^{\Phi} \tilde F_{\it 3}) { &=& }- g F_{\it 5}
\wedge F_{\it 3} \ ,
\nonumber\\
d{*}\tilde F_{\it 5} { &=& }- F_{\it 3} \wedge H_{\it 3}\ ,
\nonumber\\
R_{MN} { &=& }\frac{1}{2} \partial_M \Phi \partial_N \Phi +
\frac{e^{2\Phi}}{2} \partial_M C \partial_N C + \frac{g^2}{96}
\tilde F_{MPQRS} \tilde F_N{}^{PQRS}
\nonumber\\
&&\qquad +\frac{g}{4}(e^{-\Phi} H_{MPQ} H_N{}^{PQ} +
e^{\Phi} \tilde F_{MPQ} \tilde F_N{}^{PQ}) \nonumber\\
&&\qquad
- \frac{g}{48} G_{MN} (e^{-\Phi} H_{PQR} H^{PQR} +
e^{\Phi} \tilde F_{PQR} \tilde F^{PQR})\ . \label{feldeq}\end{aligned}$$ We use indices $M,N,\ldots$ in ten dimensions. The Bianchi identities are $$\begin{aligned}
d\tilde F_{\it 3}{ &=& }- dC \wedge H_{\it 3}
\nonumber\\
d\tilde F_{\it 5}{ &=& }- F_{\it 3} \wedge H_{\it 3}\ .\end{aligned}$$
One class of solutions is $$\begin{aligned}
ds^2 { &=& }ds^2_{\rm string} =
Z^{-1/2} \eta_{\mu\nu} dx^\mu dx^\nu + Z^{1/2} dx^m
dx^m\ ,\nonumber\\
\tilde F_{\it 5} { &=& }d\chi_{\it 4} + * d\chi_{\it 4}\ ,\quad
\chi_{\it 4} = \frac{1}{gZ} dx^0 \wedge dx^1
\wedge dx^2
\wedge dx^3\ , \nonumber\\
e^\Phi { &=& }g \ ,\quad C = \frac{\theta}{2\pi}\ ,\label{backg}\end{aligned}$$ with $g$ and $\theta$ constant and other fields vanishing. Here $\mu,\nu = 0,1,2,3$, and $m,n = 4,\ldots,9$. Also, $Z$ is any harmonic function of the $x^m$, $\partial_m \partial_m Z = 0$. For $AdS_5
\times S^5$, $$Z = \frac{R^4}{r^4}\ ,\quad R^4 = 4\pi g N\alpha'^2\ .$$ This fails to be harmonic at the origin, but this is a horizon, dual to a D3-brane source at the origin. More generally a nonharmonic $Z$ corresponds to a distributed D3-brane source.
We will need to expand the field equations around this solution. The equations for linearized $F_{\it 3}$ and $H_{\it 3}$ perturbations are conveniently written in terms of $$G_{\it 3} = F_{\it 3} - \hat\tau H_{\it 3}\ .$$ Here $$\tau = C + i e^{-\Phi}\ ,$$ and a $\hat{}$ denotes unperturbed fields, so that $$\hat\tau = \frac{\theta}{2\pi} + \frac{i}{g}\ .$$ The linearized field and Bianchi equations in a general background are $$\begin{aligned}
d \hat{*} G_{\it 3} + i G_{\it 3} \wedge \hat{\tilde F}_{\it 5} { &=& }0\ ,
\nonumber\\
d G_{\it 3} { &=& }0\ .\end{aligned}$$ We will only be interested in the transverse ($mnp$) components of $G_{\it 3}$. For the background (\[backg\]) and a transverse 3-form field, $$\hat{*} G_{\it 3} = Z^{-1} {*_6} G_{\it 3} \wedge
dx^0 \wedge dx^1 \wedge dx^2 \wedge dx^3 \ .$$ where the dual $*_6$ acts in the six-dimensional transverse space with respect to the flat metric $\delta_{mn}$. Then, in the solution (\[backg\]) with general $Z$, the field equation for a transverse 3-form field can be written simply as $$d [ Z^{-1} ( *_6 G_{\it 3} - i G_{\it 3})] = 0\ . \label{field}$$
The duality of the field strengths implies that the 7-form field strength is $$- {*} \tilde F_{\it 3}
= dC_{\it 6} - H_{\it 3} \wedge C_{\it 4}\ . \label{f7}$$ This is parallel in form to the other field strengths [(\[fstrens\])]{}. The relative sign of the two terms on the right can be deduced by noting that the D5-brane action, which we will write in section IV.A, and the field strength are both invariant under $\delta C_{\it 4} =
d\chi_{\it 3}$ provided that $\delta C_{\it 6} = - H_{\it 3} \wedge
\chi_{\it 3}$. The relative sign of the two sides is obtained by acting with $d$ and comparing with the field equation [(\[feldeq\])]{}.
For the 6-form we write $$d(B_{\it 6} - \hat\tau C_{\it 6}) = \frac{i}{g} {*}G_{\it 3} + C_{\it 4}
\wedge G_{\it 3}\ .\label{a6a}$$ The imaginary part of this equation is just [Eq. [(\[f7\])]{}]{}, while the real part defines $B_{\it 6}$; the meaning of $B_{\it 6}$ will become clear in the section IV.B. For the background [(\[backg\])]{} this becomes $$d(B_{\it 6} - \hat\tau C_{\it 6}) = \frac{i}{gZ} ({*_6}G_{\it 3} - i
G_{\it 3}) \wedge dx^0 \wedge dx^1 \wedge dx^2 \wedge dx^3
\ .\label{a6b}$$
Fermion Masses and Tensor Spherical Harmonics
---------------------------------------------
The theory has Weyl fermions $\lambda_\alpha$ transforming as a [**4**]{} of the $SO(6)$ $R$-symmetry. We will add a mass term $$m^{\alpha\beta} \lambda_\alpha \lambda_\beta + {\rm h.c.}$$ (spinor indices suppressed), which we can assume to be diagonal, $m^{\alpha\beta} = m_\alpha \delta^{\alpha\beta}$. When one of the masses, say $m_4$, vanishes, the Hamiltonian has an supersymmetric completion, as given by the superpotential [(\[supmass\])]{}. The fermion $\lambda_4$ is then the gluino.[^8]
The fermion bilinear transforms as the $({\bf 4} \times {\bf 4})_{\rm
sym} = {\bf 10}$ of $SO(6)$, and the mass matrix as the ${ \hspace{1pt}\overline{\hspace{-1pt}\bf 10
\hspace{-1pt}}\hspace{1pt} }$. The $\bf 10$ and ${ \hspace{1pt}\overline{\hspace{-1pt}\bf 10
\hspace{-1pt}}\hspace{1pt} }$ are imaginary-self-dual antisymmetric 3-tensors, $$*_6 T_{mnp} \equiv \frac{1}{3!} \epsilon_{mnp}{}^{qrs} T_{qrs} = \pm i
T_{mnp}
\ ,\label{dual}$$ with $+$ for the ${\bf 10}$ and $-$ for the ${ \hspace{1pt}\overline{\hspace{-1pt}\bf 10
\hspace{-1pt}}\hspace{1pt} }$. The indices again run from 4 to 9.
To relate the fermion mass to a tensor, it is convenient to adopt complex coordinates $z^i$: $$z^1 = \frac{x^4 + i x^7}{\sqrt 2}\ ,\quad z^2 = \frac{x^5 + i x^8}{\sqrt 2}\
,\quad z^3 = \frac{x^6 + i x^9}{\sqrt 2}\ .$$ Under a rotation $z^i \to e^{i\phi_i} z^i$ the spinors in the [**4**]{} transform $$\begin{aligned}
\lambda_1 { &\to& }e^{i(\phi_1 - \phi_2 - \phi_3)/2} \lambda_1\ , \nonumber\\
\lambda_2 { &\to& }e^{i(-\phi_1 + \phi_2 - \phi_3)/2} \lambda_2\ , \nonumber\\
\lambda_3 { &\to& }e^{i(-\phi_1 - \phi_2 + \phi_3)/2} \lambda_3\ , \nonumber\\
\lambda_4 { &\to& }e^{i(\phi_1 + \phi_2 + \phi_3)/2} \lambda_4\ .\end{aligned}$$ From this it follows that a diagonal mass term transforms in the same way as the form $$T_{\it 3} = m_{1} dz^1 \wedge d\bar z^2 \wedge d\bar z^3 +
m_{2} d\bar z^1 \wedge dz^2 \wedge d\bar z^3 +
m_{3} d\bar z^1 \wedge d\bar z^2 \wedge dz^3 +
m_{4} dz^1 \wedge dz^2 \wedge dz^3 \ . \label{tenso}$$ In language, $m_{4}$ is a gluino mass and the other $m_{\alpha}$ are chiral superfield masses. In the supersymmetric case the nonzero components are $$T_{1\bar2\bar3} = m_1\ ,\quad
T_{\bar1 2\bar3} = m_2\ ,\quad
T_{\bar1\bar2 3} = m_3 \label{susyt}$$ and permutations, and in the equal-mass case $$T_{\bar \imath\bar \jmath k} =T_{i\bar \jmath \bar k} =
T_{\bar \imath j\bar k} = m
\epsilon_{ijk} \ . \label{tensor}$$ These satisfy $*_6 T = -iT$.
One might guess, correctly, that the fermion mass is associated with the lowest spherical harmonic of the field $G_{\it 3}$ [@witads; @KRv; @gunmar]. To make a 3-tensor field transforming in the same way as any given tensor $T$, we can use the constant $T$ itself, or combine it with the radius vector to form $$V_{mnp} = \frac{x^q}{r^2} ( x^m T_{qnp} + x^n T_{mqp} + x^pT_{mnq})$$ where $r^2 = x^m x^m$. Define the forms $$\begin{aligned}
T_{\it 3} { &=& }\frac{1}{3!} T_{mnp} dx^m\wedge dx^n\wedge dx^p \ ,\quad
V_{\it 3} = \frac{1}{3!} V_{mnp} dx^m\wedge dx^n\wedge dx^p \ ,
\nonumber\\
S_{\it 2} { &=& }\frac{1}{2} T_{mnp} x^m dx^n\wedge dx^p\ .\end{aligned}$$ One then finds $$\begin{aligned}
dS_{\it 2} { &=& }3T_{\it 3}\ ,\quad d(\ln r)\wedge S_{\it 2} = V_{\it 3}\ ,
\quad
d(r^p S_{\it 2}) = r^p(3 T_{\it 3} + p V_{\it 3})\ ,\nonumber\\
dT_{\it 3} { &=& }0\ ,\quad dV_{\it 3} = -3 d(\ln r) \wedge T_{\it 3}\ ,\end{aligned}$$ and $${*_6} T_{\it 3} = \pm i T_{\it 3}\ ,\quad
{*_6} V_{\it 3} = \pm i (T_{\it 3}-V_{\it 3})\ . \label{uvdual}$$
Linearized Solutions
--------------------
We specialize to perturbations on the $AdS_5 \times S^5$ case $Z = R^4
/r^4$, which is invariant under the transverse $SO(6)$. This will be applicable to the probe calculation of the next section. A general form for the perturbation is $$G_{\it 3} = r^p (\alpha T_{\it 3} +
\beta V_{\it 3} )\ ,$$ where for now we take $T$ to be an arbitrary constant tensor in the [**10**]{} or ${ \hspace{1pt}\overline{\hspace{-1pt}\bf 10
\hspace{-1pt}}\hspace{1pt} }$. The Bianchi identity gives $$0 = dG_{\it 3} = (p\alpha - 3\beta) d(\ln r) \wedge S_{\it 2}
\ \Rightarrow\ \beta = p\alpha/3\ ,$$ corresponding to $$G_{\it 3} = (\alpha/3) d ( r^p S_{\it 2})\ .$$ Using the duality properties [(\[uvdual\])]{} we then have $${*_6} G_{\it 3} - i G_{\it 3}=
-ir^p (\alpha/3)
[ (3 \mp p \mp 3) T_{\it 3} + (p\pm p) V_{\it 3} ]\ ,$$ and so the equation of motion [(\[field\])]{} gives $$p^2 -10 p + (12 \mp 12) = 0\ .$$
For the lower sign, the ${ \hspace{1pt}\overline{\hspace{-1pt}\bf 10
\hspace{-1pt}}\hspace{1pt} }$, there are two solutions: $$\begin{aligned}
p { &=& }-4\ ,\quad G_{\it 3} = \alpha r^{-4} (T_{\it 3} - 4V_{\it 3}/3)
\ ,\nonumber\\
p { &=& }-6\ ,\quad G_{\it 3} = \alpha r^{-6} (T_{\it 3} - 2V_{\it 3})\ .
\label{perts}\end{aligned}$$ In interpreting these, note that a factor $Z^{-3/4} = (r/R)^3$ must be included to translate the tensors to an inertial frame. These solutions then have the falloffs appropriate to the nonnormalizable and normalizable solutions for a operator of $\Delta=3$. The former thus corresponds to the perturbation of $m$, and the latter to the vev of $\bar\lambda\bar\lambda$. The mass perturbation therefore corresponds at first order to $$G_{\it 3} = \frac{\zeta}{g}
\biggl( \frac{R}{r} \biggr)^4 (T_{\it 3} - 4V_{\it 3}/3)
= d\Biggl[ \frac{\zeta}{3g}
\biggl( \frac{R}{r} \biggr)^4 S_{\it 2} \Biggr]
\label{first}$$ with $T_{\it 3}$ given in [Eq. [(\[tenso\])]{}]{}. The factors of $R$ are necessary for the dimensions, and the factor of $g^{-1}$ arises from the overall $g_{\rm YM}^{-2}$ in the superpotential. The numerical coefficient $\zeta$ appearing in the relation between the fermion bilinear and the supergravity field will eventually be determined to take the value $\zeta
= -3\sqrt 2$. Note also that as a consequence of the equation of motion [(\[field\])]{}, $$Z^{-1} ({*_6} G_{\it 3} - i G_{\it 3}) = \frac{2i\zeta}{3g} T_{\it 3}
= \frac{2i\zeta}{9g} d S_{\it 2}
\label{dufirst}$$ is exact.
For fields in the ${\bf 10}$, the upper sign, there are again two solutions: $$\begin{aligned}
p { &=& }0\ ,\quad G_{\it 3} = \alpha T_{\it 3}
\ ,\nonumber\\
p { &=& }-10\ ,\quad G_{\it 3} = \alpha
r^{-10} (T_{\it 3}- 10 V_{\it 3}/3)\ .\end{aligned}$$ The first of these corresponds to the coefficient of $\bar\lambda\bar \lambda
F^2$, and the second to the vev of $\lambda \lambda F^2$.
Five-brane Probes
=================
In this section we consider probes in the background given by $AdS_5
\times S^5$ plus the linear $G_{\it 3}$ perturbation. The probes are 5-branes with world-volume $\RR^4 \times S^2$ and D3-brane charge $n
\ll N$, with $n\gg \sqrt{gN}$. We consider first D5-brane probes, and then use duality to extend to a general $(c,d)$ 5-brane. For all such probes we find that there is a supersymmetric minimum at nonzero $AdS$ radius $r$.
The D5 Probe Action
-------------------
The relevant terms in the action for a D5-brane are [@lido; @dbranes; @bergd] $$S = -\frac{\mu_5}{g}
\int d^6\xi\, \Bigl[-\det (G_\parallel)
\det(g^{-1/2}e^{\Phi/2}G_\perp + 2\pi\alpha'{\cal F})\Bigr]^{1/2}
+ \mu_5\int ( C_{\it 6} + 2\pi\alpha'{\cal F}_{\it 2} \wedge
C_{\it 4} )
\ , \label{d5act}$$ where $$2\pi\alpha'{\cal F}_{\it 2} = 2\pi\alpha'F_{\it 2} - B_{\it
2}\ .$$ Here $G_\parallel$ is the metric in the $\RR^4$ directions of the world-volume and $G_\perp$ is the metric in the $S^2$ directions, pulled back from spacetime. It is convenient to note that $\det G_\parallel=Z^{-2}$ and that $\det{\cal F} = \half {\cal
F}_{ab}{\cal F}^{ab}\det G_\perp$.
The D3-brane charge of the probe is $n$, so that $$\int_{S^2} F_{\it 2} = 2\pi n\ . \label{quant}$$ This is assumed in this section to be small compared to $N$ so that the effect of the probe on the background can be ignored. If the internal directions are a sphere, rotational symmetry and the quantization (\[quant\]) give $F_{\theta\phi} = \frac{1}{2} n
\sin\theta$, or $F_{ab} F^{ab} = n^2/2Zr^4$.
Let us first consider the action in the absence of the $G_{\it 3}$ background so in particular ${\cal F}_{ab} = F_{ab}$. The first term in the Born-Infeld action is dominated by the second, since for $Z=R^4/r^4$ $$4 \pi^2 \alpha'^2 F_{ab} F^{ab} =
2 \pi^2 \alpha'^2 n^2/2R^4
\sim n^2/gN \gg 1$$ That the field strength dominates reflects the physical input that the D3-brane charge dominates. It is then useful to write $$\begin{aligned}
\sqrt{\det(G_\perp + 2\pi\alpha'{ F})}
{ &=& }2\pi\alpha'\sqrt{\det{ F}}
\left[1 + {1\over(2\pi\alpha')^2 { F}_{ab}{ F}^{ab}}\right]
\nonumber\\
{ &=& }2\pi\alpha'\sqrt{\det{ F}} +
{\det G_\perp \over4\pi\alpha'\sqrt{\det { F}}}\ .\end{aligned}$$ If the D5-brane is a sphere in the $x_\perp$ directions, then in spherical coordinates $\det G_\perp = Zr^4\sin^2 \theta$. Since a D3-brane probe feels no force from D3-branes, there is a large cancellation between the Born-Infeld and Chern-Simons terms. The leading nonvanishing term in the D5-action gives a potential density of the form $$\frac{\mu_5}{g} \int_{S^2} d^2 \xi \,
{\sqrt{\det G_\parallel} \det G_\perp \over
4\pi \alpha' \sqrt{\det { F}}}
= \frac{\mu_5}{g} \int_{S^2} d\cos\theta\, d\phi\,{r^4\over
2\pi \alpha' n} =\frac{\mu_5}{g} \frac{2r^4}{n\alpha'}\ ,
\label{quarticnew}$$ where in the last two equations we have assumed the 5-brane is a two-sphere in the $x_\perp$ directions. Notice the $Z$ factors cancel explicitly; if the metric takes the form in [Eq. [(\[backg\])]{}]{}, the energy density of the 5-brane goes as $r^4$. This is consistent with the fact that the D3-branes see this energy as coming from the square of a commutator term, $([\Phi,\Phi^\dagger])^2$.
Now let us add the perturbation back in. For the linear perturbation, [Eq. [(\[first\])]{}]{} immediately gives the potentials (up to an irrelevant gauge choice) as $$C_{\it 2} - \hat\tau B_{\it 2} = \frac{\zeta}{3g} \biggl( \frac{R}{r}
\biggr)^4 S_{\it 2} \ . \label{potent}$$ For the 6-form, Eqs. [(\[dufirst\])]{} and [(\[a6b\])]{} then give $$C_{\it 6} = \frac{2\zeta}{9g}
dx^0 \wedge dx^1 \wedge dx^2 \wedge dx^3 \wedge {\rm Im}( S_{\it 2})
\label{dupotent}
\ ,$$ up to gauge choice.
The effect of $B_2$ in the D5-brane action is subleading and can be ignored. Using the flux [(\[quant\])]{} and the potential [(\[potent\])]{}, one finds the ratio of the two terms in ${\cal F}_{ab}$ is $${B_{ab}}/{2\pi\alpha' F_{ab}} \sim \frac{mR^4}{r^3} \biggl/
\frac{\alpha' n}{r^2} \sim \frac{mgN\alpha'}{nr}\ . \label{terms}$$ Looking ahead, the minimum of interest is located at $$r \sim m n \alpha' \ ,\label{rapp}$$ and so the ratio [(\[terms\])]{} becomes $gN / n^2$ which is just the small parameter. Thus, at the $AdS$ radii [(\[rapp\])]{} or greater, the field strength term in ${\cal F}_{ab}$ dominates: ${\cal F}_{ab}
\approx { F}_{ab}$. The cancellation between the Born-Infeld and Chern-Simons terms is unaffected; $B$ need merely be inserted in [Eq. [(\[quarticnew\])]{}]{}, where it is negligible.
Inserting the perturbed $C_{\it 6}$ from [Eq. [(\[dupotent\])]{}]{} into the D5-brane action gives an additional potential density $$-\frac{\Delta S}{V} = -\frac{\mu_5}{g}
\int_{S^2} \frac{2\zeta}{9}
{\rm Im}( S_{\it 2})
\ .\label{cubicnew}$$ which is cubic in $r$, linear in $m$, and independent of $Z$.
The two terms in Eqs. [(\[quarticnew\])]{} and [(\[cubicnew\])]{} can be identified with the quartic $\phi^4$ and cubic $m\phi^3$ terms in the supersymmetric potential, as we will see in more detail in section IV.C. For consistency we must also keep the term of order $m^2\phi^2$. This arises from the second-order perturbations of the dilaton, metric, and four-form potential. In fact, supersymmetry makes it possible to write the second-order term in the potential directly: $$\begin{aligned}
-\frac{S}{V} { &=& }\frac{\mu_5}{g}\Biggl\{ \int_{S^2} d^2 \xi \,
{\sqrt{\det G_\parallel} \det G_\perp \over
4\pi \alpha' \sqrt{\det F}}
- \frac{\zeta}{9}\int_{S^2}
{\rm Im}( T_{mnp} x^m dx^n\wedge dx^p)
\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad
+ \frac{\pi\alpha'\zeta^2}{18}
T_{i\bar \jmath \bar k}{ \hspace{1pt}\overline{\hspace{-1pt}T
\hspace{-1pt}}\hspace{1pt} }_{\bar l j k} \int_{S^2} F_{\it 2} z^i \bar
z^{\bar l}
\Biggr\}
\ . \label{pot2}\end{aligned}$$ The form of this term is readily understood. The integral $\int_{S^2} F_{\it
2}$ essentially sums over D3-branes, while the tensor structure gives the scalar mass $\sum_i |m_i|^2 |\phi_i|^2$. The coefficient will be deduced in section IV.C.
Before we go on, let us address two puzzles. The first is the expansion around $AdS_5 \times S^5$, and why we need to keep terms precisely through second order. A measure of the square of the size of the perturbation is the ratio of the energy density in the perturbation $|F_{\it 3}|^2$ with that in the unperturbed $|F_{\it 5}|^2$: $${|F_{\it 3}|^2}/{|F_{\it 5}|^2} \sim
\frac{m^2 R^2}{g^2 r^2} \biggl/\frac{1}{ g^2 Z^{1/2} r^2}
\sim \frac{m^2 gN\alpha'^2}{r^2} \sim \frac{gN}{n^2}\ , \label{enrat}$$ which is the controlling small parameter, basically the effective ratio of brane charge densities $\sigma_5^2/\sigma_3^2$. The three terms in the potential [(\[pot2\])]{} are respectively of zeroth, first, and second order in the perturbation. The zeroth order term is the remainder after cancellation between the Born-Infeld and Chern-Simons terms, and, since the D5 and D3 tensions add in quadratures, is of order $$\sqrt{\sigma_5^2 + \sigma_3^2} - \sigma_3 \sim \frac{\sigma_5^2}{\sigma_3}\
.
\label{magnit}$$ The linear perturbation is of order $\sigma_5/\sigma_3$ and couples to $\sigma_5$, so the first order term is again of magnitude [(\[magnit\])]{}. The second order perturbation is felt by the D3-branes and so this term is of order $\sigma_3
(\sigma_5/\sigma_3)^2$, again the same. Note that this analysis does not use supersymmetry, and so will apply to the case as well.
The second puzzle is that the second order term in the potential [(\[pot2\])]{} makes reference to complex coordinates in spacetime, and these are not intrinsic. In particular, when all four fermion masses are nonvanishing () there is no special complex structure. The point[^9] is that the supergravity equations have [*homogeneous*]{} second order solutions, corresponding to the traceless scalar bilinear $A_m A_n - \frac{1}{6} \delta_{mn} A_pA_p$. The coefficients of these solutions are determined by boundary conditions, so the inhomogeneous solution with $(G_{\it 3})^2$ as source determines only the trace part $A_m A_m$. Thus, the general form for the second order term, not imposing supersymmetry, is given by replacing $$T_{i\bar \jmath \bar k}{ \hspace{1pt}\overline{\hspace{-1pt}T
\hspace{-1pt}}\hspace{1pt} }_{\bar l j k} z^i z^{\bar l} \to
T_{mnp} { \hspace{1pt}\overline{\hspace{-1pt}T
\hspace{-1pt}}\hspace{1pt} }_{mnp} \frac{r^2}{18} + \mu_{mn} x^m x^n
\label{ambig}$$ with arbitrary traceless $\mu_{mn}$. Note that both $T_{mnp}$ and $\mu_{mn}$ are intrinsic (determined by the boundary conditions).
The $(c,d)$ Probe Action
------------------------
A given background can also be given in an $S$-dual description, $$\tau' = \frac{a \tau + b}{c\tau + d}\ .$$ Specifically, $$\begin{aligned}
g' { &=& }g |M|^2\ ,\quad G_{MN}' = G^{\vphantom{\prime}}_{MN} |M|\ ,
\quad C_{\it 4}' = C^{\vphantom{\prime}}_{\it 4}\ ,\nonumber\\
G_{\it 3}' { &=& }G^{\vphantom{\prime}}_{\it 3} M^{-1}\ ,
\quad B_{\it 6}' -\hat\tau' C_{\it 6}' = (B_{\it 6} -\hat\tau C_{\it 6})
M^{-1}\ ,
\label{doof}\end{aligned}$$ where $M = c\tau + d$. A D5-brane in the primed description has the action $$-{S} = \mu_5\int d^4x \Biggl\{ \frac{1}{g'}\int_{S^2} d^2 \xi \,
{\sqrt{\det G'_\parallel} \det G'_\perp \over
4\pi \alpha' \sqrt{\det F}}
- \int_{S^2} C'_{\it 6} + O(T^2)
\Biggr\}
\ .$$ Under the duality (\[doof\]), this translates into $$\begin{aligned}
-\frac{S}{V} { &=& }\frac{\mu_5}{g}\Biggl\{ |M|^2 \int_{S^2} d^2 \xi \,
{\sqrt{\det G_\parallel} \det G_\perp \over
4\pi \alpha' \sqrt{\det F}}
- \frac{\zeta}{9} \int_{S^2} {\rm Im}({ \hspace{1pt}\overline{\hspace{-1pt}M
\hspace{-1pt}}\hspace{1pt} } T_{mnp} x^m dx^n\wedge dx^p)
\nonumber\\ &&\qquad\qquad\qquad\qquad
+ \frac{\pi\alpha'\zeta^2}{18}
T_{i\bar \jmath \bar k}{ \hspace{1pt}\overline{\hspace{-1pt}T
\hspace{-1pt}}\hspace{1pt} }_{\bar l j k} \int F_{\it 2} z^i \bar
z^{\bar l}
\Biggr\}
\ .\label{cdact}\end{aligned}$$
The probe couples to $$C'_{\it 6} = -g' {\rm Im}(B'_{\it 6} - \hat\tau C'_{\it 6})
= -g{\rm Im}({ \hspace{1pt}\overline{\hspace{-1pt}M
\hspace{-1pt}}\hspace{1pt} } [B_{\it 6} - \hat\tau C_{\it 6}])
= B_{\it 6} c + C_{\it 6} d\ . \label{bccd}$$ This is the coupling of a $(c,d)$ 5-brane, a bound state of $c$ NS5-branes and $d$ D5-branes. In the first term of the potential, the factor $|c\tau + d|^2$ is the tension-squared of the $(c,d)$ 5-brane, squared from the addition in quadratures in the Born-Infeld term. The second is the coupling to the background [(\[bccd\])]{}. The final term has again been added by hand in the form required by supersymmetry, which is in fact independent of $(c,d)$. This is because it is the interaction of the D3-brane charge with the second-order background, and so does not depend on the 5-brane quantum numbers. The duality transformation only gives relatively prime $(c,d)$, but the result holds generally, by superposition.
The Probe Potential and Minima
------------------------------
We now focus on the $SO(3)$-invariant equal-mass case. The general $SO(3)$-invariant brane configuration is $$z^i = z e^i\ ,\quad e^i = { \hspace{1pt}\overline{\hspace{-1pt}e^{i}
\hspace{-1pt}}\hspace{1pt} }\ ,\quad e^i e^i = 1\ .
\label{config}$$ This is a sphere of coordinate radius $|z|/\sqrt{2}$, obtained from the sphere $(x^4)^2 + (x^5)^2 + (x^6)^2 = \frac{1}{2}|z|^2$ by a simultaneous phase rotation of the $z^i$. Rotational symmetry and the quantization (\[quant\]) give $F_{\theta\phi} = \frac{1}{2} n \sin\theta$, or $F_{ab} F^{ab} =
n^2/2Zr^4$. Inserting this configuration into the action (\[cdact\]) gives $$\begin{aligned}
-\frac{S}{V} { &=& }\frac{\mu_5}{g} \Biggl[
\frac{8}{\alpha' n} |M|^2 |z|^4 + \frac{8\pi\zeta}{3} {\rm Im}({ \hspace{1pt}\overline{\hspace{-1pt}M
\hspace{-1pt}}\hspace{1pt} }\bar
z^2 mz) + \frac{2\pi^2 n \alpha' \zeta^2}{9} |m|^2|z|^2
\Biggr] \nonumber\\
{ &=& }\frac{4}{\pi g n} |M \phi^2 + i \zeta m n \phi /12 |^2\ .
\label{finpot}\end{aligned}$$ Here $\phi = z/2\pi\alpha'$ is the normalization of the gauge theory scalar relative to the D3-brane collective coordinate. This is of the form required by supersymmetry; the second order term was normalized to give this result.
For $M = 1$, the D5-brane, we can compare to the classical potential. We can use the Ansatz $$\Phi_i = \frac{2}{n} \Phi L_i\ ,$$ where $\Phi$ is a scalar (not a matrix) complex superfield, so that $\sum_i \Phi_i \Phi_i = \Phi^2{\bf 1}$. The Kähler potential and superpotential are then $$K = \frac{n}{2\pi g} { \hspace{1pt}\overline{\hspace{-1pt}\Phi
\hspace{-1pt}}\hspace{1pt} }\Phi \ , \quad
W = \frac{mn}{4\pi g} \Phi^2 + \frac{i \sqrt{2} }{3\pi g} \Phi^3\ .
\label{KW}$$ The potential then agrees with that found in the brane calculation provided $\zeta = -3 \sqrt 2$. This could be checked by various independent means, such as the fermionic terms in the D3-brane action in a $G_{\it 3}$ background.
Returning to general $M$, there is supersymmetric minimum at $$z = \frac{ \pi \alpha' i m n}{\sqrt 2M}\ .$$ For a D5-brane, $(c,d) = (0,1)$ and $z= {i \pi \alpha' m n/\sqrt 2}$. For illustration let $m$ be real. The $i$ reflects the fact that the two-sphere lies in the 789-directions, where $\tilde F_{\it 3}$ is maximized. For an NS5-brane, taking $C = 0$ for convenience, $z= {\pi \alpha' m g n/\sqrt 2}$. This is smaller by $g$, and lies in the 456-directions where $H_{\it 3}$ is maximized. Note that the potential in each case has another minimum at $z = 0$, where the probe has dissolved into the source branes; our approximation is not valid at $z=0$, but it is valid far enough to show that the potential becomes attractive at small $z$.
We can also introduce several probes of arbitrary types, and each will independently sit at the minimum of its own potential. Note that in the $AdS$ geometry we should not think of these as concentric, but rather arranged along the $AdS$ coordinate $r$ while wrapped at various angles on equators of the $S^5$.
An $S^2$ on $S^5$ can be contracted to a point, but it is energetically unfavorable to do so. The first term in the potential vanishes in this limit (since $\det G_\perp$ goes to zero), and the second does as well, leaving only the positive third term. This is because the pointlike D5-brane retains only its D3 charge, which feels a positive potential.
The Full Problem
================
We now consider the fields and self-energy of the full set of $N$ D3-branes, when these are in the configuration $\RR^4 \times S^2$ (or a sum of several two-spheres) with 5-brane charges. As an intermediate step we consider a probe moving in such a background. One might expect these calculations to be much harder that the previous probe problem, as the symmetry is greatly reduced. Remarkably, however, all of the work has already been done. The expanded brane configuration is reflected in a less symmetric warp factor $Z$, but we will see that this drops out of all terms in the potential.
In this section we also work out the first-order correction to the background. In addition we show that our approximation breaks down close to the 5-brane shell, and give the corrected form.
The Warped Geometry
-------------------
Consider $N$ D3-branes spread on a two-sphere of $AdS$ radius $r_0$ in some 3-plane in the six transverse dimensions. This Coulomb branch background is again of the form (\[backg\]), with the $Z$-factor given by harmonic superposition. The $Z$-factor at any point can depend only on its radii $w$ in the 3-plane and $y$ in the orthogonal 3-plane: $$\begin{aligned}
Z { &=& }\frac{1}{2} \int_{-1}^1 d\cos\theta\, \frac{R^4}{(w^2 + y^2 + r_0^2 -
2r_0 w \cos\theta)^2} \nonumber\\
{ &=& }\frac{R^4}{(y^2 + [w + r_0]^2)(y^2 + [w - r_0]^2)}\ .
\label{zwarp}\end{aligned}$$ This is normalized to agree with the $AdS$ $Z$-factor at large $w,y$. When the D3-brane charge is divided among several two-spheres, then $Z$ is a sum of such terms, with total coefficient $R^4$. At $r=0$ this $Z$ goes to a constant, so for $w,y\ll r_0$ we find flat ten-dimensional spacetime, with no nontrivial topology.
To next order we consider linearized $G_{\it 3}$ fields in this background. The field equation is again $$d [ Z^{-1} ( *_6 G_{\it 3} - i G_{\it 3})] = 0\ , \label{field2}$$ and the Bianchi identity is $dG_{\it 3} = 0$. The origin is now a smooth point and the perturbation will be nonsingular there. It has a specified nonnormalizable behavior at infinity, corresponding to the perturbation of the gauge theory Hamiltonian, and a specified source at the 5-branes. Note that this is a magnetic source, appearing in the Bianchi identity but not the field equation. Note also that $$d {*_6} [ Z^{-1} ( *_6 G_{\it 3} - i G_{\it 3})] =
d [-i Z^{-1} ( *_6 G_{\it 3} - i G_{\it 3})] = 0\ .$$ Thus, the combination $Z^{-1} ( *_6 G_{\it 3} - i G_{\it 3})$ is annihilated by both $d$ and $d {*}_6$. Further, at infinity it approaches the constant value [(\[dufirst\])]{} which is just governed by the boundary condition on the nonnormalizable solution: $$Z^{-1} ( *_6 G_{\it 3} - i G_{\it 3}) \to - i\frac{2\sqrt{2}}{g} T_{\it 3} \ .$$ It follows that it takes this constant value everywhere, independent of the warp factor $Z$ and of the configuration of the brane.
The field $G_{\it 3}$ itself does depend on the brane configuration, and we will determine it in section V.C, but it is not relevant here. The brane dominantly couples only to the integral of the potential $B_{\it 6} - \hat \tau C_{\it 6}$ which is already determined by [Eq. [(\[a6b\])]{}]{} to be independent of $Z$. Thus it too is independent of the brane configuration.
The Potential and Solutions
---------------------------
Let us consider again a probe, but now moving in the warped geometry just described. The potential felt by the D3-brane charge of the probe is again zero, for the usual supersymmetric reasons, so the Born-Infeld and Chern-Simons terms again nearly cancel, leaving behind the first term in the potential [(\[pot2\])]{}. As we noted, this term is independent of $Z$. The second term in the potential comes from the coupling to $B_{\it 6} - \hat \tau C_{\it 6}$, and we have found that this too is independent of $Z$. The third term, given by supersymmetry, must then also be $Z$-independent. Thus, [*a probe feels exactly the same potential in the warped geometry formed by $\RR^4 \times S^2$ sources, as when all the sources are at the origin.*]{}
Now consider the potential felt by the full set of $N$ D3-branes with 5-brane charges. As is familiar from electrostatics, we cannot simply take the coupling of the branes to their self-field. Rather, we must think of dividing them into infinitesimal fractions and assembling the configuration by bringing these together one at a time; in electrostatics this produces the familiar factor of $\frac{1}{2}$. In the present case, however, there is no ‘charging up’ effect because as just shown the potential felt by each fractional ‘probe’ is unaffected by the distribution of the earlier fractions. Thus the potential is the same as in the probe case. If the brane configuration consists of two-spheres of respective D3-charges $n_I$ (with $\sum_I n_I = N$), 5-brane charges $(c_I,d_I)$, and radii and orientations $z_I = 2\pi\alpha'
\phi_I$, the potential is $$-\frac{S}{V} = \sum_I \frac{4}{\pi g n_I} |M_I \phi_I^2 - i m n_I \phi_I /2\sqrt
2 |^2\ .
\label{fullpot}$$ Thus, for every collection of 5-branes of total D3-brane charge $N$ there is a solution with nonzero radii, $$z_I = \frac{\pi\alpha' i m n_I }{M_I \sqrt 2} \ .
\label{finrad}$$ For a D5 sphere this is $AdS$ coordinate radius $r = \pi\alpha' m n_I$. For an NS5-sphere it is $r = \pi\alpha' m gn_I$, smaller by a factor $g$ (when $C=0$).
It is important to check the validity of these solutions. We have already argued that for all $N$ D3-branes in a single D5 or NS5 two-sphere the solution is valid in the entire supergravity regime. Now let us consider the problematic case discussed in section II.B, namely $p$ D5-branes each of charge $q$, which is supposed to represent the same state as $q$ NS5-branes each of charge $p$. For the former solution, each D5-brane has charge $N/p = q$ and so the central condition [(\[condit\])]{} becomes $$\frac{q^2}{gN} = \frac{q}{gp} \gg1\ . \label{dcon}$$ For the NS5-brane solution we can simply interchange $g \leftrightarrow 1/g$ and $p \leftrightarrow q$ via $S$-duality to obtain $$\frac{gp}{q} \gg1\ . \label{nscon}$$ The conditions [(\[dcon\])]{} and [(\[nscon\])]{} are beautifully complementary, so that only one solution is valid at a time. At weak coupling the state is described by a D5-brane and at strong coupling by an NS5-brane.
This example also provides the evidence that the condition [(\[condit\])]{} is a necessity, not a convenience: if the solutions persisted beyond this range there would be too many, as compared to the known vacua of the gauge theory. Thus, we require that for each sphere $$\frac{n_I}{g |M_I|^2} \gg 1\ .$$
It would be extremely interesting to understand the crossover between the D5 and NS5 representations of the above phase. At a minimum this will require the full nonlinear supergravity solutions, but it may involve nonperturbative brane dynamics beyond this. Note that at the crossover coupling the D5 and NS5 two-spheres have the same $AdS$ radii but different and nonoverlapping orientations.
There should be a similar story for the minima of the potential at $\phi = 0$. These are outside the range of validity of the approximation, and should not correspond to true solutions because these would again have no duals in the gauge theory. Rather, a 5-brane at small $\phi$ should transmute into a different kind of 5-brane.
As another example consider the oblique solutions $(c,d) = (1,s)$. The condition that the 5-brane energy density, added in quadratures, be much less than the D3-brane energy density, is \[see [Eq. [(\[enrat\])]{}]{}\] $$\frac{1}{\alpha'} \biggl( \frac{1}{g^4} + \frac{s^2}{g^2} \biggr)^{1/2}
\ll \frac{N^{1/2}}{g^{3/2}\alpha'}\quad \Rightarrow\quad 1+g^2 s^2 \ll gN\ .
\label{pqcon}$$ For small $s$ this is valid in most of the supergravity regime, but for $s \sim
N$ it is valid nowhere. This resolves the overcounting, that $(1,s)$ and $(1,s+N)$ represent the same state. Note that for $s \gg 1$ there is a range of $g$ where supergravity is valid but the $(1,s)$ brane solution is not; the duality (which acts on these vacua in an intricate way) gives other candidate brane configurations.
There is one final issue connected with the stability of the brane solutions. Let us focus on the D5-brane. At opposite points on the two-sphere, the D5 world-volumes are antiparallel. Intuition from flat space D5-branes [@dbranes] would suggest that this configuration is not supersymmetric, but this must be wrong. The supersymmetry transformation related to the D5 charge must be offset by the effect of the background on the much larger D3 charge.
We leave the analysis of supersymmetry for the future, but do address a related point: the self-force of the D5-brane. Again, intuition suggests that there should be an attractive force between opposite sides of the two-sphere, rendering the state unstable, but if the configuration is supersymmetric then this must vanish. Let us see how this works. In the D5-brane action [(\[d5act\])]{}, the strongest couplings to bulk fields are those of the D3-brane charge to $G_\parallel$ and to $C_{\it 4}$. The self-force from these cancels as usual due to the supersymmetry of D3-branes. The next strongest coupling is of the D5-brane charge to $G_{\it 3}$. It is this that might give an attractive force, but in fact it does not: [Eq. [(\[a6b\])]{}]{} shows that the field sourced by the D5-brane does not act back on the D5-brane. The $C_{\it 4}$ background induces mixing between $F_{\it
3}$ and $H_{\it 3}$ in such a way that the self-force cancels for any orientation![^10] Finally, the dilaton and metric couple to the quadrature term; this is second order in $\sigma_5/\sigma_3$, and so the exchange force would be fourth order. In the supersymmetric case this should actually vanish, but because it is in any event small we will not show this. Moreover, even for a nonsupersymmetric perturbation the arguments for the vanishing of the forces from $G_\parallel$, $C_{\it 4}$, and $G_{\it 3}$ continue to hold, so only the small residue from the dilaton and $G_\perp$ remains. This is too small to destabilize the solution, as the potential [(\[fullpot\])]{} is a second order effect.
First Order $G_{\it 3}$ Background
----------------------------------
Here we work out the first order correction to the background, which appears only in the field $G_{\it 3}$. In addition to the earlier result [(\[a6b\])]{}, $$*_6 G_{\it 3} - i G_{\it 3} = -i\frac{2\sqrt{2}}{g} Z T_{\it
3}\ , \label{a62}$$ we have the Bianchi identity with magnetic source, $$dG_{\it 3} = J_{\it 4}\ . \label{biaj}$$ Let us adopt a coordinate system in which the brane is a sphere of radius $r_0$ in the $w^{1,2,3}$ directions and at the origin in the $y^{1,2,3}$ directions. Then $$J_{\it 4} = 4\pi^2\alpha' M \delta^3(y) \delta(w-r_0) dw \wedge d^3y\ ,$$ where $w$ is the radius in the $w$-plane, $d^3 y = dy^1 \wedge dy^2 \wedge
dy^3$, and the factor $4\pi^2\alpha'$ arises as $2 \kappa^2 \mu_5 / g^2$. Note that the quantum numbers $M$ appear in a simple way. In place of [Eq. [(\[a62\])]{}]{}, we can use its exterior derivative, $$d{*_6} G_{\it 3} = i J_{\it 4} -i\frac{2\sqrt{2}}{g} dZ \wedge T_{\it
3}\ . \label{deq}$$ This and the Bianchi identity determine $G_{\it 3}$; they can be solved in terms of potentials.
Write $$G_{\it 3} = {*_6} d\omega_{\it 2} + i d\omega_{\it 2} + d \eta_{\it 2}
\label{gsol}$$ with the gauge choice $$d {*_6} \omega_{\it 2} = d {*_6} \eta_{\it 2} = 0\ .$$ Then $$\begin{aligned}
\partial_m \partial_m \omega_{\it 2} { &=& }{*_6} J_{\it 4}
= \frac{2\pi^2\alpha' M}{r_0} \delta^3(y) \delta(w-r_0)
\epsilon_{ijk} w^i dw^j \wedge dw^k\ , \nonumber\\
\partial_m \partial_m \eta_{\it 2} { &=& }- \frac{2i\sqrt{2}}{g} {*_6} (dZ \wedge
T_{\it 3})
= -\frac{\sqrt{2}}{g} T_{mnp} \partial_m Z dx^n \wedge dx^p\ .\end{aligned}$$ The solutions are $$\begin{aligned}
\omega_{\it 2} { &=& }-\frac{\alpha' M}{4w^3}
\epsilon_{ijk} w^i dw^j \wedge dw^k
\partial_t \biggl( \frac{1}{t} \ln \frac{y^2 + w^2 + r_0^2 + 2 r_0 wt}{y^2
+ w^2 +
r_0^2 - 2 r_0 wt}
\biggr)\biggr|_{t = 1} \ ,
\nonumber\\
\eta_{\it 2} { &=& }\frac{R^4}{8 g r_0 \sqrt{2}} T_{mnp} dx^n \wedge dx^p
\partial_m \biggl( \frac{1}{w} \ln \frac{y^2 + [w+r_0]^2}{y^2 + [w-r_0]^2}
\biggr)\ .
\label{onsol}\end{aligned}$$ These do not seem very enlightening, but we can obtain their forms at large $r^2$: $$\begin{aligned}
\omega_{\it 2} &\approx& -\frac{8\alpha' M r^3_0}{3 r^6}
\epsilon_{ijk} w^i dw^j \wedge dw^k
\ ,
\nonumber\\
\eta_{\it 2} &\approx& -\frac{R^4}{g \sqrt{2}r^4 } T_{mnp} x^m dx^n \wedge
dx^p
\ . \label{onasy}\end{aligned}$$ These scale as the normalizable and nonnormalizable solutions respectively. The latter, $\eta_2$, matches the boundary condition [(\[first\])]{}.
The Near-Shell Solution
-----------------------
Our small parameter guarantees that our solution is good over most of spacetime, but it must break down as we approach the 5-brane shell. The metric in the directions parallel to the 5-brane and orthogonal to the D3-branes expands, diluting the D3-brane charge so that close to the 5-brane it no longer dominates. One also sees this in the ratio of energy densities, where the metric has the same effect. Since this occurs only close to the 5-brane, we can approximate the solution in this region by a flat 5-brane+D3-brane solution. Specializing to $p$ D5-branes,[^11] $$\begin{aligned}
ds^2_{\rm string} { &=& }\frac{\alpha' u}{agp} \Bigl[ \eta_{\mu\nu} dx^\mu dx^\nu
+ h(d\tilde x^4 d\tilde x^4 + d\tilde x^5 d\tilde x^5) \Bigr] +
\frac{\alpha' a gp}{u} (du^2 + u^2 d\Omega_3^2)\ , \nonumber\\
e^{2\Phi} { &=& }g^2 \frac{a^2 u^2}{1 + a^2 u^2}\ ,
\quad ds^2 = g^{1/2}e^{-\Phi/2} ds^2_{\rm string}\ ,
\label{aosj}\end{aligned}$$ where $$h = (1 + a^2 u^2)^{-1}
\ .$$ Let us compare with the near-shell metric based on the harmonic function [(\[zwarp\])]{}, near the point $(w_1, w_2, w_3) =
(0,0,r_0)$: $$ds^2_{\rm string} = \frac{2r_0 \rho}{R'^2} \eta_{\mu\nu} dx^\mu dx^\nu
+ \frac{R'^2}{2r_0 \rho} (dw \cdot dw + dy \cdot dy)\ , \label{mid}$$ where $$\rho^2 = (w_3 - r_0)^2 + y^2\ .$$ We have also defined $R'^4 = 4\pi g n \alpha'^2$ to include the case that the shell does not carry the full D3 charge $N$; we do not assume that $n$ is small. The metrics agree away from the shell, $au \gg 1$, provided that $$u = \frac{\rho}{\alpha'}\ ,\quad
a = \frac{R'^2}{2 g p r_0}\ ,\quad
\tilde x^{4,5} = \frac{R'^8}{16 g^3 p^2 r_0^4 \alpha'^2} w^{1,2}\ .$$
With these identifications, the solution [(\[aosj\])]{} gives the continuation to $au < 1$. As a check, the crossover distance $au = 1$ is $$\rho_{\rm c} = \alpha' a^{-1} = \frac{2 g p r_0 \alpha'}{R'^2}
\sim \frac{p g^{1/2}}{n^{1/2}} r_0 \sim p m (gn\alpha')^{1/2}\ .$$ Thus the shell is indeed thin: $\rho_{\rm c}$ is smaller than the radius $r_0$ by $p(g/n)^{1/2}$, which is precisely our controlling parameter [(\[dcon\])]{} for the D5 solution. As a reminder, $r_0 = m\pi \alpha'n/p$ for this shell. In summary, the components of the metric tangent to the two-sphere, and the dilaton, are multiplied by a factor $
{\rho^2}/(\rho^2 + \rho_c^2),
$ $$\begin{aligned}
ds_{\rm string}^2 { &=& }\frac{2r_0 \rho}{R'^2} \eta_{\mu\nu} dx^\mu dx^\nu
+ \frac{R'^2\rho}{2r_0(\rho^2 + \rho_c^2)}
(dw^1 dw^1 + dw^2 dw^2)
+ \frac{R'^2}{2r_0 \rho} (dw^3 dw^3 + dy \cdot dy)\ , \nonumber\\
e^{2\Phi} { &=& }g^2 \frac{\rho^2}{\rho^2 + \rho_c^2}\ ,
\quad ds^2 = g^{1/2}e^{-\Phi/2} ds^2_{\rm string}\ . \label{dshell}\end{aligned}$$ This interpolates between the D3- and D5-brane metrics.
Similarly for $q$ NS5-branes, the solution interpolates between the D3- and NS5- solutions. The crossover radius is now $$\rho'_{\rm c} = \frac{2 r_0 q \alpha'}{R'^2}
\sim \frac{q}{(gn)^{1/2}} r_0 \sim q m (gn\alpha')^{1/2}\ ,$$ the $AdS$ radius is $r_0 = m\pi \alpha'gn/q$ for this shell, and the solution is $$\begin{aligned}
ds^2_{\rm string} { &=& }\frac{2r_0 (\rho^2 + \rho'^2_{\rm c})^{1/2}}{R'^2}
\eta_{\mu\nu} dx^\mu dx^\nu + \frac{R'^2}{2r_0 (\rho^2 + \rho'^2_{\rm
c})^{1/2}} (dw^1 dw^1 + dw^2 dw^2) \nonumber\\
&&\qquad\qquad
+\frac{R'^2 (\rho^2 + \rho'^2_{\rm c})^{1/2}}{2r_0 \rho^2} (dw^3 dw^3 + dy
\cdot dy)\ , \nonumber\\
e^{2\Phi} { &=& }g^2 \frac{\rho^2 + \rho'^2_c}{\rho^2}\ ,
\quad ds^2 = g^{1/2}e^{-\Phi/2} ds^2_{\rm string}\ . \label{nshell}\end{aligned}$$ For $\rho<\rho_c'$ the metric develops the usual throat for $q$ NS5-branes [@CHS2]. The string coupling becomes strong at $\rho/\rho_c'\sim g$, a proper distance $\ln 1/g$ from the crossover region.
It is important to see where the supergravity solution is valid. A crude but simple measure is that the radius of a transverse sphere (fixed $\rho$) must be large in string units. (We assume $g \leq 1$ so that the F-string scale is the relevant one.) At the crossover point, the D5 and NS5 radii-squared are respectively $$gp\alpha'\ ,\quad q\alpha'\ .$$ The NS5 solution is valid for $q \gg 1$ and marginal for $q = 1$ (these properties continue to hold down the throat, until the dilaton diverges). The D5 solution has a limited range of validity for $p \gg 1$ but none for $p=1$ (not even $g \gg1$, because the dual string theory is strongly curved). Thus the low energy physics of the Higgs phase is given by the dual field theory description.
The Complete Metric and Dilaton
-------------------------------
The pieces of our solution are scattered through this paper. The zeroth order solution is the D3-brane background [(\[backg\])]{} with harmonic function [(\[zwarp\])]{}, with the brane locations and orientations [(\[finrad\])]{}. The first order correction is given by Eqs. [(\[gsol\])]{} and [(\[onsol\])]{}. The correction near the brane is given in Eqs. [(\[dshell\])]{} and [(\[nshell\])]{}. For convenience we give here the full solution for the metric and dilaton in a form that interpolates between the zeroth order solution and the near-shell solution. We emphasize that these have overlapping ranges of validity, $\rho > \rho_{\rm c}, \rho'_{\rm c}$ versus $\rho < r_0$.
We focus on a single shell of D5 or NS5 type, but the generalization is straightforward. The solution is be conveniently written using coordinates $x^\mu$ for spacetime, $w^i$ for the three coordinates in which the brane is embedded, and $y^i$ for the other three. Write $w, \Omega_w$ as spherical coordinates for the $w^i$, and similarly for the $y^i$. Both the Higgs and confining metrics, in string frame, can be conveniently written $$Z_{x}^{-1/2} \eta_{\mu\nu}dx^\mu dx^\nu
+
Z_{y}^{1/2} (dy^2 + y^2 d\Omega_y^2 + dw^2)
+
Z_\Omega^{1/2} w^2 d\Omega_w^2\ .$$ For the Higgs (D5) vacuum, the $w^i$ are $x^{7,8,9}$ and the $y^i$ are $x^{4,5,6}$; for the confining (NS5) vacuum at $\theta = 0$ this is reversed.
For the D5 brane we have $$\label{hmetric}
Z_x = Z_y = Z_0 \equiv { R^{4}\over\rho_+^2\rho_-^2}
\ , \quad
Z_\Omega = Z_0 \left[{\rho_-^2\over \rho_-^2 + \rho_c^2}\right]^2 \ ,$$ where $$R^4 = 4\pi g N\ ,\quad\rho_\pm = (y^2 + [w \pm r_0]^2) \ ,\quad \rho_c =
{2gr_0\alpha'\over R^2}
\ ,\quad r_0 = \pi \alpha' mN\ .$$ The dilaton is $$\label{hdilaton}
e^{2\Phi} = g^2 {\rho_-^2\over \rho_-^2 + \rho_c^2} \ .$$
For the NS5-brane, we have $$\label{cmetric}
Z_x=Z_\Omega = Z_0 {\rho_-^2\over \rho_-^2 + \rho^2_c}
\ ,\quad
Z_y = Z_0 {\rho_-^2 + \rho^2_c\over \rho_-^2} \ ,$$ where $$\rho_c = {2r_0\alpha'\over R^2}
\ , \quad r_0 = \pi \alpha' mgN\ .$$ Meanwhile the dilaton is $$\label{cdilaton}
e^{2\Phi} = g^2 {\rho_-^2 + \rho_c^2\over \rho_-^2}\ .$$ Note $\rho_c = m R^2 / 2 \sim m\sqrt{gN}\alpha'$ for both branes.
Gauge Theory Physics
====================
In this section, we consider some of the non-perturbative objects in the field theory — strings, baryon vertices, domain walls, condensates, instantons and glueballs, — and discuss their appearance in the supergravity representation. Although objects of this type have appeared in a number of previous incarnations [@highTa; @highTb; @highTc; @ewbaryons; @grossooguri; @gktII; @dibaryons], they arise here in novel forms. We will also consider a vacuum with massive fundamental matter and mention some of its amusing properties.
Flux Tubes: A First Pass
------------------------
Many of the vacua of the field theory have stable flux tubes. At weak coupling, the Higgs vacuum, where the $SU(N)/\ZZ_N$ gauge group is completely broken, has semiclassical vortex solitons in which certain components of the adjoint scalars wind at infinity. The topological charge associated with this winding takes values in $\pi_1(SU(N)/\ZZ_N) = \ZZ_N$; it measures the magnetic flux carried by the vortex. The confining vacuum has electric flux tubes carrying flux in the $\ZZ_N$ center of $SU(N)$. These become semiclassical solitons in the $S$-dual description of the theory as $\tau\to0$. Similar statements apply for the oblique confining vacua. In the other massive vacua [@rdew] there are both electric and magnetic flux tubes, and in the Coulomb vacua there may or may not be any stable flux tubes. We will return to these cases in a later section. For the moment we focus our attention on the strings of the Higgs and confining vacua.
One of the surprising features of Maldacena’s duality is that it relates string theory to a conformal rather than a confining gauge theory. Unconfined electric flux lines between two charged sources in the conformal field theory are represented by a string in the gravity dual [@reyyee; @maldastring]. The string in question droops into the $AdS_5$ space, rather than lying at a fixed $AdS$ radius $r$. Since small $r$ corresponds to large distances in the field theory, the drooping string represents flux lines which spread out in the region between the sources, as expected in a nonconfining theory. The symmetries of $AdS$ space suffice to show that the energy of the string scales as a constant plus a term inversely proportional to the separation of the sources.
In the realization of confining gauge theories via high-temperature five-dimensional field theories [@highTa; @highTb; @highTc], the temperature provides an IR cutoff on $r$. The flux between two charged sources in the field theory now is represented by a string which droops only part way into the $AdS_5$ space, becoming stuck at a radius of order the temperature $R^2 T$; consequently the string represents flux lines trapped in a physical string-like object, of definite tension and width. In this way the confinement of this theory, which is hoped to be in the same universality class as asymptotically free Yang-Mills theory, was established. The same happens in our dual description of gauge theory.
Before treating the supergravity picture carefully, we begin with an intuitive argument. Let us assume, as we will shortly show, that a $(p,q)$ string, with its world-sheet oriented in the $x^\mu$ directions, can bind to a $(p,q)$ 5-brane with D3-brane charge, in a state of finite width and nonzero tension. We claim that this object is a confining flux tube of the gauge theory; since its $AdS$ radius is by construction constant, it certainly has a definite tension. Let us consider $p=0$, $q=1$, the Higgs vacuum. The potential between charged electric sources, given by suspending a fundamental string from two points on the $AdS$ boundary, is highly suppressed: the string can split into two strings joining the D5-brane to the boundary, meaning there is little energy cost to moving the endpoints of the string apart. By contrast, a D1-brane cannot end on the D5-brane. However, it can link up with our putative D1-D5/D3 bound state. This makes the potential between two magnetic sources linear in the distance between them, with a coefficient set by the tension of the bound state. Note also that any $(p,q)$ string with $q\neq0$ is similarly confined — its $p$ F1 charges ending on the D5, its $q$ charges connected to $q$ flux tubes (or a bound state of such tubes) on the D5 brane. It follows that monopoles and dyons, represented by strings with D-charge, are confined in the Higgs vacuum, while electric charges are screened. This is as expected on general grounds from the field theory.
By $S$-duality, the confining vacuum sports F1-NS5 bound states. All strings except those having only D1-charge will bind to the D3-NS5-brane. These bound states are the electric flux tubes of the gauge theory. In this vacuum it is fundamental string charge which is confined and D-charge which is screened, in agreement with expectations. Similar conclusions hold in the oblique confining vacua.
We now turn to the supergravity description of this physics, and demonstrate that these bound states truly exist. In our solutions the function $Z$, given in [Eq. [(\[zwarp\])]{}]{}, diverges at the branes, so all strings can lower their tensions by drooping inward toward one of the branes. However, we have seen that there is a crossover point near each brane, where the universal D3-brane behavior ceases to hold and 5-brane behavior takes over. An F-string, representing electric flux, couples to the string metric. The string stretches in a noncompact direction, so the relevant metric component is $G_{\mu\nu}$. In the D5-solution [(\[dshell\])]{} this still goes to zero at $\rho = 0$, so electric flux is unconfined. In the NS5-solution [(\[nshell\])]{} it takes the minimum value $ r_0^2 q / \pi g n \alpha' = \pi \alpha' m^2
g n q$, so for the confining phase, where $n=N$ and $q =1$, the tension is $$\tau_{\rm e} = \pi \alpha' m^2 g N\frac{1}{2\pi\alpha'}
= \frac{m^2 gN}{2}\ .$$ This satisfies ’t Hooft scaling, as expected in a confining vacuum. The F-string lowers its tension, but only by a finite amount, by binding to the NS5-brane.
A D-string, representing magnetic flux, couples to $e^{-\Phi}$ times the string metric. For the NS5-brane this now vanishes at the brane,[^12] but for the D5-brane there is a minimum value $r_0^2 p / \pi n \alpha' = \pi \alpha' m^2 n q$. The Higgs phase magnetic tension is then $$\tau_{\rm m} = \pi \alpha' m^2 N \frac{1}{2\pi\alpha'g} =
\frac{m^2 N }{2 g} \ . \label{tmag}$$ The $g$ and $N$ scaling appears to be the same as for a classical Nielsen-Oleson vortex. The action scales as $N^3/g$, and the change in the field, which appears squared, is presumably of order $1/N$ for a $\ZZ_N$ vortex. The D-string is bound to the D5-brane.
Though satisfying, these results are partly outside the range of validity of the supergravity description. We have seen in section V.D that for D5-branes this description breaks down before the crossover point, while for NS5-branes it is marginal (we assume $g < 1$; for $g
> 1$ the $S$-dual is true.)
Let us discuss the bound state more carefully in the D5 case, by considering the limit in which the D5-brane is flat. Note first that D1-branes outside a D5-brane are BPS saturated and not attracted to the D5-brane. By contrast, D1-branes outside and parallel to a set of D3-branes are attracted to the D3-branes; upon reaching the D3-branes they appear as tubes of magnetic flux inside an gauge theory, which, since flux is unconfined, expand to infinite radius. Combining the D5 and D3 branes, the D1-brane is attracted to the D5/D3 object but upon reaching it cannot expand to arbitrary size. Its behavior within the D3-brane field theory is determined by the semiclassical calculation of the vortex soliton which confines magnetic flux. Alternatively, it should be a semiclassical instanton of the noncommutative field theory on the 5-brane [@ncinst].
Another intuitive way to see the bound state is to use $T$-duality. Begin with a D1-brane extended in the 01-directions, a 012345 D5-brane, and 0123 D3-branes which are distributed in the 45-directions with density $\sigma$. A $T$-duality in the 5-direction converts the D1-brane into a 015 D2-brane and the D5/D3 system into a D4 brane, which fills 0123 plus a line in the 45-plane. The line makes an angle $\theta$ with the 5-axis, where $$\tan \theta = (4\pi^2\alpha'\sigma)^{-1}\ .$$ If $\theta=\pi/2$ ($\sigma = 0$), the D2 and D4 are perpendicular and BPS, so there is no force on the D2. If $\theta=0$ ($\sigma \to
\infty$), then the D2 can be absorbed by the D4. But if $0<\theta<\pi/2$ the D2 brane is attracted to the D4 but is misaligned with it, and so cannot be completely absorbed. Instead, only a part of it is absorbed, leading to a D2-D4 bound state of finite energy and size. As shown in figure 1, the tension is reduced by a factor $\sin\theta$.
In fact, this bound state is supersymmetric: as one sees from figure 1, after the flux dissolves to the maximal extent, the remaining state is a D2-brane ending on a D4-brane, a familiar supersymmetric configuration. The BPS bound [@dbranes] is $$\tau \geq \Bigl[(V \tau_{\rm D4} + \tau_{\rm D2})^2 + V \tau_{\rm
D4}^2 \cot^2\theta \Bigr]^{1/2} - \Bigl[V \tau_{\rm D4}^2 + V \tau_{\rm
D4}^2 \cot^2\theta \Bigr]^{1/2} = \frac{\tau_{\rm D_2} }{\sin\theta}\ ,$$ where $V$ is a large regulator volume in the 123 directions.
Having established that a BPS state exists in this limit, we should now verify that a nearly-BPS state of essentially the same mass is still present when the D5-brane has the shape of a two-sphere. The D1-D5/D3 bound state is in a rather difficult region of parameter space because the effect of gravity is large (because the D3 charge is large) but the gravity description is not valid everywhere (because the D1 and D5 charges are small). At large $r$ the supergravity description is valid, while at small $r$ the effective description is the field theory on the brane, as in examples in ref. [@IMSY]. It appears that a correct treatment requires that we match these two descriptions, in the spirit of the correspondence principle [@corr]. By the logic of section V.D, the crossover between the two descriptions occurs at a radius $$\hat\rho = \eta \frac{\alpha' r_0}{R^2}\ .$$ We will see that it is interesting to retain an undetermined constant $\eta$ in the crossover point. At the crossover, the metrics in the 0123 and the 45 ($w^{1,2}$) directions are $$\frac{2 \eta\alpha' r_0^2}{R^4} \eta_{\mu\nu} dx^\mu dx^\nu
+ \frac{R^4}{2 \eta\alpha' r_0^2}(dw^1 dw^1 + dw^2 dw^2)\ . \label{cormet}$$ The area of the two-sphere is then $$4\pi r_0^2 \frac{R^4}{2 \eta\alpha' r_0^2} = \frac{8\pi^2 g N \alpha'}{\eta}\ ,$$ giving $$4\pi^2\alpha' \sigma = \frac{\eta}{2 g }\ .$$ Combining the D1 tension, the rescaling of $G_{\mu\nu}$, and the effect shown in figure 1 gives the tension $$\tau_{\rm m} = \frac{1}{2\pi\alpha' g} \frac{2 \eta\alpha' r_0^2}{R^4}
\frac{2 g }{\eta} = \frac{ m^2 N }{2 g}\ .$$
This is the same as the estimate [(\[tmag\])]{} which came from the purely gravitational picture; it is independent of the precise crossover $\eta$ (a necessary, though not sufficient, condition for correspondence arguments to give a correct numerical value); and, one gets the same result if one ignores the gravitational effect entirely and takes unit coefficients in the metric [(\[cormet\])]{}.
It has been suggested that the $\ZZ_N$ strings of supersymmetric QCD might be nearly BPS saturated in the limit of infinite $N$. In this hope is realized, although we see that large $gN$ is necessary for this to be the case. But we have not yet explained why the strings carry charges which are conserved only mod $N$. To do so, we turn to the construction of the baryon vertex.
Baryon Vertex
-------------
To put $N$ sources in the fundamental representation into a gauge invariant configuration requires a baryon vertex. In the $AdS_5\times
S^5$ supergravity dual of Yang-Mills this vertex is given by a D5-brane which wraps the entire $S^5$ [@ewbaryons; @grossooguri]. We will see that in , by contrast, the baryon vertex is a D3-brane with the topology of a ball $B^3$, whose boundary is the two-sphere of the 5-branes which form the vacuum. The link between these two pictures is the Hanany-Witten brane-creation mechanism [@ahew].
One way to derive the nature of the baryon vertex in the theory is to begin in the ultraviolet. The ultraviolet theory is supersymmetric and the spacetime is approximately $AdS_5\times S^5$. Let us consider the confining vacuum, represented by an NS5-brane two-sphere. A baryon vertex joining charged sources in a small spatial region corresponds to a D5-brane wrapping the $S^5$ near the $AdS_5$ boundary, with $N$ fundamental strings joining the boundary and the D5-brane. Now let the region containing the sources grow comparable to the IR cutoff distance $m^{-1}$ (we will give a more precise estimate in section VI.H). The $AdS_5$ radius of the D5-brane decreases until it eventually crosses the NS5-brane. Since drawing the sources further apart than $m^{-1}$ should lead to a large energy cost, something dramatic must happen at this scale. And indeed, it does: the crossing of the D5- and NS5-brane produces a D3-brane which connects the two.
To see this, consider the configuration more carefully. The brane-creation process is local, so let us consider a nearly-flat portion of the NS5-brane, which extends in the $12345$-directions. The D5-brane locally extends in the $45678$-directions. The distance vector between the two branes lies in the $6$-direction. In this arrangement, the crossing of the branes leads to the creation of a D3-brane which fills the dimensions $456$. The transition is shown in figure 2.
Looking globally at the two-sphere, we see that the D3-brane fills the part of the NS5-brane two-sphere which lies outside the D5-brane. But the space inside the NS5-brane is topologically flat; the radius of the five-sphere shrinks to zero inside. The D5-brane therefore is topologically unstable and can be shrunk to zero radius, leaving a D3-brane which fills the entire two-sphere inside the NS5-brane. Like the D5-brane which created it, this D3-brane is a particle in the four-dimensional spacetime; more precisely, it is a localized object whose size is of order $m^{-1}$. If the charged sources are taken to lie further apart than this, then they will connect to the D3-brane not directly but through F1-NS5 flux tubes; thus the baryon vertex behaves dynamically as we would expect in a confining theory.
The D5-brane baryon vertex in the theory has no preferred size or energy. Here, the D3-brane actually represents a physical excitation of definite size and mass. The mass is $$\begin{aligned}
\frac{\mu_3}{g} \int_{B^3} d^3x\, e^{-\Phi} {G^{1/2}_{\rm string}}
{ &=& }\frac{\mu_3 R^2}{g} \int_0^{r_0} dw\, \frac{4\pi w^2}{(r_0^2 +
\rho'^2_{\rm c} - w^2)}
\nonumber\\
&\approx& N\frac{m \sqrt{gN}}{2 \pi^{3/2}} \ln (gN)
\ .\end{aligned}$$ Note that this diverges, due to a net factor $Z^{1/2}$ in the integrand, until the near-shell form is taken into account. The result is a factor of $N$ times ’t Hooft scaling, as would be expected.
To see directly that the D3-ball is a baryon vertex, note that the NS5-brane world-volume action includes a Chern-Simons term $$\int F_{\it 2} \wedge F_{\it 2} \wedge B_{\it 2} \ .$$ This is the $S$-dual of the term $$\int F_{\it 2} \wedge F_{\it 2} \wedge C_{\it 2}$$ in the D5-brane action, which is familiar as it implies that a world-volume gauge instanton is a dissolved D1-brane. The D3-brane ending on the NS5-brane is a magnetic monopole source for $F_{\it 2}$ (the $S$-dual of a familiar fact for D3- and D5-branes), while the dissolved D3-branes become $N$ units of $F_{\it 2}$. Under $\delta B_{\it 2} = d\chi_{\it 1}$, $$\delta \int F_{\it 2}(\mbox{monopole}) \wedge F_{\it 2}
(\mbox{dissolved}) \wedge B_{\it 2}
= - \int dF_{\it 2}(\mbox{monopole}) \wedge F_{\it 2}
(\mbox{dissolved}) \wedge
\chi_{\it 1} \ .$$ This violation, proportional to the number of dissolved D3-branes, must be offset by $N$ fundamental strings ending at the D3-NS5 junction.
Note that this now also explains the $\ZZ_N$ quantum numbers of the flux tubes. If we place $N$ flux tubes close and parallel to each other, pair-creation of these D3-branes can occur. This allows the flux tubes to annihilate in groups of $N$.
Application of $S$-duality allows us to form the same construction for other similar vacua. Notice that the baryon vertex is always a D3-brane. However, the 5-brane on which it ends determines its properties. For example, if we are in the Higgs vacuum, the magnetic flux and its magnetic baryon form the $S$-dual of what we just considered. By contrast, the electric baryon is completely shielded: the $N$ fundamental string sources, which in the confining vacuum were forced to end on the D3-brane, are no longer forced to do so, since they may end anywhere on the D5-brane. This is of course consistent with field theory expectations.
Now let us consider some other vacua. Suppose that we take a vacuum where the classical unbroken gauge group was $SU(N/k)$, with $k\ll\sqrt N$ a small divisor of $N$. Since only the $SU(N/k)$ confines, and since a fundamental representation of the $SU(N)$ parent breaks up into $k$ copies of the fundamental representation of $SU(N/k)$, we should expect that $N$ sources would now be joined by not one but $k$ different baryon vertices. To see this in the supergravity is straightforward. The relevant vacuum is given by $k$ coincident NS5-branes, so when the D5-brane baryon vertex of crosses the NS5-branes, $k$ D3-branes are created. Each of these carries $N/k$ units of string charge (since each NS5-brane has $N/k$ units of D3-brane charge) and so $N/k$ strings must end on each of them. On the other hand, the $k$ D3-branes are not bound together and may be separated spatially from one another. Each one represents a separate, dynamical, massive baryon vertex of $SU(N/k)$. Note also that pair creation of these objects ensures that the electric flux tubes in this vacuum carry only $\ZZ_{N/k}$ quantum numbers.
Flux Tubes: A Second Pass
-------------------------
Here we will look at Coulomb vacua to understand how the baryons and strings behave, and obtain the correct flux tube quantum numbers.
We have already noted the flux tubes present when the vacuum is massive — that is, when the classical vacuum is given by $k$ copies of the $N/k$-dimensional representation of $SU(2)$. The baryons ensured that the electric flux tubes carry flux in $\ZZ_{k}$ and the magnetic or dyonic flux tubes have charge in $\ZZ_{N/k}$.
Let us consider instead a general Coulomb vacuum, given by choosing $p_i$ copies of the $q_i$-dimensional representation, with $\sum
p_iq_i= N$. The unbroken gauge group is $[U(p_1)\times U(p_2)\times
...\times U(p_k)] / U(1)$, corresponding to $p_i$ D5-branes of $AdS$ radius proportional to $q_i$. Let $r$ to be the greatest common divisor of the $p_i$, and $s=\gcd(q_i)$. Simple field theoretic arguments then determine the properties of possible flux tubes. The topology of the breaking pattern of the gauge group permits magnetic flux tubes to carry a $\ZZ_s$ charge, while the massive $W$-bosons of the theory will break all electric flux tubes down to those carrying a $\ZZ_r$ charge.
How do we see these flux-charges in the supergravity? For the magnetic flux, it is straightforward. Consider a collection of $k$ magnetic flux tubes. A magnetic flux tube can be moved with impunity from one D5-brane to another, since two flux tubes on different D5-branes can be connected by a D1-string in the radial direction, corresponding to a magnetic gauge boson. A magnetic baryon vertex connecting to a D5-brane of radius $q_1$ can remove or add $q_1$ flux tubes from the $k$ that we started with. Since we may move all the flux tubes from the first group of D5-branes to the second group, we may also remove any multiple of $q_2$ flux tubes from our collection. Removing $q_i$ flux tubes in any combination, we are left with a number $\hat k$ with $0\leq \hat k<\gcd(\{q_i\})$. This confirms what we set out to prove.
To see the charges of the electric flux tubes requires $S$-duality, which in not understood for the general Coulomb vacuum. However, we conjecture that the $\tau\to -1/\tau$ transformation acts in a simple way in an important subclass of the vacua. In particular, consider those classes of vacua [*where all $\{p_i\}$ are distinct integers and all $\{q_i\}$ are distinct integers.*]{} In this case we claim that the $S$-dual of this vacuum is that with $q_i$ NS5-branes of radius $p_i$. This is of course consistent with the known transformation of the massive vacua [@rdew], for which $p_1q_1=N$. The $S$-dual of the argument in the previous paragraph then shows that the electric flux tubes for these vacua is indeed $\ZZ_r$. Indeed, this is our main evidence for the conjecture.
If the $p_i$ or the $q_i$ are not distinct integers, then the $S$-duality transformation we have suggested is ambiguous. We do not know what happens in this case, either in field theory or in supergravity.
Domain Walls
------------
Since the theory has many isolated vacua, it also has a large number of domain walls which can separate two spatial regions in different vacua. If the walls are spatially uniform then they may be BPS saturated [@dvalishif; @ewMQCD].
Between the oblique confining vacuum represented by a $(1,1)$ 5-brane sphere and the confining vacuum represented by an NS5-brane, there must be a BPS domain wall which carries off one unit of D5-brane charge. We may therefore conjecture that a BPS junction of three 5-branes — the NS5-brane sphere for $x^1>0$, the $(1,1)$ 5-brane sphere for $x^1<0$, and a D5-brane at $x^1=0$ which fills the two-sphere — describes this domain wall. That is, the world-volume of the D5-brane is the $023$-plane of the domain wall times the three-ball spanning the two-sphere. At small $g$ the NS5-brane and the $(1,1)$ brane are nearly coincident (their $AdS$ radius and orientation on the $S^5$ differ only at order $g$) and the effect of the D5-brane on the much denser NS5-brane is very small.
We can see that this reproduces some known properties of the domain wall. First [@dvalishif; @ewMQCD], the flux tubes of the theory (F1 strings) obviously can end on the domain wall (a D5-brane). Furthermore, consider dragging a $(1,1)$ dyonic string, representing a dyonic source in the gauge theory, across the wall. For $x^1<0$, the dyon is screened; it can end happily on the $(1,1)$ 5-brane. For $x^1>0$, the dyon is confined; its monopole charge ends on the NS5-brane, but its electric charge must join onto a flux tube — an F1-NS5 bound state — which in turn ends on the D5-brane domain wall. Finally, note that we may dissolve an $N$-string vertex (a D3-brane) into this domain wall (a D5-brane), leaving $N$ strings which end on the wall and are free to move around on it. If we then permit this domain wall to annihilate with an antidomain wall with no strings attached, then the annihilation will leave a D3-brane behind on which the strings may end, as in the well-known process described in [@Sen].
More generally, if for $x^1<0$ the system is in the phase corresponding to a $(c,d)$ 5-brane, and for $x^1 > 0$ it is in the phase corresponding to a $(c',d')$ 5-brane, then a $(c-c',d-d')$ 5-brane must fill the 2-sphere where they meet. In general the branes on the right and left have different orientations and radii, and so must bend as they meet as depicted in figure 3.
When the left and right phases involve multiple spheres, there will be a more complicated domain wall, constructed from multiple triple 5-brane junctions.
To discuss the domain wall tensions quantitatively we need the kinetic term for the collective coordinate $z = 2\pi\alpha' \phi$. This arises in the Born-Infeld action, from $$G_{\mu\nu}(\mbox{induced}) = G_{\mu\nu} + G_{mn} \partial_{\mu} x^m
\partial_{\nu} x^n\ .$$ Then $$\frac{S}{V} = -\frac{\mu_5}{2g} 2\pi\alpha' \int d^2\xi\, G_{\perp}^{1/2}
(F_{ab} F^{ab})^{1/2}
\eta^{\mu\nu} \partial_\mu x^m \partial_\nu x^n
= -\frac{n}{2\pi g} \eta^{\mu\nu}\partial_\mu \bar\phi \partial_\nu \phi\ .$$ This gives the Kähler potential $K = n\bar\Phi \Phi/2\pi g$. This is the same as [Eq. [(\[KW\])]{}]{} for the classical gauge theory, but by an transformation one can show that it holds for all $(c,d)$. This makes sense, as the main kinetic effect comes from the D3-branes, which are self-dual. With this normalization the potential [(\[fullpot\])]{} implies the superpotential $$W = \frac{1}{4\pi g} (i\frac{4\sqrt{2}}{3M} \Phi^3 + mn\Phi^2)\ .$$ as in [Eq. [(\[KW\])]{}]{}. At the nonzero stationary point this takes the value $$W \to -\frac{m^3 n^3}{96 \pi g M^2}\ .
\label{oursup}$$ For a multi-brane configuration it is summed over $I$.
We cannot rule out an additional additive contribution to this superpotential, although from our semiclassical reasoning we know that any such contribution must be subleading in the Higgs vacuum and others containing only large D5-branes.[^13] Field theory also suffers from the same ambiguity [@doreykumar]. Up to these additive contributions, the field theory and supergravity agree. Consider the massive vacuum corresponding to $p$ D5 branes of radius $q$; for this vacuum $M = p$. Using [@doreya], it is easy to obtain a slight generalization of [@doreykumar] (adjusted to match our conventions, and with the function $A(\tau,N)$ in Eq. (5) of [@doreykumar] set to zero) $$W = {m^3N^2\over 24 g_{{\rm YM}}^2}\left[E_2(\tau) -
{q\over p}E_2\left({q\over p}\tau\right)\right]
\to {m^3N^2E_2(\tau)\over 96 \pi g}-{m^3N^3\over 96 \pi gp^2} \ .$$ Here $E_2$ is the second Eisenstein series, and we have used [Eq. [(\[dcon\])]{}]{} and $E_2(i\infty) = 1$. Note the first term is $p$ independent and is subleading for $p\ll\sqrt{N}$. For the massive vacuum given by $q$ $(1,k)$ 5-branes of radius $p$, $k\ll p$, in which $M = q(\tau +k)$, the formula is $$W = {m^3N^2\over 24 g_{{\rm YM}}^2}\left[E_2(\tau) -
{q\over p}E_2\left({q\over p}(\tau+k)\right)\right]\ ,$$ but since Eqs. [(\[nscon\])]{} applies and $x E_2(x) = x^{-1}E_2(-x^{-1}) + (6i/\pi)$, $$W \to {m^3N^2E_2(\tau)\over 96 \pi g}
- {m^3N^3\over 96\pi g q^2(\tau+k)^2} \ .$$ The first term in this expression is the same as in the D5-brane vacua, so field theory and supergravity agree up to a classically-subleading $M$-independent function.
For a supersymmetric domain wall between two phases, the tension is $$\tau_{\rm DW} = 2|\Delta W|\ =
\frac{ |m^3|}{48\pi g} \left| \sum_{I\,\rm left}\frac{n_I^3}{M_I^2} -
\sum_{J\,
\rm right}
\frac{n_J'^3}{M_J'^2}
\right| \ . \label{dwtens}$$ Let us consider two examples, to see that the 5-brane junction construction reproduces this tension. For both examples we take $g \ll 1$.
The first is the domain wall described above, between the confining and first oblique confining phase. The general result [(\[dwtens\])]{} becomes $$\tau_{\rm DW} =
\frac{ |m^3| N^3}{48 \pi g} \Bigl| (ig^{-1})^{-2} - (ig^{-1} + 1)^{-2}
\Bigr|
\approx \frac{ |m^3| g^2 N^3}{24 \pi}
\ .$$ In the brane picture, the tension comes from the spanning D5-brane. This has three transverse and three longitudinal dimensions and so feels no warp factor, giving simply $$\tau_{\rm DW} =
\frac{4\pi r_0^3}{3} \cdot \frac{\mu_5}{g} = \frac{ |m^3| g^2 N^3}{24
\pi}
\ .$$ The agreement is quite beautiful, given the very different physics that has gone into the two calculations.
The second example is the domain wall between the confining and Higgs phases: a junction between an NS5-brane and a D5-brane, spanned by a (1,1) 5-brane. The NS5-brane is at much smaller radius than the D5-brane (by a factor $g$) and has much greater tension, so the predominant effect is that the D5-brane bends down to join the NS5-brane. The bending is described by the BPS equation $$\partial_1 \phi = \Omega { \hspace{1pt}\overline{\hspace{-1pt}\frac{\partial W}{\partial \phi}
\hspace{-1pt}}\hspace{1pt} } \label{dwbps}$$ with $\Omega$ any phase. For $m$ real, $\Omega = -1$ gives a solution that passes through the origin and the nonzero stationary point, approximating to order $g$ the solution needed. The tension comes primarily from this bending, $$\tau_{\rm DW} = \int_0^\infty dx^1\,
\biggl(\frac{N}{2\pi g} |\partial_1 \phi|^2 + \frac{2\pi g}{N} |W_\phi|^2
\biggr)\ .$$ In this case the general result [(\[dwtens\])]{} follows by construction $$\tau_{\rm DW} =
\frac{ |m^3| N^3}{48\pi g} \ ,$$ where the superpotential in the confining phase is smaller by $O(g^2)$. It will be an interesting exercise to show that the brane construction reproduces [Eq. [(\[dwtens\])]{}]{} in the general case, without assuming small $g$.
Note that a domain wall is the same as a baryon vertex extended in two additional directions. By analogy, we might expect the D5-brane three-ball which acts as a domain wall to be associated with passing a D7-brane through the NS5-brane. This suggests that D7-branes should be reexamined in the original $AdS_5\times S^5$ context.
QCD-like vacua
--------------
It is amusing that is rich enough to permit us to study a theory similar to QCD with heavy quarks. Suppose we consider a vacuum with a D5-brane of radius $n$ and one NS5-brane of radius $N-n$. Here we assume the usual condition on $n$, $n^2\gg gN$, but take $n\sim
gN$, so the D5-branes have comparable $AdS$ radius to the NS5-brane. In the field theory this corresponds to a vacuum with a broken $SU(n)$ sector, a $U(1)$ vector multiplet, and a confining $SU(N-n)$ sector. Among the massive vector multiplets are spin-1 bosons, along with fermions and scalars, charged as $(\bf{\bar n}, \bf{N-n})$ under $SU(n)\times SU(N-n)$. These are strings connecting the D5-brane to the NS5-brane. We will refer to these as ‘quarks’. Clearly these theories have no free quarks: the D5-NS5 strings cannot exist in isolation, since they cannot actually end on the NS5-brane, and instead must be connected to a flux tube. Note that the $SU(n)$ acts as a sort of flavor group for the quarks (analogous to broken weak isospin), and we will refer to its massive adjoint representation as ‘flavor’ gauge multiplets.
In order that the supergravity solution be valid, we must have $n \gg
(gN)^{1/2}$. When $n=gN$, so that the D5 and NS5 sit at equal $AdS$ radii, the quark has mass of order $$(\alpha')^{-1}\int \sqrt{G_{00}G_{yy}r_0^2} d\zeta = r_0/\alpha' = mn$$ which agrees with field theory. However, it is easy to see that if $n$ is smaller than $gN$, the quark retains mass $\sim mgN$; indeed, as an extreme, note that if a D3-brane sits exactly at $r=0$, corresponding to $(gN)^{1/2} \ll n\ll gN$, the quark mass is just proportional to the coordinate length of the string, $mgN$. We therefore see signs that the physical ‘constituent’ masses of the quarks can be much larger than their current values. Of course the quarks are never light compared to $m$.
We can see easily that QCD-like theories have no stable flux tubes due to quark pair production. Recall that we measure the potential $V(L)$ between two electric sources by hanging a probe string by its ends from the $AdS$ boundary, with the ends a distance $L$ apart. Take $L\gg\ m^{-1}(gN)^{-1/2}$; then in the absence of the D5-brane, the probe would bind to the NS5-brane forming a confining flux tube between the sources. In the presence of the D5-brane, however, pair production of the D5-NS5 strings can occur. This breaks the flux tube, which shrinks away allowing the quark to screen the source. The probe string ends up as two strings a distance $L$ apart, each attached to the D5-brane. Note however that if we take $n\gg
gN$, the quarks become very heavy, and the time scale for their pair production becomes very long. In this limit the confining flux tubes are metastable.
Low-lying quark-antiquark mesons are not stable in this theory. Highly excited mesons are represented by two D5-NS5 strings joined by a long F1-NS5 flux tube. However, as the mesons deexcite by emission of glueballs (either supergravity or string states), it eventually becomes energetically preferable for them to decay to a D5-D5 string, bypassing the NS5-brane altogether. In short, the lowest lying mesons between two quarks always mix with and decay to a massive, but lighter, flavor particle, in a vector multiplet of the broken $SU(n)$ group. (Indeed this almost happens in nature; charged pions decay through isospin gauge multiplets, although not because those gauge bosons are light but because they couple to light leptonic states — which could also be represented here, if there were a need.)
Baryons, on the other hand, carry a conserved charge and are both stable and interesting. $N-n$ D5-NS5 strings can end on a D3-brane filling the NS5-brane two-sphere, forming an object whose mass can be computed. If we arrange for a more complicated spectrum of quarks by choosing to use multiple D5-branes of various radii, then there are processes by which baryons can be built from quarks of different masses, and can decay by emission of flavored mesons (or the corresponding gauge multiplets of the ‘flavor’ group.) Scattering of baryons, or of baryons and antibaryons, could also be studied. In addition, it is possible that these baryons have residual attractive short range interactions (different from the physical case in that they are dominated by the ‘flavor’ gauge multiplets) which can cause them to form nuclei. It would be amusing to look for such baryon-baryon bound states. Furthermore, these baryons and nuclei carry $U(1)$ gauge charges, and in some vacua there are lepton-like objects which presumably can combine with them to form atoms.
We cannot resist mentioning one more possibility, although it admittedly may not be realized. Namely, our baryons act as D0 branes in spacetime, and our domain walls as D2-branes (their structure in the extra dimensions is identical, so we suppress it.) While we are not used to thinking of baryons as places where strings can end, this is quite natural if there are no light quarks; if all quarks are heavy then short flux tubes are stable, and physical baryons can be linked by them. Turning on condensates of these flux tubes makes the positions of the baryons noncommuting (note these branes have no massless world-volume gauge fields, but still have massless scalars), and through the Kabat and Taylor mechanism [@ktsphere], an assembly of baryons can be arranged into a spherical domain wall! Thus the properties of the gauge theory recapitulate the method we have used to solve it. In practise, one should try to implement this process physically, through Myers’ mechanism [@myers]. Here we have a difficulty, as the required three-form potential is a massive state, a glueball which couples to domain walls [@dvakakgab], so we cannot create a long-range field to induce a dipole charge. However, there may be ways to circumvent this problem, and create this effect as a thought experiment or even in a lattice simulation, where hints of domain walls have been observed [@montvay].
Condensates
-----------
With the naked singularity banished, the coefficients of the normalizable terms in the supergravity fields, and so the dual condensates, become calculable. We have already determined the superpotential in our discussion of domain walls, and in principle the condensates can be determined directly from this function. The full field theoretic superpotential, and corresponding condensates, are also known [@doreykumar]. However there are subtleties [@gubs; @ADK], and our understanding is only partial.
The condensates of the operators $\lambda\lambda$, $\tr
[\Phi_1,\Phi_2]\Phi_3$ and $m_i\tr \Phi_i\Phi_i$ are all related by the chiral anomaly and operator mixing. A linear combination of these must couple, by the $AdS$/CFT correspondence, to the mode of $G_{\it 3}$ which falls off as $1/r^3$ in invariant units, which we have identified in [Eq. [(\[perts\])]{}]{}. More generally, higher modes of $G_{\it 3}$ should give the expectation values of all of the chiral operators $\lambda\lambda\phi^k + \cdots$. For the lowest mode of $G_{\it 3}$, [Eq. [(\[onasy\])]{}]{} for $\omega_2$ gives the $m$ and phase dependent parts as $$M r_0^3 \propto \frac{m^3 N^3}{M^2}\ .$$ For multiple shells, superposition gives $$m^3 \sum_I \frac{n_I^3}{M_I^2}\ .
\label{multiplevev}$$ Note that there are two $SO(3)$-invariant fermion bilinears, namely $\sum_{i=1}^3 \lambda_i \lambda_i$ and $\lambda_4 \lambda_4$. These correspond to the polarization tensors $\epsilon_{i\bar\jmath\bar k}$ and $\epsilon_{ijk}$, which have equal overlap with the actual field $\epsilon_{w^1 w^2 w^3}$.
In the Higgs vacuum, and other vacua with only large D5-branes, all condensates can be described semiclassically. The expectation values for $\tr [\Phi_1,\Phi_2]\Phi_3$ and $m_i\tr \Phi_i\Phi_i$ are known, and their $m$, $n$ and $M$ scaling agrees with [(\[multiplevev\])]{}. In the confining cases, however, the situation is more subtle, since these operators have condensates of different sizes and since the gluino bilinear is also expected to play a role. We note the following facts. First, careful examination of the field theory superpotential given in [@doreya; @doreykumar] reveals no obvious linear combination of the operators whose expectation values would have this property — except the second term in the superpotential itself, as discussed in section VII.D. Second, there is every indication from our study of domain walls that the second term in the superpotential, proportional to the two-sphere volume times the 5-brane’s charge under $*G_{\it 3}$, measures the dipole moment of the 5-branes. Naturally, the lowest normalizable mode of $G_{\it 3}$ couples to the lowest allowable 5-brane multipole moment, which is indeed a magnetic dipole. We therefore speculate that perhaps the $ijk$ components of $G_{\it 3}$ couple to this part of the superpotential, and to the worldsheet instantons which we will discuss in section VII.G below.
In any case, the supergravity clearly shows that there are expectation values for some dimension-three operators which classically would have had vanishing vevs. It also shows these vevs differ from one confining vacuum to the next. These qualitative features certainly agree with field theory.
The chiral operators $F^2 \phi^k$ are determined by the dilaton background. The dilaton is nontrivial and is obtained from $$\nabla^2 \Phi = \frac{2\pi g r_0^4 |M|^2}{NR^2} \delta^3(y)
\delta(w-r_0)
+ \frac{g^2}{12}
{\rm Re}( G_{mnp} {G}^{mnp}) \ . \label{dilback}$$ The first term comes from the coupling of the dilaton to the 5-brane through the Born-Infeld action. The second is directly from the coupling to the bulk fields. Taking the value of $r_0$ appropriate to a single shell of quantum numbers $M$, and integrating over a volume of order $r_0^6$, both terms are of order $m^6 R^2 r_0^4$ and must be retained. One can argue that they must cancel in the dilaton monopole, which gives the expectation value of the derivative of the Lagrangian with respect to the coupling; this must vanish in a supersymmetric vacuum. Many of the higher operators $F^2 \phi^k$ are also highest components of superfields, and for them a similar argument should apply. We have not yet shown this cancellation directly for the background [(\[dilback\])]{}, and it is possible that there are subtleties. It is a challenge to include this varying dilaton in the full nonlinear treatment of supergravity.
Instantons
----------
At weak coupling, many quantities in Yang-Mills theory receive contributions from instantons. Holomorphic objects can be written as an instanton expansion, given by an infinite series in powers of the parameter $q = e^{2\pi i \tau}$. This expansion is not useful at strong coupling, but in that case the same objects can typically be reexpressed, using a modular transformation, in terms of $\tilde q =
e^{-2\pi i/\tau}$ (or some other $SL(2,\ZZ)$ variant.) In string theory, both in perturbation theory for flat D3 branes and in the $AdS_5\times S^5$ language, D-instantons \[D-($-1$) branes\], whose action is $2\pi/g$, play the role of these field theory instantons. For large $g$ one uses an expansion in magnetic D-instantons, with action $2\pi g$, or more generally in dyonic D-instantons.
In , the situation is slightly different. Consider first weak ’t Hooft coupling, such that the strong coupling scale $\Lambda$ is much less than $m$. In this case the Higgs vacuum has a superpotential which is an expansion in $q = e^{2\pi i \tau}$, but the superpotential of the confining vacuum has an expansion in $q^{1/N}$. The $1/N$ in the exponent is responsible, in the $m\to\infty$ limit, for $SU(N)$ Yang-Mills having $N$ vacua, related by $\theta\to\theta+2\pi k$. It has long been suggested that this behavior implies the existence of fractional instantons, carrying $1/N$ units of instanton charge, which, unlike instantons themselves, remain important in the large $N$ limit. Some evidence for these objects has been found in MQCD [@brodie] (note that these are distinct from similar objects which require compactification of a dimension of spacetime for their existence) and there is even a claim that they have been seen in lattice nonsupersymmetric Yang-Mills [@narayanan].
We might hope to find such objects here, but for the same reason that $\Lambda\sim m$, we cannot do so, for $q^{1/N}$ is of order one, and the fractional-instanton expansion fails. We also cannot use the magnetic instanton expansion, since $g\ll 1$. But remarkably, as can be inferred from the results of [@doreykumar], there is yet another expansion, one which is dual, in the sense of $gN\leftrightarrow
(gN)^{-1}$, to the expansion in fractional instantons.
In particular, Dorey and Kumar show that the superpotential for the confining vacuum is proportional to $$\label{Etwos}
E_2(\tau) - {1\over N} E_2\left(\tau\over N\right)$$ where $E_2$ is the second Eisenstein series. For large imaginary arguments, $E_2(z)$ can be written as an expansion in $e^{2\pi iz}$. The first term in [(\[Etwos\])]{} can therefore be interpreted as a sum over ordinary instantons. The second term, by contrast, cannot be expanded in this way, since $|\tau/N|\ll 1$. However, since $N/\tau$ is large and imaginary, we may make progress as in [@doreykumar] by using the anomalous modular transformation $$\label{Etwoinvert}
E_2(z) = {1\over z^2} E_2\left(-{1\over z}\right) + {6i\over \pi z} \ .$$ from which we learn that the second term in [Eq. [(\[Etwos\])]{}]{} dominates the first and that it can be expanded in power of $e^{-2\pi iN/\tau} =
e^{-2\pi gN}$. This is an expansion in a small quantity, and we need only provide an interpretation for it.
This is not difficult to obtain, for an NS5-brane of the form $S^2$ times Minkowski space permits the string world-sheet to wrap the $S^2$, producing instantons. From the metric [(\[cmetric\])]{} the proper area of the NS5-brane sphere times the tension of the fundamental string is minimized by $$(4\pi r_0^2) {R^2\over2 r_0 \rho_c} {1\over 2 \pi \alpha'}=
{R^4\over 2 \alpha'^2} = 2 \pi g N$$ which is just as required to explain the expansion in the superpotential. As a check, note that if we repeat the calculation for a vacuum with $q$ coincident spheres, both the area of the spheres and the exponent in the field theory are reduced by a factor $q$. For the Higgs vacuum, the metric and dilaton in Eqs. [(\[hmetric\])]{} and [(\[hdilaton\])]{} imply the area of the D5-sphere times the tension of a D1 brane goes at small $\rho$ to $$\frac{R^4 }{2g^2 \alpha'^2} = 2\pi N/g \ .$$ Since the Higgs vacuum superpotential is proportional to [@doreykumar] $$\label{Etwosh}
E_2(\tau) - {N} E_2\left(N\tau\right)$$ we may again interpret the second term (now much smaller than the first term, except for its leading $\tau$-independent contribution) as an expansion in D1-instantons wrapping the D5-brane two-sphere.
It would be interesting to find a string theory interpretation for the coefficients in the expansion of of the superpotential, and especially for the anomalous term in the modular transformation of $E_2$.
Glueballs and Other Particle States
-----------------------------------
The spectrum of states in this theory is complicated, and we do not yet have a physical understanding of its features. We will outline its structure and point out some puzzles and problems which must be solved in future.
First, there is already surprising structure in the theory in the vacuum with $Z$ given by [Eq. [(\[zwarp\])]{}]{}, where the D3-branes form a 2-sphere of radius $r_0$. The typical warp factor in the region $r \sim r_0 \sim mN(g)\alpha'$ is $Z \sim R^4/r_0^4$. A supergravity state then has typical $k_\mu$ given by [@amandajoe] $$G^{\mu\nu} k_\mu k_\nu \sim G^{mn} k_m k_n$$ so that $$k_\mu \sim Z^{-1/2} k_m \sim
\frac{r_0}{R^2}
\sim m g^{\pm 1/2} N^{1/2} \ . \label{coulombstates}$$ where the minus (plus) sign applies for the D3-sphere radius appropriate to the Higgs (confining) vacuum. Immediately we have a puzzle. The semiclassical field theory of this vacuum would have led us to expect physics from the $W$-bosons whose masses lie between the scales $m$ and $Nm$. There are also monopoles of masses $m/g$ and $Nm/g$. These states are present on the supergravity side as F- and D-strings stretched between the D3-branes. But there is no sign of gauge theory states with masses given in the previous equation. The situation is not improved by consideration of excited string states in the bulk, for which one has similarly $$G^{\mu\nu} k_\mu k_\nu \sim 1/\alpha' \label{kmass}$$ and $$k_\mu \sim Z^{-1/4} \alpha'^{-1/2} \sim
\frac{r_0}{R \alpha'^{1/2}}
\sim m g^{\pm 1/2} N^{1/2} \times (gN)^{1/4}\ ,\label{coulombstring}$$ an odd-looking scale.
Now, what changes when the D5- or NS5-brane charge is added? For the D5-brane, essentially nothing happens to these arguments. This is despite the fact that a magnetic flux tube has formed, with a tension whose square root is given by [Eq. [(\[coulombstates\])]{}]{} with the minus sign. Presumably the details shifts around slightly, and of course the massless photons of the D3-branes now develop mass of order $m$, but apparently the spectrum is otherwise little changed.
For the NS5-brane, the situation is more subtle. The electric flux tube has a tension whose square root is given, as in the D5 case by [Eq. [(\[coulombstates\])]{}]{}, now with the plus sign. But in addition, unlike the D5-brane, the NS5-brane has a throat region. If we consider a vacuum with several coincident NS5-branes, then the throat region is reasonably well understood, since it can be described in conformal field theory in the region where the string coupling is small [@CHS2]. From this it is known that supergravity states get string-scale masses from the coupling to the throat geometry. Using the metric [(\[nshell\])]{} in the calculation [(\[kmass\])]{} gives $$k_\mu \sim r_0^{1/2} \rho_{\rm c}'^{1/2} R^{-1} \alpha'^{-1/2} \sim
\frac{r_0}{R^2}
\sim m g^{1/2} N^{1/2}\ .$$ This applies to both supergravity and excited string states. Notice that this is the same scale as for supergravity states in the bulk, and for the string tension.
Unfortunately, the confining vacuum has only one NS5-brane, and there is a long-standing controversy over this object, reflecting the difficulty of doing any reliable calculations in its presence [@throat]. There are disputes over whether the throat and its region of strong coupling even exist. In any case, neither supergravity nor conformal field theory is reliable, and we simply do not know what the spectrum will do in this regime. We note this may hint at a profound obstacle to using string theory as a practical computational tool in QCD.
In both the D5 and NS5 cases, there is one more class of light states, arising from the massless gauge fields on the 5-brane. Relative to the bulk states, the magnetic field on the D5-brane reduces the velocity of open string states by a factor of the dimensionless field, $v \sim(N/g)^{-1/2}$: restoring $F_{\mu\nu} F^{\mu \nu}$ and $F_{\mu
a} F^{\mu a}$ to the brane action, one finds that the latter is multiplied by $v^2$. The mass gap is reduced by the corresponding factor, and so is simply $$k_\mu \sim m \ .$$ This agrees with the classical result, since these are the $W$ bosons and this is the scale of $SU(N)$ breaking. Meanwhile, the NS5-brane has a normalizable zero mode [@CHS], which gives rise to the massless vector required by $S$-duality to the D5-brane. We assume that this mode survives when D3-branes are dissolved in the NS5-brane. Then $$G^{\mu\nu} k_\mu k_\nu \sim G^{w^1 w^1} k_{w^1} k_{w^1}\ ,$$ and $$k_\mu \sim \frac{r_0 \rho_{\rm c}'}{R^2} k_{w^1} \sim \frac{\rho_{\rm
c}'}{R^2} \sim m \ .$$ Again we find a lighter branch of states localized at the brane. By analogy to the Higgs vacuum, one might interpret them as massive magnetic gluons, but in the NS system they are not open strings but rather closed string states in localized wavefunctions on the throat. Most of them carry $SO(3)$ quantum numbers and are therefore not seen in Yang-Mills theory. Some or all of the remainder may mix with bulk states, as we discuss below.
Before doing so, we note that all of the masses we have obtained in the confining vacuum are consistent with ’t Hooft scaling, as they are proportional to $m$ times a power of $gN$. This is pleasing, although it means of course that all of the states merge and mix as $gN$ is taken small, making any quantitative computations in this regime essentially irrelevant for Yang-Mills.
Let us note one important interplay between the open and closed string states. There would appear to be one unbroken $U(1)$ gauge group per (noncoincident) brane, from the world-sheet gauge field on each brane. For an $SU(2)$ representation given by the sum of $k$ distinct irreducible blocks, $SU(N)$ is broken to $U(1)^{k-1}$. On the string side there are $k$ D5-spheres and so apparently a $U(1)^k$. In fact, one $U(1)$ should be lifted by the coupling to the bulk states.[^14] We have not understood all the details, but will indicate the ingredients. There is a massless tensor in $3+1$ dimensions, with the field $B_{\mu\nu}$ independent of $x^m$. Its kinetic term is $$\begin{aligned}
\int d^{10}x\, \sqrt{G} H_{\mu\nu\lambda} H^{\mu\nu\lambda}
= \int d^{4}x^\mu\,
\eta^{\mu\mu'} \eta^{\nu\nu'} \eta^{\lambda\lambda'}
H_{\mu\nu\lambda} H_{\mu'\nu'\lambda'} \int d^6 x^m\, Z^2\ .\end{aligned}$$ The $x^m$ integral converges both at the branes and at infinity, so this is a discrete state. By itself, the coupling of this field to $F_{\mu\nu}$ in the Born-Infeld action would generate a mass for one linear combination of $U(1)$’s via the Higgs mechanism. However, the field $C_{\mu\nu}$ also has a zero mode, seemingly lifting a second $U(1)$. The actual story must be more complicated, with the bulk Chern-Simons term playing a role, because from the point of view of a single brane this would be simultaneous electric and magnetic Higgsing, an impossibility.
Extensions
==========
Unequal Masses
--------------
Now we consider the general case, three masses not necessarily equal. Examination of the classical $F$-term equations [(\[fterm\])]{} suggest use of the coordinates $$\begin{aligned}
z^1 { &=& }\sqrt{m_2 m_3}\, \chi \cos\theta \
,\nonumber\\
z^2 { &=& }\sqrt{m_1 m_3}\, \chi \sin\theta
\cos\phi\ , \nonumber\\
z^3 { &=& }\sqrt{m_1 m_2}\, \chi
\sin\theta \sin \phi \ . \label{ellips}\end{aligned}$$ We will study the potential with the Ansatz that $\chi$ is constant and also that $F_{\theta\phi} = \frac{1}{2}n \sin\theta$. We insert this into the potential [(\[cdact\])]{}. Noting that $$\begin{aligned}
\det G_\perp { &=& }4\tilde m |m_1 m_2 m_3 \chi^4| Z \sin^2\theta\ ,
\nonumber\\
\tilde m &\equiv& |m_1| \cos^2 \theta + |m_2| \sin^2 \theta \cos^2 \phi
+ |m_3| \sin^2 \theta \cos^2 \phi \ , \nonumber\\
T_{mnp} x^m dx^n\wedge dx^p { &=& }|m_1 m_2 m_3| (|m_1| + |m_2| + |m_3|) \chi
\bar \chi^2\ , \nonumber\\
T_{i\bar \jmath \bar k}{ \hspace{1pt}\overline{\hspace{-1pt}T
\hspace{-1pt}}\hspace{1pt} }_{\bar l j k} z^i \bar
z^{\bar l} { &=& }2 \tilde m |m_1 m_2 m_3 \chi^2|\ ,\end{aligned}$$ the potential becomes $$-\frac{S}{V} = \frac{4|m_1 m_2 m_3|}{ \pi g n
(2\pi\alpha')^4 }\frac{|m_1| + |m_2| + |m_3|}{3} | M \chi^2 - 2\pi
\alpha' in\chi/2\sqrt{2} |^2\ .$$ This has a supersymmetric minimum at $\chi = 2\pi
\alpha' in/2\sqrt{2}M$. The two-sphere is now an ellipsoid with axes $$\frac{\pi \alpha' n}{|M|} \sqrt{|m_2 m_3|}\ ,\quad
\frac{\pi \alpha' n}{|M|} \sqrt{|m_1 m_3|}\ ,\quad
\frac{\pi \alpha' n}{|M|} \sqrt{|m_1 m_2|}\ .$$
When $m_3\to 0$ with $m_1=m_2$ fixed, we obtain Yang-Mills; here the ellipsoid degenerates into a line of length $m_1$. This is very easy to understand, classically, for the Higgs vacuum. Yang-Mills with a massive hypermultiplet has a moduli space, which classically is just given by the positions of D3-branes (suitably modified so that they can only move in two dimensions — this remains to be understood in our present context.) On this space is a single point where there are $N-1$ massless [*electrically*]{} charged particles, from the hypermultiplet. At this point, the $N$ D3-branes are arranged in a line. From the classical equations, it is easy to see that an breaking mass parameter for the vector multiplet causes this line to become a noncommutative ellipsoid analogous to the sphere of [@ktsphere]. In addition, the electromagnetic dual of this transition corresponds to a well-known fact, both in field theory [@nsewone; @klytaps; @mdss] and in MQCD [@ewMQCD; @ucbmqcd; @hsz; @deBOz], concerning the breaking of pure Yang-Mills to . The moduli space of has an associated Seiberg-Witten auxiliary Riemann surface of genus $g$. There are $N$ special isolated points on the moduli space where $N-1$ mutually local dyons become massless, and the Riemann surface completely degenerates. In this degeneration, the $N$ handles of the surface join along a singular line segment. The addition of a mass parameter breaking supersymmetry to engenders a transition whereby the $N$ handles join to form a single surface of genus zero: the line segment opens up into a closed curve. In M theory this is represented by a multigenus M5-brane making a transition to a genus zero M5-brane. Here we see signs of a similar phenomenon; the 3-branes which represent the moduli space of the theory presumably align along a line segment, then join and expand to form an NS5-brane ellipsoid.
By contrast, when $m_1=m_2\to 0$ with $m_3$ fixed, the ellipsoid becomes a disk while its overall size shrinks to zero. In the field theory, one expects an infrared fixed point, and indeed the supergravity should go over smoothly to the kink solution of [@ir3] and at small $r$ to the ten-dimensional space of [@pwtend]. Presumably the $G_{\it 3}$ background in [@pwtend] is related to the linearized one we have obtained in [Eq. [(\[susyt\])]{}]{}, although we have not checked this.
It would be interesting to understand these connections in more detail, but we should note that our approximations will break down in these limits. We have seen that the 5-brane shell has a finite thickness, and we need this to be less than the shortest axis of the ellipsoid for our linearized approach to be consistent.
We note also that most of our results generalize easily to this case. The superpotential, the domain wall tensions and the condensates, are all related, as is the dipole moment of the 5-brane, to the volume of the ellipsoid $\propto m_1m_2m_3$. The flux tubes will presumably show signs of the extra metastable Regge trajectories seen in [@mdss] by localizing on the ellipsoid, along the lines observed in MQCD in [@hsz].
Let us make a few brief remarks about the nonsupersymmetric case, $m_4 = m'$ with $m_1 = m_2 = m_3 = m$ kept equal. The potential is now $$-\frac{S}{V} = \frac{4}{ \pi g n
(2\pi\alpha')^4 } \Biggl\{
|M|^2 |z|^4 + \frac{ 2\pi
\alpha' n}{3\sqrt{2}} {\rm Im}\Bigl[ (3 m z \bar z^2 + m' z^3) { \hspace{1pt}\overline{\hspace{-1pt}M
\hspace{-1pt}}\hspace{1pt} }
\Bigr] +
\frac{ (2\pi
\alpha' n)^2 }{8}O(z^2)
\Biggr\}\ .$$ The quadratic term depends on the boundary conditions, as discussed at the end of section IV.A. Its general form is given by $$O(z^2) = \frac{1}{3} |z|^2 \sum_{i=1}^4 |m_i|^2 + (L=2)\ .
\label{nzquad}$$ In the absence of supersymmetry we must make a particular choice of the $L=2$ harmonic $\mu_{mn}$, defined in [Eq. [(\[ambig\])]{}]{}, which represents a traceless combination of masses for the scalar bilinears. This harmonic reduces the masses of some of the scalars, and if too large it can cause the gauge symmetry to break. However, it represents an adjustable parameter in the Lagrangian, so we are completely free to choose it in such a way that it preserves a stable vacuum, assuming such a choice exists. Further, if we maintain $SO(3)$ invariance, the choices are greatly reduced.
Since $\mu_{mn}=0$ in the $SO(3)$-invariant supersymmetric case, and since the vacua in the supersymmetric case are stable, it is evident that for small $m'/m$ the continued use of $\mu_{mn}=0$ leads to stable vacua at nonzero $z$. Of course, the vacua need no longer be degenerate. Whether the single NS5-brane is the preferred vacuum is less obvious, although it seems likely, since it is an extreme case among the vacua. In such a vacuum the spectrum would be altered and the stable domain walls would be lost, but most of the other features of the confining vacuum — flux tubes, baryon vertices, condensates and instantons — would be qualitatively unchanged. The new features would be the appearance of gluon condensates, such as $\tr F^2$, which must be zero in the supersymmetric case, and nontrivial dependence on the phase of $m'/m$.
While the situation is less clear when $m'\sim m$, there are reasons to expect, on purely physical grounds, that a confining vacuum does in fact exist. It seems a worthwhile challenge to seek it in supergravity.
Orbifolds and QCD
-----------------
Many authors have studied supersymmetric and nonsupersymmetric orbifolds of the Yang-Mills theory, its D3-brane representation and its supergravity dual. (A list of references may be found in [@magoo].) Here we briefly consider two orbifolds of the results we have obtained above.
In the theory, a $\ZZ_2$ orbifold on four coordinates leaves an $SU(N)\times SU(N)$ theory with $({\bf N},{ \hspace{1pt}\overline{\hspace{-1pt}\bf N
\hspace{-1pt}}\hspace{1pt} }) +
({ \hspace{1pt}\overline{\hspace{-1pt}\bf N
\hspace{-1pt}}\hspace{1pt} },{\bf N})$ hypermultiplets. We can combine this action with the mass perturbation in two distinct ways. The first is a simultaneous rotation by $\pi$ in the 45- and 78-planes. This is part of the $SU(2)$ that acts on the chiral superfields, and so commutes with the supersymmetry and leaves it unbroken. The second is a rotation by $\pi$ in the 45-plane and $-\pi$ in the 78-plane. Since it differs from the first rotation by a $2\pi$ rotation in the 78-plane, we can think of it as the first rotation times $(-1)^F$, with $F$ being spacetime fermion number. The first rotation commutes with the supersymmetry generator, so the second anticommutes with it and leaves a nonsupersymmetric theory. Note that the two rotations are conjugate to one another, and so in the absence of the mass term would give equivalent theories.
We are assuming that the $\ZZ_2$ acts on the Chan-Paton factors as $$\biggl[ \begin{array}{cc} I_N & 0 \\ 0 & -I_N \end{array} \biggr]\ .$$ More generally the blocks could be different sizes, leading to an $SU(N)
\times SU(M)$ gauge theory; this gives rise to a twisted state tadpole and so is more complicated [@quiver]. For the supersymmetric orbifold, the massless fields that survive are $$\begin{aligned}
A_\mu { &=& }\biggl[ \begin{array}{cc} A_\mu & 0 \\ 0 & A'_\mu \end{array}
\biggr]\ ,\qquad
\Phi_3 = \biggl[ \begin{array}{cc} \Phi & 0 \\ 0 & \Phi' \end{array}
\biggr]\ ,\nonumber\\
\Phi_1 { &=& }\biggl[ \begin{array}{cc} 0 & Q_1 \\ \tilde Q_1 & 0
\end{array}
\biggr]\ ,\qquad
\Phi_2 = \biggl[ \begin{array}{cc} 0 & Q_2 \\ \tilde Q_2 & 0 \end{array}
\biggr]\ . \label{spectrum}\end{aligned}$$ Thus there are $SU(N) \times SU(N)$ vector multiplets, chiral multiplets $\Phi$ and $\Phi'$ in the respective adjoints, and bifundamental chiral multiplets $Q_1$, $Q_2$ in the $({\bf N},{ \hspace{1pt}\overline{\hspace{-1pt}\bf
N
\hspace{-1pt}}\hspace{1pt} })$ and $\tilde Q_1$, $\tilde Q_2$ in the $({ \hspace{1pt}\overline{\hspace{-1pt}\bf N
\hspace{-1pt}}\hspace{1pt} },{\bf N})$. The superpotential is $W\propto \phi (Q_1\tilde Q_2 - Q_2\tilde
Q_1) + \hat\phi(\tilde Q_1 Q_2-\tilde Q_2 Q_1)$. The $AdS$ description of this theory is $AdS^5 \times S^5/\ZZ_2$. There is a fixed plane at $x^4=x^5=x^7=x^8=0$, which in the supergravity is $AdS_5\times S^1$ where the second factor is a fixed $S^1$ on the $S^5$. We may preserve supersymmetry and add the superpotential $m
(\phi^2+\hat\phi^2 + \tilde Q_1 Q_1+\tilde Q_2 Q_2)$. In the gravity description, this is simply the perturbation we studied earlier. The low energy theory consists of two separate Yang-Mills theories with no massless matter. There are two gluino bilinears operators, one even and one odd under the $\ZZ_2$.
The second rotation differs by $(-1)^F$, so the action on the bosons is the same: they are the same as for the theory [(\[spectrum\])]{}. The action on the fermions is opposite, so the fermionic partners are all absent, while fermions appear in all the blocks with 0’s. Thus the low-energy field theory in this case is nonsupersymmetric $SU(N)\times
SU(N)$ gauge theory with a Dirac fermion $\Psi$ in the $({ \hspace{1pt}\overline{\hspace{-1pt}\bf N
\hspace{-1pt}}\hspace{1pt} },{\bf
N})$. This is a $\ZZ_2$ orbifold of Yang-Mills [@schmaltz]. It has no fermion bilinear odd under the $\ZZ_2$. Note that for $N=3$ it is QCD with three massless quarks and with the vector $SU(3)$ of the flavor group gauged. This gauging removes all but a $\ZZ_3$ axial symmetry, or more generally a $\ZZ_N$.
In both cases, the renormalization group flow is from an $SU(N)\times SU(N)$ supersymmetric theory in the ultraviolet to a gauge theory with massless fermions but no massless scalars. In the limit $m\to\infty$, with $N$ and $\Lambda^3 \equiv
m^3\exp(-8\pi^2/g^2_{{\rm YM}}N)$ fixed, the supersymmetric orbifold has $N^2$ vacua; these confining vacua, in which the two gluino bilinears have separate condensates, should survive to the finite $m$ case. The presence of the massive matter in the bifundamental representation assures, however, that there is only one type of $\ZZ_N$ flux tube, and only one type of baryon vertex. Only the condensates can distinguish vacua separated by $N$ consecutive domain walls. By contrast, the nonsupersymmetric orbifold has only one fermion bilinear, with a consequently unique expectation value. As we mentioned, there is an accidental $\ZZ_N$ axial symmetry in the limit $m\to\infty$, $N,\Lambda$ fixed [*if $\Psi$ is held massless*]{} — more on this below. In this limit we expect the $\ZZ_N$ to be broken by a fermion bilinear (note this is consistent with the dynamics of physical QCD) so we expect $N$ confining vacua. Again there is only one $\ZZ_N$ string and one baryon vertex. (In both theories we expect physical baryon states; however they are rather similar in the two cases, differing in mass only slightly.)
On the string side of the duality, all of our brane configurations are invariant under $\ZZ_2$ and so survive in the orbifold theories. We do not see a mechanism for new phases to arise geometrically, so the extra vacua in the supersymmetric case are presumably associated with expectation values of fields in the twisted sector. Certainly the difference of the gluino condensates (and corresponding scalar operators) is in the twisted sector, while the sum is discussed in section V.F. The two supergravity orbifolds differ in the behavior of fermionic fields under the reflection, so that the spherical harmonics, and spectra, will be different in the two cases. The physics involving strings and baryons is essentially the same as before, but it would be interesting to understand how the domain walls are different.
The nonperturbative condensate in the unorbifolded theory survives, as a gluino bilinear expectation value, to small $gN$, where it can break the $\ZZ_N$ nonanomalous $R$-symmetry of Yang-Mills. Unfortunately we cannot make the same claim for the condensate in the nonsupersymmetric orbifold. Only if $\Psi$ is massless for $m\to\infty$, $N$ and $\Lambda$ fixed, does the theory have a discrete axial symmetry, which can be broken by a $\bar\Psi\Psi$ condensate. However, while a gluino is always massless by supersymmetry, no such symmetry protects the fermion $\Psi$. We must therefore assume that $\Psi$ can obtain a mass in perturbation theory through nonplanar graphs. Fine tuning is required to obtain the axial symmetry and the massless fermion at small $gN$, and thus any connection between chiral symmetry breaking in QCD and the large-$gN$ condensate is tenuous.
As an aside, we emphasize the physical interest of these questions. Nonsupersymmetric $SU(N)\times SU(N)$ with a single massless bifundamental fermion, treated as a function of $N$ and the ratio of gauge couplings, is a very interesting theory worthy of further attention. First, if the gauge couplings are very different, the physics between the two strong coupling scales approximates physical QCD. Second, the theory exhibits both confinement and chiral symmetry breaking, with the breaking of a discrete axial symmetry and interesting domain walls. As such, it closely resembles supersymmetric Yang-Mills theory. Third, in contrast to Yang-Mills, the low-energy theory has only vector-like fermions and can be investigated with relative ease on the lattice. To our knowledge it has not been previously studied. Of course, lattice studies of weakly broken Yang-Mills are not impossible [@montvay]; and one may hope, by comparing the $SU(N)\times SU(N)$ theory with a [ massive]{} bilinear fermion to broken Yang-Mills, to study the extent to which nonperturbative physics survives orbifolding in the large $N$ limit. In short, this theory is physically interesting, tractable on the lattice, qualitatively related to supersymmetric Yang-Mills theory even for small $N$, and perhaps quantitatively related to it at large $N$.
Discussion and Future Directions
================================
As with all dualities, our work has implications in both directions — for supergravity and string theory, and for gauge theory.
Strings, gravity, and singularities
-----------------------------------
We have found one more example of a recurrent pattern, the resolution of a naked singularity by brane physics. Earlier examples are the nonconformal D$p$ geometries [@IMSY], the Coulomb branch singularities [@coul1; @coul3; @coul4], and the enhancon [@jpp].
It is not clear how earlier work on the theory is related to ours, because it was all in the context of five-dimensional supergravity. The solutions of ref. [@gppz] might lift to ours, but this requires that a great deal of physics, the entire brane configuration, be hidden in the ‘consistent oxidation’ of the five-dimensional solution. We note that the solutions [@gppz] all have constant dilaton while our 5-branes will produce a locally varying dilaton even if not a dilaton monopole moment. As far as is known, the oxidation cannot produce such an effect. In ref. [@gubs] a general criterion was proposed for identifying physically acceptable naked singularities. Again this was expressed in terms of the five-dimensional theory and so cannot be applied to our solutions without substantial additional work.
There are a number of obvious loose ends in our work. We have not obtained the full supergravity background. We had the good fortune that we could obtain the relevant physics by working only to first order, and partly to second order, in the mass perturbation. It remains at least to solve the supergravity equations to second order. We have observed that the equations of the supergravity have many simplifications in our situation, but we have not understood their origin. It seems extremely likely that these will extend to the full second order calculation. It may even be possible, with the guidance from our approximate solution, to find the exact supergravity solutions. A related question is the analysis of the supersymmetry properties of our solutions, to establish why the supersymmetry is preserved. This involves the 5-brane world-volume as well as the bulk supergravity.
We should remind the reader that we have not fully explained a key point, namely the fact that there are far more brane configurations than there are field theory vacua: many brane configurations represent the same vacuum. For example, we have not explained why the massive vacuum with a $(1,q)$ 5-brane is the same as that with a $(1,q+N)$ 5-brane; we have merely shown that the question never arises purely within the supergravity regime. To understand the transition between these and other descriptions as $\tau$ is varied remains an interesting challenge. A related issue is the minimum of the brane potential at $z=0$. This is outside the range of validity of our approximation, but would naively correspond to an unexpanded brane and so a singular spacetime. Since there exist expanded brane representations for all ground states, we presume that a correct interpretation involves transmutation into a different brane configuration. A complete accounting of these transitions is greatly to be desired.
Another related fact is that many brane configurations (such as $p$ coincident D5-branes, $p\ll\sqrt N$) represent multiple vacua, which are only split from one another by strong coupling $SU(p)$ physics that we cannot see in supergravity because $p$ is small. This physics will have to be understood separately, and perhaps is only described by field theory.
Perturbed $AdS$ spacetime is relevant to the generation of hierarchies in Randall-Sundrum compactification [@RS; @verl]. In the latter paper it was argued that a large number of D3-branes localized in a compact space will generate an effective $AdS$ throat. That work was in the context of an compactification and so the throat was stable: no relevant perturbation exists. The situation becomes more interesting in more realistic cases. If the D3-branes are localized on a space that leaves unbroken, and the compactification is generic, the $G_{\it 3}$ mode that we have considered will have a nonzero boundary value, proportional to $m$, which is of order order one in four-dimensional Planck units. Since this is a relevant perturbation it becomes nonlinear essentially immediately, and the throat disappears. If for some reason the perturbation is anomalously small, then of course, as in any supersymmetric field theory, the ratio $m/m_{\rm Pl}$ is quantum mechanically stable and the throat will be larger. Our work shows that the incipient throat is capped by an expanded brane. It is also possible to avoid the instability using, for example, a discrete symmetry $G_{\it 3} \to -G_{\it
3}$.[^15]
It would be most interesting to study marginally relevant interactions. These may not exist in the theory, but are present in various elaborations. On general grounds, not requiring supersymmetry, such perturbations would naturally become nonlinear only well down the $AdS$ throat, where again an expanded brane would presumably form. This behavior would bear some similarity to the hierarchy mechanism of ref. [@GW]. Examples of such interactions were considered in [@klebnek; @klebtseyt], and there is strong reason to expect an expanded brane there as well.
Our work bears directly on recent proposals regarding the vanishing of the cosmological constant [@stanford], which involve a naked singularity in the compact dimensions. There have been many criticisms, published and unpublished, of the assumptions made in ref. [@stanford] regarding the properties of the singularity, but we can add to these the specific example of what string theory does in our case. In terms of our notation from section III.B, the authors of refs. [@stanford] assume that the parameter $b$ can take arbitrary values. That is, they integrate in the coordinate $r$, and assume that any singularity encountered gives an acceptable spacetime. Further, they must assume that $b$ is a fixed parameter, rather than a dynamical quantity. As we see from our discussion, this is inconsistent with the requirements of $AdS$/CFT duality, and it is not what happens in our case: $b$ takes discrete values, depending on the particular vacuum, and can make dynamical transitions from one value to another.
Thus the singularity in the end is replaced by an ordinary physical object, like a hydrogen atom. For hydrogen, too, one can integrate the wave equation inward in $r$ for any energy, obtaining a generally singular solution. The experimental discreteness of the spectrum indicates that nature abhors a generic singularity.
It is notable that our system has a large number of ground states,[^16] of order $e^{\sqrt{N}}$. In the supersymmetric case these are all degenerate, but once supersymmetry is weakly broken they will form a closely spaced, near-continuum of discrete states. If a singularity is of this type, the system may have vacua of exceedingly small cosmological constant, and the mechanism of [@brownteit] may be realizable. In order for such a mechanism to solve the problem, these states must be sufficiently metastable, and there must be a dynamical mechanism to select a state with a small net cosmological constant (the same problems are in any event present in the continuum case); for further discussion of the latter issue see [@raph].
Finally, we should note that the Myers dielectric effect will arise in many other situations, such as perturbations of other conformal and nonconformal theories. As one example, consider the perturbation of the BFSS matrix theory given by the dimensional reduction of the mass term. This has been used as a means of analyzing the structure of the matrix theory bound state [@matpert]. Now we see that this deformation has a physical interpretation in its own right: it is the matrix theory for M theory with a nontrivial boundary condition on the 4-form field strength. In this background the graviton will blow up into a finite-sized M2-brane sphere. This same mechanism, the Myers expansion of a highly boosted graviton in a background field strength, has recently led to a remarkable explanation of the stringy exclusion principle [@SEP].
Gauge theory
------------
The brane solution gives a beautiful representation of a confining gauge theory and many of the associated phenomena. There are many artifacts of the massive matter, so this is still far from QCD, but we emphasize that it is a confining four-dimensional gauge theory in its own right. We should perhaps note that we use the Maldacena duality freely in the entire range $1/N < g < N$. Discussion often focuses on large $N$ with fixed $gN$, but this is only one part of the interesting range.
Without too much additional work, it should be possible to understand the breaking of to Yang-Mills. Much of the moduli space, including the part corresponding to the repulson singularity studied in [@jpp], should be visible in supergravity. We might hope that, as in [@ewelliptic], the Seiberg-Witten Riemann surface will appear in a natural way. However, where the surface degenerates (at points with light charged particles) we expect there will be physics lying outside the supergravity regime. Since the transition from to is described simply in MQCD, we may hope that this breakdown will be described simply using semiclassical branes, giving a picture of the physics studied in [@mdss]. We also expect a breakdown at Argyres-Douglas fixed points [@AD], where the physics, presumably a supergravity kink connecting one $AdS$ region to another, may not be described in a linear approximation.
Once the supersymmetric gravity background is completely understood, it should also be possible to fully explore the nonsupersymmetric case. Since even the confining vacuum is a small perturbation of an Coulomb branch configuration, we do not believe that small breaking of supersymmetry is likely to have a large impact on the theory. New condensates and new $\theta$ dependence, and a loss of the degeneracy between vacua, should certainly be seen, but otherwise we see no reason why the nonsupersymmetric confining vacuum must appear much different from the supersymmetric one. However, while this will be true if the supersymmetry breaking scale $m'$ is small compared to $m$, it will surely be false if $m'\gg m$. The quantitative question of where the transition lies, as a function of $gN$, remains to be explored. Even assuming, however, that $m\sim m'$ is within the supergravity regime, this does not make the study of pure nonsupersymmetric Yang-Mills any easier, since it is a long way from $gN\gg1$ to the required regime.
Assuming the nonsupersymmetric case can be studied to a degree, an obvious next step in studying the gravity–gauge theory correspondence is to add massless charged fermions. It would be most interesting to see chiral symmetry breaking and to determine if and when the properties of pions lie within the supergravity regime. Adding supersymmetric charged matter is unfortunately not of great use, since $SU(N)$ with $N_f\ll N$ massless chiral multiplets in the fundamental plus antifundamental representation has no stable vacuum, while if $N_f\sim N$ there is as yet no dual string description. The low flavor case might be studied by taking type IIB on an orientifold, which gives an $Sp(N)$ gauge theory with a hypermultiplet ${\cal A}$ in the antisymmetric tensor representation and four ${\cal
F}_i$ in the fundamental representation [@QandA; @AFM]. As in , the ultraviolet theory is finite, and could be perturbed by supersymmetric masses $m$ for the adjoint chiral multiplet and ${\cal
A}$, and $\mu_i$ for the ${\cal F}_i$. When $\mu_i\sim m$, the physics probably resembles , but the theory will exit the supergravity regime as any one of the $\mu_i$ go to zero. Breaking the supersymmetry, leaving only the fermions in ${\cal F}_i$ and the gauge bosons massless, would be a challenge, but might be tractable.
In this paper we have not pursued the obvious and important connections of our work with noncommutative field theory in two additional dimensions. Similar relations are present already in MQCD. We are especially intrigued that the confining flux tube of large $N$ Yang-Mills is a nearly-BPS instanton string of a sphere-compactified six-dimensional noncommutative gauge theory (supersymmetric or not.) Note that the baryon vertex is also an interesting object in this theory. The massive vacua with multiple coincident 5-branes, and their solitons, may be of considerable interest for the study of nonabelian noncommutative field theory. This may also be true for lower-dimensional cases, where the classical vacua are still described by spherical D(p+2) branes.
It is important to note that many of our results bear some resemblance to those seen in MQCD, which is an unusual compactification of the $(2,0)$ theory on an M5-brane. Both the breaking of to Yang-Mills [@ewelliptic] and the breaking of to [@ewMQCD] have been considered, although the full model has not been constructed in MQCD. It is interesting to consider some of the similarities and differences with the supergravity dual of . In our picture, the vacuum is represented by an NS5/D3 hybrid; in MQCD the vacuum is given by a multisheeted M5-brane, which, when the radius $R_{10}$ of the M-theory circle is small, is an NS5/D4 hybrid. Second, the flux tubes in are F1 strings bound to the NS5/D3 hybrid; in MQCD they are M2-branes bound to the M5 brane. The baryon vertex in MQCD is an M2 brane which extends off of but has a boundary on the M5 brane, analogous to our D3 brane whose boundary is the NS5-brane two-sphere. Domain walls are in both approaches are 5-branes mediating transitions between different 5-brane vacua. There are a number of important differences that this list understates; but the most interesting difference, perhaps, is the instantons at large $N$. As we have seen, the fractional instantons noted in [@brodie] are resummed at large $gN$ into strings wrapping the NS5-brane two-sphere. All of these connections hint at the usual duality between M theory on a torus and type IIB string theory, but the connection between the two pictures is not as simple as this, given the absence of a torus in both constructions. It would be interesting to make these connections precise.
One of the most interesting aspects of the theory, and any similar gravity dual of broken to , is that it can be used to great effect to understand more deeply the connection of field theory with gravity and string theory. Our work and that of [@doreykumar; @ADK] points in this direction. The holomorphic properties of these field theories can be completely understood using field theory and/or M theory, as in [@rdew; @ewelliptic; @doreya], for all values of $g$ and all values of $gN$. Consequently, one can distinguish clearly those regimes of parameter space in which the theory is well-described by electric variables, by magnetic variables, or by IIB string variables respectively. The various universal and quasiuniversal physical properties of the theory are given different descriptions in each regime (see for example our discussion of the instanton expansion of the superpotential, section VI.G,) and there are surely many things to be learned by studying how one regime converts to another. In particular, the possibility of [*quantitatively*]{} matching one regime to another deserves attention. It is probable that this will be required if one is to study QCD, which lies mostly outside the supergravity regime. If such matching is only possible for holomorphic and BPS quantities, as so far seems likely, it poses yet another obstacle in the quest for a description of the strong interactions. However, even if the goal of doing calculations in large-$N$ QCD, using string variables, is shown to be unrealizable, the fact that field theories exhibit behavior which differs greatly from the physics we have observed so far in nature is of potentially great importance, in that it may open new avenues for solving old problems.
We would like to thank O. Aharony, N. Arkani-Hamed, I. Bena, M. Berkooz, K. Dasgupta, S. Dimopoulos, E. Gimon, S. Gubser, A. Hashimoto, S. Hellerman, N. Itzhaki, D. Kabat, N. Kaloper, R. Myers, N. Nekrasov, K. Pilch, L. Randall, S. Sethi, E. Silverstein, W. Taylor, N. Warner, E. Witten, and many participants of the ITP Program on Supersymmetric Gauge Theories for helpful discussion of various aspects of this work. M.J.S. is especially grateful to D. Kabat for discussions and assistance at early stages of this work. We also especially thank O. Aharony for pointing out important deficiencies in our discussion of the superpotential and the condensates. The work of J.P. was supported by National Science Foundation grants PHY94-07194 and PHY97-22022. The work of M.J.S. was supported by National Science Foundation grant NSF PHY95-13835 and by the W.M. Keck Foundation.
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[^1]: Institute for Theoretical Physics, University of California, Santa Barbara CA 93106-4030
[^2]: School of Natural Sciences, Institute for Advanced Study, Olden Lane, Princeton NJ 08540
[^3]: The Kähler potential is normalized $(2/ g_{\rm YM}^{2}) \tr\, { \hspace{1pt}\overline{\hspace{-1pt}\Phi
\hspace{-1pt}}\hspace{1pt} }_i
\Phi_i$.
[^4]: It would be very interesting to find this integrable system in the supergravity dual description of this theory.
[^5]: In section VI.C we will show that some of the Coulomb vacua are transformed in a simple way by certain elements of . However, we will not obtain the full story.
[^6]: The absence of an overall center-of-mass $U(1)$ in the brane configuration, in parallel with the absence of a $U(1)$ in the gauge theory, is not completely understood, although we will comment on it in section VI.H.
[^7]: This estimate [(\[densest\])]{} ignores the warping of the geometry by the expanded brane, but should be correct in order of magnitude almost everywhere. In fact, very close to the surface of the two-sphere the effect of the D5-brane dominates. However, in this regime we can match onto the exact solution for a flat D5-brane with D3-brane charge, as we develop further in section V.D.
[^8]: Even when all four masses are nonvanishing, this operator is still chiral and has a supersymmetric completion to linear order in $m$. However, the Hamiltonian at order $m^2$ is nonsupersymmetric. This case will be discussed in section VII.B.
[^9]: See also section 5 of ref. [@ir3].
[^10]: This might seem to contradict claims that there is a large-$N$ limit of $AdS$ space which gives flat-spacetime physics [@smat], since nonparallel D5-branes do attract in flat spacetime. The point is that this large-$N$ limit includes going to small $AdS$ distances. This would bring us into the ‘near-shell’ region of the D5-brane (to be discussed in section V.D), where the above no-force analysis does not apply.
[^11]: See for example Eq. (6), and for the NS5 brane Eq. (35), of Ref. [@AOS-J]. Note that these equations arise after taking the limit where a noncommutative gauge theory describes the 5-brane dynamics. We will return to this issue briefly in our conclusions.
[^12]: This ‘magnetic screening’ is required both by physical intuition and by $S$-duality, but notice that it requires that the string coupling diverge in the NS5-brane throat. For multiple coincident NS5-branes, this can be seen in supergravity alone. For one NS brane, however, the very nature of the throat is in dispute [@throat] and it is not clear whether supergravity, worldsheet CFT, or semiclassical brane physics gives a good description. In any case, the D-string must dissolve in the NS5-brane, one way or another.
[^13]: We thank O. Aharony for pointing out an error in our orginal approach.
[^14]: This was suggested by E. Witten.
[^15]: This has also been noted by O. Aharony.
[^16]: N. Arkani-Hamed, S. Dimopoulos, J. Feng, S. Gubser, N. Kaloper, E. Silverstein and F. Wilczek have emphasized the importance of this point.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'M. Dziemiańczuk'
title: 'Counting Bipartite, k-Colored Multi and Directed Acyclic Multi Graphs Through F-nomial coefficients'
---
[<span style="font-variant:small-caps;">Counting Bipartite, k-Colored\
and Directed Acyclic Multi Graphs Through F-nomial coefficients</span>]{}\
[Maciej Dziemiańczuk]{}
[Gdańsk University Student, the Institute of Computer Science]{}
[PL-80-952 Gdańsk, st. Wita Stwosza 57, Poland]{}
[e-mail: [email protected]]{}
**Abstract**
F-nomial coefficients encompass among others well-known binomial coefficients or Gaussian coefficients that count subsets of finite set and subspaces of finite vector space respectively. Here, the so called F-cobweb tiling sequences $N(\alpha)$ are considered. For such specific sequences a new interpretation with respect to Kwaśniewski general combinatorial interpretation of $F$-nomial coefficients is unearhed.
Namely, for tiling sequences $F = N(\alpha)$ the $F$-nomial coefficients are equal to the number of labeled special bipartite multigraphs denoted here as $\alpha$-multigraphs $G(\alpha,n,k)$.
An explicit relation between the number of $k$-colored $\alpha$-multigraphs and multi $N(\alpha)$ -nomial coefficients is established. We also prove that the unsigned values of the first row of inversion matrix for $N(\alpha)$ -nomial coefficients considered here are equal to the numbers of directed acyclic $\alpha$-multigraphs with $n$ nodes.
AMS Classification Numbers: 05A19 , 11B39, 15A09.
Keywords: bigraphs, $k$-colored graphs, DAG, multigraphs, f-nomial coefficients
Affiliated to The Internet Gian-Carlo Polish Seminar:
*http://ii.uwb.edu.pl/akk/sem/sem\_rota.htm*
Introduction
============
The notation from [@akk1; @akk2] is being here taken for granted.
**Comment 1**
For the mnemonic efficiency of Kwaśniewski up-side-down notation see Appendix in [@akk4] and references therein and consult recent [@akk5], [@akk7]. With this Kwaśniewski “upside down notation” inspired by Gauss and applying reasonings almost just repeated with “$k_F$” numbers replacing $k$ - natural numbers one gets in the spirit of Knuth [@knuth] clean results also in this report. And more ad “upside down notation”: concerning Gauss and Knuth - see remarks in [@knuth] also on Gaussian binomial coefficients.
Let any $F$-cobweb admissible sequence then $F$-nomial coefficients are defined as follows $${ {{n} \choose {k}}_{\!\!F} } = \frac{n_F!}{k_F!(n-k)_F!}
= \frac{n_F\cdot(n-1)_F\cdot ...\cdot(n-k+1)_F}{1_F\cdot 2_F\cdot ... \cdot k_F}
= \frac{n^{\underline{k}}_F}{k_F!}$$ while $n,k\in \mathbb{N}$ and $0_F! = n^{\underline{0}}_F = 1$.
Let us denote by $\mathcal{T}_\lambda$ a family of natural numbers’ valued sequences $F\equiv\{n_F\}_{n\geq 0}$ constituted by $n$-th coefficients of the generating function $F(x)$ expansion i.e. $n_F = [x^n]F(x)$ (in Wilf’s notation [@wilf]), where $$F(x) = 1_F\cdot \frac{x}{(1 - \alpha x)(1 - \beta x)}$$ for $1_F\in\mathbb{N}$ and $\alpha,\beta\in \mathbb{R}$.
It was shown in [@md3] that any $F\in\mathcal{T}_\lambda$ is *cobweb-tiling* sequence.
$N(\alpha) = F = \{n_F\}_{n\geq 0}\in\mathcal{T}_\lambda$ means that $\alpha=\beta$ and $1_F=1$.
$N(\alpha)$ is a cobweb tiling sequence, of course.
Let us recall some of the sequence $F \equiv \{n_F\}_{n\geq 0} = N(\alpha)$ properties following [@md3]. Let at first $n>1$ and $1_F\in\mathbb{N}$ be given, then
$$\label{eqAlpha}
n_F = 1_F \cdot n \cdot \alpha^{n-1}$$
Then the following recurrence relation for any $m,k\in \mathbb{N}$ takes place
$$n_F = (k+m)_F = \alpha^m k_F + \alpha^k m_F$$
Hence the corresponding $F$-nomial coefficients do satisfy
$${ {{n} \choose {k}}_{\!\!F} } = \alpha^m { {{n-1} \choose {k-1}}_{\!\!F} } + \alpha^k { {{n-1} \choose {k}}_{\!\!F} }$$
while ${ {{n} \choose {n}}_{\!\!F} } = { {{n} \choose {0}}_{\!\!F} } = 1$.
Combinatorial interpretation of $N(\alpha)$ tiling sequences $N(\alpha)$-nomials
================================================================================
$F$-cobweb admissible sequence
------------------------------
At first, let us refer to the base joint Kwaśniewski combinatorial interpretation of $F$-nomial coefficients of all at once cobweb-admissible sequences. Then we recall some special properties of the sequence $N(\alpha)$ following [@md3; @md4].
\[fact:1\] For $F$-cobweb admissible sequences $F$-nomial coefficient ${ {{n} \choose {k}}_{\!\!F} }$ is the number of max-disjoint *equipotent* copies $\sigma P_{n-k}$ of the layer ${\langle\Phi_{k+1} \! \to\! \Phi_{n}\rangle}$. ([@akk1; @akk2] and references therein)
**Specifically important**: $F$-cobweb tiling sequences’ $F$-nomial coefficients are the numbers of max-disjoint equipotent copies $\sigma P_{n-k}$ - **the layer ${\langle\Phi_{k+1} \! \to\! \Phi_{n}\rangle}$ is tiled with**.
Because of that we consider only just these sequences $N(\alpha)\in \mathcal{T}_\lambda$, in what follows.
**Recall.** The sequences $F=N(\alpha)\in \mathcal{T}_\lambda$ have several combinatorial interpretations. For example, if $a=2$ then $n_F$ equals the number of edges of an $n$-dimensional hypercube and the number of 132-avoiding permutations of $[n+2]$ containing exactly one $123$ pattern (sequence $A001787$ [@oeis])
**Examples of the sequences $N(\alpha) = \{ n_{N(\alpha)}\}_{n\geq{0}}$**
1. $N(1) = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... $
2. $N(2) = 0, 1, 4, 12, 32, 80, 192, 448, 1024, 2304, 5120, ... $
3. $N(3) = 0, 1, 6, 27, 108, 405, 1458, 5103, 17496, 59049, 196830, ... $
4. $N(4) = 0, 1, 8, 48, 256, 1280, 6144, 28672, 131072, 589824, 2621440,... $
**Observe.** $F= N(\alpha)$. Let $\alpha \in \mathbb{N}$ be given, then $n_F=n\cdot \alpha^{n-1}$ i.e. it is equal to the number of sequences $(s_1,s_2,...,s_n)$ such that one of terms is equal to zero i.e. $\exists! k : (1\leq k \leq n \wedge s_k=0)$, and the rest of them $s_j\in [\alpha]$.
Counting bipartite and $K$-colored $\alpha$-multigraphs
-------------------------------------------------------
Recall. Labeled bipartite $\alpha$-multigraph $G(\alpha,n,k)$ is a bipartite graph with $n$ vertices ($k$ of them is in one two disjoint vertices’ sets) with multiedges, such that any two vertices might be connected by at most $(\alpha-1)$ edges. We define $K$-colored $\alpha$-multigraph in a similar way.
Note. If $\alpha=1$ then the graph $G(\alpha,n,k) = (V_1 \cup V_2,E)$ has no edges i.e. $E=\emptyset$. Therefore it might be considered as a $k$-subset of $n$-set i.e. $G(1,n,k) \equiv V_1 \subseteq [n]\wedge |V_1|=k$ . Then labeled $K$-colored $1$-multigraph is considered as a partition of set $[n]$ into $k$-nonempty blocks where each of them represents a set of vertices’ indices of $G$ with the same color.
\[obs:1\] The number of labeled bipartite $\alpha$-multigraphs $G(\alpha,n,k)$ denoted by $\beta_{\alpha,n,k}$ is $$\beta_{\alpha,n,k} = {n \choose k}\cdot \alpha^{k(n-k)}$$
[*[**Proof.**]{}*]{} Take any $\alpha\in\mathbb{N}$. If $\alpha=1$ then we have no edges between nodes, therefore we need to count all $k$-subsets of vertices set $n$. If $\alpha>1$ then there is additionally $\alpha^{k(n-k)}$ possibilities to create at most $(\alpha-1)$ edges between any two vertices from disjoint sets $V_1$ and $V_2$ $\blacksquare$
Note, $\beta_{2,n,k}$ is equal to the number of bipartite graphs $G(2,n,k)$ of $n$ nodes where $k$ of them belong to one of disjoint vertices’ sets.
\[cor:1\] Let $F$ be a cobweb tiling sequence $N(\alpha) \in \mathcal{T}_{\lambda}$, such that $1_F=1$. Then the $F$-nomial coefficient is equal to the number $\beta_{a,n,k}$ of labeled bipartite graphs $G(\alpha,n,k)$ $$\label{eqFib}
{ {{n} \choose {k}}_{\!\!F} } = \beta_{\alpha,n,k}$$ for $n,k\in\mathbb{N}\cup\{0\}$.
[*[**Proof.**]{}*]{} Consider the sequence $F\equiv\{n_F\}_{n\geq 0} = N(\alpha)$ i.e. such that $1_F=1$ and $n_F = n\cdot \alpha^{n-1}$. We only need to show that the number of bipartite $\alpha$-multigraphs $G(\alpha,n,k)$ is $\beta_{\alpha,n,k}$ (Observation \[obs:1\]). For that to see take any $n,k\in\mathbb{N}$, and just check that $${ {{n} \choose {k}}_{\!\!F} } = \frac{n\alpha^{n-1} \cdot (n-1)\alpha^{n-2} \cdot ... \cdot (n-k+1)\alpha^{n-k}}{1\cdot\alpha^0 \cdot 2\alpha^1 \cdot ... \cdot k\alpha^{k-1}} =$$ $$= \frac{n^{\underline{k}}}{k!} \cdot \frac{\alpha^{k\frac{2n-k-1}{2}}}{\alpha^{k\frac{k-1}{2}}}
= {n \choose k} \cdot \alpha^{k(n-k)} \blacksquare$$
In another words, ${ {{n} \choose {k}}_{\!\!F} } = { {{n} \choose {k}}_{\!\!N(\alpha)} }$ is equal to the number of $2$-colored $\alpha$-multigraphs with $n$ vertices, while $k$ of them are colored by one color and $n-k$ by another one.
**Note.** Consider $F=N(2)$. The number $\gamma_{n,2}$ (see: [@finch]) of all 2-colored graphs is equal to
$$\gamma_{n,2} = \sum_{k\geq 0} {n\choose k} \cdot 2^{k(n-k)} = \sum_{k\geq 0} { {{n} \choose {k}}_{\!\!N(2)} }$$
In general, for $\alpha\in\mathbb{N}$ the sum $\sum_{n\geq 0}{ {{n} \choose {k}}_{\!\!N(\alpha)} }$ is equal to the number of all $2$-colored $\alpha$-multigraphs.
**Comment 2** *On F-binomiality and 2-colored $\alpha$-multigraphs.*
(Due to A. Krzysztof Kwaśniewski, see [@akk6] and consult for notations also references \[7,8,9\] therein).
In view of the final Remark in [@akk6], the combinatorics fundamental logarithmic Fib-Binomial Formula ([@akk6], Section 4)
$$\phi_n^{(t)}(x +_F a) \equiv \left[ \mathrm{exp}\{a\partial_F\}\phi_n^{(t)} \right](x)
= \sum_{k\geq 0}\left[\!\! \begin{array}{c} n \\ k \end{array}\!\!\right]_F \phi_{n-k}^{(t)}(a)x^k$$ $$t=0,1; \ \ \ |x| < a; \ \ \ n\in \mathbb{Z}$$
may be considered as $F$-Binomial for any natural numbers valued sequence $F$ with $F_0=1$ (the class considered in [@akk6] is much broader). For special $F$-sequences known as $F$-cobweb posets admissible sequences the $F$-nomial coefficients
$$\label{eq:hybrid}
\left[\!\! \begin{array}{c} n \\ k \end{array}\!\!\right]_F = { {{n} \choose {k}}_{\!\!F} }$$
for $n,k\geq 0$ acquire Kwaśniewski joint combinatorial interpretation (Fact \[fact:1\]).
Now put in $F$-binomial formula above $t=0$ and $a=x=1$ and pay attention to - that according to the definition of $F$-hybrid binomial coefficients one has (\[eq:hybrid\]) for $n,k\geq 0$. Then we get (known at least since Morgan Ward famous Calculus of Sequences [@ward]) the clean - appealing in Kwaśniewski notation formula:
$$\left( 1 +_F 1 \right)^n \equiv \sum_{n\geq 0}{ {{n} \choose {k}}_{\!\!F} }$$
In particular and rephrasing: the number $\gamma_{n,2}$ of all 2-colored graphs as in [@finch] is now equal to
$$\gamma_{n,2} = \sum_{k\geq 0} {n\choose k} \cdot 2^{k(n-k)} = \sum_{k\geq 0} { {{n} \choose {k}}_{\!\!N(2)} } \equiv \left( 1 +_F 1 \right)^n$$
Let us show up some values of $\{\gamma_{n,2}\}_{i\geq 0}$ (sequence $A047863$ [@oeis])
$\{\gamma_{n,2}\}_{i\geq 0} = {1, 2, 6, 26, 162, 1442, 18306, 330626, 8488962, ... }$
And here is the matrix $\mathbf{M}=[m_{ij}]$, $m_{ij} = { {{i} \choose {j}}_{\!\!N(2)} }$ of the sequence $N(2)$, where $0\leq i,j\leq 7$.
$$M = \left[
\begin{array}{llllllll}
1 \\
1 & 1 \\
1 & 4 & 1 \\
1 & 12 & 12 & 1 \\
1 & 32 & 96 & 32 & 1 \\
1 & 80 & 640 & 640 & 80 & 1 \\
1 & 192 & 3840 & 10240 & 3840 & 192 & 1 \\
1 & 448 & 21504 & 143360 & 143360 & 21504 & 448 & 1
\end{array}
\right]$$
Due to Corollary \[cor:1\] and thanks to general properties of $F$-cobweb admissible tiling sequences we infer families of identities. For example recurrence relation for the number $\beta_{\alpha,n,k}$ of labeled bipartite $\alpha$-multigraphs $G(\alpha,n,k)$ is given “for free” and it reads
$$\beta_{\alpha,n,k} = { {{n} \choose {k}}_{\!\!N(\alpha)} }
= \alpha^{n-k}\cdot { {{n-1} \choose {k-1}}_{\!\!N(\alpha)} } + \alpha^k\cdot { {{n-1} \choose {k}}_{\!\!N(\alpha)} }$$
It has got combinatorial proof in cobweb posets language as it is the case with all cobweb-tiling sequences [@md3].
It might be expressed also in bipartite graphs terms with the use of a standard counting rule. We fix the last $n$-th vertex and separate family of all graphs $G(\alpha,n,k)$ into two disjoint classes depending on that, where the vertex is assigned.
Let $c_\alpha(\vec{b})$ be the number of labeled $k$-colored $\alpha$-multigraphs $G(\alpha,\vec{b})$ with $n$ vertices, where $\vec{b} = \langle b_1,b_2,...,b_k \rangle$ such that $b_1$ vertices are colored by first color, next $b_2$ vertices by another one and so on. Then $c_\alpha(\vec{b})$ is
$$c_\alpha(\vec{b}) = {n \choose {b_1,b_2,...,b_k}} \cdot \alpha^{\frac{1}{2}\left(n^2 - b_1^2 - b_2^2 - ... - b_k^2\right)}$$
while $n = b_1 + b_2 + ... + b_k$.
[*[**Proof.**]{}*]{} Take any such vector $\vec{b} = \langle b_1,b_2,...,b_k \rangle$. The coloring of $n$ vertices might be chosen on ${n \choose {b_1,b_2,...,b_k}}$ ways. Additionally, if $\alpha = 1$ then a graph $G(\alpha,\vec{b})$ has no edges but if $\alpha > 1$, then there might be created at most $(\alpha - 1)$ edges on $\alpha^{b_i\cdot b_j}$ ways between any two vertices from disjoint vertices’ sets $V_i$, $V_j$ where $i\neq j$. Therefore the overall number of all possibilities is $$\alpha^{\sum_{1\leq i < j \leq k} b_i\cdot b_j}
= \alpha^{b_1 b_2 + b_1 b_3 + ... + b_{k-1} b_k}
= \alpha^{\frac{1}{2}\left(n^2 - b_1^2 - b_2^2 - ... - b_k^2\right)}$$
Hence the thesis $\blacksquare$
\[cor:2\] Let $F$ be a cobweb tiling sequence $N(\alpha) \in \mathcal{T}_\lambda$ such that $1_F=1$. Then the multi $F$-nomial coefficient is equal to the number $c_\alpha(\vec{b})$ of labeled bipartite $\alpha$-multi graphs $G(\alpha,\vec{b})$ i.e.
$${ {{n} \choose {b_1,b_2,...,b_k}}_{\!\!F} } = c_\alpha(\vec{b})$$
where $\vec{b} = \langle b_1,b_2,...,b_k \rangle$ and $b_1 + b_2 + ... + b_k = n$
**Note.** Let $\gamma_{\alpha,n,k}$ be the number of all $k$-colored $\alpha$-multigraphs with $n$ vertices. Then
$$\gamma_{\alpha,n,k} = \sum_{{b_1+...+b_k=n \atop b_1,...,b_k\geq 0}} { {{n} \choose {b_1,b_2,...,b_k}}_{\!\!N(\alpha)} }$$
The case of $\alpha=2$ i.e. when $G(\alpha,\vec{b})$ is a $k$-colored graph without multiple edges was already considered in [@finch].
Counting labeled directed acyclic $\alpha$-multigraphs ($\alpha$-DAGs)
=======================================================================
A directed acyclic graph with $\alpha$-multiple edges i.e. any two vertices might be connected by at most $(\alpha-1)$ directed edges is called acyclic $\alpha$-multi digraph ($\alpha$-DAG for short).
\[lem:1\] Let $A_\alpha(n)$ denotes the number of acyclic $\alpha$-multi digraphs ($\alpha$-DAGs) with $n$ labeled nodes. Then for $n\in\mathbf{N}$
$$\label{eq:dags}
A_\alpha(n) = \sum_{k\geq 1} (-1)^{k+1} { {{n} \choose {k}}_{\!\!N(\alpha)} }\cdot A_\alpha(n-k)$$
while $A_\alpha(0) = 1$ and $\alpha\in\mathbb{N}$.
[*[**Proof.**]{}*]{} The main idea of the proof comes from [@robin] (see also [@robin0; @robin1]) where particular case of $\alpha=2$ is considered with the help of inclusion-exclusion principle. One shows that any directed acyclic multi-graph with no cyclic paths has at least one vertex with in-degree equal to zero (such vertices are so-called *out-points* [@robin]).
Take $\alpha\in\mathbb{N}$ and a graph $\alpha$-DAG with $n\in\mathbb{N}$ nodes. Denote by $X_i$ a family of $\alpha$-DAGs, such that $i$-th point is an out-point for $1\leq i \leq n$. Therefore $A_\alpha(n) = \left| \bigcup_{i=1}^{n} X_i \right|$ and from inclusion-exclusion principle
$$A_\alpha(n) = \left| \bigcup_{i=1}^{n} X_i \right|
= \sum_{k=1}^{n} (-1)^{k+1} \sum_{1\leq b_1<...<b_k\leq n} \left| X_{b_1} \cap ... \cap X_{b_k} \right|$$
Let us consider the number $k\in[n]$ of out-points. We can label them on ${n\choose k}$ ways. Next, there are $\alpha^{k(n-k)}$ possibilities to eventually create $\alpha$-multiple edges from these $k$ points to the rest $(n-k)$ ones for which we can create $A_\alpha(n-k)$ $\alpha$-DAGs, thus $$\sum_{1\leq b_1<...<b_k\leq n} \left| X_{b_1} \cap ... \cap X_{b_k} \right|
= {n \choose k}\alpha^{k(n-k)}\cdot A_\alpha(n-k)$$
Therefore
$$A_\alpha(n) = \sum_{k=1}^{n} (-1)^{k+1} {n \choose k} \alpha^{k(n-k)} \cdot A_\alpha(n-k)$$
and according to Corollary \[cor:1\]
$$A_\alpha(n) = \sum_{k=1}^{n} (-1)^{k+1} { {{n} \choose {k}}_{\!\!N(\alpha)} } \cdot A_\alpha(n-k)$$
Hence the thesis $\blacksquare$
Let $F$-cobweb admissible sequence be given. Then an inversion formula for $F$-nomial coefficients derived in [@md4] is of the form
$${ {{n} \choose {k}}_{\!\!F} }^{-1} = { {{n} \choose {k}}_{\!\!F} } { {{n-k} \choose {0}}_{\!\!F} }^{-1}, \ \ \ \ { {{n} \choose {n}}_{\!\!F} }^{-1} = 1$$ $${ {{n} \choose {0}}_{\!\!F} }^{-1} = \sum_{s=1}^k (-1)^s \sum_{k_1+...+k_s=n \atop k_1,...,k_s\geq 1} { {{n} \choose {k_1,k_2,...,k_s}}_{\!\!F} }$$
Here is the inversion matrix $\mathbf{M}^{-1} \equiv { {{i} \choose {j}}_{\!\!F} }^{-1}$ of matrix $\mathbf{M}$ from previous section example i.e. for $F=N(2)$.
$$M = \left[
\begin{array}{llllllll}
{\scriptstyle}1 \\
{\scriptstyle}-1 & {\scriptstyle}1 \\
{\scriptstyle}3 & {\scriptstyle}-4 & {\scriptstyle}1 \\
{\scriptstyle}-25 & {\scriptstyle}36 & {\scriptstyle}-12 & {\scriptstyle}1 \\
{\scriptstyle}543 & {\scriptstyle}-800 & {\scriptstyle}288 & {\scriptstyle}-32 & {\scriptstyle}1 \\
{\scriptstyle}-29281 & {\scriptstyle}43440 & {\scriptstyle}-16000 & {\scriptstyle}1920 & {\scriptstyle}-80 & {\scriptstyle}1 \\
{\scriptstyle}3781503 & {\scriptstyle}-5621952 & {\scriptstyle}2085120 & {\scriptstyle}-256000 & {\scriptstyle}11520 & {\scriptstyle}-192 & {\scriptstyle}1 \\
\end{array}
\right]$$
Let $F$ be a cobweb tiling sequence $F=N(\alpha)\in\mathcal{T}_\lambda$, such that $1_F=1$ and let $A_\alpha(n)$ denotes the number of labeled acyclic $\alpha$-multi digraphs with $n$ vertices. Then for $n\in\mathbb{N}$
$$A_\alpha(n) = (-1)^n { {{n} \choose {0}}_{\!\!F} }^{-1} = \left| { {{n} \choose {0}}_{\!\!F} }^{-1} \right|$$
where $A_\alpha(0) = 1$ and ${ {{n} \choose {k}}_{\!\!F} }^{-1}$ stays for an inversion matrix of $F$-nomial coefficients.
[*[**Proof.**]{}*]{} Take any $n\in\mathbb{N}$. If Lemma \[lem:1\] is taken into account, then $$A_\alpha(n) = \sum_{1\leq k_1\leq n} (-1)^{k_1+1} { {{n} \choose {k_1}}_{\!\!F} }\cdot A_\alpha(n-k_1)$$
While expanding the above we set new sums’ variables as $k_1,k_2,...,k_n$ and there are at the most $n$ variables $k_s$ with each of them equal one, according to the conditions $k_1+...+k_n = n$, with $k_i \geq 0$
$$\begin{aligned}
\lefteqn{
A_\alpha(n) =
\sum_{k_1=1}^{n} (-1)^{k_1+1}{ {{n} \choose {k_1}}_{\!\!F} }
\sum_{k_2=1}^{n-k_1} (-1)^{k_2+1}{ {{n-k_1} \choose {k_2}}_{\!\!F} }
\cdots
} & &
\nonumber\\
& & \cdots \sum_{k_n=1}^{n-k_1-...-k_{n-1}} (-1)^{k_n+1}{ {{n-k_1-...-k_{n-1}} \choose {k_n}}_{\!\!F} }
\cdot A_\alpha(0) \end{aligned}$$
and consequently
$$A_\alpha(n) = \sum_{I}
(-1)^{k_1+k_2+...+k_n + S}
{ {{n} \choose {k_1,k_2,...,k_n}}_{\!\!F} }$$
where $S$ is the number of variables $k_1,k_2,...,k_n$ with positive value and $$I = \left\{ \begin{array}{l}
{\scriptstyle}0\leq k_1 \leq n \\
{\scriptstyle}0\leq k_2 \leq n-k_1 \\
{\scriptstyle}... \\
{\scriptstyle}0\leq k_n \leq n-k_1-...-k_{n-1}
\end{array} \right.$$
Now, let us rearrange the sum into two summations as follows
$$A_\alpha(n) = \sum_{s=1}^n (-1)^s \sum_{k_1+...+k_s=n \atop k_1,...,k_s\geq 1} \!\!\!\!\!(-1)^n { {{n} \choose {k_1,...,k_s}}_{\!\!F} }
= (-1)^n { {{n} \choose {0}}_{\!\!F} }^{-1}$$
The value of $A_\alpha(n)$ is positive for any natural $n$, hence the thesis $\blacksquare$
**Acknowledgements**
I would like to thank Professor A. Krzysztof Kwaśniewski - who initiated my interest in his cobweb poset concept - for his very helpful comments and improvements of this note.
[99]{}
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study the spectral sequence associated to the filtration by powers of the augmentation ideal on the (twisted) equivariant chain complex of the universal cover of a connected CW-complex $X$. In the process, we identify the $d^1$ differential in terms of the coalgebra structure of $H_*(X,{\Bbbk})$, and the ${\Bbbk}\pi_1(X)$-module structure on the twisting coefficients. In particular, this recovers in dual form a result of Reznikov, on the mod $p$ cohomology of cyclic $p$-covers of aspherical complexes. This approach provides information on the homology of all Galois covers of $X$. It also yields computable upper bounds on the ranks of the cohomology groups of $X$, with coefficients in a prime-power order, rank one local system. When $X$ admits a minimal cell decomposition, we relate the linearization of the equivariant cochain complex of the universal abelian cover to the Aomoto complex, arising from the cup-product structure of $H^*(X,{\Bbbk})$, thereby generalizing a result of Cohen and Orlik.'
address:
- 'Institute of Mathematics Simion Stoilow, P.O. Box 1-764, RO-014700 Bucharest, Romania'
- 'Department of Mathematics, Northeastern University, Boston, MA 02115, USA'
author:
- Stefan Papadima$^1$
- 'Alexander I. Suciu$^2$'
title: The spectral sequence of an equivariant chain complex and homology with local coefficients
---
[^1]
[^2]
Introduction {#sect:intro}
============
The equivariant chain complex {#intro:eq chain}
-----------------------------
In his pioneering work from the late 1940s, J.H.C. Whitehead established the category of CW-complexes as the natural framework for much of homotopy theory. In [@Wh], he highlighted the key role played by the cellular chain complex of the universal cover, ${\widetilde{X}}$, of a connected CW-complex $X$. Among other things, Whitehead showed that a map $f\colon X\to Y$ is a homotopy equivalence if and only if the induced map between equivariant chain complexes, $\tilde{f_*}\colon C_{\bullet} ({\widetilde{X}},{\mathbb{Z}})\to C_{\bullet} ({\widetilde{Y}},{\mathbb{Z}})$, is an equivariant chain-homotopy equivalence. For $2$-dimensional complexes, the fundamental group, together with the equivariant chain-homotopy type of $C_{\bullet} ({\widetilde{X}},{\mathbb{Z}})$, constitute a complete set of homotopy type invariants: if $X$ and $Y$ are two such spaces, with $\pi_1(X) \cong \pi_1(Y)$ and $C_{\bullet} ({\widetilde{X}},{\mathbb{Z}})\simeq
C_{\bullet} ({\widetilde{Y}},{\mathbb{Z}})$, then $X\simeq Y$.
As noted by S. Eilenberg [@Ei], the equivariant chain complex is tightly connected to homology with twisted coefficients: Given a linear representation, $\rho\colon \pi_1(X) \to \operatorname{GL}(n,{\Bbbk})$, the homology of $X$ with coefficients in the local system ${}_{\rho}{\Bbbk}^n$ is $H_* ({}_{\rho}{\Bbbk}^n \otimes_{{\Bbbk}\pi_1(X)}
C_{\bullet}({\widetilde{X}},{\Bbbk}))$.
In this paper, we revisit these classical topics, drawing much of the motivation from recent work on the topology of complements of hyperplane arrangements, and the study of cohomology jumping loci. One of our main goals is to give tight upper bounds for the twisted Betti ranks, computable in terms of much simpler data, involving only ordinary cohomology. The basic tool for our approach is a spectral sequence, which we now proceed to describe.
The equivariant spectral sequence {#intro:thmA}
---------------------------------
Let ${\Bbbk}\pi$ be the group ring of a group $\pi$, over a coefficient ring ${\Bbbk}$, and let $M$ be a right ${\Bbbk}\pi$-module. The successive powers of the augmentation ideal, $I=I_{{\Bbbk}}(\pi)$, determine a filtration on $M$; the associated graded object, $\operatorname{gr}(M)=\bigoplus_{n\ge 0} MI^n /MI^{n+1}$, is a module over the ring $\operatorname{gr}({\Bbbk}\pi)$.
Now let $X$ be a connected CW-complex, with fundamental group $\pi=\pi_1(X)$, and let $C_{\bullet}(X, M)=M\otimes_{{\Bbbk}\pi} C_{\bullet}({\widetilde{X}},{\Bbbk})$ be the equivariant chain complex of $X$ with coefficients in $M$, discussed in §\[sec:equivchain\]. The $I$-adic filtration on $C_{\bullet}(X, M)$ is compatible with the boundary maps, and thus gives rise to a spectral sequence, $E^{\bullet}(X, M)$, as explained in §\[sec:spectral\]. Under some mild conditions, we identify in §\[sec:e1 page\] the differential $d^1\colon E^1\to E^1$, solely in terms of the coalgebra structure of $H_*(X,{\Bbbk})$ and the $\operatorname{gr}({\Bbbk}\pi)$-module structure on $\operatorname{gr}(M)$.
\[thm:A\] There is a second-quadrant spectral sequence, $\{E^r(X,M),d^r\}_{r\ge 1}$, with $E^1_{-p,p+q}(X,M)=H_{q}(X,\operatorname{gr}^p(M))$. If ${\Bbbk}$ is a field, or ${\Bbbk}={\mathbb{Z}}$ and $H_*(X,{\mathbb{Z}})$ is torsion-free, then $E^1_{-p,p+q}(X,M)=\operatorname{gr}^p(M) \otimes_{{\Bbbk}} H_{q}(X,{\Bbbk})$, and the $d^1$ differential decomposes as $$\xymatrixcolsep{42pt}
\xymatrixrowsep{10pt}
\xymatrix{ \operatorname{gr}^p(M) \otimes_{{\Bbbk}} H_q \ar^(.4){\operatorname{id}\otimes \nabla_X}[r]
& \operatorname{gr}^p(M)\otimes_{{\Bbbk}} (H_1\otimes_{{\Bbbk}} H_{q-1}) \ar^{\cong}[d]
\\
& (\operatorname{gr}^p(M)\otimes_{{\Bbbk}} \operatorname{gr}^1({\Bbbk}\pi))\otimes_{{\Bbbk}} H_{q-1}
\ar^(.57){\operatorname{gr}(\mu_M) \otimes \operatorname{id}}[r]
& \operatorname{gr}^{p+1}(M) \otimes_{{\Bbbk}} H_{q-1}\,,
}$$ where $\nabla_X$ is the comultiplication map on $H_*=H_*(X,{\Bbbk})$, and $\mu_M\colon M\otimes_{{\Bbbk}} {\Bbbk}\pi\to M$ is the multiplication map.
Full details are given in Theorem \[thm:d1map\]. In the case when $X$ is of finite type, $d^1$ is determined by the cup-product structure in $H^*(X,{\Bbbk})$ and the map $\operatorname{gr}(\mu_M)$.
The idea to use powers of augmentation ideals to define a second quadrant spectral sequence in terms of group presentations goes back to J. Stallings [@St]. For more on the Stallings spectral sequence, see [@Gr], [@Wa].
In §\[sec:convergence\], we study the convergence properties of the spectral sequence from Theorem \[thm:A\]. Under fairly general assumptions, $E^{\bullet}(X,M)$ has an $E^{\infty}$ term. Yet, as we show in Example \[ex:non conv\], there are finite CW-complexes $X$ for which $E^{\bullet}(X,{\Bbbk}\pi)$ does not converge.
Base change {#intro:base change}
-----------
To obtain more structure in the spectral sequence, we restrict in §\[sect:change rings\] to a special situation. Suppose $\nu\colon \pi{\twoheadrightarrow}G$ is an epimorphism onto a group $G$; then the group ring ${\Bbbk}{G}$ becomes a right ${\Bbbk}\pi$-module, via extension of scalars. The resulting spectral sequence, $E^{\bullet}(X,{\Bbbk}{G}_{\nu})$, is a spectral sequence in the category of left $\operatorname{gr}_J({\Bbbk}{G})$-modules, where $J$ is the augmentation ideal of ${\Bbbk}{G}$. In Proposition \[prop:d1 kg\], we describe the differential $d^1_G$, solely in terms of the induced homomorphism, $\nu_*\colon H_1(X,{\Bbbk}) \to H_1(G,{\Bbbk})$, and the comultiplication map, $\nabla_X$.
Note that $H_*(X,{\Bbbk}{G}_{\nu})=H_*(Y,{\Bbbk})$, where $Y\to X$ is the Galois $G$-cover defined by $\nu$. The homology groups of $Y$ support two natural filtrations: $F^{\bullet}$, coming from the spectral sequence, and $J^{\bullet}$, by the powers of the augmentation ideal. We then have an inclusion, $J^k \cdot H_*(Y,{\Bbbk}) \subseteq F^{k} H_*(Y,{\Bbbk})$. Equality holds for $G={\mathbb{Z}}$, as we show in Lemma \[lem:equal filt\], but in general the two filtrations do not agree, even when $G={\mathbb{Z}}^2$.
In §\[sec:completions\] we study the more general situation when $\nu\colon \pi{\twoheadrightarrow}G$ is an epimorphism to an abelian group, making use of the general machinery developed by J.P. Serre in [@Ser]. Assuming $X$ is of finite type and ${\Bbbk}$ is a field, the spectral sequence $E^{\bullet}(X,{\Bbbk}{G}_{\nu})$ converges, and computes the $J$-adic completion of $H_*(X,{\Bbbk}{G}_{\nu})$. Moreover, if $X$ is a finite CW-complex, the spectral sequence collapses in finitely many steps.
In the case when $X$ is a $K(\pi,1)$ space, $G={\mathbb{Z}}_p$, and ${\Bbbk}={\mathbb{F}}_p$, discussed separately in §\[sec:kzp\], the spectral sequence $E^{\bullet}(X,{\Bbbk}{G}_{\nu})$ is the homological version of a spectral sequence first considered by A. Reznikov in [@Re].
Monodromy action {#intro:thmB}
----------------
In §\[sec:mono\], we focus exclusively on infinite cyclic covers, analyzing the homology groups $H_{q}(X,{\Bbbk}{\mathbb{Z}}_{\nu})$, viewed as modules over the Laurent polynomial ring, ${\Bbbk}{\mathbb{Z}}={\Bbbk}[t^{\pm 1}]$. We assume ${\Bbbk}$ is a field, so that ${\Bbbk}[t^{\pm 1}]$ is a PID. Given an element $a\in H^1(X,{\Bbbk})$ with $a^2=0$, left-multiplication by $a$ turns the cohomology ring of $X$ into a cochain complex, $(H^*( X,{\Bbbk}),\cdot a)\colon H^0(X,{\Bbbk}) \xrightarrow{a}
H^1(X,{\Bbbk}) \xrightarrow{a} H^2(X,{\Bbbk}) \xrightarrow{a} \cdots$.
\[thm:B\] Let $X$ be a connected, finite-type CW-complex, $\nu\colon \pi_1(X){\twoheadrightarrow}{\mathbb{Z}}$ an epimorphism, and $\nu_{{\Bbbk}}\in H^1(X,{\Bbbk})$ the corresponding cohomology class. Then, for all $q\ge 0$:
1. \[B1\] The $\operatorname{gr}_J({\Bbbk}{\mathbb{Z}})$-module structure on $E^{\infty}(X, {\Bbbk}{\mathbb{Z}}_{\nu})$ determines $P^q_0$ and $P^{q}_{t-1}$, the free and $(t-1)$-primary parts of $H_q(X, {\Bbbk}{\mathbb{Z}}_{\nu})$, viewed as ${\Bbbk}[t^{\pm 1}]$-modules.
2. \[B2\] The monodromy action of ${\mathbb{Z}}$ on $P^{j}_0 \oplus
P^{j}_{t-1}$ is trivial for all $j\le q$ if and only if $H^j(H^*( X,{\Bbbk}) , \cdot\nu_{{\Bbbk}})=0$, for all $j\le q$.
For a more detailed statement, see Propositions \[prop:monospsq\] and \[prop:trivial action\]. Part has an analog, for $\nu\colon \pi_1(X) {\twoheadrightarrow}{\mathbb{Z}}_p$ and ${\Bbbk}= {\mathbb{F}}_p$ ($p\ne 2$); see Proposition \[prop:jordan block\].
Particularly interesting is the case of a smooth manifold $X$ fibering over the circle, with $\nu\colon \pi {\twoheadrightarrow}{\mathbb{Z}}$ the homomorphism induced on $\pi_1$ by the projection map, $p\colon X\to S^1$. The homology of the resulting infinite cyclic cover was studied by J. Milnor in [@Mil2]. This led to another spectral sequence, introduced by M. Farber in [@Fa85], and further developed by S.P. Novikov in [@Nov]. The Farber-Novikov spectral sequence has $(E_1, d_1)$-page dual to our $(E^1(X,{\Bbbk}{\mathbb{Z}}_{\nu}), d^1_{\nu})$-page, and higher differentials given by certain Massey products, see [@Fa]. Their spectral sequence, though, converges to the free part of $H_*(X,{\Bbbk}{\mathbb{Z}}_\nu)$, and thus misses the information on the $(t-1)$-primary part captured by the equivariant spectral sequence. The spectral sequence $E^{\bullet}(X,{\Bbbk}{\mathbb{Z}}_{\nu})$ was also investigated by G. Denham in [@De], in the special case when $X$ is the complement of a complexified arrangement of real hyperplanes through the origin of ${\mathbb{R}}^{\ell}$, and $p$ is the Milnor fibration.
Theorem \[thm:B\] has already found several applications in the literature. In [@PST], we analyze the monodromy action on the homology of Galois ${\mathbb{Z}}$–covers, for toric complexes associated to finite simplicial complexes. Using Part of the Theorem, we obtain a combinatorial criterion for the full triviality of this action, up to a given degree. In [@PS08], Theorem \[thm:B\] yields a new formality criterion, and a purely topological proof of a basic result on the monodromy action, for the homology of Milnor fibers of plane curve singularities.
Bounds on twisted Betti ranks {#intro:thmD}
-----------------------------
Computing cohomology groups with coefficients in a rank $1$ local system can be an arduous task. It is thus desirable to have efficient, readily computable bounds for the ranks of these groups. We supply such bounds in §§\[sect:betti bounds\]–\[sect:aomoto bounds\].
Given a non-zero complex number $\zeta\in {\mathbb{C}}^{\times}$ and a homomorphism $\nu\colon \pi\to {\mathbb{Z}}$, define a representation $\rho\colon \pi\to {\mathbb{C}}^{\times}$ by $\rho(g)=\zeta^{\nu(g)}$. If $\zeta$ is a $d$-th root of unity, such a homomorphism $\rho$ is called a rational character (of order $d$).
\[thm:D\] Let $X$ be a connected, finite-type CW-complex, and let $\rho\colon \pi_1(X)\to {\mathbb{C}}^{\times}$ be a rational character, of prime-power order $d=p^r$. Then, for all $q\ge 0$, $$\dim_{{\mathbb{C}}} H^q(X, {}_{\rho}{\mathbb{C}}) \le
\dim_{{\mathbb{F}}_p} H^q(X,{\mathbb{F}}_p).$$ If, moreover, $H_*(X,{\mathbb{Z}})$ is torsion-free, $$\dim_{{\mathbb{C}}} H^q(X, {}_{\rho}{\mathbb{C}}) \le
\dim_{{\mathbb{F}}_p} H^q(H^{*}(X,{\mathbb{F}}_p),\nu_{{\mathbb{F}}_p}).$$
The proof is given in Theorems \[thm:bettibound\] and \[thm:cohobound\]. Neither of these two inequalities can be sharpened further: we give examples showing that both the prime-power order hypothesis on $d$ and the torsion-free hypothesis on $H_*(X,{\mathbb{Z}})$ are really necessary.
The second inequality above generalizes a result of D. Cohen and P. Orlik ([@CO Theorem 1.3]), valid only when $X$ is the complement of a complex hyperplane arrangement, and $r=1$. This inequality is used in a crucial way in [@MP]. It yields a purely combinatorial description of the monodromy action on the degree $1$ rational homology of the Milnor fiber, for arbitrary subarrangements of classical Coxeter arrangements.
Minimality and linearization {#intro:thmE}
----------------------------
Using the equivariant spectral sequence of a Galois cover, we give in §\[sect:minilin\] an intrinsic meaning to linearization of equivariant (co)chain complexes, in the case when the CW-complex $X$ admits a cell structure with minimal number of cells in each dimension.
Pick a basis $\{ e_1,\dots, e_n \}$ for $H_1=H_1(X,{\Bbbk})$, and identify the symmetric algebra $S=\operatorname{Sym}(H_1)$ with the polynomial ring ${\Bbbk}[e_1,\dots, e_n]$. The [*universal Aomoto complex*]{} of $H^*=H^*(X,{\Bbbk})$ is the cochain complex of free $S$-modules, $$\xymatrix{H^0 \otimes_{{\Bbbk}} S \ar^{D^0}[r] &
H^1 \otimes_{{\Bbbk}} S \ar^{D^1}[r] &
H^2 \otimes_{{\Bbbk}} S \ar^(.6){D^2}[r] &
\cdots},$$ with differentials $D(\alpha \otimes 1)= \sum_{i=1}^{n}
e_i^* \cdot \alpha \otimes e_i$.
\[thm:E\] Let $X$ be a minimal CW-complex, and assume ${\Bbbk}={\mathbb{Z}}$, or a field.
1. \[E1\] Let $\nu\colon \pi{\twoheadrightarrow}G$ be an epimorphism. Then the linearization of the equivariant chain complex, $(C_{\bullet}(X,{\Bbbk}G_{\nu}), \tilde{\partial}_{\bullet}^G)$, is equal to the $E^1$-term of the equivariant spectral sequence, $(E^1(X,{\Bbbk}G_{\nu}), d_G^1)$.
2. \[E2\] The linearization of the equivariant cochain complex of the universal abelian cover of $X$ coincides with the universal Aomoto complex of $H^*(X,{\Bbbk})$.
Part —suitably interpreted with the aid of Theorem \[thm:A\]—was proved in [@DP1 Theorem 20], in the case when $\nu =\operatorname{id}$ and ${\Bbbk}= {\mathbb{Z}}$, and under the additional assumption that the cohomology ring $H^*(X, {\mathbb{Z}})$ is generated in degree one.
Part generalizes [@CO Theorem 1.2], valid only for complements of complex hyperplane arrangements, and ${\Bbbk}={\mathbb{C}}$. Earlier results in this direction, also within the confines of arrangement theory, were obtained in [@CS99] and [@C]. The result in Part was recently proved by M. Yoshinaga [@Yo], under an additional condition, satisfied by arrangement complements, but not by arbitrary minimal CW-complexes. Theorem \[thm:E\] shows that this condition is unnecessary, thereby answering a question raised by Yoshinaga in Remark 15, preprint version of [@Yo].
The equivariant chain complex of a CW-complex {#sec:equivchain}
=============================================
In this section, we review a well-known construction, going back to J.H.C. Whitehead [@Wh] and S. Eilenberg [@Ei]: the chain complex of the universal cover of a cell complex $X$, with coefficients in a ${\Bbbk}\pi_1(X)$-module $M$. We start with some basic algebraic notions.
Associated graded rings {#subsec:gring}
-----------------------
Let $R$ be a ring, and $J$ a two-sided ideal. The successive powers of $J$ determine a descending filtration on $R$, called the [*$J$-adic filtration*]{}. The [*associated graded ring*]{}, $\operatorname{gr}_J(R)$, is defined as $$\label{eq:gr R}
\operatorname{gr}_J(R)=\bigoplus_{n\ge 0} J^n /J^{n+1}.$$ On homogeneous components, the multiplication map $\operatorname{gr}_J^n(R) \otimes \operatorname{gr}_J^m(R) \to \operatorname{gr}_J^{n+m}(R)$ is induced from multiplication in the ring $R$.
We will be mainly interested in the following situation. Let $\pi$ be a group, and let ${\Bbbk}$ be a commutative ring with unit $1$. The [*group ring*]{} of $\pi$, denoted ${\Bbbk}\pi$, consists of all finite linear combinations of group elements with coefficients in ${\Bbbk}$. Multiplication in ${\Bbbk}\pi$ is induced from the group operation on $\pi$; the unit (denoted by $1$) is $1\cdot \iota$, where $\iota$ is the identity of $\pi$. The [*augmentation ideal*]{}, $I=I_{{\Bbbk}}\pi$, is the kernel of the ring homomorphism $\epsilon \colon {\Bbbk}\pi \to {\Bbbk}$, $\sum n_g g\mapsto \sum n_g$; as a ${\Bbbk}$-module, $I$ is freely generated by the elements $g-1$, for $g\ne \iota$.
The augmentation ideal defines the $I$-adic filtration on ${\Bbbk}\pi$; let $\operatorname{gr}({\Bbbk}\pi)=\operatorname{gr}_I({\Bbbk}\pi)$ be the associated graded ring. Note that $\operatorname{gr}({\Bbbk}\pi)$ is always generated as a ring in degree $1$. Clearly, $\operatorname{gr}^0({\Bbbk}\pi) \cong {\Bbbk}$. Moreover, the map $\pi \to I$, $g\mapsto g-1$ factors through an isomorphism $H_1(\pi,{\Bbbk})\to I/I^2$ (see [@HS]). Thus, $$\label{eq:gr1}
\operatorname{gr}^1({\Bbbk}\pi)\cong H_1(\pi,{\Bbbk}).$$
As an example, consider the free abelian group $\pi={\mathbb{Z}}^n$. The group ring ${\Bbbk}{\mathbb{Z}}^n$ can be identified with the Laurent polynomial ring ${\Bbbk}[t_1^{\pm 1},\dots ,t_n^{\pm 1}]$, while $\operatorname{gr}({\Bbbk}{\mathbb{Z}}^n)$ can be identified with ${\Bbbk}[x_1,\dots ,x_n]$, the polynomial ring in $n$ variables, via the map $t_i-1\mapsto x_i$.
Associated graded modules {#subsec:func1}
-------------------------
A similar construction applies to modules: if $M$ is a left ${\Bbbk}\pi$-module, then $M$ is filtered by the submodules $\{ I^n M \}_{n\ge 0}$. The associated graded object, $\operatorname{gr}_I(M)=\bigoplus_{n\ge 0} I^n M / I^{n+1} M$, inherits a graded module structure over $\operatorname{gr}({\Bbbk}\pi)$. This module is generated in degree $0$ by the module of coinvariants, $M/{I M}$.
All these constructions are functorial. For example, if $\alpha\colon \pi\to \pi'$ is a group homomorphism, then the linear extension to group rings, $\bar\alpha\colon {\Bbbk}\pi\to {\Bbbk}\pi'$, is a ring map, preserving the respective $I$-adic filtrations. Thus, $\bar\alpha$ induces a degree $0$ ring homomorphism, $\operatorname{gr}(\alpha)\colon \operatorname{gr}({\Bbbk}\pi)\to \operatorname{gr}({\Bbbk}\pi')$.
Given a ${\Bbbk}\pi$-module $M$ and a ${\Bbbk}\pi'$-module $M'$, a ${\Bbbk}$-linear map $\phi\colon M\to M'$ is said to be equivariant with respect to the ring map $\bar\alpha\colon {\Bbbk}\pi\to {\Bbbk}\pi'$ if $\phi(gm)=\alpha(g) \phi(m)$, for all $g\in \pi$ and $m\in M$. Such a map $\phi$ preserves $I$-adic filtrations, and thus induces a map $\operatorname{gr}(\phi)\colon \operatorname{gr}_I(M)\to \operatorname{gr}_I(M')$, equivariant with respect to $\operatorname{gr}(\alpha)$.
Completely analogous considerations apply to right ${\Bbbk}\pi$-modules.
Chains on the universal cover {#subsec:equivchain}
-----------------------------
Let $X$ be a connected CW-complex, with skeleta $\{X^q\}_{q\ge 0}$. Up to homotopy, we may assume $X$ has a single $0$-cell, call it $e_0$, which we will take as the basepoint. Moreover, we may assume all attaching maps $(S^q,*) \to (X^q,e_0)$ are basepoint-preserving.
Fix a commutative ring ${\Bbbk}$ with unit, and denote by $C_{\bullet}(X, {\Bbbk})=(C_q(X, {\Bbbk}),\partial_q)_{q\ge 0}$ the cellular chain complex of $X$, with coefficients in ${\Bbbk}$. Recall $C_q(X, {\Bbbk})=H_q(X^q,X^{q-1},{\Bbbk})$ is a free ${\Bbbk}$-module, with basis indexed by the $q$-cells of $X$.
Let $p\colon {\widetilde{X}}\to X$ be the universal cover. The cell structure on $X$ lifts to a cell structure on ${\widetilde{X}}$. Fixing a lift $\tilde{e}_0\in p^{-1}(e_0)$ identifies the fundamental group $\pi=\pi_1(X,e_0)$ with the group of deck transformations of ${\widetilde{X}}$, which permute the cells. Therefore, we may view $$\label{eq:equiv}
C_{\bullet}({\widetilde{X}}, {\Bbbk})=(C_q({\widetilde{X}}, {\Bbbk}),\tilde{\partial}_q)_{q\ge 0}$$ as a chain complex of left-modules over the group ring ${\Bbbk}\pi$. We shall call $C_{\bullet}({\widetilde{X}},{\Bbbk})$ the [*equivariant chain complex*]{} of $X$, over ${\Bbbk}$.
Using the action of $\pi$ on ${\widetilde{X}}$, we may identify $$\label{eq:tensor}
C_q({\widetilde{X}},{\Bbbk}) \cong {\Bbbk}\pi \otimes_{\Bbbk}C_q(X,{\Bbbk})$$ as left ${\Bbbk}\pi$-modules. Under this identification, a basis element of the form $1\otimes e$ from the right hand side corresponds to the unique lift $\tilde{e}$ of the $q$-cell $e$ sending the basepoint $*\in D^{q}$ to $\tilde{e}_0$.
These constructions are functorial, in the following sense. Suppose $f\colon X \to X'$ is a map between connected CW-complexes. By cellular approximation, we may assume $f$ respects the CW-structures; in particular, $f(e_0)=e'_0$. Let $f_*\colon C_{\bullet} (X,{\Bbbk}) \to C_{\bullet}(X',{\Bbbk})$ be the induced map on cellular chain complexes, and let $f_{\sharp}\colon \pi \to \pi'$ be the induced homomorphism on fundamental groups. By covering space theory, the map $f$ lifts to a cellular map, $\tilde{f}\colon {\widetilde{X}}\to {\widetilde{X}}'$, uniquely specified by the requirement that $\tilde{f}(\tilde{e}_0)=\tilde{e}'_0$. The induced chain map, $\tilde{f}_*\colon C_{\bullet} ({\widetilde{X}},{\Bbbk}) \to C_{\bullet}({\widetilde{X}}',{\Bbbk})$, is equivariant with respect to the ring homomorphism $\bar{f}_{\sharp} \colon {\Bbbk}\pi\to {\Bbbk}\pi'$.
The first differentials {#subsec:diff}
-----------------------
Let $\epsilon \colon {\Bbbk}\pi \to {\Bbbk}$ be the augmentation homomorphism. Under identification , the induced homomorphism $p_*\colon C_{q}({\widetilde{X}},{\Bbbk})\to C_{q}(X,{\Bbbk})$ coincides with $\epsilon \otimes \operatorname{id}\colon {\Bbbk}\pi \otimes_{{\Bbbk}}
C_q(X,{\Bbbk}) \to {\Bbbk}\otimes_{{\Bbbk}} C_q(X,{\Bbbk})$.
The first boundary map, $\tilde{\partial}_1$, is easy to write down. Let $e_1$ be a $1$-cell of $X$. Recall we assume $X$ has a single $0$-cell, $e_0$; thus, $e_1$ is a loop at $e_0$, representing an element $x=[e_1]\in \pi$. Let $\tilde{e}_1$ be the lift of $e_1$ at $\tilde{e}_0$. Clearly, $\tilde{\partial}_1(\tilde{e}_1)= (x-1) \tilde{e}_0$. Thus, $\tilde{\partial}_1\colon
{\Bbbk}\pi \otimes_{{\Bbbk}} C_1(X,{\Bbbk}) \to {\Bbbk}\pi \otimes_{{\Bbbk}} C_0(X,{\Bbbk})={\Bbbk}\pi$ is given by $$\label{eq:bdry1}
\tilde{\partial}_1(1 \otimes e_1) = x-1.$$
The second boundary map, $\tilde{\partial}_2\colon
{\Bbbk}\pi \otimes_{{\Bbbk}} C_2(X,{\Bbbk}) \to {\Bbbk}\pi \otimes_{{\Bbbk}} C_1(X,{\Bbbk})$, can be written by means of Fox derivatives [@Fox]. If $\{e^i_1\}_i$ are the $1$-cells of $X$, and $e_2$ is a $2$-cell, then $$\label{eq:fox der}
\tilde{\partial}_2 (1\otimes e_2) =
\sum_{i} \Big(\frac{\partial r}{\partial x_i}\Big)^{\phi} \otimes e^i_1,$$ where $r$ is the word in the free group $F$ on generators $x_i$, determined by the attaching map of the $2$-cell, and $\bar{\phi}\colon {\Bbbk}{F} \to {\Bbbk}\pi$ is the extension to group rings of the projection map $\phi \colon F{\twoheadrightarrow}\pi$. As for the higher boundary maps $\tilde{\partial}_{q}$, $q>2$, there is no general procedure for computing them, except in certain very specific situations.
If $X$ is an Eilenberg-MacLane $K(\pi,1)$ space, i.e., if ${\widetilde{X}}$ is contractible, then the augmented chain complex $\widetilde{C}_{\bullet} \colon
C_{\bullet}({\widetilde{X}},{\Bbbk}) \to {\Bbbk}$ is a free ${\Bbbk}\pi$-resolution of the trivial module ${\Bbbk}$. For instance, if $T^n=K({\mathbb{Z}}^n,1)$ is the $n$-torus, then $\widetilde{C}_{\bullet}$ is the Koszul complex over the ring ${\Bbbk}{\mathbb{Z}}^n={\Bbbk}[t_1^{\pm 1},\dots ,t_n^{\pm 1}]$.
Building up CW-complexes {#subsec:discuss}
------------------------
Given a group $\pi$, and a matrix $D$ over ${\mathbb{Z}}\pi$, it is possible to construct a CW-complex $X$ having $D$ as a block in one of the equivariant boundary maps in $C_{\bullet}({\widetilde{X}},{\mathbb{Z}})$.
\[ex:construction\] Let $X_0$ be a connected CW-complex, with fundamental group $\pi= \pi_1(X_0,e_0)$, and denote by $X_0\vee S^{n}$ the wedge sum of $X_0$ with the $n$-sphere, $n\ge 2$. Note that $\pi_1(X_0\vee S^n)=\pi$. The inclusion $S^n {\hookrightarrow}X_0\vee S^n$ defines an element $[S^n]\in \pi_n(X_0\vee S^n)$.
Given an element $x$ in ${\mathbb{Z}}\pi$, construct a new CW-complex, $X$, by attaching an $(n+1)$-cell along a map $\phi_x\colon S^n \to X_0\vee S^{n}$ representing $x \cdot [S^n] \in \pi_n(X_0\vee S^n)$. Clearly, $\pi_1(X)=\pi_1(X_0)$, and $C_{\bullet}(\widetilde{X},{\mathbb{Z}})= C_{\bullet}({\widetilde{X}}_0,{\mathbb{Z}}) \oplus C(n,x)$, where $C(n,x)$ denotes the elementary chain complex $C_{n+1}\to C_n$, with differential $\cdot x\colon {\mathbb{Z}}\pi\to {\mathbb{Z}}\pi$.
More generally, for any integer $n\ge 2$, and any ${\mathbb{Z}}\pi$-linear map $D\colon ({\mathbb{Z}}\pi)^{m} \to ({\mathbb{Z}}\pi)^{\ell}$, a construction much as above produces a CW-complex $X$, with $\pi_1(X)=\pi_1(X_0)$, and $C_{\bullet}({\widetilde{X}},{\mathbb{Z}})= C_{\bullet}({\widetilde{X}}_0,{\mathbb{Z}}) \oplus C(n,D)$, where $C(n,D)$ denotes the chain complex concentrated in degrees $n+1$ and $n$, with differential equal to $D$.
Coefficient modules {#subsec:modules}
-------------------
Let $X$ be a connected CW-complex, with fundamental group $\pi$. Suppose $M$ is a right ${\Bbbk}\pi$-module. The cellular chain complex of $X$ with coefficients in $M$ is defined as $$\begin{aligned}
\label{eq:cxm}
C_{\bullet}(X, M)&=\big( C_q(X, M) ,\: \tilde{\partial}^M_q\big)_{q\ge 0}
\\
\notag &=\big(M \otimes_{{\Bbbk}\pi} C_q({\widetilde{X}}, {\Bbbk}) ,\:
\operatorname{id}_M \otimes_{{\Bbbk}\pi}\, \tilde{\partial}_q\big)_{q\ge 0} . \end{aligned}$$ In the particular case when $M$ is the free ${\Bbbk}\pi$-module of rank one, $C_{\bullet}(X, {\Bbbk}\pi)$ coincides with the equivariant chain complex $C_{\bullet}({\widetilde{X}}, {\Bbbk})$.
Similarly, if $M$ is a left ${\Bbbk}\pi$-module, define the cellular cochain complex of $X$ with coefficients in $M$ as $$\begin{aligned}
\label{eq:cochains}
C^{\bullet}(X, M)&=\big( C^q(X, M) ,\: \tilde{\delta}^q_M\big)_{q\ge 0}
\\
\notag &=\big(\operatorname{{Hom}}_{{\Bbbk}\pi} (C_q({\widetilde{X}}, {\Bbbk}),M) ,\:
\operatorname{{Hom}}_{{\Bbbk}\pi}( \tilde{\partial}_{q}, M)
\big)_{q\ge 0} . \end{aligned}$$
\[ex:locsyst\] Let ${\Bbbk}$ be a field, and let $\rho\colon \pi\to \operatorname{GL}(n,{\Bbbk})$ be a linear representation of $\pi$. Then $M={\Bbbk}^n$ acquires a right module structure over ${\Bbbk}\pi$, via $mg = \rho(g^{-1})(m)$; such a module is called a rank $n$ local system on $X$, with monodromy $\rho$. We write $H_*(X, {}_{\rho}{\Bbbk}^n):=H_*(C_{\bullet}(X, M))$ for the homology of $X$ with coefficients in this local system. When $n=1$, there is no need to turn a representation into an anti-representation; in this case, we simply define $H_*(X, {\Bbbk}_{\rho})$ to be $H_*(C_{\bullet}(X, M))$, where $M={\Bbbk}$ is viewed as a [*right*]{} ${\Bbbk}\pi$-module, via $mg = \rho(g)(m)$.
\[ex:covers\] Let $Y\to X$ be a Galois cover, with group of deck transformations $G$ and classifying map $\nu\colon \pi\to G$. The cell structure on $X$ lifts in standard fashion to a cell structure on $Y$. Let ${\Bbbk}{G}_{\nu}$ denote the group-ring of $G$, viewed as a right ${\Bbbk}\pi$-module, via $h\cdot g =h\nu(g)$. Similarly, let ${}_{\nu}{\Bbbk}{G}$ denote the same group-ring, viewed as a left ${\Bbbk}\pi$-module, via $g\cdot h =\nu(g) h$. Then $C_{\bullet}(Y,{\Bbbk})=C_{\bullet}(X,{\Bbbk}{G}_{\nu})$, as chain complexes of left ${\Bbbk}{G}$-modules, and $C^{\bullet}(Y,{\Bbbk})=C^{\bullet}(X,{}_{\nu}{\Bbbk}{G})$, as cochain complexes of right ${\Bbbk}{G}$-modules.
Notable is the case of the universal abelian cover, $X^{\operatorname{{ab}}} \to X$, defined by the abelianization map, $\operatorname{{ab}}\colon \pi {\twoheadrightarrow}\pi_{\operatorname{{ab}}}$. The homology groups $H_q(X^{\operatorname{{ab}}},{\Bbbk})$, viewed as modules over the ring $\Lambda={\Bbbk}\pi_{\operatorname{{ab}}}$, are called the [*Alexander invariants*]{} of $X$, while the boundary maps $\widetilde{\partial}^{\operatorname{{ab}}}_q=
\operatorname{id}_{\Lambda} \otimes_{{\Bbbk}\pi} \widetilde{\partial}_q$ are called the [*Alexander matrices*]{} of $X$.
The equivariant spectral sequence {#sec:spectral}
=================================
We now set up the spectral sequence associated to the $I$-adic filtration on the equivariant chain complex $C_{\bullet}(X, M)$, and analyze some of its properties.
A spectral sequence {#subsec:ss}
-------------------
We use [@CE] and [@Sp] as standard references for spectral sequences. Given an increasing filtration $F_{\bullet}=\{F_{-n}\}_{n\ge 0}$ on a chain complex $C_{\bullet}=(C_q,\partial_q)$ over a coefficient ring ${\Bbbk}$, there is a spectral sequence $E^{\bullet}=\{E^r_{s,t}, d^r\}_{r\ge 1}$. The $E^1$ term is defined as $E^1_{s,t}=H_{s+t}(F_s/F_{s-1})$, while the $d^1$ differential is the boundary operator in the homology exact sequence associated to the triple $(F_s,F_{s-1},F_{s-2})$. Each term $E^r$ is a bigraded ${\Bbbk}$-module, the differentials $d^r$ have bidegree $(-r,r-1)$, and $E^{r+1}=H(E^r, d^r)$.
Now let $X$ be a connected CW-complex as in §\[subsec:equivchain\], with fundamental group $\pi=\pi_1(X)$, and augmentation ideal $I=I_{{\Bbbk}}(\pi)$. Let $M$ be a right ${\Bbbk}\pi$-module, and let $C_{\bullet}(X, M)$ be the cellular chain complex of $X$ with coefficients in $M$.
The $I$-adic filtration on $M$ yields a descending filtration, $F^0\supset F^1 \supset F^2 \supset \cdots$, on $C_{\bullet}(X, M)$. The $n$-th term of this filtration is given by $$\label{eq:filtcxm}
F^n(C_{\bullet}(X, M))= M\cdot I^n \otimes_{{\Bbbk}\pi} C_{\bullet}({\widetilde{X}},{\Bbbk}).$$ Clearly, the differentials $\tilde{\partial}^M$ of $C_{\bullet}(X, M)$ preserve this filtration. Set $F_{-n} = F^n$. Then $F_{\bullet}=\{F_{-n}\}_{n\ge 0}$ is an increasing filtration on $C_{\bullet}(X,M)$, bounded above by $F_{0}= C_{\bullet}(X,M)$.
\[def:ss\] The [*equivariant spectral sequence*]{} of the CW-complex $X$, with coefficients in the ${\Bbbk}\pi_1(X)$-module $M$ is the spectral sequence associated to the $I$-adic filtration $F_{\bullet}$ on the chain complex $C_{\bullet}(X,M)$, $$\label{eq:sscw}
\{E^r(X,M),d^r\}_{r\ge 0},$$ with differentials $d^r \colon E^r_{s,t} \to E^r_{s-r,t+r-1}$.
In order to analyze this spectral sequence, we need some preliminary facts.
The associated graded chain complex {#subsec:asscc}
-----------------------------------
Under the identification from , the terms of the chain complex can be written as $$\label{eq:equiv again}
C_{q}(X,M)=M\otimes_{{\Bbbk}} C_{q}(X,{\Bbbk}).$$ The $I$-adic filtration on this chain complex is then given by $$\label{eq:filtcxm again}
F^n(C_{\bullet}(X, M))= M\cdot I^n \otimes_{{\Bbbk}} C_{\bullet}(X,{\Bbbk}).$$
Clearly, $F^n/F^{n+1}=\operatorname{gr}^n(M)\otimes_{{\Bbbk}} C_{\bullet}(X, {\Bbbk})$, where $\operatorname{gr}(M)=\bigoplus_{n\ge 0} M I^n / M I^{n+1}$. Now recall that the boundary maps in $C_{\bullet}(X,M)$ have the form $\tilde{\partial}^M=\operatorname{id}_M \otimes_{{\Bbbk}\pi} \tilde{\partial}$.
\[lem:grd\] For each $q\ge 0$, we have $\operatorname{gr}(\operatorname{id}_M \otimes_{{\Bbbk}\pi} \tilde{\partial}_q)=
\operatorname{id}_{\operatorname{gr}(M)} \otimes_{{\Bbbk}} \partial_q$.
Let $e$ be a $q$-cell of $X$. Then $$p_* \tilde{\partial}_q (1\otimes e)= \partial_q p_* (1\otimes e) =
\partial_q(e)= p_* (1 \otimes \partial_q e).$$ Hence, $\tilde{\partial}_q (1 \otimes e) -
1\otimes \partial_q (e)$ belongs to $I\otimes_{{\Bbbk}} C_{q-1}(X,{\Bbbk})$.
Now let $m$ be an arbitrary element in $M\cdot I^n$. By the above, $$\big ( \operatorname{id}_{M}\otimes_{{\Bbbk}\pi} \tilde{\partial}_q \big )
( m \otimes e ) - m \otimes \partial_q (e) \in
M \cdot I^{n+1} \otimes_{{\Bbbk}} C_{q-1}(X,{\Bbbk}).$$ The conclusion follows at once.
It follows that the graded chain complex associated to filtration has the form $$\label{eq:grequiv}
\operatorname{gr}(C_{\bullet}(X,M))=(\operatorname{gr}(M) \otimes_{\Bbbk}C_q(X,{\Bbbk}),
\operatorname{id}_{\operatorname{gr}(M)} \otimes\, \partial_q)_{q\ge 0}.$$ This chain complex is typically much easier to handle than the equivariant chain complex . Here is an illustration.
\[ex:grm easy\] Consider a local system $M$ as in Example \[ex:locsyst\]. Suppose there is an element $g\in \pi$ such that $\rho(g)$ does not admit $1$ as an eigenvalue—this happens whenever $M$ is a non-trivial, rank $1$ local system. Then clearly $MI=M$. Thus, $\operatorname{gr}(M)=0$ and $\operatorname{gr}(C_{\bullet}(X,M))$ is the zero complex in this instance.
\[rem:grlin\] In the case when $M={\Bbbk}\pi$, the $I$-adic filtration on $C_{\bullet}(X,M)=C_{\bullet}({\widetilde{X}},{\Bbbk})$ is simply $C_{\bullet}({\widetilde{X}},{\Bbbk}) \supseteq I \cdot C_{\bullet}({\widetilde{X}},{\Bbbk})
\supseteq \cdots \supseteq I^n \cdot C_{\bullet}({\widetilde{X}},{\Bbbk})
\supseteq \cdots$, while the associated graded chain complex takes the form $$\label{eq:grcoeff}
\operatorname{gr}(C_{\bullet}({\widetilde{X}},{\Bbbk}))=(\operatorname{gr}({\Bbbk}\pi) \otimes_{\Bbbk}C_q(X,{\Bbbk}),
\operatorname{id}_{\operatorname{gr}({\Bbbk}\pi)} \otimes\, \partial_q)_{q\ge 0}.$$ Notice that the differentials in this chain complex are $\operatorname{gr}({\Bbbk}\pi)$-linear.
The first pages {#subsec:e0e1}
---------------
The $E^0$ term of the equivariant spectral sequence of $X$ with coefficients in $M$ is defined in the usual manner, as the associated graded of the filtration $F_{\bullet}$ on $C_{\bullet}(X,M)$. Using , we find: $$\label{eq:e0}
E^0_{-p,q}(X,M)= \operatorname{gr}^{p} (M) \otimes_{{\Bbbk}} C_{q-p}(X,{\Bbbk}) ,$$ if $p\ge 0$ and $q\ge p$, and otherwise $E^0_{-p,q}(X,M)=0$. Hence, the spectral sequence is concentrated in the second quadrant. Under this identification, and as a consequence of Lemma \[lem:grd\], the differential $d^0\colon E^0_{-p,q}(X,M) \to E^0_{-p,q-1}(X,M)$ takes the form $d^0= \operatorname{id}\otimes\, \partial$. Hence, $$\label{eq:e1}
E^1_{-p,q}(X,M)= H_{q-p}(X, \operatorname{gr}^{p} (M) ).$$
Functoriality properties {#subsec:func}
------------------------
Let $f\colon (X,e_0) \to (X',e'_0)$ be a cellular map, and suppose $\phi\colon M\to M'$ is equivariant with respect to $f_{\sharp}$. The resulting ${\Bbbk}$-linear map, $$\label{eq:phistar}
\phi \otimes \tilde{f}_* \colon C_{\bullet}(X, M)=
M \otimes_{{\Bbbk}\pi} C_{\bullet}({\widetilde{X}},{\Bbbk})
\to M' \otimes_{{\Bbbk}\pi'} C_{\bullet}({\widetilde{X}}',{\Bbbk})= C_{\bullet}(X', M'),$$ is a chain map, preserving $I$-adic filtrations. Consequently, $\phi \otimes \tilde{f}_*$ induces a morphism between the corresponding $I$-adic spectral sequences, $$\label{eq:morfismos}
E^r(\phi \otimes \tilde{f}_*)\colon E^r(X,M) \to E^r(X',M').$$
In particular, $\phi \otimes \tilde{f}_*$ induces a chain map between the associated graded chain complexes, $$\label{eq:grftilde}
\operatorname{gr}(\phi \otimes \tilde{f}_*)\colon
\operatorname{gr}(M)\otimes_{{\Bbbk}} C_{\bullet}(X,{\Bbbk}) \to
\operatorname{gr}(M') \otimes_{{\Bbbk}} C_{\bullet}(X',{\Bbbk}).$$
\[lem:grf\] For each $q\ge 0$, we have $\operatorname{gr}(\phi \otimes \tilde{f}_q)=\operatorname{gr}(\phi ) \otimes f_q$.
It is readily seen that, under the identification $\operatorname{gr}^0({\Bbbk}\pi) ={\Bbbk}$, the map $\operatorname{gr}^0(\tilde{f}_q)$ corresponds to the map $f_q\colon C_q(X,{\Bbbk})\to C_q(X',{\Bbbk})$.
Let $m$ be an element of $MI^n$, and let $e$ be a $q$-cell of $X$, with lift $\tilde{e}$. By the above, $$\tilde{f}_q(\tilde{e}) - 1\otimes f_q(e)\in I'\cdot C_q({\widetilde{X}}',{\Bbbk}).$$ Hence, $(\phi \otimes \tilde{f}_q) (m\otimes \tilde{e})=
\phi(m)\otimes \tilde{f}_q (\tilde{e})$ equals $\phi(m)\otimes f_q(e)$ modulo $F'^{n+1}$. The conclusion follows.
As a simple example, take $f=\operatorname{id}_X$. Then any morphism $\phi\colon M\to M'$ of right ${\Bbbk}\pi$-modules induces a morphism of spectral sequences, $E^{\bullet}(\phi)\colon
E^{\bullet}(X,M)\to E^{\bullet}(X,M')$, with $E^0(\phi)=\operatorname{gr}(\phi)\otimes \operatorname{id}$.
Homotopy invariance {#subsec:hinv}
-------------------
Let $f\colon (X,e_0)\to (X',e'_0)$ be a cellular homotopy equivalence, and let $M$ be a right ${\Bbbk}\pi$-module. Use the isomorphism $f_{\sharp}\colon \pi\to \pi'$ to view $M$ as a right ${\Bbbk}\pi'$-module.
\[cor:uniqueness\] The spectral sequences $\{E^r(X,M)\}_{r\ge 1}$ and $\{E^r(X',M)\}_{r\ge 1}$ are isomorphic.
View $\operatorname{id}_M$ as an equivariant map with respect to $f_{\sharp}$. Using Lemma \[lem:grf\] and identification , we see that the map $E^1(\operatorname{id}_M \otimes \tilde{f}_*)$ coincides with the induced homomorphism $H_*(f)\colon H_*(X,\operatorname{gr}(M)) \to H_*(X',\operatorname{gr}(M))$. Hence, the maps $E^r(\operatorname{id}_M \otimes \tilde{f}_*)$ are isomorphisms, for all $r\ge 1$.
\[cor:cw space\] Let $X$ be a path-connected topological space having the homotopy type of a CW-complex, and let $M$ be a right ${\Bbbk}\pi_1(X)$-module. Then there is a well-defined second quadrant $I$-adic spectral sequence $\{ E^r (X,M), d^r \}$, starting at $r=1$.
Identifying the $d^1$ differential {#sec:e1 page}
==================================
Throughout this section, $X$ is a CW-complex with a single $0$-cell $e_0$ and with basepoint-preserving attaching maps, $\pi=\pi_1(X,e_0)$ is the fundamental group, and $C_{\bullet}(X,M)$ is the cellular chain complex, with coefficients in a right ${\Bbbk}\pi$-module $M$. Our goal is to identify the first page of the equivariant spectral sequence of $X$, in terms of the coalgebra structure of $H_*(X,{\Bbbk})$ and the module structure of $M$.
The first page {#subsec:e1 page}
--------------
Let $\{ E^r (X,M),d^r\}_{r\ge 1}$ be the spectral sequence from Definition \[def:ss\]. We wish to analyze the differentials $$\label{eq:dq diff}
d^1\colon E^1_{-p,q}(X,M) \to E^1_{-p-1,q}(X,M),$$ for all $p\ge 0$ and $q\ge p$. Schematically, the $E^1$ page looks like $$\xymatrixrowsep{6pt}
\xymatrix{
E^1_{-2,2} & E^1_{-1,2} \ar_(.4){d^1}[l] & E^1_{0,2}
\ar_(.4){d^1}[l]\\
& E^1_{-1,1} & E^1_{0,1} \ar_(.4){d^1}[l]\\
& & E^1_{0,0}
}$$
To identify these maps in terms of cohomological data, we need to assume the following: Either ${\Bbbk}={\mathbb{Z}}$ and $H_*(X,{\mathbb{Z}})$ is torsion-free, or ${\Bbbk}$ is a field. Using this assumption, formula , and the Universal Coefficients Theorem, we find $$\label{eq:e1 new}
E^1_{-p,q}(X,M)= \operatorname{gr}^{p} (M) \otimes_{{\Bbbk}} H_{q-p}(X,{\Bbbk}),$$ for each $p\ge 0$ and $q\ge p$.
The $d^1$ differentials enjoy the following naturality property. Suppose $f\colon X\to X'$ is a cellular map, and $\phi\colon M\to M'$ is a morphism of modules, equivariant with respect to $\bar{f}_{\sharp}\colon {\Bbbk}\pi_1(X)\to {\Bbbk}\pi_1(X')$. From Lemmas \[lem:grd\] and \[lem:grf\], we deduce $E^1(\phi \otimes \tilde{f}_*)= \operatorname{gr}(\phi) \otimes H_*(f)$. Using the naturality property from , we obtain a commuting diagram, $$\label{eq:E1nat}
\xymatrix{
\operatorname{gr}^p(M) \otimes_{{\Bbbk}} H_q(X,{\Bbbk}) \ar[rr]^{d^1}
\ar[d]^{\operatorname{gr}(\phi) \otimes H_*(f)}
&& \operatorname{gr}^{p+1}(M) \otimes_{{\Bbbk}} H_{q-1}(X,{\Bbbk})
\ar[d]^{\operatorname{gr}(\phi)\otimes H_*(f)} \\
\operatorname{gr}^p(M') \otimes_{{\Bbbk}} H_q(X',{\Bbbk}) \ar[rr]^{d'^1}
&& \operatorname{gr}^{p+1}(M') \otimes_{{\Bbbk}} H_{q-1}(X',{\Bbbk})
}$$
Cohomological interpretation {#subsec:d1diff}
----------------------------
The main result of this section is the following Theorem, which identifies the differentials on the $E^1$ page in terms of the comultiplication in $H_*(X, {\Bbbk})$, and the ${\Bbbk}\pi$-module structure on $M$, given by the multiplication map, $\mu_M\colon M\otimes_{{\Bbbk}} {\Bbbk}\pi \to M$, $m \otimes g\mapsto mg$. Fix integers $p\ge 0$ and $q\ge 1$.
\[thm:d1map\] Let $X$ be a connected CW-complex, and $M$ a right ${\Bbbk}\pi$-module. Assume either ${\Bbbk}={\mathbb{Z}}$ and $H_*(X,{\mathbb{Z}})$ is torsion-free, or ${\Bbbk}$ is a field. Then the differential $d^1\colon E^1_{-p,p+q}(X,M) \to E^1_{-p-1,p+q}(X,M)$ can be decomposed as $$\xymatrix{
\operatorname{gr}^p(M) \otimes_{{\Bbbk}} H_q(X,{\Bbbk}) \ar^{d^1}[r]
\ar^{\operatorname{id}\otimes \nabla_X}[d]&
\operatorname{gr}^{p+1}(M) \otimes_{{\Bbbk}} H_{q-1}(X,{\Bbbk}) \\
\operatorname{gr}^p(M) \otimes_{{\Bbbk}} (H_1(X,{\Bbbk}) \otimes_{{\Bbbk}} H_{q-1}(X,{\Bbbk}) )
\ar^{\cong}[r]
&( \operatorname{gr}^p(M) \otimes_{{\Bbbk}} \operatorname{gr}^1({\Bbbk}\pi) ) \otimes_{{\Bbbk}} H_{q-1}(X,{\Bbbk})\, .
\ar_{\operatorname{gr}(\mu_M)\otimes \operatorname{id}}[u]
}$$
Here, $\nabla_X$ is the comultiplication map, defined as the composite $$\label{eq:comult}
\xymatrix{
H_q(X,{\Bbbk}) \ar[r]^(.46){\Delta_*} \ar[drr]_{\nabla_X}
& H_q(X\times X,{\Bbbk}) \ar[r]^(.36){\cong} &
{\displaystyle}{\bigoplus_{i=0}^{q} H_i(X,{\Bbbk}) \otimes_{{\Bbbk}} H_{q-i}(X,{\Bbbk})}
\ar[d]^(.53){\operatorname{pr}} \\
&& H_1(X,{\Bbbk}) \otimes_{{\Bbbk}} H_{q-1}(X,{\Bbbk}),
}$$ where the first arrow is the homomorphism induced by the diagonal map, the second arrow is the Künneth isomorphism, and the third arrow is projection onto direct summand. Under the identification $H_i(X,{\Bbbk})^* \cong H^i(X,{\Bbbk})$, the map $\nabla_X$ is the dual of the cup-product map $\cup_X \colon H^1(X,{\Bbbk})\otimes_{{\Bbbk}} H^{q-1}(X,{\Bbbk}) \to H^q(X,{\Bbbk})$, provided $X$ is a finite-type CW-complex.
The proof of Theorem \[thm:d1map\] will occupy the rest of this section.
Reducing to the case $M={\Bbbk}\pi$ {#subsec:mtor}
-----------------------------------
Using the functoriality of the spectral sequence with respect to coefficient modules, we first reduce the proof to the case when $M$ is a free ${\Bbbk}\pi$-module of rank $1$.
\[lem:reduction\] If the conclusion of Theorem \[thm:d1map\] holds for the coefficient module ${\Bbbk}\pi$, then it holds for any coefficient module $M$.
Let $\psi\colon N \to M$ be a homomorphism of right ${\Bbbk}\pi$-modules. Write $H_q=H_q(X,{\Bbbk})$, and $\otimes=\otimes_{{\Bbbk}}$. We then have the following cube diagram: $$\label{eq:cdm}
\xymatrixcolsep{-20pt}
\xymatrixrowsep{32pt}
\xymatrix{
& \operatorname{gr}^p(N) \otimes (H_1 \otimes H_{q-1}) \ar^(.43){\cong}[rr]
\ar'[d]^(.5){\operatorname{gr}(\psi) \otimes \operatorname{id}}[dd]
&& ( \operatorname{gr}^p(N) \otimes \operatorname{gr}^1({\Bbbk}\pi)) \otimes H_{q-1}
\ar^{\operatorname{gr}(\mu_N)\otimes \operatorname{id}}[dl]
\ar^(.35){\operatorname{gr}(\psi) \otimes \operatorname{id}}[dd]
\\
\operatorname{gr}^p(N) \otimes H_q \ar^(.52){d^1_{N}}[rr]
\ar^(.35){\operatorname{gr}(\psi) \otimes \operatorname{id}}[dd]
\ar^{\operatorname{id}\otimes \nabla}[ur] &&
\operatorname{gr}^{p+1}(N) \otimes H_{q-1}
\ar^(.35){\operatorname{gr}(\psi) \otimes \operatorname{id}}[dd]
\\
& \operatorname{gr}^p(M) \otimes (H_1 \otimes H_{q-1}) \ar'[r]^(.9){\cong}[rr]
&& ( \operatorname{gr}^p(M) \otimes \operatorname{gr}^1({\Bbbk}\pi)) \otimes H_{q-1}
\ar^{\operatorname{gr}(\mu_M)\otimes \operatorname{id}}[dl]
\\
\operatorname{gr}^p(M) \otimes H_q \ar^(.52){d^1_{M}}[rr]
\ar^{\operatorname{id}\otimes \nabla}[ur]
&&\operatorname{gr}^{p+1}(M) \otimes H_{q-1}
}$$ All side squares of the cube commute: the front one by naturality of $d^1$, as explained in , the right one by functoriality of $\operatorname{gr}$, and the other two for obvious reasons.
Now suppose the top square commutes for $N={\Bbbk}\pi$. Let $[x] \otimes h$ be an additive generator of $\operatorname{gr}^p(M) \otimes H_q$, where $x=y a$, with $y\in M$ and $a\in I^p$. Let $\psi\colon {\Bbbk}\pi \to M$ be the ${\Bbbk}\pi$-linear map sending the unit of $\pi$ to $y$. Chasing the cube diagram, we see that the bottom square commutes, with $[x] \otimes h=(\operatorname{gr}(\psi)\otimes \operatorname{id})([a]\otimes h) $ as input. Hence, the bottom square commutes.
The case $M={\Bbbk}\pi$ {#subsec:sskpi}
-----------------------
The $E^1$-term of the $I$-adic spectral sequence of $X$ with coefficients in ${\Bbbk}\pi$, $$E^1(X,{\Bbbk}\pi)=\operatorname{gr}({\Bbbk}\pi) \otimes_{{\Bbbk}} H_*(X,{\Bbbk}),$$ is a left $\operatorname{gr}({\Bbbk}\pi)$-module, freely generated by a ${\Bbbk}$-basis for $1 \otimes H_*(X,{\Bbbk})$. We want to show $$\label{eq:d1 nabla}
d^1=(\operatorname{gr}(\mu_{{\Bbbk}\pi})\otimes \operatorname{id}) \circ
(\operatorname{id}\otimes \nabla_X).$$ Clearly, the map on the right side is $\operatorname{gr}({\Bbbk}\pi)$-linear. In the next Lemma, we verify that $d^1$ is $\operatorname{gr}({\Bbbk}\pi)$-linear, too (a more general result will be proved in Lemma \[lem:dr lin\]).
\[lem:d1lin\] The differential $d^1\colon E^1_{-p,p+q}(X,{\Bbbk}\pi) \to
E^1_{-p-1,p+q}(X,{\Bbbk}\pi)$ is $\operatorname{gr}({\Bbbk}\pi)$-linear.
Let $F^s= I^s C_{\bullet}({\widetilde{X}},{\Bbbk})$ be the $I$-adic filtration on $C_{\bullet}(X, {\Bbbk}\pi)$. By definition, the differential $d^1$ is the connecting homomorphism in the homology exact sequence of $$\xymatrix{0\ar[r]& F^{s+1}/F^{s+2} \ar[r]& F^{s}/F^{s+2}
\ar[r]& F^{s}/F^{s+1} \ar[r]& 0}.$$
Let $z\in C_q(X,{\Bbbk})$ such that $\partial_q(z)=0$. Then $d^1(1\otimes [z]) \equiv \tilde{\partial}(1\otimes z),\,
\bmod\, F^2$. More generally, if $x\in I^p$, then $d^1([x]\otimes [z]) \equiv \tilde{\partial}(x\otimes z),\,
\bmod\, F^{p+2}$. On the other hand, $\tilde{\partial}(x\otimes z)= x
\tilde{\partial} (1\otimes z)$. Combining these formulas implies $d^1([x]\otimes [z]) = [x] d^1(1\otimes [z])$. Clearly, this implies the claim of the lemma.
Thus, to verify , it is sufficient to check equality on free $\operatorname{gr}({\Bbbk}\pi)$-generators, i.e., on a ${\Bbbk}$-basis for $1 \otimes H_*(X,{\Bbbk})$. In other words, to identify the differentials on the $E^1$ page, we only need to show that, upon identifying $E^1_{0,q}=H_q(X,{\Bbbk})$ and $E^1_{-1,q}= H_1(X,{\Bbbk}) \otimes_{{\Bbbk}} H_{q-1}(X,{\Bbbk})$, the map $d^1\colon E^1_{0,q}\to E^1_{-1,q}$ coincides with the comultiplication map $\nabla_X$, for all $q\ge 1$.
Further reductions {#subsec:further reduce}
------------------
Clearly, we may assume $X$ is a finite CW-complex. Indeed, an arbitrary homology class in $H_q(X,{\Bbbk})$ is represented by a cycle supported on a finite subcomplex of $X$. Dualizing, we are left with proving the following Proposition.
\[prop:deltacup\] Let $X$ be a connected, finite-type CW-complex, and assume either ${\Bbbk}={\mathbb{Z}}$ and $H_*(X,{\mathbb{Z}})$ is torsion-free, or ${\Bbbk}$ is a field. Then, the dual $\delta^1=(d^1)^{*}\colon (E^1_{-1,q})^* \to (E^1_{0,q})^*$ coincides with the cup-product map $\cup_X\colon
H^1(X,{\Bbbk}) \otimes_{{\Bbbk}} H^{q-1}(X,{\Bbbk}) \to H^{q}(X,{\Bbbk})$.
In the above, we may suppose ${\Bbbk}$ is actually a prime field. Indeed, in the first case, both $\delta^1$ and $\cup_X$ are defined over ${\mathbb{Z}}$, and $H^*(X, {\mathbb{Z}})$ injects into $H^*(X, {\mathbb{Q}})$, so we may replace ${\Bbbk}={\mathbb{Z}}$ by ${\mathbb{Q}}$. In the second case, both maps are defined over the prime field of ${\Bbbk}$, so we may replace ${\Bbbk}$ by its prime field.
We will reduce the proof to a special class of spaces, using the functoriality properties of the spectral sequence. Let us first state those properties, in the form we will need.
Let $f\colon X\to X'$ be a cellular map, and consider diagram , for the morphism $\phi=\bar{f}_{\sharp}\colon {\Bbbk}\pi\to {\Bbbk}\pi'$. In degree $p=0$, this diagram simplifies to $$\label{eq:e1d0}
\xymatrixcolsep{50pt}
\xymatrix{
H_q(X,{\Bbbk}) \ar[r]^(.4){d^1}
\ar[d]^{H_*(f)}
& H_1(X,{\Bbbk}) \otimes_{{\Bbbk}} H_{q-1}(X,{\Bbbk})
\ar[d]^{H_*(f)\otimes H_*(f)} \\
H_q(X',{\Bbbk}) \ar[r]^(.4){d'^1}
& H_1(X',{\Bbbk}) \otimes_{{\Bbbk}} H_{q-1}(X',{\Bbbk})
}$$ Dualizing, and writing $f^*=H^*(f)$, we obtain the commuting diagram $$\label{eq:e1cup}
\xymatrixcolsep{45pt}
\xymatrix{
H^1(X,{\Bbbk}) \otimes_{{\Bbbk}} H^{q-1}(X,{\Bbbk})
\ar[r]^(.6){\delta^1} &H^q(X,{\Bbbk})
\\
H^1(X',{\Bbbk}) \otimes_{{\Bbbk}} H^{q-1}(X',{\Bbbk})
\ar[u]_{f^{*}\otimes f^{*}}
\ar[r]^(.65){\delta'^1} &H^q(X',{\Bbbk})
\ar[u]_{f^*}
}$$
A computation with Eilenberg-MacLane spaces {#subsec:kpin}
-------------------------------------------
Recall ${\Bbbk}$ is a prime field. Let $R={\mathbb{Z}}$ if ${\Bbbk}={\mathbb{Q}}$, and $R={\mathbb{Z}}_p$ if ${\Bbbk}={\mathbb{F}}_p$. For an integer $r\ge 1$, let $K_r$ be an Eilenberg-MacLane space of type $K(R,r)$. We may assume $K_r$ is a finite-type CW-complex, obtained from $S^r$ by attaching cells of dimension $r+1$ and higher, $$K_r=e_0\cup e_r \cup e^{1}_{r+1}\cup \cdots \cup e^{m}_{r+1}
\cup \cdots$$ Let ${[\![ e_r ]\!]}\in H_r(K_r,R)$ be the homology class represented by the $r$-cell, and let ${[\![ e_r ]\!]}^*\in H^r(K_r,{\Bbbk})$ be its dual over ${\Bbbk}$.
Now let $K'_r$ be another copy of $K_r$, with cells $e'_j$, and consider the product CW-complex $K=K_1\times K'_{r}$. Let $\operatorname{pr}$ and $\operatorname{pr}'$ be the projections of $K$ onto the two factors. Define cohomology classes $u_1=\operatorname{pr}^*({[\![ e_1 ]\!]}^*) \in H^1(K,{\Bbbk})$ and $u'_r=\operatorname{pr}'^*({[\![ e'_r ]\!]}^*) \in H^r(K,{\Bbbk})$.
\[lem:del cup\] For the CW-complex $K$ defined above, $\delta^1(u_1\otimes u'_{r}) = u_1 \cup u'_{r}$.
Consider the map $\delta^1\colon
H^1(K) \otimes H^{r}(K) \to H^{r+1}(K)$, with ${\Bbbk}$ coefficients. By Kronecker duality, we only need to verify: $$\label{eq:kron}
\langle d^1(z), u_1\otimes u'_{r} \rangle =
\langle z, u_1\cup u'_{r} \rangle,$$ for all $z\in H_{r+1}(K)$. By the Künneth formula, $$\label{eq:kunneth}
H_{r+1}(K)=(H_0(K_1)\otimes H_{r+1}(K'_{r})) \oplus
(H_1(K_1)\otimes H_{r}(K'_{r}))\oplus
(H_{r+1}(K_1)\otimes H_{0}(K'_{r})).$$ By construction, $u_1\cup u'_{r} = {[\![ e_1 ]\!]}^* \times {[\![ e'_r ]\!]}^*$. It follows that $ \langle z, u_1\cup u'_{r} \rangle= 1$ if $z={[\![ e_1 ]\!]} \times {[\![ e'_r ]\!]}$ and vanishes if $z$ belongs to one of the other two summands in .
Similarly, the left hand side of vanishes if $z$ belongs to one of those two summands; this follows from the naturality of $d^1$, as expressed in .
Now suppose $r>1$. Note that $\pi_1(K_1)$ is a cyclic group, generated by $[e_1]$, whereas $\pi_1(K'_{r})=0$. Moreover, $\widetilde{K}=\widetilde{K}_1 \times K'_{r}$. We compute: $$\tilde{\partial} (\tilde{e}_1\times e'_{r}) =
\tilde{\partial}(\tilde{e}_1)\times e'_{r} -
\tilde{e}_1\times \partial(e'_{r}) =
([e_1]-1) \tilde{e}_0 \times e'_{r},$$ and so $d^1({[\![ e_1\times e'_{r} ]\!]}) =
{[\![ e_1\times e'_0 ]\!]} \otimes {[\![ e_0\times e'_{r} ]\!]}$. It follows that $\langle d^1({[\![ e_1\times e'_{r} ]\!]}), u_1\otimes u'_{r} \rangle =1$.
If $r=1$, then $\widetilde{K}=\widetilde{K}_1
\times \widetilde{K}'_{1}$. We compute: $$\tilde{\partial} (\tilde{e}_1\times \tilde{e}'_1) =
\tilde{\partial}(\tilde{e}_1)\times \tilde{e}'_1 -
\tilde{e}_1\times \tilde{\partial}(\tilde{e}'_1) =
([e_1]-1) \tilde{e}_0 \times \tilde{e}'_1
-([e'_1]-1) \tilde{e}_1 \times \tilde{e}'_0,$$ and so $d^1({[\![ e_1\times e'_1 ]\!]}) =
{[\![ e_1\times e'_0 ]\!]} \otimes {[\![ e_0\times e'_{1} ]\!]} -
{[\![ e_0\times e'_1 ]\!]} \otimes {[\![ e_1\times e'_0 ]\!]}$. It follows that $\langle d^1({[\![ e_1\times e'_{1} ]\!]}), u_1\otimes u'_{1} \rangle =1$.
Proof of Proposition \[prop:deltacup\] {#subsec:proofdeltacup}
--------------------------------------
First assume $q=1$. Let $e$ be a $1$-cell of $X$, and ${[\![ e ]\!]}\in H_1(X,{\Bbbk})$ the homology class it represents. From the identification $I/I^2=\operatorname{gr}^1({\Bbbk}\pi)$, and formula , we get $d^1({[\![ e ]\!]})={[\![ e ]\!]}$. Thus, $d^1\colon E^1_{0,1}\to E^1_{-1,1}$ coincides with $\nabla_X=\operatorname{id}\colon H_{1}(X,{\Bbbk}) \to H_{1}(X,{\Bbbk})$.
Now assume $q>1$. It is enough to show that $\delta^1(v_1 \otimes v_{q-1}) = v_1 \cup v_{q-1}$ over ${\Bbbk}$, for each $v_1\in H^1(X,R)$ and $v_{q-1}\in H^{q-1}(X,R)$. Let $K_r=K(R,r)$ be an Eilenberg-MacLane space as above. By obstruction theory, there is a map $f_r\colon X\to K_r$ such that $v_r=f_r^*({[\![ e_r ]\!]}^*)$. Now set $K=K_1\times K'_{q-1}$, where $K'_{q-1}$ is a copy of $K_{q-1}$, and define $f$ to be a cellular approximation of the map $F=(f_1,f'_{q-1})\colon X\to K$. With notation as above, $v_1=f^*(u_1)$ and $v_{q-1}=f^*(u'_{q-1})$, over ${\Bbbk}$. Using diagram and the naturality of cup-products, the conclusion follows from Lemma \[lem:del cup\].
Convergence issues {#sec:convergence}
==================
In this section, we discuss the convergence properties of the equivariant spectral sequence $E^{\bullet}(X,M)$. Throughout this section, ${\Bbbk}$ will denote a fixed field.
The $E^{\infty}$ term {#subsec:conv}
---------------------
Let $C_{\bullet}=(C_q,\partial_q)$ be a chain complex over ${\Bbbk}$, endowed with an increasing filtration, $\cdots \subset F_{-2}\subset F_{-1} \subset F_{0}=C$, and let $E^{\bullet}=\{E^r_{s,t}, d^r\}_{r\ge 1}$ be the spectral sequence associated to $F_{\bullet}$. Suppose the following condition holds: For each $s$ and $t$, the ${\Bbbk}$-vector space $E^1_{s,t}$ is finite-dimensional. Then $E^r_{s,t}=E^{r+1}_{s,t}$ for all $r\ge r_0$. In this case, the $E^{\infty}$ term is defined as $E^{\infty}_{s,t}=E^{r_0}_{s,t}$, and the inclusion $$\label{eq:inftyb}
E^{\infty}_{s,t} \supseteq \operatorname{gr}^s (H_{s+t}(C))\, ,$$ holds for all $s, t$.
The spectral sequence is said to be [*convergent*]{} if $E^{\infty}_{s,t} = \operatorname{gr}^s (H_{s+t}(C))$, for all $s, t$. For short, we write $E^1_{s,t}\Rightarrow H_{s+t}(C)$. Recall $Z^r_s=\{z\in F_s \mid \partial z \in F_{s-r}\}$, and write $Z^{\infty}_s=\{z\in F_s \mid \partial z =0\}$. In this setup, convergence of $E^{\bullet}$ is equivalent to $$\label{eq:weak conv}
\bigcap_{r} (Z^r_s + F_{s-1}) \subseteq Z^{\infty}_s + F_{s-1},
\quad \text{for all $s$.}$$
The $E^{\infty}$ term of the equivariant spectral sequence {#subsec:Iadic conv}
----------------------------------------------------------
Now let $X$ be a connected CW-complex, with fundamental group $\pi=\pi_1(X)$. Let $I=I_{\Bbbk}(\pi)$ be the augmentation ideal of ${\Bbbk}\pi$, and let $M$ be a right ${\Bbbk}\pi$-module. Under fairly general assumptions, the $I$-adic spectral sequence $E^{\bullet}(X,M)$ has an $E^{\infty}$ term. Its convergence, though, is a rather delicate matter.
\[prop:conv ss\] Suppose $\dim_{{\Bbbk}} H_q(X,{\Bbbk})<\infty$, for all $q\ge 0$, and $\dim_{{\Bbbk}} M/MI < \infty$. Then, the spectral sequence $E^{\bullet}(X,M)$ has an $E^{\infty}$ term.
Recall $\operatorname{gr}(M)$ is generated as a $\operatorname{gr}({\Bbbk}\pi)$-module by $\operatorname{gr}^0(M)=M/MI$, and $\operatorname{gr}({\Bbbk}\pi)$ is generated as a ring by $\operatorname{gr}^1 ({\Bbbk}\pi)= H_1(X, {\Bbbk})$. We infer that $\dim_{{\Bbbk}} \operatorname{gr}^s(M)< \infty$, for all $s$. Hence, for any fixed $s$ and $t$, the vector space $E^1_{-s,t}= \operatorname{gr}^s(M)\otimes_{{\Bbbk}} H_{t-s}(X,{\Bbbk})$ is finite-dimensional. The conclusion follows.
\[rem:einf\] Let $X$ be a finite-type CW-complex, and suppose $M={\Bbbk}\pi$. Then $M/MI={\Bbbk}$, and so the $E^{\infty}$ term exists. More generally, suppose $\nu\colon \pi{\twoheadrightarrow}G$ is an epimorphism. Take $M={\Bbbk}{G}_{\nu}$, and let $J=I_{\Bbbk}{G}$. Then $M/MI={\Bbbk}{G}/J={\Bbbk}$, and so $E^{\infty}$ exists.
A non-convergent spectral sequence {#subsec:non conv}
----------------------------------
We now give an example of a CW-complex $X$ and a ${\Bbbk}\pi$-module $M$ for which the assumptions of Proposition \[prop:conv ss\] are satisfied, yet the spectral sequence $E^{\bullet}(X,M)$ does not converge. More precisely, for each integer $m\ge 3$, we construct a finite CW-complex $X$ of dimension $m$ for which the convergence condition fails for the coefficient module $M={\Bbbk}\pi$, in filtration degree $s=0$ and total degree $m$.
![Hillman’s $2$-component link[]{data-label="fig:hillman link"}](2lk){height="1.8in" width="1.8in"}
\[ex:non conv\] Let $L$ be the $2$-component link in $S^3$ depicted in Figure \[fig:hillman link\],[^3] and let $\pi=\pi_1(S^3\setminus L)$ be the fundamental group of its complement. In [@Hil], J. Hillman showed that this group is not residually nilpotent; that is, if $\{\Gamma_n(\pi)\}_{n\ge 1}$ is the lower central series of $\pi$, then $\Gamma_{\omega}(\pi):=\bigcap_{n\ge 1} \Gamma_n(\pi)$ is non-trivial. Pick an element $1\ne g\in \Gamma_{\omega}(\pi)$, and set $x=g-1\in I$. From [@Q], we know that the map $\pi{\hookrightarrow}{\Bbbk}\pi$, $h\mapsto h-1$, sends $\Gamma_n(\pi)$ to $I^n$, for all $n\ge 1$. Hence, $x$ belongs to the $I$-adic radical, $I^{\omega}:=\bigcap_{n\ge 1} I^n$.
The link complement has the homotopy type of a finite $2$-complex, say $X_0$. Fix an integer $m\ge 3$. Using the construction from Example \[ex:construction\], we may define a CW-complex $$X=(X_0\vee S^{m-1} ) \cup_{\phi_x} e_{m},$$ with $[\phi_x]=x\cdot [S^{m-1}] \in \pi_{m-1}(X_0\vee S^{m-1})$. Clearly, $\pi_1(X)=\pi$. We know that $C_{\bullet}({\widetilde{X}},{\Bbbk})= C_{\bullet}({\widetilde{X}}_0,{\Bbbk})\oplus C(m-1, x)$, and the differential of $C(m-1, x)$, $\tilde{\partial}_m \colon {\Bbbk}\pi \to {\Bbbk}\pi$ , sends $1\in {\Bbbk}\pi$ to $x \in {\Bbbk}\pi$.
From the definitions, $Z^r_{0,m}=\{ z \in {\Bbbk}\pi\mid \tilde{\partial}_m(z) \in I^r\}$; hence, $1\in \bigcap_{r} Z^r_{0,m}$. On the other hand, $Z^{\infty}_{0,m} =\ker (\tilde{\partial}_m)$. Thus, if were to hold, we would have $$1\in \ker (\tilde{\partial}_m)+I.$$
Now, $\ker (\tilde{\partial}_m)=\{y \in {\Bbbk}\pi \mid yx=0\}$, and this subspace vanishes. Indeed, every link group is locally indicable, by [@HSh], and the group algebra of a locally indicable group has no zero divisors, by [@Hig]. We conclude that $1\in I$, a contradiction.
Base change and filtrations on homology {#sect:change rings}
=======================================
We now turn to the special case when the module $M$ is actually a ring $R$, with right ${\Bbbk}\pi$-module structure given by extension of scalars. We analyze the $d^1$ differential of $E^1(X,R)$, and two natural filtrations on $H_*(X,R)$.
Base change {#subsec:base change}
-----------
Let $\pi$ be a group, and ${\Bbbk}$ a commutative, unital ring. Suppose we are given a ring $R$, and a ring homomorphism $\rho\colon {\Bbbk}\pi\to R$ (also known as a ‘base change’ or ‘extension of scalars’). Then $R$ becomes a right ${\Bbbk}\pi$-module by setting $r\cdot x =r \rho(x)$, for all $x\in {\Bbbk}\pi$ and $r\in R$. We will denote by $J$ the two-sided ideal of $R$ generated by $\rho(I)$, where $I$ is the augmentation ideal of ${\Bbbk}\pi$.
A particular case arises when we are given a group homomorphism, $\nu\colon \pi\to G$. Then, the linear extension of $\nu$ to group rings, call it $\bar{\nu}\colon {\Bbbk}\pi \to {\Bbbk}{G}$, is a ring homomorphism. As in Example \[ex:covers\], we will denote the resulting ${\Bbbk}{\pi}$-module by ${\Bbbk}{G}_{\nu}$.
\[lem:irj\] With notation as above, suppose $R$ is a commutative ring, or $R={\Bbbk}{G}_{\nu}$, for some epimorphism $\nu\colon \pi{\twoheadrightarrow}G$. Then:
1. \[i1\] $R I^{n}=J^{n}$, for all $n\ge 0$. If $R={\Bbbk}{G}_{\nu}$, then $J=I_{\Bbbk}{G}$.
2. \[i2\] $\operatorname{gr}_I(R)=\operatorname{gr}_J(R)$.
By definition, $J=R \rho(I) R$, and so the inclusion $R I^{n}\subset J^{n}$ always holds. If $R$ is commutative, the reverse inclusion is obvious.
Now suppose $R={\Bbbk}{G}_{\nu}$. Recall $I=I_{\Bbbk}\pi$ is generated (as a ${\Bbbk}$-module) by all elements of the form $g-1$, with $g\in \pi$. Since $\nu$ is surjective, it follows that $\bar{\nu} (I)=I_{\Bbbk}{G}$. In particular, $\bar{\nu} (I)$ is a two-sided ideal of $R$, so $J=I_{\Bbbk}{G}$ and $J^{n} \subset R I^{n}$.
Follows from .
The spectral sequence with $R$-coefficients {#subsec:exr}
-------------------------------------------
Let $X$ be a connected CW-complex with a single $0$-cell $e_0$, and fundamental group $\pi=\pi_1(X,e_0)$. Suppose $R$ is a ring, endowed with the right ${\Bbbk}\pi$-module structure given by a ring homomorphism $\rho\colon {\Bbbk}\pi\to R$. Then the $I$-adic spectral sequence $E^{\bullet}(X, R)$ is a spectral sequence in the category of left $R$-modules. Indeed, $R$ acts on itself by left-multiplication, and this action extends in a natural way to each term $E^{r}(X, R)$. Furthermore, all differentials $d^r_R\colon E^{r} \to E^{r}$ are $R$-linear.
Now assume $R$ satisfies one of the two conditions of Lemma \[lem:irj\]. As above, let $J$ be the two-sided ideal of $R$ generated by $\rho (I)$. Consider the $R$-chain complex $C_{\bullet}(X, R)= R\otimes_{{\Bbbk}} C_{\bullet}(X,{\Bbbk})$, with differential $\tilde{\partial}^R=\operatorname{id}_R \otimes_{{\Bbbk}\pi} \tilde{\partial}$, and filtration $F^{\bullet}$ given by . By Lemma \[lem:irj\], the terms of this filtration can be expressed as $$\label{eq:jfilt}
F^n C_{\bullet}(X, R)= J^n \otimes_{{\Bbbk}} C_{\bullet}(X,{\Bbbk}),$$ for all $n\ge 0$. Hence, by , $$\label{eq:e1stxr}
E^1_{-s, t}(X, R)= H_{t-s}(X, \operatorname{gr}^s_J (R)).$$
\[lem:dr lin\] Suppose $R$ is a commutative ring, or $R={\Bbbk}{G}_{\nu}$, for some epimorphism $\nu\colon \pi{\twoheadrightarrow}G$. Then $E^{\bullet}(X, R)$ is a spectral sequence in the category of left $\operatorname{gr}_J(R)$-modules.
Let $Z^r_{-s}=\{z\in F_{-s} \mid \tilde{\partial}^R z \in F_{-s-r}\}$. By construction, the terms of the spectral sequence are given by $$\label{eq:ers}
E^r_{-s}= Z^r_{-s}/ (Z^{r-1}_{-(s+1)} +
\tilde{\partial}^R Z^{r-1}_{-(s+1-r)}).$$
From , we find that $J^n \cdot Z^r_{-s} \subset Z^r_{-s-n}$, for all $n\ge 0$. Using these inclusions, together with , we infer that $$J^n \cdot E^r_{-s} \subset E^r_{-s-n}, \quad\text{for all $n\ge 0$}.$$ This allows us to define a natural $\operatorname{gr}_J(R)$-module structure on ${}_qE^r= \bigoplus_{s\ge 0} E^r_{-s, s+q}$, for all $q\ge 0$, and thus on $E^r=\bigoplus_{q\ge 0} {}_qE^r$, for all $r\ge 1$. (The $\operatorname{gr}_J(R)$-module structure on ${}_qE^1= \bigoplus_{s\ge 0} H_q(X, \operatorname{gr}_J^s(R))$ is induced by left-multiplication.)
Using once again , it is readily checked that the differentials $d^r_R\colon {}_qE^r \to {}_{q-1}E^r$ act $\operatorname{gr}_J(R)$-linearly, for all $r\ge 1$.
The differential of $E^1(X, {\Bbbk}{G}_{\nu})$ {#subsec:d1 again}
----------------------------------------------
In this subsection, we assume that either ${\Bbbk}= {\mathbb{Z}}$ and $H_*(X, {\mathbb{Z}})$ is torsion-free, or ${\Bbbk}$ is a field. In this case, ${}_q E^1 = \operatorname{gr}_J(R) \otimes_{{\Bbbk}} H_q(X, {\Bbbk})$, by ; in particular, ${}_q E^1$ is a free $\operatorname{gr}_J(R)$-module, for all $q\ge 0$.
The above Lemma tells us that, in order to describe $d^1_R$ completely, it is enough to identify its effect on free $\operatorname{gr}_J (R)$-module generators. We will do this now, in the case when the ring $R$ is a group-ring, ${\Bbbk}{G}$, with ${\Bbbk}\pi$-module structure given by $\bar{\nu}\colon {\Bbbk}\pi{\twoheadrightarrow}{\Bbbk}{G}$, the linear extension of an epimorphism $\nu\colon \pi{\twoheadrightarrow}G$.
As above, let $J= I_{{\Bbbk}} G$ be the two-sided ideal of ${\Bbbk}{G}$ generated by $\bar{\nu}(I)$, where $I=I_{{\Bbbk}}\pi$. Let $$\nu_*\colon H_1(X,{\Bbbk}) \longrightarrow H_1(G,{\Bbbk})$$ be the homomorphism induced by $\nu$ in homology with coefficients in ${\Bbbk}$. In view of Lemma \[lem:dr lin\], the next result describes the differential $d^1_G$ of $E^1(X, {\Bbbk}{G}_{\nu})$, solely in terms of $\nu_*$ and the comultiplication map $\nabla_X$ in $H_*(X,{\Bbbk})$.
\[prop:d1 kg\] The restriction of $d^1_{G}$ to $1\otimes H_q(X, {\Bbbk})$ is the composite $$\xymatrixcolsep{30pt}
\xymatrix{H_q(X, {\Bbbk}) \ar^(.36){\nabla_X}[r]
& H_1(X, {\Bbbk})\otimes_{{\Bbbk}} H_{q-1}(X, {\Bbbk})
\ar^{\nu_{*} \otimes \operatorname{id}}[r] & H_1(G, {\Bbbk})\otimes_{{\Bbbk}} H_{q-1}(X, {\Bbbk})}.$$
Let $\mu\colon {\Bbbk}{G} \otimes_{{\Bbbk}} {\Bbbk}\pi \to {\Bbbk}{G}$ the multiplication map of the ${\Bbbk}\pi$-module ${\Bbbk}{G}$, given on group elements by $\mu(g \otimes x)= g\nu(x)$. In view of Theorem \[thm:d1map\], we only need to identify the map $$\operatorname{gr}(\mu) \colon \operatorname{gr}^0_J ({\Bbbk}{G})\otimes_{{\Bbbk}} \operatorname{gr}^1_I ({\Bbbk}\pi)
\to \operatorname{gr}^1_J ({\Bbbk}{G}).$$
Under the identification $\operatorname{gr}^0_J({\Bbbk}{G})={\Bbbk}$, this map coincides with $\operatorname{gr}^1 (\bar{\nu})\colon \operatorname{gr}^1_I ({\Bbbk}\pi) \to
\operatorname{gr}^1_J ({\Bbbk}{G})$. In turn, under the identifications $\operatorname{gr}^1_I ({\Bbbk}\pi) = H_1(X,{\Bbbk})$ and $\operatorname{gr}^1_J ({\Bbbk}{G}) = H_1(G,{\Bbbk})$ provided by , the map $\operatorname{gr}^1 (\bar{\nu})$ coincides with $\nu_{*}$. This finishes the proof.
The next corollary describes (under some mild hypothesis) the dual of the differential $d^1_G$, solely in terms of the cup-product structure on $H^*(X,{\Bbbk})$, and the homomorphism $\nu^*\colon H^1(G,{\Bbbk}){\hookrightarrow}H^1(X,{\Bbbk})$ induced in cohomology by the epimorphism $\nu\colon \pi{\twoheadrightarrow}G$.
\[cor:d1 transp\] Suppose $X$ is a finite-type CW-complex, and $H_1(G,{\Bbbk})$ is a free ${\Bbbk}$-module. Let $\delta^1_G$ be the transpose of the restriction of $d^1_G$ to $1\otimes H_q(X, {\Bbbk})$. We then have a commuting triangle $$\xymatrixcolsep{25pt}
\xymatrix{
H^1(X, {\Bbbk})\otimes_{{\Bbbk}} H^{q-1}(X, {\Bbbk}) \ar^(.6){\cup_X}[rr]&&
H^q(X, {\Bbbk})\\
H^1(G, {\Bbbk})\otimes_{{\Bbbk}} H^{q-1}(X, {\Bbbk}) \ar@{^{(}->}^{\nu^* \otimes \operatorname{id}}[u]
\ar_(.55){\delta^1_G}[urr] &
}$$
Filtrations on homology {#subsec:two filt}
-----------------------
Let $X$ be a connected CW-complex as before, ${\Bbbk}$ a commutative ring, and let $M$ be a right ${\Bbbk}\pi$-module. The $I$-adic filtration on $C_{\bullet}(X,M)$ naturally defines a descending filtration on $H_*(X,M)$. The $n$-th term of the filtration is given by $$\label{eq:filthom}
F^{n} H_r(X,M) =
\operatorname{im}\big(H_r(F^{n} C_{\bullet}(X, M)) \longrightarrow
H_r(X,M)\big).$$ Clearly, $F^0 H_r(X, M)=H_r(X, M)$. As usual, we will write $F_{-n}=F^n$. The associated graded object of this filtration will be denoted by $\operatorname{gr}_F (H_*(X,M))$.
The filtration need not be separated, even when the spectral sequence converges. We shall illustrate this phenomenon in Example \[ex:p3\].
Now consider the case when $M$ is a ring $R$, with right ${\Bbbk}\pi$-module structure given by a base change $\rho\colon {\Bbbk}\pi\to R$. Let $J$ be the two-sided ideal generated by $\rho(I)$. As before, we will assume that either $R$ is commutative, or $R={\Bbbk}{G}$, and $\rho=\bar{\nu}$, for some epimorphism $\nu\colon \pi{\twoheadrightarrow}G$.
Using , we find that the spectral sequence filtration on $H_{*}(X, R)$, as defined in , is given by $$\label{eq:ffilt}
F^{n}H_{*}(X, R)=\big( \ker \tilde{\partial}^R \cap
J^n C_{\bullet}(X,R)+\operatorname{im}\tilde{\partial}^R \big)/\operatorname{im}\tilde{\partial}^R.$$ This puts a natural $\operatorname{gr}_J(R)$-module structure on $\operatorname{gr}_F H_*(X, R)$, since plainly $$\label{eq:hfilts}
J^k\cdot (J^n C_{\bullet}(X,R) \cap \ker \tilde{\partial}^R)
\subset J^{k+n} C_{\bullet}(X,R) \cap \ker \tilde{\partial}^R,
\quad \forall k, n.$$
\[rem:gr einf\] Suppose $X$ is of finite type, ${\Bbbk}$ is a field, and $\dim_{{\Bbbk}}R/J< \infty$ (this happens automatically when $R={\Bbbk}{G}_{\nu}$, in which case $R/J={\Bbbk}$). Fix an integer $q\ge 0$. Applying Proposition \[prop:conv ss\], and making use of , we obtain an inclusion $$\label{eq:inftygr}
\operatorname{gr}_F H_q(X, R)\subset {}_qE^{\infty}=
\bigoplus_{s\ge 0}E^{\infty}_{-s, s+q} ,$$ which is compatible with the respective $\operatorname{gr}_J(R)$-actions.
Comparing the two filtrations {#subsec:comp filt}
-----------------------------
We now have two filtrations on $H_q(X, R)$: the spectral sequence filtration $F^{\bullet}H_q(X, R)$, given by , and the $J$-adic filtration, $J^{\bullet}\cdot H_q(X, R)$. The two filtrations are related as follows.
\[lem:if\] $J^k \cdot F^{n} H_*(X,R) \subseteq F^{k+n} H_*(X,R)$, for all $k, n \ge 0$.
Our claim follows from and .
\[cor:two filtrations\] The $J$-adic filtration on $H_*(X,R)$ is finer than the filtration defined by $I$-adic spectral sequence: $$J^k \cdot H_*(X,R) \subseteq F^{k} H_*(X,R), \quad
\text{for all $k \ge 0$}.$$
In general, the two filtrations differ, even in the case when $R={\Bbbk}{A}$, with $A$ an abelian group, as the next example shows. On the other hand, we will see in Lemma \[lem:equal filt\] that the two filtrations coincide for $A={\mathbb{Z}}$.
\[ex:f2\] Consider the space $X=S^1 \vee S^1$, and its universal abelian cover, $X^{\operatorname{{ab}}}\to X$, classified by the map $\operatorname{{ab}}\colon \pi_1(X) \to {\mathbb{Z}}^2$. The chain complex $C_{\bullet}(X^{\operatorname{{ab}}}, {\Bbbk})$ takes the form $\tilde{\partial}_1 \colon \Lambda^2 \to \Lambda$, where $\Lambda={\Bbbk}[t_1^{\pm 1}, t_2^{\pm 1}]$ and $\tilde{\partial}_1=\left(\begin{smallmatrix}
t_1-1\\ t_2-1\end{smallmatrix}\right)$. Hence, $$H_1(X^{\operatorname{{ab}}},{\Bbbk}) = \ker \tilde{\partial}_1=
\operatorname{im}(t_2-1\, , 1-t_1) \cong \Lambda.$$ It follows that $J^s \cdot H_1(X^{\operatorname{{ab}}},{\Bbbk})\cong J^s$. On the other hand, $F_{-s}H_1(X^{\operatorname{{ab}}},{\Bbbk}) \cong J^{s-1}$. Thus, the two filtrations on $H_1(X^{\operatorname{{ab}}},{\Bbbk})$ do not coincide.
The homological Reznikov spectral sequence {#sec:kzp}
==========================================
In this section, we specialize to the case where the coefficient module is the group-ring ${\Bbbk}G$ of a cyclic group of prime-power order $p^r$, over a field ${\Bbbk}$ of characteristic $p$. When $X=K(\pi,1)$ is an Eilenberg–MacLane space and $G={\mathbb{Z}}_p$, the resulting spectral sequence is the homological version of a cohomology spectral sequence described by Reznikov in [@Re].
A convergent spectral sequence {#subsec:rez ss}
------------------------------
Let $X$ be a connected CW-complex, and let $\pi=\pi_1(X)$. Assume we have an epimorphism $\nu\colon \pi{\twoheadrightarrow}{\mathbb{Z}}_{p^r}$. Fix a field ${\Bbbk}$ of characteristic $p$, and denote by $({\Bbbk}{\mathbb{Z}}_{p^r})_{\nu}$ the group-ring of ${\mathbb{Z}}_{p^r}$, viewed as a right ${\Bbbk}\pi$-module via the linear extension $\bar{\nu}\colon {\Bbbk}\pi{\twoheadrightarrow}{\Bbbk}{\mathbb{Z}}_{p^r}$. Let $I=I_{\Bbbk}(\pi)$ and $J=I_{\Bbbk}( {\mathbb{Z}}_{p^r} )$ be the respective augmentation ideals.
\[lem:conv rez\] The $I$-adic filtration on $C_{\bullet}(X,({\Bbbk}{\mathbb{Z}}_{p^r})_{\nu})$ is finite.
By , $F^n C_{\bullet}(X,({\Bbbk}{\mathbb{Z}}_{p^r})_{\nu}) =J^n \otimes _{{\Bbbk}}
C_{\bullet}(X,{\Bbbk})$, for all $n\ge 0$. Now identify ${\Bbbk}{\mathbb{Z}}_{p^r}={\Bbbk}[t]/(t^{p^r}-1)$, and note that $J$ is the ideal generated by $(t-1)$. By the binomial formula, $(t-1)^{p^r}=t^{p^r}-1$ over ${\Bbbk}$. Thus, $J^{p^r} =0$. Putting things together, we conclude that $F^{p^r} C_{\bullet}(X,({\Bbbk}{\mathbb{Z}}_{p^r})_{\nu}) =0$.
Therefore, $E^{\bullet}(X,({\Bbbk}{\mathbb{Z}}_{p^r})_{\nu})$ converges. In fact, the spectral sequence collapses in finitely many steps: $\bigoplus_{s+t=q} E^{p^r}_{s,t}(X,({\Bbbk}{\mathbb{Z}}_{p^r})_{\nu})=
H_q (X,({\Bbbk}{\mathbb{Z}}_{p^r})_{\nu})$. For another situation when the spectral sequence converges, see Proposition \[prop:ss kzhat\] below.
Nevertheless, the spectral sequence filtration, $F^{\bullet} H_*(X,({\Bbbk}{\mathbb{Z}}_{p^r})_{\nu})$, does not always coincide with the $J$-adic filtration, $J^{\bullet}\cdot H_*(X,({\Bbbk}{\mathbb{Z}}_{p^r})_{\nu})$, as the following example shows.
\[ex:rez filt\] Let $X=S^1$, and consider the canonical projection $\nu\colon \pi_1(S^1)={\mathbb{Z}}{\twoheadrightarrow}{\mathbb{Z}}_p$. Identifying ${\mathbb{F}}_p {\mathbb{Z}}_p$ with $\Lambda_p={\mathbb{F}}_p[t]/(t^p-1)$, the chain complex $C_{\bullet}(S^1,({\mathbb{F}}_p {\mathbb{Z}}_p)_{\nu})$ takes the form $\tilde{\partial}_1 \colon \Lambda_p \to \Lambda_p$, with $\tilde{\partial}_1=(t-1)$. It follows that $H_1(S^1, \Lambda_p)=\ker( \tilde\partial_1)$ is the ideal generated by the norm element, $N=1+\cdots +t^{p-1}$. Hence, $J \cdot H_1(S^1, \Lambda_p) =0$.
On the other hand, $N$ augments to $0$ in ${\mathbb{F}}_p$, and so $N\in J\cap \ker(\tilde{\partial}_1)=F^1 H_1(S^1, \Lambda_p )$. Thus, $J \cdot H_1(S^1, \Lambda_p) \ne F^1 H_1(S^1, \Lambda_p )$.
The Reznikov spectral sequence {#subsec:rez e1}
------------------------------
We now specialize to the case when $G={\mathbb{Z}}_p$ and ${\Bbbk}={\mathbb{F}}_p$. Identify ${\mathbb{F}}_p{\mathbb{Z}}_p$ with ${\mathbb{F}}_p[t]/(t^p-1)$ and $\operatorname{gr}_J ({\mathbb{F}}_p {\mathbb{Z}}_p)$ with ${\mathbb{F}}_p [x]/(x^p)$, where $x= t-1$. Let $\nu\colon \pi{\twoheadrightarrow}{\mathbb{Z}}_p$ be an epimorphism, and consider the spectral sequence $E^{\bullet}(X,({\mathbb{F}}_p{\mathbb{Z}}_p)_{\nu})$. By , the first term is $$\label{eq:e1kzp}
E^1_{-s,t} = \operatorname{gr}^s_J({\mathbb{F}}_p{\mathbb{Z}}_p) \otimes_{{\mathbb{F}}_p}
H_{t-s}(X,{\mathbb{F}}_p) = H_{t-s}(X,{\mathbb{F}}_p),$$ for $t-s\ge 0$ and $0\le s\le p-1$, and $0$ otherwise. Here we used that $\operatorname{gr}^s_J({\mathbb{F}}_p{\mathbb{Z}}_p)={\mathbb{F}}_p$, with basis $x^s$, in the specified range.
To identify the $d^1$ differential, let $\nu_{*}\colon
H_1(X,{\mathbb{F}}_p) \to H_1({\mathbb{Z}}_p,{\mathbb{F}}_p)={\mathbb{F}}_p$ be the homomorphism induced by $\nu$ in homology, and let $\nu_{{\mathbb{F}}_p}\in H^1(X,{\mathbb{F}}_p)$ be the corresponding cohomology class.
\[prop:d1zp\] Under the identification from , the differential $d^1\colon E^1_{-s,s+q} \to
E^1_{-s-1,s+q}$ is the composite $$\xymatrixcolsep{28pt}
\varrho_q\colon \xymatrix{
H_q(X,{\mathbb{F}}_p) \ar[r]^(.34){\nabla_X} & H_1(X,{\mathbb{F}}_p) \otimes_{{\mathbb{F}}_p}
H_{q-1}(X,{\mathbb{F}}_p) \ar[r]^(.55){\nu_{*} \otimes \operatorname{id}}
& {\mathbb{F}}_p\otimes_{{\mathbb{F}}_p}H_{q-1}(X,{\mathbb{F}}_p)
}.$$ If, moreover, $X$ is of finite type, then the dual $\delta^1=(d^1)^{*}$ coincides with the left-multiplication map $$\cdot \nu_{{\mathbb{F}}_p} \colon H^{q-1}(X,{\mathbb{F}}_p) \to H^{q}(X,{\mathbb{F}}_p).$$
For the first statement, use Proposition \[prop:d1 kg\] with $G= {\mathbb{Z}}_p$ and ${\Bbbk}={\mathbb{F}}_p$, noting that under the identifications ${\mathbb{F}}_p \cong H_1({\mathbb{Z}}_p, {\mathbb{F}}_p) \cong J/J^2 \cong {\mathbb{F}}_p \cdot x$, the unit $1$ corresponds to $x$. The second statement follows from Corollary \[cor:d1 transp\].
In the particular case when $X=K(\pi,1)$, we recover the spectral sequence from [@Re Theorem 13.1], in homological form.
\[cor:rez ss\] Let $1\to K \to \pi \xrightarrow{\nu} {\mathbb{Z}}_p \to 1$ be an exact sequence of groups. There is then a spectral sequence $$E^1_{-s,s+q} = H_{q} (\pi,{\mathbb{F}}_p)\Rightarrow H_{q} (K, {\mathbb{F}}_p),$$ $q\ge 0$ and $0\le s\le p-1$, with differential $d^1 \colon H_q(\pi,{\mathbb{F}}_p) \to H_{q-1}(\pi,{\mathbb{F}}_p)$ equal to $\varrho_q=(\nu_{*} \otimes \operatorname{id}) \circ \nabla_{K(\pi,1)}$, and converging in finitely many steps.
Monodromy action {#subsec:rez mono}
----------------
In view of Proposition \[prop:d1zp\], the first page of the spectral sequence $E^{\bullet}(X,({\mathbb{F}}_p{\mathbb{Z}}_p)_{\nu})$ looks like $$\xymatrixrowsep{8pt}
\xymatrix{
\bullet & \bullet \ar_{\varrho_q}[l] & \bullet
\ar_{\varrho_{q+1}}[l] & \bullet \ar[l] & \bullet \ar[l] \\
& \bullet & \bullet \ar_{\varrho_q}[l] & \bullet
\ar_{\varrho_{q+1}}[l] & \bullet \ar[l] \\
& & \bullet& \bullet \ar_{\varrho_q}[l] & \bullet
\ar_{\varrho_{q+1}}[l] \\
& & & \bullet& \bullet \ar_{\varrho_q}[l]
}$$ with columns running right-to-left from $E^1_{0,*}$ to $E^1_{-p+1,*}$. Thus, if $p\ne 2$, we have a chain complex, $$\xymatrix{ \cdots \ar[r]
& H_{q+1}(X,{\mathbb{F}}_p) \ar^(.52){\varrho_{q+1}}[r]
& H_{q}(X,{\mathbb{F}}_p) \ar^(.45){\varrho_{q}}[r]
& H_{q-1}(X,{\mathbb{F}}_p) \ar[r] & \cdots}.$$
\[prop:jordan block\] If the chain complex $(H_*(X,{\mathbb{F}}_p), \varrho_*)$ is acyclic in degree $*=q$, then $J^2\cdot H_q(X,({\mathbb{F}}_p {\mathbb{Z}}_p)_{\nu})=0$.
Let $F^n = F^n H_q(X,({\mathbb{F}}_p{\mathbb{Z}}_p)_{\nu})$ be the spectral sequence filtration. Our hypothesis forces $E^2_{-s,q+s}=0$, for $0<s<p-1$. Hence, $E^{\infty}_{-s,q+s}=0$, for $0<s<p-1$, which implies $F^1 = \cdots = F^{p-1}$.
Now recall from Lemma \[lem:if\] that $J^k\cdot F^n \subseteq F^{k+n} $. Hence: $$J^2\cdot H_q(X,({\mathbb{F}}_p {\mathbb{Z}}_p)_{\nu}) = J^2 F^0 \subseteq
J F^1 = J F^{p-1} \subseteq F^p =0,$$ and this finishes the proof.
Completion and homology of abelian covers {#sec:completions}
=========================================
In this section, we study the more general situation when the coefficient module is the group-ring of an abelian group $A$, with emphasis on the good convergence properties of the corresponding equivariant spectral sequence. In the process, we use standard completion tools from commutative algebra, as described in [@Mat] and [@Ser].
Convergence {#subsec:jadic}
-----------
Let ${\Bbbk}$ be a field. Suppose we are given an epimorphism $\nu\colon \pi{\twoheadrightarrow}A$ onto an abelian group. We may then view the ring $\Lambda:={\Bbbk}{A}$ as a right ${\Bbbk}\pi$-module, via the ring map $\bar{\nu}\colon {\Bbbk}\pi{\twoheadrightarrow}{\Bbbk}{A}$. We shall denote this module by $\Lambda_{\nu}$, and will consider the spectral sequence $E^{\bullet}(X,\Lambda_{\nu})$, associated to the Galois cover of the CW-complex $X$ corresponding to $\nu$.
As usual, write $I=I_{{\Bbbk}}\pi$ and $J=\bar{\nu}(I) \Lambda$, and recall that $J$ is the augmentation ideal of $\Lambda={\Bbbk}{A}$. Also, note that $J$ is a maximal ideal of the Noetherian ring $\Lambda$. Let $\iota \colon \Lambda \to \widehat{\Lambda}$ be the $J$-adic completion. Set $\widehat{J}:= \iota (J) \widehat{\Lambda}$, and note that $\widehat{\Lambda}$ is a Noetherian local ring, with maximal ideal $\widehat{J}$, hence a Zariski ring. The ring morphism $\hat{\nu}:= \iota \circ \bar{\nu}$ defines another ${\Bbbk}\pi$-module, which we will denote by $\widehat{\Lambda}_{\hat{\nu}}$. We thus obtain a morphism of spectral sequences, $$\xymatrixcolsep{32pt}
\xymatrix{E^{\bullet} (X, \Lambda_{\nu})
\ar^{E^{\bullet}(\iota)}[r]& E^{\bullet} (X, \widehat{\Lambda}_{\hat{\nu}})
}.$$
We will frequently make use of the fact that, for any finitely generated $\Lambda$-module $M$, there is a natural $\widehat{\Lambda}$-isomorphism, $$\label{eq:jcompl}
\xymatrix{
M \otimes_{\Lambda} \widehat{\Lambda} \ar^(.6){\cong}[r]& \widehat{M}},$$ and the completion filtration on the $J$-adic completion $\widehat{M}$ coincides with the $\widehat{J}$-adic filtration.
The good convergence properties of both spectral sequences follow easily from the Artin-Rees Lemma, as explained by Serre in [@Ser].
\[prop:ss kzhat\] Let $X$ be a connected CW-complex of finite type. With notation as above, we have:
1. $E^1_{s,t} (X, \Lambda_{\nu})\Rightarrow H_{s+t}(X, \Lambda_{\nu})$ and $E^1_{s,t} (X, \widehat{\Lambda}_{\hat{\nu}})\Rightarrow
H_{s+t}(X, \widehat{\Lambda}_{\hat{\nu}})$.
2. The filtration $F_{\bullet}H_{*}(X, \widehat{\Lambda}_{\hat{\nu}})$ is separated.
3. If $X$ is a finite complex, the spectral sequence $E^{\bullet} (X, \Lambda_{\nu})$ collapses in finitely many steps.
See [@Ser pp. 22–24].
The next lemma shows that $E^{\bullet} (X, \Lambda_{\nu})$ computes $H_{*}(X, \Lambda_{\nu}) \otimes_{\Lambda}
\widehat{\Lambda}$, the $J$-adic completion of $H_{*}(X, \Lambda_{\nu})$.
\[lem:eiota\] Let $X$ be a connected CW-complex of finite type. Then the maps $$E^r(\iota)\colon E^r(X, \Lambda_{\nu}) \to
E^r(X, \widehat{\Lambda}_{\hat{\nu}})\quad {\it and} \quad
\operatorname{gr}_F (\iota)\colon \operatorname{gr}_F H_{q}(X, \Lambda_{\nu})
\to \operatorname{gr}_F H_{q}(X, \widehat{\Lambda}_{\hat{\nu}})$$ are isomorphisms, for all $1\le r \le \infty$ and $q\ge 0$.
Set $C_{\bullet}:= C_{\bullet}(X, \Lambda)$ and $\widehat{C}_{\bullet}:= C_{\bullet}(X, \widehat{\Lambda})$. Note that, by construction, $\widehat{C}_{\bullet}$ is the $J$-adic completion of the $\Lambda$-chain complex $C_{\bullet}$. By , $$F^k C_{\bullet}(X, \Lambda)= J^k \cdot C_{\bullet}
\quad\text{and}\quad
F^k C_{\bullet}(X, \widehat{\Lambda})=
\widehat{J}^k \cdot \widehat{C}_{\bullet},$$ for all $k$. Since $\iota \colon C_{\bullet} \to \widehat{C}_{\bullet}$ is the completion map, it follows from that $E^0(\iota)$ is an isomorphism. Therefore, $E^r(\iota)$ is an isomorphism, for $1\le r <\infty$. The remaining assertions follow from convergence.
Cyclic quotients {#subsec:cycli quotient}
----------------
We now specialize further, to the case when $A={\mathbb{Z}}$. Let $\nu\colon \pi{\twoheadrightarrow}{\mathbb{Z}}$ be an epimorphism, and ${\Bbbk}$ a commutative ring. Identify the group-ring ${\Bbbk}{A}$ with the Laurent polynomial ring $\Lambda={\Bbbk}[t^{\pm 1}]$, and note that the augmentation ideal of $\Lambda$ is principal: $J= (t-1)\Lambda$. Identify also the associated graded ring $\operatorname{gr}({\Bbbk}{\mathbb{Z}})$ with the polynomial ring ${\Bbbk}[x]$, where $x=t-1$. Assuming either ${\Bbbk}={\mathbb{Z}}$ and $H_*(X, {\mathbb{Z}})$ is torsion-free, or ${\Bbbk}$ is a field, it follows from and Lemma \[lem:irj\] that $$\label{eq:e1kz}
E^1_{-s,t}(X,\Lambda_{\nu}) = \operatorname{gr}^s_J({\Bbbk}{\mathbb{Z}}) \otimes_{{\Bbbk}} H_{t-s}(X,{\Bbbk})=
{\Bbbk}\otimes_{{\Bbbk}} H_{t-s}(X,{\Bbbk}) = H_{t-s}(X,{\Bbbk}),$$ where we used that $\operatorname{gr}^s_J({\Bbbk}{\mathbb{Z}})$ is freely generated by $x^s$.
The induced homomorphism $\nu_{*}\colon H_1(\pi,{\Bbbk})\to H_1({\mathbb{Z}},{\Bbbk})={\Bbbk}$ determines a cohomology class $\nu_{{\Bbbk}}\in H^1(X,{\Bbbk})$. Proceeding as in the proof of Proposition \[prop:d1zp\], we obtain the following.
\[cor:d1z\] Let $X$ be a connected CW-complex and let $\nu\colon \pi_1(X){\twoheadrightarrow}{\mathbb{Z}}$ be an epimorphism. Assume either that ${\Bbbk}={\mathbb{Z}}$ and $H_*(X, {\mathbb{Z}})$ is torsion-free, or ${\Bbbk}$ is a field. With identifications as in , the differential $d^1\colon E^1_{-s,s+q} \to E^1_{-s-1,s+q}$ is the composite $$\xymatrixcolsep{30pt}
\xymatrix{
H_q(X,{\Bbbk}) \ar[r]^(.35){\nabla_X} & H_1(X,{\Bbbk}) \otimes_{{\Bbbk}}
H_{q-1}(X,{\Bbbk}) \ar[r]^(.45){\nu_* \otimes \operatorname{id}}
& {\Bbbk}\otimes_{{\Bbbk}} H_{q-1}(X,{\Bbbk}) = H_{q-1}(X,{\Bbbk})
}\!.$$ If, moreover, $X$ is of finite type, then the dual $\delta^1=(d^1)^{*}$ coincides with the left-multiplication map $$\cdot \nu_{{\Bbbk}} \colon H^{q-1}(X,{\Bbbk}) \to H^{q}(X,{\Bbbk}).$$
Comparing the two filtrations {#subsec:comp filt again}
-----------------------------
Next, identify the $J$-adic completion of $\Lambda={\Bbbk}{\mathbb{Z}}$ with the power-series ring $\widehat{\Lambda}= {\Bbbk}[[x]]$, so that the map $\iota \colon \Lambda
\to \widehat{\Lambda}$ sends $t$ to $x+1$. Note that the extension of the ideal $J$ is principal: $\widehat{J}= x \widehat{\Lambda}$.
Let $\nu\colon \pi {\twoheadrightarrow}{\mathbb{Z}}$ be an epimorphism. We have seen in §\[subsec:exr\] that each piece of the corresponding spectral sequences, ${}_qE^r_*$ and ${}_q\widehat{E}^r_*$, supports a natural graded module structure over the ring $\operatorname{gr}_J (\Lambda)= \operatorname{gr}_{\widehat{J}} (\widehat{\Lambda})$. Recall from Corollary \[cor:two filtrations\] that, for any commutative ring $R$, viewed as a right ${\Bbbk}\pi$-module via a ring map $\rho \colon {\Bbbk}\pi \to R$, we have an inclusion $J^s \cdot H_q(X, R) \subseteq F_{-s} H_q(X, R)$, where $J$ is the ideal of $R$ generated by $\rho (I)$.
\[lem:equal filt\] Let $X$ be a connected CW-complex and ${\Bbbk}$ a commutative ring. Set $R={\Bbbk}{\mathbb{Z}}_\nu$ or $\widehat{{\Bbbk}{\mathbb{Z}}}_{\hat{\nu}}$. Then
1. \[l1\] For each $r$ and $q$, the $\operatorname{gr}(R)$-module ${}_qE^r_*$ is generated by ${}_qE^r_0$.
2. \[l2\] The spectral sequence filtration on $H_*(X, R)$ coincides with the $(x)$-adic filtration.
Part . It is enough to check that $Z^r_{-s}= x^s\cdot Z^r_0$, for all $r,s\ge 0$. Recall from that $F^k C_{\bullet}(X, R)= x^k \cdot C_{\bullet}(X, R)$. If $z=x^s y$ and $\tilde{\partial}(y)= x^r y'$, then $z\in F^s$ and $\tilde{\partial}(z)= x^{s+r} y'\in F^{s+r}$, hence $z\in Z^r_{-s}$. Conversely, if $z=x^s y$ and $\tilde{\partial}(z)= x^s \tilde{\partial}(y)= x^{r+s} y'$, then $\tilde{\partial}(y)= x^{r} y'$, and therefore $y\in Z^r_0$, as needed.
Part follows from the equality $Z^{\infty}_{-s}=
x^s\cdot Z^{\infty}_0$, which is checked in a similar way.
Nevertheless, the filtration $F_{\bullet}(X,{\Bbbk}{\mathbb{Z}})$ need not be separated, as the next example shows.
\[ex:p3\] Let $X=S^1\times (S^1 \vee S^1)$, and identify $\pi=\pi_1(X)$ with ${\mathbb{Z}}\times F_2=\langle a,b,c \mid a \text{ central}\rangle$. The epimorphism $\nu\colon \pi {\twoheadrightarrow}{\mathbb{Z}}$ that sends $a\mapsto 2$, $b\mapsto 1$, $c\mapsto 1$ defines a Galois ${\mathbb{Z}}$-cover, $Y\to X$. Identify the group ring ${\Bbbk}{\mathbb{Z}}$ with $\Lambda={\Bbbk}[t^{\pm 1}]$, and consider the $\Lambda$-module $N:=H_1(Y,{\Bbbk})=H_1(X, {\Bbbk}{{\mathbb{Z}}}_{\nu})$. Then $$N=\begin{cases}
\big( \Lambda/(1-t) \big)^2 \oplus \Lambda/(1+t) & \text{if $\operatorname{char}{\Bbbk}\ne 2$},
\\[3pt]
\Lambda/(1-t)\oplus \Lambda/(1-t)^2 & \text{if $\operatorname{char}{\Bbbk}=2$}.
\end{cases}$$ Thus, $$\bigcap_{s\ge 0} J^s N =
\begin{cases}
\Lambda/(1+t) & \text{if $\operatorname{char}{\Bbbk}\ne 2$}, \\
0 & \text{if $\operatorname{char}{\Bbbk}=2$}.
\end{cases}$$ where $J$ denotes the augmentation ideal of $\Lambda$. By Lemma \[lem:equal filt\], the filtration $\{F_{s} N\}$ is not separated, if $\operatorname{char}{\Bbbk}\ne 2$. In this situation, the spectral sequence $E^{\bullet} (X,{\Bbbk}{\mathbb{Z}})$ converges, yet we cannot recover $N=H_1(X,{\Bbbk}{\mathbb{Z}})$ from the spectral sequence, even additively.
Monodromy action and the Aomoto complex {#sec:mono}
=======================================
In the previous section, we showed that the equivariant spectral sequence of a finite type CW-complex $X$, starting at $E^{1}_{s,t} (X,{\Bbbk}{\mathbb{Z}}_{\nu})$, converges to $H_{s+t}(X,{\Bbbk}{\mathbb{Z}}_{\nu})$, the homology groups of the cover determined by an epimorphism $\nu\colon \pi_1(X){\twoheadrightarrow}{\mathbb{Z}}$, with coefficients in a field ${\Bbbk}$. Using this fact, we now investigate the monodromy action of ${\mathbb{Z}}$ on $H_{*}(X,{\Bbbk}{\mathbb{Z}}_{\nu})$. In the process, we relate the triviality of the action to the exactness of the Aomoto complex defined by $\nu_{{\Bbbk}}\in H^1(X,{\Bbbk})$.
Modules over ${\Bbbk}{\mathbb{Z}}$ {#subsec:kz mod}
----------------------------------
Start by identifying the group-ring ${\Bbbk}{\mathbb{Z}}$ with $\Lambda={\Bbbk}[t^{\pm 1}]$. Since ${\Bbbk}$ is a field, the ring $\Lambda$ is a PID, and so every finitely-generated $\Lambda$-module $P$ decomposes as a direct sum of a free module, $P_0=\Lambda^{\operatorname{rank}P}$, with finitely many modules of the form $$\label{eq:primary}
P_f=\bigoplus _{i> 0} (\Lambda/f^i)^{e_{f,i}(P)},$$ indexed by irreducible polynomials $f \in \Lambda$. We call $P_f$ the $f$-primary part of $P$, and $\Lambda/f^i$ an $f$-primary Jordan block of size $i$.
Now let $X$ be a finite-type CW complex, and $\nu\colon \pi_1(X){\twoheadrightarrow}{\mathbb{Z}}$ an epimorphism. For each $q\ge 0$, the homology group $P^q:=H_q(X,{\Bbbk}{\mathbb{Z}}_{\nu})$, viewed as a module over $\Lambda={\Bbbk}{\mathbb{Z}}$, decomposes as $$\label{eq:hdecomp}
H_q(X,{\Bbbk}{\mathbb{Z}}_{\nu}) = P^q_0 \oplus P^q_{t-1} \oplus
\bigoplus_{f(1)\ne 0} P^q_f,$$ with $P^q_0=\Lambda^{r_q}$, where $r_q$ is the rank of $P^q$, and $P^q_{t-1}=
\bigoplus_{i>0} (\Lambda/(t-1)^i)^{e^q_i}$, where $e^q_i=e_{t-1,i}(P^q)$ is the number of $(t-1)$-primary Jordan blocks of size $i$.
Our first result in this section identifies precisely the information that the $I$-adic spectral sequence carries about the ${\Bbbk}{\mathbb{Z}}$-module structure on $H_*(X, {\Bbbk}{\mathbb{Z}}_{\nu})$, or, equivalently, the monodromy action of the group ${\mathbb{Z}}=\langle t\rangle$ on the homology groups of the covering space of $X$ determined by $\nu$. Set $J=(t-1){\Bbbk}{\mathbb{Z}}$.
\[prop:monospsq\] For each $q\ge 0$, the $\operatorname{gr}_J({\Bbbk}{\mathbb{Z}})$-module structure on ${}_qE^{\infty}(X, {\Bbbk}{\mathbb{Z}}_{\nu})$ described in determines the free and $(t-1)$-primary parts of $H_q(X, {\Bbbk}{\mathbb{Z}}_{\nu})$, viewed as a module over ${\Bbbk}{\mathbb{Z}}$.
Identify $\operatorname{gr}_J({\Bbbk}{\mathbb{Z}})= {\Bbbk}[x]$, where $x=t-1$. Proposition \[prop:ss kzhat\], together with Lemma \[lem:equal filt\] and , guarantee that the ${\Bbbk}[x]$-modules ${}_qE^{\infty}$ and $\operatorname{gr}_J H_q(X, {\Bbbk}{\mathbb{Z}}_{\nu})$ are isomorphic. From , we obtain a decomposition $$\label{eq:kx}
\operatorname{gr}_J H_q(X, {\Bbbk}{\mathbb{Z}}_{\nu}) = {\Bbbk}[x]^{r_q} \oplus
\bigoplus_{i>0} \big( {\Bbbk}[x]/x^i \big)^{e^q_i} ,$$ as ${\Bbbk}[x]$-modules. Hence, the $\operatorname{gr}_J({\Bbbk}{\mathbb{Z}})$-module structure of ${}_qE^{\infty}$ determines the numbers $r_q$ and $e^q_i$, that is, the free and $(t-1)$-primary parts of $H_q(X, {\Bbbk}{\mathbb{Z}}_{\nu})$.
Aomoto complex {#subsec:aomoto}
--------------
As before, denote by $\nu_{{\mathbb{Z}}}$ the class in $H^1(X,{\mathbb{Z}})$ determined by an arbitrary homomorphism $\nu\colon \pi\to {\mathbb{Z}}$. Let $\nu_{*}\colon H_1(X,{\Bbbk})\to H_1({\mathbb{Z}},{\Bbbk})={\Bbbk}$ be the induced homomorphism. The corresponding cohomology class, $\nu_{{\Bbbk}}\in H^1(X,{\Bbbk})$, is the image of $\nu_{{\mathbb{Z}}}$ under the coefficient homomorphism ${\mathbb{Z}}\to {\Bbbk}$. We then have: $$\label{eq:nusquare}
\nu_{{\Bbbk}} \cup \nu_{{\Bbbk}} = 0 \text{ in } H^2(X,{\Bbbk}).$$ Indeed, by obstruction theory, there is a map $f\colon X\to S^1$ and a class $\omega\in H^1(S^1,{\mathbb{Z}})$ such that $\nu_{{\mathbb{Z}}}=f^*(\omega)$. Hence, $\nu_{{\mathbb{Z}}}\cup\nu_{{\mathbb{Z}}}= f^*(\omega\cup\omega) =0$. Formula then follows by naturality of cup products with respect to coefficient homomorphisms.
As a consequence, left-multiplication by $\nu_{{\Bbbk}}$ turns the cohomology ring $H^*( X, {\Bbbk})$ into a cochain complex, $$\label{eq:aomoto}
(H^*( X, {\Bbbk}) , \cdot\nu_{{\Bbbk}})\colon
\xymatrix{
H^0(X,{\Bbbk}) \ar^{\nu_{{\Bbbk}}}[r] &
H^1(X,{\Bbbk}) \ar^{\nu_{{\Bbbk}}}[r] &
H^2(X,{\Bbbk}) \ar[r] & \cdots
}$$ which we call the [*Aomoto complex*]{} of $H^*(X, {\Bbbk})$, with respect to $\nu_{{\Bbbk}}$.
\[def:aomoto betti\] The [*Aomoto Betti numbers*]{} of $X$, with respect to the cohomology class $\nu_{{\Bbbk}}\in H^1(X,{\Bbbk})$, are defined as $$\label{eq:aomoto betti}
\beta_q(X, \nu_{{\Bbbk}}) :=
\dim_{{\Bbbk}} H^q(H^*( X, {\Bbbk}) , \cdot\nu_{{\Bbbk}}).$$
Clearly, $\beta_q(X, \nu_{{\Bbbk}}) \le \dim_{{\Bbbk}} H^q( X, {\Bbbk})$. In general though, the inequality is strict.
\[ex:exterior\] Let $X=T^n$, the $n$-dimensional torus. The cohomology ring $H^*( X, {\Bbbk})$ is simply the exterior algebra on $H^1(X,{\Bbbk})={\Bbbk}^n$. If $\nu$ is onto, $\nu_{{\Bbbk}} \ne 0$, and so the complex is exact. Thus, $\beta_q(X, \nu_{{\Bbbk}})=0$, for all $q\ge 0$, though, of course, $b_q(X)=\binom{n}{q}$.
Monodromy action {#subsec:mono}
----------------
We are now ready to state and prove the second result of this section. Let $P^q=H_q(X,{\Bbbk}{\mathbb{Z}}_{\nu})$, viewed as a module over ${\Bbbk}{\mathbb{Z}}$ via the epimorphism $\nu\colon \pi{\twoheadrightarrow}{\mathbb{Z}}$, with decomposition as in .
\[prop:trivial action\] For each $k\ge 0$, the following are equivalent:
1. \[m0\] For each $q\le k$, the vector space $H_q(X,{\Bbbk}{\mathbb{Z}}_{\nu})$ is finite-dimensional, and contains no $(t-1)$-primary Jordan blocks of size greater than $1$.
2. \[m1\] For each $q\le k$, the monodromy action of ${\Bbbk}{\mathbb{Z}}$ on $P^q_0 \oplus
P^{q}_{t-1}$ is trivial.
3. \[m2\] $\beta_0(X,\nu_{{\Bbbk}})=\cdots =\beta_k(X,\nu_{{\Bbbk}})=0$.
The equivalence $\Leftrightarrow$ is an immediate consequence of decompositions –. To prove the equivalence $\Leftrightarrow$ , recall we have a spectral sequence $E^1_{-p,q}=\operatorname{gr}^p({\Bbbk}{\mathbb{Z}})\otimes H_{q-p}(X,{\Bbbk})\Rightarrow
H_q(X,{\Bbbk}{\mathbb{Z}}_\nu)$, whose differential $d^1\colon
H_q(X,{\Bbbk})\to H_{q-1}(X,{\Bbbk})$ is the transpose of $\cdot\nu_{{\Bbbk}}$.
$\Rightarrow$ . The condition $\beta_{\le k}(X,\nu_{{\Bbbk}})=0$ is equivalent to $E^2_{-p,q}=0$, for $q-p\le k$ and $p>0$. Consequently, $E^{\infty}_{-p,q}=0$, in the same range. By convergence of the spectral sequence and Lemma \[lem:equal filt\], this means: $$\frac{(t-1)^p \cdot H_{q-p}(X,{\Bbbk}{\mathbb{Z}}_{\nu})}
{(t-1)^{p+1} \cdot H_{q-p}(X,{\Bbbk}{\mathbb{Z}}_{\nu})}=0, \quad
\text{for $q-p\le k$ and $p>0$}.$$ Taking $p=1$ in the above, and recalling the discussion from §\[subsec:kz mod\], we obtain $$\frac{(t-1) \cdot (P^q_0 \oplus P^{q}_{t-1})}
{(t-1)^{2} \cdot (P^q_0 \oplus P^{q}_{t-1})}=0,$$ for all $q\le k$. This implies $r_q=0$ (since $\operatorname{gr}^1(\Lambda)\ne 0$) and $e^{q}_{i}=0$, for $i>1$ (since $\dim_{\Bbbk}\operatorname{gr}^1(P^{q}_{t-1})=\sum_{i>1} e^q_i$), for all $q\le k$, which is equivalent to $t-1=0$ on $P^q_0 \oplus P^{q}_{t-1}$.
$\Rightarrow$ . Induction on $k$. For $k=0$, we have $$\beta_0(X,\nu_{{\Bbbk}})= \dim_{\Bbbk}\ker(\cdot \nu_{{\Bbbk}} \colon
H^0(X,{\Bbbk})\to H^1(X,{\Bbbk}))=0.$$ Now assume $\beta_{\le {k-1}}(X,\nu_{{\Bbbk}})=0$. Then $E^2_{-1,k+1}=\cdots = E^{\infty}_{-1,k+1}$. Moreover, we always have $\beta_q(X,\nu_{{\Bbbk}})=\dim_{\Bbbk}E^2_{-p,q+p}$, for all $p\ge 1$. In our situation, we have $$\beta_k(X,\nu_{{\Bbbk}})= \dim_{\Bbbk}E^{\infty}_{-1,k+1}
=
\dim_{\Bbbk}\frac{(t-1) \cdot (P^k_0 \oplus P^{k}_{t-1})}
{(t-1)^{2} \cdot (P^k_0 \oplus P^{k}_{t-1})},$$ and this vanishes, by assumption. The induction step is thus proved.
Bounds on twisted cohomology ranks: I {#sect:betti bounds}
=====================================
In this section, we give upper bounds on the ranks of the cohomology groups with coefficients in a rank $1$ local system defined by a rational character of prime-power order.
Twisted Betti numbers {#subs:twist betti}
---------------------
We start with some definitions. Let $X$ be a connected, finite-type CW-complex. Given a field ${\Bbbk}$, the [*${\Bbbk}$-Betti numbers*]{} of $X$ are defined as $b_q(X, {\Bbbk}) := \dim _{{\Bbbk}} H^q( X, {\Bbbk})$. More generally, if $R$ is a Noetherian ring, define $b_q(X, R)$ to be the minimal number of generators of the $R$-module $H^q(X,R)$.
Now suppose $\rho \colon \pi\to {\mathbb{C}}^{\times}$ is a homomorphism from $\pi=\pi_1(X)$ to the multiplicative group of non-zero complex numbers. The [*twisted Betti numbers*]{} of $X$ corresponding to the character $\rho$ are defined by $$\label{eq:twisted betti}
b_q(X, \rho) := \dim _{{\mathbb{C}}} H^q( X, {}_{\rho}{\mathbb{C}}),$$ where ${}_{\rho}{\mathbb{C}}={\mathbb{C}}$, viewed as a left module over ${\mathbb{C}}\pi$, via $\rho$. By duality, $b_q(X, \rho) = \dim _{{\mathbb{C}}} H_q( X, {\mathbb{C}}_{\rho})$, where ${\mathbb{C}}_{\rho}={\mathbb{C}}$, viewed as a right module over ${\mathbb{C}}\pi$, via $\rho$.
Of particular importance to us are characters of the form $\rho(g)=\zeta^{\nu(g)}$, where $\nu\colon \pi \to {\mathbb{Z}}$ is a homomorphism, $d$ is a positive integer, and $\zeta$ is a primitive $d$-th root of unity. In this case, we say $\rho$ is a [*rational character*]{} of order $d$, and write $$\label{eq:nud betti}
b_q(X, \nu/d) := b_q(X,\rho).$$ It follows from elementary Galois theory that the twisted Betti numbers $b_q(X, \nu/d)$ do not depend on the choice of primitive $d$-th root of unity, thereby justifying the notation.
Cyclotomic polynomials {#subs:cyclo}
----------------------
For each positive integer $d$, the $d$-th cyclotomic polynomial is defined as $\Phi_d(t)=\prod_{\zeta} (t-\zeta)$, where $\zeta$ ranges over all primitive $d$-th roots of unity. If $d=p^r$, with $p$ a prime, then $\Phi_{d}(t)=(t^{p^r}-1)/(t^{p^{r-1}}-1)$, and so $\Phi_{d}(1)=p$. If $d$ is not a prime power, and $d>1$, then $\Phi_{d}(1)=1$.
\[lem:cyclo\] Let $Q\in {\mathbb{Z}}[t]$ be a polynomial with integer coefficients. Suppose $Q(\zeta)=0$, for some root of unity $\zeta\in {\mathbb{C}}$ of prime-power order $p^r$. Then $Q(1)=0 \pmod{p}$.
The minimal polynomial of $\zeta$ is the cyclotomic polynomial $\Phi_{p^r}(t)$. Thus, $\Phi_{p^r}\mid Q$. But $\Phi_{p^r}(1)=p$, and so $p\mid Q(1)$.
The following Corollary will be useful in the sequel. Let $A$ be a matrix with entries in ${\mathbb{Z}}[t^{\pm 1}]$. We will denote by $A(z)$ the evaluation of $A$ at a non-zero complex number $z\in {\mathbb{C}}^{\times}$. For $p$ a prime, we may also view $A(1)$, after reduction modulo $p$, as a matrix with entries in ${\mathbb{F}}_p$.
\[cor:minors\] Let $\zeta$ be a root of unity of order a power of a prime $p$. Then $$\operatorname{rank}_{\mathbb{C}}A(\zeta) \ge \operatorname{rank}_{{\mathbb{F}}_p} A(1).$$
Suppose $m(t)$ is a minor of $A$, and $m(\zeta)=0$. Then $t^k m(t)\in {\mathbb{Z}}[t]$, for some $k\ge 0$, and $\zeta^k m(\zeta)=0$. Hence, by Lemma \[lem:cyclo\], $m(1)=0$ in ${\mathbb{F}}_p$. The conclusion follows.
Modular Betti bounds {#subs:mod bb bound}
--------------------
The following result relates the two kinds of Betti numbers defined above, under a prime-power assumption on the order of the rational character.
\[thm:bettibound\] Let $X$ be a connected, finite-type CW-complex. Let $\nu\colon \pi_1(X)\to {\mathbb{Z}}$ be a homomorphism, and $p$ a prime. Then, for all $r\ge 1$ and $q\ge 0$, $$\label{eq:bettibound}
b_q(X,\nu/p^r) \le b_q(X,{\mathbb{F}}_p).$$
Let $C_{\bullet}(X,{\mathbb{Z}})=(C_q,\partial_q)_{q\ge 0}$ be the cellular chain complex of $X$, and let ${\mathbb{Z}}{\mathbb{Z}}_{\nu}$ be the ring ${\mathbb{Z}}{\mathbb{Z}}={\mathbb{Z}}[t^{\pm 1}]$, viewed as a module over ${\mathbb{Z}}\pi_1(X)$ via the ring map $\bar{\nu}\colon {\mathbb{Z}}\pi_1(X) \to {\mathbb{Z}}{\mathbb{Z}}$. The equivariant chain complex $C_{\bullet}(X,{\mathbb{Z}}{\mathbb{Z}}_{\nu})$ has chains $C_q(X,{\mathbb{Z}}{\mathbb{Z}}_\nu) = {\mathbb{Z}}[t^{\pm 1}] \otimes C_q$, and differentials $$\label{eq:del nu}
{\partial^{\nu}}_q:=\tilde{\partial}^{{\mathbb{Z}}{\mathbb{Z}}_{\nu}}_q\colon
{\mathbb{Z}}[t^{\pm 1}] \otimes C_q\to{\mathbb{Z}}[t^{\pm 1}] \otimes C_{q-1}.$$ Note that ${\partial^{\nu}}_q(1)=\partial_q$. From the definition of twisted Betti numbers, we have $$\label{eq:bettitwist}
b_q(X, \nu/p^r) = \operatorname{rank}C_q -\operatorname{rank}_{\mathbb{C}}{\partial^{\nu}}_q(\zeta)
-\operatorname{rank}_{\mathbb{C}}{\partial^{\nu}}_{q+1}(\zeta),$$ where $\zeta$ is a primitive root of unity of order $p^r$.
Now consider the chain complex $C_{\bullet}(X,{\mathbb{F}}_p)$. By definition, $C_q(X,{\mathbb{F}}_p)=C_q\otimes {\mathbb{F}}_p$. Clearly, the differential $\partial_q \otimes \operatorname{id}_{{\mathbb{F}}_p}$ equals the reduction mod $p$ of ${\partial^{\nu}}_q(1)$. Thus, $$\label{eq:betti p}
b_q(X,{\mathbb{F}}_p) = \operatorname{rank}C_q -\operatorname{rank}_{{\mathbb{F}}_p} {\partial^{\nu}}_q(1)
-\operatorname{rank}_{{\mathbb{F}}_p} {\partial^{\nu}}_{q+1}(1).$$
The desired inequality follows from , , and Corollary \[cor:minors\].
\[rem:knots\] Given a knot $K$ in $S^3$, let $X=S^3 \setminus K$ be the knot complement, $\pi=\pi_1(X)$ the knot group, and $\nu=\operatorname{{ab}}\colon \pi {\twoheadrightarrow}{\mathbb{Z}}$ the abelianization map. The [*Alexander polynomial*]{} of the knot, $\Delta_K(t)\in {\mathbb{Z}}[t^{\pm 1}]$, is the greatest common divisor of the codimension $1$ minors of the Alexander matrix, $\partial_2^{\nu}$, defined as in . Let $\rho \colon \pi\to {\mathbb{C}}^{\times}$ be a non-trivial character. Writing $\rho(g)=z^{\nu(g)}$, for some $z\in {\mathbb{C}}^{\times}$, we have: $$\label{eq:alex root}
b_1(X,\rho)\ne 0 {\Longleftrightarrow}\Delta_K(z)=0$$ (see [@DPS07] for a much more general statement). In particular, $b_1(X,\nu/d)\ne 0$ if and only if $\Delta_K(\zeta)=0$, for some primitive $d$-th root of unity $\zeta$.
Examples and discussion {#subsec:sharp bettibound}
-----------------------
We conclude this section by discussing the necessity of the hypothesis in Theorem \[thm:bettibound\], and the sharpness of inequality .
We start with the prime-power hypothesis. Suppose $d$ is a positive integer so that $d\ne p^r$, for any prime $p$. One may wonder whether an inequality of the form $$\label{eq:betti general}
b_q(X,\nu/d) \le b_q(X,R)$$ holds, for some suitable choice of Noetherian ring $R$. The following example shows that this is not possible, in general.
\[ex:knots1\] By assumption, $d$ is not a prime-power integer; thus, $\Phi_d(1)=1$. On the other hand, $\Phi_d(t^{-1})\equiv \Phi_d(t)$, up to units in ${\mathbb{Z}}[t^{\pm 1}]$. Hence, by a classical result of Seifert [@Se], there is a knot $K_d$ in $S^3$ with Alexander polynomial $\Delta_{K_d}(t)=\Phi_d(t)$.
Fix an integer $n>1$, and let $K=\sharp^n K_d$ be the connected sum of $n$ copies of $K_d$. Denote by $X_d =S^3 \setminus K_d$ and $X =S^3 \setminus K$ the corresponding knot complements. By additivity of Alexander invariants under connected sums of knots (see for instance [@Ro]), we have an isomorphism of ${\mathbb{C}}{\mathbb{Z}}$-modules, $H_1(X, {\mathbb{C}}{\mathbb{Z}}_{\nu})\cong H_1(X_d, {\mathbb{C}}{\mathbb{Z}}_{\nu_d})^n$, where the twisted coefficients are defined by abelianization. Let ${\mathbb{C}}_{\zeta}={\mathbb{C}}$, viewed as a ${\mathbb{C}}{\mathbb{Z}}$-module via evaluation of Laurent polynomials at a primitive $d$-root of unity $\zeta$. Then $$\begin{aligned}
b_1(X, \nu/d)&= \dim_{{\mathbb{C}}} ({\mathbb{C}}_{\zeta} \otimes_{{\mathbb{C}}{\mathbb{Z}}}
H_1(X, {\mathbb{C}}{\mathbb{Z}}_{\nu})) \\
&= n \dim_{{\mathbb{C}}} ({\mathbb{C}}_{\zeta} \otimes_{{\mathbb{C}}{\mathbb{Z}}} H_1(X_d, {\mathbb{C}}{\mathbb{Z}}_{\nu_d})) \\
& = n\, b_1(X_d, \nu_d/d). \end{aligned}$$ But $\Delta_{K_d}(\zeta)=\Phi_d(\zeta)=0$, and so, as noted in Remark \[rem:knots\], $b_1(X_d, \nu_d/d)\ge 1$. Thus, $b_1(X, \nu/d)\ge n$. On the other hand, $b_1(X,R)=1$, for any ring $R$.
Next, we show that inequality from Theorem \[thm:bettibound\] may fail to be an equality.
\[ex:knots2\]
Let $X=S^3\setminus K$ be a knot complement, $\nu\colon \pi_1(X){\twoheadrightarrow}{\mathbb{Z}}$ the abelianization map, and $d=p^r$. If $b_1(X,\nu/d)$ were non-zero, then $\Delta_K(\zeta)$ would vanish, and so $\Phi_d(t)$ would divide $\Delta_K(t)$. But $\Phi_d(1)=p$, while $\Delta_K(1)=\pm 1$. Hence, we have $b_1(X,\nu/d)=0$, yet $b_1(X,{\mathbb{F}}_p)=1$.
Finally, we show that the bound cannot be improved.
\[ex:sharp\] Let $X$ be any connected, finite-type CW-complex with $H_q(X,{\mathbb{Z}})$ torsion-free, and let $\nu\colon \pi_1(X)\to {\mathbb{Z}}$ be the zero map. Then $b_q(X,\nu/p^r)=b_q(X,{\mathbb{F}}_p)=b_q(X)$, for all primes $p$.
Bounds on twisted cohomology ranks: II {#sect:aomoto bounds}
======================================
Continuing the theme from the previous section, we sharpen the bounds on the twisted Betti numbers $b_q(X,\nu/p^r)$, by using the Aomoto Betti numbers $\beta_q(X,\nu_{{\mathbb{F}}_p})$ instead of the usual Betti numbers $b_q(X,{\mathbb{F}}_p)$, under an additional freeness assumption on $H_*(X,{\mathbb{Z}})$.
Decomposing the boundary map {#subsec:homology}
----------------------------
Let $X$ be a connected, finite-type CW-complex. Assume ${\Bbbk}={\mathbb{Z}}$ and $H_*(X,{\mathbb{Z}})$ is torsion-free, or ${\Bbbk}$ is a field. Let $C_q=C_q(X,{\Bbbk})$ be the the group of cellular $q$-chains over ${\Bbbk}$, and $\partial_q\colon C_q \to C_{q-1}$ the boundary map. Writing $Z_q=\ker \partial_q$ and $B_{q-1}=\operatorname{im}\partial_{q}$, we have a split exact sequence $$0\to Z_q \to C_q \xrightarrow{\partial_q} B_{q-1}\to 0.$$ By assumption, $H_q=Z_q/B_q$ is a free ${\Bbbk}$-module. Thus, $$\label{eq:chain decomp}
C_q \cong Z_q \oplus B_{q-1} \cong B_q \oplus N_q,$$ where $N_q \cong H_q\oplus B_{q-1}$. The next Lemma follows at once.
\[eq:block\] With respect to the direct sum decompositions $C_q=Z_q \oplus B_{q-1}$ and $C_{q-1}=B_{q-1} \oplus N_{q-1}$, the matrix of the differential $\partial_q\colon C_q \to C_{q-1}$ takes the block-matrix form $$\label{eq:del decomp}
\partial_q =\begin{pmatrix}
0 & \operatorname{id}\\
0 & 0
\end{pmatrix}.$$
\[rem:uct\] Suppose $H_*(X,{\mathbb{Z}})$ is torsion-free. Then, by standard homological algebra, $Z_q(X,{\Bbbk})=Z_q(X,{\mathbb{Z}})\otimes {\Bbbk}$, $B_q(X,{\Bbbk})=B_q(X,{\mathbb{Z}})\otimes {\Bbbk}$, and $H_q(X,{\Bbbk})=H_q(X,{\mathbb{Z}})\otimes {\Bbbk}$. It easily follows that the block-matrix decomposition of $\partial_q$ is compatible with the canonical coefficient homomorphism ${\mathbb{Z}}\to {\Bbbk}$.
Aomoto bounds on the twisted Betti numbers {#subsec:aomoto bound}
------------------------------------------
We are now ready to state the main result of this section.
\[thm:cohobound\] Let $X$ be a connected, finite-type CW-complex, with $H_*(X,{\mathbb{Z}})$ torsion-free. Let $\nu\colon \pi_1(X)\to {\mathbb{Z}}$ be a homomorphism, and $p$ a prime. Then, for all $r\ge 1$ and $q\ge 0$, $$\label{eq:cohobound}
b_q(X, \nu/p^r) \le \beta_q (X,\nu_{{\mathbb{F}}_p}).$$
Without loss of generality, we may assume $\nu$ is surjective. Indeed, if $\nu =0$ then clearly $$b_q(X, \nu/p^r) = b_q(X, {\mathbb{C}}) \le
b_q(X, {\mathbb{F}}_p) =\beta_q (X,\nu_{{\mathbb{F}}_p}).$$ On the other hand, if the image of $\nu$ has index $m$ in ${\mathbb{Z}}$, then plainly $b_q(X, \nu/p^r) = b_q(X, \nu'/p^s)$, for an epimorphism $\nu'\colon \pi_1(X){\twoheadrightarrow}{\mathbb{Z}}$ such that $\nu =m \nu'$. (Here $p^s$ is the order of $\zeta^m$, where $\zeta$ is a primitive root of unity of order $p^r$.) If $p$ divides $m$, then $\nu_{{\mathbb{F}}_p}= 0$ and claim becomes inequality . Otherwise, $s\ge 1$ and $\beta_q (X,\nu_{{\mathbb{F}}_p})= \beta_q (X,\nu'_{{\mathbb{F}}_p})$. Hence, the result for $\nu'$ implies the claim for $\nu$.
Consider the equivariant chain complex $C_{\bullet}(X,{\mathbb{Z}}{\mathbb{Z}}_\nu)$, viewed as a chain complex over $\Lambda={\mathbb{Z}}[t^{\pm 1}]$. Identify $C_q(X,{\mathbb{Z}}{\mathbb{Z}}_\nu) = \Lambda \otimes C_q$, where $C_q=C_q(X,{\mathbb{Z}})$, and denote by ${\partial^{\nu}}_q\colon \Lambda\otimes C_q\to
\Lambda \otimes C_{q-1}$ the boundary maps.
Since ${\partial^{\nu}}_q(1)=\partial_q$ and $\left. \partial_q \right|_{Z_q}=0$, the restriction of ${\partial^{\nu}}_q$ to $\Lambda\otimes Z_q$ takes values in $J\otimes C_{q-1}$, where $J=(t-1)\Lambda$. As in §\[subsec:homology\], write $C_q= Z_q \oplus B_{q-1}$ and $C_{q-1} = N_{q-1}\oplus B_{q-1}$. Using formula , we see that ${\partial^{\nu}}_q$ takes the block-matrix form $$\label{eq:dq}
{\partial^{\nu}}_q =\begin{pmatrix}
(t-1) {\mathsf{P}}& {\mathsf{Q}}\\
(t-1) {\mathsf{R}}& {\mathsf{S}}\end{pmatrix},$$ where ${\mathsf{Q}}(1)$ is the identity, and ${\mathsf{S}}(1)$ is the zero matrix. For future use, define the block-matrix ${A}_q:=\left(\begin{smallmatrix} {\mathsf{P}}& {\mathsf{Q}}\\ {\mathsf{R}}& {\mathsf{S}}\end{smallmatrix}\right)$, and note that $\operatorname{rank}_{\mathbb{C}}{\partial^{\nu}}_q(z) = \operatorname{rank}_{\mathbb{C}}{A}_q(z)$, for any $z\in {\mathbb{C}}\setminus \{0,1\}$.
Recall from that $b_q(X, \nu/p^r) = \operatorname{rank}C_q -\operatorname{rank}_{\mathbb{C}}{\partial^{\nu}}_q(\zeta)
-\operatorname{rank}_{\mathbb{C}}{\partial^{\nu}}_{q+1}(\zeta)$, where $\zeta$ is a primitive root of unity of order $p^r$. Hence, $$\label{eq:bqa}
b_q(X, \nu/p^r) =\operatorname{rank}C_q -\operatorname{rank}_{\mathbb{C}}{A}_q(\zeta)
-\operatorname{rank}_{\mathbb{C}}{A}_{q+1}(\zeta).$$
Let us estimate the right side. By Corollary \[cor:minors\], $$\label{eq:aqz}
\operatorname{rank}_{\mathbb{C}}{A}_q(\zeta) \ge \operatorname{rank}_{{\mathbb{F}}_p} {A}_q(1).$$ By Lemma \[eq:block\] and Remark \[rem:uct\], $$\label{eq:arb}
\operatorname{rank}_{{\mathbb{F}}_p} {A}_q(1)=\operatorname{rank}_{{\mathbb{F}}_p} {\mathsf{R}}_q(1)
+ \operatorname{rank}B_{q-1}.$$ Combining , , and with the equality $b_q(X)= \operatorname{rank}C_q -\operatorname{rank}B_q -\operatorname{rank}B_{q-1}$, we obtain $$\label{eq:bqnupr}
b_q(X, \nu/p^r)
\le b_q(X) -\operatorname{rank}_{{\mathbb{F}}_p} {\mathsf{R}}_q(1) -\operatorname{rank}_{{\mathbb{F}}_p} {\mathsf{R}}_{q+1}(1).$$
We now turn to the Aomoto-Betti numbers. By definition, $$\label{eq:betaqnufp}
\beta_q(X,\nu_{{\mathbb{F}}_p}) = b_q(X,{\mathbb{F}}_p)-\operatorname{rank}_{{\mathbb{F}}_p} \nu^q_{{\mathbb{F}}_p}
-\operatorname{rank}_{{\mathbb{F}}_p} \nu^{q+1}_{{\mathbb{F}}_p},$$ where $\nu^q_{{\mathbb{F}}_p} \colon H^{q-1}(X,{\mathbb{F}}_p)\to
H^{q}(X,{\mathbb{F}}_p)$ denotes left-multiplication by $\nu_{{\mathbb{F}}_p}\in H^1(X,{\mathbb{F}}_p)$. By Corollary \[cor:d1z\], this map is dual to the differential $d^1_q\colon E^1_{0,q} \to E^1_{-1,q}$, where $E^1_{0,q} = \operatorname{gr}^0({\mathbb{F}}_p{\mathbb{Z}})\otimes_{{\mathbb{F}}_p} H_q(X,{\mathbb{F}}_p)$ and $E^1_{-1,q} = \operatorname{gr}^1({\mathbb{F}}_p{\mathbb{Z}})\otimes_{{\mathbb{F}}_p} H_{q-1}(X,{\mathbb{F}}_p)$. Using the identifications $\operatorname{gr}^0({\mathbb{F}}_p{\mathbb{Z}})={\mathbb{F}}_p$ and $\operatorname{gr}^1({\mathbb{F}}_p{\mathbb{Z}})=x\cdot {\mathbb{F}}_p\cong {\mathbb{F}}_p$, where $x=t-1$, we may view this differential as a map $d^1_q\colon H_q(X,{\mathbb{F}}_p) \to H_{q-1}(X,{\mathbb{F}}_p)$.
Clearly, $d^1_q$ has the same rank as $\delta^1_q=\iota \circ d^1_q \circ \pi$, where $\pi\colon Z_q(X,{\mathbb{F}}_p) {\twoheadrightarrow}H_q(X,{\mathbb{F}}_p)$ is the projection, and $\iota \colon H_{q-1}(X,{\mathbb{F}}_p) {\hookrightarrow}N_{q-1}(X,{\mathbb{F}}_p)$ is the inclusion. Hence, $$\label{eq:betaqd1}
\beta_q(X,\nu_{{\mathbb{F}}_p}) = b_q(X)-\operatorname{rank}_{{\mathbb{F}}_p} \delta^1_q
-\operatorname{rank}_{{\mathbb{F}}_p} \delta^1_{q+1}.$$
In view of and , it suffices to show that $\delta^1_q={\mathsf{R}}_q(1)$. Let $z\in Z_q(X,{\mathbb{F}}_p)$. Using formula —with everything reduced mod $p$—we find: $$\begin{aligned}
\delta^1_q(z)
& = \iota( [{\partial^{\nu}}_q(1 \otimes z) \bmod J^2] ) \\
&= (t-1) {\mathsf{R}}_q(t) z \bmod J^2 \\
&=x {\mathsf{R}}_q(1) z \bmod J^2\\
&\equiv {\mathsf{R}}_q(1) z\end{aligned}$$ where at the last step we used the identification $\operatorname{gr}^1({\mathbb{F}}_p{\mathbb{Z}})=J/J^2=x\cdot {\mathbb{F}}_p\cong {\mathbb{F}}_p$. This finishes the proof.
Necessity of the hypothesis {#subsec:hypo}
---------------------------
We now give examples showing that the two hypotheses in Theorem \[thm:cohobound\] are necessary. We start with the prime-power hypothesis.
Given a Noetherian ring $R$, let $\nu_R\in H^1(X,R)$ be the cohomology class determined by the homomorphism $\pi_1(X)\xrightarrow{\nu} {\mathbb{Z}}\xrightarrow{\iota} R$, where $\iota(1)=1$. Define $\beta_q(X, \nu_R)$ to be the minimal number of generators of the $R$-module $H^q(H^*( X,R) , \cdot\nu_R)$.
Now suppose $d$ is a positive integer so that $d\ne p^r$, for any prime $p$. One may wonder whether an inequality of the form $b_q(X,\nu/d) \le \beta_q(X,\nu_R)$ holds, for some suitable choice of Noetherian ring $R$. The following example shows that this is not possible.
\[ex:coho torus\] Let us start by recalling an old result of R. Lyndon (see [@Bi Thm. 3.6]). If $v_1,\dots,v_n$ are elements in $\Lambda= {\mathbb{Z}}{\mathbb{Z}}^n$ satisfying $\sum_{i=1}^n v_i(t_i-1)=0$, then there is a word $r\in F'_n$ such that $(\partial r/\partial x_i)^{\operatorname{{ab}}}=v_i$, for all $i$. Hence, if $\pi=\langle x_1,\dots, x_n\mid r\rangle$ is the corresponding $1$-relator group, and $X$ is the presentation $2$-complex, then the chain complex $C_{\bullet}(X^{\operatorname{{ab}}},{\Bbbk})$ has boundary maps $\widetilde{\partial}^{\operatorname{{ab}}}_2\colon \Lambda \to \Lambda^n$ and $\widetilde{\partial}^{\operatorname{{ab}}}_1\colon \Lambda^n \to \Lambda$ given by $$\widetilde{\partial}^{\operatorname{{ab}}}_2(1 \otimes e_2)
=\sum_{i=1}^n v_i e^i_1
\quad\text{and}\quad
\widetilde{\partial}^{\operatorname{{ab}}}_1(e^i_1) =
(t_i -1) e_0.$$
Now take $v_1=\Phi_d(t_1) (t_2-1)$ and $v_2=\Phi_d(t_1) (1-t_1)$, and let $X$ be the $2$-complex constructed above. Let $\nu\colon \pi{\twoheadrightarrow}{\mathbb{Z}}$, $\nu(x_1)=\nu(x_2)=1$, and fix a primitive $d$-th root of unity $\zeta$. The chain complex $C_{\bullet}(X,{\mathbb{C}}_{\rho})\colon {\mathbb{C}}\to {\mathbb{C}}^2 \to {\mathbb{C}}$ corresponding to the character $\rho\colon \pi\to {\mathbb{C}}^{\times}$, $\rho(g)=\zeta^{\nu(g)}$, has boundary maps $\partial^{\rho}_2=0$ and $\partial^{\rho}_1 =\Big(\begin{smallmatrix}\zeta-1\\[2pt]
\zeta-1\end{smallmatrix}\Big)$. Hence, $H_1(X,{\mathbb{C}}_{\rho})={\mathbb{C}}$, and so $b_1(X,\nu/d)=1$.
Next, let $R$ be a Noetherian ring, and set $J=I_R{\mathbb{Z}}^2$. The differential $d^1\colon H_2(X,R) \to H_1(X,R)\otimes H_1(X,R)$ on $E^1(X,R{\mathbb{Z}}^2_{\operatorname{{ab}}})$ is given by: $$\begin{aligned}
d^1( {[\![ e_2 ]\!]})
&= \widetilde{\partial}^{\operatorname{{ab}}}_2(1\otimes e_2) \mod J^2 \\
&=\Phi_d(t_1) (t_2-1) \otimes {[\![ e^1_{1} ]\!]} +
\Phi_d(t_1) (1-t_1) \otimes {[\![ e^2_{1} ]\!]} \mod J^2 \\
&=\Phi_d(1) (t_2-1) \otimes {[\![ e^1_{1} ]\!]} +
\Phi_d(1) (1-t_1) \otimes {[\![ e^2_{1} ]\!]} \mod J^2 \\
&\equiv \Phi_d(1) \left( {[\![ e^2_{1} ]\!]} \otimes {[\![ e^1_{1} ]\!]} -
{[\![ e^1_{1} ]\!]} \otimes {[\![ e^2_{1} ]\!]}\right). \end{aligned}$$ Hence, by Corollary \[cor:d1z\], the map $\cdot \nu_R \colon H^1(X,R) \to H^2(X,R)$ sends ${[\![ e^{1}_{1} ]\!]}^*\mapsto \Phi_d(1){[\![ e_2 ]\!]}^*$ and ${[\![ e^{2}_{1} ]\!]}^*\mapsto -\Phi_d(1){[\![ e_2 ]\!]}^*$. On the other hand, the map $\cdot \nu_R \colon H^0(X, R) \to H^1(X, R)$ sends ${[\![ e_{0} ]\!]}^*\mapsto {[\![ e^1_1 ]\!]}^*+{[\![ e^2_1 ]\!]}^*$. Recall we assumed $d$ is not a prime power, i.e., $\Phi_d(1)=1$. Therefore, $H^1(H^*(X,R),\nu_R) = 0$.
To recap, we showed that $b_1(X,\nu/d)=1$, yet $\beta_1(X,\nu_R)=0$. Let us note that $X$ is a minimal CW-complex, in the sense of the definition from §\[subsect:mini\] below; indeed, $\epsilon(v_1)=\epsilon(v_2)=0$, and so the boundary maps in $C_{\bullet}(X,{\mathbb{Z}})$ vanish. Furthermore, Corollary \[cor:d1 transp\] implies that $X$ has the same cohomology ring as $S^1\times S^1$.
Next, we show that the hypothesis that $H_*(X,{\mathbb{Z}})$ be torsion-free is really necessary.
\[ex:tors free\] Start with $Y=S^1 \vee S^2$, and identify $\pi_1(Y)={\mathbb{Z}}$, with generator $t$, and set $\Lambda={\mathbb{Z}}[t^{\pm 1}]$. Using the construction from Example \[ex:construction\], build the CW-complex $X=Y\cup_{\phi_{1+t}} e_3$. The equivariant chain complex of $X$ can be written as $$\label{eq:equiv2}
C_{\bullet}({\widetilde{X}},{\mathbb{Z}})\colon
\xymatrix{\Lambda \ar[r]^{1+t} & \Lambda \ar[r]^{0}
& \Lambda \ar[r]^{t-1} &\Lambda}.$$ Consequently, $C_{\bullet}(X,{\mathbb{Z}})$ has the form ${\mathbb{Z}}\xrightarrow{2} {\mathbb{Z}}\xrightarrow{0} {\mathbb{Z}}\xrightarrow{0} {\mathbb{Z}}$, and so $H_2(X,{\mathbb{Z}})={\mathbb{Z}}_2$.
Now take $\nu\colon \pi_1(X)\to {\mathbb{Z}}$ to be the identity, and pick the prime $p=2$. The chain complex $C_{\bullet}(X,{\mathbb{C}}_{\rho})$ corresponding to the rational character $\rho(t)=-1$ has the form ${\mathbb{C}}\xrightarrow{0} {\mathbb{C}}\xrightarrow{0}
{\mathbb{C}}\xrightarrow{-2} {\mathbb{C}}$, and so $b_3(X,\nu/2)=1$.
On the other hand, it follows from that all boundary maps of $C_{\bullet}(X,{\mathbb{F}}_2)$ are $0$. Hence, $H_i(X,{\mathbb{F}}_2)={\mathbb{F}}_2$, generated by ${[\![ e_i ]\!]}$, for $0\le i\le 3$. Moreover, $\nu_{{\mathbb{F}}_2}={[\![ e_1 ]\!]}^*$, the generator of $H^1(X,{\mathbb{F}}_2)={\mathbb{F}}_2$. We also know from that $\tilde{\partial}_3(1\otimes e_3)= (1+t) \otimes e_2$. Hence, the differential $d^1\colon H_3(X,{\mathbb{F}}_2)\to H_2(X,{\mathbb{F}}_2)$ of $E^1 (X, {\mathbb{F}}_2 {\mathbb{Z}}_{\nu})$ is given by ${[\![ e_3 ]\!]} \mapsto {[\![ e_2 ]\!]}$. By Corollary \[cor:d1z\], the map $\cdot \nu_{{\mathbb{F}}_2} \colon H^2(X,{\mathbb{F}}_2) \to H^3(X,{\mathbb{F}}_2)$ takes ${[\![ e_2 ]\!]}^*$ to ${[\![ e_3 ]\!]}^*$. Hence, $\beta_3(X,\nu_{{\mathbb{F}}_2})=0$.
Sharpness of the bound {#subsec:sharp}
----------------------
Inequality from Theorem \[thm:cohobound\] may fail to be an equality, as we now show.
\[ex:coho strict\]
Let $G=\langle x, y \mid [x,y]^p \rangle$, let $X$ be the associated presentation $2$-complex, and let $\nu\colon G\to {\mathbb{Z}}$ be the diagonal character, sending both $x$ and $y$ to $1$. We then have $b_1(X,\nu/p)=0$, whereas $\beta_1(X,\nu_{{\mathbb{F}}_p})=1$.
On the other hand, the bound cannot be improved, as the next example shows.
\[ex:coho sharp\]
Let ${{\mathcal{A}}}$ be a subarrangement of a complexified reflection arrangement of type $A$, $B$, or $D$, and let $X$ be the complement of ${{\mathcal{A}}}$. Choose $\nu\colon \pi_1(X) \to {\mathbb{Z}}$ to be the diagonal character, sending each oriented meridian to $1$. In this case, the bound in Theorem \[thm:cohobound\] is attained at all primes, for $q=1$ and $r=1$; see [@MP Theorem C].
Galois covers, minimality, and linearization {#sect:minilin}
============================================
In this section, we analyze in detail the first page of the equivariant spectral sequence of an arbitrary Galois cover. Using this approach, we give an intrinsic meaning to the linearization of the equivariant chain complex, in the important case of minimal CW-complexes. Throughout, ${\Bbbk}$ will denote the integers ${\mathbb{Z}}$, or a field.
Minimal cell complexes {#subsect:mini}
----------------------
We start by reviewing a notion discussed in [@PS]; see also [@DP1; @DP2] for various applications. Let $X$ be a connected, finite-type CW-complex. We say the CW-structure on $X$ is [*minimal*]{} if the number of $q$-cells of $X$ coincides with the (rational) Betti number $b_q(X)$, for every $q\ge 0$. Equivalently, the boundary maps in the cellular chain complex $C_{\bullet}(X,{\mathbb{Z}})$ are all the zero maps. In particular, $X$ has a single $0$-cell, call it $e_0$.
If $X$ is a minimal cell complex, the homology groups $H_*(X,{\mathbb{Z}})$ are all torsion-free. In particular, $H_q(X,{\Bbbk})=H_q(X,{\mathbb{Z}})\otimes {\Bbbk}$ and $H^q(X,{\Bbbk})=H_q(X,{\Bbbk})^*$, for ${\Bbbk}={\mathbb{Z}}$ or ${\Bbbk}$ a field.
Even if the homology groups of $X$ are torsion-free, the space $X$ need not be minimal. For example, if $K$ is a knot in $S^3$, with complement $X$, then $H_*(X,{\mathbb{Z}})=H_*(S^1,{\mathbb{Z}})$, yet $X$ does not admit a minimal CW-structure, unless $K$ is the trivial knot.
Examples of spaces admitting minimal CW-structures are: spheres $S^n$, tori $T^n$, orientable Riemann surfaces, and complex Grassmanians—in fact, any compact, connected smooth manifold admitting a perfect Morse function. If ${{\mathcal{A}}}$ is a complex hyperplane arrangement, then its complement, $X$, has a minimal cell decomposition; see [@DP1].
Linearizing the equivariant boundary maps {#subsec:lin diff}
-----------------------------------------
Let $X$ be a minimal CW-complex, with fundamental group $\pi=\pi_1(X)$. As usual, let $(C_{\bullet}(X,{\Bbbk}), \partial)$ be the cellular chain complex of $X$, and let $(C_{\bullet}({\widetilde{X}},{\Bbbk}), \tilde{\partial})$ be the equivariant chain complex, with filtration $F^n= I^n \cdot C_{\bullet} ({\widetilde{X}}, {\Bbbk})$, where $I=I_{{\Bbbk}} \pi$. Let $\nu\colon \pi {\twoheadrightarrow}G$ be an epimorphism, and consider the chain complex $(C_{\bullet}(X, {\Bbbk}{G}_{\nu}), \tilde{\partial}^G)$, with filtration $F^{n} C_{\bullet}(X, {\Bbbk}{G}_{\nu}) =
J^n \otimes_{{\Bbbk}} C_{\bullet}(X,{\Bbbk})$, where $J=\bar{\nu}(I) =I_{{\Bbbk}} G$.
\[lem:minfilt\] $\tilde{\partial}^G F^n C_{\bullet}(X, {\Bbbk}{G}_{\nu})
\subset F^{n+1} C_{\bullet}(X, {\Bbbk}{G}_{\nu})$, for all $n\ge 0$.
Recall from §\[subsec:diff\] that $p\circ \tilde{\partial} =
\partial \circ p$, where $p\colon C_{\bullet}({\widetilde{X}},{\Bbbk}) {\twoheadrightarrow}C_{\bullet}({\widetilde{X}},{\Bbbk})/
F^1 C_{\bullet}(X,{\Bbbk})$ is the quotient map. Consequently, $\tilde{\partial} F^0 \subset F^1$, by minimality of $X$.
From the discussion in §\[subsec:func\], we know that $(\nu\otimes \operatorname{id})\circ \tilde{\partial} \tilde{\partial}^G\circ (\nu\otimes \operatorname{id})$, and that $\nu \otimes \operatorname{id}$ preserves filtrations. Since $\nu \otimes \operatorname{id}$ is onto, we infer that $\tilde{\partial}^G F^0 \subset F^1$. The conclusion follows from ${\Bbbk}{G}$-linearity of $\tilde{\partial}^G$.
\[def:lin\] The [*linearization*]{} of the boundary map $\tilde{\partial}^G$ is the map induced by $\tilde{\partial}^G$ at the associated graded level, $$\label{eq:del lin}
\partial_G^{\operatorname{\,lin}} \colon \operatorname{gr}^*_F C_{\bullet} (X, {\Bbbk}{G}_{\nu})
\to \operatorname{gr}^{*+1}_F C_{\bullet} (X, {\Bbbk}{G}_{\nu}).$$
Now use again the minimality of $X$ to identify $C_q(X,{\Bbbk}) = H_q(X,{\Bbbk})$ and $C_{\bullet} (X, {\Bbbk}{G}_{\nu})= {\Bbbk}{G} \otimes_{{\Bbbk}} H_q(X,{\Bbbk})$. From the construction of the equivariant spectral sequence, we obtain immediately that $$\label{eq:freelin}
E^1( X, {\Bbbk}G_{\nu}) = E^0( X, {\Bbbk}G_{\nu}),\quad \text{and}
\quad d^1_G= \partial_G^{\operatorname{\,lin}} .$$
At this point, it is easy to give a concrete interpretation of linearization, in terms of matrices and natural ${\Bbbk}$-bases provided by cells. For each $q\ge 1$, denote by $\operatorname{{Mat}}(\tilde{\partial}_q^G)$ the matrix corresponding to the boundary map $$\tilde{\partial}_q^G \colon {\Bbbk}{G} \otimes_{{\Bbbk}} H_q(X,{\Bbbk})
\to {\Bbbk}{G} \otimes_{{\Bbbk}} H_{q-1}(X,{\Bbbk}).$$ By Lemma \[lem:minfilt\], all the entries of $\operatorname{{Mat}}(\tilde{\partial}_q^G)$ belong to the ideal $J=I_{{\Bbbk}}G$. Reducing those entries modulo $J^2$, we obtain a new matrix, denoted by $\operatorname{{Mat}}(\tilde{\partial}_q^G )\bmod J^2$, with entries in $J/J^2 \cong H_1(G, {\Bbbk})$.
\[cor:matlin\] Let $X$ be a minimal CW-complex, and let $\nu\colon \pi_1(X){\twoheadrightarrow}G$ be an epimorphism. Then, for all $q\ge 1$, $$\operatorname{{Mat}}(\tilde{\partial}_q^G ) \bmod J^2= \operatorname{{Mat}}\big( \!
\xymatrixcolsep{50pt}
\xymatrix{H_q(X, {\Bbbk}) \ar^(.36){(\nu_{*}\otimes \operatorname{id})\circ \nabla_X}[r] &
H_1(G, {\Bbbk})\otimes_{{\Bbbk}} H_{q-1}(X, {\Bbbk})}\! \big) .$$
With the identifications discussed above, $\operatorname{{Mat}}(\tilde{\partial}_q^G)
\bmod J^2$ is the matrix of $\partial_G^{\operatorname{\,lin}}=d^1_G$. The conclusion follows from Proposition \[prop:d1 kg\].
\[rem:nuid\] The above equality was proved in [@DP1 Theorem 20] for $\nu =\operatorname{id}$ and ${\Bbbk}= {\mathbb{Z}}$, provided $H^*(X, {\mathbb{Z}})$ is generated as a ring in degree one.[^4] When ${\Bbbk}$ is a field, there is a connection between the chain complex $(E^1(X, {\Bbbk}\pi), d^1)$ and the well-known Koszul complex from homological algebra. We refer to [@DP1 Proposition 22] for a class of Koszul resolutions coming from linearization, and to [@DP1 Theorem 23] for an application to the computation of higher homotopy groups.
Linearizing the equivariant cochain complex {#subsec:linearize cochains}
-------------------------------------------
Let $X$ be a connected CW-complex, with $\pi=\pi_1(X)$, and let $\nu\colon \pi {\twoheadrightarrow}G$ be an epimorphism. Recall from Example \[ex:covers\] that $$C^{\bullet}(X, {}_{\nu}{\Bbbk}{G})=
( \operatorname{{Hom}}_{{\Bbbk}\pi}(C_{\bullet}({\widetilde{X}},{\Bbbk}), {}_{\nu}{\Bbbk}{G}), \,
\tilde{\delta}^{\bullet}_G)$$ denotes the cochain complex of $X$, with coefficients in the left ${\Bbbk}\pi$-module ${}_{\nu}{\Bbbk}{G}$. This is a cochain complex of (right) ${\Bbbk}{G}$-modules, endowed with a decreasing filtration $F^{\bullet}$, with $n$-th term given by $$F^n= \operatorname{{Hom}}_{{\Bbbk}\pi}(C_{\bullet}({\widetilde{X}},{\Bbbk}), J^n),$$ with $J=I_{{\Bbbk}}G$ viewed as a left ${\Bbbk}\pi$-module via $\nu$. Alternatively, if we identify $C^q(X, {}_{\nu}{\Bbbk}{G})$ with $\operatorname{{Hom}}_{{\Bbbk}} (C_q(X,{\Bbbk}), {\Bbbk}{G})$, then $F^n$ corresponds to $\operatorname{{Hom}}_{{\Bbbk}} (C_q(X,{\Bbbk}), J^n)$.
\[lem:mindfilt\] Suppose $X$ is a minimal CW-complex. Then $\tilde{\delta}_G F^n \subset F^{n+1}$, for all $n\ge 0$.
Follows immediately from the definition of $\tilde{\delta}_G$ and Lemma \[lem:minfilt\] applied to $\nu= \operatorname{id}$.
Thus, the coboundary map $\tilde{\delta}^{\bullet}_G$ induces a (dual) linearization map, $\operatorname{gr}^*_F C^{\bullet} (X, {}_{\nu}{\Bbbk}{G})\to
\operatorname{gr}^{*+1}_F C^{\bullet} (X, {}_{\nu}{\Bbbk}{G})$, or, modulo standard identifications, $$\label{eq:deltalin}
\delta_G^{\operatorname{\,lin}} \colon C^{\bullet}(X,{\Bbbk})\otimes_{{\Bbbk}} \operatorname{gr}^*_J ({\Bbbk}{G})
\longrightarrow C^{\bullet}(X,{\Bbbk})\otimes_{{\Bbbk}} \operatorname{gr}^{*+1}_J ({\Bbbk}{G}) .$$ It is a routine matter to verify that the map $\delta_G^{\operatorname{\,lin}}$ above is $\operatorname{gr}_J ({\Bbbk}{G})$-linear.
Now fix pairs of natural dual bases for $H_q(X, {\Bbbk})$ and $H^q(X, {\Bbbk})$, for each $q\ge 1$. It is easy to check that $\operatorname{{Mat}}(\tilde{\delta}_G^q)=\operatorname{{Mat}}(\tilde{\partial}^G_q)^{{{\scriptstyle{\mathsf{T}}}}}$, where $(\cdot)^{{{\scriptstyle{\mathsf{T}}}}}$ denotes the transpose. By Corollary \[cor:matlin\], then, $$\label{eq:dmatlin}
\operatorname{{Mat}}(\tilde{\delta}_G^q)\bmod J^2= \operatorname{{Mat}}\!\big(\!\xymatrixcolsep{22pt}
\xymatrix{H_q(X, {\Bbbk})\ar^(.38){(\nu_{*}\otimes \operatorname{id})\circ \nabla_X}[rr] &&
H_1(G, {\Bbbk})\otimes_{{\Bbbk}} H_{q-1}(X, {\Bbbk})}\!\big)^{{{\scriptstyle{\mathsf{T}}}}}.$$
Linearization and the Aomoto complex {#subsec:lin aomoto}
------------------------------------
Finally, we describe the dual linearization map in more familiar terms, in the case when $\nu\colon \pi {\twoheadrightarrow}\pi_{\operatorname{{ab}}}$ is the abelianization homomorphism. It turns out that this may be done in terms of the universal Aomoto complex, so we begin by reviewing this notion.
Let $H^*$ be a graded ${\Bbbk}$-algebra. We need to assume that $H^1$ is a free, finitely-generated ${\Bbbk}$-module, and $a^2=0$, for all $a\in H^1$. (These conditions are satisfied by the cohomology rings of minimal CW-complexes, with arbitrary coefficients.) Pick a ${\Bbbk}$-basis $\{ e_1^*,\dots, e_n^* \}$ of $H^1$, and denote by $\{ e_1,\dots, e_n \}$ the dual basis of $H_1:= \operatorname{{Hom}}_{{\Bbbk}}(H^1, {\Bbbk})$. Denote by $S$ the symmetric algebra $\operatorname{Sym}(H_1)$, and identify it with the polynomial ring ${\Bbbk}[e_1,\dots, e_n]$. The [*universal Aomoto complex*]{} of $H$ is the cochain complex of free $S$-modules, $$\label{eq:univaom}
{\mathbb{A}}^{\bullet} (H)\colon
\xymatrix{H^0 \otimes_{{\Bbbk}} S \ar^{D^0}[r] &
H^1 \otimes_{{\Bbbk}} S \ar^{D^1}[r] &
H^2 \otimes_{{\Bbbk}} S \ar^(.6){D^2}[r] &
\cdots},$$ where the differentials are defined by $$\label{eq:aomoto diff}
D^{q-1}(\alpha \otimes 1)= \sum_{i=1}^{n}
e_i^* \cdot \alpha \otimes e_i$$ for $\alpha\in H^{q-1}$, and then extended by $S$-linearity. Our hypothesis on (strong) anti-commutativity of $H^*$ in degree one easily implies that $D\circ D= 0$.
The terminology is motivated by the following universal property of ${\mathbb{A}}^{\bullet}$. Pick any element $z\in H^1= \operatorname{{Hom}}_{{\Bbbk}}(H_1, {\Bbbk})$, and denote by $\operatorname{ev}_z \colon S\to {\Bbbk}$ the change of rings given by evaluation at $z$. This leads to a specialization of ${\mathbb{A}}^{\bullet}$, namely to the ${\Bbbk}$-cochain complex ${\mathbb{A}}^{\bullet} (z):= {\mathbb{A}}^{\bullet}\otimes_S {\Bbbk}$. It is easy to check that ${\mathbb{A}}^{\bullet} (z)$ coincides with the Aomoto complex of $H$ with respect to $z$, as defined in §\[subsec:aomoto\]. We are now in position to state the last result of this paper.
\[thm:linaom\] Let $X$ be a minimal CW-complex. Then the linearization of the equivariant cochain complex of the universal abelian cover of $X$, with coefficients in ${\Bbbk}={\mathbb{Z}}$ or a field, coincides with the universal Aomoto complex of the cohomology ring $H^*(X,{\Bbbk})$.
The dual linearized complex is described by and . More precisely, for each $q\ge 1$, the transposed matrix of $$\delta^{\operatorname{\,lin}}_{\pi_{\operatorname{{ab}}}} \colon H^{q-1}(X, {\Bbbk})\otimes_{{\Bbbk}} S
\to H^{q}(X, {\Bbbk})\otimes_{{\Bbbk}} S ,$$ coincides with the matrix of $$\label{eq:matnabla}
\nabla_X \colon H_q(X, {\Bbbk})\to H_1(X, {\Bbbk})\otimes_{{\Bbbk}} H_{q-1}(X, {\Bbbk})\, .$$ On the other hand, $\nabla_X$ is the transpose of $\cup_X$. Thus, $\delta^{\operatorname{\,lin}}_{\pi_{\operatorname{{ab}}}}$ coincides with the differential $D^{q-1}$ from .
This work was started while the first author visited Northeastern University, in Spring, 2006. He thanks the Northeastern Mathematics Department for its support and hospitality during this visit. A substantial portion of the work was done while both authors visited the Abdus Salam International Centre for Theoretical Physics in Trieste, Italy, in Fall, 2006. We thank ICTP for its support and excellent facilities.
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S. Papadima, A. I. Suciu, [*Algebraic monodromy and obstructions to formality*]{}, [`arxiv:0901.0105`]{}, to appear in Forum Math. (2010).
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[^1]: $^1$Partially supported by the CEEX Programme of the Romanian Ministry of Education and Research, contract 2-CEx 06-11-20/2006
[^2]: $^2$Partially supported by NSF grant DMS-0311142
[^3]: This picture was drawn with the help of the Mathematica package [KnotTheory]{}, by Dror Bar-Natan, and the graphical package [Knotilus]{}, by Ortho Flint and Stuart Rankin.
[^4]: The sign in that Theorem comes from considering $C_{\bullet}(X, {\mathbb{Z}}\pi)$ as a right ${\mathbb{Z}}\pi$-module instead of a left ${\mathbb{Z}}\pi$-module: see equations (10) and (13) from [@DP1].
|
{
"pile_set_name": "ArXiv"
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---
author:
- |
$^{1,3}$, M. Battisti$^{1,3}$, A. Belov$^{4}$, M.E. Bertaina$^{1,3}$, F. Bisconti$^1$, R. Bonino$^{1,3}$, S. Blin-Bondil$^{5}$, F. Cafagna$^{6}$, G. Cambiè$^{7,8}$, F. Capel$^9$, M. Casolino$^{7,8,10}$, A. Cellino$^{1,2}$, I. Churilo$^{11}$, G. Cotto$^{1,3}$, A. Djakonow$^{12}$, T. Ebisuzaki$^{10}$, F. Fausti$^{1,3}$, F. Fenu$^{1,3}$, C. Fornaro$^{13}$, A. Franceschi$^{14}$, C. Fuglesang$^{9}$, D. Gardiol$^2$, P. Gorodetzky$^{15}$, F. Kajino$^{16}$, P. Klimov$^{4}$, L. Marcelli$^{7}$, W. Marsza[ł]{}$^{12}$, M. Mignone$^{1}$, A. Murashov$^{5}$, T. Napolitano$^{14}$, G. Osteria$^{17}$, M. Panasyuk$^{4}$, E. Parizot$^{15}$, A. Poroshin$^{4}$, P. Picozza$^{7,8}$, L.W. Piotrowski$^{10}$, Z. Plebaniak$^{12}$, G. Prévôt$^{15}$, M. Przybylak$^{12}$, E. Reali$^{8}$, M. Ricci$^{14}$, N. Sakaki$^{10}$, K. Shinozaki$^{1,3}$, G. Suino$^{1,3}$, J. Szabelski$^{12}$, Y. Takizawa$^{10}$, M. Traïche$^{18}$, and S. Turriziani$^{12}$ for the JEM-EUSO Collaboration\
$^1$INFN Turin, Italy; $^2$OATo - INAF Turin, Italy; $^3$University of Turin, Italy; $^4$SINP, Lomonosov Moscow State University, Moscow, Russia; $^5$Omega, Ecole Polytechnique, CNRS/IN2P3, Palaiseau, France, $^6$INFN Bari, Italy; $^7$INFN Roma Tor Vergata, Italy; $^8$University of Roma Tor Vergata, Italy; $^9$KTH Royal Institute of Technology, Stockholm, Sweden; $^{10}$RIKEN, Wako, Japan; $^{11}$Russian Space Corporation Energia, Moscow, Russia; $^{12}$National Centre for Nuclear Research, Lodz, Poland; $^{13}$UTIU Rome, Italy; $^{14}$INFN - Laboratori Nazionali di Frascati, Italy; $^{15}$APC, Univ Paris Diderot, CNRS/IN2P3, CEA/Irfu, Obs de Paris, Sorbonne Paris Cité, France; $^{16}$Konan University, Kobe, Japan; $^{17}$INFN Naples, Italy; $^{18}$Centre for Development of Advanced Technologies (CDTA), Algiers, Algeria\
title: Space Debris detection and tracking with the techniques of cosmic ray physics
---
Introduction
============
Over the last 60 years, since man began to explore space, several thousand tons of satellites and missiles have been launched and there are about 18,000 objects in orbit; 1100 of them (6%) are still in operation, while the remaining (94%) can be classified as SD [@ref:SD], i.e., derelict satellites, parts of rockets and space vehicles, no longer in use, and that remain in orbit around the Earth. These objects travel at high speeds, of the order of 7-9 km/s near the Low Earth Orbit, and can collide with spacecraft such as the ISS or other manned or unmanned spacecrafts, damaging them and in turn producing new debris. The great majority of these objects are not catalogued and, even if they were catalogued, usually tracking data are not precise enough. Moreover, most of them are cm-sized, that makes their detection even more difficult.
The aim of this project is to study the feasibility of SD detection and tracking with techniques usually employed in cosmic-rays physics. We started a feasibility study investigating the performance of already existing instrumentation borrowed from the JEM-EUSO (Joint Experiment Missions for Extreme Universe Space Observatory) project [@ref:JEM-EUSO], a concept of new generation space telescopes for Extreme Energy Cosmic Ray (EECR) detection. We benefited from the presence of the Mini-EUSO EM, a prototype of Mini-EUSO telescope [@ref:Mini-EUSO], in our lab in February and March 2018 and we performed several key tests.
SD itself do not emit the light but a Mini-EUSO-like detector can detect the reflected light from the SD illuminated by a laser or by the Sun light at sunrise and sunset (see left part of Fig. \[Fig:SDtriggerSchem\]). In such a way, SD can be detected as tracks crossing the Field of View (FoV) of the detector, enabling us to identify and track the SD. This feasibility study will be also useful to verify the possibility of using an EUSO-class telescope in combination with a high energy laser for SD remediation [@ref:Toshi]. To verify this idea, we performed extended simulations and dedicated experiments at the TurLab facility located in the Physics department of the University of Turin and in open-sky conditions.
Simulation
==========
We performed simulations to estimate the range of distances and SD dimensions detectable by a Mini-EUSO-like telescope. The Focal Surface (FS) of such a telescope consist of a Photo-Detector-Module (PDM), which consists of 36 Hamamatsu 64-ch Multi-Anode PhotoMultiplier Tubes (MAPMTs), resulting in a readout of 2304 pixels. For SD detection, we used a time resolution of 40.96 ms (= 1 Level 3 Gate Time Unit, 1 L3\_GTU), which corresponds to the time resolution of the Mini-EUSO level 3 (L3) data. In this time resolution, Mini-EUSO records and store a continuous “movie” data through the entire observation time.
We simulated the light track of a SD with ESAF (EUSO Simulation and Analysis Framework), an end-to-end simulation of the phenomenon from the light emission at the source, the propagation through the environment, to the simulation of the detector response and its reconstruction algorithms (see another contribution in this conference [@ref:ESAF-fenu] for the details). We also developed the detection strategy, by testing different trigger algorithms. The selected algorithm works offline for the moment, but could also be implemented in a Field Programmable Gate Array (FPGA) for real-time detection and active debris mitigation.
Trigger algorithm for the SD detection {#sec:Trigger}
--------------------------------------
Fig. \[Fig:SDtriggerSchem\] shows a schematic view of the trigger logic for the SD detection by a Mini-EUSO-like detector. Defining 25 “virtual” Elementary Cells (ECs) (middle part of Fig. \[Fig:SDtriggerSchem\]), the trigger scans the entire PDM and looks for an excess in neighboring pixels, which is lasting 5 consecutive L3\_GTUs (right part of Fig. \[Fig:SDtriggerSchem\]). One EC consists of 4 MAPMTs. Neighbouring ECs are overlapping each other by 2 PMTs for vertical or horizontally, or by 1 PMT for diagonally. With a threshold of pixel count, which is 3 $\sigma$ above the average background in the pixel, $\mu_{pix}+3\times\sigma_{bkg}$, the fake trigger rate becomes low enough as $<3\times10^{-6}$ Hz.
![ Conceptual figure for the SD detection by a Mini-EUSO-like detector. The telescope detects the reflected light from SD illuminated by the Sun (Left). Definition of virtual Elementary Cell (EC) on the PDM in the trigger algorithm (Middle). The trigger scans each ECs to find an excess (> $\mu_{pix}+3\times\sigma_{bkg}$) in neighboring pixels lasting 5 consecutive L3\_GTUs (Right). \[Fig:SDtriggerSchem\] ](SDtriggerLogicSchem.png){width="0.94\hsize" height="6cm"}
Maximum Detection Distance
--------------------------
Using ESAF simulation and applying externally the trigger algorithm described above to the simulated data, we estimated the maximum distance that a Mini-EUSO-like telescope can detect SD. The plot in the Fig. \[Fig:maxDistanceSat\] at the end of this paper shows the estimated maximum detection distance \[km\] as a function of the size of SD in radius \[cm\]. Here the value 0.5 for the SD reflectance is employed in the simulation, which is confirmed to be more or less appropriate by the lab and open-sky tests as described in later sections.
Estimation of the expected number of SD detection by Mini-EUSO
--------------------------------------------------------------
After estimating the maximum distance for SD detection as a function of SD size, we estimated the expected number of SD detection by a Mini-EUSO-like telescope using MASTER [@ref:MASTER], a simulation software developed by ESA which provides us with the density of SD with parameters such as altitude and inclination. We integrated the total number of SD provided by MASTER within the telescope FoV ($\pm18^{\circ}$ at the orbital inclination of $\pm51.64^{\circ}$ with an altitude of $\sim$400km, and the depth of maximum distances described above) multiplied by the observation time. Here we employed 5 min per orbit for the observation time, which corresponds to the twilight time. For the smallest size of SD (r=0.5\[cm\]) simulated here a time resolution of 320 $\mu$s is used instead of 40.96 ms, which corresponds to the one for Mini-EUSO Level 2 data, as it can be detected only when it is relatively closer to the telescope.
Table \[tb:nDet\] shows the result of the calculation.
[|c|c|c|c|c|]{}
[c]{} SD radius\
\
&
[c]{} altitude\
\
&
[c]{} distance from\
Mini-EUSO\
\
&
---------------------------------
ave. N$_{SD}[/km^3]$ for
declination $\pm51.64[^\circ]$,
refl=0.5
---------------------------------
: Maximum detection distance and expected number of the SD detection by Mini-EUSO, estimated with . For the minimum size of SD (r=0.5cm), the time resolution of Mini-EUSO level 2 data (=320 $\mu$s) is used instead of the one for level 3 (=40.96 ms). \[tb:nDet\]
&
---------------
N$_{det}/day$
---------------
: Maximum detection distance and expected number of the SD detection by Mini-EUSO, estimated with . For the minimum size of SD (r=0.5cm), the time resolution of Mini-EUSO level 2 data (=320 $\mu$s) is used instead of the one for level 3 (=40.96 ms). \[tb:nDet\]
\
5.0 & 300 & 100 & 1.91E-09 & 1.47E-01\
4.5 & 312 & 88 & 5.68E-13 & 3.61E-05\
4.0 & 323 & 77 & 6.33E-13 & 3.61E-05\
3.5 & 334 & 66 & 7.17E-13 & 2.56E-05\
3.0 & 344 & 56 & 3.28E-11 & 8.42E-04\
2.5 & 352 & 48 & 5.02E-10 & 9.47E-03\
2.0 & 365 & 35 & 8.93E-10 & 8.96E-03\
1.5 & 373 & 27 & 1.41E-08 & 8.42E-02\
1.0 & 385 & 15 & 1.53E-08 & 2.81E-03\
0.5 & 396 & 4 & 3.42E-07 & 4.48E-02\
& & & TOTAL(N$_{det}$/day)& 3.08E-01\
& & & TOTAL(N$_{det}$/yr) & 112\
The obtained value above is not taking into account the effect of increasing background level due to the twilight. During the night-sky observation described in section \[Sec:NightSkyObservation\], the background level is increased by a factor of 3. This fact directly affects the maximum detection distance, lowering it by a factor of 1/$\surd{3}$. Therefore, the final value for the expected number of SD detection taking into account this effect would be $\sim$45 debris/yr.
Tracking SD
-----------
After obtaining triggered events, we tried to implement the SD tracking using the trigger output data, and to predict the SD position some time later. The trigger output file provides us the position of SD on the detector FS, together with the time stamp in a unit of L3\_GTU (=40.96 ms). Then we derive the track of SD as a linear function of a time (L3\_GTU). Fig. \[Fig:SDtracking\] shows examples of the tracking and prediction method applied to the data simulated by ESAF. The dark blue line along the track is the calculated track from the trigger output file, while the continued light blue line shows the predicted track in the following L3\_GTUs. We tried several kinds of events with different crossing angles and altitudes of SD.
The study of the estimation of the maximum and minimum distance of an SD, as well as the estimation of the minimum number of L3\_GTUs required to predict the track sufficiently well, is currently ongoing.
{width="0.97\hsize"}
Mini-EUSO EM tests at TurLab and open-sky conditions
====================================================
We performed some laboratory measurements at the TurLab facility as well as in an open-sky condition at the observatory in Pino-Torinese (see other contributions to this conference [@ref:Bisconti], [@ref:EUSOatTurLab] for the details). We prepared the experimental setup at the TurLab, that is equipped with a rotating tank ($\phi$ = 5 m, h = 1 m) and represents an ideal condition for testing SD detection, being much darker than the night sky by several orders of magnitude.
For these tests, we used the rotating tank with a series of configurations to reproduce the Earth views. Mini-EUSO EM was hung from the ceiling above the tank pointing downward to mimic the detector’s view from the ISS. At that time, the Mini-EUSO EM had only 4 central MAPMTs and we used a 1" $\phi$ plano-convex lens instead of the Fresnel lenses for all the measurements described in this paper. In this configuration, the telescope has a FoV of 0.5$^{\circ}$/pix instead of 0.8$^{\circ}$/pix for the Mini-EUSO telescope. Two strips of cold-white LEDs illuminating the ceiling were used to reproduce the diffused light above those materials. In this way, we could perform the measurements in a controlled way, for different levels of the background and reflected light from the materials, depending on the detector configuration or the conditions we wanted to test.
SD measurement with Mini-EUSO EM at TurLab
------------------------------------------
Fig. \[Fig:SDatTurLab\] shows the setup for reproducing SD detection principle. Mini-EUSO EM is hung on the ceiling above the TurLab tank, with a “Sun visor” to avoid the direct light from the high power LED which is mimicking the Sun light. A balled Aluminium under the telescope is attached to the edge of a stand which is fixed to the bottom of the tank. As the tank rotates, the balled Aluminium moves within the detector FoV being illuminated by the LED, while the Mini-EUSO EM FS remains in the shade of the Sun visor. The right plot of Fig. \[Fig:SDevent\] shows the image of the track of the balled Aluminium moving within the detector FoV, during the time (in units of L3\_GTU=40.96 ms) highlighted in the light curve on the left. The image is integrated in a way similar to our trigger to obtain the track, i.e., keeping the counts in each pixel when it is exceeding 3$\sigma$ above the one in the previous L3\_GTU.
{height="6.8cm"}
{height="6.8cm"}
SD reflectivity measurement
---------------------------
Different materials such as polished Aluminium foil, canned Aluminium, unpolished Aluminium foil, mirror, copper foil, white paper, balled Aluminium foil, a sample of Kevlar$^{\footnotesize{\textregistered}}$, electronics board, often make the SD, were placed on the bottom of the tank, within the Mini-EUSO EM FoV, in order to measure the relative reflectances of these materials as seen by the detector. For the rough estimation, assuming that the reflectivity of polished Aluminium is about 0.92 [@ref:PrecisionOptics], we observed the reflectance of the different materials varying from 0.17 (electronics board) to 0.92 (fixed reference value for the polished Aluminium foil). The mean value $\sim$0.5 was used for the ESAF simulation. It also seems consistent with a rocket body detection during the open-sky measurement as described in the following section.
Night-sky observation and a rocket body detection {#Sec:NightSkyObservation}
-------------------------------------------------
We also performed night-sky observations at the Astronomy Observatory in Pino-Torinese, using Mini-EUSO EM in March 2018. A rocket body, as well as several stars, has been detected during the observation period. Scaling down the size and distance of the detected rocket body, we could estimate the actual performance of Mini-EUSO EM against SD-like objects. With such a scaling, we could see if the simulation is reasonable or not. The detected object turned out to be Meteor 1-31 Rocket, a cylindrical object 2.8 m $\times$ 2.6 m on a $\sim$530 km orbit. The red line in the Fig. \[Fig:maxDistanceSat\] shows the altitude (distance from the telescope) and size of the detected rocket body scaled to an object in the range of the plot, considering the lens size, pixel FoV and the absorption by the air. For example, the detected photon counts of this rocket body with its size and altitude corresponds to an object with the size of 5.6 cm if it were at the distance of 100 km, or to the size of 2.8 cm if it were at a distance of 50 km from the telescope.
Also, the case for the reflectance of 0.1 is estimated by scaling the result of 0.5 as shown as the green line in the Fig. \[Fig:maxDistanceSat\], as our detection strategy purely depends on the luminosity which is proportional to the area of SD. One may see that the reflectance could be a bit smaller than 0.5 but certainly much higher than 0.1. These results imply that our simulation and estimation of the reflectance are conceivable with the actual situation (see Fig. \[Fig:maxDistanceSat\]).
{width="0.9\hsize" height="8cm"}
Summary
=======
We performed the feasibility study with simulations, laboratory and open-air measurements of the prototype Mini-EUSO telescope. The ESAF and MASTER simulations applying with our trigger algorithm indicated that we are capable of observing SD, and for the case of Mini-EUSO, we estimate that we will detect $\sim$45 SD per year.
The SD tracking and prediction algorithm is also studied using the ESAF simulation and its trigger output file. We found that as a first step, the estimated track and prediction are well matching to the SD tracks. Further studies in different conditions, limit of the tracking and prediction in the same logic are currently ongoing. We also performed reflectance measurements of various SD materials, that provides us with the mean value of about 0.5.
We detected a rocket body during a night-sky observation with Mini-EUSO EM. Conversion of the detected object to the size and distance, makes it comparable to SD with the size of 5.6 cm in radius, at the distance of 100 km, or to the SD with the size of 2.8 cm in radius, at the distance of 50 km from the telescope. Such values are well along the result of maximum SD detection distance in the case of SD reflectance is 0.5 which implies that at least the simulations are predicting detections in the right range of sizes and distances.
Acknowledgments {#acknowledgments .unnumbered}
===============
The support received by the Astronomical Observatory of Turin is deeply acknowledged.\
The authors acknowledge support from Compagnia di San Paolo with the project “New techniques for the detection of space debris”; Id Project: CSTO164394.\
This work was partially supported by Basic Science Interdisciplinary Research Projects of RIKEN and JSPS KAKENHI Grant (22340063, 23340081, and 24244042), by the Italian Ministry of Foreign Affairs and International Cooperation, by the Italian Space Agency through the ASI INFN agreement n. 2017-8-H.0, by NASA award 11-APRA-0058 in the USA, by the French space agency CNES, by the Deutsches Zentrum für Luft- und Raumfahrt, the Helmholtz Alliance for Astroparticle Physics funded by the Initiative and Networking Fund of the Helmholtz Association (Germany), by Slovak Academy of Sciences MVTS JEM-EUSO, by National Science Centre in Poland grant (2015/19/N/ST9/03708), by Mexican funding agencies PAPIIT-UNAM, CONACyT and the Mexican Space Agency (AEM), as well as VEGA grant agency project 2/0132/17, and by State Space Corporation ROSCOSMOS and Russian Foundation for Basic Research (grant 16-29-13065).
[99]{} https://www.esa.int/Our\_Activities/Space\_Safety/Space\_Debris M. Bertaina et al (JEM-EUSO Coll.), Search for the Ultra High Energy Cosmic Rays from Space - The JEM-EUSO program, PoS(ICRC2019) [**192**]{}. F. Capel et al., Mini-EUSO: A high resolution detector for study of terrestrial and cosmic UV emission from the International Space Station, Advances in Space Research - [**62**]{} (2018) 2954 T. Ebisuzaki, M. N. Quinn, S. Wada, et al. Demonstration designs for the remediation of space debris from the International Space Station. Acta Astronautica, 112:102-113, 2015. https://laserbeamproducts.wordpress.com/2014/06/19/reflectivity-of-aluminium-uv-visible-and-infrared/ F. Fenu et al (JEM-EUSO Coll.), Simulations for the JEM-EUSO program with ESAF PoS(ICRC2019) [**252**]{}. https://sdup.esoc.esa.int/ http://www.turlab.ph.unito.it/ F. Bisconti et al (JEM-EUSO Coll.), Mini-EUSO engineering model: tests in open-sky condition, PoS(ICRC2019) [**198**]{}. H. Miyamoto et al (JEM-EUSO Coll.), The EUSO@Turlab Project: Tests of Mini-EUSO Engineering Model, PoS(ICRC2019) [**194**]{}.
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{
"pile_set_name": "ArXiv"
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---
abstract: |
An analytic solution for Bragg grating with linear chirp in the form of confluent hypergeometric functions is analyzed in the asymptotic limit of long grating. Simple formulas for reflection coefficient and group delay are derived. The simplification makes it possible to analyze irregularities of the curves and suggest the ways of their suppression. It is shown that the increase in chirp at fixed other parameters decreases the oscillations in the group delay, but gains the oscillations in the reflection spectrum. The conclusions are in agreement with numerical calculations.
PACS [42.81.Wg; 78.66.-w]{}
author:
- |
O.V. Belai, E.V. Podivilov, D. A. Shapiro\
Institute of Automation and Electrometry,\
Russian Academy of Sciences, Siberian Branch,\
Novosibirsk, 630090 Russia
title: Group delay in Bragg grating with linear chirp
---
INTRODUCTION {#sect:intro}
============
Optical filters based on fiber gratings attract particular interest because of their applications in high-speed lightware communications [@ptTAPC00], fiber lasers [@D01] and sensors [@FOS91]. The Bragg reflector is based on periodic modulation of the refractive index along the line [@OK99; @RK99]. Gratings that have a nonuniform period along their length are known as chirped. The theory of linearly chirped grating holds the central place in the fiber optics. Chirped grating is of importance because of its applications as a dispersion-correcting or compensating devices [@olO87]. The study of linearly chirped grating is also helpful for approximate solution of more general problem of complex Gaussian modulation [@ocSL04]. The group delay as a function of wavelength is a linear function with additional oscillations. For applications the problem is to minimize the amplitude of regular oscillations and the ripple resulting from the errors of manufacturing [@ofcrSF05].
The purpose of this work is to present and study a solution of the equations for amplitudes of coupled waves in quasi-sinusoidal grating with quadratic phase modulation. The solution of coupled-wave equations is derived in terms of the confluent hypergeometric functions. Their asymptotic expansion in terms of Euler $\Gamma$-functions makes it possible to obtain relatively simple formulas for reflectivity and group delay. The simplification enables analysis of irregularities of the curves and suggestions on the ways of their suppression.
The paper is organized as follows. The equations for amplitudes in the grating with quasi-sinusoidal modulated refractive index are derived in Sec. \[sect:2\]. Their analytic solutions are obtained and compared with numerical results in Sec. \[sect:3\]. The asymptotic behavior is treated in Sec. \[sect:4\]. Some estimations and qualitative explanations are presented in Sec. \[sect:5\]. Possible methods to suppress the oscillations are summarized in Sec. \[sect:6\].
Equations for slow amplitudes {#sect:2}
=============================
Consider a single-mode fiber with the weakly modulated refractive index $n+\delta
n(z)$. Steady-state electric field $E(z)$ satisfies one-dimensional Helmholtz equation $$\label{Helmholtz}
\frac{d^2E}{dz^2}+k^2\left[1+\frac{2\delta n(z)}{n}
+\left(\frac{\delta n(z)}{n}\right)^2\right]E=0,
\quad k=\frac{\omega n}{c},$$ where $z$ is the coordinate, $k$ is the wavenumber in glass outside the grating, where $\delta n(z)=0$, $\omega,c$ are the frequency and speed of light. The addition to mean refractive index may be a function with phase and amplitude modulation. A family of analytical solutions for amplitude modulation was obtained in [@ocS03]. Below we treat a case of phase modulation $$\label{quasi-sinusoidal}
\frac{\delta n(z)}{n}=2\beta\cos\theta(z),$$ where $\theta(z)$ is the phase, constant $\beta$ is the modulation depth. Since $\beta\ll1$ we neglect the quadratic term in (\[Helmholtz\]). The phase is general quadratic function $$\label{quadratic}
\theta(z)=\alpha z^2/2+\kappa z +\theta_0,$$ where $\kappa$ is the frequency of spatial modulation at $z=0$, $\theta_0$ is the constant phase shift. The condition of slow phase variation is $$\label{slow-variation}
\left\vert\frac{d\theta}{dz}-\kappa\right\vert\ll\kappa.$$
![The statement of the scattering problem.[]{data-label="fig:envelope"}](reflection){width="70.00000%"}
Let us introduce complex amplitudes $a,b$ of waves running in positive and negative directions $$E=a\e^{\i kz}+b\e^{-\i kz}.$$ Keeping only resonant terms and neglecting the parametric resonance of higher orders at the detuning $$\label{resonance-condition}
q=k-\kappa/2\ll k_0=\kappa/2,$$ we get the equations for coupled waves $$\label{interaction}
a'=\i k_0\beta \e^{-2\i k z+\i\theta(z)}b,\quad
b'=-\i k_0\beta\e^{2\i k z-\i\theta(z)}a,$$ where prime denotes $z$-derivative.
Set (\[interaction\]) conserves $|a|^2-|b|^2$, since the signs in right-hand sides of equations are different. The same equations with identical signs conserve the sum of populations $|a|^2+|b|^2$ and describe the amplitudes of probability in two-state quantum system. The exact solutions in this case are of importance in quantum optics, then they are studied in details [@AE87; @praCM00]. Within the limits of resonance approximation (\[resonance-condition\]) we replace $k$ in front of exponents (\[interaction\]) by $k_0$.
Finding the derivatives of (\[interaction\]) with respect to $z$ we get complex conjugated second-order equations $$\label{second-order}
a''-\i\alpha\left(z-z_0\right)a'-k_0^2\beta^2 a=0,\quad
b''+\i\alpha\left(z-z_0\right)b'-k_0^2\beta^2 b=0.$$ Here $z_0=(2k-\kappa)/\alpha=2q/\alpha$ is the coordinate of resonance point for the wave with wavenumber $k$. It is the turning point where the wave with given $q$ is reflected. The parametric resonance for central wavenumber $q=k-\kappa/2=0$ occurs at $z=0$. Let $\alpha>0$, then for the red detuning $q<0$ we have $z_0<0$, in opposite case $q>0$ of blue detuning $z_0>0$.
Consider the Bragg grating written in the interval $-L\leqslant z\leqslant L$. The problem of left reflection coefficient calculation is illustrated by Fig. \[fig:envelope\]. Boundary conditions are defined by the scattering problem statement. We set amplitude $b$ at the right end equal to zero $$\label{boundary}
b(L)=0$$ and get the reflection and transmission coefficients $$\label{transmission}
r=\frac{b(-L)}{a(-L)},\quad t=\frac{a(L)}{a(-L)}.$$ The chirp is weak and satisfy (\[slow-variation\]), then the equations for complex amplitudes are valid when $$\label{slow}
\alpha L\ll\kappa.$$
Note that set (\[second-order\]) is symmetric under transformation $\alpha\to-\alpha, q\to-q,a\leftrightarrow
b$. Then the right reflection coefficient can be obtained from the expression for left one by changing signs of parameters $\alpha$ and $q$.
Solution {#sect:3}
========
Equations (\[second-order\]) are reduced to the confluent hypergeometric form by the substitution $t=\i\alpha(z-z_0)^2/2$: $$t\Ddot{a}+\left(\frac12- t\right)\Dot{a}+\i\eta
a=0,$$ where dot denotes the derivative with respect to new variable $t$, $\eta=\beta^2k_0^2/2\alpha$ is the adiabatic parameter. The equation for second amplitude $b$ is complex conjugated. The general solutions at $-L<z<L$ are linear combinations $$\label{a-first}
a(z)=A_1u_1(z)+A_2u_2(z),\quad
b(z)=B_1u_1^*(z)+B_2u_2^*(z)$$ of the Kummer confluent hypergeometric functions [@BE53]: $$\begin{aligned}
u_1=F\left(-\i\eta;{\scriptstyle\frac12};\i\alpha(z-z_0)^2/2\right),\nonumber\\
u_2=(z-z_0) F\left({\scriptstyle\frac12}
-\i\eta;{\scriptstyle\frac32};\i\alpha(z-z_0)^2/2\right);
\label{u12}\\
F(a;c;x)=1+\frac{a}{c}\frac{x}{1!}
+\frac{a(a+1)}{c(c+1)}\frac{x^2}{2!}+\dots,\nonumber\end{aligned}$$ where $A_1,A_2,B_1,B_2$ are constants and the asterisk denotes the complex conjugation. The solution was obtained in [@osaMHW75] for optical waveguide. The solution for coupled-wave equations with identical signs has been obtained in the context of nonadiabatic population inversion in two-level system [@aplH75].
The relations between constants can be obtained from set (\[interaction\]) near resonance point $z=z_0$ where $a=A_1+A_2(z-z_0)+O(z-z_0)^2, b=B_1+B_2(z-z_0)+O(z-z_0)^2$: $$\label{constants}
\frac{A_2}{B_1}=\i k_0\beta\e^{\i\theta_0-\i\alpha
z_0^{2}/2},\quad
\frac{B_2}{A_1}=-\i
k_0\beta\e^{-\i\theta_0+\i\alpha z_0^{2}/2}.$$ The right boundary condition (\[boundary\]) yields the ratio of coefficients $A_1$ and $A_2$ $$\label{rho}
\rho=\frac{A_1}{A_2}=\frac1{k_0^2\beta^2}\frac{B_2}{B_1}=-\frac
{
F\left(\i\eta;{\scriptstyle\frac12};-\i\alpha(L-z_0)^2/2\right)
}
{\beta^2k_0^2(L-z_0) F\left({\scriptstyle\frac12}
+\i\eta;{\scriptstyle\frac32};-\i\alpha(L-z_0)^2/2\right)}.$$ The left reflection and transmission coefficients (\[transmission\]) can be expressed in terms of confluent hypergeometric functions $$\label{reflection-general}
r=\frac{\e^{-\i\theta_0+\i\alpha z_0^{2}/2}}{\i k_0\beta}
\frac{ u_1^*(-L)+\beta^2k_0^2\rho u_2^*(-L)}
{\rho u_1(-L)+u_2(-L)},\quad
t=\frac{\rho u_1(L)+u_2(L)}{\rho u_1(-L)+u_2(-L)},$$ where functions $u_{1,2}$ are defined by (\[u12\]) and $\rho$ is defined by (\[rho\]).
![(a) Reflection spectrum $R(q)$ at $\alpha=600~\mbox{cm}^{-2}, L=0.5~\mbox{cm},
k_0=6\times10^{4}~\mbox{cm}^{-1}$, from the top down $\beta=\beta_0, \beta_0/2, \beta_0/4$, where $\beta_0=0.67\times10^{-3}$. (b) A part of the lower curve $\beta=\beta_0/4$, crosses denote the numerical result. []{data-label="fig:spectrum"}](r2_amp_dep "fig:"){width="75.00000%"} ![(a) Reflection spectrum $R(q)$ at $\alpha=600~\mbox{cm}^{-2}, L=0.5~\mbox{cm},
k_0=6\times10^{4}~\mbox{cm}^{-1}$, from the top down $\beta=\beta_0, \beta_0/2, \beta_0/4$, where $\beta_0=0.67\times10^{-3}$. (b) A part of the lower curve $\beta=\beta_0/4$, crosses denote the numerical result. []{data-label="fig:spectrum"}](inset1 "fig:"){width="75.00000%"}
The reflection spectrum, i.e., the reflectivity $R=|r|^2$ as a function of detuning $q$, is shown in Fig. \[fig:spectrum\] (a). The central frequency of the spectrum comes to resonance at $z=0$, in the middle of grating. The central part has a flat top at high adiabatic parameter, as the upper curve shows, and the maximal reflectivity is close to 1. The width of central part is proportional to the length $L$. The reflectivity is relatively high if the turning point $z_0$ lies inside the grating $|z_0|<L$. This inequality gives the bandwidth $|q|<\alpha L/2$. There is no parametric resonance at higher detuning, when $|z_0|>L$, and the reflectivity is small. Fig. \[fig:alpha-dependence\] (a) shows how the bandwidth grows up with the chirp parameter $\alpha$ at fixed modulation depth $\beta$. The adiabatic parameter decreases with $\alpha$, then the reflectivity in the center decreases from curve to curve.
![(a) Reflection spectrum from the top down at $L=0.5~\mbox{cm}, \beta=0.33\times10^{-3},
k_0=6\times10^{4}~\mbox{cm}^{-1}$ and $\alpha=\alpha_0,
2\alpha_0, 3\alpha_0,$, where $\alpha_0=600~\mbox{cm}^{-2}$. (b) A part of the lower curve $\alpha=3\alpha_0$, crosses denote the numerical result. []{data-label="fig:alpha-dependence"}](r2_alp_dep "fig:"){width="75.00000%"} ![(a) Reflection spectrum from the top down at $L=0.5~\mbox{cm}, \beta=0.33\times10^{-3},
k_0=6\times10^{4}~\mbox{cm}^{-1}$ and $\alpha=\alpha_0,
2\alpha_0, 3\alpha_0,$, where $\alpha_0=600~\mbox{cm}^{-2}$. (b) A part of the lower curve $\alpha=3\alpha_0$, crosses denote the numerical result. []{data-label="fig:alpha-dependence"}](inset2 "fig:"){width="75.00000%"}
The spectrum was recalculated numerically by $T$-matrix approach. The initial Helmholtz equation (\[Helmholtz\]) was solved numerically with neither approximation of slow envelope, nor quadratic term $(\delta n/n)^2$ neglecting. The number of layers per period of spatial modulation was fixed at $N=32$, then the step varied along the grating. The spectra for $n=1.5$ and the same parameters are shown in Fig. \[fig:spectrum\] (b) by crosses. The numerical results are very close to analytical, since both dimensionless parameters controlling the validity of coupled-wave approximation are small: $\beta\sim10^{-3},
\alpha L/\kappa\sim 5\times10^{-3}$. At higher parameter $\alpha$ the deviation of coupled-wave equations solutions from that of Helmholtz equation increases, but not dramatically, as shown in Fig. \[fig:alpha-dependence\] (b). The origin of the deviation is resonance approximation (\[resonance-condition\]). We replace $k$ by $k_0$ in coupled-mode equations (\[interaction\]), while the Helmholtz wave equation (\[Helmholtz\]) involves $k$. Comparing Fig. \[fig:spectrum\] (a) and Fig. \[fig:alpha-dependence\] (b) we see that the latter involves higher detuning, then the deviation is greater at higher $q=k-k_0$.
![(a) Group delay (ps) as a function of detuning $q$ (cm$^{-1}$) at $\alpha=600~\mbox{cm}^{-2}, L=0.5~\mbox{cm},
k_0=6\times10^{4}~\mbox{cm}^{-1}$, from the top down $\beta=\beta_0, \beta_0/2, \beta_0/4$, where $\beta_0=0.67\times10^{-3}$. (b) Comparing with numerical calculation denoted by dots.[]{data-label="fig:groupdelay"}](delay "fig:"){width="75.00000%"} ![(a) Group delay (ps) as a function of detuning $q$ (cm$^{-1}$) at $\alpha=600~\mbox{cm}^{-2}, L=0.5~\mbox{cm},
k_0=6\times10^{4}~\mbox{cm}^{-1}$, from the top down $\beta=\beta_0, \beta_0/2, \beta_0/4$, where $\beta_0=0.67\times10^{-3}$. (b) Comparing with numerical calculation denoted by dots.[]{data-label="fig:groupdelay"}](inset3 "fig:"){width="75.00000%"}
The group delay found from analytical solution (\[reflection-general\]) is plotted in Fig. \[fig:groupdelay\] (a) at the same parameters as the reflection spectrum in Fig. \[fig:spectrum\]. The deviation of curves from the linear dependence, the group delay ripple, manifests itself as oscillations with variable frequency. The frequency grows up towards the blue end of spectrum in agreement with results from [@ECOC97; @RK99]. For the negative chirp (or when the incident light enters from the right) the frequency grows up towards the red edge of spectrum. The maximum deviation from the averaged slope decreases with decreasing modulation depth $\beta$. Meanwhile, the ripple in reflectivity increases for small $\beta$. A fragment of group delay characteristics is shown in Fig. \[fig:groupdelay\] (b) along with numerical calculations. Dots obtained from numerical calculation are very close to the curve given by analytical formula.
It is difficult to analyze the solution in its general form. In particular the cumbersome expression for group delay, the derivative of (\[reflection-general\]) with respect to the detuning, is not presented here. Let us simplify expressions using the asymptotics of Kummer functions in the next section.
Asymptotics {#sect:4}
===========
The asymptotic expressions for the reflection coefficient can be obtained from (\[reflection-general\]) in two cases. The first case is the resonance condition at the left end, namely, detuning $q=-\alpha L/2$ for which $z_0=-L$. In this case it follows from (\[u12\]) that $u_1(-L)=1,u_2(-L)=0$, and then from (\[reflection-general\]) $$\begin{aligned}
\label{tanh}
r\approx\frac{\e^{-\i\theta_0+\i\alpha L^2/2}}{\i k\beta\rho}
\approx-\frac{\e^{-\i\theta_0+\i\alpha L^2/2}}{\sqrt{\i\eta}}
\frac{\Gamma(1/2-\i\eta)}{\Gamma(-\i\eta)},\\
R=|r|^2\approx\tanh\pi\eta.\label{semi-infinite}\end{aligned}$$
The other case is when the resonance point $z_0$ being far from both ends inside the grating: $|q|<\alpha L/2$ and $\alpha(z_0\pm L)^2/2\gg1$. The asymptotic expression of the confluent hypergeometric functions [@BE53] at $|\mathrm{arg}\,x|<\pi$ $$\label{Kummer-asymptotics}
F(a;c;x)\approx\frac{\Gamma(c)}{\Gamma(c-a)}
\left(\frac{\e^{\i\pi\epsilon}}{x}\right)^a
+
\frac{\Gamma(c)}{\Gamma(a)}\e^xx^{a-c},\quad \epsilon=
\begin{cases}
+1, & {\textrm{Im}\,}x>0,\\
-1, & {\textrm{Im}\,}x<0
\end{cases}$$ allows one to simplify expression (\[reflection-general\]).
![The reflectivity $R$ as a function of adiabatic parameter $\eta$ at $z_0=-L$ (solid line) and at $z_0=0$ (dashed).[]{data-label="fig:saturation"}](saturation){width="55.00000%"}
The reflection coefficient can be written using (\[u12\]) $$\begin{aligned}
r\approx-\sqrt{R_0}\e^{\i\phi-\i\pi/4+2\i q^2/\alpha-\i\theta_0}
\left[\frac\alpha2\left(L+
\frac{2q}\alpha\right)^2\right]^{-2\i\eta}\times\nonumber\\
\frac
{
1+\e^{+\i\pi/4-\i\phi}\sqrt{\frac{\eta}{R_0}}
\left(
\psi_+^{2\i\eta-1/2}\e^{-\i\psi_+}+
\psi_-^{2\i\eta-1/2}\e^{-\i\psi_-}
\right)
}
{
1+\sqrt{\eta R_0}
\left(
\psi_+^{-2\i\eta-1/2}\e^{\i\psi_++\i\phi-\i\pi/4}+
\psi_-^{2\i\eta-1/2}\e^{-\i\psi_--\i\phi+\i\pi/4}
\right)
},
\label{reflection-simple}\end{aligned}$$ where $\psi_\pm(q)=\alpha(L\pm2q/\alpha)^2/2$, $\phi=\mbox{arg}[\Gamma(\i\eta)\Gamma(1/2+\i\eta)]$, $R_0=1-\e^{-4\pi\eta}$ and we omit terms of the order of $1/\alpha L^2\ll1$. The enumerator and denominator of the fraction in the second line of (\[reflection-simple\]) are close to 1, if $\psi_\pm\gg1$. Then the formula for reflectivity becomes simple $$\label{4tanh}
R\approx R_0=1-\e^{-4\pi\eta}.$$ Both curves (\[semi-infinite\]) and (\[4tanh\]) are shown in Fig. \[fig:saturation\]. One can see their saturation, moreover, when the turning point $z_0=-L$, the saturation occurs later than when the turning point is far from both ends.
The group delay obtained from (\[reflection-simple\]) is $$\begin{aligned}
f=\frac{d\arg r}{d \omega}=\frac{n}{c}\frac{d{\textrm{Im}\,}\ln r}{d q}
\approx\frac{n}{c}
\Bigl\{\frac{4q}{\alpha}
-2\sqrt{\frac{2\eta}{R_0\alpha}}\times\nonumber\\
\times
\Bigl[
(1+R_0)\cos\left(\phi-\pi/4+\psi_+-
2\eta\ln\psi_+\right)+\nonumber\\
+(1-R_0)\cos\left(\phi-\pi/4+\psi_--
2\eta\ln\psi_-\right)
\Bigr] \Bigr\},\label{group_delay}\end{aligned}$$ where we neglect terms of the order of $1/\alpha L^2\ll1$. Expression (\[group\_delay\]) involves three terms. The first (in the first line) gives the averaged slope. It is a linear function within the bandwidth. Its slope depends on parameter $\alpha$. At $1-R_0\ll1$ the second term (the second line) gives the ripple, chirped oscillations. The frequency of these oscillations is double distance from left end of the grating to the reflection point $z_0$. Their frequency $\psi_+'(q)=2L+4q/\alpha$ grows up towards the blue edge of the spectrum. When reflectivity $R_0$ becomes smaller, the last term (the third line in Eq.\[group\_delay\]) proportional to $T_0=1-R_0$ comes into effect. It gives the additional oscillations with variable frequency $\psi_-'(q)=2L-4q/\alpha$ that grows up towards the red edge of the spectrum. It is precisely the sum of two chirped oscillations with significantly different frequencies that the left part of the lower curve in Fig. \[fig:groupdelay\] (a) displays. Magnified view of the corresponding fragment is also shown in Fig. \[fig:groupdelay\] (b). If we change the sign of chirp parameter $\alpha$, then functions $\psi_\pm$ switch their roles: $\psi_+\leftrightarrow\psi_-$. Therefore at high reflectivity $1-R_0\ll1$ the spatial frequency of leading oscillations $\psi_+'=2L+4q/\alpha$ decreases towards the shorter wavelengths.
The amplitude of oscillations in group delay (\[group\_delay\]) increases when $R_0$ tends to unity, while that in the spectrum decreases. Formula for the reflection inside the bandwidth can be obtained from (\[reflection-simple\]) with the accuracy to the next order of transparency $T_0=1-R_0$ $$\begin{aligned}
R\approx R_0+2\sqrt{\eta R_0}(1-R_0)\times\nonumber\\
\left[
\frac{\cos(\phi-\pi/4+\psi_+-2\eta\ln\psi_+)}{\sqrt{\alpha/2}(L+2q/\alpha)}+
\frac{\cos(\phi-\pi/4+\psi_--2\eta\ln\psi_-)}{\sqrt{\alpha/2}(L-2q/\alpha)}
\right].
\label{modulation}\end{aligned}$$ At high reflectivity $R_0\to1$ oscillations (\[modulation\]) are suppressed. The first term in square brackets describes oscillations with frequency $2L+4q/\alpha$, their amplitude gains towards the red edge of the spectrum. The second term corresponds to oscillations with frequency $2L-4q/\alpha$ with amplitude growing towards the blue edge. Both approximate formulas (\[modulation\]) and (\[group\_delay\]) for oscillations are plotted in Fig. \[fig:oscillations\] and Fig. \[fig:asymptotics\], respectively. As figures illustrate, the asymptotic expressions nearly coincide with exact Kummer solutions. The departure of the simple formula from the Kummer solution (left edge in Fig. \[fig:oscillations\] and both edges in Fig. \[fig:asymptotics\]) occurs when we get the limit of applicability of the asymptotic expansion. The turning point should be located far from the ends of grating, i.e., $|L\pm2q/\alpha|\gg L_{eff}=(2\pi/|\alpha|)^{1/2}$. Asymptotics are broken when the turning point occurs too close to the end.
![The group delay calculated according to asymptotic formula (\[group\_delay\]) at $\alpha=600~\mbox{cm}^{-2}, L=0.5~\mbox{cm},
k_0=6\times10^{4}~\mbox{cm}^{-1}$ $\beta=0.67\times10^{-3}$. Dots denote the exact Kummer solution.[]{data-label="fig:oscillations"}](GDR){width="75.00000%"}
![Reflection spectrum calculated by asymptotic formula (\[modulation\]) at $\alpha=600\mbox{
cm}^{-1}, \beta= 0.17\times10^{-3}, k_0 =
6\times10^{4}\mbox{ cm}^{-1}$. Dots denote the exact Kummer solution.[]{data-label="fig:asymptotics"}](modulation){width="75.00000%"}
The dependence on parameters $\alpha,\beta$ in Fig. \[fig:spectrum\] and \[fig:alpha-dependence\] can also be explained by the asymptotic expressions. At fixed chirp parameter $\alpha$ the adiabatic parameter $\eta=(k_0\beta)^2/2\alpha$ in (\[4tanh\]) decreases with decreasing the modulation depth $\beta$. Then reflectivity $R_0$ at $q=0$ is relatively small and oscillations with amplitude $1-R_0$ in the spectrum become noticeable. At fixed $\beta$, on the contrary, the adiabatic parameter decreases with increasing $\alpha$. It is the reason of the most evident oscillation in the spectrum corresponding to the higher chirp parameter $\alpha$.
Discussion {#sect:5}
==========
The reflectivity is maximal at $k=k_0=\kappa/2=\pi/\Lambda$, where $\Lambda$ is the period of modulation in the middle of the grating, at $z=0$. The spatial frequency of modulation $\theta'(z)=\alpha z+\kappa$ depends on coordinate $z$. Then at some distance from the center the wave with $k=k_0$ comes out from the resonance. The dephasing occurs when $\theta=\alpha z^2/2\sim\pi$, i.e., at distance $z=L_{\textrm{eff}}\sim (2\pi/\alpha)^{1/2}$. The effective number of strokes along length $L_{\textrm{eff}}$ should be large $M_{\textrm{eff}}=L_{\textrm{eff}}/\Lambda=(2\pi/\alpha)^{1/2}\Lambda^{-1}\gg1$. Moreover, to provide the high reflectivity it should satisfy the stricter limitation of dense grating $M_{\textrm{eff}}\beta\gtrsim1$. From here we get a condition for adiabatic parameter $$\eta=\frac{\beta^2k_0^2}{2\alpha}\gtrsim1/4\pi.$$
The bandwidth of the reflection spectrum is $\alpha L$, as shown in Sec. \[sect:3\]. The fronts of spectrum are determined by the effective length $L_{\textrm{eff}}$. When point $z_0=2q/\alpha$ is placed outside the grating at distance $\sim L_{\textrm{eff}}$ from the end, the reflection almost vanishes. The width of fronts is $\delta
q=\alpha L_{\textrm{eff}}=1/L_{\textrm{eff}}$. The fronts are steep while $L_{\textrm{eff}}\ll L$, i.e., in the limit of long grating.
Phase modulation $\theta(z)$ provides the parametric resonance condition for different wavelengths. The shorter waves meet their resonance at longer distance $z_0=2q/\alpha$, and then the group delay of blue light is more than that of red one, Fig. \[fig:groupdelay\]. The linear dependence of the average group delay (\[group\_delay\]) upon the detuning has also simple explanation. The delay $\tau=f/v_{\textrm{gr}}$ is defined by double distance from starting point to the resonance for given wavenumber $f\approx2z_0=4(k-k_0)/\alpha$. Here $v_{\textrm{gr}}$ is the group velocity of light. If the chirp $\alpha$ is negative, then the sign of delay characteristics becomes negative.
![Configuration of compound cavity: left “mirror” $z=-L$ is the left edge of grating, right “mirror” $z=+L$ is the right edge of grating. Middle variable “mirror”, the turning point $z=z_0$, is located at different positions depending on the wavelength. Then the ripple frequencies are determined by the variable lengths of sub-cavities I and II.[]{data-label="fig:compound"}](compound){width="55.00000%"}
The ripple outside the reflection spectrum bandwidth, Fig. \[fig:spectrum\],\[fig:alpha-dependence\], with period $\pi/L$ are the Gibbs oscillations originated by steep boundaries, i.e., reflection from the grating edges. The aperiodic oscillation inside the bandwidth arise from the triple-mirror cavity with moving middle mirror, Fig. \[fig:compound\]. The wave reflected to the left from turning point $z_0$ could reflect back to the right from the left end of the grating. Then the cavity appears between $z=-L$ and $z=z_0$; its effective length is $l=z_0+L=2q/\alpha+L$. It results in oscillations with period $\pi/(L+2q/\alpha)$. The cavity with variable “mirror” is longer for blue spectrum and shorter for red, then the frequency of oscillations increases with $q$, as mentioned in paper [@ECOC97]. At $R_0\to1$ these oscillations are suppressed in the reflection spectrum, but remain in the group delay characteristics. If the reflectivity is not close to 1, the additional oscillations come into effect due to the “right” cavity with variable “left mirror”. Their period $\pi/(L-2q/\alpha)$ on the contrary is longer for red spectrum. These oscillations are suppressed at $R_0\to1$ both in reflection spectrum and group delay characteristics.
Conclusions {#sect:6}
===========
Thus, the analysis of the reflection spectrum and group delay of linearly chirped grating becomes simple if the turning point $z_0=2q/\alpha$ is far from both ends of the grating compared to the effective length $L_{\textrm{eff}}=(2\pi/\alpha)^{1/2}$. Formulas for reflectivity demonstrate the irregular oscillations in the reflection spectrum when the adiabatic parameter is not large. The oscillations are aperiodic and their amplitude slowly increases from the center of spectrum. The nature of the oscillations is reflection in compound cavity with a mobile middle “mirror”. There are two terms in asymptotic expression. The first has a period $\pi/(L+z_0)$ (round trip in the left sub-cavity), the second — $\pi/(L-z_0)$ (round trip in the right sub-cavity). The oscillations in group delay characteristics have the same origin. The difference is that the right sub-cavity takes a negligible part in forming the oscillations of group delay characteristics at $R_0\to1$.
The amplitude of oscillations is suppressed at high chirp parameter $\alpha$ even at fixed reflectivity. The conservation of high reflectivity with increasing $\alpha$ requires increasing parameter $\beta$. In order to suppress both oscillations one must choose as high the modulation depth as possible, but the limitation exists in fiber Bragg grating manufacturing. The alternative method to diminish the unwanted echo might be to provide the signal dephasing by apodization, i.e., smoothing the grating profile [@RK99].
Acknowledgments
===============
Authors are grateful to S.A. Babin for fruitful discussions. The work is partially supported by the CRDF grant RUP1-1505-NO-05 and the Government support program of the leading research schools (NSh-7214.2006.2).
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
PACS-CS Collaboration : ${}^{a}$[^1], S. Aoki${}^{b,c}$, N. Ishii${}^{a}$, K.-I. Ishikawa${}^{d}$, N. Ishizuka${}^{a,b}$, T. Izubuchi${}^{c,e}$, D. Kadoh${}^{a}$, K. Kanaya${}^{b}$, Y. Kuramashi${}^{a,b}$, M. Okawa${}^{d}$, Y. Taniguchi${}^{a,b}$, A. Ukawa${}^{a,b}$, N. Ukita${}^{a}$, T. Yoshié${}^{a,b}$\
Center for Computational Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan\
Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan\
Riken BNL Research Center, Brook-haven National Laboratory, Upton, New York 11973, USA\
Graduate School of Sciences, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, Japan\
Institute for Theoretical Physics, Kanazawa University, Kanazawa, Ishikawa 920-1192, Japan
title: 'Charm quark system in $2+1$ flavor lattice QCD using the PACS-CS configurations'
---
Introduction {#section:introduction}
============
Precise determination of physical quantities for heavy quark systems provides us with an opportunity to search for new physics beyond the standard model. For this purpose lattice QCD should be a powerful tool. However, the use of conventional lattice quark actions are problematic because of large cutoff errors due to the heavy quark masses. So far several approaches to avoid this problem have been employed for the study of heavy quark physics on the lattice[@Gamiz]. Our choice is to employ the relativistic heavy quark action of Ref. [@akt]. This formalism allows us to take the continuum limit in which $m_Q a$ corrections are controlled by a smooth function. In fact the cutoff errors are reduced from $O((m_Q a)^n)$ to $O(f(m_Q a)(a \Lambda_{QCD})^2)$ where $f(m_Q a)$ is an analytic function around $m_Q a = 0$.
Simulation parameters {#sec:param}
=====================
We simulate the charm quark system with the relativistic heavy quark action of Ref. [@akt] on the 2+1 flavor lattice QCD configurations which are generated by the PACS-CS Collaboration employing the nonperturbatively $O(a)$-improved Wilson quark action with $c_{\rm SW}^{\rm NP}=1.715$[@csw_np] and the Iwasaki gauge action. The lattice size is $32^3\times 64$ whose spatial extent is $L=2.9$ fm with the lattice spacing of $a=0.09$ fm. The dynamical up-down quark mass ranges from 67 MeV down to 3.5 MeV which is close to the the physical value. The details of the configuration production and light hadron physics that emerges from them are described in Refs. [@pacscs; @kura_lat08; @ukita_lat08; @kadoh_lat08]. Table \[tab:stat\] summarizes the simulation parameters and the statistics of the configuration sets we have used for the heavy quark measurements. The number of the source points is to be quadrupled. We emphasize that the data point at ($\kappa_{\rm ud},\kappa_{\rm s}$)=(0.137785,0.13660) is almost on the physical point: the up-down quark mass is only 30% heavier and the strange quark mass is almost exactly at the physical value.
The relativistic heavy quark action proposed in Ref. [@akt] is given by $$\begin{aligned}
S_Q & = & \sum_{x,y}\overline{Q}_x D_{x,y} Q_y,\\
D_{x,y} &=& \delta_{xy}
- \kappa_{\rm h}
\sum_i \left[ (r_s - \nu \gamma_i)U_{x,i} \delta_{x+\hat{i},y}
+(r_s + \nu \gamma_i)U_{x,i}^{\dag} \delta_{x,y+\hat{i}}
\right]
\nonumber \\
&&- \kappa_{\rm h}
\left[ (r_t - \nu \gamma_i)U_{x,4} \delta_{x+\hat{4},y}
+(r_t + \nu \gamma_i)U_{x,4}^{\dag} \delta_{x,y+\hat{4}}
\right]
\nonumber \\
&&- \kappa_{\rm h}
\left[
c_B \sum_{i,j} F_{ij}(x) \sigma_{ij}
+ c_E \sum_i F_{i4}(x) \sigma_{i4}
\right],\end{aligned}$$ where we are allowed to choose $r_t=1$, while the other four parameters $\nu$, $r_s$, $c_B$, $c_E$ should be adjusted in the mass dependent way. We use the one-loop perturbative values for $r_s$, $c_B$ and $c_E$ evaluated in Ref. [@param]. For the clover coefficients $c_{B}$ and $c_{E}$ we incorporate the nonperturbative contributions at the massless limit adopting the procedure $c_{B,E}=(c_{B,E}(m_Q a) - c_{B,E}(0))^{\rm PT} + c_{\rm SW}^{\rm NP}$. At each simulation point, the parameter $\nu$ is nonperturbatively determined from the dispersion relation for the spin-averaged $1S$ state of the charmonium: $$E({\vec p})^2
= E({\vec p})^2+c_{\rm eff}^2 |{\vec p}|^2,$$ where $\nu$ is adjusted such that the effective speed of light $c_{\rm eff}$ becomes unity. Figure \[fig:nu\_np\] shows an example of the nonperturbative tuning of $\nu$ with $\kappa_{\rm h}=0.11022$ at ($\kappa_{\rm ud},\kappa_{\rm s}$)=(0.13770,0.13640). In order to search for the physical charm quark mass point, we employ two values of the hopping parameter of a heavy quark $\kappa_{\rm h}$. The values of $\kappa_{\rm h}$ are chosen to sandwich the physical charm quark mass.
---------- --------- --------- --------- ---------- --------------- ---
\[MeV\] \[MeV\] \#source
measured total/MD time
0.13770 0.13640 12.3(2) 90(1) 400 800/2000 1
0.13781 0.13640 3.5(2) 87(1) 100 198/990 1
0.137785 0.13660 3.5(1) 73(1) 90 200/1000 1
2.53(5) 72.7(8)
---------- --------- --------- --------- ---------- --------------- ---
: Simulation parameters. Quark masses are perturbatively renormalized in the ${{\overline {\rm MS}}}$ scheme. The renormalization scale is $\mu=1/a$ for $\kappa_{\rm ud}\le 0.137785$ and $\mu=2$ GeV for the physical point.[]{data-label="tab:stat"}
![ Nonperturbative tuning of $\nu$ with $\kappa_{\rm h}=0.11022$ at ($\kappa_{\rm ud},\kappa_{\rm s}$)=(0.13770,0.13640). A red symbol denotes the result for the nonperturbative $\nu$. []{data-label="fig:nu_np"}](fig1.eps){width="75mm"}
Heavy-heavy system {#section:heavy_heavy}
==================
Let us first investigate the charmonium spectrum. At each combination of ($\kappa_{\rm ud},\kappa_{\rm s}$) we determine the physical point of the charm quark by the condition that the mass of the spin-averaged $1S$ state reproduces the experimental value, $M(1S) = (M_{\eta_c} + 3 M_{J/\psi})/4
= 3.0677(3)$ \[GeV\][@PDG]. For this purpose we linearly interpolate the results for $M(1S)$ at two values of the hopping parameter $\kappa_{\rm h}$. Figure \[figure:kappa\_heavy\_inv-m\_PS\_V\_nu2\] illustrates this procedure for the case of ($\kappa_{\rm ud},\kappa_{\rm s}$)=(0.13770,0.13640).
In Fig. \[figure:bare\_m\_ud\_AWI-m\_3\_P\_1\_minus\_m\_V\] we plot results for the mass of the orbital excitation $m_{\chi_{c1}}(1P) - m_{J/\psi}(1S)$ at the physical point of the charm quark mass. It is hard to detect the dynamical quark mass dependence within the errors in the range of $3.5 {\rm MeV}\simlt m_{\rm ud}\simlt 12{\rm MeV}$ and $73 {\rm MeV}\simlt m_{\rm s}\simlt 90{\rm MeV}$. A very short chiral extrapolation is made employing a linear function of the up-down and the strange quark masses: $$m_{\chi_{c1}}(1P) - m_{J/\psi}(1S) = \alpha + \beta m_{\rm ud}
+ \gamma m_{\rm s}.$$ In Fig. \[figure:bare\_m\_ud\_AWI-m\_3\_P\_1\_minus\_m\_V\] we find that the extrapolated value and its error are almost identical to the result at ($\kappa_{\rm ud},\kappa_{\rm s}$)=(0.137785,0.13640). This illustrates how close our simulation points are to the physical point. The result at the physical point is consistent with the experimental value within the error.
Figure \[figure:bare\_m\_ud\_AWI-m\_V\_minus\_m\_PS\] shows the results for the hyperfine splitting $m_{J/\psi}-m_{\eta_c}$. As in the orbital excitation we find little dynamical quark mass dependence. We extrapolate the results to the physical point employing a linear function of the dynamical quark masses. The extrapolated value, which is essentially determined by the result at ($\kappa_{\rm ud},\kappa_{\rm s}$)=(0.137785,0.13640), shows a 10% deficit from the experimental value. Possible sources of the discrepancy are $O(g^2 a)$ effects in the relativistic heavy quark action, dynamical charm quark effects and disconnected loop contributions. A recent 2+1 flavor lattice QCD calculation with the highly improved staggered quarks[@hfs_hpqcd] shows a similar value to ours.
We compare our results for the hyperfine splitting in $N_f=2+1$ QCD with the previous $N_f=0,2$ results[@kura_lat05; @RHQ-N_f_0_2] in Fig. \[figure:comparison\_of\_m\_V\_minus\_m\_PS\]. We observe a clear trend that the results become closer to the experimental value as the number of the flavor is increased. The dynamical quarks give significant contributions to the hyperfine splitting.
![ Interpolation of the spin-averaged $1S$ charmonium mass to the physical point as a function of $\kappa_{\rm h}$. Errors are within symbols.[]{data-label="figure:kappa_heavy_inv-m_PS_V_nu2"}](fig2.eps){width="75mm"}
![ Orbital excitation $m_{\chi_{c1}}-m_{J/\psi}$ as a function of $m_{\rm ud}^{AWI}$. Vertical dotted line denotes the physical point.[]{data-label="figure:bare_m_ud_AWI-m_3_P_1_minus_m_V"}](fig3.eps){width="75mm"}
![ Hyperfine splitting of the charmonium as a function of $m_{\rm ud}^{AWI}$. A vertical dotted line denotes the physical point.[]{data-label="figure:bare_m_ud_AWI-m_V_minus_m_PS"}](fig4.eps){width="75mm"}
![ Comparison of hyperfine splittings in $N_f=0$[@kura_lat05], 2[@RHQ-N_f_0_2] and $2+1$, together with the experimental value. All the lattice results are obtained at $a^{-1}\approx 2$ GeV.[]{data-label="figure:comparison_of_m_V_minus_m_PS"}](fig5.eps){width="75mm"}
Heavy-light system {#section:heavy_light}
==================
For the heavy-light system we focus on the $D$ and $D_s$ mesons and their decay constants measured at ($\kappa_{\rm ud},\kappa_{\rm s}$)=(0.137785,0.13660). This data point is so close to the physical point that $m_{\rm ud}$ corrections in the results could be smaller than the statistical errors. We employ perturbative values for the renormalization factor and the improvement coefficients of the axial vector current evaluated in Ref. [@zvza]. For $c_{A_4}^{+}$ we incorporate the nonperturbative contribution at the massless limit by $c_{A_4}^{+}=(c_{A_4}^{+}(m_Q a) - c_{A_4}^{+}(0))^{\rm PT}
+ c_{A}^{\rm NP}$ with $c_{A}^{\rm NP} = -0.03876106$[@NP-c_A]. Figure \[fig:mps\_hl\] compares our results for the $D$ and $D_s$ meson masses with the experimental values [@CLEO]. They are consistent within the errors. It is noteworthy that the physical charm quark mass determined from the heavy-heavy system successfully reproduces the heavy-light meson masses. The results for the decay constants are shown in Fig. \[fig:fps\_hl\], where we also plot the recent 2+1 flavor lattice QCD results with relativistic heavy quark actions[@hpqcd; @fnal] for comparison. Although we find a sizable discrepancy between our result and the experimental value for $f_{D_s}$, we should analyze the full configuration set and improve statistics before we derive any conclusions.
In Fig. \[figure:bare\_m\_ud\_AWI-f\_D\_s\_over\_f\_K\] we plot the ratios of $f_{D_s}$ to $f_D$ and $f_{D_s}$ to $f_K$ in which uncertainties coming from the perturbative renormalization factors and the lattice cutoff should cancel out. For both cases our results show larger values than the experimental ones, which originate from the discrepancy found in Fig. \[fig:fps\_hl\].
![ $D$ (left) and $D_s$ (right) meson masses as a function of $m_{\rm ud}^{\rm AWI}$. A vertical dotted line denotes the physical point.[]{data-label="fig:mps_hl"}](fig6a.eps "fig:"){width="75mm"} ![ $D$ (left) and $D_s$ (right) meson masses as a function of $m_{\rm ud}^{\rm AWI}$. A vertical dotted line denotes the physical point.[]{data-label="fig:mps_hl"}](fig6b.eps "fig:"){width="75mm"}
![$f_D$ (left) and $f_{D_s}$ (right) meson masses as a function of $m_{\rm ud}^{\rm AWI}$. A vertical dotted line denotes the physical point.[]{data-label="fig:fps_hl"}](fig7a.eps "fig:"){width="75mm"} ![$f_D$ (left) and $f_{D_s}$ (right) meson masses as a function of $m_{\rm ud}^{\rm AWI}$. A vertical dotted line denotes the physical point.[]{data-label="fig:fps_hl"}](fig7b.eps "fig:"){width="75mm"}
![ Ratios of $f_{D_s}$ to $f_{D}$ (left) and $f_{D_s}$ to $f_K$ (right). A vertical dotted line denotes the physical point. []{data-label="figure:bare_m_ud_AWI-f_D_s_over_f_K"}](fig8a.eps "fig:"){width="75mm"} ![ Ratios of $f_{D_s}$ to $f_{D}$ (left) and $f_{D_s}$ to $f_K$ (right). A vertical dotted line denotes the physical point. []{data-label="figure:bare_m_ud_AWI-f_D_s_over_f_K"}](fig8b.eps "fig:"){width="75mm"}
Numerical calculations for the present work have been carried out on the PACS-CS computer under the “Interdisciplinary Computational Science Program” of Center for Computational Sciences, University of Tsukuba. This work is supported in part by Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology (Nos. 16740147, 17340066, 18104005, 18540250, 18740130, 19740134, 20340047, 20540248, 20740123, 20740139 ).
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[^1]: E-mail: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Andrew D. Box'
title: Flavour Violating Interactions of Supersymmetric Particles
---
Introduction {#intro}
============
In most supersymmetric models, the soft SUSY breaking parameters are determined by high scale physics. Using an appropriate high scale ansatz, weak scale SUSY couplings relevant for phenomenology can be calculated using the renormalization group equations (RGEs). This procedure also yields predictions for flav-our violation at the weak scale, assuming of course we have a theory (or at least an ansatz) of flavour at a high energy scale.
To accurately obtain the RG prediction at the two loop level, effects of thresholds at one loop must be incorporated. These include the decoupling of heavy particles and the concomitant splitting between couplings that are equal in the SUSY limit (the gauge and gaugino couplings, for instance). These new couplings will be identified with a tilde and labelled with the scalar which appears in the operator, e.g. the $\tilde{B}u\tilde{u}_{R}$ coupling is denoted by ${(\tilde{g}'_{u_R})}$.
The fact that gaugino couplings can become distinct from their corresponding SM gauge counterparts means that they can even develop flavour off-diagonal terms. These couplings therefore play an important role in flavour physics. It is also important to remember that threshold effects must be included in the running of the soft masses and trilinear SSB parameters.
The RGEs are constructed to describe a collection of effective theories (with varying particle content) valid at different scales. We approximate the changing particle content via step functions which remove the influence of the heavy particles at the appropriate scales [@cas2; @dedes]. Each threshold is denoted by a $\theta$ and labelled with the relevant particle. Above all thresholds, in the unbroken regime, all $\theta$-functions are unity, and the running is as given by the MSSM [@martv]. Below all sparticle and additional Higgs thresholds, these step functions all vanish and the running is purely SM [@cas1].
The importance of considering these effects can be shown by re-examining previous estimates for the flavour changing decay $\tilde{t}_{1}\rightarrow c\tilde{Z}_{1}$, which requires a solution of the RGEs in order to estimate $\tilde{t}-\tilde{c}$ mixing. This has previously been estimated [@HK] by integrating the RGEs in a single step. We will show that the results are significantly different when the RGEs are solved numerically, and the various effects discussed above are taken into account.
Deriving and solving the threshold RGEs {#sec:der}
=======================================
The general procedure for inserting thresholds begins with RGEs for a general theory [@machv] which do not depend on the particle structure of the theory. The RGEs in [@machv] are written for dimensionless parameters. In [@luo] they are rewritten for complex fermions and expanded to include RGEs for dimensionful parameters which are necessary for supersymmetric theories. We write the interaction of 2-component Majorana fermions ($\psi_{p}$) with real scalars ($\phi_{a}$) via Yukawa couplings ($Y^{a}_{pq}$) as $$\label{eq:2complag}
\mathcal{L}_{int}=-\left(\frac{1}{2}Y^{a}_{pq}\psi^{T}_{p}\zeta\psi_{q}\phi_{a}+\mathrm{h.c.}\right),$$ using $\zeta$ as the metric which makes the fermion bilinear Lorentz invariant. With this interaction, the correct form of the general Yukawa RGE is $$\begin{aligned}
\nonumber\lefteqn{\left(4\pi\right)^2\left.\frac{d{\mathbf{Y}}^{a}}{dt}\right|_{1-loop}=}\qquad\qquad \\
\nonumber&&\frac{1}{2}\left[{\mathbf{Y}}^T_2(F){\mathbf{Y}}^a+{\mathbf{Y}}^a{\mathbf{Y}}_2(F)\right]+2{\mathbf{Y}}^b{\mathbf{Y}}^{\dagger a}{\mathbf{Y}}^b\\
\nonumber&&+{\mathbf{Y}}^b\mathrm{Tr}\left\{\frac{1}{2}\left({\mathbf{Y}}^{\dagger b}{\mathbf{Y}}^a+{\mathbf{Y}}^{\dagger a}{\mathbf{Y}}^b\right)\right\}\\
&&-3g^2\left\{\mathbf{C}_2(F),{\mathbf{Y}}^a\right\}, \label{eq:2comprge}\end{aligned}$$ in which ${\mathbf{Y}}_2(F)={\mathbf{Y}}^{\dagger b}{\mathbf{Y}}^b$ and $\mathbf{C}_2(F)=\mathbf{t}^A\mathbf{t}^A$. An example, rewritten in the more phenomenologist friendly 4-component language with complex scalars and both Majorana and Dirac fermions, is shown in Appendix \[sec:4comp\]. It should be stressed that though the equations look much lengthier when written in 4-component form, much work needed to derive the RGEs has already been done, so their larger size is also indicative of ease of use.
To facilitate decoupling, interactions of the fields need to be written in the (approximate) mass basis. In the Higgs sector the real neutral fields $h$, $G^{0}$, $H$ and $A$ are combined into two complex Higgs fields, $\mathsf{h}$ and $\mathcal{H}$ (using a different font to differentiate the complex fields from the real fields) such that $$\begin{aligned}
\mathsf{h}&=\frac{h+iG^{0}}{\sqrt{2}}\\
\mathcal{H}&=\frac{-H+iA}{\sqrt{2}}.\end{aligned}$$ $\mathsf{h}$ and $\mathcal{H}$ along with the complex charged fields $G^{+}$ and $H^{+}$, are obtained from the usual Higgs doublets (in the notation of [@wss]) through a field rotation: $$\begin{aligned}
\label{eq:hrot}\left(\begin{array}{c}G^{+}\\\mathsf{h}\end{array}\right)&={s}\left(\begin{array}{c}h^{+}_{u}\\[5pt]h^{0}_{u}\end{array}\right)+{c}\left(\begin{array}{c}h^{-*}_{d}\\[5pt]h^{0*}_{d}\end{array}\right)\\[5pt]
\label{eq:Hrot}\left(\begin{array}{c}H^{+}\\\mathcal{H}\end{array}\right)&={c}\left(\begin{array}{c}h^{+}_{u}\\[5pt]h^{0}_{u}\end{array}\right)-{s}\left(\begin{array}{c}h^{-*}_{d}\\[5pt]h^{0*}_{d}\end{array}\right),\end{aligned}$$ where ${s}=\sin{\beta}$ and ${c}=\cos{\beta}$. We consider the effects of Higgs boson thresholds in the approximation $m_{A}>>M_{Z}$, in which case $m_{H}\sim m_{A}\sim m_{H^{+}}$, with $m_{h}\sim M_{Z}$[^1]. The doublet in (\[eq:Hrot\]) then decouples at a scale $Q\sim m_{H}$ while that in (\[eq:hrot\]) remains in the low energy theory.
For the neutralino and chargino thresholds, we ignore gaugino-Higgsino mixing (an excellent approximation if either of the conditions $\left|\mu\right|>>M_W$ or $\left|M_{1,2}\right|>>M_W$ are satisfied[^2]) so that the Higgsino mass eigenstates $$\frac{\psi_{h_{d}}\pm\psi_{h_{u}}}{\sqrt{2}}$$ are decoupled at the scale $\left|\mu\right|$, while the bino and wino states are decoupled at $Q=M_{1}$ and $Q=M_{2}$ respectively.
Below $Q\sim m_{H}$ the theory contains just the SM Higgs scalar (although it may still contain Higgsinos), and the MSSM Yukawas ($f$) are replaced by their SM equivalents ($\lambda$) using the conditions: $\lambda_{u}=\sin{\beta}f_{u}$ and $\lambda_{\{d,e\}}=\cos{\beta}f_{\{d,e\}}$. Note that if $\left|\mu\right|<m_{H}$, Higgsinos would still couple fermions to sfermions, an example of which is the term $\Psi_{h^{0}_{u}}u\tilde{u}_{R}$ with coupling ${(\tilde{f}^{u_R}_u)}$.
The general approach to solving the RGEs is as follows:
- Begin with weak scale gauge couplings and quark masses. Rotate the Yukawas to the current basis using four rotation matrices chosen to make sure the KM matrix is correct.
- Run the gauge couplings and Yukawas up to the high scale.
- Input the high scale ansatz which can depend on the Yukawa flavour stucture,
- Run the RGEs down in the presence of thresholds. As various thresholds are passed, remove the particles from the theory, with special attention given to the matching at $m_{H}$ as mentioned above.
Since the location of the thresholds is not known at the beginning of the process, it is necessary to begin with an estimate of their location and iterate this procedure until the solution converges to the required accuracy.
In this manner, it is possible to obtain the whole set of couplings at a scale appropriate to the problem – for example $Q=m_{\tilde{t}_{1}}$ in the case of stop decay. The inputs are: weak scale gauge couplings and squark masses; and high scale SUSY parameters governed by a particular model.
Additional Couplings when SUSY is broken {#sec:tilde}
========================================
When SUSY is broken by the decoupling of one or more of the SUSY particles, there are many couplings which, although equal in the supersymmetric limit, have different RGEs. This causes the couplings themselves to become different. As a simple example we can compare the quark-quark-gauge coupling to the squark-quark-gaugino coupling. These couplings are equal when SUSY is exact, but can be conceptually and numerically different below the scale where some sparticles have decoupled from the theory.
To one loop, the running of the gauge coupling does not depend on any other couplings and the only effect of thresholds is to change the numerical factor in the RGE. On the other hand, the running of the gaugino coupling contains many extra terms, some depending on various gaugino couplings and others depending on Yukawa matrices.
The extra Yukawa terms in the right-handed up-type gaugino coupling running are: $$\begin{aligned}
\nonumber&1.\quad&\left[{s}^2{\theta_h}+{c}^2{\theta_H}\right](f_u)^T_{ik}(f_u)^*_{kl}{(\tilde{g}'_{u_R})}_{lj}\\
\nonumber&2.\quad&-3\left[{s}^2{\theta_h}+{c}^2{\theta_H}\right]{\theta_{\tilde{B}}}{\theta_{\tilde{h}}}(f_u)^T_{ik}{(\tilde{f}^{u_R}_u)}^*_{kj}{(\tilde{g}'_{h_u})}\\
\nonumber&3.\quad&{\theta_{\tilde{h}}}{\theta_{{\tilde{Q}_k}}}{(\tilde{f}^Q_u)}^T_{ik}{(\tilde{f}^Q_u)}^*_{kl}{(\tilde{g}'_{u_R})}_{lj}\\
\nonumber&4.\quad&-{\theta_{\tilde{h}}}{\theta_{{\tilde{Q}_k}}}{(\tilde{f}^Q_u)}^T_{ik}{(\tilde{g}'_Q)}_{kl}{(\tilde{f}^{u_R}_u)}^*_{lj}\\
&5.\quad&2{\theta_{\tilde{h}}}{\theta_{{\tilde{u}_k}}}{(\tilde{g}'_{u_R})}_{ik}{(\tilde{f}^{u_R}_u)}^T_{kl}{(\tilde{f}^{u_R}_u)}^*_{lj}\end{aligned}$$ In the exact SUSY limit, the extra Yukawa terms cancel and the gaugino couplings are equal to the gauge coupling so that the RGEs are identical. When SUSY is broken this is no longer the case. If the heavy Higgs particles decouple first, for example, $\theta_{H}=0$ and several terms disappear from the running. This means that the cancellation of the Yukawa terms will no longer be exact, leading to a non-zero overall Yukawa factor in the RGE. The coupling can develop non-zero off-diagonal flavour changing terms which are not present in the gauge coupling. Thus, if the squarks and gauginos are lighter than $m_{H}$ the RGE for this coupling will have a Yukawa component at a scale where it is still important for phenomenology.
In the U(1) gaugino running, the decoupling of other gaugino terms can also result in a different value for the diagonal elements, as shown in Figure \[fig:ggtild\]. Here we show the 1-loop and 2-loop running of $g_{1}$ in the dotted curve and dot-dashed curve respectively. The solid line shows the running of the $g_{1}$ coupling when thresholds are switched on. It is clear that at $Q=m_{t}$ the effect of thresholds on the coupling is comparable to the difference between the 1- and 2-loop lines. The dashed line running upwards from the gluino threshold is the $\tilde{g}'_{u_{R}}$ coupling which is important until the up-right squarks decouple, shown by the central dotted line. We can see that the gaugino coupling at this point is very different from its SM counterpart.
![The evolution of the U(1) gauge coupling to 1-loop order (dotted curve) and 2-loop order (dot-dashed curve). Also the U(1) gauge coupling with thresholds (solid). The upper line (dashed) is the U(1) gaugino coupling to a right-handed up-type squark ($\tilde{g}_{1}$). Vertical dotted lines indicate the presence of thresholds.[]{data-label="fig:ggtild"}](threshtildenew.eps){width="34.80000%" height="45.00000%"}
There are many terms in the full threshold RGEs which can provide an unexpected dependence like the Yukawa terms in the gaugino coupling RGE. Another such example is in the running of the CP-conserving gaugino mass, $M_{2}$ [@wss], for which $$\left(4\pi\right)^{2}\frac{dM_{2}}{dt}\ni2M_{2}{s}{c}\left(-\theta_{h}+\theta_{H}\right)\theta_{\tilde{h}}\left[\mu^{*}\tilde{g}_{h_{d}}\tilde{g}_{h_{u}}+\mathrm{c.c.}\right],$$ It is clear that this term does not contribute in the SUSY limit (when all $\theta=1$) but can appear in the RGE if $m_{H}>\left|\mu\right|$.
We have obtained the full system of threshold RGEs for the gauge couplings and Yukawas, in addition to threshold RGEs for $\mu$, gaugino masses, soft masses and trilinear SSB parameters [@abfut] and plan to incorporate our code into ISAJET [@ijet].
Flavour changing top squark decay {#sec:HK}
=================================
The two body decay of the lighter stop ($\tilde{t}_{1}$) into a charm quark and the lightest neutralino ($\tilde{Z}_{1}$) occurs at the one loop level. This decay becomes important if tree level two body decays are kinematically forbidden, although it may still compete with three body decays [@porod].
Two body flavour changing stop decay was first studied by Hikasa and Kobayashi [@HK], where the off-diagonal elements of the up-squark SSB matrix, specifically $\tilde{t}_{\{L,R\}}-\tilde{c}_{L}$ mixing elements, were estimated under the approximation that the RGEs could be integrated using a single step and assuming the LSP was a photino[^3]. The partial decay width can then be readily obtained. Although their estimate was adequate for the very light stop case they were interested in, it is not valid for $m_{\tilde{t}_{1}}$ in the range of interest today.
To this end, the RG method was used to calculate the mixing numerically. This removes the need for the one-step approximation and also allows us to obtain the full weak scale couplings for use in the decay calculation. The decay width was re-derived from the Lagrangian, keeping the difference between sparticle and SM couplings, which allows for the possibility of extra flavour changing terms coming from sources other than the up-squark mass matrix.
Table \[tab:comp\] compares this partial width calculated using the various methods. The Hikasa-Kobayashi result clearly overestimates the width, illustrating the importance of our improvement. In a random sample of 10,000 mSUGRA points, over 300 were found with a stop mass above $100\ \mathrm{GeV}$ and with kinematically forbidden two body decays. In these cases the one-step approximation overestimated the width by between a factor of 15-25 although in a few cases the difference was as large as a factor of 35.
\[tab:comp\]
[lc]{} Method & Width\
Hikasa-Kobayashi & $\sim18$\
1-loop (all thresholds at $m_{H}$) & 1.04\
2-loop (all thresholds at $m_{H}$) & 1.15\
2-loop (realistic thresholds) & 1.19\
2-loop (realistic thresholds and tilde terms) & 1.31\
Looking at the rest of the results in the table, we can see that introducing thresholds to the equations produces a result which, although less than the difference between the 1-loop and 2-loop running, is of similar order. This indicates that the thresholds are important for claiming true 2-loop accuracy. The introduction of tilde terms produces a sizeable change, which shows that the difference between sparticle and SM couplings is also important.
Clearly the single step integration may give a qualitatively misleading value of the $\tilde{t}_{1}$ decay branching ratios in the event there are competing modes. The other effects are also important for obtaining a quantitative projection of stop decay patterns.
Conclusions {#sec:conc}
===========
We have constructed a closed system of RG equations including threshold decoupling for sparticles and heavy Higgs particles. This includes SSB trilinear couplings and masses in addition to the gauge and Yukawa couplings. A code has been developed to solve the RGEs which will be incorporated into the Isajet event generator.
When SUSY is broken, the RGEs for sparticle couplings can become different from the RGEs for their SM counterparts, i.e. the gauge and Yukawa couplings. This not only results in different diagonal elements at the weak scale, but many sparticle couplings *also develop additional flavour off-diagonal terms*.
Our consideration of the flavour changing decay of the stop finds the partial width to have been over-estimated by a factor of 15 to 25 as a result of the one-step approximation. We also find that threshold effects in both the standard couplings and the so-called tilde terms are important for more quantitative flavour violation calculations.
### Acknowledgements {#acknowledgements .unnumbered}
This research was carried out in collaboration with Xerxes Tata and was supported in part by a grant from the US Department of Energy. The University of Hawaii Graduate Student Organisation provided partial support for accommodation during the Conference.
4-component Yukawa RGE {#sec:4comp}
======================
The Lagrangian in (\[eq:2complag\]) can be recast into 4-component form by combining spinors into 4-component Majorana fermions given by $\Psi_{M i}={\left(\psi_M,-\zeta\psi^*_M\right)}_i$ and Dirac fermions, $\Psi_{D i}={\left(\psi_L,-\zeta\psi^*_R\right)}_i$: $$\begin{aligned}
\nonumber\mathcal{L}_c\ni&&-\left(U^{1a}_{jk}\overline{\Psi}_{Dj}P_L\Psi_{Dk}\Phi_a+U^{2a}_{jk}\overline{\Psi}_{Dj}P_L\Psi_{Dk}\Phi^\dagger_a\right.\\
\nonumber&&+V^a_{jk}\overline{\Psi}_{Dj}P_L\Psi_{Mk}\Phi_a+W^a_{jk}\overline{\Psi}_{Mj}P_L\Psi_{Dk}\Phi^\dagger_a\\
\nonumber&&+\frac{1}{2}X^{1a}_{jk}\overline{\Psi}_{Mj}P_L\Psi_{Mk}\Phi_a+\frac{1}{2}X^{2a}_{jk}\overline{\Psi}_{Mj}P_L\Psi_{Mk}\Phi^\dagger_a\\
&&\left.+\mathrm{h.c.}\right)\end{aligned}$$ where the scalar $\phi_{a}$ is now complex.
Using this new Lagrangian, the RGEs for $U^{1a}$, $U^{2a}$, $V^{a}$, $W^{a}$, $X^{1a}$ and $X^{2a}$ can be obtained from (\[eq:2comprge\]). In the case of the MSSM, the Lagrangian does not contain any terms where the Higgs fields ($h$) connect Dirac and Majorana fermions, so $W^{h}$ and $V^{h}$ vanish. Here $h$ can stand for any of the fields $\mathsf{h}$, $\mathcal{H}$, $G^{+}$ or $H^{+}$ since the general form of the RGE is the same for all Higgs fields.
The RGE for $U^{1h}$, which we show as an example, is $$\begin{aligned}
\nonumber\lefteqn{\left(4\pi\right)^2\left.\frac{d{\mathbf{U}}^{1h}}{dt}\right|_{1-loop}=} \qquad \\
\nonumber&&\frac{1}{2}\left[\left({\mathbf{U}}^{1b}{\mathbf{U}}^{1b\dagger}+{\mathbf{U}}^{2b}{\mathbf{U}}^{2b\dagger}+{\mathbf{V}}^{b}{\mathbf{V}}^{b\dagger}\right){\mathbf{U}}^{1h}\right.\\
\nonumber&&\left.\qquad+{\mathbf{U}}^{1h}\left({\mathbf{U}}^{1b\dagger}{\mathbf{U}}^{1b}+{\mathbf{U}}^{2b\dagger}{\mathbf{U}}^{2b}+{\mathbf{W}}^{b\dagger}{\mathbf{W}}^{b}\right)\right]\\
\nonumber&&+2\left[{\mathbf{U}}^{1b}{\mathbf{U}}^{2h\dagger}{\mathbf{U}}^{2b}+{\mathbf{U}}^{2b}{\mathbf{U}}^{2h\dagger}{\mathbf{U}}^{1b}+{\mathbf{V}}^{b}{\mathbf{X}}^{2h\dagger}{\mathbf{W}}^{b}\right]\\
\nonumber&&+{\mathbf{U}}^{1b}\mathrm{Tr}\left\{\left({\mathbf{U}}^{1b\dagger}{\mathbf{U}}^{1h}+{\mathbf{U}}^{2h\dagger}{\mathbf{U}}^{2b}\right)\right.\\
\nonumber&&\left.\qquad\quad\ +\frac{1}{2}\left({\mathbf{X}}^{1b\dagger}{\mathbf{X}}^{1h}+{\mathbf{X}}^{2h\dagger}{\mathbf{X}}^{2b}\right)\right\}\\
\nonumber&&+{\mathbf{U}}^{2b}\mathrm{Tr}\left\{\left({\mathbf{U}}^{2b\dagger}{\mathbf{U}}^{1h}+{\mathbf{U}}^{2h\dagger}{\mathbf{U}}^{1b}\right)\right.\\
\nonumber&&\left.\qquad\quad\ +\frac{1}{2}\left({\mathbf{X}}^{2b\dagger}{\mathbf{X}}^{1h}+{\mathbf{X}}^{2h\dagger}{\mathbf{X}}^{1b}\right)\right\}\\
&&-3g^2\left[{\mathbf{U}}^{1h}\mathbf{C}^L_2(F)+\mathbf{C}^R_2(F){\mathbf{U}}^{1h}\right],\end{aligned}$$ where the final term has been constructed by separating $\mathbf{C}_2(F)$ into separate terms for left-handed and right-handed fields. The other RGEs will appear in [@abfut].
[999]{}
D.J. Castaño, E.J. Piard, P. Ramond, Phys. Rev. D **49**, (1994) 4882
A. Dedes, A.B. Lahanas, K. Tamvakis, Phys. Rev. D **53**, (1996) 3793
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[^1]: If instead $m_{A}\sim M_{Z}$, all Higgs bosons have a roughly common mass, and threshold effects in this sector are unimportant.
[^2]: If neither of these conditions are met, the gauginos and Higgsinos will all have similar mass so they will all decouple at about the same scale. It will therefore still be approximately valid to use a single scale to decouple the Higgsinos.
[^3]: The Hikasa-Kobayashi result was later modified to allow for arbitrary composition of the LSP.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In our previous articles, we have presented a class of endomorphisms of the Cuntz algebras which are defined by polynomials of canonical generators and their conjugates. We showed the classification of some case under unitary equivalence by help of branching laws of permutative representations. In this article, we construct an automaton which is called the Mealy machine associated with the endomorphism in order to compute its branching law. We show that the branching law is obtained as outputs from the machine for the input of information of a given representation.'
author:
- Katsunori Kawamura
date: 'xxx,xxx, 2006'
---
[^1]
[**]{}\
\[section:first\] In [@PE01; @PE02], we introduced a class of endomorphisms of the Cuntz algebra $\con$ which are called permutative endomorphisms. They are given by noncommutative polynomials in canonical generators of $\con$. Such endomorphisms were motivated by an interest of the following endomorphism $\rho_{\nu}$ of $\co{3}$ discovered by Noboru Nakanishi: $$\label{eqn:nakeq}
\left\{
\begin{array}{c}
\rho_{\nu}(s_{1})\equiv s_{2}s_{3}s_{1}^{*}+s_{3}s_{1}s_{2}^{*}+s_{1}s_{2}s_{3}^{*},\\
\\
\rho_{\nu}(s_{2})\equiv s_{3}s_{2}s_{1}^{*}+s_{1}s_{3}s_{2}^{*}+s_{2}s_{1}s_{3}^{*},\\
\\
\rho_{\nu}(s_{3})\equiv s_{1}s_{1}s_{1}^{*}+s_{2}s_{2}s_{2}^{*}+s_{3}s_{3}s_{3}^{*}
\end{array}
\right.$$ where $s_{1},s_{2},s_{3}$ are canonical generators of $\co{3}$. Because $\rho_{\nu}(s_{1}),\rho_{\nu}(s_{2}),\rho_{\nu}(s_{3})$ satisfy the relation of canonical generators of $\co{3}$, we can verify that $\rho_{\nu}$ is an endomorphism of $\co{3}$. $\rho_{\nu}$ is very concrete but its property is not so clear. In Theorem 1.2 of [@PE01], we proved that $\rho_{\nu}$ is irreducible but not an automorphism by using branching laws of $\rho_{\nu}$ with respect to permutative representations. Especially, $\rho_{\nu}$ is not unitarily equivalent to the canonical endomorphism of $\co{3}$.
In general, representations of C$^{*}$-algebras do not have unique decomposition (up to unitary equivalence) into sums or integrals of irreducibles. However, the permutative representations of $\con$ do [@BJ; @DaPi2; @DaPi3]. Because a representation arising from the right transformation of a permutative representation by a permutative endomorphism is also a permutative representation, their branching laws make sense. By such branching laws, permutative endomorphisms are characterized and classified effectively.
\[defi:first\] Let $s_{1},\ldots,s_{N}$ be canonical generators of $\con$ and $({\cal H},\pi)$ be a representation of $\con$.
1. $({\cal H},\pi)$ is a [*permutative representation*]{} of $\con$ if there is a complete orthonormal basis $\{e_{n}\}_{n\in\Lambda}$ of ${\cal H}$ and a family $f=\{f_{i}\}_{i=1}^{N}$ of maps on $\Lambda$ such that $\pi(s_{i})e_{n}=e_{f_{i}(n)}$ for each $n\in\Lambda$ and $i\edot$.
2. For $J=(j_{i})_{i=1}^{k}\in\nset{k}$, $({\cal H},\pi)$ is $P(J)$ if there is a unit cyclic vector $\Omega\in {\cal H}$ such that $\pi(s_{J})\Omega=\Omega$ and $\{\pi(s_{j_{i}}\cdots s_{j_{k}})\Omega\}_{i=1}^{k}$ is an orthonormal family in ${\cal H}$ where $s_{J}\equiv s_{j_{1}}\cdots s_{j_{k}}$.
3. $({\cal H},\pi)$ is a [*cycle*]{} if there is $J\in\nset{k}$ such that $({\cal H},\pi)$ is $P(J)$.
For any $J\in \nset{k}$, $P(J)$ exists uniquely up to unitary equivalence. In Theorem 1.3 of [@PE02], we showed the following:
\[Thm:mainzero\] Let ${\goth S}_{N,l}$ be the set of all permutations on the set $\nset{l}$. For $\sigma\in {\goth S}_{N,l}$, let $\psi_{\sigma}$ be the endomorphism of $\con$ defined by $$\label{eqn:end}
\psi_{\sigma}(s_{i})\equiv u_{\sigma}s_{i}\quad(i\edot)$$ where $u_{\sigma}\equiv \sum_{J\in \nset{l}}s_{\sigma(J)}(s_{J})^{*}$. If a representation $({\cal H},\pi)$ of $\con$ is $P(J)$ for $J\in\nset{k}$ and $\sigma\in {\goth S}_{N,l}$, then there are $J_{1},\ldots,J_{M}\in\bigcup_{m\geq 1}\nset{m}$ and subrepresentations $\pi_{1},\ldots,\pi_{M}$ of $\pi\circ \psi_{\sigma}$ such that $$\label{eqn:branching}
\pi\circ \psi_{\sigma}=\pi_{1}\oplus\cdots\oplus\pi_{M},$$ $\pi_{i}$ is $P(J_{i})$ and $J_{i}\in \coprod_{n=1}^{N^{l-1}}\nset{nk}$ for $i=1,\ldots,M$. Further $1\leq M\leq N^{l-1}$.
$\psi_{\sigma}$ in (\[eqn:end\]) is called the [*permutative endomorphism*]{} of $\con$ by $\sigma$. The canonical endomorphism of $\con$ and $\rho_{\nu}$ in (\[eqn:nakeq\]) are permutative endomorphisms.
By the uniqueness of decomposition of permutative representation, the rhs in (\[eqn:branching\]) is unique up to unitary equivalence. When $({\cal H},\pi)$ is $P(J)$ and $\rho\in {\rm End}\con$, we denote $({\cal H},\pi\circ \rho)$ by $P(J)\circ \rho$ simply. Then (\[eqn:branching\]) can be rewritten as follows: $$\label{eqn:simpdeco}
P(J)\circ \psi_{\sigma}=P(J_{1})\oplus\cdots\oplus P(J_{M}).$$ We call (\[eqn:simpdeco\]) by the [*branching law*]{} of $\psi_{\sigma}$ with respect to $P(J)$. The branching law of $\psi_{\sigma}$ is unique up to unitary equivalence of $\psi_{\sigma}$. Concrete such branching laws are already given in [@PE01; @PE02] by direct computation. These branching laws are interesting subjects themselves and they are useful to classify endomorphisms effectively. On the other hand, an automaton is a typical object to consider algorithm of computation in the computer science [@Eilenberg; @Ginz; @hopul; @Mealy]. An automaton is a machine which changes the internal state by an input. A Mealy machine is a kind of automaton with output.
In this article, we show a better algorithm to compute branching law, that is, an algorithm to seek $J_{1},\ldots,J_{M}$ from a given $J$ in (\[eqn:simpdeco\]) by reducing the problem to a semi-Mealy machine $\mms$ as an input ($=J$) and outputs ($=J_{1},\ldots,J_{M}$):
(1001,650)(0,50) (2000,0)
If $J=J_{0}^{r}$, that is, $J$ is a sequence of $r$-times repetition of a sequence $J_{0}\in\nset{k^{'}}$ and $r\geq 2$, then there are $z_{1},\ldots,z_{r}\in U(1)$ such that $P(J)=\bigoplus_{j=1}^{r}P(J_{0})\circ \gamma_{z_{j}}$ where $\gamma$ is the gauge action on $\con$ by Theorem 2.4 (iv) in [@PE02]. Because $\gamma_{z}\circ \psi_{\sigma}=\psi_{\sigma}\circ \gamma_{z}$ for each $z$, the branching law of $P(J)\circ \psi_{\sigma}$ is reduced to that of $P(J_{0})\circ \psi_{\sigma}$. Therefore it is sufficient to show the case that $J$ is [*nonperiodic*]{}, that is, $J$ is impossible to be written as $J_{0}^{r}$ for $r\geq 2$. Hence we assume that $J$ is nonperiodic.
For $\sigma\in {\goth S}_{N,l}$ with $l\geq 2$ and $J\in\nset{l}$, we define $\sigma_{1}(J),\ldots,\sigma_{l}(J)\in\nset{}$ by $\sigma(J)=(\sigma_{1}(J),\ldots,\sigma_{l}(J))$ and let $\sigma_{n,m}(J)\equiv (\sigma_{n}(J),\ldots,
\sigma_{m}(J))$ for $1\leq n<m\leq l$. Define $\nset{0}\equiv \{0\}$ for convenience.
\[defi:mealy\] For $\sigma\in {\goth S}_{N,l}$, a data $\mms\equiv(Q,\Sigma,\Delta,\delta,\lambda)$ is called the [*semi-Mealy machine by $\sigma$*]{} if $Q,\Sigma,\Delta$ are finite sets, $$Q\equiv\{q_{K}:K\in \nset{l-1}\},\quad
\Sigma\equiv\{a_{j}\}_{j=1}^{N},\quad
\Delta\equiv \{b_{j}\}_{j=1}^{N}$$ and two maps $\delta:Q\times\Sigma^{*}\to Q$, $\lambda:Q\times \Sigma^{*}\to \Delta^{*}$ are defined by $$\delta(q_{K},a_{i})\equiv
\left\{
\begin{array}{ll}
q_{0} \quad&(l=1),\\
&\\
q_{(\sigma^{-1})_{2,l}(K,i)}\quad&(l\geq 2),\\
\end{array}
\right.\quad \!\!
\lambda(q_{K},a_{i})\equiv
\left\{
\begin{array}{ll}
b_{\sigma^{-1}(i)} \quad&(l=1),\\
&\\
b_{(\sigma^{-1})_{1}(K,i)}\quad&(l\geq 2)\\
\end{array}
\right.$$ for $i\edot$ and $K\in\nset{l-1}$ where $\Sigma^{*}$ and $\Delta^{*}$ are free semigroups generated by $\Sigma$ and $\Delta$, respectively.
We posteriori define $\delta(q,wa)\equiv \delta(\delta(q,w),a)$ and $\lambda(q,wa)\equiv \lambda(q,w)\lambda(\delta(q,w),a)$ for $q\in Q$, $w\in \Sigma^{*}$ and $a\in \Sigma$. For a given $J=(j_{i})_{i=1}^{k}\in\nset{k}$, define $Q_{J}\equiv \{q\in Q:\mbox{there exists } n\in{\bf N}\,s.t.\,
\delta(q,(a_{J})^{n})=q\}$ where $a_{J}\equiv a_{j_{1}}\cdots a_{j_{k}}\in \Sigma^{*}$ and define an equivalence relation $\sim$ in $Q_{J}$ by $q\sim q^{'}$ if there is $n\in {\bf N}$ such that $\delta(q,(a_{J})^{n})=q^{'}$. Define $[q]\equiv\{q^{'}\in Q_{J}:q\sim q^{'}\}$. Then $[q]$ is a cyclic component of $Q_{J}$ with respect to the iteration of the right action of $a_{J}$ by $\delta$. There are $p_{1},\ldots,p_{M}\in Q_{J}$ such that the set $Q_{J}$ of periodic points is decomposed into orbits as follows: $$\label{eqn:qdeco}
Q_{J}=[p_{1}]\sqcup\cdots \sqcup [p_{M}].$$ Under these preparations, the main theorem is given as follows:
\[Thm:mainone\] If $J$ is nonperiodic, then $J_{1},\ldots,J_{M}$ in (\[eqn:simpdeco\]) are obtained by $$b_{J_{i}}=\lambda(p_{i},(a_{J})^{r_{i}}) \quad (i=1,\ldots,M)$$ where $p_{1},\ldots,p_{M}\in Q_{J}$ are taken as (\[eqn:qdeco\]) and $r_{i}\equiv \#[p_{i}]$ for $i=1,\ldots,M$.
In Theorem \[Thm:mainone\], if $p_{1}^{'},\ldots,p_{M}^{'}$ satisfy (\[eqn:simpdeco\]) and $[p_{i}^{'}]=[p_{i}]$ for each $i$, then the associated $J_{1}^{'},\ldots,J_{M}^{'}$ satisfy that $P(J_{i}^{'})=P(J_{i})$ for each $i$. We show a more practical algorithm to compute branching laws by using the Mealy diagram as follows:
The [*transition diagram (Mealy diagram)*]{} ${\cal D}(\mathsf{M})$ of a semi-Mealy machine $\mathsf{M}
=(Q,\Sigma,\Delta,\delta,\lambda)$ is a directed graph with labeled edges, which has a set $Q$ of vertices and a set $E\equiv \{(q,\delta(q,a),a)\in Q\times Q\times \Sigma:q\in Q,\, a\in \Sigma\}$ of directed edges with labels. The meaning of $(q,\delta(q,a),a)$ is an edge from $q$ to $\delta(q,a)$ with a label “$a/\lambda(q,a)$" for $a\in\Sigma$:
(1001,450)(99,-250) (300,-50)[$\delta(q,a)=p$, $\lambda(q,a)=b$ $\Longleftrightarrow$]{} (3200,0)
For $\rho_{\nu}$ in (\[eqn:nakeq\]), we compute branching laws by the semi-Mealy machine. Define $\sigma_{0}\in{\goth S}_{3,2}$ by ${\small \begin{array}{c|ccccccccc}
J&11&12&13&21&22&23&31&32&33\\ \hline
\sigma_{0}(J)&23&31&12&32&13&21&11&22&33\\
\end{array}}$. Then $\rho_{\nu}= \psi_{\sigma_{0}}$ and $\mathsf{M}_{\sigma_{0}}=(\{q_{1},q_{2},q_{3}\},
\{a_{1},a_{2},a_{3}\}$, $\{b_{1},b_{2},b_{3}\},
\delta,\lambda)$ is given as follows:
[$$\begin{array}{c|c|c|c|c|c|c}
p&\delta(p,a_{1})&\delta(p,a_{2})&\delta(p,a_{3})
&\lambda(p,a_{1})&\lambda(p,a_{2})&\lambda(p,a_{3})\\
\hline
q_{1}&q_{1}&q_{3}&q_{2}&b_{3}&b_{1}&b_{2}\\
q_{2}&q_{3}&q_{2}&q_{1}&b_{2}&b_{3}&b_{1}\\
q_{3}&q_{2}&q_{1}&q_{3}&b_{2}&b_{1}&b_{3}\\
\end{array}$$ ]{}
From this, ${\cal D}(\mathsf{M}_{\sigma_{0}})$ is as follows:
\#1[(-50,-20)[$q_{#1}$]{}]{} \#1\#2[(0,0)[$a_{#1}/b_{#2}$]{}]{}
(3011,2120)(-2299,-360) (0,0) (0,0) (0,0) (-20,1620) (850,-100) (-1055,-130)
According to Theorem \[Thm:mainone\], we compute branching laws for $\rho_{\nu}$ by ${\cal D}(\mathsf{M}_{\sigma_{0}})$. When the input word is $a_{1}$, $\delta(q_{1},a_{1})=q_{1}$, $\delta(q_{2},a_{1})=q_{3}$, $\delta(q_{3},a_{1})=q_{2}$. Therefore $Q_{1}=[q_{1}]\sqcup[q_{2}]$, $r_{1}=1$, $r_{2}=2$ and there are two cycles $q_{1}$ and $q_{2}q_{3}$ in $Q$ with respect to $a_{1}$. From this, we have output words, $\lambda(q_{1},a_{1})=b_{3}$ and $\lambda(q_{2},(a_{1})^{2})=b_{2}b_{1}$. Hence $P(1)\circ \rho_{\nu}=P(3)\oplus P(21)=P(3)\oplus P(12)$. where we use a fact that $P(j_{p(1)},\ldots,j_{p(k)})=P(j_{1},\ldots,j_{k})$ for each $p\in {\bf Z}_{k}$. Further the following holds:
[$$\begin{array}{c|c|c|c}
\mbox{input}&\mbox{cycles}&\mbox{outputs}
&\mbox{branching law}\\
\hline
a_{1}&q_{1},q_{2}q_{3}
&b_{3},b_{2}b_{1}&P(1)\circ \rho_{\nu}=P(3)\oplus P(12)\\
\hline
a_{1}a_{2}&q_{1}q_{1}q_{3}q_{2}q_{2}q_{3}
&b_{3}b_{1}b_{1}b_{3}b_{2}b_{2}
&P(12)\circ \rho_{\nu}=P(113223)\\
\hline
a_{1}a_{2}a_{3}&q_{1}q_{1}q_{3}q_{3}q_{2}q_{2},q_{2}q_{3}q_{1}
&b_{3}b_{1}b_{3}b_{1}b_{3}b_{1},b_{2}b_{2}b_{2}
&P(123)\circ \rho_{\nu}=P(131313)\oplus P(222)\\
\hline
a_{1}a_{3}a_{2}&q_{1}q_{1}q_{2}q_{2}q_{3}q_{3},q_{3}q_{2}q_{1}
&b_{3}b_{2}b_{3}b_{2}b_{3}b_{2},b_{1}b_{1}b_{1}
&P(132)\circ \rho_{\nu}=P(232323)\oplus P(111)\\
\end{array}$$ ]{}
In $\S$\[section:third\], we rewrite branching laws by branching function systems and their transformations, and we review known facts about endomorphisms. $\S$\[section:fourth\] is devoted to prove Theorem \[Thm:mainone\] by branching function systems. In $\S$\[section:fifth\], we show examples of Mealy diagram of the semi-Mealy machine $\mms$ and branching laws of $\psi_{\sigma}$ for concrete $\sigma\in {\goth S}_{N,l}$.
\[section:third\] In order to compute branching laws of endomorphisms, we introduce branching function systems and their transformations by permutations. Let $\nset{*}_{1}\equiv \bigcup_{k\geq 1}\nset{k}$. For $J\in \nset{*}_{1}$, the [*length*]{} of $J$ is defined by $k$ when $J\in \nset{k}$. For $J_{1}=(j_{1},\ldots,j_{k}),J_{2}=(j_{1}^{'},\ldots,j_{l}^{'})$, let $J_{1}\cup J_{2}\equiv(j_{1},\ldots,j_{k},j_{1}^{'},\ldots,j_{l}^{'})$. Especially, we define $(i,J)\equiv (i)\cup J$ for convenience. For $J$ and $k\geq 2$, $J^{k}=J\cup\cdots\cup J$ ($k$-times). For $J=(j_{1},\ldots,j_{k})$ and $\tau\in {\bf Z}_{k}$, define $\tau(J)\equiv (j_{\tau(1)}, \ldots,j_{\tau(k)})$. For $J_{1},J_{2}\in \nset{*}_{1}$, $J_{1}\sim J_{2}$ if there are $k\geq 1$ and $\tau\in {\bf Z}_{k}$ such that $J_{1},J_{2}\in\nset{k}$ and $\tau(J_{1})=J_{2}$. For $J_{1}=(j_{1},\ldots,j_{k}),J_{2}=(j_{1}^{'},\ldots,j_{k}^{'})$, $J_{1}\prec J_{2}$ if $\sum_{l=1}^{k}(j_{l}^{'}-j_{l})N^{k-l}\geq 0$. $J\in\nset{*}_{1}$ is [*minimal*]{} if $J\prec J^{'}$ for each $J^{'}\in\nset{*}_{1}$ such that $J\sim J^{'}$. Define $[1,\ldots,N]^{*}\equiv \{J\in\nset{*}_{1}:
J\mbox{ is minimal and nonperiodic}\}$. $[1,\ldots,N]^{*}$ is in one-to-one correspondence with the set of all equivalence classes of nonperiodic elements in $\nset{*}_{1}$ with respect to the equivalence relation $\sim$.
Let $\Lambda$ be an infinite set and $N\geq 2$. $f=\{f_{i}\}_{i=1}^{N}$ is a [*branching function system on $\Lambda$*]{} if $f_{i}$ is an injective transformation on $\Lambda$ for $i\edot$ such that a family of their images coincides a partition of $\Lambda$. Let $\bfsnl$ be the set of all branching function systems on $\Lambda$. $f=\{f_{i}\}_{i=1}^{N}\in{\rm BFS}_{N}(\Lambda_{1})$ and $g=\{g_{i}\}_{i=1}^{N}\in{\rm BFS}_{N}(\Lambda_{2})$ are [*equivalent*]{} if there is a bijection $\varphi$ from $\Lambda_{1}$ to $\Lambda_{2}$ such that $\varphi\circ f_{i}\circ \varphi^{-1}=g_{i}$ for $i\edot$. For $f=\{f_{i}\}_{i=1}^{N}$, we denote $f_{J}\equiv f_{j_{1}}\circ\cdots\circ f_{j_{k}}$ when $J=(j_{1},\ldots,j_{k})\in \nset{k}$ and define $f_{0}\equiv id$. For $x,y\in \Lambda$, $x\sim y$ (with respect to $f$) if there are $J_{1},J_{2}\in\nset{*}$ and $z\in \Lambda$ such that $f_{J_{1}}(z)=x$ and $f_{J_{2}}(z)=y$. For $x\in \Lambda$, define $A_{f}(x)\equiv \{y\in \Lambda:x\sim y\}$. $f=\{f_{i}\}_{i=1}^{N}\in\bfsnl$ is [*cyclic*]{} if there is an element $x\in \Lambda$ such that $\Lambda=A_{f}(x)$. $\{n_{1},\ldots,n_{k}\}\subset \Lambda$ is a [*cycle*]{} of $f$ if there is $J=(j_{1},\ldots,j_{k})$ such that $f_{j_{1}}(n_{1})=n_{k},f_{j_{2}}(n_{2})=n_{1},\ldots,f_{j_{k}}(n_{k})=n_{k-1}$. $f$ has a [*cycle*]{} if there is a cycle of $f$ in $\Lambda$.
Let $\Xi$ be a set and $\Lambda_{\omega}$ be an infinite set for $\omega\in\Xi$. For $f^{[\omega]}=\{f_{i}^{[\omega]}\}_{i=1}^{N}\in{\rm BFS}_{N}(\Lambda_{\omega})$, $f$ is the [*direct sum*]{} of $\{f^{[\omega]}\}_{\omega\in\Xi}$ if $f=\{f_{i}\}_{i=1}^{N}\in {\rm BFS}_{N}(\Lambda)$ for a set $\Lambda\equiv \coprod_{\omega\in\Xi}\Lambda_{\omega}$ which is defined by $f_{i}(n)\equiv f^{[\omega]}_{i}(n)$ when $n\in\Lambda_{\omega}$ for $i\edot$ and $\omega\in\Xi$. For $f\in\bfsnl$, $f=\bigoplus_{\omega\in\Xi}f^{[\omega]}$ is a [*decomposition*]{} of $f$ into a family $\{f^{[\omega]}\}_{\omega\in\Xi}$ if there is a family $\{\Lambda_{\omega}\}_{\omega\in\Xi}$ of subsets of $\Lambda$ such that $f$ is the direct sum of $\{f^{[\omega]}\}_{\omega\in\Xi}$. For each $f=\{f_{i}\}_{i=1}^{N}\in\bfsnl$, there is a decomposition $\Lambda=\coprod_{\omega\in\Xi}\Lambda_{\omega}$ such that $\#\Lambda_{\omega}=\infty$, $f|_{\Lambda_{\omega}}\equiv \{f_{i}|_{\Lambda_{\omega}}\}_{i=1}^{N}
\in{\rm BFS}_{N}(\Lambda_{\omega})$ and $f|_{\Lambda_{\omega}}$ is cyclic for each $\omega\in\Xi$.
\[defi:pj\]
1. For $J\in \nset{k}$, $f\in\bfsnl$ is $P(J)$ if $f$ is cyclic and has a cycle $\{n_{1},\ldots,n_{k}\}$ such that $f_{J}(n_{k})=n_{k}$.
2. For $f\in\bfsnl$ and $J\in\nset{*}_{1}$, $g$ is a $P(J)$-component of $f$ if $g$ is a direct sum component of $f$ and $g$ is $P(J)$.
For $f\in\bfsnl$ and $\Lambda_{1},\Lambda_{2}\subset \Lambda$, if $f|_{\Lambda_{i}}$ is $P(J_{i})$ for $i=1,2$, then either $\Lambda_{1}\cap \Lambda_{2}=\emptyset$ or $\Lambda_{1}=\Lambda_{2}$.
Recall ${\goth S}_{N,l}$ in Theorem \[Thm:mainzero\]. For $\sigma\in {\goth S}_{N,l}$ and $f=\{f_{i}\}_{i=1}^{N}\in\bfsnl$, define $f^{(\sigma)}=\{f^{(\sigma)}_{i}\}_{i=1}^{N}\in\bfsnl$ by $$\label{eqn:sigmaf}
f^{(\sigma)}_{i}\equiv f_{\sigma(i)}\quad(l=1),\quad
f^{(\sigma)}_{i}(f_{J}(n))\equiv f_{\sigma(i,J)}(n)\quad(l\geq 2)$$ for $n\in\Lambda$, $i\edot$ and $J\in \nset{l-1}$. If $\sigma\in {\goth S}_{N}={\goth S}_{N,1}$ and $f\in\bfsnl$ is $P(J)$, then $f^{(\sigma)}$ is $P(J_{\sigma^{-1}})$ where $J_{\sigma^{-1}}\equiv\left(\sigma^{-1}(j_{1}),\ldots,\sigma^{-1}(j_{k})\right)$ for $J=(j_{1},\ldots,j_{k})$. For any $J\in\nset{*}_{1}$, there is $f\in\bfsnl$ for some set $\Lambda$ such that $f$ is $P(J)$. In this case, for $\sigma\in {\goth S}_{N,l}$, there is $1\leq M\leq N^{l-1}$ such that $f^{(\sigma)}$ is decomposed into a direct sum of $M$ cycles by Lemma 2.2 in [@PE02]. Furthermore, the length of each cycle is a multiple of that of $J$.
For $N\geq 2$, let $\con$ be the [*Cuntz algebra*]{} [@C], that is, the C$^{*}$-algebra which is universally generated by $s_{1},\ldots,s_{N}$ satisfying $s^{*}_{i} s_j=\delta_{ij}I$ for $i,j\edot$ and $s_1 s^{*}_1+\cdots+s_N s^{*}_N=I$. In this article, any representation and endomorphism are assumed unital and $*$-preserving.
$(l_{2}(\Lambda),\pi_{f})$ is the [*permutative representation of $\con$ by $f=\{f_{i}\}_{i=1}^{N}\in\bfsnl$*]{} if $\pi_{f}(s_{i})e_{n}\equiv e_{f_{i}(n)}$ for $n\in\Lambda$ and $i\edot$. For $J\in \nset{*}_{1}$, $P(J)$ in Definition \[defi:first\] is irreducible if and only if $J$ is nonperiodic. For $J_{1},J_{2}\in \nset{*}_{1}$, $P(J_{1})\sim P(J_{2})$ if and only if $J_{1}\sim J_{2}$ where $P(J_{1})\sim P(J_{2})$ means the unitary equivalence of two representations which satisfy the condition $P(J_{1})$ and $P(J_{2})$, respectively. $[1,\ldots,N]^{*}$ is in one-to-one correspondence with the set of equivalence classes of irreducible permutative representations of $\con$ with a cycle. If $f\in\bfsnl$ and $g\in{\rm BFS}_{N}(\Lambda^{'})$ satisfy $f\sim g$, then $(l_{2}(\Lambda),\pi_{f})\sim(l_{2}(\Lambda^{'}),\pi_{g})$. If $f$ is cyclic, then $(l_{2}(\Lambda),\pi_{f})$ is cyclic. If $f$ is $P(J)$, then $(l_{2}(\Lambda),\pi_{f})$ is $P(J)$. If $\Lambda=\Lambda_{1}\sqcup \Lambda_{2}$ and $f^{(i)}\equiv f|_{\Lambda_{i}}\in{\rm BFS}_{N}(\Lambda_{i})$ for $i=1,2$, then $(l_{2}(\Lambda),\pi_{f})\sim
(l_{2}(\Lambda_{1}),\pi_{f^{(1)}})\oplus(l_{2}(\Lambda_{2}),\pi_{f^{(2)}})$.
Let $\enda$ be the set of all unital $*$-endomorphisms of a unital $*$-algebra ${\cal A}$. For $\rho\in\enda$, $\rho$ is [*proper*]{} if $\rho({\cal A})\ne {\cal A}$. $\rho$ is [*irreducible*]{} if $\rho({\cal A})^{'}\cap {\cal A}={\bf C}I$ where $\rho({\cal A})^{'}\cap {\cal A}
\equiv \{x\in {\cal A}:\mbox{for all } a\in{\cal A},\,\rho(a)x=x\rho(a)\}$. $\rho$ and $\rho^{'}$ are [*equivalent*]{} if there is a unitary $u\in{\cal A}$ such that $\rho^{'}={\rm Ad}u\circ \rho$. In this case, we denote $\rho\sim\rho^{'}$. Let ${\rm Rep}{\cal A}$ ([*resp.*]{} ${\rm IrrRep}{\cal A}$) be the set of all unital ([*resp.*]{} irreducible) $*$-representations of ${\cal A}$. We simply denote $\pi$ for $({\cal H},\pi) \in{\rm Rep}{\cal A}$. If $\rho,\rho^{'}\in {\rm End}{\cal A}$ and $\pi,\pi^{'}\in {\rm Rep}{\cal A}$ satisfy $\rho\sim\rho^{'}$ and $\pi\sim \pi^{'}$, then $\pi\circ \rho\sim \pi^{'}\circ \rho^{'}$. Assume that ${\cal A}$ is simple. If there is $\pi\in {\rm IrrRep}{\cal A}$ such that $\pi\circ \rho\in {\rm IrrRep}{\cal A}$, then $\rho$ is irreducible. If there is $\pi\in {\rm Rep}{\cal A}$ such that $\pi\circ\rho\not\sim \pi\circ\rho^{'}$, then $\rho\not\sim\rho^{'}$. If there is $\pi\in {\rm IrrRep}{\cal A}$ such that $\pi\circ\rho\not\in {\rm IrrRep}{\cal A}$, then $\rho$ is proper.
For $\psi_{\sigma}$ in (\[eqn:end\]), define $$\label{eqn:edef}
E_{N,l}\equiv\{\psi_{\sigma}\in\endcon:\sigma\in {\goth S}_{N,l}\}\quad(l\geq 1).$$ If $\sigma\in {\goth S}_{N}$, then $\psi_{\sigma}$ is an automorphism of $\con$ which satisfies $\psi_{\sigma}(s_{i})=s_{\sigma(i)}$ for $i\edot$. Especially, if $\sigma=id$, then $\psi_{id}=id$. If $\sigma\in {\goth S}_{N,2}$ is defined by $\sigma(i,j)\equiv (j,i)$ for $i,j\edot$, then $\psi_{\sigma}$ is just the canonical endomorphism of $\con$. For $\sigma\in {\goth S}_{N,l}$ and $f\in\bfsnl$, $\pi_{f}\circ \psi_{\sigma}=\pi_{f^{(\sigma)}}$ where $f^{(\sigma)}$ is in (\[eqn:sigmaf\]). If $\rho$ is a permutative endomorphism and $({\cal H},\pi)$ is a permutative representation of $\con$, then $\pi\circ\rho$ is also a permutative representation.
A representation $({\cal H},\pi)$ of $\con$ has a [*$P(J)$-component*]{} if $({\cal H},\pi)$ has a subrepresentation $({\cal H}_{0},\pi|_{{\cal H}_{0}})$ which is $P(J)$. A component of a representation $P(J)\circ \rho$ of $\con$ means a subrepresentation of $({\cal H},\pi)$ which is equivalent to $P(J^{'})$ for some $J^{'}$.
For comparison of the method to find $(J_{i})_{i=1}^{M}$ in (\[eqn:simpdeco\]) for a given $J$, we show the usual method to determine $(J_{i})_{i=1}^{M}$ as follows: (a) Prepare a representation $({\cal H},\pi)$ which is $P(J)$. We often take ${\cal H}=\ltn$ and $\pi=\pi_{f}$ for suitable branching function system $f$ on ${\bf N}$. (b) Compute $\pi(\psi_{\sigma}(s_{i}))e_{n}$ for each $n\in {\bf N}$ and $i\edot$. By the proof of Lemma 2.2 in [@PE02], we see that it is sufficient to check for $1\leq n\leq N^{l-1}k$ when $|J|=k$. (c) Find all cycles in ${\cal H}$ by using results in (b). In this way, the direct computation of branching law is too much of a bother because of a great number of calculated amount when $N,k,l$ are large.
\[section:fourth\] In this section, we assume that $\sigma\in{\goth S}_{N,l}$, $l\geq 2$, $J=(j_{i})_{i=1}^{k}\in\nset{k}$ and $J$ is nonperiodic. For $r\geq 2$, extend $J=(j_{i})_{i=1}^{k}$ as $(j_{n})_{n=1}^{r\cdot k}$ by $j_{k(c-1)+i}\equiv j_{i}$ for each $c=1,\ldots,r$ and $i=1,\ldots,k$ for convenience.
\[lem:direct\] Let $f\in \bfsnl$ be $P(J)$, $f^{(\sigma)}$ be in (\[eqn:sigmaf\]) and let $M_{\sigma}=(Q,\Sigma,\Delta,\delta,\lambda)$ be in Definition \[defi:mealy\]. For $p\in Q_{J}$, define $r_{J}(p)\in{\bf N}$ by $r_{J}(p)\equiv \#[p]$.
1. For $p\in Q_{J}$ and $\alpha\equiv r_{J}(p)\cdot k$, define $p_{1},\ldots,p_{\alpha}\in Q$ and $T=(t_{i})_{i=1}^{\alpha}\in\nset{\alpha}$ by $p_{1}\equiv p$, $b_{t_{1}}=\lambda(p_{\alpha},a_{j_{\alpha}})$ and $$p_{i}\equiv \delta(p_{i-1},a_{j_{i-1}}),\quad
b_{t_{i}}=\lambda(p_{i-1},a_{j_{i-1}})\quad(i=2,\ldots,\alpha),$$ then there is $\Lambda(p)\subset \Lambda$ such that $f^{(\sigma)}|_{\Lambda(p)}$ is $P(T)$.
2. In (i), define $T^{'}\in\nset{\alpha}$ by $b_{T^{'}}=\lambda(p,a_{J}^{r_{J}(p)})$. Then $f^{(\sigma)}|_{\Lambda(p)}$ is $P(T^{'})$.
3. If there is $\Lambda_{0}$ such that $f^{(\sigma)}|_{\Lambda_{0}}$ is $P(T)$ for $T=(t_{i})_{i=1}^{\alpha}\in\nset{\alpha}$, then there is $p\in Q_{J}$ such that $\Lambda_{0}$ is equal to $\Lambda(p)$ in (i).
4. In (i), $p\sim p^{'}$ if and only if $\Lambda(p)=\Lambda(p^{'})$.
5. Choose $p_{1},\ldots,p_{M}$ as (\[eqn:qdeco\]). Then the decomposition $f^{(\sigma)}=f^{[1]}\oplus\cdots\oplus f^{[M]}$ holds as a branching function system where $f^{[i]}\equiv f^{(\sigma)}|_{\Lambda(p_{i})}$ for each $i$.
Let $n_{0}\in\Lambda$ such that $f_{J}(n_{0})=n_{0}$. Because $J$ is nonperiodic, such $n_{0}$ is unique in $\Lambda$.
\(i) Let $r\equiv r_{J}(p)$. There is a sequence $(I_{1},\ldots,I_{\alpha})$ in $\nset{l-1}$ such that $p_{i}=q_{I_{i}}$ for each $i$. By definition of $\delta$ and $\lambda$ and assumption, $$\label{eqn:its}
\sigma(t_{1},I_{1})=(I_{\alpha},j_{\alpha}),\sigma(t_{2},I_{2})=(I_{1},j_{1}),
\ldots,\sigma(t_{\alpha},I_{\alpha})=(I_{\alpha-1},j_{\alpha-1}).$$ Define $m(p)\equiv f_{\sigma(t_{1},I)}(n_{0})\in\Lambda$. Then $m(p)=f_{I_{\alpha}}(f_{j_{\alpha}}(n_{0}))$. By this and definition of $f^{(\sigma)}$, we can verify that $f^{(\sigma)}_{T}(m(p))=m(p)$. Define $$m_{\alpha}\equiv m(p),\quad m_{\alpha-1}\equiv f^{(\sigma)}_{t_{\alpha}}(m(p)),
\ldots,m_{1}\equiv f^{(\sigma)}_{(t_{1},\ldots,t_{\alpha})}(m(p))$$ and $\Lambda(p)\equiv \{f^{(\sigma)}_{K}(m(p)):K\in\nset{*}_{1}\}$. It is sufficient to show that $m_{i}\ne m_{j}$ when $i\ne j$. By definition, $$m_{i}=f^{(\sigma)}_{t_{i+1}}(m_{i+1})
=f_{(I_{i},j_{i})}(f_{(j_{i+1},\ldots,j_{\alpha})}(n_{0}))
\quad(i=1,\ldots,\alpha-1)\quad m_{\alpha}=f^{(\sigma)}_{t_{1}}(m_{1}).$$ Assume that $m_{i}=m_{i^{'}}$ and $c\equiv i^{'}-i\geq 0$. This implies that $m_{\tau(i)}=m_{\tau(i^{'})}$ for each $\tau\in{\bf Z}_{\alpha}$. From this, $(I_{\tau(i)},j_{\tau(i)})=(I_{\tau(i^{'})},j_{\tau(i^{'})})$ and $f^{(\sigma)}_{(t_{i+1},\ldots,t_{\alpha})}(m(p))
=f^{(\sigma)}_{(t_{i^{'}+1},\ldots,t_{\alpha})}(m(p))$. This implies that $f_{(I_{\alpha},j_{\alpha})}(n_{0})
=f_{(I_{c},j_{c})}(f_{(j_{c+1},\ldots,j_{\alpha})}(n_{0}))$. Therefore $n_{0}=f_{(j_{c+1},\ldots,j_{\alpha})}(n_{0})$. By the uniqueness of the cycle in $\Lambda$ with respect to $f$, $c=k(d-1)$ for $1\leq d\leq r$. Hence $I_{\tau(i)}=I_{\tau(i+k(d-1))}$ for each $\tau$. Therefore $p_{\tau(i)}=q_{I_{\tau(i)}}=q_{I_{\tau(i+k(d-1))}}=p_{\tau(i+k(d-1))}$ for each $\tau$. By the choice of $r$, $d=1$ and $i=i^{'}$. Hence the statement holds.
\(ii) We see that $t_{1}^{'}=t_{\alpha},
t_{2}^{'}=t_{1},\ldots,t_{\alpha}^{'}=t_{\alpha-1}$. Hence $P(T)\sim P(T^{'})$ by definition.
\(iii) Fix $\tau\in {\bf Z}_{\alpha}$. Define $T^{'}=(t_{i}^{'})^{\alpha}_{i=1}\in\nset{\alpha}$ by $$\label{eqn:newt}
t_{i}^{'}\equiv t_{\tau^{-1}(i)}\quad(i=1,\ldots,\alpha).$$ Then $f^{(\sigma)}|_{\Lambda_{0}}$ is also $P(T^{'})$ and there is $m_{0}\in\Lambda_{0}$ such that $f_{T^{'}}^{(\sigma)}(m_{0})=m_{0}$. Define $m_{\alpha}\equiv m_{0}$ and $m_{i}\equiv f^{(\sigma)}{(t_{i+1},\ldots,t_{\alpha})}(m_{0})$ for $i=1,\ldots,\alpha-1$. Then $m_{i}\ne m_{i^{'}}$ when $i\ne i^{'}$. By definition of $f$, there are $n^{'}\in\Lambda$, $I_{0}\in\nset{l-1}$ and $u_{0}\in\nset{}$ such that $m_{\alpha}=f_{(I_{0},u_{0})}(n^{'})$. Define a sequence $(I_{i}^{'})_{i=1}^{\alpha}$ in $\nset{l-1}$ and $U=(u_{i})_{i=1}^{\alpha}\in\nset{\alpha}$ by $$I_{\alpha}^{'}\equiv I_{0},\quad u_{\alpha}\equiv u_{0},\quad
(I_{i}^{'},u_{i})\equiv \sigma(t_{i+1}^{'},I_{i+1}^{'})\quad
(i=\alpha-1,\alpha-2,\ldots,1).$$ By assumption, we see that $f_{(I_{\alpha}^{'},u_{\alpha})}(n^{'})
=f_{\sigma(t_{1}^{'},I_{1}^{'})}(f_{U}(n^{'}))$. By definition of $f$, $(I_{\alpha}^{'},u_{\alpha})=\sigma(t_{1}^{'},I_{1}^{'})$ and $n^{'}=f_{U}(n^{'})$. By the uniqueness of cycle in $\Lambda$ with respect to $f$, $U\sim J^{r}$. Hence there is $\tau^{'}\in{\bf Z}_{\alpha}$ such that $j_{i}=u_{\tau^{'}(i)}$ for $i=1,\ldots,\alpha$. Here choose $\tau$ in (\[eqn:newt\]) by $\tau\equiv \tau^{'}$ and define $I_{i}\equiv I_{\tau(i)}^{'}$ for each $i$. Then (\[eqn:its\]) holds. From this, we can verify that $p\equiv q_{I_{1}}$ belongs to $Q_{J}$. Define $m(p)\equiv f_{(I_{\alpha},j_{\alpha})}(n_{0})$ as (i). Then $n_{0}= f_{(j_{1},\ldots,j_{\tau^{-1}(\alpha)})}(n^{'})$ and $m_{\alpha}=f^{(\sigma)}_{(t_{\tau^{-1}(1)},\ldots,t_{\alpha})}(m(p))$. Therefore $m_{\alpha}\in\Lambda(p)$. Since $m_{\alpha}\in \Lambda_{0}\cap\Lambda(p)$, $\Lambda_{0}=\Lambda(p)$.
\(iv) If $p\sim p^{'}$, then there is $c$ such that $p^{'}=p_{kc+1}$ in (i) and we can verify that $m(p^{'})=f^{(\sigma)}_{(t_{1+kc},\ldots,t_{\alpha})}(m(p))\in\Lambda(p)$. Since $m(p^{'})\in \Lambda(p^{'})\cap\Lambda(p)$, $\Lambda(p^{'})=\Lambda(p)$.
Assume that $\Lambda(p)=\Lambda(p^{'})$. Let $m(p),m(p^{'})\in \Lambda$ be in the proof of (i). Then there are $T,T^{'}\in\nset{*}_{1}$ such that $f^{(\sigma)}_{T}(m(p))=m(p)$ and $f^{(\sigma)}_{T^{'}}(m(p^{'}))=m(p^{'})$. Then $f^{(\sigma)}|_{\Lambda(p)}$ is $P(T)$ and $f^{(\sigma)}|_{\Lambda(p^{'})}$ is $P(T^{'})$. Since $f^{(\sigma)}|_{\Lambda(p)}=f^{(\sigma)}|_{\Lambda(p^{'})}$, $T^{'}\sim T$. Assume that $T=(t_{i})_{i=1}^{\alpha}$ and $T^{'}=(t_{i}^{'})_{i=1}^{\alpha}$. Let $\{m_{i}\}_{i=1}^{\alpha}$ be the cycle in $\Lambda(p)$ of $f^{(\sigma)}$ in (i). By the uniqueness of the cycle in $\Lambda(p)$ with respect to $f^{(\sigma)}$, $\{m_{i}\}_{i=1}^{\alpha}$ is also the cycle in $\Lambda(p^{'})$ of $f^{(\sigma)}$. By the proof of (i), $m(p^{'})\in\{m_{i}\}_{i=1}^{\alpha}$. Hence there is $\tau\in {\bf Z}_{\alpha}$ such that $m(p^{'})=m_{\tau(\alpha)}$. From this, $t_{i}^{'}=t_{\tau(i)}$ for $i=1,\ldots,\alpha$. Because $T\sim T^{'}$, $r_{J}(p^{'})=r_{J}(p)$. Let $r\equiv r_{J}(p)$. Assume that $p=q_{I_{1}}$ and $p^{'}=q_{I_{1}^{'}}$. By definition of $m(p)$ and $m(p^{'})$ and their relation, we see that $I_{1}^{'}=I_{\tau(1)}$. Therefore $p^{'}=q_{I_{\tau(1)}}$. By choice of $p$ and $p^{'}$, $\delta(p,a_{J}^{r})=p$ and $\delta(p^{'},a_{J}^{r})=p^{'}$. Because $J$ is nonperiodic, $\tau(i)=i+kc$ for a certain $c$ modulo $\alpha$. Therefore $p^{'}=q_{I_{\tau(1)}}=q_{I_{1+kc}}=\delta(p,a_{J}^{c})$. Therefore $p^{'}\sim p$.
\(v) If $i\ne j$, then $\Lambda(p_{i})\ne \Lambda(p_{j})$ by (iv). Hence $\Lambda(p_{i})\cap \Lambda(p_{j})=\emptyset$. Therefore $\Lambda(p_{1})\sqcup\cdots \sqcup\Lambda(p_{M})\subset \Lambda$. By (iii) and the decomposability of the branching function $f^{(\sigma)}$, $\Lambda(p_{1})\sqcup\cdots \sqcup\Lambda(p_{M})=\Lambda$. This implies the statement. [\
]{}
[*Proof of Theorem \[Thm:mainone\].*]{} Assume that $J=(j_{i})_{i=1}^{k}\in\nset{k}$. When $l=1$, $Q_{J}=\{q_{0}\}$. Let $J_{\sigma^{-1}}\equiv(\sigma^{-1}(j_{1}),\ldots, \sigma^{-1}(j_{k}))$. Then we can check that $\lambda(q_{0},a_{J})=b_{J_{\sigma^{-1}}}$ and $P(J)\circ \psi_{\sigma}=P(J_{\sigma^{-1}})$ independently. Hence the assertion is verified. Assume that $l\geq 2$. By applying the correspondence between branching function systems and permutative representations, we see that the decomposition in Lemma \[lem:direct\] (v) implies that in (\[eqn:simpdeco\]). By definition of $J_{i}$ and applying Lemma \[lem:direct\] (i), (ii) to each component in the decomposition, the statement holds. [\
]{}
By Theorem \[Thm:mainone\], it is not necessary for computation of branching law (\[eqn:simpdeco\]) to prepare any representation space. Further Theorem \[Thm:mainone\] implies the following:
\[prop:connected\] If the Mealy diagram of $\mms$ has $M$ connected components, then $P(J)\circ \psi_{\sigma}$ has $M$ components of direct sum at least for each $J$.
\[section:fifth\] We show examples of permutative endomorphism of $\con$ and compute their branching laws by using the Mealy diagram according to Theorem \[Thm:mainone\]. Recall $E_{N,l}$ in (\[eqn:edef\]). Here we often denote $(j_{1},\ldots,j_{k})$ by $j_{1}\cdots j_{k}$ simply. \[subsection:etwo\] In [@PE01], we show that there are $16$ equivalence classes in $E_{2,2}$ and there are $5$ irreducible and proper classes ${\cal E}$ in them. We treat $3$ elements in ${\cal E}$ here. For each $\sigma\in {\goth S}_{2,2}$, $\mms=(Q,\Sigma,\Delta,\delta,\lambda)$ consists of $Q=\{q_{1},q_{2}\}$, $\Sigma=\{a_{1},a_{2}\}$ and $\Delta=\{b_{1},b_{2}\}$.
Define a transposition $\sigma\in {\goth S}_{2,2}$ by $\sigma(1,1)\equiv (1,2)$. Then $\psi_{\sigma}$ and the Mealy diagram ${\cal D}(\mms)$ of $\mms$ are as follows:
\#1[(-50,-30)[$q_{#1}$]{}]{} \#1\#2[(0,0)[$a_{#1}/b_{#2}$]{}]{}
(3011,620)(-3099,-300) (0,0) (0,0) (0,0) (850,10) (-1075,-20) (-3500,-50)[$\left\{
\begin{array}{ll}
\psi_{\sigma}(s_{1})\equiv s_{1}s_{2}s_{1}^{*}+s_{1}s_{1}s_{2}^{*},\\
&\\
\psi_{\sigma}(s_{2})\equiv s_{2},\\
\end{array}
\right.$]{}
$\psi_{\sigma}$ is irreducible and proper (Table II in [@PE01]). We denote $\psi_{\sigma}$ by $\psi_{12}$ in convenience. We show several branching laws by $\psi_{12}$: [$$\begin{array}{c|c|c|c}
\mbox{input}&\mbox{cycles}&\mbox{outputs}
&\mbox{branching law}\\
\hline
a_{1}&q_{1}q_{2}&b_{1}b_{2}&P(1)\circ
\psi_{12}=
P(12)\\
\hline
a_{2}&q_{1},q_{2}&b_{1},b_{2}
&P(2)\circ \psi_{12}=
P(1)\oplus P(2)\\
\hline
a_{1}a_{2}&q_{1}q_{2}q_{2}q_{1}
&b_{1}b_{2}b_{2}b_{1}
&P(12)\circ \psi_{12}=
P(1122)\\
\hline
a_{1}a_{1}a_{2}a_{2}
&q_{1}q_{2}q_{1}q_{1},q_{2}q_{1}q_{2}q_{2}
&b_{1}b_{2}b_{1}b_{1},b_{2}b_{1}b_{2}b_{2}
&P(1122)\circ \psi_{12}=P(1112)\oplus P(1222)\\
\end{array}$$ ]{}
Focusing attention on closed paths in ${\cal D}(\textsf{M}_{\sigma})$, we can verify the following:
\[prop:psonetwo\] For each $J\in\{1,2\}^{*}_{1}$, there are $J_{1},J_{2}$ or $J_{3}$ such that $$P(J)\circ \psi_{12}=
\left\{
\begin{array}{ll}
P(J_{1})\oplus P(J_{2})\quad &
(n_{1}(J)=\mbox{ even} ),\\
&\\
P(J_{3})\quad &(n_{1}(J)= \mbox{ odd})\\
\end{array}
\right.$$ where $n_{1}(J)\equiv \sum_{l=1}^{k}(2-j_{l})$ for $J=(j_{1},\ldots,j_{k})\in\{1,2\}^{k}$.
Let $\sigma\in {\goth S}_{2,2}$ be a transposition defined by $\sigma(1,1)\equiv (2,1)$. Then $\psi_{\sigma}$, ${\cal D}(\mms)$ and branching laws of $\psi_{\sigma}$ are given as follows:
\#1[(-50,-30)[$q_{#1}$]{}]{} \#1\#2[(0,0)[$a_{#1}/b_{#2}$]{}]{}
(3011,500)(-3099,-200) (0,0) (0,0) (0,0) (850,10) (-1075,-20) (-3500,-50)[$\left\{
\begin{array}{ll}
\psi_{\sigma}(s_{1})\equiv s_{2}s_{1}s_{1}^{*}+s_{1}s_{2}s_{2}^{*},\\
&\\
\psi_{\sigma}(s_{2})\equiv s_{1}s_{1}s_{1}^{*}+s_{2}s_{2}s_{2}^{*},\\
\end{array}
\right.$]{}
[$$\begin{array}{c|c|c|c}
\mbox{input}&\mbox{cycles}&\mbox{outputs}
&\mbox{branching law}\\
\hline
a_{1}&q_{1}&b_{2}&P(1)\circ\psi_{\sigma}=P(2)\\
\hline
a_{2}&q_{2}&b_{2}&P(2)\circ\psi_{\sigma}=P(2)\\
\hline
a_{1}a_{2}&q_{2}q_{1}&b_{1}b_{2}&P(12)\circ\psi_{\sigma}=P(11)\\
\hline
a_{1}a_{1}a_{2}&
q_{2}q_{1}q_{1}&b_{1}b_{2}b_{1}&P(112)\circ\psi_{\sigma}=P(112)\\
\hline
a_{1}a_{2}a_{2}&
q_{2}q_{1}q_{2}&b_{1}b_{1}b_{2}&P(122)\circ\psi_{\sigma}=P(112)\\
\end{array}$$ ]{}
Let $\sigma\in {\goth S}_{2,2}$ be defined by $\sigma(1,1)\equiv (2,2)$, $\sigma(1,2)\equiv (1,1)$, $\sigma(2,1)\equiv (2,1)$, $\sigma(2,2)\equiv (1,2)$. Then $\psi_{\sigma}$, ${\cal D}(\mms)$ and branching laws are as follows:
\#1[(-50,-30)[$q_{#1}$]{}]{} \#1\#2[(0,0)[$a_{#1}/b_{#2}$]{}]{}
(3011,920)(-3499,-290) (0,0) (0,0) (0,0) (-3500,0)[$\left\{
\begin{array}{ll}
\psi_{\sigma}(s_{1})\equiv s_{2}s_{2}s_{1}^{*}+s_{1}s_{1}s_{2}^{*},\\
&\\
\psi_{\sigma}(s_{2})\equiv s_{2}s_{1}s_{1}^{*}+s_{1}s_{2}s_{2}^{*},\\
\end{array}
\right.$]{}
[$$\begin{array}{c|c|c|c}
\mbox{input}&\mbox{cycles}&\mbox{outputs}
&\mbox{branching law}\\
\hline
a_{1}&q_{1}q_{2}&b_{1}b_{2}&P(1)\circ
\psi_{\sigma}=
P(12)\\
\hline
a_{2}&q_{1}q_{2}&b_{2}b_{1}&P(2)\circ
\psi_{\sigma}=
P(12)\\
\hline
a_{1}a_{2}&q_{1}q_{2},q_{2}q_{1}
&b_{1}b_{1},b_{2}b_{2}&P(12)\circ
\psi_{\sigma}=
P(11)\oplus P(22)\\
\end{array}$$ ]{} \[subsection:fifthtwo\] Note that $\#E_{2,2}=2^{2}!=24$ and $\#E_{3,2}=3^{2}!\sim 3.6\times 10^{5}$. Hence it is difficult to classify every element in $E_{3,2}$ by computing its branching laws in comparison with the case $E_{2,2}$. We see that $\mms=(\{q_{1},q_{2},q_{3}\},
\{a_{1},a_{2},a_{3}\}$, $\{b_{1},b_{2},b_{3}\},
\delta,\lambda)$ for each $\sigma\in {\goth S}_{3,2}$. $\rho_{\nu}$ in (\[eqn:nakeq\]) belongs to $E_{3,2}$.
Let $\sigma\in {\goth S}_{3,2}$ be a transposition by $\sigma(1,1)\equiv (1,2)$. Then $\psi_{\sigma}$, ${\cal D}(\mms)$ and branching laws are as follows:
\#1[(-50,-20)[$q_{#1}$]{}]{} \#1\#2[(0,0)[$a_{#1}/b_{#2}$]{}]{}
(3011,1960)(-2899,-300) (0,0) (0,0) (0,0) (-20,1620) (850,-100) (-1055,-130) (-3155,830)[$
\left\{
\begin{array}{rl}
\psi_{\sigma}(s_{1})\equiv& s_{12,1}
+s_{11,2}+s_{13,3},\\
&\\
\psi_{\sigma}(s_{2})\equiv &s_{2},\\
&\\
\psi_{\sigma}(s_{3})\equiv &s_{3},\\
\end{array}
\right.
$]{}
[$$\begin{array}{c|c|c|c}
\mbox{input}&\mbox{cycles}&\mbox{outputs}
&\mbox{branching law}\\
\hline
a_{1}&q_{1}q_{2}&b_{1}b_{2}&
P(1)\circ \psi_{\sigma}=
P(12)\\
\hline
a_{2}&q_{1},q_{2}&b_{1},b_{2}
&P(2)\circ \psi_{\sigma}=
P(1)\oplus P(2)\\
\hline
a_{3}&q_{3}&b_{3}
&P(3)\circ \psi_{\sigma}=
P(3)\\
\end{array}$$ ]{}
where $s_{ij,k}\equiv s_{i}s_{j}s_{k}^{*}$. From this, we see that $\psi_{\sigma}^{n}$ is proper and irreducible for each $n\geq 1$, and $\psi_{\sigma}$ and $\rho_{\nu}$ are not equivalent. \[subsection:fourththree\] Define $\sigma\in {\goth S}_{4,2}$ by
[$$\begin{array}{c|cccccccccccccccc}
J&11&12&13&14&21&22&23&24&31&32&33&34&41&42&43&44\\
\hline
\sigma(J)&11&21&31&41&12&22&43&42&32&23&13&33&44&24&14&34\\
\end{array}$$ ]{}
Then $\psi_{\sigma}$ and ${\cal D}(\mms)$ are as follows: $$\begin{array}{ll}
\psi_{\sigma}(s_{1})\equiv
s_{11,1}+s_{21,2}+s_{31,3}+s_{41,4},
&
\psi_{\sigma}(s_{2})\equiv
s_{12,1}+s_{22,2}+s_{43,3}+s_{42,4},\\
&\\
\psi_{\sigma}(s_{3})\equiv
s_{32,1}+s_{23,2}+s_{13,3}+s_{33,4},
&
\psi_{\sigma}(s_{4})\equiv
s_{44,1}+s_{24,2}+s_{14,3}+s_{34,4},\\
\end{array}$$
\#1[(-70,-30)[$q_{#1}$]{}]{} \#1\#2[(-70,0)[$a_{#1}/b_{#2}$]{}]{}
(2756,2406)(-1799,-500) (0,0) (-250,-100) (100,-100) (0,500) (250,-100) (650,-100)[(-1,0)[1300]{}]{} (450,50) (-450,40)
When $J=(1)$, $\delta(q_{i},a_{1})=q_{i}$ and $\lambda(q_{i},a_{1})=b_{1}$ for each $i=1,2,3,4$. Therefore $P(1)\circ \psi_{\sigma} =P(1)\oplus P(1)\oplus P(1)\oplus P(1)$. In the same way, we have $$P(2)\circ \psi_{\sigma}=P(2)\oplus P(2)\oplus P(2), \quad
P(4)\circ \psi_{\sigma}=P(4)\oplus P(444).$$ This is an example of Proposition \[prop:connected\].
\[subsection:fourthfour\] The Mealy diagram associated with the canonical endomorphism $\rho$ of $\con$ (see $\S$\[section:third\]) is given as follows:
\#1[(500,300)(-60,-30)[$q_{#1}$]{}]{} \#1\#2[(0,0)[$a_{#1}/b_{#2}$]{}]{}
(3011,1000)(-3899,-150) (0,0) (0,0) (0,100) (-1600,100) (-4000,300)[$\rho(x)\equiv
s_{1}xs_{1}^{*}+\cdots
+s_{N}xs_{N}^{*}$,]{}
In this case, there is no transition among different states. We see that $P(J)\circ\rho=P(J)^{\oplus N}$ for each $J\in\nset{*}_{1}$ where $P(J)^{\oplus N}$ is the direct sum of $N$ copies of $P(J)$. In general, $\pi\circ \rho=\pi^{\oplus N}$ for any representation $\pi$ of $\con$.
\[subsection:fourthfive\] Let $\sigma\in {\goth S}_{2,3}$ be a transposition by $\sigma(1,1,1)\equiv (1,2,1)$. Then $\psi_{\sigma}\in E_{2,3}$, ${\cal D}(\mms)$ and branching laws are as follows:
$$\left\{
\begin{array}{l}
\psi_{\sigma}(s_{1})\equiv
s_{121}s_{11}^{*}+s_{112}s_{12}^{*}+s_{111}s_{21}^{*}+s_{122}s_{22}^{*},\\
\\
\psi_{\sigma}(s_{2})\equiv s_{2},
\end{array}
\right.$$
\#1[(300,280)(-90,-30)[$q_{#1}$]{}]{} \#1\#2[(0,0)[$a_{#1}/b_{#2}$]{}]{}
(3011,1420)(-2000,-100) (0,0) (0,0) (0,0) (860,-150)
[$$\begin{array}{c|c|c|c}
\mbox{input}&\mbox{cycles}&\mbox{outputs}
&\mbox{branching law}\\
\hline
a_{1}&q_{11}q_{21}
&b_{1}b_{2}&
P(1)\circ \psi_{\sigma}=P(12)\\
\hline
a_{2}&q_{22}
&b_{2}&
P(2)\circ \psi_{\sigma}=P(2)\\
\hline
a_{1}a_{2}&q_{12}q_{11}
&b_{1}b_{1}&
P(12)\circ \psi_{\sigma}=P(11)\\
\hline
a_{1}a_{1}a_{2}
&q_{12}q_{11}q_{21}
&b_{1}b_{1}b_{2}
&P(112)\circ \psi_{\sigma}=P(112)\\
\end{array}$$ ]{} We see that $\psi_{\sigma}^{n}$ is irreducible and proper for each $n\geq 1$.
Let $\sigma\in {\goth S}_{2,3}$ be defined by the product $\sigma=\sigma^{'}\circ \sigma^{''}$ of two transpositions $\sigma^{'}$ and $\sigma^{''}$ defined by $\sigma^{'}(1,1,1)\equiv (1,2,1)$ and $\sigma^{''}(1,1,2)\equiv (1,2,2)$, respectively. In this case $\psi_{\sigma}=\psi_{12}\in E_{2,2}$ in $\S$\[subsection:etwo\]. ${\cal D}(\mms)$ is as follows:
\#1[(300,280)(-90,-20)[$q_{#1}$]{}]{} \#1\#2[$a_{#1}/b_{#2}$]{}
(3011,1720)(-2099,-300) (0,0) (0,0) (0,0) (860,-150) (840,1180)
We can verify that branching laws of $\psi_{\sigma}$ coincide with those of $\psi_{12}$. \[subsection:fourthsix\] Define a transposition $\sigma\in {\goth S}_{2,4}$ by $\sigma(1,1,1,1)\equiv (1,2,1,1)$. Then $\psi_{\sigma}\in E_{2,4}$, ${\cal D}(\mms)$ and branching laws are given as follows: [$$\!\psi_{\sigma}(s_{1})\equiv
s_{1211}s_{111}^{*}+
s_{1112}s_{112}^{*}+
s_{112}s_{12}^{*}+
s_{1111}s_{211}^{*}+
s_{1212}s_{212}^{*}+
s_{122}s_{22}^{*},\quad
\psi_{\sigma}(s_{2})\equiv s_{2},$$ ]{}
\#1[(300,280)(-120,-20)[$q_{#1}$]{}]{} \#1\#2[$a_{#1}/b_{#2}$]{}
(3011,2320)(100,-200) (0,0) (0,0) (0,0) (4570,-150)
[$$\begin{array}{c|c|c|c}
\mbox{input}&\mbox{cycles}&\mbox{outputs}
&\mbox{branching law}\\
\hline
a_{1}&q_{111}q_{211}
&b_{1}b_{2}&
P(1)\circ \psi_{\sigma}=P(12)\\
\hline
a_{2}&q_{222}
&b_{2}&
P(2)\circ \psi_{\sigma}=P(2)\\
\hline
a_{1}a_{2}&q_{212}q_{121}
&b_{2}b_{1}&
P(12)\circ \psi_{\sigma}=P(12)\\
\hline
a_{1}a_{1}a_{2}&q_{112}q_{121}q_{111}
&b_{1}b_{1}b_{1}&
P(112)\circ \psi_{\sigma}=P(111)\\
\end{array}$$ ]{}
[**Acknowledgement:**]{} The author would like to thank Takeshi Nozawa for useful comment on this article.
[99]{} O. Bratteli and P. E. T. Jorgensen, [*Iterated function systems and permutation representations of the Cuntz algebra*]{}, Memoirs Amer. Math. Soc. [**139**]{} (1999), no.663. J. Cuntz, [*Simple $C^*$-algebras generated by isometries*]{}, Comm. Math. Phys. [**57**]{}, 173-185 (1977). K. R. Davidson and D. R. Pitts, [*The algebraic structure of non-commutative analytic Toeplitz algebras*]{}, Math. Ann. 311, 275-303 (1998). K. R. Davidson and D. R. Pitts, [*Invariant subspaces and hyper-reflexivity for free semigroup algebras*]{}, Proc. London Math. Soc. (3) 78 (1999) 401-430. S. Eilenberg, [*Automata, languages and machines*]{}, vol A, Academic Press (1974). A. Ginzburg, [*Algebraic theory of automata*]{}, Academic Press (1968). J. E. Hopcroft and J. D. Ullman, [*Introduction to automata theory, languages and computation*]{}, Addison-Wesley Publishing Co. Inc. Reading, Massachusetts, U.S.A. (1979). K. Kawamura, [*Polynomial endomorphisms of the Cuntz algebras arising from permutations. I —General theory—*]{}, Lett. Math. Phys. [**71**]{}, 149-158 (2005). ———, [*Branching laws for polynomial endomorphisms of Cuntz algebras arising from permutations*]{}, Lett. Math. Phys., to appear. G. J. Mealy, [*A method for synthesizing sequencial circuits*]{}, Bell System Technical J. [**34**]{}: 5, 1045-1079 (1955).
[^1]:
|
{
"pile_set_name": "ArXiv"
}
|
addtoreset[equation]{}[section]{}
\
We compute the SUSY-breaking soft terms in a magnetized D7-brane model with MSSM-like spectrum, under the general assumption of non-vanishing auxiliary fields of the dilaton and Kähler moduli. As a particular scenario we discuss SUSY breaking triggered by ISD or IASD 3-form fluxes.
Introduction
============
Recently a simple intersecting D-brane model was proposed with massless chiral spectrum close to that of the MSSM [@cim1]. In this model the SM fields lie at the intersections of four sets of D6-branes wrapping an (orientifolded) toroidal compactification of Type IIA string theory. The same model may be equivalently described in terms of different T-dual configurations, e.g. in terms of a Type IIB orientifold with (magnetized) D9-branes and D5-branes [@cim2]. Recently [@ms] it has been shown how this type of D-brane configurations may be promoted to a fully 1 SUSY tadpole-free model (see also [@Cvetic:2004nk]) by embedding it into a $\Z_2\times \Z_2$ Type IIB orientifold along the lines suggested in [@csu]. A number of results for the effective Lagrangian in such type of D-brane models is known by now. The Yukawa couplings among chiral fields were computed in [@cim1; @cvetic; @abel; @cim2] and other aspects of the effective action may be found in [@Cim1; @Cim2; @gauge; @lmrs; @lrs1; @jl; @lrs2; @kn]. For an up-to-date review, see [@blum].
One interesting point to address is the structure of possible SUSY-breaking soft terms in this model. It has been recently realized that fluxes of antisymmetric R-R and NS-NS fields in Type IIB orientifolds may provide a source of such terms [@Grana; @ciu1; @ggjl; @ciu2; @jl; @iflux; @msw]. It was also realized [@ciu1] that SUSY breaking from imaginary self-dual (ISD) 3-form fluxes correspond to a non-vanishing vev for the auxiliary field of the overall modulus $T$ and imaginary anti-self-dual (IASD) correspond to a non vanishing vev for the auxiliary field of the complex dilaton $S$. On the other hand, a possible phenomenological application of these ideas was proposed in [@iflux]. In particular, if one assumes that the SM particles correspond to geometric D7-brane moduli, a simple set of SUSY-breaking soft terms may be shown to arise from ISD fluxes.
In this article we would like to present explicit results for the SUSY breaking soft terms in the MSSM-like model of ref. [@cim1] as a function of the vevs of the auxiliary fields of the Kähler moduli $T_i$ and/or the complex dilaton $S$. As particular examples we consider vevs induced by ISD and/or IASD 3-form fluxes. We construct the model [@cim1] in terms of 3 stacks of intersecting D7-branes, one of them containing a constant magnetic field (leading to chirality and family replication). We then use the effective supergravity Lagrangian approach in order to obtain soft terms, as in ref. [@soft; @bim2].
A previous detailed analysis of these soft terms including the effect of the non-vanishing magnetic flux was presented by Lüst, Reffert and Stieberger in ref. [@lrs2], based on the Kähler metrics of matter fields computed in ref. [@lmrs; @lrs1]. These included the effect of magnetic fluxes. Some phenomenological analysis of those results was described in [@kane]. Soft terms in the T-dominance case were briefly discussed in [@iflux] following [@imr], in which the effect of magnetic fluxes was not included. In the present paper we revisit previous results taking into account a proper normalization of the matter fields. We also make use of the Kähler metrics for chiral fields discussed in ref. [@lmrs; @lrs1]. Including a factor implied by the analysis of [@lmrs] the SUSY-breaking soft terms simplify considerably. One of the motivations of the present work was to find the connection with the analogous results obtained in [@imr] in the absence of magnetic fluxes. Indeed, we find that in the limit of diluted magnetic fluxes those results are recovered.
It is known that NS-NS and R-R fluxes on [*toroidal* ]{} settings induce soft terms on D3-brane fields of order $M_s^2/M_{Pl}$, so that one can obtain a hierarchy of scales by lowering the string scale [@ciu1]. However, in the case of intersecting D7-branes, as in the model at hand, one cannot lower the string scale without making the SM gauge couplings unacceptably small. Therefore, in toroidal/orbifold models with intersecting D7-branes the fluxed-induced soft terms are typically of order the string scale. This fact is due to the simplicity of toroidal compactifications in which the compact space is flat and the fluxes are distributed uniformly. In a generic Calabi-Yau (CY) compactification this is not going to be the case and there may be regions in the CY in which fluxes are concentrated and others in which fluxes are diluted. This possibility was considered e.g. in [@gkp; @kklt] in order to obtain hierarchies. Thus, for generic CY compactifications the size of soft terms will actually depend on the detailed geometry of the fluxes in the CY.
The local set of branes leading to a MSSM-like spectrum introduced in [@cim1] is nevertheless likely to be more generic than the toroidal setting in which it was first proposed (see e.g. [@bbkl]). In particular, it has recently been shown [@schellekens] that there are many thousands of models with the 4-stacks of branes structure of the model in [@cim1] (these are labeled Type-4 models in ref. [@schellekens]). Therefore, one may expect to obtain this MSSM structure in CY orientifold models beyond the toroidal setting. In these more general models the size of induced soft terms may not be tied to the string scale and could be much lower. In our effective field theory analysis below we will not commit ourselves to a particular scale for the soft terms. Instead, following [@soft; @bim2], we will assume that the effect of SUSY-breaking is encoded in non-vanishing vevs for the auxiliary fields of the complex dilaton and Kähler moduli. However, we also discuss the case in which the source of SUSY-breaking are constant IASD or/and IASD 3-form fluxes, which correspond to a particular choice for the auxiliary fields.
A MSSM-like model from magnetized D7-branes {#sec:setup}
===========================================
We will construct the model in [@cim1] in terms of three sets of intersecting Type IIB D7-branes (see e.g.[@ms]). We consider type IIB string theory compactified on a factorized six-torus $\T^6= \otimes_{i=1}^{3} \T^2_i$. We will further do an orientifold projection by $\Omega (-1)^F I_6$, $\Omega $ being the world-sheet parity operator and $I_6$ a simultaneous reflexion of the six toroidal coordinates. We also include sets of ${\rm D}9_a$-branes and allow for possible constant magnetic fluxes across any of the three 2-tori \_[\^2\_i]{} F\_a\^i = n\_a\^i , \[nmdef\] where $F_a^i$ is the world-volume magnetic field. For each group of branes the state of magnetization is thus characterized by the integers $(n_a^i, m_a^i)$, where $m_a^i$ is the wrapping number and $n_a^i$ is the total magnetic flux. It is useful to introduce the angles \_a\^i = 2F\_a\^i= \[psidef\] where $(2\pi)^2A_i$ is the area of the $\T^2_i$. The magnetized ${\rm D}9_a$-branes [@bachas; @bgkl; @aads] preserve the same supersymmetry of the orientifold planes provided that [@blt; @cu] \_[i=1]{}\^[3]{} \_a\^i = 2 [mod]{} 2 . \[susycon\] Note that in this scheme lower dimensional branes are described setting $m_a^i=0$ for all $i$ transverse to the brane. For example, a D3-brane has $(n_a^i,m_a^i)=(1,0)$, $i=1,2,3$. Notice in particular that the D3-brane satisfies (\[susycon\]). T-duality along the horizontal direction in each $\T^2_i$ gives the dual picture of D6-branes at angles. For example, for square $\T^2_i$, $A_i=R_{ix} R_{iy}$, and the dual angle is $\vartheta_a^i = \arctan (n_a^i R_{ix}/m_a^i R_{iy})$.
In order to reproduce the structure of the MSSM-like model of ref. [@cim1; @cim2] one introduces three sets of $\D7_i$-branes $i=1,2,3$ which are characterized by being transverse to the $i$-th 2-torus. In particular the relevant magnetic data is
------------------------ ----------------- ----------------- ----------------- -------------------------------------------------------
Branes $(n_a^1,m_a^1)$ $(n_a^2,m_a^2)$ $(n_a^3,m_a^3)$ $(\psi_a^1, \psi_a^2, \psi_a^3)$
\[0.2ex\] ${\rm D}7_1$ (1,0) $(g,1)$ $(g,-1)$ $(\displaystyle{\frac{\pi}2}, \pi\d_2, \pi -\pi\d_3)$
${\rm D}7_2$ (0,1) $(1,0)$ $(0,-1)$ $(0, \displaystyle{\frac{\pi}2}, \pi)$
${\rm D}7_3$ (0,1) $(0,-1)$ $(1,0)$ $(0, \pi, \displaystyle{\frac{\pi}2})$
------------------------ ----------------- ----------------- ----------------- -------------------------------------------------------
§ . \[sbs\] where $\pi\d_i=\arctan (\ap g/A_i)$. We will take $\D7_1$-branes to come in four copies so that generically the associated gauge group will be $U(4)$. Branes $\D7_2$ and $\D7_3$ come only in one copy and are located on top of the orientifold plane at the origin so that they give rise to a gauge group $Sp(2)\times Sp(2)\simeq SU(2)\times SU(2)$. Altogether the overall gauge group is $U(4)\times SU(2)\times SU(2)$. It may be shown that the $U(1)$ (which corresponds to $(3B+L)$) is anomalous and becomes massive in the usual way by the Stückelberg mechanism. In fact, one can further make the breakings $SU(4)\rightarrow SU(3)_c\times U(1)_{B-L}$ and $SU(2)_R\rightarrow U(1)_R$ by e.g. Wilson lines. Thus the final gauge group is just $SU(3)\times SU(2)\times U(1)_Y\times U(1)_{B-L}$. The final chiral spectrum is displayed in Table 1 for the choice $g=3$ which leads to three quark/lepton generations.
Intersection Matter fields Rep. $Q_{B-L}$ Y
--------------- --------------- ----------------- ----------- ------
$\D7_1-\D7_2$ $Q_L$ $3(3,2)$ 1 1/6
$\D7_1-\D7_3$ $U_R$ $3({\bar 3},1)$ -1 -2/3
$\D7_1-\D7_3$ $D_R$ $3({\bar 3},1)$ -1 1/3
$\D7_1-\D7_2$ $E_L$ $3(1,2)$ -1 1/2
$\D7_1-\D7_3$ $E_R$ $3(1,1)$ 1 -1
$\D7_1-\D7_3$ $N_R$ $3(1,1)$ 1 0
$\D7_2-\D7_3$ $H$ $(1,2)$ 0 1/2
$\D7_2-\D7_3$ ${\bar H}$ $(1,2)$ 0 -1/2
: Chiral spectrum of the MSSM-like model.[]{data-label="mssm"}
The ${\rm D}7_2$ and ${\rm D}7_3$ do not have magnetic flux and do verify the supersymmetric condition (\[susycon\]). However, for the case of the ${\rm D}7_1$-branes, with opposite magnetic fields turned on in the second and third $\T^2$, to be supersymmetric we need to impose A\_2=A\_3=A , \[susya\] so that $\d_2=\d_3=\d$ and = . \[deldef\] Clearly, the condition (\[susycon\]) also guarantees that any two sets of branes preserve a common supersymmetry. Notice that the relative angles \^i\_[ab]{} = \^i\_b - \_a\^i \[reltheta\] automatically satisfy $\sum_{i} \theta^i_{ab} = 0 \, {\rm ºmod} \, 2\pi$.
Departures from the equality $A_2=A_3$ may be shown [@cvetic; @Cim1; @Cim3; @cgqu] to correspond to a non-vanishing Fayet-Iliopoulos term for the anomalous $U(1)_{3B+L}$. In fact, for $A_2=(A_3\ +\ \epsilon)$ and small $\epsilon $ one finds [@Cim1; @Cim3] \_[FI]{} = \[fi\] where the D-term potential is of the form V\_[FI]{}(\_n) = [1 2 g\_[U(1)]{}\^2]{} (\_n q\_n|\_n|\^2 + \_[FI]{})\^2. \[potentialFI2\] and $\phi_n$ runs over squarks and sleptons. Left-handed and right-handed chiral fields have positive and negative $U(1)_{3B+L}$ charge respectively so that a non-vanishing $\epsilon$ may induce further symmetry breaking. Note that this potential as it stands does not prefer $\xi_a = 0$ (and hence the SUSY condition $A_2=A_3$) as sometimes claimed in the literature, since a non-vanishing $\xi_a $ may always be compensated with a vev for a right-handed scalar field (e.g. the right-handed sneutrino). On the other hand, in the presence of soft masses for the chiral fields, as shown to appear in the next sections, a vanishing FI-term is dynamically preferred and so is the SUSY condition $A_2=A_3$. As we will see, this implies in turn the unification of $SU(2)_L$ and $SU(2)_R$ gauge couplings.
Let us finally comment that, as it stands, this brane configuration has R-R tadpoles so some additional (‘hidden’) brane system should be added. This can be done in a way consistent with 1 SUSY if we embed this brane configuration in a $\Z_2\times \Z_2$ orientifold [@csu] as recently shown in [@ms] (although in this case one cannot do the breaking $SU(2)_R\rightarrow U(1)_R$ via Wilson lines [@ms]). Since we are only interested in the structure of soft terms for the MSSM fields we will not deal here with these global issues of the compactification. Our results will still hold for those global generalizations.
Massless fields and effective supergravity action {#sec:action}
=================================================
Let us now turn to the effective supergravity action in this model. We will compile general formulas for the Kähler potential, matter metrics and gauge kinetic function and we will apply them to the specific D-brane model at hand.
We begin with the field content. In the closed string sector, in addition to the supergravity multiplet, one has the dilaton $S$ plus the Kähler and complex structure moduli. We use conventions such that the complex dilaton is given by S= e\^[-\_[10]{}]{} + i a\_0 , \[sdef\] where $a_0$ is the R-R 0-form. Recall that the string coupling constant is $g_s=e^{\phi_{10}}$.
For the metric moduli we will restrict for simplicity here to the diagonal fields $U_j$, $T_j$, $j=1,2,3$. Note that in any case the off-diagonal fields would not be present in a $\Z_2\times \Z_2$ embedding of the present model. To be more concrete, let the $\T^2_j$ lattice vectors be denoted $e_{jx}$, $e_{jy}$. Then, the geometric toroidal moduli are \_j & = & 1[e\^2\_[jx]{}]{}(A\_j + i e\_[jx]{} e\_[jy]{})\
\_j & = & A\_j + i a\_j , \[gmod\] where the axions $a_j$ arise from the R-R 4-form. In type IIB, the toroidal complex structure $\tau_j$ is equal to the moduli field $U_j$ that appears in the 4 supergravity action. However, the correct Kähler moduli field $T_j$ is not $\rho_j$. One way to see this is to realize that the gauge coupling squared of [*unmagnetized*]{} ${\rm D}7_j$-branes should be equal to $2\pi/\re T_j$. Then, since, e.g. a ${\rm D}7_1$ wraps $\T^2_2$ and $\T^2_3$, from the Born-Infeld action it follows that $\re T_1 = e^{-\phi_{10}} A_2 A_3/\a^{\prime \, 2}$. In general T\_i = e\^[-\_[10]{}]{} + i a\_i ; j=k=i . \[tdef\] For later convenience we define s=S+ ; t\_i=T\_i + \_i ; u\_i=U\_i + \_i . \[minimod\] The 4 gravitational coupling is $G_N=\kappa^2/8\pi$ where \^[-2]{} = = e\^[-2\_[10]{}]{} = . \[mplanck\] It is also useful to introduce the T-duality invariant four-dimensional dilaton, namely \_4 = \_[10]{} - (A\_1 A\_2 A\_3/\^[ 3]{}) . \[phi4\] Notice that $\kappa^{-2} = e^{-2\phi_4}/\pi \ap$. The string scale is $M_s=1/\sqrt{\ap}$.
Open strings give rise to charged fields. We call ‘untwisted’ the states corresponding to open strings beginning and ending on the same stack of branes, whereas ‘twisted’ refers to the chiral fields lying at the intersection of two different stacks of D7-branes. For the content of branes in (\[sbs\]), and assuming supersymmetry is preserved, the untwisted sectors ${\rm D}7_i$-${\rm D}7_i$, $i=1,2,3$, give a gauge multiplet of a group $G_i$ and 3 massless chiral multiplets, denoted $C_j^{7_i}$, $j=1,2,3$, transforming in the adjoint of $G_i$. The $C_j^{7_i}$ are the ${\rm D}7_i$-brane moduli, $C_i^{7_i}$ gives the position of the brane in the transverse $\T^2_i$ whereas $C_j^{7_i}$, $j\not= i$, correspond to Wilson lines on the two internal complex dimensions parallel to the $\D7_i$-brane. From the twisted sectors ${\rm D}7_i$-${\rm D}7_j$ there are only chiral massless multiplets, denoted $C^{7_i 7_j}$, transforming as bifundamentals of $G_i \times G_j$.
The low-energy dynamics of the massless fields is governed by a 4, 1 supergravity action that depends on the Kähler potential, the gauge kinetic functions and the superpotential. In particular, the F-part of the scalar potential is V = e\^[\^2 K]{} , \[sugrapot\] where $D_A W = \partial_A W + \kappa^2 \partial_A K W$ and $K^{\bar A B}$ is the inverse of $K_{\bar A B}= \partial_{\bar A} \partial_B K$. Recall that the auxiliary field of a chiral superfield $\Phi_A$ is |F\^[|A]{} = \^2 e\^[\^2 K/2]{} K\^[|A B]{} D\_BW . \[fterm\] We now describe the functions $K(\phi; \bar \phi)$, $f_i(\phi)$ and $W(\phi)$ in our setup, in which $\Phi_A = \{M, C_I\}$, with $M=\{S, T_i, U_i\}$ and $C_I=\{C_j^{7_i}, C^{7_i 7_j} \}$.
Kähler potential
----------------
The Kähler potential has the structure K = K(M,|M) + \_[I,J]{} K\_[I |J]{}(M,|M) C\_I |C\_J + \_[I,J]{} \[ Z\_[IJ]{}(M,|M) C\_I C\_J + c.c.\] + . \[kpot\] The contribution of the closed string moduli is \^2 K(M,|M) = -s -\_[i=1]{}\^3 t\_i -\_[i=1]{}\^3 u\_i . \[kmodu\] Below we describe in detail the Kähler metrics of matter fields.
The $\tilde K_{I \bar J}$ for unmagnetized branes were deduced in [@imr] using T-duality arguments. For generic magnetized branes they have been obtained in [@lmrs; @lrs1] from a computation of string scattering amplitudes. These metrics vanish when $J \not= I$. Below we present the diagonal entries for all possible cases with the brane content of (\[sbs\]). To streamline notation we write $\tilde K_{i,j \bar \jmath}= \tilde K_{C_j^{7_i} \bar C_j^{7_i}}$ in untwisted sectors, and $\tilde K_{ij,C \bar C}=
\tilde K_{C^{7_i 7_j} \bar C^{7_i 7_j}}$ in twisted sectors. For untwisted fields we have
${\rm D}7_3$-${\rm D}7_3$ (untwisted, unmagnetized) \^2 K\_[3,1 |1]{} = 1[u\_1 t\_2]{} ; \^2 K\_[3,2 |2]{} = 1[u\_2 t\_1]{} ; \^2 K\_[3,3 |3]{} = 1[u\_3 s]{} . \[stst\]
${\rm D}7_2$-${\rm D}7_2$ (untwisted, unmagnetized) \^2 K\_[2,1 |1]{} = 1[u\_1 t\_3]{} ; \^2 K\_[2,2 |2]{} = 1[u\_2 s]{} ; \^2 K\_[2,3 |3]{} = 1[u\_3 t\_1]{} . \[sdsd\]
${\rm D}7_1$-${\rm D}7_1$ (untwisted, magnetized) \^2 K\_[1,1 |1]{} = 1[u\_1 t\_1 s]{}(g\^2 s + t\_1) ; \^2 K\_[1,2 |2]{} = 1[u\_2 t\_2]{} ; \^2 K\_[1,3 |3]{} = 1[u\_3 t\_3]{} . \[susu\] To obtain these results we start from the general expressions in the geometric basis given in [@lrs1]. In our notation these are \^2 K\_[1,1 |1]{} & = & |A\_3 m\_1\^3 + i n\_1\^3| |A\_2 m\_1\^2 + i n\_1\^2| ,\
\^2 K\_[1,2 |2]{} & = & | | , \[sususb\]\
\^2 K\_[1,3 |3]{} & = & | | . We then substitute the values of the $(n_1^i, m_1^i)$ given in (\[sbs\]), use the supersymmetry condition (\[susya\]) and also \^[ 2]{} t\_1 = s A\^2 \[agf\] that follows from (\[tdef\]). When $m_1^i=0$, (\[sususb\]) gives the metric of a ${\rm D}3$-${\rm D}3$ sector. When $n_1^i=0$, we just obtain the metric of unmagnetized ${\rm D}7_1$-${\rm D}7_1$.
For the metrics of twisted fields one has
${\rm D}7_2$-${\rm D}7_3$ (twisted, unmagnetized) \^2 K\_[23,C |C]{} = 1[(u\_2 u\_3 s t\_1)\^[1/2]{}]{} \[sdst\] .
${\rm D}7_1$-${\rm D}7_2$ (twisted, magnetized) \^2 K\_[12,C |C]{} = 1[(s t u\_1)\^[1/2]{} u\_2\^[1/2 + ]{} u\_3\^[1-]{}]{} , \[susd\] where $t=t_2=t_3$. Notice that $\d$ depends implicitly on $s$ and $t_1$. From (\[deldef\]) and (\[agf\]), = g (s/t\_1)\^[1/2]{} . \[tanpid\] Observe that $0 \leq \d < \oh$. To derive (\[susd\]) we start from \^2 K\_[12,C |C]{} = e\^[\_4]{} \_[j=1]{}\^3 u\_j\^[-\_j]{} , \[kori\] where the $\nu_j$, computed from $\hat \nu_j= \theta^j_{12}/\pi$, are such that $0 \leq \nu_j < 1$ and $\nu_1 + \nu_2 + \nu_3 = 2$. To determine the $\nu_j$, observe first that $\hat \nu_1 + \hat \nu_2 + \hat \nu_3 = 0$. Assuming $\hat \nu_j \not=0$ then implies that one or two of the $\hat \nu_j$ are negative. In the first case start instead from $\hat \nu_j= \theta^j_{21}/\pi$. Now two of the $\hat \nu_j$ are negative by construction. Finally, define $\nu_j=1 + \hat \nu_j$ if $\hat \nu_j$ is negative, otherwise $\nu_j=\hat \nu_j$. In this case $\nu=(\oh, \oh + \d, 1-\d)$.
To arrive at (\[susd\]) we use (\[tanpid\]) and the relation = . \[gprop\] In (\[susd\]) we can take the limit $\d \to 0$ and recover the metric of unmagnetized ${\rm D}7_1$-${\rm D}7_2$, provided we drop $u_3$ that would have exponent -1. Using again (\[tanpid\]) and (\[gprop\]) we can also take the limit $\d \to \oh$ and, dropping $u_2$ now with exponent -1, retrieve the metric of ${\rm D}3$-${\rm D}7_2$, namely $\kappa^2 \tilde K_{C \bar C} = (t_1 t_3 u_1 u_3)^{-1/2}$. The fact that moduli $u_j$ with would be exponent -1 do not appear in the metric also occurs in twisted sectors of heterotic orbifolds [@dkl].
Eq. (\[kori\]), including the prefactor $e^{\phi_4}$, follows putting together results found in [@lmrs]. In the field basis this prefactor can be recast as $2(s t_1 t_2 t_3)^{-1/4}$. The square root of Gamma functions, with the arguments as shown in (\[kori\]), is determined by the differential equation that dictates the dependence on the Kähler moduli [@lmrs]. Finally, for the remaining twisted sector the metric is
${\rm D}7_1$-${\rm D}7_3$ (twisted, magnetized) \^2 K\_[13,C |C]{} = 1[(s t u\_1)\^[1/2]{} u\_2\^[1-]{} u\_3\^[1/2+ ]{}]{} . \[sust\]
Gauge kinetic functions
-----------------------
The gauge kinetic functions $f_i$ for the groups arising in the ${\rm D}7_i$-${\rm D}7_i$ sectors are f\_1 = T\_1 + g\^2 S ; f\_2 = T\_2 ; f\_3= T\_3 . \[efes\] In general [@Cim1; @lmrs], f\_i = \_[j = i]{} |m\_i\^j A\_j + in\_i\^j| . \[realfgen\] Substituting the values of the $(n_i^j, m_i^j)$ given in (\[sbs\]) leads to (\[efes\]).
Note that if the SUSY condition (\[susya\]) is verified, one has $\re f_2= \re f_3$ and the $SU(2)_L$ and $SU(2)_R$ gauge couplings are unified. Note also that if the complete model has additional branes (as in e.g., [@ms]) the SUSY conditions may involve in general also the area $A_1$ of the first torus and imply further unification constraints.
Concerning the axions, one can check that the linear combination $(a_2-a_3)$ becomes massive combining with the anomalous $U(1)_{3B+L}$ gauge boson through the Stückelberg mechanism. On the other hand, one can also check that the linear combination $(9a_0-a_1)$ has axionic couplings with the QCD gauge bosons. This may help in solving the strong CP problem.
Further inspection of the $f_i$ reveals an interesting bound on the string coupling constant in the D-brane model. From (\[tanpid\]) and the $SU(4)$ gauge coupling $\a_1$ we deduce the relation \^2 = . \[gsbound\] This has a number of consequences. To have three generations, $g=3$. Hence, the above relation implies $\a_1 \leq g_s/18$. For $\a_1(M_s) \sim 1/24$ this is consistent with $g_s < 1$. However, we already see that to get values of coupling constants of the order of (extrapolated) known gauge couplings, the string coupling constant $g_s$ approaches the non-perturbative regime. In fact, in the present specific toroidal model in addition to the chiral spectrum there are massless chiral adjoints which will make the gauge interactions asymptotically non-free. Thus, the $\alpha_i$ will be larger than $\sim 1/24$ and hence the above bound will give $g_s >O(1)$. A second (related) implication of eq.(\[gsbound\]) is that in order to accommodate gauge couplings consistent with experiment but still stay within the string perturbative regime with $g_s<1$, the value of $\delta$ (and hence the magnetic flux) will be substantial. Thus, e.g. for $\alpha _1=1/24$ one finds $\pi\delta \sim 60^0$. All this tells us that in order to have consistency with the observed values of gauge couplings in this class of intersecting D7-brane models we would probably need to go to a non-perturbative F-theory (or perhaps simply non-toroidal) versions of them. Constructing such class of F-theory models would be a rather non-trivial task. In what follows we will not further deal with these issues and simply assume that we still remain in a perturbative regime, hoping that a proper fit of experimentally measured gauge couplings does not substantially modify our soft term results.
Superpotential
--------------
The superpotential can be written as W(M, C\_I) = W(M) + \_[I,J]{} \_[IJ]{}(M) C\_I C\_J + 16 \_[I,J,L]{} Y\_[IJL]{}(M) C\_I C\_J C\_L + . \[superw\] For the brane content of (\[sbs\]), the cubic couplings allowed are of the form [@bl] C\_1\^[7\_i]{} C\_2\^[7\_i]{} C\_3\^[7\_i]{} ; d\_[ijk]{} C\_i\^[7\_j]{} C\^[7\_j 7\_k]{} C\^[7\_j 7\_k]{} ; C\^[7\_1 7\_2]{} C\^[7\_2 7\_3]{} C\^[7\_3 7\_1]{} , \[cubic\] where $d_{ijk}=1$ if $i\not=j\not=k$, otherwise $d_{ijk}=0$. Note that for the case at hand the first type of couplings in (\[cubic\]) correspond to standard 4 Yukawa couplings among adjoints. Concerning the second type of couplings, only one of type $XH{\bar H}$ is present in the model. Here $X$ is a linear combination of $C_1^2,C_1^3$, the latter corresponding to Wilson line chiral fields of the branes $\D7_2$, $\D7_3$ in the first complex plane. Note that a vev for $X$ would render the Higgs multiplets massive so $X$ behaves as a $\mu $-term in the effective Lagrangian[^1]. Finally, the third type of superpotential couplings corresponds to the regular Yukawa couplings between the chiral generations and the Higgs multiplets. All in all, the perturbative superpotential among the chiral open string multiplets in this model has the general expression (we write it for the extended gauge group $SU(4)\times SU(2)_L\times SU(2)_R$ for simplicity of notation) W\_[Yukawa]{} = \_[i]{}C\_1\^[7\_i]{} C\_2\^[7\_i]{} C\_3\^[7\_i]{} +X +\_[,]{} h\_ L\_R\_ , \[yukis\] where $\ch$, $L_\a$ and $R_\b$ are the Higgs and chiral fermions transforming as $(1,2,2)$, $(4,2,1)$ and $({\bar 4},1,2)$ respectively ($\a,\b=1,2,3$ are generation labels).
The superpotential couplings $h_{\a\b}$ for this toroidal model have been computed in [@cim1; @cim2] and are given by h\_ = (3\_2, 3U\_2) (3\_3, 3U\_3 ) , \[ahuevo\] where $\z_i$ are certain combinations of singlet $C_i^{7_j}$ fields (see [@cim1; @cim2]) and $\vt $ are Jacobi theta functions. For our purposes the only relevant thing to point out is that these superpotential couplings only depend on the complex structure moduli $U_2$, $U_3$, and not on the Kähler moduli nor the dilaton.
In principle we can use the above results to compute soft terms under the general assumption that the auxiliary fields of the moduli and dilaton are non-vanishing, in the spirit of refs.[@soft; @bim2]. We would not need then to specify the microscopic source of this values. On the other hand, lately we have learned that such microscopic source of SUSY-breaking may be provided by fluxes in Type II theory. Hence in addition to the above perturbative chiral couplings, a moduli dependent superpotential $\hat W(M)$ may be present. In particular, it is known that antisymmetric R-R and NS-NS fluxes $F_3,H_3$ generate a superpotential [@gvw] W\_f = G\_3 \[wflux\] that depends on $S$ and the $U_i$. Here $G_3 = F_3 - i S H_3$ and $\Omega$ is the holomorphic (3,0) form of $\T^6$. Besides, there may be non-perturbative interactions (like e.g. those from gaugino condensation) giving rise to a generic superpotential $W_{np}$. Thus the total moduli-dependent superpotential will have the general form W(M) = W\_f(S,U\_i) + W\_[np]{}(S,U\_i,T\_i) . \[wmodu\] In the absence of $W_{np}$ the equations of motion require $G_3$ to be imaginary self-dual (ISD), meaning that $G_3$ is a combination of (0,3) and (2,1) fluxes [@gkp]. In this case $D_S W_f =0$ and $D_{U_i} W_f =0$ but $D_{T_i} W_f \not=0$ because $W_f$ does not depend on the $T_i$ and a (0,3) piece in $G_3$ generates $W_f \not= 0$.
Soft Terms {#sec:soft}
==========
Armed with all the above data for the low-energy effective action we can now compute the SUSY-breaking soft terms. To this purpose we will follow the approach in [@soft; @bim2] and assume that the auxiliary fields $F_{T_i},F_S$ of the Kähler moduli and the complex dilaton acquire non-vanishing expectation values. We will later consider the particular case in which the microscopic origin of such non-vanishing values is provided by ISD and IASD three-form fluxes. The standard results for the normalized soft parameters may be found e.g. in [@bim2] and read M\_i & = & 1[2 f\_i]{} F\^M \_M f\_i ,\
m\^2\_I & = & m\^2\_[3/2]{} + V\_0 - \_[M,N]{} |F\^[|M]{} F\^N \_[|M]{} \_N (K\_[I |I]{}) , \[softt\]\
A\_[IJL]{} & = & F\^M \[ K\_M + \_M (Y\_[IJL]{}) - \_M (K\_[I |I]{} K\_[J |J]{} K\_[L |L]{}) \] . Here $V_0$ is the vev of the scalar potential and the gravitino mass is m\_[3/2]{}= e\^[\^2 K/2]{} |W| . \[mgravi\] Note that, as pointed out above, the $Y_{IJL}$ superpotential couplings do not depend on the $S$ and $T_i$ fields, so that the second contribution to $A_{IJL}$ in (\[softt\]) vanishes identically. The expressions (\[softt\]) are valid when $\tilde K_{I \bar J} \propto \delta_{I\bar J}$ which is our case.
The vevs of the auxiliary fields are conveniently parametrized as [@bim2] F\^S & = & 3 s C m\_[3/2]{} e\^[-i\_S]{}\
F\^[T\_i]{} & = & 3 t\_i \_i C m\_[3/2]{} e\^[-i\_i]{} , \[vevstino\] where the goldstino angle $\theta$ and the $\eta_i$, with $\sum_i \eta_i^2 =1$, control whether $S$ or the $T_i$ dominate SUSY breaking. We further assume that $F^{U_i}=0$. Then, substituting in (\[sugrapot\]) gives C\^2= 1 + . \[cconst\] We will now present the soft terms for the D-brane model.
Masses for gauginos of the group $G_i$ arising in the $\D7_i$-$\D7_i$ sector are denoted $M_i$ or $M_{G_i}$. Then, M\_1 & = & M\_[SU(4)]{} = 3 C m\_[3/2]{} ,\
M\_2 & = & M\_[SU(2)\_L]{} = 3 C m\_[3/2]{} e\^[-i\_2]{} \_2 ,\
\[gmasses\] M\_3 & = & M\_[SU(2)\_R]{} = 3 C m\_[3/2]{} e\^[-i\_3]{} \_3 .
.
The Higgs multiplets appear in a twisted sector (unmagnetized) from $\D7_2$-$\D7_3$ intersections. One finds then the simple result m\^2\_[H]{} = m\^2\_[3/2]{} + V\_0 - 32 C\^2 m\^2\_[3/2]{} (\^2 + \_1\^2 \^2 ) . \[smasstu\] On the other hand, the chiral quark/lepton fields appear on magnetized and twisted sectors $\D7_1$-$\D7_2$ and $\D7_1$-$\D7_3$. In particular, all three generations of left-handed quarks and leptons come from the $\D7_1$-$\D7_2$ sector and have soft masses m\^2\_[L\_]{} & = & m\^2\_[3/2]{} + V\_0 - 32 C\^2 m\^2\_[3/2]{} (\^2 + \_3\^2 \^2 ) + C\^2 m\^2\_[3/2]{} \^2 2B\_1() ||\^2\
& + & C\^2 m\^2\_[3/2]{} 2 , \[smasstm\] where we have defined & = & e\^[-i\_S]{} - e\^[-i\_1]{} \_1 ,\
B\_0() & = & \_0(1-) - \_0( -) ,\
\[atajo\] B\_1() & = & \_1(1-) - \_1( -) . Here $\psi_0(z)= \Gamma^\prime(z)/\Gamma(z)$ and $\psi_1(z)= \psi_0^\prime(z)$.
The three generations of right-handed quarks and leptons come from the magnetized and twisted $\D7_1$-$\D7_3$ sector. The soft masses $m^2_{R_{\a}}$ have the same form as (\[smasstm\]) except for the replacements $u_2\leftrightarrow u_3$, $\eta_2\leftrightarrow \eta_3$. Note that the limit $\d \to 0$ just yields the scalar mass for unmagnetized twisted fields like the Higgs multiplet, as it should. This also agrees with the results obtained for unmagnetized branes in [@imr].
As a check on the results we can use \_0( -) = \_0( +) - , ; \_1( -) = \^2 \^2 - \_1( +) , \[psiprops\] to take the limit $\d \to \oh$. This corresponds to infinite magnetic flux in the 2nd and 3rd torus. In this limit the magnetized $\D7_1$-brane behaves as a D3-brane. Thus, taking $\d \to \oh$ in (\[smasstm\]) should give the mass squared parameter of a scalar in a ${\rm D}3$-${\rm D}7_2$ type of sector. In this way we obtain m\^2\_[C\^[37\_2]{}]{} = m\^2\_[3/2]{} + V\_0 - 32 C\^2 m\^2\_[3/2]{} (1- \_2\^2) \^2 , \[mass37\] in accordance with the expected outcome [@imr].
.
The coupling $H L_{\alpha} R_{\beta}$ is of type $C^{7_1 7_2} C^{7_2 7_3} C^{7_3 7_1}$. The trilinear term turns out to be A\_[HLR]{} = 2 C m\_[3/2]{} . \[atwi\] When $\d \to 0$, the result agrees with that in [@imr]. One can also take the limit when $\d \to \oh$ which should correspond to a coupling of type $C^{3 7_2} C^{7_2 7_3} C^{3 7_3}$. It indeed follows that A\_[C\^[3 7\_2]{} C\^[7\_2 7\_3]{} C\^[3 7\_3]{}]{} = 2 C m\_[3/2]{} , \[tri37\] also in agreement with [@imr].
The other relevant trilinear coupling involving SM fields is of the form $C_1^{7_j} C^{7_2 7_3} C^{7_2 7_3}$, $j=2,3$. For those couplings one gets trilinear terms $A=-M_j$. In our case the only such coupling is $XHH$. Then, A\_[XHH]{} = - 3 C m\_[3/2]{} e\^[-i\_2]{} \_2 = - M\_[SU(2)]{} . \[trili\]
.
Together with the chiral MSSM-like spectrum, there are three chiral multiplets in the adjoint of the gauge group coming from untwisted ${\rm D}7_j$-${\rm D}7_j$ sectors. We set $\Phi_{ij}=C_i^{7_j}$ and recall that $\Phi_{jj}$ parametrizes the position of each $\D7_j$-brane in transverse space, whereas the $\Phi_{ij}$ correspond to Wilson lines on the two complex dimensions inside the $\D7_j$-brane worldvolume. For the unmagnetized $\Phi_{i2}$ we find m\^2\_[12]{} & = & m\^2\_[3/2]{} + V\_0 - 3 C\^2 m\^2\_[3/2]{} \_3\^2 \^2 ,\
m\^2\_[22]{} & = & m\^2\_[3/2]{} + V\_0 - 3 C\^2 m\^2\_[3/2]{} \^2 ,\
\[smassu\] m\^2\_[32]{} & = & m\^2\_[3/2]{} + V\_0 - 3 C\^2 m\^2\_[3/2]{} \_1\^2 \^2 . For the $\Phi_{i3}$ there are analogous results. For the magnetized $\Phi_{i1}$ the masses are instead m\^2\_[11]{} & = & m\^2\_[3/2]{} + V\_0 - 3 C\^2 m\^2\_[3/2]{} (\^2 + \_1\^2 \^2 ) + |M\_1|\^2 ,\
m\^2\_[21]{} & = & m\^2\_[3/2]{} + V\_0 - 3 C\^2 m\^2\_[3/2]{} \_2\^2 \^2 ,\
\[smassum\] m\^2\_[31]{} & = & m\^2\_[3/2]{} + V\_0 - 3 C\^2 m\^2\_[3/2]{} \_3\^2 \^2 . In all cases there is a sum rule m\^2\_[1j]{} + m\^2\_[2j]{} + m\^2\_[3j]{} = 2V\_0 + |M\_j|\^2 º . \[sumrule\] Besides scalar masses there are also trilinear terms associated to the superpotential couplings $\Phi_{1i} \Phi_{2i} \Phi_{3i}$. They are given by $A_i = -M_i$.
.
Clearly, the structure of soft terms strongly depends on which auxiliary fields, either $F_{T_i}$ or $F_S$, dominate SUSY-breaking. As a general property one must emphasize that in all cases the results for scalar masses are flavor independent. The trilinear terms involving squarks and sleptons are also flavor diagonal. This is an interesting property which was not always present in heterotic models and is welcome in order to suppress too large flavor changing neutral currents (FCNC). Concerning gaugino masses, they are equal for the $SU(2)_L\times SU(2)_R$ sector of the theory but different for the $SU(3+1)$ gauginos which get an extra contribution proportional to $F_S$ due to the presence of magnetic flux in the $\D7_1$-brane.
One can easily check that in the diluted magnetic flux limit the results for soft terms agree with those found in [@imr]. Of particular interest are the extremes in which either the overall modulus $T$ or the dilaton $S$ auxiliary field dominate SUSY-breaking. T-dominance $(\cos \theta = 1)$ appears for SUSY-breaking induced by ISD fluxes and we will discuss it in detail in the next section. Concerning dilaton dominance $(\sin \theta = 1)$, eq. (\[smasstm\]) shows that it is potentially dangerous since squarks and sleptons typically become tachyonic for small $\d$. However, we already mentioned that to stay within the string perturbative regime the value of $\delta$ cannot be too small, c.f. eq.(\[gsbound\]). Thus, dilaton dominance could still be consistent if magnetic fluxes are substantial, in fact for $\d \gtrsim 0.258$ when $u_2=u_3$. One may argue that there are scalars whose masses become tachyonic in the limit $\sin \theta =1$ and have no dependence on $\delta$, since they are related to unmagnetized branes. This is the case of the Higgs and the non-chiral scalars $\Phi_{22}$ and $\Phi_{33}$ parameterizing the position of branes $\D7_2$, $\D7_3$. However, for both there could be extra contributions to their masses which would render them non-tachyonic. In the case of the Higgs multiplet we already mentioned that they get an additional SUSY mass term for $\langle X \rangle\not=0$. Concerning the $\Phi_{22}$, $\Phi_{33}$ scalars, they may also have SUSY $\mu $-terms. For example, in flux induced SUSY-breaking, fluxes of type $(2,1)$ or $(1,2)$ could give rise to such terms (see [@ciu2]). All in all, whereas T-dominance always leads to a non-tachyonic structure of soft terms, in the case of dilaton dominance substantial magnetic fluxes and additional positive contribution for Higgsses and some adjoints are required to avoid tachyons.
Soft terms induced by fluxes
----------------------------
We now wish to discuss the situation in which the moduli superpotential is just given by $W_f$, c.f. (\[wflux\]). Then, fluxes are the only source of SUSY breaking.
.
A concrete realization of T-dominance arises when the flux $G_3$ is generic ISD. In this case, $\langle F^S \rangle = \langle F^{U_i} \rangle= 0$ and only $\langle F^{T_i} \rangle \not= 0$. The cosmological constant vanishes automatically since $W_f$ is independent of $T_i$ and $\hat K$ is of no-scale form. Thus, $V_0=0$ and the other relevant vevs are m\^2\_[3/2]{} = ; |F\^[|T\_i]{} = - = e\^[i\_T]{} t\_i m\_[3/2]{} , \[isd\] where $P=s \prod_i t_i u_i$. Comparing with (\[vevstino\]) and (\[cconst\]) shows that $C=1$, $\cos \theta=1$, $\eta_i=1/\sqrt3$, $\g_i=\g_T$, $\forall i$. The soft terms follow substituting these values in the general expressions. The results are collected in Table 2 [^2].
-------------------------------------------------------------------------------------------------
$m^2_{L_\a}$ $\frac12 \ -
\frac1{8\pi} \sin 2\pi\d (2 + \cos 2\pi\d)[\log{\frac{u_3}{u_2}} + B_0(\d)]
+ \frac1{16\pi^2} \sin^2 2\pi\d B_1(\d)$
------------------- -----------------------------------------------------------------------------
$m^2_{R_\b}$ $\frac12 \ -
\frac1{8\pi} \sin 2\pi\d (2 + \cos 2\pi\d)[\log{\frac{u_2}{u_3}} + B_0(\d)]
+ \frac1{16\pi^2} \sin^2 2\pi\d B_1(\d)$
$m^2_{H} $ $ \frac {1}{2} $
$ M_{SU(3+1)}$ $e^{-i\gamma _T} \cos^2\pi \delta $
$ M_{SU(2)_L}$ $e^{-i\gamma _T}$
$ M_{SU(2)_R}$ $e^{-i\gamma _T} $
$A_{HL_\a R_\b}$ $e^{-i\gamma _T}\ \left[ -\frac32 \ +
\frac1{2\pi} \sin 2\pi\d B_0(\d) \right]$
$A_{XHH}$ $ -e^{-i\gamma _T}$
$m^2_{\Phi_{jj}}$ $|M_j|^2$
$m^2_{\Phi_{ij}}$ 0
-------------------------------------------------------------------------------------------------
: Soft terms for T-dominant ISD fluxes. Results are given in $m_{3/2}$ units.
\[resuT\]
It is straightforward to expand the soft terms near $\d=0$. For example, m\^2\_[L\_]{} & = & m\^2\_[3/2]{} (12 - 34 - 3 \^2 + ) ,\
A\_[HL\_R\_]{} & = & e\^[-i\_T]{} m\_[3/2]{} (-32 + 2 2 + 3 \^2 + ) . \[softexp\] Using (\[psiprops\]) we can also take the limit $\d \to \oh$ in which $\D7_1 \to \D_3$. We find, $m^2_{L_\a} \to 0$ and $A_{HL_\a R_\b} \to -\frac12 e^{-i\g_T}\, m_{3/2}$, matching results of [@imr].
Note that the structure of soft terms in this subsection corresponds to SUSY-breaking induced by the auxiliary field of the overall Kähler modulus $T$, and the fact that it may be induced by fluxes plays no role in the obtained results. A few comments on the structure of soft terms in this simple case are in order.
- The structure of scalar soft terms is not universal, i.e. the scalar masses of left-handed sfermions, right-handed sfermions and Higgsses are different. However they are [*flavor independent*]{}. This is due to the origin of family replication in this class of models. The massless chiral fermions come in identical replicas with diagonal kinetic terms. Concerning gaugino masses, as we said the $SU(2)_L$ and $SU(2)_R$ gaugino masses are identical but that of $SU(3+1)$ is different.
- Note that if at the end of the day a non-vanishing FI-term (\[fi\]) is present, an extra contribution to squark/slepton masses with opposite signs for left and right-handed fields will be added. This contribution will not be present for the Higgs fields which are neutral under the anomalous $U(1)$.
- In the formal limit $\delta \rightarrow 0$ corresponding to diluted magnetic fluxes one obtains particularly simple and universal results for soft terms : m\^2\_[L\_]{} & = & m\^2\_[R\_]{} = m\^2\_[H]{} = 12 m\_[3/2]{}\^2\
M\_[SU(3+1)]{} & = & M\_[SU(2)\_L]{} = M\_[SU(2)\_R]{} = e\^[-i\_T]{} m\_[3/2]{}\
A\_[HL\_R\_]{} & = & - m\_[3/2]{} e\^[-i\_T]{} \
A\_[XHH]{} & = & e\^[-i\_T]{} m\_[3/2]{} \[softguay\] These results correspond to those advanced in section 6 of [@iflux] for $\mu = \langle X \rangle$ and $\xi =1/2$.
.
Let us now analyze the mass parameters generated by the presence of IASD fluxes. Some words of caution should first be given. Care should be taken in comparing the results below for IASD fluxes to those for dilaton dominance ($\sin \theta =1$) in the previous section. Indeed in the analysis of that section the values of $V_0$, $\sin \theta$, $m_{3/2}$, appear as independent parameters. Thus, in principle one can conceive a situation with $\sin \theta =1$, $V_0=0$ and $m_{3/2}\not =0$ with SUSY broken in Minkowski space. However, with IASD (3,0) fluxes one can show that $V_0\not = 0$ and $m_{3/2}=0$, so we have broken SUSY in de Sitter space. We will still provide the generated mass terms for completeness.
If we consider $W_f$ as our only source of SUSY-breaking dilaton dominance appears when the flux $G_3$ is IASD of (3,0) type. In the absence of a $W_{np}$ term this background is not in general a solution of the Type IIB equations of motion. However, it is a simple example of S-dominance because $\langle F^{T_i} \rangle = \langle F^{U_i} \rangle= 0$ but $\langle F^S \rangle \not= 0$. In this case $m_{3/2} = 0$ automatically and, as we said, there is a cosmological constant $V_0\not= 0$. In terms of $Y_f=\int {\bar G_3} \wedge
\Omega$ one finds V\_0 = ; |F\^[|S]{} = - = e\^[i\_S]{} s . \[iasd\] Hence, the soft terms can be obtained from the general expressions setting $\sin \theta = 1$ and $C m_{3/2} \to \sqrt{V_0/3}$. Results are displayed in Table 3. Note that the mass parameters are in general not tachyonic. This is not in contradiction with our results discussed at the end of section 3, since there we assumed arbitrary $V_0$ and $m_{3/2}\not= 0$, whereas IASD $(3,0)$ fluxes lead to $V_0\not=0$ and $m_{3/2}=0$.
-------------------------------------------------------------------------------------------------
$m^2_{L_\a}$ $\frac12 \ +
\frac1{8\pi} \sin 2\pi\d (2 - \cos 2\pi\d)[\log{\frac{u_3}{u_2}} + B_0(\d)]
+ \frac1{16\pi^2} \sin^2 2\pi\d B_1(\d)$
------------------- -----------------------------------------------------------------------------
$m^2_{R_\b}$ $\frac12 \ +
\frac1{8\pi} \sin 2\pi\d (2 - \cos 2\pi\d)[\log{\frac{u_2}{u_3}} + B_0(\d)]
+ \frac1{16\pi^2} \sin^2 2\pi\d B_1(\d)$
$m^2_{H} $ $ \frac {1}{2} $
$ M_{SU(3+1)}$ $e^{-i\gamma _S} \sin^2\pi \delta $
$ M_{SU(2)_L}$ 0
$ M_{SU(2)_R}$ 0
$A_{HL_\a R_\b}$ $e^{-i\gamma _S}\ \left[ \frac12 \ -
\frac1{2\pi} \sin 2\pi\d B_0(\d) \right]$
$A_{XHH}$ 0
$m^2_{\Phi_{jj}}$ $|M_j|^2$
$m^2_{\Phi_{ij}}$ 1
-------------------------------------------------------------------------------------------------
: Soft terms for S-dominant IASD fluxes. Results are given in $V_0$ units.
\[resuS\]
When ISD and IASD fluxes are turned on simultaneously the auxiliary fields, $m_{3/2}$ and $V_0$ are just given by (\[isd\]) and (\[iasd\]). Then, $\eta_i=1/\sqrt3$, $\g_i=\g_T$, $\forall i$, $3 \tan^2 \theta=V_0/3m_{3/2}$, and $C=\sec \theta$. The soft terms can be found substituting these values in the general expressions. In most cases it suffices to add the entries in Tables 2 and 3.
Let us end this section with some comments concerning the mass scales in this model. Note that as it stands, in a [*toroidal* ]{} model like this, the string scale $M_s$ should be of order (or slightly smaller) than the Planck scale. Indeed, both scales are related by eq.(\[mplanck\]). Although one may think that one can make $M_{Pl}>>M_s$ by taking $t_i$ very large, that would make the SM gauge couplings unacceptably small, as shown by eqs.(\[efes\]). On the other hand, the size of SUSY-breaking soft terms depends on the value of the gravitino mass in these theories. In general, if the source of SUSY-breaking is not specified, as in section 3, one can assume that $m_{3/2}$ may be small, i.e. of order the electroweak scale, as in the canonical approach to gravity mediated SUSY-breaking models. This was our general philosophy in the first part of section 4. On the other hand, if one insists that the microscopic source of SUSY-breaking is some ISD flux in a [*toroidal* ]{} setting , then the gravitino mass is given by eq.(\[isd\]). In that case, since the $t_i$ fields cannot be too large, the gravitino mass is of order the string scale (which is only slightly smaller than $M_{Pl}$) and hence too large to lead to a solution of the hierarchy problem. This is the fact already mentioned in the introduction. However, as we said, this is a particular property of toroidal settings in which fluxes are distributed uniformly in extra dimensions. One can conceive an embedding of the MSSM-like brane setting in [@cim1] into a CY/F-theory compactification in which the distribution of fluxes in extra dimensions is not constant and hierarchically small soft terms may appear.
Conclusions {#sec:fin}
===========
In this paper we have computed the SUSY-breaking soft terms for the MSSM-like model introduced in [@cim1] under the assumption of generic vevs for the auxiliary fields $F_{T_i}$ and $F_S$. We provide the soft terms as explicit functions of the gravitino mass, goldstino angle and a parameter $\delta $ that characterizes the magnetic flux in one of the brane stacks. We find that the case of isotropic $T$-dominance is particularly interesting since it always leads to simple results with no tachyons. For dilaton dominance there is the risk of getting some tachyonic masses for SM fields unless magnetic fluxes are large and additional sources for masses of non-chiral fields are present. The case of isotropic $T$-dominance appears in particular when SUSY-breaking is triggered by ISD antisymmetric Type IIB fluxes. We argue that although in a toroidal setting the soft terms induced by fluxes are typically too large, they may be hierarchically small in more general CY/F-theory embeddings of this MSSM-like brane configuration.
The results for soft terms in $T$-dominance are summarized in Table 2, and take an even simpler form (\[softguay\]) in the dilute flux limit $\delta \rightarrow
0$. They are flavor universal and depend only on the values of $m_{3/2}$, $\delta $ and a complex phase $\gamma_T$.
[**Acknowledgments**]{}
We thank P.G. Cámara, D. Cremades, F. Marchesano, F. Quevedo, S. Theisen, and A. Uranga for useful discussions. This work has been partially supported by the European Commission under the RTN European Program MRTN-CT-2004-503369 and the CICYT (Spain).
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[^1]: In the T-dual version in terms of intersecting D6-branes, $\langle X \rangle$ corresponds to the distance between the $SU(2)_L$ and $SU(2)_R$ D6-branes in the first complex plane.
[^2]: When the ISD flux has both (2,1) and (0,3) components there is an induced supersymmetric $\mu$ term and an extra soft bilinear parameter for the $\Phi_{ii}$ scalars [@ciu2].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We report an infinite class of discrete hierarchies which naturally generalize familiar discrete KP one.'
---
=-.5in =-.7in =.1in
Introduction
============
The interrelation between discrete and differential integrable hierarchies plays crucial role in obtaining solutions to the discrete multi-matrix models [@bonora1], [@bonora2], [@aratyn]. At a level of KP-type differential hierarchies the discrete structure of multi-matrix models is captured by the Darboux–Bäcklund (DB) transformations. In turn partition functions of multi-matrix models turns out to be $\tau$-functions of differential hierarchies and are constructed as DB orbits of certain simple initial conditions [@aratyn]. The well known discrete KP (1-Toda lattice) hierarchy [@ueno] together with its reductions can be viewed as a container for a set of KP-type differential hierarchies whose solutions are generated by DB transformations.
This paper is designed to exhibit certain class of discrete hierarchies which generalize discrete KP and show the relationship with general (unconstrained) differential KP. This relationship yields bi-infinite sequences of differential KP equipped with two compatible gauge transformations. We believe that these results would be of potential interest from the physical point of view.
$n$th discrete KP
=================
Given the shift operator $\Lambda = (\delta_{i,j-1})_{i,j\in{\bf Z}}$ one considers the Lie algebra of pseudo-difference operators $${\cal D} = \left\{
\sum_{-\infty<k\ll\infty}\ell_k\Lambda^k
\right\} = {\cal D}_{-} + {\cal D}_{+}$$ with usual splitting into “negative" and “positive" parts: $${\cal D}_{-} = \left\{
\sum_{-\infty<k\leq -1}\ell_k\Lambda^k
\right\}\;\; {\rm and}\;\;
{\cal D}_{+} = \left\{
\sum_{0<k\ll\infty}\ell_k\Lambda^k
\right\}.$$ We assume that entries of bi-infinite diagonal matrices $\ell_k
\equiv (\ell_k(i))_{i\in{\bf Z}}$ may depend on “spectral" parameter $z$ and multi-time $t\equiv(t_1\equiv x, t_2, t_3,...)$. In what follows $\partial\equiv \partial/\partial x$ and $\partial_p\equiv\partial/
\partial t_p$.
Let us define[^1] $$Q = \Lambda + a_0z^{n-1}\Lambda^{1-n} + a_1z^{2(n-1)}\Lambda^{1-2n} + ...\;\;
\in{\cal D},\;\;
n\in{\bf N}
\label{matrix}$$ with $a_k = (a_k(i))_{i\in{\bf Z}}$ being functions on $t$ only.
[**Proposition 1.**]{} [*Lax equations of $Q$-deformations $$z^{p(n-1)}\partial_p Q = [Q_{+}^{pn}, Q],\;\;
p = 1, 2,...
\label{LAX}$$ make sense.*]{}
[**Proof.**]{} One needs to use standard simple arguments to prove correctness of Eqs. (\[LAX\]). It is enough to show that $[Q_{+}^{pn}, Q] =
- [Q_{-}^{pn}, Q]$ is of the same form as l.h.s. of (\[LAX\]). $\Box$
We will refer to (\[LAX\]) as $n$th discrete KP hierarchy. Let us represent $Q$ as a dressing up of $\Lambda$ by a “wave" operator $$W = I + w_1z^{n-1}\Lambda^{-n} + w_2z^{2(n-1)}\Lambda^{-2n} + w_3z^{3(n-1)}
\Lambda^{-3n} + ... \in I + {\cal D}_{-}.$$ Then $Q$-deformations are induced by $W$-deformations $$\begin{array}{c}
\displaystyle
z^{p(n-1)}\partial_p W = Q_{+}^{pn}W - W\Lambda^{pn}, \\[0.4cm]
\displaystyle
z^{p(n-1)}\partial_p (W^{-1})^T = (W^{-1})^T\Lambda^{-pn} - (Q_{+}^{pn})^T(W^{-1})^T.
\end{array}
\label{SW}$$ Define $\chi(t, z)=(z^ie^{\xi(t, z)})_{i\in{\bf Z}}$, $\chi^{*}(t, z)=(z^{-i}e^{-\xi(t, z)})_{i\in{\bf Z}}$ with $\xi(t, z)\equiv \sum_{p=1}^{\infty}t_pz^p$ and wave vectors $$\Psi(t, z) = W\chi(t, z),\;\;
\Psi^{*}(t, z) = (W^{-1})^T\chi^{*}(t, z).
\label{wv}$$ Discrete linear system $$\begin{array}{c}
\displaystyle
Q\Psi(t, z) = z\Psi(t, z),\;\;Q^T\Psi^{*}(t, z) = z\Psi^{*}(t, z), \\[0.4cm]
\displaystyle
z^{p(n-1)}\partial_p\Psi = Q_{+}^{pn}\Psi,\;\;
z^{p(n-1)}\partial_p\Psi^{*} = - (Q_{+}^{pn})^T\Psi^{*}
\end{array}
\label{DLS1}$$ are evident consequence of (\[SW\]) and (\[wv\]). Making use of obvious relations $z\chi = \Lambda\chi$ and $\chi_i = \partial^{i-j}\chi_j$ with $i$ and $j$ being arbitrary integers, we deduce $$\Psi_i(t, z) =
z^i(1 + w_1(i)z^{-1} + w_2(i)z^{-2} + ...)e^{\xi(t, z)}$$ $$= z^i(1 + w_1(i)\partial^{-1} + w_2(i)\partial^{-2} + ...)e^{\xi(t, z)} \equiv
z^i\hat{w}_i(\partial)e^{\xi(t, z)} \equiv
z^i\psi_i(t, z).$$
What we are going to do next is to establish equivalence of $n$th discrete KP to bi-infinite sequence of differential KP copies “glued" together by two compatible gauge transformations one of which can be recognized as DB transformation mapping ${\cal Q}_i \equiv \hat{w}_i\partial
\hat{w}_i^{-1}$ to ${\cal Q}_{i+n} \equiv \hat{w}_{i+n}\partial\hat{w}_{i+n}^{-1}$. By straightforward calculations one can prove
[**Proposition 2.**]{}
*The following three statements are equivalent*
\(i) The wave vector $\Psi(t, z)$ satisfies discrete linear system $$Q\Psi(t, z) = z\Psi(t, z),\;\;
z^{n-1}\partial\Psi = Q_{+}^n\Psi;
\label{DLS2}$$
\(ii) The components $\psi_i$ of a vector $\psi \equiv (\psi_i =
z^{-i}\Psi_i)_{i\in{\bf Z}}$ satisfy $$G_i\psi_i(t, z) = z\psi_{i+n-1}(t, z),\;\;
H_i\psi_i(t, z) = z\psi_{i+n}(t, z)
\label{system1}$$ with $H_i\equiv \partial - \sum_{s=1}^na_0(i+s-1)$ and $$G_i\equiv \partial - \sum_{s=1}^{n-1}a_0(i+s-1) + a_1(i+n-1)H_{i-n}^{-1} +
a_2(i+n-1)H_{i-2n}^{-1}H_{i-n}^{-1} + ... ;$$
\(iii) For sequence of dressing operators $\hat{w}_i$ following equations $$G_i\hat{w}_i = \hat{w}_{i+n-1}\partial,\;\;
H_i\hat{w}_i = \hat{w}_{i+n}\partial
\label{system2}$$ hold.
Consistency condition of (\[DLS2\]) is given by Lax equation $$z^{n-1}\partial Q = [Q_{+}^n, Q]
\label{first}$$ which in explicit form looks as $$\begin{array}{c}
\partial a_k(i) = a_{k+1}(i+n) - a_{k+1}(i) \\[0.4cm]
\displaystyle
+ a_k(i)\left(
\sum_{s=1}^na_0(i+s-1) - \sum_{s=1}^na_0(i+s-(k+1)n)\right),\;\;
k\geq 0.
\end{array}
\label{motion}$$
[**Remark.**]{} One-field reductions of the systems (\[motion\]) lead to Bogoyavlenskii lattices [@bogoyavlenskii] $$\partial r_i = r_i\left(
\sum_{s=1}^{n-1}r_{i+s} - \sum_{s=1}^{n-1}r_{i-s}\right),
\;\; r_i \equiv a_0(i)$$ including well known Volterra lattice $\partial r_i = r_i(r_{i+1} - r_{i-1})$ in the case $n=2$.
Consistency condition of (\[system2\]) is given by relations $$G_{i+n}H_i = H_{i+n-1}G_i,\;\;
i\in{\bf Z}
\label{relations}$$ which in fact are equivalent to (\[first\]).
[**Proposition 3.**]{} [*By virtue of (\[system2\]) and its consistency condition, Lax operators ${\cal Q}_i$ are connected with each other by two invertible compatible gauge transformations $${\cal Q}_{i+n-1} = G_i{\cal Q}_iG_i^{-1},\;\;
{\cal Q}_{i+n} = H_i{\cal Q}_iH_i^{-1}.
\label{similarity1}$$* ]{}
[**Proof.**]{} By virtue of (\[system2\]), we have $${\cal Q}_{i+n-1} = \hat{w}_{i+n-1}\partial\hat{w}_{i+n-1}^{-1} =
(G_i\hat{w}_{i}\partial^{-1})\partial(\partial\hat{w}_{i}^{-1}G_i^{-1})$$ $$= G_i\hat{w}_{i}\partial\hat{w}_{i}^{-1}G_i^{-1} = G_i{\cal Q}_iG_i^{-1}.$$ The similar arguments are applied to show second relation in (\[similarity1\]). The mapping ${\cal Q}_i\rightarrow\tilde{{\cal Q}}_i = {\cal Q}_{i+n-1}$ we denote as $s_1$, while $s_2$ stands for transformation ${\cal Q}_i\rightarrow\overline{{\cal Q}}_i = {\cal Q}_{i+n}$. As for compatibility of $s_1$ and $s_2$, by virtue of (\[relations\]), we have $${\cal Q}_{i+2n-1} = G_{i+n}{\cal Q}_{i+n}G_{i+n}^{-1} =
G_{i+n}H_i{\cal Q}_{i}H_i^{-1}G_{i+n}^{-1}$$ $$= H_{i+n-1}G_{i}{\cal Q}_iG_i^{-1}H_{i+n-1}^{-1} =
H_{i+n-1}{\cal Q}_{i+n-1}H_{i+n-1}^{-1}.$$ So we can write $s_1\circ s_2 = s_2\circ s_1$. The inverse maps $s_1^{-1}$ and $s_2^{-1}$ are well defined by the formulas ${\cal Q}_{i-n+1} = G_{i-n+1}^{-1}{\cal Q}_iG_{i-n+1}$ and ${\cal Q}_{i-n} = H_{i-n}^{-1}{\cal Q}_iH_{i-n}$. $\Box$
It is obvious that relation $s_1^n = s_2^{n-1}$ holds. Indeed the l.h.s. and r.h.s. of this relation correspond to the same mapping ${\cal Q}_i\rightarrow {\cal Q}_{i+n(n-1)}$. The abelian group generated by $s_1$ and $s_2$ we denote by symbol ${\cal G}$.
Rewrite second equation in (\[system1\]) as $
z^{n-1}H_i\Psi_i(t, z) = \Psi_{i+n}(t, z) = (\Lambda^n\Psi)_i.
$ >From this we derive $$z^{k(1-n)}(\Lambda^{kn}\Psi)_i = H_{i+(k-1)n}...H_{i+n}H_i\Psi_i,$$ $$z^{k(n-1)}(\Lambda^{-kn}\Psi)_i = H_{i-kn}^{-1}...H_{i-2n}^{-1}H_{i-n}^{-1}\Psi_i.$$ These relations make connection between matrices of the form $$P = \sum_{k\in{\bf Z}}z^{k(1-n)}p_k(t)\Lambda^{kn}$$ and sequences of pseudo-differential operators $\{{\cal P}_i,\; i\in{\bf Z}\}$ mapping the upper triangular part of given matrix (including main diagonal) into the differential parts of ${\cal P}_i$’s and the lower triangular part of the matrix to the purely pseudo-differential parts. More exactly, we have $(P\Psi)_i = {\cal P}_i\Psi_i$, $(P_{-}\Psi)_i = ({\cal P}_i)_{-}\Psi_i$ and $(P_{+}\Psi)_i = ({\cal P}_i)_{+}\Psi_i$, where $${\cal P}_i =
\sum_{k > 0}p_{-k}(i, t)H_{i-kn}^{-1}...H_{i-2n}^{-1}H_{i-n}^{-1} +
\sum_{k\geq 0}p_{k}(i, t)H_{i+(k-1)n}...H_{i+n}H_{i} = ({\cal P}_i)_{-} +
({\cal P}_i)_{+}.$$
[**Proposition 4.**]{} [*Equations $z^{p(n-1)}\partial_p\Psi = Q_{+}^{pn}\Psi,\;p=2, 3,...$ lead to $\partial_p\psi_i = ({\cal Q}_i^p)_{+}\psi_i,\;p=2, 3,...$.* ]{}
[**Proof.**]{} We have $$z^{p(1-n)}(Q^{pn}\Psi)_i = z^p\Psi_i = z^{i+p}\hat{w}_ie^{\xi(t, z)} =
z^i\hat{w}_i\partial^p e^{\xi(t, z)}$$ $$= z^i\hat{w}_i\partial^p\hat{w}_i^{-1}
\psi_i = z^i{\cal Q}_i^p\psi_i = {\cal Q}_i^p\Psi_i.$$ Thus $$z^{p(n-1)}\partial_p\Psi_i = z^{i+p(n-1)}\partial_p\psi_i =
(Q_{+}^{pn}\Psi)_i = z^{p(n-1)}({\cal Q}_i^p)_{+}\Psi_i =
z^{i+p(n-1)}({\cal Q}_i^p)_{+}\psi_i.$$ The latter proves proposition. $\Box$
Let us establish equations managing $G_i$- and $H_i$-evolutions with respect to KP flows. Differentiating l.h.s. and r.h.s. of (\[system2\]) by virtue of Sato–Wilson equations $\partial_p\hat{w}_i = ({\cal Q}_i^p)_{+}\hat{w}_i
- \hat{w}_i\partial^p$ formally leads to evolution equations $$\begin{array}{c}
\displaystyle
\partial_p G_{i} = ({\cal Q}_{i+n-1}^p)_{+}G_{i} -
G_{i}({\cal Q}_{i}^p)_{+}, \\[0.4cm]
\displaystyle
\partial_p H_{i} = ({\cal Q}_{i+n}^p)_{+}H_{i} -
H_{i}({\cal Q}_{i}^p)_{+}.
\end{array}
\label{leadsto}$$
Standard arguments can be used to show that Eqs. (\[leadsto\]) are properly defined individually. Let us show that permutation relations (\[relations\]) are invariant under the flows given by equations (\[leadsto\]). We have $$\partial_p(H_{i+n-1}G_i)$$ $$= \{({\cal Q}_{i+2n-1}^p)_{+}H_{i+n-1} - H_{i+n-1}({\cal Q}_{i+n-1}^p)_{+}\}G_i + H_{i+n-1}\{({\cal Q}_{i+n-1}^p)_{+}G_{i} - G_{i}({\cal Q}_{i}^p)_{+}\}$$ $$= ({\cal Q}_{i+2n-1}^p)_{+}H_{i+n-1}G_i - H_{i+n-1}G_i({\cal Q}_{i}^p)_{+}
= ({\cal Q}_{i+2n-1}^p)_{+}G_{i+n}H_i - G_{i+n}H_i({\cal Q}_{i}^p)_{+}$$ $$= \{({\cal Q}_{i+2n-1}^p)_{+}G_{i+n} - G_{i+n}({\cal Q}_{i+n}^p)_{+}\}H_i +
G_{i+n}\{({\cal Q}_{i+n}^p)_{+}H_{i} - H_{i}({\cal Q}_{i}^p)_{+}\}
= \partial_p(G_{i+n}H_i).$$ Hence we proved that evolution equations (\[leadsto\]) are consistent.
Define $\Phi_i = \Phi_i(t)$ via $H_i\Phi_i = 0$ or equivalently through equation $$\partial \Phi_i = \Phi_i\sum_{s=1}^na_0(i+s-1).$$ Takig into consideration second equation in (\[leadsto\]), we have $$\partial_p(H_i\Phi_i) = ({\cal Q}_{i+n}^p)_{+}H_i\Phi_i -
H_i({\cal Q}_{i}^p)_{+}\Phi_i + H_i\partial_p\Phi_i = 0.$$ >From this we derive $\partial_p\Phi_i = ({\cal Q}_{i}^p)_{+}\Phi_i +
\alpha_i\Phi_i$ where $\alpha_i$’s are some constants. Commutativity condition $\partial_p\partial_q\Phi_i =
\partial_q\partial_p\Phi_i$ leads to evolution equations for KP eigenfunctions $\partial_p\Phi_i = ({\cal Q}_{i}^p)_{+}\Phi_i$, i.e. $\alpha_i = 0$. Thus the relations ${\cal Q}_{i+n} = H_i{\cal Q}_iH_i^{-1}$ defines DB transformations with eigenfunctions $\Phi_i = \tau_{i+n}/\tau_i$. It should perhaps to recall that arbitrary eigenfunction of Lax operator ${\cal Q}$ contains information about DB transformation $\tau \rightarrow \overline{\tau} = \Phi\tau$ while the identity [^2] $$\{\tau(t - [z^{-1}]), \overline{\tau}(t)\} +
z(\tau(t - [z^{-1}])\overline{\tau}(t) - \overline{\tau}(t - [z^{-1}])
\tau(t)) = 0$$ holds
So, we have shown that $n$th discrete KP is equivalent to sequence of differential KP linked with each other by two compatible gauge transformations one of which, namely, $s_2 : {\cal Q}_i\rightarrow {\cal Q}_{i+n}$ are nothing but Darboux–Bäcklund transformation. The problem which can be addressed is to describe $n$th discrete KP in the language of bilinear identities by analogy as was done for ordinary discrete KP [@adler].
Acknowledgments {#acknowledgments .unnumbered}
---------------
Many thanks to the organizers for the invitation to participate Fourth Conference “Symmetry in Nonlinear Mathematical Physics".
This research has been partially supported by INTAS grant 2000-15.
[99]{}
Bonora L., Xiong C.S., An alternative approach to KP hierarchy in matrix models, [*Phys. Lett.*]{}, 1992, V. B285, P. 191-198.
Bonora L., Xiong C.S., Multi-matrix models without continuum limit, [*Nucl. Phys.*]{}, 1993, V. B405, P. 191.
Aratyn H., Nissimov E., Pacheva S., Constrained KP hierarchies: additional symmetries, Darboux–Bäcklund solutions and relations to multi-matrix models, [*Int. J. Modern Physics A*]{}, 1997, V. 12, P. 1265-1340.
Ueno K., Takasaki K., Toda lattice hierarchy, [*Adv. Studies in Pure Math.*]{}, 1984, V. 4, P. 1-95.
Bogoyavlenskii O.I., Algebraic constructions of integrable dynamical systems — extensions of Volterra system, [*Russian Math. Surveys*]{}, 1991, V. 46, P. 1-46.
Adler M., van Moerbeke P. Vertex operator solutions to the discrete KP-hierarchy, [*Comm. Math. Phys.*]{}, 1999, V. 203, P. 185-210.
[^1]: where $z$ acts as component-wise multiplication.
[^2]: here conventional notations $\{f, g\} = \partial f\cdot g - \partial g\cdot f$ and $[z^{-1}] = (1/z, 1/(2z^2),...)$ are used.
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abstract: 'Intermolecular electron transfer reaction often occurs over long range distances (i.e. up to several tens of angstroms) and plays a key role in various physical, chemical and biological processes. In these reactions the rate constant of long range electron transfer depends upon electronic coupling between the donor and acceptor. The coupling between donor and acceptor may increases by the atoms located between them which form a kind of bridge for electron tunneling. By using exact analytical method we calculated the value of electronic coupling for the above said processes in which the interaction of an electron with the donor, acceptor are represented as Dirac delta functions and conjugated bridge is represented by finite square well.'
address: 'School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, Himachal Pradesh 175001, India'
author:
- 'Ashish Kumar, Diwaker, Anirudhha Chakraborty'
title: 'Exact solution of long-range electron transfer through conjugated molecular bridge.'
---
electron transfer rate , electronic coupling, distance between donor and acccetor.
Introduction
============
Intermolecular electron transfer (i.e either between different freely diffusing, donor and acceptor molecules or between Donor (D) and acceptor (A) subunits of supermolecule through bridging (B) subunit of supermolecule) which occurs over long distances ranging over several 10 of angstrom plays vital role in various scientific phenomenons which include physical chemical and biological processes. Intermolecular electron transfer (IT) is one of the fundamental process of molecular electronics and applied research field. Natural Photosynthesis is the one of the best example, in which light harvesting molecules gather and transfer energy to reaction centers, where cascades of electron transfer reactions take place. Redox reactions and respiration are also some of the well cited examples in literature[@David; @Moore; @Satyam]. Such kind of long range electron transfer reactions are termed as Bridge mediated electron transfer (IT)reactions which involve the tunneling of an electron from a localized donor state to acceptor state by an intervening bridge that connects the donor and the acceptor. The internuclear distance between donor and acceptor and electronic structure of bridge component in D-B-A system are well known to play a crucial role in determining the electron transfer rate[@McConnell; @Barbara; @Wasielewski; @William; @B]. The electron transfer rate has exponential behavior with the internuclear distance between donor and acceptor or length of bridge molecules in most of D-B-A systems studied by the many authors[@B; @J; @V.P.Zhdanov]. In biological processes the bridge is often a protein molecule and distance which electron had to transfer from donor to acceptor is usually greater than 6 angstrom. In intermolecular electron transfer processes the electron transfer rate explicitly depends on communication between donor and acceptor i.e. electronic coupling ($ G_{DA}$). The current work deals with the calculation of exact analytical formula for the electronic coupling between donor and acceptor where the electronic coupling is treated as a function of internuclear distance R between the donor and the acceptor and length(L) of conjugated bridge. The donor and acceptor are both represented as Dirac delta Potentials while the conjugated chain of atoms (i.e. bridge) which is responsible for electron transfer is represented as a finite square well. There are different experimental, analytical and computational approaches used by different authors to explain the electron transfer process through bridge between donor and acceptor in intermolecular electron transfer processes. For example in molecular electronics, Bo Albinsson and co-authors[@B] studied the electron and energy transfer mediated by the $\pi$-conjugated molecular bridges in D-B-A system and specifically studied the influence of donor, bridge structure and distance between donor and acceptor on the electronic coupling by using different bridge molecules. Yinxi Yu and co-workers have studied bridge-mediated inter-valence electron transfer coupling in different metallocene complexes[@Y] by using computational approach. In his work he used constrained density functional theory to study the impact of the bridge length on the electronic coupling between the metal centers by using different models. Jean-Luc Brendas and Co-workers[@J] also studied the charge and energy transfer in $\pi$-conjugated oligomers and polymers. V.P. Zhdanov[@V.P.Zhdanov] in his finding used an analytical approach in which he derived an asymptotic expression for $G_{DA}$ for bridge-mediated electron process where he represents the bridge by single Dirac-delta function potential. Natalie Gorczak et.al[@N.; @G.] gives different mechanisms for electron and hole transfer along identical oligo-p-phenylene molecular bridges of increasing length. The bridge-mediated electron transfer processes studied by many other authors[@S.Larsson; @J.Wolfgang; @I.A; @A.Onipko; @S.C; @A; @D.J] and they all devolved different approaches to calculate the electronic coupling($G_{DA}$) which is the key parameter of electron transfer rate in bridge-mediated electron transfer processes. In our model we have considered the electron transfer through conjugated bridge because conjugated bridge has an advantage over non-conjugated one[@Y]. Our model for long-range electron transfer through conjugated bridge has an advantages over earlier discussed models which we will represent in results section. Furthermore the rate of this non adiabatic electron transfer(IT) can be calculated for any system by Fermi Golden rule[@N.S; @J.J; @J.I; @M.Bixon] The schematic diagram for our model is shown in Figure 1. $$R_{IT}= 2\pi|G_{DA}|^2(FC)$$ where $G_{DA}$ is the electron coupling between the donor and the acceptor, (FC) is franck Condon factor[@R.A] with $\hbar=1$, m=1
Formulation of the Problem
==========================
We consider one dimension electron transfer from donor (located at $ x=-\eta$ ($\eta\equiv R/2$) to acceptor (located at $x=\eta$ ($\eta\equiv R/2$) and conjugated bridge formed by particles centered at $x=0$. R is the distance between donor and acceptor. We assumed that electron weakly interact with bridge in comparison to donor and acceptor. The electronic potential energy is represented by the following equation $$V(x) = -\alpha\delta(x+\eta)-\alpha\delta(x-\eta)+U(x)$$ The first and second terms in the above equation represents the position of donor and acceptor as represented by Dirac delta potentials and third term represents bridge by the finite square well potential because we are dealing with the conjugated molecular bridge in our model and finite square well potential can be used to represent the conjugated chain of atoms. Furthermore $$U(x) = \left\{\begin{array}{ll}
-V_0 &;for -a<x<a \\
0 &;$ elsewhere$
\end{array}\right.$$\
where $V_0 $ is depth of the well and $\alpha$ is the strength of Dirac delta functions and $ L=2a $is the width of the square well. The time-independent Schrödinger equation for the above potential can be written as $$-\frac{1}{2}\frac{d^2\psi}{dx^2}+V(x)\psi=E\psi$$ The solution of the above equation for bound states can be written as For region $x>\eta$, V(x)= 0 so $$\frac{d^2\psi}{dx^2}=-2E\psi=k^2\psi,$$ where $$k=\sqrt{-2E}$$ the general solution of Eq.(5) can be written as $$\psi(x)=A\exp[-kx]+B\exp[kx]$$ for x$\rightarrow\infty$,B=0 hence $$\psi(x)=A\exp[-kx]$$ For region $a<x<\eta$,V(x)= 0 the solution of Eq. (5) is $$\psi(x)=C\exp[-kx]+D\exp[kx]$$ For region $-a<x<a$, V(x) = U(x) the solution of Eq. (4) is $$\psi(x)=I\cos(lx)+F\sin(lx)$$\
where l is wave vector and is given by $$l = \sqrt{2(E+V_0)}$$ For region $-\eta<x<-a$,V(x) = 0 hence the solution of Eq. (5) is $$\psi(x)=G\exp[-kx]+H\exp[kx]$$ For region $x<-\eta $V(x) = 0. the solution of Eq. (5) is $$\psi(x)=J\exp[kx]$$ Now by using the two boundary conditions i.e. wave function is continuous every where and its derivative is also continuous at every where except at the points where potential is infinite and the fact that since used potential is a symmetric function, so we can assume with no loss of generality that the solution are either even or odd and apply the boundary conditions on either side ( say, at $x=\eta , x= a$) or to the other side ( say, at $x= -\eta , x= -a$). Firstly the solution are calculated for ( i.e symmetric wave function)in case of above potential and hence the wave function in collective form can be written as $$\psi_{S}(x)\propto\left\{\begin{array}{ll}
\exp[-kx] &;x>\eta\\
C\exp[-kx]+D\exp[kx] &; a<x<\eta\\
I\cos(lx) &;-a<x<a\\
C\exp[kx]+D\exp[-kx] &;-\eta<x<-a\\
\exp[kx] &;x<-\eta
\end{array}\right.$$ continuity of $\psi_s(x)$ and discontinuity of derivative of $\psi_s(x)$ at $x=\eta$ , gives $$\exp[- k\eta]= C \exp[-k \eta]+D \exp[K \eta]$$ and $$-k\exp[-k\eta]-[-C k\exp[-k \eta]+D k\exp[k \eta]= -2\alpha\exp[-k \eta]$$ further simplification of these equations will give me $$1-C-D\exp[2 k \eta]=0$$ and $$\left(1-\frac{2\alpha}{k}\right)-C+D\exp[2k\eta]=0$$ similarly at points x=a, another set of equations can be written as $$C\exp[-ka]+D\exp[ka]=I\cos(la)$$ and $$-kC\exp[-ka]+kD\exp[ka]=-lI\sin(la)$$ Then from Eq. (17) and Eq. (18) we obtained. $$D=\frac{\alpha}{k}\exp[-2k\eta]$$ and $$C=\left(\frac{k-\alpha}{k}\right)$$ and by Dividing Eq. (20) by Eq. (19), we get $$\tan(la)=\frac{k}{l}\left(\frac{C\exp[-ka]-D\exp[ka]}{C\exp[-ka] + D\exp[ka]}\right)$$ Now by using Eq. (22) and Eq. (21) in above equation we get the analytical formula $$\tan(la)=\frac{k}{l}\left(1-\frac{2\alpha\exp[2k(a-\eta)]}{k-\alpha+\alpha\exp[2k(a-\eta)]}\right)$$ This is the formula for allowed energies, since k and $l$ are both function of Energy. To simplify above expression we make the following substitution for our convenience: Let\
$$z_s=la \;and \;z_0=a\sqrt{2V_0},\;
(k^2+l^2)=2V_0, and\; ka=\sqrt{z_0^2-z_s^2}$$ Using these substitutions Eq. (24) can be written as $$\tan(z_s)=\sqrt{\left(\frac{z_0}{z_s}\right)^2-1}\left(1-\frac{2\alpha a\exp{[2(\sqrt{z_0^2-z_s^2}-k\eta)}]}{\sqrt{z_0^2-z_s^2}-\alpha a+\alpha a\exp{[2(\sqrt{z_0^2-z_s^2}-k\eta)}]}\right)$$ further $$\tan(z_s)=\sqrt{\left(\frac{z_0}{z_s}\right)^2-1}\left(1-\frac{\alpha L \exp[2\sqrt{z_0^2-z_s^2}(1-\frac{R}{L})]}{\sqrt{z_0^2-z_s^2}-\frac{\alpha L}{2}+\frac{\alpha L}{2}\exp[2\sqrt{z_0^2-z_s^2}(1-\frac{R}{L})]}\right)$$ Here $L=2a$ is the length of conjugated chain of atoms between donor and acceptor (i.e bridge).\
This is the transcendental equation for $z_s$(and hence for $E_s$) as function of $z_0$ (which is a measure of the size of the well). The above equation is solved graphically to get the value of $z_s$ and hence of energy(E) for symmetric states.\
Similarly we can adopt the same methodology for antisymmetric states as mentioned below. The antisymmetric wave function is represented as $$\psi_{u}(x)\propto\left\{\begin{array}{ll}
\exp[-kx] &;x>\eta\\
C\exp[-kx]+D\exp[kx] &; a<x<\eta\\
F\sin(lx) &;-a<x<a\\
-C\exp[kx]-D\exp[-kx] &;-\eta<x<-a\\
-\exp[kx] &;x<-\eta
\end{array}\right.$$ applying the boundary conditions at $x=\eta$ and $x=a$ we get corresponding transcendental equation for $z_u$ (and hence for $E_u$ ). $$-cot(z_u)=\sqrt{\left(\frac{z_0}{z_u}\right)^2-1}\left(1-\frac{\alpha L \exp[2\sqrt{z_0^2-z_u^2}(1-\frac{R}{L})]}{\sqrt{z_0^2-z_u^2}-\frac{\alpha L}{2}+\frac{\alpha L}{2}\exp[2\sqrt{z_0^2-z_u^2}(1-\frac{R}{L})]}\right)$$ where\
$$z_u=la \hspace{10pt} \& \hspace{10pt} z_0=a\sqrt{2V_0}$$ This is another transcendental equation for $z_u$(and hence for $E_u$) as function of $z_0$ which is solved by graphical method to get the value of energy(E) for antisymmetric states.\
The electronic coupling between the donor and the acceptor equals to half the energy differences of lowest antisymmetric and symmetric states[@V.P.Zhdanov].i.e $$G_{DA} =\frac{\left(E_u-E_s\right)}{2}$$ using the values from Eq. (11), (25) and (29) we get, $$E_u=\frac{z_u^2}{2a^2}-V_0\;and\;E_s=\frac{z_s^2}{2a^2}-V_0$$ hence Eq. (30) becomes:\
$$G_{DA} =\frac{\left(z_u^2-z_s^2\right)}{L^2}$$ This is expression for electronic coupling for system having conjugated bridge. Now we analyses the case where energy of the bridge level is appreciably higher than the energies of lowest antisymmetric and symmetric states.
Results
=======
We choose the parameter $ \alpha = \frac{1}{\sqrt{2}}, LV_0 = \frac{2}{\sqrt{3}}$ such that energy of donor, acceptor and bridge is comparable to the binding energies in real electron transfer reactions[@V.P.Zhdanov] and analyze our formalism for electronic coupling vs internuclear distances between donor and acceptor ranging from (16 - 50) atomic units. The results for electronic coupling are shown in Figure (2a, 2b, 3a, 3c). Figure 2a shows the exponential variation of electronic coupling vs internuclear distances between donor and acceptor calculated by us for conjugated bridge mediated electron transfer processes while solving Eq. (26) and Eq. (28) by graphical method and plugging the results into Eq. (30). The reason for choosing this range is that electron transfer reactions in proteins and other conjugated systems generally occurs within this range[@David]. Figure 2b shows the logarithmic variation of electronic coupling vs internuclear distances between donor and acceptor. The reason for taking log values is to compare our exact analytical results with the already published results for such type of coupling by others i.e Figure 3a[@V.P.Zhdanov]. From this comparison we conclude that the value of electronic coupling calculated by other authors comes in the range from $10^{-4}\; to \;10^{-14}$ atomic units by varying the internuclear distances from (16-50) atomic units, however using our exact analytical results this values ranges from $10^{-1}\; to \;10^{-2}$ atomic units over same range of internuclear distances between donor and acceptor thereby showing that the value of electronic coupling is better in our case. Figure 3b shows the area of direct overlapping( i.e. the area of overlapping between donor and acceptor wave functions shown in Figure 4) in atomic units vs internuclear distances between donor and acceptor for the bridge mediated electron transfer processes which shows that if we move below the 16 atomic units then the electronic transfer may occurs by direct overlapping instead of bridge between the donor and the acceptor. The figure 3c shows that how the electronic coupling ( which is the exponential function of internuclear distances(R) between donor and acceptor) is decreasing with increasing length of different conjugated bridges having different length. Our results which shown in figure 2e are qualitatively similar to the earlier published experimental and computational results in Fig.14[@B] for different conjugated bridges ( i.e. Oligoethynylenes (OE), Phenyle-Oligoethynylenes (Ph-OE) and Oligo-p-phenyleneethynylene (OPE)) having different length for same donor and acceptor(Pentacene) bridge-mediated electron transfer systems. In both cases value of electronic coupling ($G_{DA}$) decreases as length of bridge increases. Our results have same behavior as the earlier published computational results shown in Table 6.[@Y] by Yinxi Yu and coworkers, which shows that the value of electronic coupling decreases with increases the length of bridge for bridge-mediated intervalence electron transfer in different metallocene complexes. Our model for bridge-mediated electron transfer processes has advantages over earlier model because our model is universal model and it can be use to calculate or estimate the value of electronic coupling ($ G_{DA}$) without doing any computational or experimental work for any conjugate bridge-mediated electron transfer processes if we know the values of binding energies of donor/acceptor molecule and bridge molecule of the systems( e.g. supermolecules) involve electron transfer processes. Hence we can calculate the rate of electron transfer for bridge-mediated electron transfer processes for such systems.
Conclusion
==========
On comparison of results obtained by our exact analytical method in which we assumed that electron transfer through conjugated bridge in electron transfer processes with the already published results for such type of coupling by author[@V.P.Zhdanov] we conclude results obtained by our method are better and qualitatively similarity of our results with the earlier published experimental and computational results by authors[@B; @Y] we conclude that our model is easier model to calculate the value of electronic coupling($G_{DA}$) hence rate of electron transfer for conjugated bridge-mediated electron transfer processes. Thus our exact analytical method is better to calculate the long-range conjugated bridge-mediated electron transfer rate for real physical, chemical and biological processes.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, Himachal Pradesh 175001, India for financial support and providing necessary facilities to accomplish this work.
References {#references .unnumbered}
==========
[00]{} V. L. Davidson , Acc. Chem. Res. 33 (2000) 87. G. R.Moore, Williams, R. J. P. Coord. Chem. Rev. 18 (1976) 125. S. Priyadarsh, S. S. Skourtis, J. Chem. Phys. 104 (1996) 9473. H.M. McConnell , J.Chem.Phys. 35 (1961) 508. P.F. Barbara, T.J. Meyer, and M.A. Ratner, J.Phys.Chem. 100 (1996) 13148. M.R. Wasieleswski, Chem.Rev. 92 (1992) 435. B. D. William, A. S. Walter, A. R. Marker, R. W. Michael. Nature 396 (1998). B. Albinsson, M.P. Eng, K. Pettersson, M.U. Winters, Phys. Chem. Chem. Phys. 9 (2007) 5847. Y. Yu, H. Wang, S. Chen, J. Theor. Comp. Chem. 11 (2012) 1341. J. L. Brends, D. Beljonne, V. Coropceanu, J. Cornil, Chem. Rev. 104 (2004) 4971. V. P. Zhdanov, Europhys. Lett. 59 (2002) 681. N. Gorczak, S. Tarkuc, N. Renaud, A. J. Houtepen, R. Eelkema, L. D. A. Siebbeles, F. C. Grozema, J. Phys. Chem. A 118 (2014) 3891. S.Larsson , J.Am.Chem.Soc. 103 (1981) 4034. J. Wolfgang , S. M. Risser, S. Priyadarshy, D.N. Beratan, J.Phy.Chem. B 101 (1997) 2986. I.A. Balabin, J.N Onuchic, J.Phys.Chem. B 102 (1998) 7497. A. Onipko, Y. Klymenko, J.Phy.Chem. A 102 (1998) 4246. S. Creager, C. J. Yu, C. Bamdad, S. O’Connor, T. MacLean, E. Lam, Y. Chong, G.T. Olsen, J. Luo, M. Gozin, J.F. Kayem J. Am. Chem. Soc. 121 (1999) 1059. A.A. Voityuk, Phys. Chem. Chem. Phys. 14 (2012) 13789. D.J. Bicout, F.Varchon, E. Kats, Europhys. Lett. 70 (2005) 457. N.S. Hush, Trans. Faraday Soc. 57 (1961) 155. J.J. Hopfield, Proc.Natl. Acad. Sci. U.S.A. 71 (1974) 3640. M. Bixon, J. Jortner, J.Chem.Phys. 48 (1968) 715. J.Jortner, ibid. 64 (1976) 4860. R.A. Marcus, J.Chem.Phys. 24 (1956) 979.
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abstract: 'We investigate the orbital stability and instability of standing waves for two classes of Klein-Gordon equations in the semi-classical regime.'
address:
- 'Dipartimento di Matematica Applicata Università di Pisa Via F. Buonarroti 1/c, 56127 Pisa Italia'
- 'Institut de Mathématiques de Toulouse, Université Paul Sabatier 118 route de Narbonne, 31062 Toulouse Cedex 9 France'
- 'Dipartimento di Informatica Università di Verona Strada Le Grazie 15, 37134 Verona Italia'
author:
- Marco Ghimenti
- Stefan Le Coz
- Marco Squassina
bibliography:
- 'biblio.bib'
title: 'On the stability of standing waves of Klein-Gordon equations in a semiclassical regime'
---
Introduction and results
========================
The nonlinear Klein-Gordon equation $$\label{kg}
\varepsilon^2 u_{tt}-{\varepsilon}^2\Delta u+mu-|u|^{p-1}u=0 \qquad (t,x)\in{{\mathbb R}}^+\times {{\mathbb R}}^N,$$ where ${\varepsilon},m>0$, $p>1$ for $N=1,2$ and $1<p<(N+2)/(N-2)$ for $N\geq 3$, is one of the simplest nonlinear partial differential equations invariant for the Poincaré group. We are interested in the study of the nonlinear Klein Gordon equation in presence of a potential depending on the space variable. Two different choices are viable. We can simply add a potential term $W(x)u$ to equation . This case has been studied, for the linear wave equation, for example, by Beals and Strauss in [@BS]. Otherwise, if we want to fully preserve the invariance for the Poincaré group of , we have to change the temporal derivative ${\varepsilon}^2\partial_{tt}$ with a covariant derivative, depending on the potential $D^2_{tt}$, where $D_t= {\varepsilon}\partial_t +{{\rm i}}V(x)$. This second approach is classical, in quantum electrodynamics, when considering electromagnetic waves. The first approach leads us to consider the equation $$\label{seconmod}
{\varepsilon}^2 u_{tt}-{\varepsilon}^2\Delta u+mu-Wu-|u|^{p-1}u=0, \qquad\text{in }{{\mathbb R}}^N,$$ while the second one to investigate the equation $$\label{primod}
{\varepsilon}^2 u_{tt}+2{{\rm i}}{\varepsilon}Vu_t-{\varepsilon}^2\Delta u+mu-V^2u-|u|^{p-1}u=0, \qquad\text{in }{{\mathbb R}}^N.$$ In this paper, we treat simultaneously the two previous Klein-Gordon equations by studying $$\label{genequation}
{\varepsilon}^2 u_{tt}+2{{\rm i}}{\varepsilon}Vu_t-{\varepsilon}^2\Delta u+mu-Wu-|u|^{p-1}u=0, \qquad \text{ in }{{\mathbb R}}^N,$$ where $u:{{\mathbb R}}^N\times{{\mathbb R}}\to{{\mathbb C}}$, ${\varepsilon}>0$ and $V,W$ are real valued potential functions. Equation formally yields to for the choice $W=V^2$ as well as to when $V=0$. We shall study the stability of standing waves of this equation in the semiclassical regime ${\varepsilon}\to 0$. It admits standing waves, namely solutions of the form $u(x,t)=e^{{{\rm i}}\omega t/{\varepsilon}}\varphi_\omega(x/{\varepsilon})$, where $\omega\in{{\mathbb R}}$ and $\varphi_\omega$ satisfies $$\label{ellip-omega}
-\Delta \varphi_\omega+\big(m-\omega^2-2\omega V({\varepsilon}y)-W({\varepsilon}y)\big)\varphi_\omega-|\varphi_\omega|^{p-1}\varphi_\omega=0, \qquad\text{in }{{\mathbb R}}^N.$$ To ensure existence of solutions to for ${\varepsilon}$ close to $0$, we assume the following. Let $V$ and $W$ be $\mathcal{C}^2$. For the function $$Z(y)=m-\omega^2-2\omega V(y)-W(y),\,\,\ y\in{{\mathbb R}}^N$$ there exists $x_0\in{{\mathbb R}}^N$ such that $$\label{criticality}
\nabla Z(x_0) = 0,\qquad \nabla^2 Z(x_0)\text{ is non-degenerate.}$$ Furthermore, we assume that $$\label{positivity}
Z(x_0)=m-\omega^2-2\omega V(x_0)-W(x_0)>0.$$
Under these hypotheses, it is well-know (see e.g. [@AmBaCi97] or [@AmMa06 Section 8.2]) that when ${\varepsilon}$ is close to $0$ the equation admits a family of positive, exponentially decaying, solutions $\varphi_\omega\subset H^1({{\mathbb R}}^N)$ (hiding the dependance upon ${\varepsilon}$). More precisely, there exists $\xi_{\varepsilon}\in{{\mathbb R}}^N$ and $\psi_\omega\in H^1({{\mathbb R}}^N)$ such that $\varphi_{\omega}(\cdot)= \psi_\omega(\cdot-\xi_{\varepsilon})+{{\mathcal O}}({\varepsilon}^2)$ in $H^1({{\mathbb R}}^N)$ as ${\varepsilon}\to 0$, where $\xi_{\varepsilon}=o({\varepsilon})$ and $$-\Delta \psi_\omega+Z(x_0)\psi_\omega=|\psi_\omega|^{p-1}\psi_\omega, \qquad\text{in ${{\mathbb R}}^N$}.$$
We are interested in the (orbital) stability or instability of standing waves when ${\varepsilon}$ goes to $0$.
A standing wave of is said to be *(orbitally) stable* if any solution of starting close to the standing wave remains close for all time, up to the invariances of the equation. More precisely, we say that $e^{\frac{{{\rm i}}\omega t}{{\varepsilon}}}\varphi_\omega\left(\frac{x}{{\varepsilon}}\right)$ is stable if for all $\eta>0$ there exists $\delta>0$ such that for all $u_0\in H^1({{\mathbb R}}^N)$ verifying $\nhu{u_0-\varphi_\omega}<\delta$ the solution $u(x,t)$ of with initial data $u_0$ satisfies $$\sup_{t\in{{\mathbb R}}}\inf_{\theta\in{{\mathbb R}}}\nhu{u-e^{i\theta}\varphi_\omega}<\eta.$$ Since the pioneering works [@BeCa81; @CaLi82; @GrShSt87; @GrShSt90; @We83; @We85], the study of orbital stability for standing waves of dispersive PDE has attracted a lot of attention. Among many others, one can refer to [@JeLe06; @JeLe09; @LeFuFiKsSi08]; see also the books and surveys [@Ca03; @Le09; @St08; @Ta09] and the references therein. Relatively few works [@IaLe09; @LiWe08; @Oh89] are concerned with stability at the semi-classical limit for Schrödinger type equations. For Klein-Gordon equations, after the ground works [@ShSt85; @Sh85], there has been a recent interest for stability by blow-up [@OhTo05; @OhTo06; @OhTo07; @LiOhTo07].
To study stability, we first rewrite in Hamiltonian form $$\label{eq:hamilt}
{\varepsilon}\frac{\partial U}{\partial t}=JE'(U),$$ where $U=\begin{pmatrix}u\\v\end{pmatrix}$, $J=\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}$, and $$\begin{gathered}
E(U)=\frac12\nldd{v-i Vu}+\frac12\nldd{\nabla u}+m\frac12\nldd{u}\\-\frac12\int_{{{\mathbb R}}^N}W|u|^2dx-\frac{1}{p+1}\nlpsps{u}.\end{gathered}$$ It is easy to see that if $u$ solves and $v$ is defined by $v:={\varepsilon}u_t+i V u$, then $U=\begin{pmatrix}u\\v\end{pmatrix}$ solves . The charge $Q(U)$ is defined by $$Q(U)=\Im\int_{{{\mathbb R}}^N}\bar{u}vdx.$$ In particular, for a standing wave $u=e^{i\omega t/{\varepsilon}}\varphi_\omega(x/{\varepsilon})$, the charge is given by $$\label{eq:defQ}
Q(\varphi_\omega):=Q(U)={\varepsilon}^N\left(\omega\nldd{\varphi_\omega}+\int_{{{\mathbb R}}^N} V({\varepsilon}y)|\varphi_\omega|^2\right).$$
According to the theory developed in [@GrShSt87; @GrShSt90], a standing wave $e^{\frac{{{\rm i}}\omega t}{{\varepsilon}}}\varphi_\omega\left(\frac{x}{{\varepsilon}}\right)$ is stable if two conditions are satisfied.
- The *Slope Condition*: $\displaystyle \frac{\partial}{\partial \omega}Q(\varphi_\omega)<0$.
- The *Spectral Condition*: $L_{\varepsilon}:=-\Delta+Z({\varepsilon}y)-p|\varphi_\omega|^{p-1}$ has exactly one negative eigenvalue and is non degenerate.
On the other hand, denote by $n(L_{\varepsilon})$ the number of negative eigenvalues of $L_{\varepsilon}$ and set $p(\omega) =
0$ if $\displaystyle \frac{\partial}{\partial \omega}Q(\varphi_\omega)>0$, $p(\omega) = 1$ if $\displaystyle \frac{\partial}{\partial \omega}Q(\varphi_\omega)<0$. Then the standing wave is unstable if $n(L_{\varepsilon})-p(\omega)$ is odd.
Then we have the following
\[mainthm\] Assume that conditions - hold. Then, we have the following facts.
1. If $p<1+4/N$, then the Slope Condition $\frac{\partial}{\partial\omega} Q(\varphi_\omega)<0$ is fulfilled if $$Z(x_0)<(\omega+V(x_0))^2\Big(\frac{4}{p-1}-N\Big) \qquad\text{(non-critical case)}$$ or if $$\left\{
\begin{array}{l}
Z(x_0)=(\omega+V(x_0))^2\Big(\frac{4}{p-1}-N\Big), \\
\left(\Delta Z(0)-\Delta V(0)\left(1+\frac{2(\omega+V(0))}{Z(0)}\right)\right)<0,
\end{array}
\right.
\qquad\text{(critical case).}$$
2. If $p\geq 1+4/N$, then we always have $\frac{\partial}{\partial\omega} Q(\varphi_\omega)>0$.
3. We have the equality $n(L_{\varepsilon})=n(\nabla^2Z(x_0))+1$, where $n(\nabla^2Z(x_0))$ is the number of negative eigenvalues of $\nabla^2Z(x_0)$.
In particular, the standing waves $e^{i\omega t}\varphi_\omega$ are stable if $x_0$ is non-degenerate local minimum of $Z$, $p<1+4/N$ and $$Z(x_0)<(\omega+V(x_0))^2\Big(\frac{4}{p-1}-N\Big).$$
Note that, conversely to what was happening in the case of Schrödinger equations studied in [@LiWe08], the values of the potentials $V$ and $W$ at $x_0$ comes into play for the Slope Condition even in the noncritical case. Note also that only the local behavior of $Z$ around $x_0$ influences the stability or instability.
Notations : Most of the objects we consider will depend both on ${\varepsilon}$ and $\omega$. We will emphasize the most important parameter by indicating it as a subscript, the dependence in the other parameter being understood.
Proof of Theorem \[mainthm\]
============================
In this section, we prove Theorem \[mainthm\]. We start be focusing on the Slope Condition and then we study the Spectral Condition. For the sake of simplicity in notations and without loss of generality, in the rest of this section we assume that $x_0=0$.
The Slope Condition
-------------------
We start with the noncritical case.
### Noncritical case
We assume that $$Z(0)\neq(\omega+V(0))^2\Big(\frac{4}{p-1}-N\Big).$$
We first rewrite $Q(\varphi_\omega)$ by expanding $V({\varepsilon}y)$ and using the exponential decay of $\varphi_\omega$: $$Q(\varphi_\omega)={\varepsilon}^N\left( \omega+V(0) \right)\nldd{\varphi_\omega}+{{\mathcal O}}({\varepsilon}^{N+1}).$$ Therefore, since $$\label{Q1}
\frac{\partial}{\partial\omega} Q(\varphi_\omega)={\varepsilon}^N
\|\varphi_\omega\|_{L^2({{\mathbb R}}^N)}^2+{\varepsilon}^N(\omega+V(0) )\frac{\partial}{\partial\omega}\|\varphi_\omega\|_{L^2({{\mathbb R}}^N)}^2+{{\mathcal O}}({\varepsilon}^{N+1}),$$ to evaluate the sign of the map $\omega\mapsto \frac{\partial}{\partial\omega} Q(\varphi_\omega)$ one should compute the quantity $$\label{Q2}
\frac{\partial}{\partial\omega}\|\varphi_\omega\|_{L^2({{\mathbb R}}^N)}^2
=2\int_{{{\mathbb R}}^N} R_\omega\varphi_\omega,$$ where $R_\omega(x):=\frac{\partial\varphi_\omega}{\partial\omega}(x)$.
We remark that differentiation of with respect to $\omega$ easily yields $$\label{identitt}
L_{\varepsilon}R_\omega=2(\omega +V({\varepsilon}y))\varphi_\omega.$$ If we now introduce the rescaling $\varphi_\omega(x)=\lambda^{\frac{1}{p-1}}\varphi_\lambda(\sqrt{\lambda}x)$, it follows that $\varphi_\lambda$ satisfies $$\label{ellip-lambda}
-\Delta \varphi_\lambda+
\lambda^{-1}Z\left(\frac{{\varepsilon}y}{\sqrt{\lambda}}\right)\varphi_\lambda
-|\varphi_\lambda|^{p-1}\varphi_\lambda=0, \qquad\text{in ${{\mathbb R}}^N$}.$$ Now, differentiating equation with respect to $\lambda$ and denoting $T_\lambda=\frac{\partial\varphi_\lambda}{\partial\lambda}_{|\lambda=1}$ yields $$\label{oplin}
L_{\varepsilon}T_\lambda-
Z({\varepsilon}y)\varphi_\omega
-\frac{1}{2} {\varepsilon}y\cdot\nabla Z({\varepsilon}y)\varphi_\omega=0.$$
Since $0$ is a critical point of $Z$, a Taylor expansion gives $$\begin{gathered}
\label{sviluppo1}
Z({\varepsilon}y)=Z(0)+{{\mathcal O}}({\varepsilon}^2|y|^2),\\
\label{sviluppo2}
\frac{1}{2} {\varepsilon}y\cdot\nabla Z({\varepsilon}y)={{\mathcal O}}({\varepsilon}^2|y|^2).\end{gathered}$$ Then, from , as ${\varepsilon}\to 0$ we have $$\label{oplin-appr}
L_{\varepsilon}T_\lambda=Z(0)\varphi_\omega+{{\mathcal O}}({\varepsilon}^2|y^2|\varphi_\omega),\quad\text{in ${{\mathbb R}}^N$}.$$ Then, in turn, taking into account identity we get $$\begin{aligned}
Z(0)\int_{{{\mathbb R}}^N} R_\omega\varphi_\omega&=\int_{{{\mathbb R}}^N} R_\omega L_{\varepsilon}T_\lambda+{{\mathcal O}}({\varepsilon}^2) \nonumber\\
&=\int_{{{\mathbb R}}^N} L_{\varepsilon}R_\omega T_\lambda+{{\mathcal O}}({\varepsilon}^2) \nonumber \\
&=\int_{{{\mathbb R}}^N} 2(\omega +V({\varepsilon}y))\varphi_\omega T_\lambda+{{\mathcal O}}({\varepsilon}^2) \label{eq:nondegcase}\\
&=2(\omega+V(0))\int_{{{\mathbb R}}^N}\varphi_\omega T_\lambda+{{\mathcal O}}({\varepsilon}) \nonumber\\
&=(\omega+V(0))\frac{\partial}{\partial\lambda}{\|\varphi_\lambda\|_{L^2({{\mathbb R}}^N)}^2}_{|\lambda=1}+{{\mathcal O}}({\varepsilon}) \nonumber\\
&=(\omega+V(0))\Big(\frac{N}{2}-\frac{2}{p-1}\Big)\|\varphi_\omega\|_{L^2({{\mathbb R}}^N)}^2+{{\mathcal O}}({\varepsilon}). \nonumber\end{aligned}$$ In conclusion, by combining , and , we have $$\frac{\partial}{\partial\omega} Q(\varphi_\omega)={\varepsilon}^N
\left(1+\frac{(\omega+V(0))^2}{Z(0)}\Big(N-\frac{4}{p-1}\Big)\right)
\|\varphi_\omega\|_{L^2({{\mathbb R}}^N)}^2+{{\mathcal O}}({\varepsilon}^{N+1}).$$ Then, taking into account the fact that $Z(0)>0$ and that $\varphi_\omega$ converges to $\psi_\omega$ in $L^2({{\mathbb R}}^N)$ as ${\varepsilon}\to 0$, the sign of $\frac{\partial}{\partial\omega} Q(\varphi_\omega)$ is the sign of $$Z(0)-(\omega+V(0))^2\Big(\frac{4}{p-1}-N\Big).$$
### Critical case
We assume now that $$\label{eq:degencase}
Z(0)=(\omega+V(0))^2\Big(\frac{4}{p-1}-N\Big).$$ In the critical case, the term of order ${\varepsilon}^N$ in the expansion of $\frac{\partial }{\partial \omega}Q(\varphi_\omega)$ vanishes and we need to calculate the expansion at a higher order. We first refine -. $$\begin{gathered}
Z({\varepsilon}y)=Z(0)+\frac{{\varepsilon}^2}{2}\nabla^2Z(0)(y,y)+{{\mathcal O}}({\varepsilon}^3|y|^3)\\
\frac{1}{2} {\varepsilon}y\cdot\nabla Z({\varepsilon}y)=\frac{{\varepsilon}^2}{2} \nabla^2 Z(0)(y,y)+{{\mathcal O}}({\varepsilon}^3|y|^3).\end{gathered}$$ Then gives $$L_{\varepsilon}T_\lambda=Z(0)\varphi_\omega+{\varepsilon}^2\nabla^2 Z(0)(y,y)\varphi_\omega+{{\mathcal O}}({\varepsilon}^3|y^3|)\varphi_\omega.$$ Now, we have $$\label{eq:hessian1}
Z(0)\int_{{{\mathbb R}}^N}R_\omega\varphi_\omega=\int_{{{\mathbb R}}^N}R_\omega L_{\varepsilon}T_\lambda-{\varepsilon}^2\int_{{{\mathbb R}}^N}\nabla^2 Z(0)(y,y)R_\omega\varphi_\omega+{{\mathcal O}}({\varepsilon}^3).$$ From , we obtain $$\label{eq:comeback}
\int_{{{\mathbb R}}^N}R_\omega L_{\varepsilon}T_\lambda=\int_{{{\mathbb R}}^N}L_{\varepsilon}R_\omega T_\lambda=\int_{{{\mathbb R}}^N} 2(\omega+V({\varepsilon}y))\varphi_\omega T_\lambda.$$ Expanding the potential $V$ we get $$\begin{gathered}
\label{eq:Vexpansion}
\int_{{{\mathbb R}}^N} 2V({\varepsilon}y)\varphi_\omega T_\lambda=\int_{{{\mathbb R}}^N} 2V(0)\varphi_\omega T_\lambda+2{\varepsilon}\int_{{{\mathbb R}}^N}
y\cdot\nabla V(0)\varphi_\omega T_\lambda\\+{\varepsilon}^2\int_{{{\mathbb R}}^N} \nabla^2V(0)(y,y)\varphi_\omega T_\lambda+{{\mathcal O}}({\varepsilon}^3).\end{gathered}$$ Note that since $\varphi_\omega=\psi_\omega(\cdot-\xi_{\varepsilon})+{{\mathcal O}}({\varepsilon}^2)$ and $\xi_{\varepsilon}=o({\varepsilon})$, we have $$\label{eq:cancelation}
2{\varepsilon}\int_{{{\mathbb R}}^N} y\cdot\nabla V(0)\varphi_\omega T_\lambda=2{\varepsilon}\int_{{{\mathbb R}}^N} y\cdot\nabla V(0)\psi_\omega T_\lambda+o({\varepsilon}^2)=o({\varepsilon}^2)$$ where the last cancellation comes from the fact that $\psi_\omega$ is radial. Coming back to and as in , we have $$\begin{gathered}
\label{eq:hessian2}
\int_{{{\mathbb R}}^N}R_\omega L_{\varepsilon}T_\lambda
=(\omega+V(0))\Big(\frac{N}{2}-\frac{2}{p-1}\Big)\|\varphi_\omega\|_{L^2({{\mathbb R}}^N)}^2\\+{\varepsilon}^2\int_{{{\mathbb R}}^N} \nabla^2V(0)(y,y)\varphi_\omega T_\lambda+o({\varepsilon}^2).\end{gathered}$$
It remains to compute the integrals involving the Hessians in and . Since our problem is invariant by an orthonormal transformation, we can assume without loss of generality that $\nabla^2V(0)=\mathrm{diag}(b_1,\dots,b_N)$. Hence $\nabla^2V(0)(y,y)=\sum_{j=1}^Nb_jy_j^2$. Recall also that $T_\lambda$ can be computed explicitly to have $$T_\lambda=-\frac{1}{p-1}\varphi_\omega-\frac{1}{2}y\cdot\nabla\varphi_\omega.$$ Therefore, $$\int_{{{\mathbb R}}^N} b_jy_j^2\varphi_\omega T_\lambda=-\frac{b_j}{p-1}\int_{{{\mathbb R}}^N}y_j^2\varphi_\omega^2-\frac{b_j}{2}\sum_{k=1}^N\int_{{{\mathbb R}}^N}y_j^2y_k\varphi_\omega\frac{\partial\varphi_\omega}{\partial y_k}.$$ We have after integration by parts $$2\sum_{k=1}^N\int_{{{\mathbb R}}^N}y_j^2y_k\varphi_\omega\frac{\partial\varphi_\omega}{\partial y_k}=-\sum_{k=1}^N\int_{{{\mathbb R}}^N}(y_j^2+2\delta_{jk}y_j^2)\varphi_\omega^2=-(N+2)\int_{{{\mathbb R}}^N}y_j^2\varphi_\omega^2.$$ Thus $$\int_{{{\mathbb R}}^N} \nabla^2V(0)(y,y)\varphi_\omega T_\lambda=\sum_{j=1}^N\int_{{{\mathbb R}}^N} b_jy_j^2\varphi_\omega T_\lambda=-\left(\frac{1}{p-1}-\frac{N+2}{4}\right)\sum_{j=1}^Nb_j\int_{{{\mathbb R}}^N}y_j^2\varphi_\omega^2.$$ Recall the following expansion in ${\varepsilon}$ for $R_\omega$ and $\varphi_\omega$. $$\varphi_\omega=\psi_\omega+o({\varepsilon}),\qquad
R_\omega=\frac{\partial\psi_\omega}{\partial \omega}+o({\varepsilon}).$$ Therefore, since $\psi_\omega$ is radial, $$\int_{{{\mathbb R}}^N}y_j^2\varphi_\omega^2=\int_{{{\mathbb R}}^N}y_j^2\psi_\omega^2+o({\varepsilon})=\frac{1}{N}\nldd{|y|\psi_\omega}+o({\varepsilon}),$$ and so $$\begin{gathered}
\label{eq:star}
\int_{{{\mathbb R}}^N} \nabla^2V(0)(y,y)\varphi_\omega T_\lambda=\\-\left(\frac{1}{p-1}-\frac{N+2}{4}\right)\frac{1}{N}\nldd{|y|\psi_\omega}\Delta V(0)+o({\varepsilon}).\end{gathered}$$ Similarly, we have $$\begin{gathered}
\label{eq:starstar}
\int_{{{\mathbb R}}^N}\nabla^2 Z(0)(y,y)R_\omega\varphi_\omega=\\-\left(\frac{1}{p-1}-\frac{N+2}{4}\right)
\frac{1}{N}\nldd{|y|\psi_\omega}\Delta Z(0)+o({\varepsilon}).\end{gathered}$$
Summarizing, using successively , , , and we have obtained $$\begin{gathered}
\label{eq:summarize}
\int_{{{\mathbb R}}^N}R_\omega\varphi_\omega=-\frac{1}{2(\omega+V(0))}\nldd{\varphi_\omega}\\+{\varepsilon}^2\frac{(\Delta Z(0)-\Delta V(0))}{NZ(0)}\left(\frac{1}{p-1}-\frac{N+2}{4}\right)\nldd{|y|\psi_\omega}+o({\varepsilon}^2).\end{gathered}$$
Now, we compute $\frac{\partial Q(\varphi_\omega)}{\partial\omega}$. First, recall that, coming back to the definition of $Q$, we have $${\varepsilon}^{-N}\frac{\partial Q(\varphi_\omega)}{\partial\omega}=\nldd{\varphi_\omega}+2\omega\int_{{{\mathbb R}}^N}R_\omega\varphi_\omega+2\int_{{{\mathbb R}}^N}V({\varepsilon}y)R_\omega\varphi_\omega.$$ As in , , and we can expand in ${\varepsilon}$ and get $$\begin{gathered}
2\int_{{{\mathbb R}}^N}V({\varepsilon}y)R_\omega\varphi_\omega=2V(0)\int_{{{\mathbb R}}^N} R_\omega\varphi_\omega\\
-{\varepsilon}^2\left(\frac{1}{p-1}-\frac{N+2}{4}\right)\frac{1}{N}\nldd{|y|\psi_\omega}\Delta V(0)+o({\varepsilon}^2).\end{gathered}$$ Therefore, $$\begin{gathered}
{\varepsilon}^{-N}\frac{\partial Q(\varphi_\omega)}{\partial\omega}=\nldd{\varphi_\omega}+2(\omega+V(0))\int_{{{\mathbb R}}^N}R_\omega\varphi_\omega\\
-{\varepsilon}^2\left(\frac{1}{p-1}-\frac{N+2}{4}\right)\frac{1}{N}\nldd{|y|\psi_\omega}\Delta V(0)+o({\varepsilon}^2).\end{gathered}$$ Using , we finally get $$\begin{gathered}
{\varepsilon}^{-N}\frac{\partial Q(\varphi_\omega)}{\partial\omega}={\varepsilon}^2\left(\frac{1}{p-1}-\frac{N+2}{4}\right)\frac{1}{N}\nldd{|y|\psi_\omega}\times\\
\times\left(\Delta Z(0)-\Delta V(0)\left(1+\frac{2(\omega+V(0))}{Z(0)}\right)\right)+o({\varepsilon}^2).\end{gathered}$$
The Spectral Condition
----------------------
We define the operator $L_0:=-\Delta+Z(0)-p\psi_\omega^{p-1}$. It is well known (see e.g. [@AmMa06]) that the spectrum of $L_0$ consists of one negative eigenvalue, a $N$-dimensional kernel (generated by $\frac{\partial \psi_\omega}{\partial y_j}$ for $j=1,\dots,N$) and the rest of the spectrum is bounded away from $0$. When ${\varepsilon}$ is close to $0$, the spectrum of $L_{\varepsilon}$ will be close to the spectrum of $L_0$. In particular, the $0$ eigenvalue, of multiplicity $N$, will transform into $N$ possibly different eigenvalues close to $0$ but shifted either to the positive or to the negative side of the real axis, depending on the sign of the eigenvalues of the Hessian of $Z$ at $0$. More precisely, the following proposition was proved in [@LiWe08] (see [@IaLe09] for a detailed justification).
\[prop:spectrum\] The spectrum of $L_{\varepsilon}$ consists of positive spectrum away from $0$ and a set of $N+1$ simple eigenvalues $\{\lambda_0,\lambda_1,\dots,\lambda_N\}$ such that $$\lambda_0<\lambda_1\leq\cdots\leq\lambda_N.$$ As ${\varepsilon}\to0$, we have $\lambda_0<0$ and the following asymptotic expansion holds for the other eigenvalues: $$\lambda_j=c_j{\varepsilon}^2+o({\varepsilon}^2),\qquad j=1,...,N,$$ where $c_j=\frac{1}{2}\frac{\nldd{\psi_\omega}}{\nldd{\frac{\partial\psi_\omega}{\partial x_j}}}a_j$ and $\{a_1,\dots,a_N\}$ are the eigenvalues of the Hessian matrix $\nabla^2Z(0)$.
Therefore, (3) in Theorem \[mainthm\] is a direct consequence of Proposition \[prop:spectrum\]. In particular, the spectral condition for stability will be satisfied if and only if $0$ is a non-degenerate local minimum of $Z$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The jet opening angle and the bulk Lorentz factor are crucial parameters for the computation of the energetics of Gamma Ray Bursts (GRBs). From the $\sim$30 GRBs with measured or it is known that: (i) the real energetic , obtained by correcting the isotropic equivalent energy for the collimation factor $\sim\theta^2_{\rm jet}$, is clustered around $10^{50}$–$10^{51}$ erg and it is correlated with the peak energy of the prompt emission and (ii) the comoving frame and are clustered around typical values. Current estimates of and are based on incomplete data samples and their observed distributions could be subject to biases. Through a population synthesis code we investigate whether different assumed intrinsic distributions of and can reproduce a set of observational constraints Assuming that all bursts have the same and in the comoving frame, we find that and cannot be distributed as single power–laws. The best agreement between our simulation and the available data is obtained assuming (a) log–normal distributions for and and (b) an intrinsic relation between the peak values of their distributions, i.e $^{2.5}$=const. On average, larger values of (i.e. the “faster" bursts) correspond to smaller values of (i.e. the “narrower"). We predict that $\sim$6% of the bursts that point to us should not show any jet break in their afterglow light curve since they have $\sin\theta_{\rm jet}<1/$. Finally, we estimate that the local rate of GRBs is $\sim$0.3% of all local SNIb/c and $\sim$4.3% of local hypernovae, i.e. SNIb/c with broad–lines.'
author:
- |
G. Ghirlanda$^{1}$[^1], G. Ghisellini$^{1}$, R. Salvaterra$^{2}$, L. Nava$^{3}$, D. Burlon$^{4}$, G. Tagliaferri$^{1}$, S. Campana$^{1}$, P. D’Avanzo$^{1}$, A. Melandri$^{1}$\
$^{1}$INAF – Osservatorio Astronomico di Brera, via E. Bianchi 46, I-23807 Merate, Italy\
$^{2}$INAF - IASF Milano, via E. Bassini 15, I-20133 Milano, Italy\
$^{3}$APC Université Paris Diderot, 10 rue Alice Domon et Leonie Duquet, F-75205 Paris Cedex 13, France\
$^{4}$Sydney Institute for Astronomy, School of Physics, The University of Sydney, NSW 2006, Australia\
title: 'The faster the narrower: characteristic bulk velocities and jet opening angles of Gamma Ray Bursts'
---
\[firstpage\]
Gamma-ray: bursts
Introduction
============
Gamma Ray Bursts (GRBs) have extremely high energetics. The isotropic equivalent energy , released during the prompt phase, is distributed over four orders of magnitudes in the range 10$^{50-54}$ erg. correlates with , i.e. the peak of the $\nu F_{\nu}$ spectrum (Amati et al. 2002, 2009): $E_{\rm p}\propto E_{\rm iso}^{0.5}$. This holds for long duration GRBs. A similar correlation exists between the isotropic equivalent luminosity and (Yonetoku et al. 2004) obeyed also by short events (Ghirlanda et al. 2009). The scatter of the data points around the correlation, modeled with a Gaussian, has a dispersion $\sigma_{\rm sc}=$ 0.23 dex (see e.g. Nava et al. 2012 for a recent update of these correlations). This dispersion is much larger than the average statistical error $\bar{\sigma}_{E_{\rm iso}}=0.06$ dex and $\bar{\sigma}_{E_{\rm peak}}=0.10$ dex associated with and , respectively.
Since is computed assuming that GRBs emit isotropically, it is only a proxy of the real GRB energetic. GRBs are thought to emit their radiation within a jet of opening angle . If the jet opening angle is known, the [*true energy*]{} $\simeq$$\theta_{\rm jet}^2$ and the [*true GRB rate*]{} can be estimated (Frail et al. 2001).
The estimate of is made possible by the measure of the jet break time , typically observed between 0.1 to $>$10 days in the afterglow optical light curve. Although has been measured only for $\sim$30 GRBs (Ghirlanda et al. 2007) it shows that:
1. clusters around $10^{50}$ erg with a small dispersion (Frail et al. 2001; but see Racusin et al. 2009; Kocevski & Butler 2008);
2. is tightly correlated with (Ghirlanda, Ghisellini & Lazzati 2004; Ghirlanda et al. 2007) with a scatter $\sigma_{\rm sc}= 0.07$ dex (consistent with the average statistical error $\bar{\sigma}_{E_{\gamma}}=\bar{\sigma}_{E_{\rm p}}\simeq 0.1$ dex associated with and );
3. the true rate of local GRBs ranges from $\sim$ 250 Gpc$^{-3}$ yr$^{-1}$ (e.g. Frail et al. 2001) to $\sim$ 33 Gpc$^{-3}$ yr$^{-1}$ (Guetta, Piran & Waxman 2005). These different values are mainly due to the different values assumed for the collimation factor $f\propto \theta_{\rm jet}^{-2}$. The true GRB rate can be compared with the local rate of SN Ib/c (e.g. Soderberg 2006; Guetta & Della Valle 2007; Grieco et al. 2012), i.e. the candidate progenitors of long GRBs, and allows to estimate the rate of orphan afterglows (e.g. Guetta et al. 2005).
The , and correlations could enclose some underlying feature of the GRB emission mechanism (e.g. Rees & Meszaros 2005; Ryde et al. 2006; Thompson 2006; Giannios & Spruit 2007; Thompson, Meszaros & Rees 2007; Panaitescu 2009), of the GRB jet structure (e.g. Yamazaki, Ioka & Nakamura 2004; Eichler & Levinson 2005; Lamb, Donaghy & Graziani 2005; Levinson & Eichler 2005) or of the progenitor (e.g. Lazzati, Morsony & Begelman 2011). An intriguing application of these correlations is the use of GRBs as standard candles (Ghirlanda, Ghisellini & Firmani 2005; Firmani et al. 2005; Amati et al. 2009).
The presence of outliers of the correlation in the / GRB population (Band & Preece 2005; Nakar & Piran 2005; Shahmoradi & Nemiroff 2011) and in the /GBM burst sample (Collazzi et al. 2012) and the presence of possible instrumental biases (Butler et al. 2007; Butler, Kocevski & Bloom 2009; Kocevski 2012) caution about the use of these correlations either for deepening into the physics of GRBs and for cosmological purposes. Although instrumental selection effects are present, it seems that they cannot produce the correlations we see (Ghirlanda et al. 2008; Nava et al. 2008; Ghirlanda et al. 2012b). Moreover, a correlation between and is present within individual GRBs as a function of time (Firmani et al. 2009; Ghirlanda et al. 2010; 2011; 2011a), suggesting that the radiative process(es) might be the origin of the correlation. Despite these studies, the spectral energy correlations of GRBs and their possible applications are still a matter of intense debate.
A new piece of information recently added to the puzzle is that the GRB energetics ($E_{\rm iso}^\prime$, $L_{\rm iso}^\prime$ and $E_{\rm p}^\prime$) appear nearly similar in the comoving frame (Ghirlanda et al. 2012 – G12 hereafter). To measure these comoving quantities[^2] we have to know the bulk Lorentz factor , that can be estimated through the measurement of the peak time of the afterglow light curve. G12 could estimate in 30 long GRBs with known $z$ and well defined energetics, finding that:
1. ()$\propto\Gamma_0^2$ and $\propto$;
2. the comoving frame $\sim$3.5$\times10^{51}$ erg (dispersion 0.45 dex), $\sim$5$\times10^{48}$ erg s$^{-1}$ (dispersion 0.23 dex) and $\sim$6 keV (dispersion 0.27 dex).
These results imply that the and correlation are a sequence of different factors (see also Dado, Dar & De Rujula 2007).
The values of GRBs are known only for a couple of dozens of bursts (Ghirlanda et al. 2007). appears distributed as a log–normal with a typical $\sim3^\circ$ (Ghirlanda et al. 2005). By correcting the isotropic comoving frame energy by this typical jet opening angle, the comoving frame true energy results $\sim 5\times 10^{48}$ erg. In G12 we also argued that in order to have consistency between the and the correlations one must require $\theta_{\rm jet}^2\Gamma_0$ = constant. A possible anti–correlation between $\theta_{\rm jet}$ and $\Gamma_0$ is predicted by models of magnetically accelerated jets (Tchekhovskoy, McKinney & Narayan 2009; Komissarov, Vlahakis & Koenigl 2010) but, at present, only 4 GRBs have an estimate of and and well constrained spectral properties.
The measure of relies on the measure of , that in turn requires the follow up of the optical afterglow emission up to a few days after the burst explosion (Ghirlanda et al. 2007). The measurement of is difficult, not only because it requires a large investment of telescope time, but also because several are chromatic (contrary to what predicted; but see Ghisellini et al. 2009), and the jet break can be a smooth transition whose measurement requires an excellent sampling of the afterglow light curve (e.g. Van Eerten et al. 2010, 2011). Another complication is that the early afterglow emission is characterized by several breaks. For instance, the end of the plateaux phase typically observed in the X–ray light curves, if misinterpreted as a jet break, biases the distribution towards small values of (Nava, Ghisellini & Ghirlanda 2006). Finally, the measure of large is complicated by the faintness of the afterglow and its possible contamination by the host galaxy emission and the supernova associated to the burst. Several observational biases could shape the observed distribution. Among these the fact that more luminous bursts (i.e. those more easily detected) should have the smallest jet opening angles. For all these reasons the [*observed*]{} distribution of might not be representative of the real distribution of GRBs jet opening angles.
The distribution of is centered around =65 (130) in the case of a wind (uniform) density distribution of the circum–burst medium. The distribution of is broad and extends between $\sim$20 and $\sim$800. These results are still based on a sample of only 30 GRBs (G12). The difficulties of early follow–up of the optical afterglow emission could prevent the measure of very large on the one hand, while the possible contamination by flares (Burrows et al. 2005; Falcone et al. 2007) or by other (non afterglow) emission components (e.g. Ghisellini et al. 2010) at intermediate times could prevent the estimate of the low–end of the distribution. One could argue if GRBs can have of a few. While there are some hints that GRB060218 should have $\sim$5 (Ghisellini et al. 2006) the classical compactness argument, for typical GRB parameters (e.g. Piran 1999), requires that $\ge$100-200. This argument was successfully applied to few bursts observed up to GeV energies by LAT on board Fermi (e.g. Abdo et al. 2009, Ghirlanda et al. 2009) to derive lower limits of several hundreds on . If, instead, the highest energy photon detected has an energy of say $E_{\rm max}\sim3$ MeV, the lower limit derived from the classical compactness argument would be $>$ a few (i.e. $\sim 2E_{\rm max}/m_{e} c^{2}$). Therefore, also in the case of , the [*observed*]{} distribution, derived with still few events, could be not representative of the real distribution of this parameter.
The main aim of this paper is to constrain the distribution of and in GRBs using the available independent constraints. This aim can be translated into a simple question: do and follow power law distributions or do they follow some kind of peaked distribution (e.g. a broken power law or a log–normal)? In both cases the resulting distributions could be different from the observed ones since some selection effect (as discussed above) might prevent to measure very low and/or high values of and . Another scope of the present paper is to test which is (if any) the relation between and . A relation $^2$=const was assumed in G12 to explain the spectral energy correlations and a similar relation seems to arise from numerical simulations of jet accelerations (Tcheckolskoy et al. 2012). Here we use several observational constraints and test whether there is a $^a$=const relation and try to constrain its exponent $a$. One important effect that we consider in this paper for the first time is the collimation of the burst radiation when is small. In general we are led to think that given a value of the collimation corrected energy , the corresponding isotropic equivalent energy is $\sim$/$^2$. This is true if the beaming of the radiation is “dominated” by the jet opening angle, i.e. 1/$\le \sin$. However, GRBs with very low could have 1/$\ge \sin$ and in this case the isotropic equivalent energy is determined by (i.e. $\sim$$^2$ - see §. 2.2) rather than by . This effect, introduces a limit ($\propto$$^{1/3}$) in the classical plane (Fig.1) accounting for the absence of bursts with intermediate/low and large values of . This limit can also partly account for the problem of “missing jet breaks" since these bursts with 1/$\ge \sin$ should not show any jet break in their afterglow light curve (§4.5).
We rely on a GRB population synthesis code that we have recently adopted to explore the issue of instrumental selection biases on the correlation (Ghirlanda et al. 2012b).
The simulation steps are described in §2 while the observational constraints that we aim to reproduce are outlined in §3. In §4 we present our results. We summarize and discuss our findings in §5. Throughout the paper a standard flat universe with $h=\Omega_{\Lambda}=0.7$ is assumed.
Population synthesis code
=========================
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So far, the approach adopted in studying the spectral–energy correlations and the distributions of or was (i) to derive the collimation corrected correlation by correcting the isotropic energy for the collimation factor $\propto\theta_{\rm jet}^2$ (e.g. Ghirlanda et al. 2004), or (ii) to derive the comoving frame properties of GRBs by correcting, for the factor, the isotropic values , and (G12).
In this paper, we tackle the problem from the opposite side and jointly work with and : we assume that GRBs have all the same comoving frame and and simulate GRB samples with different distributions of and . This produces a population of GRBs with known energetics , peak energy and observer frame fluence $F$ and peak flux $P$. We would like to stress that our main assumption (same and for all burst) is a crude simplification. Nevertheless, our assumption can work if the real and distributions are indeed narrower that the distributions of the corresponding observed quantities. Recently, Giannios 2012 have shown that in photospheric models a comoving frame peak energy $\sim$1.5 keV is expected.
The observational constraints that we aim to reproduce (see §3) are: (i) the rate of GRBs observed by /BAT, / and /GBM, (ii) the correlation defined by the complete sample of bright bursts (Salvaterra et al. 2012; Nava et al. 2012) and (iii) the fluence and peak flux distributions of the population of bursts detected by /GBM (Goldstein et al. 2012) and / (Meegan et al. 1998).
Note that, since one of the aims of the present paper is to constrain the distributions of and we cannot adopt the observed ones (discussed in the introduction) as constraints, otherwise we would fall into a circular argument. The distributions of and that we assume in our simulations (power law, broken power law, log–normal) have all their characteristic parameters (slope, normalization, break values, width etc.) free to vary. These parameters are what we aim to constrain through our population synthesis code.
In Fig. \[fg0\] we show the rest frame peak energy versus the total energy $E$ (where $E$ here is generically used to indicate an energy, either isotropic or collimation corrected). We highlight different regions (I, II and III) that are useful to explain the simulation steps (§2.1). This plane will be one of our observational constraints: in Fig. \[fg0\] we show (black filled points) the complete sample of bursts (Salvaterra et al. 2012; Nava et al. 2012) which we aim to reproduce through our simulations.
Simulation steps
----------------
Our starting assumption is that all GRBs have the same comoving frame =1.5 keV and =1.5$\times10^{48}$ erg. This is shown by the black circle in Fig. \[fg0\]. G12 find that $\sim$const and that the observed duration $T_{90}$ does not depend on . Therefore, in the comoving frame, $T_{90}^\prime \propto \Gamma_0 T_{90} \propto \Gamma_0$. It follows that =$T'_{90}\theta_{\rm jet}^2$ is also constant if, as discussed in G12, $\theta_{\rm jet}^2$=const. Although some dispersion of the values of is present in the sample of G12, the value of that we assume here is consistent at the 2$\sigma$ level of confidence with the distribution of values reported in G12 for the wind density ISM.
The main steps of our simulation are:
1. we simulate a population of GRBs distributed in redshift $z$ between $z=0$ and $z=10$ according to the GRB formation rate (GRBFR) $\psi(z)$. This is formed by two parts: $\psi(z)=e(z) R(z)$. The first term is a cosmic evolution term, while $R(z)$ is taken from Li (2008) (which extended to higher redshifts the results of Hopkins & Beacom 2008): $$R(z)=\frac{0.0157+0.118z}{1+(z/3.23)^{4.66}}
\label{z}$$ $R(z)$ is in units of ${\rm M}_{\sun}$ ${\rm yr}^{-1}$ ${\rm Mpc}^{-3}$. Concerning $e(z)$, Salvaterra et al. (2012) derived the luminosity function of GRBs by jointly fitting the redshift distribution of a complete sample of bright GRBs detected by and the count distribution of a larger sample of bursts. They found that either the evolution of the luminosity function or the evolution of the density of GRBs is required in order to account for these data sets. We assume the same term $e(z)=(1+z)^{1.7}$ found by S12.
2. We assign to each GRB a bulk Lorentz factor extracted from a specified distribution, in the range \[1, 8000\]. The upper limit ($\Gamma_{\rm 0, max}=8000$) is somewhat arbitrary, but large enough to encompass all the values of estimated so far, and in particular the large values derived for the few GRBs detected by the LAT instrument on board , if the GeV emission is interpreted as afterglow (Ghisellini et al. 2010).
For each simulated burst the rest frame peak energy and the energy are (see G12): $$E_{\rm peak}=E'_{\rm peak}\frac{5\Gamma_{0}}{5-2\beta_{0}} \, \, ; \, \, E_{\gamma}=E'_{\gamma}\Gamma_{0}$$ where =$1/(1-\beta_{0}^{2})^{1/2}$. The simulated bursts define a correlation between and : $$E_{\rm peak} = \frac{E'_{\rm peak}}{E'_{\gamma}}
\frac{5 E_{\gamma}}{5-2\beta_{0}} \propto \frac{E_{\gamma}}{5-2\beta_{0}}
\label{lghi}$$ for $\beta_{0}\sim1$ this corresponds to the correlation in the case of a wind density profile (Nava et al. 2006). This relation is shown in Fig. \[fg0\] with the solid black line (labelled $\propto$). The simulated distribute between 1.5 keV (=1) and $\sim$20 MeV (=8000).
3. We assign to each simulated burst a jet opening angle $\in[1^{\circ}, 90^{\circ}]$ extracted from a specified distribution.
4. The probability for a burst to be observed from the Earth depends on the viewing angle between the jet axis and the line of sight of the observer. We extract randomly a viewing angle from the cumulative distribution of the probability density function $\sin$.
5. In order to compare the simulated bursts with the source count distribution of existing samples of GRBs (see §3) we compute the observer frame peak fluxes $P$ and fluences $F$. To this aim we assume a typical spectrum described by the Band function (Band et al. 1993), with low and high photon spectral indexes $\alpha=-1.0$ and $\beta=-2.3$, respectively (i.e. corresponding to the typical values observed by different instruments – e.g. Kaneko et al. 2006; Sakamoto et al. 2011)[^3]. The fluence ${F}$ of each simulated burst in a given energy range is computed by re-normalizing this spectrum through the bolometric fluence ${F_{\rm bol}}$=$(1+z)/4\pi d_{\rm L}^2$, where $ d_{\rm L}^2$ is the luminosity distance for a given redshift $z$. To derive the peak flux $P$, we assign to each burst an (observer frame) duration $T_{90}$ extracted from a distribution centered at 27.5 s and with a dispersion $\sigma_{\rm Log T_{90}}=0.35$. This distribution is truncated at $T_{90}=2$ s because we consider only long duration GRBs in this analysis. Such a duration distribution is similar to that of the /GBM GRBs (Paciesas et al. 2012; Goldstein et al. 2012) and includes also very long bursts with $T_{90}\sim$300 s. We assume that the bursts have a simple triangular light curve and derive the peak luminosity as $L_{\rm peak}=2$$(1+z)/T_{90}$. The peak flux $P$ in a given energy range is obtained by re–normalizing the spectrum through the bolometric peak flux $P_{\rm bol}=L_{\rm peak}/4\pi d_{\rm L}^2$.
Computation of
---------------
The isotropic equivalent energy of the simulated bursts can be derived from . Since $\le$90$^\circ$, simulated bursts cannot be in region II of Fig. \[fg0\] and can take values on the right hand side of the limit of Eq. \[lghi\] shown in Fig. \[fg0\]. According to the values of and assigned to each simulated bursts, the isotropic equivalent energy is: $$E_{\rm iso}=E_{\gamma}/(1-\cos\theta_{\rm jet}) \,\,\,\,\,
{\rm if} \,\,\,\,\,\, 1/\Gamma_{0}\leq \sin \theta_{\rm jet}
\label{eq1}$$ $$E_{\rm iso}=E_{\gamma}(1+\beta_{0})\Gamma_{0}^{2} \,\,\,\,\,\,\,\,\,\,\,\,\,
{\rm if}\,\,\,\,\,\, 1/\Gamma_{0}> \sin \theta_{\rm jet}
\label{eq2}$$ In the latter case is smaller than in Eq. \[eq1\]. This introduces a limit in the plane of Fig. \[fg0\] corresponding to the line: $$E_{\rm peak} \propto \left[\frac{E_{\rm iso}}{(5-2\beta_{0})^{3}(1+\beta_{0})}\right]^{1/3}
\label{gammalimit}$$ (labelled $\propto$$^{1/3}$ in Fig. \[fg0\]). For a given , bursts with a small value of will have an computed through Eq. \[eq2\] and will lie on the limiting line of region III in Fig. \[fg0\]. Their radiation is, indeed, collimated within an angle $\arcsin$(1/) which is larger than their .
Simulated bursts can populate the region delimited by boundaries (I, II and III) in Fig. \[fg0\]. This is one (among others) observational constraint that we will adopt in our simulations (§3) to constrain the distributions of and and their possible relation. According to the relative values of , and , simulated bursts are classified as: bursts “pointing to us" (PO, hereafter), i.e. those that can be seen from the Earth, with $\sin$$\le$max\[$\sin$, 1/\] and bursts pointing in other directions (NPO, hereafter), i.e. not observable from the Earth, with $\sin$$>$max\[$\sin$, 1/\]. We will compare the PO simulated bursts with our observational constraints, while the entire population of simulated bursts (i.e. PO and NPO) will be used to infer the properties of GRBs (e.g. the distributions of and and the true burst rate).
Observational constraints
=========================
In order to test whether and assume characteristic values or not we compare the population of simulated bursts with real samples of GRBs. In this section we describe our observational constraints. We consider the ensemble of GRBs detected by the Burst Alert Telescope (BAT) on board , the Gamma Burst Monitor (GBM) on board and the Burst And Transient Source Experiment (BATSE) on board the [*Compton Gamma Ray Observatory*]{} ().
The BAT complete sample
------------------------
Salvaterra et al. (2012 - S12 hereafter) constructed a sample of bright bursts consisting of 58 GRBs detected by /BAT with $P\ge P_{\rm lim}$=2.6 ph cm$^{-2}$ s$^{-1}$ (integrated in the 15–150 keV energy range). Fifty four of these events have a measured redshift $z$ so that the S12 sample is 90% complete in redshift. Forty six (out of 54) GRBs in this sample have well determined spectral properties (filled circles in Fig. \[fg0\]) and define a statistically robust correlation with rank correlation coefficient $\rho=0.76$ and chance probability $P=7\times10^{-10}$ (Nava et al. 2012, N12 hereafter)[^4]. The correlation properties (slope and normalization) of the complete sample are consistent with those defined with the incomplete larger sample of 136 bursts with known $z$ and spectral parameters (see N12). Therefore, the distribution of the complete sample (46/54 events with well constrained ) in the plane is representative of the larger (heterogeneous) population of GRBs with measured $z$ and well constrained spectral properties. The 46 GRBs of the complete sample define a correlation $E_{\rm p}\propto E_{\rm iso}^{0.61\pm0.06}$ (shown by the dot–dashed line in Fig. \[fg0\]) with a scatter (computed perpendicular to the best fit line) with a Gaussian dispersion $\sigma=$0.29 dex.
The complete sample of S12 contains $\sim$1/3 of the bursts detected by [^5] with $P\ge$2.6 ph cm$^{-2}$ s$^{-1}$. We verified that the complete sample of 54 events selected by S12 is representative of the larger population of 149 long bursts with $P\ge$: the Kolmogorov–Smirnov test on the peak flux distribution of the two samples gives a probability of 0.6 that the two distributions are drawn from the same parent population. These bursts were not included in the selection of S12 because they do not have favorable conditions for ground–based follow up.
These 149 events with $P\ge$ are the bursts detected by in $\sim$7 yrs from its launch within the (half coded) field of view of $\sim$1.4 sr of BAT. This corresponds to an average detection rate of $\mathcal{R_{\rm Swift}}\sim$15 events sr$^{-1}$ yr$^{-1}$.
The GBM sample
---------------
Another observational constraint that we consider is the population of bursts detected by the GBM on board . The spectral properties of GBM bursts have been studied in Nava et al. (2011a) and compared to those of BATSE bursts in Nava et al. (2011b). More recently, the first release of the GBM spectral catalog (Goldstein et al. 2012) provided the spectral parameters and derived quantities (i.e. peak fluxes and fluences) for 487 GRBs detected by the GBM in its first 2 years of activity. 398 bursts in this catalog are long events and have measured peak flux $P$ and fluence $F$ (both integrated in the 10 keV–1 MeV energy range)[^6].
We cut the GBM sample to $P\ge P_{\rm lim}=2.5$ ph cm$^{-2}$ s$^{-1}$, in order to account for the possible incompleteness of the sample at lower fluxes, obtaining 312 GBM bursts.
The GBM is an all sky monitor that observes on average $\sim$60–70% of the sky. Therefore, the average GBM detection rate is $\mathcal{R_{\rm GBM}}\sim$21 events sr$^{-1}$ yr$^{-1}$ with peak flux, integrated in the 10 keV–1 MeV energy range, $P\ge 2.5$ ph cm$^{-2}$ s$^{-1}$.
The sample
------------
We also consider the sample of GRBs detected by . The 4B sample (Meegan et al. 1998) contains 1540 long events and 1496 of these have their $P$ and $F$ (both integrated in the 50–300 keV energy range) measured. The sample of 1496 bursts is cut at $P\ge P_{\rm lim} = 1$ ph cm$^{-2}$ s$^{-1}$ with 716 bursts above this threshold. Considering the average portion of the sky observed by , i.e. $\sim$70% of the sky, the detection rate of is $\mathcal{R_{\rm BATSE}}\sim$16 events sr$^{-1}$ yr$^{-1}$ for GRBs with a peak flux, integrated in the 50–300 keV energy range, $P\ge$ 1 ph cm$^{-2}$ s$^{-1}$.
The lower detection rate of with respect to GBM is due to the different energy range where the peak fluxes are calculated (i.e. 10 keV–1 MeV for GBM and 50–300 keV for , respectively). We verified that by considering the GBM bursts with peak flux $P$ integrated in the same energy range of (i.e. 50–300 keV) larger than 1 ph cm$^{-2}$ s$^{-1}$ (i.e. the same threshold adopted for ), the GBM rate is equal to the one.
Extraction of results
---------------------
From each simulation we extract three populations of GRBs among the bursts pointing to us (PO):
1. the comparison sample: [*simulated*]{} GRBs with peak flux, integrated in the 15–150 keV band, larger than 2.6 ph cm$^{-2}$ s$^{-1}$. We also require that their observer frame peak energy is in the range 15 keV–2 MeV. Indeed, this is the energy range where can be measured by presently flying satellites like , Konus and .
2. the GBM comparison sample: [*simulated*]{} bursts with a peak flux, integrated in the 10 keV–1 MeV energy range, larger than 2.5 ph cm$^{-2}$ s$^{-1}$;
3. the comparison sample: [*simulated*]{} bursts with a peak flux, integrated in the 50–300 keV energy range, larger than 1.0 ph cm$^{-2}$ s$^{-1}$.
The simulation is adjusted so that the comparison sample contains 149 GRBs, i.e. the same number of bright bursts detected by (§3.1). Therefore, the rate $\mathcal{R_{\rm Swift}}$ is imposed. What we derive instead from the simulation is the rate of GBM and GRBs that we compare with the real rates of these two instruments described in §3.2 and §3.3 respectively.
We also require that the comparison sample is consistent with the complete sample of S12. To this aim we compare them in the rest frame plane and in the observer frame plane deriving a 2 dimensional Kolmogorov–Smirnov (KS) probability (one for the and one for plane). We also verify through a 1 dimensional KS test that the redshift distribution of the comparison sample is consistent with that of the complete sample. Finally we compare, through a 1D–KS test, the fluence and peak flux distributions of the GBM and comparison samples with those of the real samples of GRBs detected by these instruments and described in §3.2 and §3.3, respectively.
Since the complete sample contains only the brightest bursts it maps the high $P$ end of the peak flux distribution of GRBs. The GBM and samples that we adopt here extend the comparison sample to lower values of $P$ and ensures that our simulations reproduce also the faint end of the GRB population[^7].
For each simulation we derive the following probabilities:
- the 2D-KS probability that the comparison sample is consistent with the complete sample of S12 in the plane;
- the 2D-KS probability that the comparison sample is consistent with the complete sample of S12 in the plane;
- the 1D-KS probability that the comparison sample has a redshift distribution consistent with that of the S12 sample;
- the 1D-KS probabilities that the GBM comparison sample is consistent with the GBM sample in terms of peak flux $P$ and fluence $F$;
- the 1D-KS probabilities that the comparison sample is consistent with the sample in terms of peak flux $P$ and fluence $F$;
- we verify if the GBM rate predicted by the simulation is consistent, at 1$\sigma$, with the GBM rate $\mathcal{R_{\rm GBM}}$.
- we verify if the BATSE rate predicted by the simulation is consistent, at 1$\sigma$, with the rate $\mathcal{R_{\rm BATSE}}$.
For the KS probabilities we set a limit of 10$^{-3}$ below which we consider that two distributions (either 1D or 2D) are inconsistent at more than 3$\sigma$. Each simulation, with its assumptions on the distribution of and , is repeated 1000 times and we compute the percentage $\mathcal{P}$ of repeated simulations that produce GRB samples (i.e. , GBM and comparison samples) consistent with our observational constraints.
$
\begin{array}{cc}
\hskip -1.5truecm\includegraphics[width=9cm,trim=50 20 40 40,clip=true]{pltheta_plgamma_rest.ps} &
\hskip -0.4truecm\includegraphics[width=9cm,trim=50 20 40 40,clip=true]{pltheta_plgamma_obs.ps} \\
\hskip -1.5truecm\includegraphics[width=9cm,trim=20 10 20 20,clip=true]{logn_gbm_pltheta_plgamma.ps} &
\hskip -0.4truecm\includegraphics[width=9cm,trim=20 10 20 20,clip=true]{logn_batse_pltheta_plgamma.ps} \\
\end{array}$
Results
=======
In the following sections we present the results obtained with different possible assumptions for the distributions of and . We want to test which one among the possible intrinsic distributions of and that one can think of (e.g. power laws, broken power laws or log–normal) best reproduces the observational constraints described in the previous section.
Power law distributions of and
--------------------------------
$
\begin{array}{cc}
\hskip -1.5truecm\includegraphics[width=8.5cm,trim=50 20 40 40,clip=true]{bkntheta_bkngamma_rest.ps} &
\hskip -0.3truecm\includegraphics[width=8.5cm,trim=50 20 40 40,clip=true]{bkntheta_bkngamma_obs.ps} \\
\hskip -1.5truecm\includegraphics[width=8.5cm,trim=20 10 20 20,clip=true]{logn_gbm_bkntheta_bkngamma.ps} &
\hskip -0.3truecm\includegraphics[width=8.5cm,trim=20 10 20 20,clip=true]{logn_batse_bkntheta_bkngamma.ps} \\
\end{array}$
We assume that both and are distributed as power laws: $dN/d\theta_{\rm jet}\propto \theta_{\rm jet}^{a}$ and $dN/d\Gamma_{0}\propto \Gamma_{0}^{c}$. This corresponds to the hypothesis that and do not have a characteristic value. We consider $a\in[-2,-1]$ and $c\in[-2,-1]$.
The choice of these parameters corresponds to have most of the simulated bursts with low factors and with small values. One could think that such distributions are already excluded by the observed distributions of and (which are log–normal) discussed in §1. However, those are the [*observed*]{} distributions of and and they are subject to several biases (see §1). The intrinsic distributions might well be completely different and this motivates to start with this simplest assumption, i.e. that both and have power law distributions.
Under the hypothesis that both and have power law distributions (with free parameters $a$ and $c$ varied in the above ranges with a step 0.2 in both parameters), only in 1% of 1000 repeated simulations we can find an agreement with all our observational constraints. In order to show the inconsistency of the simulations with the observational constraints we present in Fig. \[fg1\] the results of the simulations assuming that and have power law distributions with $a=c=-1$. This case, shown as an example, corresponds to a uniform distribution of Log and Log.
The rest frame plane (top left panel in Fig. \[fg1\]) is filled uniformly with simulated bursts (yellow dots) distributed between the limit and with a minimum =1$^\circ$ (the oblique right limit to the distribution of yellow dots). The simulated GRBs pointing to us (PO) have preferentially large values (blue dots in the top left panel of Fig. \[fg1\]). The simulated bursts of the comparison sample (here represented by the smoothed density contours[^8] – red solid lines in Fig. \[fg1\]) are inconsistent with the real GRBs of the complete sample (open squares). The red contours extend at high values where there is a deficit of bursts and they also over predict the number of bursts on the right hand side of the distribution of the real bursts (i.e towards large values of for intermediate/high values of ).
Also in the observer frame plane (top right panel in Fig. \[fg1\]) the simulated comparison sample (solid contours) are inconsistent with the real bursts of the complete sample (open squares). Simulated bursts of the comparison sample tend to concentrate towards the upper part of the plane.
The bottom panels of Fig. \[fg1\] show the cumulative rate distribution of the fluence for the GBM and sample (right and left panels of Fig. \[fg1\]) compared with the predictions of the simulations (dashed regions in the bottom panels of Fig. \[fg1\]). The rate of GBM and bursts predicted by the simulation which assumes a power law distribution for both and (with index –1) is a factor $\sim$2 larger than the rate of GBM bursts. Also the distributions of the peak flux of the simulated and GBM samples are inconsistent with the real samples.
If we assume steeper power law distributions of and \[e.g. $(a,c)=(-2,-2)$\], the excess of bursts with large peak energy (both in the rest frame and in the observer frame of Fig. \[fg1\], top left and right panels respectively) is reduced but the rate of simulated GBM and BATSE bursts increases becoming more inconsistent with the real rates of GRBs detected by these two instruments (bottom panels of Fig. \[fg1\]). This result shows that all the constraints that we have adopted (§3) are relevant: the GBM and comparison sample map the low end of the peak flux/fluence distribution while the complete sample maps the bright burst tail of such distributions. The bursts of the complete sample, having their $z$ measured, map the distribution of GRBs in the rest frame plane.
Peaked distributions of and
-----------------------------
$
\begin{array}{cc}%
\hskip -1.5truecm\includegraphics[width=8.5cm,trim=50 20 40 40,clip=true]{correla_rest.ps} &
\hskip -0.3truecm\includegraphics[width=8.5cm,trim=50 20 40 40,clip=true]{correla_obs.ps} \\
\hskip -1.5truecm\includegraphics[width=8.5cm,trim=20 10 20 20,clip=true]{logn_gbm_correla.ps} &
\hskip -0.3truecm\includegraphics[width=8.5cm,trim=20 10 20 20,clip=true]{logn_batse_correla.ps} \\
\end{array}$
Since we could not find agreement between the simulations which assume power law distributions of and and our observational constraints, we now consider the case of peaked distributions of and .
The simplest assumption is that and/or are distributed as broken power laws. We first assumed that only or have a broken power law distribution, while the other parameter is distributed as a single power law. In this case we cannot find a percentage of repeated simulations larger than 2% in agreement with our observational constraints.
We then considered the case of a broken power law distribution for both and : $$\frac{dN}{d\theta_{\rm jet}} = \left\{ \begin{array}{rl}
\theta_{\rm jet}^{a} &\mbox{ if \,\, $\theta_{\rm jet}\le \theta_{*}$} \\
\theta_{\rm jet}^{b} &\mbox{ if \,\, $\theta_{\rm jet}> \theta_{*}$}
\end{array} \right.$$ $$\frac{dN}{d\Gamma_{0}} = \left\{ \begin{array}{rl}
\Gamma_{0}^{c} &\mbox{ if \,\, $\Gamma_{0}\le \Gamma_{*}$} \\
\Gamma_{0}^{d} &\mbox{ if \,\, $\Gamma_{0}> \Gamma_{*}$}
\end{array} \right.$$ For the distribution of we consider the following parameter ranges: $a\in[0.5, 2.0]$, $b\in[-2.0, -5.0]$ and $\theta_{*}\in[3^{\circ}, 12^{\circ}]$. For : $c\in[0.5, 2.0]$, $d\in[-2.0, -5.0]$ and $\Gamma_{*}\in[50, 120]$. The free parameters are varied with step 0.1 for $a$ and $b$ and 0.5$^\circ$ for $\theta_{*}$ for the broken power law distribution of and with step 0.1 for $c$ and $d$ and 10 for $\Gamma_{*}$ for the broken power law distribution of .
We find that at most $\sim$20% of the 1000 repeated simulations reproduce our observational constraints when $(a,b,\theta_{*})=(0.5,-3.0,4.5^{\circ})$ and $(c,d,\Gamma_{*})=(1.8,-3.5,70)$ with step 0.1 for $c$ and $d$ and 10 for $\Gamma_{*}$ for the broken power law distribution of . A lower percentage of agreement is obtained for any other choice of the free parameters.
We show in Fig. \[fg2\] the results of the simulations with the above parameter values for the distributions of and . We note that a better agreement is now found between the rate of the GBM and bursts (bottom panels of Fig. \[fg2\]) while the distribution of simulated bursts of the comparison sample (solid contours) are inconsistent with the bursts of the complete sample both in the rest frame plane (top left panel of Fig. \[fg2\]) and in the observer frame plane (top right panel of Fig. \[fg2\]).
The assumption of a characteristic value of corresponds to concentrate GRBs around a typical value of (see Eq. 2). In this case the narrower distribution reduces the number of simulated bursts with large values of , thus clustering the simulated GRBs of the PO class around the limit of Fig. \[fg2\] (top left panel) that was found in the case of single power laws (§4.1). A broken power law is a simple approximation of a peaked distribution. The real distribution of and could have a different shape. We then considered the case of log–normal distributions for both and , with central values of the distribution between 3$^\circ$ and 12$^\circ$ (step 0.5$^\circ$) and width between 0.3 and 0.8 (step 0.05) and central values of between 50 and 120 (step 5) and width between 0.2 and 0.8 (step 0.05). We find that, if has a log–normal distribution with a median value of 4.5$^\circ$ (with a dispersion of 0.5) and is distributed as a log–normal with median 85 (with a dispersion of 0.45), the 40% of the 1000 repeated simulations is in agreement with all our observational constraints.
The latter assumption, that seems to improve the consistency between the simulated GRB population and the observational constraints, suggests that and have log–normal distributions. However, the fact that no more than 40% of the repeated simulations can reproduce all our observational constraints, is suggesting that some ingredient is still missing. This is the subject of the next section where we study for the first time through our numerical simulations, the possibility that there is a relation between the average values of and .
The relation between and
--------------------------
By assuming a distribution with a characteristic value as in §4.2, the simulated bursts in the plane cluster around a correlation which is linear in this plane (i.e. parallel to the limit), while the correlation defined by the complete sample (and similarly by the larger, incomplete, sample of bursts with measured redshift – see N12) has a flatter slope, i.e. $\propto E_{\rm iso}^{0.6}$. In other words, for an infinitely narrow distribution of , the simulated bursts (yellow dots in Fig. \[fg2\] top left panel) would produce a linear correlation which is inconsistent with the observed correlation. This suggests that, besides the fact that and should have characteristic values (i.e. peaked distributions), [*they should also be correlated*]{}.
Indeed, G12 find that the comoving frame properties of GRBs (and in particular the fact that $\propto$ and $\propto$$^2$) can be combined to explain both the $\propto E_{\rm iso}^{0.5}$ and the $\propto$ correlation if the ansatz $^{2}$=const is valid. Several recent numerical simulations of jet acceleration in GRBs suggest that a link between and should exist, although the form of this relation depends on several assumptions of these simulations. In this section we explore, for the first time, if a relation $^{m}$=$K$ can account for the observational constraints described in §3 and in this case we constrain its free parameters ($m$ and $K$). We start from the result of the previous section, which showed that the best result (i.e. 40% of the repeated simulations are in agreement with the observations) is obtained assuming two log–normal distributions for and .
We simulate bursts with Log distributed as a Gaussians with a characteristic central value Log$\Gamma_{*}$ and a dispersion $\sigma_{\rm Log\Gamma_{0}}$. Similarly we assume a Gaussian distribution for Log centered at Log$\theta_{*,\rm jet}$ and with a dispersion $\sigma_{\rm Log\theta_{\rm jet}}$. We then assume that there is a relation between and of the form Log$\theta_{*,\rm jet} = -1/m$Log$\Gamma_{*} + q$. In this way the distribution of Log is centered on a value which is given by the assumed relation between and .
We explored the parameter space (defined by 5 free parameters) and found that 80% of our simulations are consistent with our constraints if Log$\Gamma_{*}=1.95$ with a dispersion of $\sigma_{\rm Log\Gamma_{0}}=0.65$ dex, $m=2.5$, $q=1.45$ and $\sigma_{\rm Log\theta_{\rm jet}}=0.3$ dex.
We show in Fig. \[fg3\] the results of this simulation which assumes log–normal distributions of and and a relation between these two parameters. In the plane (top left panel in Fig. \[fg3\]) and in the plane (top right in Fig. \[fg3\]) we find a good agreement between the simulated comparison sample (solid contours) and the real complete sample (open squares). Now the predicted rate of GBM and bursts is fully consistent with the real ones (bottom left and right panels in Fig. \[fg3\] respectively).
We stress that, given the assumptions of our simulation (e.g. the spectrum, duration and unique values of the comoving frame energetics of all GRBs) we do not expect to find 100% of the simulations reproducing our constraints. However, we can use our code to derive interesting properties of the population of GRBs. Indeed, in our simulations we generate a population of GRBs pointing in every direction. Only those pointing towards the Earth (PO) are then compared with existing samples of GRBs (like those described in §3). This is also the population of bursts that will be explored by future GRB detectors with better sensitivity than the present ones. We can derive the properties of the whole GRB population (i.e. all the bursts pointing in whatever direction), like the jet opening angle distribution, the bulk Lorentz factor distribution and the true GRB rate.
[lllllll]{} Distrib. &sample &$\sigma$ &$\mu$ &Mode &Mean & Median\
&ALL &0.916$\pm$0.001 &1.742$\pm$0.002 &2.47$^\circ$ &8.68$^\circ$ &5.71$^\circ$\
&PO &0.874$\pm$0.010 &3.308$\pm$0.013 &12.73$^\circ$ &40.04$^\circ$ &27.33$^\circ$\
&PO\* &0.610$\pm$0.020 &2.83$\pm$0.029 &11.68$^\circ$ &20.41$^\circ$ &16.95$^\circ$\
&PO &0.527$\pm$0.032 &1.410$\pm$0.043 &3.10$^\circ$ &4.71$^\circ$ &4.10$^\circ$\
&PO\* &0.544$\pm$0.298 &1.043$\pm$0.434 &2.11$^\circ$ &3.29$^\circ$ &2.83$^\circ$\
\
&ALL &1.475$\pm$0.002 &4.525$\pm$0.002 &11 &274 &92\
&PO &1.452$\pm$0.020 &2.837$\pm$0.025 &2 &49 &17\
&PO &0.975$\pm$0.060 &5.398$\pm$0.083 &85 &355 &221\
distribution of GRBs
---------------------
-0.5truecm ![Distribution of of GRBs. The distribution of the total sample of simulated GRBs is shown by the filled circles. The solid grey line shows the fit with a lognormal function (Eq.\[lgn\]). The subsample of GRBs pointing towards the Earth (PO) is shown by the open (blue) squares and its fit with a lognormal by the cyan line. The sample of PO GRBs and peak flux $P\ge$2.6 cm$^{-2}$ s$^{-1}$ (i.e. the comparison sample) is shown by the open (red) circles and its lognormal fit by the orange line. The dashed (grey) line shows the lognormal fit of the distribution of all the bursts (solid grey line) multiplied by $1-\cos\theta_{\rm jet}$. The green triangles show the distribution of the 27 GRBs with measured jet opening angle collected in Ghirlanda et al. (2004, 2007). []{data-label="fg4"}](theta_distrib.ps "fig:"){width="9cm"}
From the best simulation described in §4.3 we can derive the distribution of the jet opening angle of GRBs. In Fig. \[fg4\] we show the distribution of for all the simulated bursts (black points) and for the PO bursts (open cyan squares). The population of GRBs pointing towards the Earth and with a peak flux $P\ge$2.6 cm$^{-2}$ s$^{-1}$ in the 15–150 keV range (i.e. the comparison sample) is shown by the open (red) circles. All the distributions of can be modeled with a log normal function: $$N(x)=\frac{A}{x \sigma \sqrt{2\pi}}\exp\left[-\frac{(\ln x-\mu)^{2}}{2\sigma^{2}}\right]
\label{lgn}$$ where the free parameters are $(\mu,\sigma)$ and the normalization $A$. The best fit parameters $\mu$ and $\sigma$ are reported in Tab. \[tab1\]. The peak of the log–normal distribution, i.e. its mode, is $\exp(\mu-\sigma^2)$, the mean is $\exp(\mu+\sigma^{2}/2)$ and the median is $\exp(\mu)$. Since the asymmetry of the log–normal distributions can be considerably large, we report in Tab. \[tab1\] all these moments.
The of GRBs of the comparison sample (red open circles in Fig. \[fg4\]) have a mean of $\sim4.7^\circ$. This distribution is consistent with the estimated from the break of the optical light curves (Ghirlanda et al. 2004, 2007), shown by the open (green) triangles in Fig. \[fg4\].
The GRBs that point to the Earth (PO - shown by the open blue squares in Fig. \[fg4\]) have a distribution peaking at considerably larger values (40$^\circ$ - see Tab.1) than the entire GRB population. This can be easily interpreted: consider the distribution of the entire population of GRBs (black dots in Fig. \[fg4\]) which contains all bursts pointing in every direction. The probability that a burst with a certain is pointing to us is proportional to $(1-\cos \theta_{\rm jet})$. Therefore the distribution of for PO bursts is obtained from the total distribution by multiplying by $(1-\cos \theta_{\rm jet})$. This reduces the number of bursts per unit and also shifts the peak of the PO distribution towards an average larger value. This is shown in Fig. \[fg4\] by the dashed (grey) line which is obtained by multiplying the fit of the distribution of of the entire GRB population (solid gray line in Fig. \[fg4\]) by $(1-\cos \theta_{\rm jet})$ and it fits the distribution of the PO bursts (open squares in Fig. \[fg4\]).
Among the simulated bursts that are pointing towards the Earth we considered the bright bursts (i.e. selected with the same peak flux threshold of the complete sample). These bursts tend to have small jet opening angles and this accounts for their distribution peaking at $\sim$5$^\circ$ in Fig. \[fg4\] (open red circles).
Although apparently there is a similarity between the distribution of all bursts (i.e. pointing in every direction) and the distribution of the PO bright bursts, they differ by a factor 2 (1.8) in their peak values (and dispersions) which are reported in Tab.1.
The three distributions shown in Fig. \[fg4\] allow us to make some further considerations. If we could measure for all bursts pointing towards the Earth (PO in Tab. \[tab1\]), we would obtain the open (blue) square distribution of Fig. \[fg4\] with a mean $\sim40^\circ$. However, the real distribution of the population of GRBs (i.e. all the simulated bursts – black filled circles distribution in Fig. \[fg4\]) has a mean of $\sim$8.7$^\circ$ and it is more consistent with the distribution of the simulated PO bursts with large peak fluxes (the comparison sample). This suggests that the bursts distributed in the high part of the correlation, where are the bursts of the complete sample (filled black dots in Fig. \[fg3\] top left panel), properly sample the peak of the distribution of the entire GRB population.
GRBs with no jet break
----------------------
-0.5truecm ![ Distribution of of GRBs: the PO simulated GRBs (open blue squares) and the GRBs of the PO class that have $\sin \theta_{\rm jet} \le 1/\Gamma_{0}$ (open green triangles) are shown. The latter are those that should not show any jet break time in their afterglow light curve. For the PO bursts with $P\ge$2.6 cm$^{-2}$ s$^{-1}$ (open red circles) we show the subsample of bursts that have $\sin \theta_{\rm jet} \le 1/\Gamma_{0}$ (open orange stars). []{data-label="fg5"}](theta_nojet.ps "fig:"){width="9cm"}
It has been shown in §3 that if a burst has a such that $\sin \theta_{\rm jet} \le 1/\Gamma_{0}$, its is determined by (Eq. \[eq2\]) and not by . This value is lower than that computed by (Eq. \[eq1\]). In these bursts, therefore, we should not observe a jet break in their light curve since the emitted radiation is initially collimated within an angle $\arcsin$1/ larger than . Since $\Gamma$ decreases during the afterglow phase due to the deceleration of the fireball by the interstellar medium, in these bursts the jet break, corresponding to the transition 1/$\Gamma \sim$, will never happen.
The above argument contributes to explain the fact that bursts might not show an evident jet break in their afterglow light curve if 1/$\ge\sin$. However, in these bursts we expect that the afterglow light curve is declining with a typical post–break decay index $\sim -p$ (where $p$ is the shock–accelerated electron energy distribution index - e.g. Panaitescu & Kumar 2001). Other possible explanations for the lack of measurements have been proposed. Numerical simulations (e.g. Van Eerten et al. 2010), for instance, suggest that the jet break transition can be very smooth (almost difficult to be distinguished from a single power law decay with available data sets) due to a combination of the jet dynamics before and after the jet break time (and additional complications can be induced by the viewing angle effects when the observer is not on–axis). Although a detailed discussion of the missing jet breaks in GRBs is out of the scope of this paper, we notice that bursts with $\sin \theta_{\rm jet} \le 1/\Gamma_{0}$ can partly account for the explanation of the lack of measured jet breaks. This is the first time that such an argument is presented and surely deserves further studies.
Fig. \[fg5\] shows the distribution of PO bursts (open blue squares) and the subsample of bursts with no jet break (open green triangles). These amount to $\sim$6% of PO bursts. The mean of their log–normal distribution is $\sim$20$^\circ$. One testable observational prediction of our simulations is that GRBs with no jet breaks should be preferentially soft ( of few tens of keV) The open red circles in Fig. \[fg5\] correspond to PO bursts of the comparison sample while the open orange star symbols correspond to bursts with no jet break. These have a mean jet opening angle $\sim$3.3$^{\circ}$. We find that $\sim$2% of the bright bursts should not have jet break in their afterglow light curves. They could correspond to those events which do not show any evidence of a jet break in their optical light curve (e.g. Mundell et al. 2006; Grupe et al. 2007) although other observational selection effects very likely contribute to the paucity of the jet break measurements. The fit of the distributions shown in Fig. \[fg5\] with log–normal functions are reported in Tab. \[tab1\].
distribution of GRBs
---------------------
-0.5truecm ![ Distribution of of GRBs. Symbols as in Fig.\[fg4\]. The 30 GRBs with estimated from the peak of their afterglow light curves (G12) are shown with the green open triangles. []{data-label="fg6"}](gamma_distrib.ps "fig:"){width="9cm"}
From our simulation we can derive the distribution of (Fig. \[fg6\]). The total population of simulated bursts (filled circles in Fig. \[fg6\]) has a log normal distribution with a mean =274. Those pointing towards the Earth (open blue squares in Fig. \[fg6\]) have a smaller mean, =49. The PO bursts with peak flux larger than 2.6 cm$^{-2}$ s$^{-1}$, i.e. those of the comparison sample, have a typical =355. Although the distribution of factors for those bursts with a peak in their afterglow light curves (G12) is still made of few events, it agrees (open green triangles in Fig. \[fg6\]) with that predicted by our simulations (for the sample of PO bursts of the comparison sample – open red circles in Fig. \[fg6\]).
Also in the case of we note that if we were able to measure for all the bursts that point towards the Earth, we would obtain a slightly smaller peak value of with respect to that of the distribution of all the bursts (pointing in every direction.
We note that the distribution of the general population of GRBs peaks at considerably low values of . This is a result of our simulations where, as explained in §4.2, we assume a peaked logarithmic distribution of with free peak and width. If we assume a distribution of with a smaller fraction of bursts with low –values, then we cannot reproduce the flux and fluence distributions and the detection rates of GRB detection of the GBM and instruments. Therefore, our simulations predict that a considerable fraction of GRBs should have as low as a few tens. These bursts might well be detected by current instruments. While the detailed study of their prompt and afterglow properties is out of the scope of the present paper, we note that their prompt emission should hardly differ from that of bursts with larger values (except for the obvious fact that their prompt and is lower). In fact, if is low the fireball deceleration timescale (e.g. Eq.14 in Ghirlanda et al. 2011) is $t_{\rm peak}\sim 4\, E_{\rm iso,50}^{1/3} \Gamma_{0,1}^{-8/3}$ hours which is much larger than the prompt emission timescale. So, while the prompt emission of low- burst should not be influenced by the afterglow contribution, their late time afterglow onset could be a distinctive feature (typical afterglow onset timescales are of the order of few hundreds second - Ghirlanda et al. 2011).
The GRB rate
------------
-0.5truecm ![ GRB rate as a function of redshift. The sample of simulated bursts is shown by the filled black circles and the GRB formation rate assumed in our simulation (see §2) is shown by the solid grey line. This curve is normalized to the histogram. The GRB formation rate without density evolution (as derived by Li et al. 2010 – see Eq. \[z\]) is shown by the dashed grey line. The rate of bursts pointing to us (PO) is shown by the open (blue) squares and that of the PO bursts with $P\ge$2.6 cm$^{-2}$ s$^{-1}$ (i.e. the comparison sample – PO ) by the open (red) circles. For comparison is also shown the rate of GRBs of the complete sample (rescaled to match the rate of PO rate). The solid lines reported in the top of the plot are the cosmic rates of SNIb/c computed by Grieco et al. (2012) for different assumptions on the cosmic star formation rate. []{data-label="fg7"}](rate.ps "fig:"){width="9cm"}
Another consequence of our simulations is the rate of GRBs. This is shown as a function of redshift for the entire population of simulated bursts (filled circles in Fig. \[fg7\]), in units of bursts Gpc$^{-3}$ yr$^{-1}$. The GRB redshift distribution (Eq. \[z\]) assumed in our simulations is shown by the solid grey line in Fig. \[fg7\] and the observed star formation rate (Li 2008) is shown by the dashed (grey) line rescaled by an arbitrary factor to match the rate of GRBs at $z=0$. We also show the rate of PO bursts (open blue squares) and that of the PO bursts of the comparison sample (open red circles). Fig. \[fg7\] also shows the recent estimate of the rate of SNIb/c computed by Grieco et al. (2012). The different curves for SNIb/c correspond to different assumption of the cosmic star formation rate (CSFR) in that paper. As a result of our simulation, the local rate of GRBs is $\sim$0.3% that of SNIb/c.
Summary and discussion
======================
We have studied two fundamental parameters of GRBs: the jet opening angle and the bulk Lorentz factor . The first question that we aimed to answer was [*whether and have preferential values*]{}. The direct measure of through the jet break times observed in the optical light curves (Frail et al. 2001; Ghirlanda et al. 2004, 2007) shows that $\sim$5$^\circ$. The measure of from the peak of the afterglow light curve for $\sim$30 GRBs (G12) also shows a characteristic value[^9] of $\sim$60. However, the limited number of events with a direct estimate of and and the possible selection effects, related to the difficulties of measuring these two parameters (see §1), prevent us to assume them as representative of the GRB population. In particular we want to test the consistency of different possible distributions of and with a set of available observational constraints (§2). Moreover, we aim at constraining the free parameters of the distributions of and and derive if and how these two parameters are correlated.
In this paper we used a population synthesis code to simulate GRBs with different assigned distributions of and of each one with a set of free parameters that we left free to vary within certain ranges. Obviously, we did not assume the observed distributions of and as constraints to avoid circularity.
We assume that GRBs have a unique comoving frame peak energy and collimation–corrected energy (the large black dot in Fig. \[fg0\]) which are transformed into their corresponding rest frame and respectively. The assigned and allow us to derive the isotropic equivalent energy of the simulated bursts according to the relative value of and . $\sim$$/\theta_{\rm jet}^2$ if $\sin\theta_{\rm jet}>1/\Gamma_0$, while $\sim$$\Gamma_0^2$ in the opposite case. This introduces a “natural bias" in the distribution of : those bursts with a small enough will have an isotropic energy which is smaller than that one would calculate using the value of . In the plane of Fig. \[fg0\] this corresponds to a limit $\propto E_{\rm iso}^{1/3}$. Bursts with 1/$>$sin will lie along this limiting line and there should be no GRBs on the right of this line \[i.e. in region (III) in Fig. \[fg0\]\].
This is the first time that the limit mentioned above is considered within the framework of studying the distributions of GRBs in e.g. the plane. Indeed, this limiting line can account for the absence, in the observed GRB sample with measured $z$ and well constrained peak energy (i.e. the bursts used to construct the correlation), of bursts with intermediate/low peak energy and very large .
The assumed distributions of and determine the distribution of simulated bursts in the plane of Fig. \[fg0\]. We considered two types of distributions for and : (A) a power law distribution, i.e. and do not assume any preferential value or (B) both and have peaked distributions (either broken power law or a log–normal distributions).
In order to test these two hypothesis we compared the results of our simulations with three GRB samples: the complete sample of GRBs detected by BAT with measured redshifts (S12), the sample of bursts detected by the GBM in the last 2 years (Goldstein et al. 2012) and the 4th BATSE catalog of GRBs (Meegan et al. 1997). The simulations should reproduce several proprieties of these samples.
While most of the bright bursts of the complete sample of S12 have measured $z$ and provide an observational constrain in the rest frame and observer frame plane (left and right panels of Fig. \[fg1\],\[fg2\],\[fg3\], respectively), the number count distribution and rate of the and GBM populations of bursts (mostly without measured $z$) are used as additional constraints since they map the faint end of the number count distribution of GRBs.
Our main result is that we cannot reproduce all our observational constraints if the and distributions are power laws. In this case the rate of GBM and BATSE bursts predicted by our simulations is a factor $\sim$2 larger than the real one and the distribution of the simulated bursts in the and plane is inconsistent with the real complete sample of bursts.
Instead, if and have broken power law distributions (with peak values $\sim 4.5^\circ$ and $\sim 70$) or log–normal distributions (with peak values $\sim 4.5^\circ$ and $\sim 85$) a better agreement between the simulations and the observational constraints is found. However, the broken power law or log–normal case produce a linear correlation due to the assumption that the simulated bursts have a distribution with a unique peak value (see §4). This motivated us to consider the possibility that there is a relation between the peak values of the distributions of and . G12 found that among GRBs with a estimate, three new correlations are found: $\propto$$^2$, $\propto$$^2$ and $\propto$. The combination of these correlations with the assumptions that $\theta_{\rm iso}^2$=const allows to derive the three main empirical correlations of GRBs: the correlation, the correlation and the correlation.
We therefore assumed that both and have log–normal distributions and that a relation of the type $\theta_{\rm jet}^m$=const exists between the peak values of their respective log–normal distributions. We found good consistency between our simulations and the observational constraints (Fig. \[fg3\]) in the case of a log–normal distribution of with central value 90 and logarithmic dispersion of 0.65. The distribution of (also a log–normal) is in this case determined by the relation $\theta^{m}_{\rm jet}$=const which we find should have $m=2.5$. This value is what one obtains by combining the above scaling relations (between and , ) with the correlation of the complete sample which is $\propto E_{\rm iso}^{0.6}$. The existence of a relation $\theta^{m}_{\rm jet}$=const (with $m\sim1$ and const=10–40) is also predicted from recent models of magnetically accelerated jets in GRBs (e.g. Tchekhovskoy et al. 2011).
We found that the distribution that best reproduces all our observational constraints extends to low values. If we cut the distribution so to exclude such low values of we cannot reproduce the observed flux and fluence distributions and detection rates of and GBM. Therefore, we find that low– bursts should exists in populations of GRBs detected by most sensitive detectors. Although a detailed study of the prompt and afterglow properties of these events is out of the scopes of this paper, we can draw some remarks. Apart from their relatively low and (which are correlated with as found by G11), the low– bursts should have a late time afterglow onset (i.e. a few hours for typical parameters, see §4.6). Therefore, their prompt emission should not be contaminated by the afterglow while their late time afterglow onset could be one of their distinctive features.
An immediate consequence of our results is that the large scatter of the correlation can be interpreted as due to the jet opening angle distribution of GRBs. The found inverse relation between and implies that bursts with the largest bulk Lorentz factors should have a smaller average . On the other hand, bursts with relatively low average factors should also have, on average, large .
Our results depend on the assumption that all bursts have the same =1.5 keV and =1.5$\times 10^{48}$ erg. Although there could be a dispersion of these values, our results still hold if the width of this dispersion is not larger than the dispersion of the observed quantities. We note that larger values of and would move the $\propto$$^{1/3}$ (Eq. \[gammalimit\]) towards the upper part of the plane of Fig. \[fg0\]. As a consequence some of the real GRBs of the complete sample, would be cut out of the plane because they would lie in the forbidden region (III) of this plane. On the other side we could assume lower values of and . Since we do not know their real dispersion, we tried to assume =0.15 keV and =1.5$\times 10^{47}$ (i.e. a factor 10 lower than the values assumed in the simulation). Under this different assumption, for the case of log–normal distributions of both and and of an intrinsic relation between these two parameters, we find that the distribution is consistent with that found with the fiducial values of and , but with a different distribution of . Indeed, in this case we find a mean value $\sim$ a factor 3 larger than that of the present simulation. Although it is not possible at the present stage to constrain the distribution of and , these results suggest that their dispersion should be lower than a factor of $\sim$10.
Our best simulations allow us to derive the properties of three populations of GRBs: those that are pointing to us and that have a peak flux bright enough to enter in the bright sample (i.e. with the same peak flux threshold adopted for the complete sample of S12), those that are pointing to us and, finally the full population of simulated GRBs, oriented randomly in the Universe (i.e. pointing to us and not). The latter is the GRB population that we cannot study on the base of the bursts that we detect. The main advantage of our population synthesis code is that we can infer the properties (e.g. the and distribution and the true GRB rate in this work) of this population of bursts, which is unaccessible through the observations.
One immediate consequence of our simulation is the true correlation. If we consider the PO bursts and if we were in principle able to detect them all, we should find a different correlation than the one presently reported in the literature. Indeed, the fit of the PO bursts in the plane of Fig. \[fg3\] yields a correlation with slope 0.5 and normalization -27.6 while the entire GRB population, the total simulated bursts, have a correlation with slope 0.44 and normalization -20.7. This is due to the fact that PO bursts tend to populate the lower region of the plane (Fig.\[fg3\]) where the $E_{\rm iso}^{1/3}$ limit cuts their distribution in the plane. Therefore, if we could measure and for all the bursts that point to us, we should determine a flatter correlation than that observed so far in the high part of the plane with bright bursts.
Our simulation predicts that the bright bursts detected by should have a mean opening angle of $\sim$4.7$^\circ$. This value is only a factor 2 smaller than the mean of the entire GRB population that we have simulated (which has $\sim 8.7^{\circ}$). However, from Fig. \[fg4\] (open blue squares) one can see that if we were able to detect fainter GRBs and to measure their jet opening angle, we would obtain a mean of 40$^\circ$. Intriguingly we note that the present distribution of measured from the optical afterglow break times in a few bursts is representative of the distribution of the entire population of bursts. This is because the bursts that we have detected so far populate the high region of the plane where the distribution can be almost unbiasedly sampled. In fact, only the bursts at lower values of and are affected by the “natural bias" of 1/$>$sin. The low – low region is where PO bursts concentrate (they have large or small , enhancing the probability to point at us).
Our simulation predicts that there are bursts with no jet break, the ones with 1/$>$sin. Their afterglows will never have a jet break since the condition 1/$\sim$sin is never met but their afterglow light curve should have a characteristic post–jet break intermediate/steep decay slope. These should be $\sim$6% of the bursts pointing to us and $\sim$2% of the bursts detected by with P$>$.
According to our best simulation, the mean of all bursts is $\langle\Gamma_0\rangle=274$. The distribution is highly asymmetric and there is a considerable difference between its mode (i.e. the peak) the mean and the median. The simulated bursts pointing to us corresponding to the complete sample have $\langle\Gamma_0\rangle=355$. These two values are broadly consistent, as explained above, since these bursts populate the upper part of the plane where the distribution of GRBs is almost free from the “natural bias". Remarkably, if we were able to measure for all the bursts pointing to us, we would find a very low value of the mean of $\langle\Gamma_0\rangle=50$. Finally, we have found that the distribution of that we predict for the bright sample is consistent with the distribution of of the GRBs studied in G12.
We can derive from our simulations the true rate of GRBs. Previous studies of the GRB rate assumed a unique value of , typically 0.2 rad or the [*observed*]{} distribution of (e.g. Guetta et al. 2005; Grieco et al. 2012). Our simulations (§4.5) show that the peak of the intrinsic/global distribution of is a factor 2 larger than the real intrinsic distribution and has a much wider dispersion (Tab.1). Differently from existing GRB rate estimates based on the correction of the isotropic GRB rate for an [*average*]{} beaming factor (e.g. Guetta et al. 2005; Grieco et al. 2012) in our simulation the total number of simulated bursts is adjusted in order to reproduce the rate of detections of GBM and BATSE. Therefore, we have the rate of GRBs as a function of redshift independently from the value of of each single burst. If we compare this rate with that of SNIb/c (from Gireco et al. 2012) we find that the local rate of GRBs is $\sim$0.3%. Moreover, if we consider the 7% fraction of SNIb/c which produce Hypernovae events (Guetta & Della Valle 2007) we find that the rate about 4.3% of local Hypernovae should produce a GRB.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank the referee for comments and suggestions that improved the manuscript. We acknowledge ASI I/004/11/0 and the 2011 PRIN-INAF grant for financial support.
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[^1]: E–mail:[email protected]
[^2]: Primed quantities are in the comoving frame of the source.
[^3]: These values are also assumed by S12 to constrain the LF of GRBs.
[^4]: The 8 GRBs without a secure estimate of the redshift or with incomplete spectral informations are consistent with the correlation defined by the 46 GRBs discussed here, see N12 for details.
[^5]: http://swift.gsfc.nasa.gov/docs/swift/archive/grb\_table/
[^6]: $P$ and $F$ are reported in Goldstein et al. (2012) and were obtained by integrating the model that best fits the peak time resolved spectrum and the time averaged spectrum, respectively.
[^7]: The $P$ values of the GBM sample are computed on the broad 10 keV–1 MeV energy range (i.e. much broader than the 15–150 keV energy range of ). This ensures that the selected sample of the GBM bursts extends the population of GRBs to lower fluxes than those of the complete sample.
[^8]: These are obtained by staking 1000 simulations and smoothing the obtained distribution in the plane.
[^9]: This average value is obtained assuming that the circumburst medium has a wind density profile (see G12).
|
{
"pile_set_name": "ArXiv"
}
|
---
title: 'A Design-Space Exploration for Allocating Security Tasks in Multicore Real-Time Systems '
---
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Given real numbers $\beta \equiv \beta ^{\left( 4\right) }\colon \beta
_{00}$, $\beta _{10}$, $\beta _{01}$, $\beta _{20}$, $\beta _{11}$, $\beta _{02}$, $\beta _{30}$, $\beta _{21}$, $\beta _{12}$, $\beta _{03}$, $\beta _{40}$, $\beta _{31}$, $\beta _{22}$, $\beta _{13}$, $\beta
_{04}$, with $\beta _{00} >0$, the *quartic real moment problem* for $\beta $ entails finding conditions for the existence of a positive Borel measure $\mu $, supported in $\mathbb{R}^2$, such that $\beta _{ij}=\int s^{i}t^{j}\,d\mu \;\;(0\leq i+j\leq
4) $. Let $\mathcal{M}(2)$ be the $6 \times 6$ moment matrix for $\beta^{(4)}$, given by $\mathcal{M}(2)_{\mathbf{i},\mathbf{j}}:=\beta_{\mathbf{i}+\mathbf{j}}$, where $\mathbf{i},\mathbf{j} \in \mathbb{Z}^2_+$ and $\left|\mathbf{i}\right|,\left|\mathbf{j}\right|\le 2$. In this note we find concrete representing measures for $\beta^{(4)}$ when $\mathcal{M}(2)$ is nonsingular; moreover, we prove that it is possible to ensure that one such representing measure is $6$-atomic.
address:
- 'Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242'
- 'Department of Mathematics, Seoul National University, Seoul 151-742, Korea'
author:
- 'Raúl E. Curto'
- Seonguk Yoo
title: |
Concrete Solution to the Nonsingular\
Quartic Binary Moment Problem
---
[^1]
\[Int\]Introduction
===================
In this paper we find a direct proof that the *nonsingular* Quartic Binary Moment Problem always admits a finitely atomic representing measure with the minimum number of atoms, that is, six atoms. We do this in three steps:
\(i) by normalizing the given moment matrix $\mathcal{M}(2)$ to ensure that $\mathcal{M}(1)$ is the identity matrix;
\(ii) by developing a new rank-reduction tool, which allows us to decompose the normalized $\mathcal{M}(2)$ matrix as the sum of a positive semidefinite moment matrix $\widetilde{\mathcal{M}(2)}$ of rank $5$ and the rank-one moment matrix of the point mass at the origin; and
\(iii) by proving that when a moment matrix $\mathcal{M}(2)$ admits such a decomposition, and $\widetilde{\mathcal{M}(2)}$ admits a column relation subordinate to a degenerate hyperbola (i.e., a pair of intersecting lines), then $\mathcal{M}(2)$ admits a $6$-atomic representing measure (as opposed to the expected $7$-atomic measure).
To describe our results in detail, we need some notation and terminology. Given real numbers $\beta \equiv \beta ^{\left( 4\right) }\colon \beta
_{00}$, $\beta _{10}$, $\beta _{01}$, $\beta _{20}$, $\beta _{11}$, $\beta _{02}$, $\beta _{30}$, $\beta _{21}$, $\beta _{12}$, $\beta _{03}$, $\beta _{40}$, $\beta _{31}$, $\beta _{22}$, $\beta _{13}$, $\beta
_{04}$, with $\beta _{00} >0$, the *Quartic Real Moment Problem* for $\beta $ entails finding conditions for the existence of a positive Borel measure $\mu $, supported in $\mathbb{R}^2$, such that $\beta _{ij}=\int s^{i}t^{j}\,d\mu \;\;(0\leq i+j\leq
4) $. Let $\mathcal{M}(2)$ be the moment matrix for $\beta^{(4)}$, given by $\mathcal{M}(2)_{\mathbf{i},\mathbf{j}}:=\beta_{\mathbf{i}+\mathbf{j}}$, where $\mathbf{i},\mathbf{j} \in \mathbb{Z}^2_+$ and $\left|\mathbf{i}\right|,\left|\mathbf{j}\right|\le 2$; this $6 \times 6$ matrix is shown below. (As is customary, the columns of $\mathcal{M}(2)$ are labeled $\textit{1},X,Y,X^2,XY,Y^2$. In a similar way, given a collection of real numbers $\beta^{(2n)}$ one defines the associated moment matrix by $\mathcal{M}(n)_{\mathbf{i},\mathbf{j}}:=\beta_{\mathbf{i}+\mathbf{j}}$, where $\mathbf{i},\mathbf{j} \in \mathbb{Z}^2_+$ and $\left|\mathbf{i}\right|,\left|\mathbf{j}\right|\le n$.) $$\mathcal{M}(2)\equiv \begin{pmatrix}
\beta_{00} &\beta_{10} & \beta_{01} & \beta_{20} & \beta_{11} & \beta_{02} \\
\beta_{10} & \beta_{20} & \beta_{11} & \beta_{30} & \beta_{21} & \beta_{12} \\
\beta_{01} &\beta_{11} & \beta_{02} & \beta_{21} & \beta_{12} & \beta_{03} \\
\beta_{20} & \beta_{30} & \beta_{21} & \beta_{40} & \beta_{31} & \beta_{22} \\
\beta_{11} & \beta_{21} & \beta_{12} & \beta_{31} & \beta_{22} & \beta_{13} \\
\beta_{02} & \beta_{12} & \beta_{03} & \beta_{22} & \beta_{13} & \beta_{04}
\end{pmatrix}.$$ Assume now that $\mathcal{M}(2)$ is nonsingular. A straightforward consequence of Hilbert’s Theorem yields the existence of a finitely atomic representing measure, as follows. Let $\mathcal{P}_4$ be the cone of nonnegative polynomials of degree at most $4$ in $x,y$, regarded as a subset of $\mathbb{R}^{15}$. The dual cone is $\mathcal{P}_4^*:=\{\xi \in \mathbb{R}^{15}: \left\langle \xi,p \right\rangle \ge 0 \; \textrm{for all } p \in \mathcal{P}_4 \}$. If $a,b \in \mathbb{R}$ and $\xi_{(a,b)}:=(1,a,b,a^2,ab,b^2,a^3,a^2b,ab^2,b^3,a^4,a^3b,a^2b^2,ab^3,b^4) \in \mathbb{R}^{15}$, then $\left\langle \xi_{(a,b)},p\right\rangle=p(a,b) \ge 0$, for all $p \in \mathcal{P}_4$. Thus, $\xi_{(a,b)} \in \mathcal{P}_4^*$ for all $a,b \in \mathbb{R}$, and $\xi_{(a,b)}$ is also an extreme point. Consider now an arbitrary moment sequence $\beta^{(4)}$ with a nonsingular moment matrix $\mathcal{M}(2)$. Regarded as a point in $\mathbb{R}^{15}$, $\beta^{(4)}$ is in the interior of $\mathcal{P}_4^*$, since every $p \in \mathcal{P}_4$ is a sum of squares of polynomials. By the Krein-Milman Theorem and Carathéodory’s Theorem, the Riesz functional $\Lambda_{\beta^{(4)}}$ is a convex combination of evaluations $\xi_{(a,b)}$; that is, $\beta^{(4)}$ admits a finitely atomic representing measure, with at most $15$ atoms. (In recent related work, L.A. Fialkow and J. Nie [@FiNi] have obtained this result as a consequence of a more general result on moment problems.)
In this note we obtain a concrete $6$-atomic representing measure for $\mathcal{M}(2)$. The Quartic Real Binary Moment Problem admits an equivalent formulation in terms of complex numbers and representing measures supported in the complex plane $\mathbb{C}$, as follows. Given complex numbers $\gamma \equiv \gamma ^{\left( 4\right) }\colon \gamma
_{00}$, $\gamma _{01}$, $\gamma _{10}$, $\gamma _{02}$, $\gamma _{11}$, $\gamma _{20}$, $\gamma _{03}$, $\gamma _{12}$, $\gamma _{21}$, $\gamma _{30}$, $\gamma _{04}$, $\gamma _{13}$, $\gamma _{22}$, $\gamma _{31}$, $\gamma
_{40}$, with $\gamma _{ij}=\bar{\gamma}_{ji}$, one seeks necessary and sufficient conditions for the existence of a positive Borel measure $\mu $, supported in $\mathbb{C}$, such that $$\gamma _{ij}=\int \bar{z}^{i}z^{j}\,d\mu \qquad (0\leq i+j\leq 4).$$ Just as in the real case, the Quartic Complex Moment Problem has an associated moment matrix $M(2)$, whose columns are conveniently labeled $1,Z,\bar{Z},Z^2,\bar{Z}Z,\bar{Z}^2$. The most interesting case of the Singular Quartic Binary Moment Problem arises when the rank of $M(2)$ is $5$, and the sixth column of $M(2)$, labeled $\bar{Z}^2$, is a linear combination of the remaining five columns. Depending on the coefficients in the linear combination, four subcases arise in terms of the associated conic $C$ [@tcmp6 Section 5]: (i) $C$ is a parabola; (ii) $C$ is a nondegenerate hyperbola; (iii) $C$ is a pair of intersecting lines; and (iv) $C$ is a circle. In subcase (iii), it is possible to prove that the number of atoms in a representing measure (if it exists) may be $6$ [@tcmp6 Proposition 5.5 and Example 5.6]; that is, in some soluble cases the rank of $M(2)$ may be strictly smaller than the number of atoms in any representing measure.
\[degenerate\] ([@tcmp6 Proposition 5.5]) If $\mathcal{M}(2) \ge0$, if $\operatorname{rank} \mathcal{M}(2) =5$, and if $XY = 0$ in the column space of $\mathcal{M}(2)$, then $\mathcal{M}(2)$ admits a representing measure $\mu$ with $\operatorname{card} \; \operatorname{supp} \mu \le 6$.
When combined with previous work on truncated moment problems, Proposition \[degenerate\] led to the following solution to the truncated moment problem on planar curves of degree $\leq 2$. Given a moment matrix $\mathcal{M}(n)$ and a polynomial $p(x,y)\equiv \sum p_{ij} x^i y^j$, we let $p(X,Y):=\sum p_{ij} X^i Y^j$. A column relation in $\mathcal{M}(n)$ is therefore always described as $p(X,Y)=0$ for some polynomial $p$, with deg $p \le n$. We say that $\mathcal{M}(n)$ is *recursively generated* if for every $p$ with $p(X,Y)=0$ and every $q$ such that deg $pq \le n$ one has $(pq)(X,Y)=0$. In what follows, $v$ denotes the cardinality of the associated algebraic variety, defined as the intersection of the zero sets of all polynomials which describe the column relations in $\mathcal{M}(n)$.
\[quartic\] ([@tcmp9 Theorem 2.1], [@Fia4 Theorem 1.2]) Let $p \in \mathbb{R}[x,y]$, with $\deg p(x,y)\leq 2$. Then $\beta ^{(2n)}$ has a representing measure supported in the curve $p(x,y)=0$ if and only if $\mathcal{M}(n)$ has a column dependence relation $p(X,Y)=0$, $\mathcal{M}(n) \geq 0$, $\mathcal{M}(n)$ is recursively generated, and $r\leq v$.
The proof of Theorem \[quartic\] made use of affine planar transformations to reduce a generic quadratic column relation to one of four canonical types: $Y=X^2$, $XY=1$, $XY=0$ and $X^2+Y^2=1$; each of these cases required an independent result. We shall have occasion to use the affine planar transformation approach in Section \[Tool\]. To date, most of the existing theory of truncated moment problems is founded on the presence of nontrivial column relations in the moment matrix $\mathcal{M}(n)$. On one hand, when all columns labeled by monomials of degree $n$ can be expressed as linear combinations of columns labeled by monomials of lower degree, the matrix $\mathcal{M}(n)$ is flat, and the moment problem has a unique representing measure, which is finitely atomic, with exactly $\operatorname{rank} \mathcal{M}(n-1)$ atoms [@tcmp2 Theorem 1.1]. As a straightforward consequence, we conclude that for $n=1$, an invertible $\mathcal{M}(n)$ always admits a flat extension, while that is not the case for $n \ge 3$; that is, there exist examples of positive and invertible $\mathcal{M}(3)$ without a representing measure (cf. [@tcmp2 Section 4]).
When $n=2$, the idea is to extend the $6 \times 6$ moment matrix $\mathcal{M}(2)$ to a bigger $10 \times 10$ moment matrix $\mathcal{M}(3)$ by adding so-called $B$ and $C$ blocks, as follows: $$\mathcal{M}\left( 3\right) \equiv
\begin{pmatrix}
\mathcal{M}\left( 2\right) & B\left( 3\right) \\
B\left( 3\right) ^{\ast} & C\left( 3\right)
\end{pmatrix}.$$ A result of J.L. Smul’jan [@Smu] states that $\mathcal{M}(3) \ge 0$ if and only if (i) $\mathcal{M}(2) \ge 0$; (ii) $B(3)=\mathcal{M}(2)W$ for some $W$; and (iii) $C(3) \ge W^{*}\mathcal{M}(2)W$. Moreover, $\mathcal{M}(3)$ is a *flat* extension of $\mathcal{M}(2)$ (i.e., $\operatorname{rank} \mathcal{M}(3) = \operatorname{rank} \mathcal{M}(2)$) if and only if $C(3)=W^{*}\mathcal{M}(2)W$. Further, when $\mathcal{M}(2)$ is invertible, one easily obtains $W=\mathcal{M}(2)^{-1}B(3)$, so in the flat extension case $C(3)$ can be written as $B(3)^{*}\mathcal{M}(2)^{-1}B(3)$. However, writing a general formula for $\mathcal{M}(3)$ is nontrivial, even with the aid of *Mathematica*, because of the complexity of $(\mathcal{M}(2))^{-1}$ and the new moments contributed by the block $B(3)$. On the other hand, if only one column relation is present (given by $p(X,Y)=0$), then $v=+\infty$, and the condition $r \le v$, while necessary, will not suffice. One knows that the support of a representing measure must lie in the zero set of $p$, but this does not provide enough information to decipher the block $B(3)$. The situation is much more intriguing when no column relations are present; this is the nonsingular case, for which very little is known.
\[Statement\] Statement of the Main Result
==========================================
\[MainTheorem\] Assume $\mathcal{M}(2)$ is positive and invertible. Then $\mathcal{M}(2)$ admits a representing measure, with exactly $6$ atoms; that is, $\mathcal{M}(2)$ actually admits a flat extension $\mathcal{M}(3)$.
The proof of Theorem \[MainTheorem\] is constructive, in that we first prove that it is always possible to switch from the invertible $\mathcal{M}(2)$ to a related singular matrix $\widetilde{\mathcal{M}(2)}$, with $\operatorname{rank} \widetilde{\mathcal{M}(2)}=5$, for which Theorem \[quartic\] applies. Since singular positive semidefinite matrices $\mathcal{M}(2)$ always admit representing measures with $6$ atoms or less, we can then conclude that an invertible positive $\mathcal{M}(2)$ admits a representing measure with at most $7$ atoms. While this would already represent a significant improvement on the upper bound given by Carath' eodory’s Theorem ($15$ atoms), we have been able to establish that all positive invertible $\mathcal{M}(2)$’s actually have *flat* extensions, and therefore their representing measures can have exactly $6$ atoms.
\[Tool\] A New Tool
===================
We begin this section with a result that will allow us to convert a given moment problem into a simpler, equivalent, moment problem. One of the consequences of this result is the equivalence of the real and complex moment problems, via the transformation $x:=$ Re$[z]$ and $y:=$ Im$[z]$; this equivalence has been exploited amply in the theory of truncated moment problems. For us, however, this simplification will allow us to assume that the submatrix $\mathcal{M}(1)$ is the identity matrix.
We adapt the notation in [@tcmp6] to the real case. For $a,b,c,d,e,f\in \mathbb{R}$, $bf-ce \ne 0$, let $\Psi(x,y)\equiv \left(\Psi_1(x,y),\Psi_2(x,y)\right):=\left( a+bx + cy, d+ex+fy\right)$ ($x,y\in \mathbb{R}$). Given $\beta ^{(2n)}$, define $\tilde{\beta}^{\left( 2n\right) }$ by $\tilde{\beta}_{ij}:=L_{\beta }(\Psi_1^i \Psi_2^j)$ ($0\leq i+j\leq 2n$), where $L_{\beta }$ denotes the *Riesz functional* associated with $\beta $. (For $p(x,y)\equiv \sum p_{ij} x^i y^j$, the Riesz functional is given by $L_{\beta}(p):=p(\beta)\equiv \sum p_{ij} \beta_{ij}$.) It is straightforward to verify that $
L_{\tilde{\beta}}(p)=L_{\beta }\left( p\circ \Psi \right) $ for every $p$ of degree at most $n$.
\[lininv\] Let $\mathcal{M}(n)$ and $\tilde{\mathcal{M}}(n)$ be the moment matrices associated with $\beta$ and $\tilde{\beta}$, and let $J\hat{p}:=\widehat{p\circ\Psi}$. Then the following statements hold.
1. \[lininv(1)\]$\tilde{\mathcal{M}}(n)=J^{\ast}\mathcal{M}(n)J$.
2. \[lininv(2)\]$J$ is invertible.
3. \[lininv(3)\]$\tilde{\mathcal{M}}(n)\geq0\Leftrightarrow \mathcal{M}(n)\geq0$.
4. \[lininv(4)\]$\operatorname{rank}\tilde{\mathcal{M}}(n)=\operatorname{rank}\mathcal{M}(n)$.
5. \[lininv(6)\]$\mathcal{M}\left( n\right) $ admits a flat extension if and only if $\tilde{\mathcal{M}}\left( n\right) $ admits a flat extension.
We are now ready to put $\mathcal{M}(2)$ in “normalized form." Without loss of generality, we always assume that $\beta_{00}=1$. Let $d_i$ denote the leading principal minors of $\mathcal{M}(2)$; in particular, $$\begin{aligned}
d_2 &=&-\beta_{10}^2+\beta_{20}\\
d_3&=& -\beta_{02} \beta_{10}^2+2 \beta_{01} \beta_{10} \beta_{11}-\beta_{11}^2-\beta_{01}^2 \beta_{20}+\beta_{02} \beta_{20}.\end{aligned}$$ Consider now the degree-one transformation $$\Psi(x,y)\equiv \left( a+bx + cy, d+ex+fy\right),$$ where $ a:=\frac{\beta_{01}\beta_{20}-\beta_{10} \beta_{11}}{\sqrt{d_2 d_3}}$, $b:=\frac{\beta_{11}-\beta_{01} \beta_{10}}{\sqrt{d_2 d_3}}$, $c:=
- \sqrt{\frac{d_2}{d_3}}$ , $d:=-\frac{\beta_{10}}{\sqrt {d_2} }$, $e:= \frac{1}{\sqrt {d_2} }$, and $f:=0$. Note that $bf-ce=- \sqrt{\frac{1}{d_3}}\neq0$. Using this transformation, and a straightforward calculation, we can prove that any positive definite moment matrix $\mathcal{M}(2)$ can be transformed into the moment matrix $$\begin{pmatrix}
1 & 0& 0 & 1 & 0 & 1 \\
0& 1& 0 & \tilde{\beta}_{30} & \tilde{\beta}_{21}& \tilde{\beta}_{12} \\
0 & 0 & 1 & \tilde{\beta}_{21} & \tilde{\beta}_{12} & \tilde{\beta}_{03} \\
1 & \tilde{\beta}_{30} & \tilde{\beta}_{21} & \tilde{\beta}_{40} & \tilde{\beta}_{31} & \tilde{\beta}_{22} \\
0 & \tilde{\beta}_{21} & \tilde{\beta}_{12} & \tilde{\beta}_{31} & \tilde{\beta}_{22} & \tilde{\beta}_{13} \\
1 & \tilde{\beta}_{12} & \tilde{\beta}_{03} & \tilde{\beta}_{22} & \tilde{\beta}_{13} & \tilde{\beta}_{04}
\end{pmatrix}.$$ Thus, without loss of generality, we can always assume that $\mathcal{M}(1)$ is the identity matrix. We will now introduce a new tool in the study of moment matrices: the decomposition of an invertible $\mathcal{M}(2)$ as a sum of a moment matrix of rank $5$ and a rank-one moment matrix. Assume now that $\mathcal{M}(2)$ is invertible and that the submatrix $\mathcal{M}(1)$ is the identity matrix. For $u \in \mathbb{R}$ decompose $\mathcal{M}(2)$ as follows: $$\mathcal{M}(2)=\begin{pmatrix}
1 - u &0 & 0 & 1 & 0 & 1 \\
0 & 1 & 0 & \beta_{30} & \beta_{21} & \beta_{12} \\
0 &0 & 1 & \beta_{21} & \beta_{12} & \beta_{03} \\
1& \beta_{30} & \beta_{21} & \beta_{40} & \beta_{31} & \beta_{22} \\
0 & \beta_{21} & \beta_{12} & \beta_{31} & \beta_{22} & \beta_{13} \\
1 & \beta_{12} & \beta_{03} & \beta_{22} & \beta_{13} & \beta_{04}
\end{pmatrix}
+
\begin{pmatrix}
u & 0&0&0&0&0 \\
0 & 0&0&0&0&0 \\
0 & 0&0&0&0&0 \\
0 & 0&0&0&0&0 \\
0 & 0&0&0&0&0 \\
0 & 0&0&0&0&0
\end{pmatrix}.$$ Denote the first summand by $\widehat{\mathcal{M}(2)}$ and the second summand by $\mathcal{P}$. It is clear that $\mathcal{P}$ is positive semidefinite and has rank $1$ if and only if $u>0$, and in that case $\mathcal{P}$ is the moment matrix of the $1$-atomic measure $u \delta_{(0,0)}$, where $\delta_{(0,0)}$ is the point mass at the origin.
\[rankred\] Let $\mathcal{M}(2)$, $\widehat{\mathcal{M}(2)}$ and $\mathcal{P}$ be as above, and let $u_0:=\frac{ \operatorname{det} \mathcal{M}(2)}{R_{11}}$, where $R_{11}$ is the $(1,1)$ entry in the positive matrix $R:=(\mathcal{M}(2))^{-1}$. Then, with this nonnegative value of $u$, we have (i) $\widehat{\mathcal{M}(2)}\ge 0$; (ii) $\operatorname{rank} \widehat{\mathcal{M}(2)}=5$; and (iii) $\widehat{\mathcal{M}(2)}$ is recursively generated. Moreover, $u_0$ is the only value of $u$ for which $\widehat{\mathcal{M}(2)}$ satisfies (i)–(iii).
For the proof of Proposition \[rankred\] we will need to following auxiliary result, which is an easy consequence of the multilinearity of the determinant.
\[lem1\] Let $M$ be an $n \times n$ invertible matrix of real numbers, let $E_{11}$ be the rank-one matrix with $(1,1)$-entry equal to $1$ and all other entries are equal to zero, and let $u \in \mathbb{R}$. Then $\operatorname{det} (M-u E_{11})=\operatorname{det} M-u \operatorname{det} M_{\{2,3,\cdots,n\}}$, where $M_{\{2,3,\cdots,n\}}$ denotes the $(n-1) \times (n-1)$ compression of $M$ to the last $n-1$ rows and columns. In particular, if $u=\frac{\operatorname{det} M}{(M^{-1})_{11}}$, then $\operatorname{det} (M-u E_{11})=0$.
\(ii) Observe that $6={\text{rank\ }}\mathcal{M}(2) \leq {\text{rank\ }}\widehat{\mathcal{M}(2)} +{\text{rank\ }}\mathcal{P} ={\text{rank\ }}\widehat{\mathcal{M}(2)} +1$, so ${\text{rank\ }}\widehat{\mathcal{M}(2)} \geq 5.$ Since $\det \widehat{\mathcal{M}(2)}=0$, we have ${\text{rank\ }}\widehat{\mathcal{M}(2)}=5$.
\(i) Using the Nested Determinant Test starting at the lower right-hand corner of $\widehat{\mathcal{M}(2)}$, we know that $\widehat{\mathcal{M}(2)}$ is positive semidefinite since the nested determinants corresponding to principal minors of size $1$, $2$, $3$, $4$ and $5$ are all positive, and the rank of $\widehat{\mathcal{M}(2)}$ is $5$. This also implies that $1-u\geq0$. We now claim that $1-u$ is strictly positive. If $1-u=0$, then the positive semidefiniteness of $\widehat{\mathcal{M}(2)}$ would force all entries in the first row to be zero. Since this is evidently false, we conclude that $1-u>0$.
\(iii) It is sufficient to show that the first three columns of $\widehat{\mathcal{M}(2)}$ are linearly independent. Consider the third leading principal minor of $\widehat{\mathcal{M}(2)}$, which equals $1-u$, and is therefore positive. Thus, there is no linear dependence in this submatrix, and as a result the same holds in $\widehat{\mathcal{M}(2)}$.
Finally, the uniqueness of $u_0$ as the only value satisfying (i)–(iii) is clear.
\[Proof\] Proof of the Main Result
==================================
We first observe that by combining Proposition \[rankred\] with Theorem \[quartic\], it suffices to consider the case when $\widehat{\mathcal{M}(2)}$ has a column relation corresponding to a pair of intersecting lines. For, in all other cases, there exists a representing measure for $\widehat{\mathcal{M}(2)}$ with exactly five atoms; when combined with the additional atom coming from the matrix $\mathcal{P}$, we see that $\mathcal{M}(2)$ admits a $6$-atomic representing measure.
We thus focus on the case when $\widehat{\mathcal{M}(2)}$ is subordinate to a degenerate hyperbola. After applying an additional degree-one transformation, we can assume, as in Proposition \[degenerate\], that the column relation $XY=0$ is present in $\widehat{\mathcal{M}(2)}$. However, we may not continue to assume that the submatrix $\widehat{\mathcal{M}(1)}$ is the identity matrix, since the degree-one transformation that produces the column relation $XY=0$ will, in general, change the low-order moments. That is, $\widehat{\mathcal{M}(2)}$ is of the form $$\widehat{\mathcal{M}(2)}=\left(
\begin{array}{cccccc}
1 & a & b & c & 0 & d \\
a & c & 0 & e & 0 & 0 \\
b & 0 & d & 0 & 0 & f \\
c & e & 0 & g & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
d & 0 & f & 0 & 0 & h \\
\end{array}
\right).$$ In this case, the original moment matrix $\mathcal{M}(2)$ is written as $$\mathcal{M}(2) = \widehat{\mathcal{M}(2)}+u \begin{pmatrix}
1 & p & q& p^2& p\, q & q^2
\end{pmatrix}^T \begin{pmatrix}
1 & p & q& p^2& p\, q & q^2
\end{pmatrix},$$ for some $u>0$ and $p\, q\neq 0$. That is, $\mathcal{M}(2)$ is the sum of a moment matrix of rank $5$ with column relation $XY=0$ and a positive scalar multiple of the moment matrix associated with the point mass at $(p,q)$, with $pq \ne 0$. Without loss of generality, we can assume that $p=q=1$ (this requires an obvious degree-one transformation, i.e., $\tilde{x}:=\frac{x}{p}, \; \tilde{y}:=\frac{y}{q}$). As a result, the form of $\mathcal{M}(2)$ is now as follows: $$\mathcal{M}(2)=\left(
\begin{array}{cccccc}
1+u & a+u & b+u & c+u & u & d+u \\
a+u & c+u & u & e+u & u & u \\
b+u & u & d+u & u & u & f+u \\
c+u & e+u & u & g+u & u & u \\
u & u & u & u & u & u \\
d+u & u & f+u & u & u & h+u \\
\end{array}
\right)$$ We will show that $\mathcal{M}(2)$ admits a flat extension, and that will readily imply that it admits a ${\text{rank\ }}\mathcal{M}(2)$-atomic (that is, 6-atomic) representing measure. The $B(3)$-block in an extension $\mathcal{M}(3)$ can be generated by letting $\beta_{41}=\beta_{32}=\beta_{23}=\beta_{14}=u$, so that $B(3)$ can thus be written as $$\left(
\begin{array}{cccc}
e+u & u & u & f+u \\
g+u & u & u & u \\
u & u & u & h+u \\
\beta_{50} & u & u & u \\
u & u & u & u \\
u & u & u & \beta_{05} \\
\end{array}
\right).$$ As usual, let $W:= \mathcal{M}(2)^{-1} B(3)$ and let $C(3)\equiv (C_{ij}):=W^{\ast} \mathcal{M}(2) W$. Note that if $C(3)$ turns out to be Hankel, then $\mathcal{M}(3)$ is a flat extension of $\mathcal{M}(2)$. Since $C(3)$ is symmetric, to ensure that $C(3)$ is Hankel (and therefore $\mathcal{M}(3)$ is a moment matrix) we only need to solve the following system of equations: $$\begin{aligned}
\label{eq-E}
\begin{cases}
E_1:=C_{13}-C_{22} =0\\
E_2:= C_{14}-C_{23}=0\\
E_3:= C_{24}-C_{33}=0.
\end{cases}\end{aligned}$$ This is a system of equations involving quadratic polynomials with $2$ unknown variables (the new moments $\beta_{50}$ and $\beta_{05}$). A straightforward calculation shows that $E_1=0$, $E_3=0$, and that $$\begin{aligned}
E_2 = 0~ \ \iff \!\!\!\!\!\!\!&&(c^2 - a e) (d^2 - b f)
\beta_{50} \beta_{05}
+(c^2 - a e) (f^3 - 2 d f h + b h^2 - d^2 u + b f u)\beta_{50} \\
&&+(d^2 - b f) (e^3 - 2 c e g + a g^2 - c^2 u + a e u) \beta_{05} \\
&&+ (e^3 - 2 c e g + a g^2 - c^2 u + a e u) (f^3 - 2 d f h + b h^2 -d^2 u + b f u)=0 \\
\ \iff \!\!\!\!\!\!\!&&\kappa \lambda \beta_{50} \beta_{05}+\kappa \mu \beta_{50}+\lambda \nu \beta_{05}+\nu \mu =0 ,\end{aligned}$$ where $\kappa$, $\lambda$, $\mu$ and $\nu$ have the obvious definitions. If $\kappa, \lambda \ne 0$, then $\beta_{05}=\frac{-\mu \nu+\kappa \mu \beta_{50}}{\kappa \lambda \beta_{50}+\lambda \nu}$ (for $\beta_{50} \ne -\frac{\nu}{\kappa}$), which readily implies that $E_2=0$ admits infinitely many solutions. When $\kappa = 0$ and $\lambda \ne 0$, we see that $E_2=\lambda \nu \beta_{05}+\mu \nu$, from which it follows that a solution always exists (and it is unique when $\nu \ne 0$). A similar argument shows that $\kappa \ne 0$ and $\lambda =0$ also yields a solution (which is unique when $\mu \ne 0$). We are thus left with the case when both $\kappa \equiv c^2-ae$ and $\lambda \equiv d^2-bf$ are equal to zero. Since $c$ and $d$ are in the diagonal of a positive semidefinite matrix, they must be positive. Thus, all of $a,$ $b$, $e$, and $f$ are nonzero and we can set $e:=c^2/a$ and $f:=d^2/b$. In this case, the moment matrix is $$\begin{aligned}
\mathcal{M}(2)=\left(
\begin{array}{cccccc}
1+u & a+u & b+u & c+u & u & d+u \\
a+u & c+u & u & \frac{c^2}{a}+u & u & u \\
b+u & u & d+u & u & u & \frac{d^2}{b}+u \\
c+u & \frac{c^2}{a}+u & u & g+u & u & u \\
u & u & u & u & u & u \\
d+u & u & \frac{d^2}{b}+u & u & u & h+u
\end{array}
\right)\end{aligned}$$ Let $k:=\det \mathcal{M}(2)/ \det \mathcal{M}(2)_{\{2,3,4,5,6\}}$. As in the proof of Proposition \[rankred\], we see that $k=\frac{-b^2 c-a^2 d+c d}{c d}>0$ and the first summand in the following decomposition of $\mathcal{M}(2)$ has rank $5$ and is positive semidefinite (note that the $(1,1)$-entry is $1+u-k$): $$\mathcal{M}(2)=\left(
\begin{array}{cccccc}
\frac{b^2 c+a^2 d+c d u}{c d} & a+u & b+u & c+u & u & d+u \\
a+u & c+u & u & \frac{c^2+a u}{a} & u & u \\
b+u & u & d+u & u & u & \frac{d^2+b u}{b} \\
c+u & \frac{c^2+a u}{a} & u & g+u & u & u \\
u & u & u & u & u & u \\
d+u & u & \frac{d^2+b u}{b} & u & u & h+u
\end{array}
\right)+
\left(
\begin{array}{cccccc}
k & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{array}
\right)$$ The only column relation in the first summand is $$\label{cr1}
XY=\frac{c d}{-b c-a d+c d}\textit{1} -\frac{a d}{-b c-a d+c d} X -\frac{b c }{-b c-a d+c d}Y=:\xi 1-\eta X - \theta Y.$$ Unless $\eta \theta=-\xi$, the conic that represents this column relation is a nondegenerate hyperbola, and therefore the moment sequence associated to the moment matrix has a $5$-atomic measure, by Theorem \[quartic\]. In the case when the conic in (\[cr1\]) is a pair of intersecting lines (i.e., $(x+\theta)(y+\eta)=0$), we must have $c=a$ or $d=b$.
Thus, the remaining two specific cases to cover are $\mathcal{M}(2)$ with $c=a$ or $d=b$. Since $\mathcal{M}(2)$ is invertible, for any $B(3)$ block we will be able to find $W$ such that $\mathcal{M}(2)W=B(3)$. We propose to use a $B(3)$ block with new moments $\beta_{32}=\beta_{23}=\beta_{14}=0$, and to then extend $\mathcal{M}(2)$ to $\mathcal{M}(3)$ using Smul’jan’s Lemma, that is, we will define $C(3):=W^*B(3)$. The goal is to establish that $C(3)$ is a Hankel matrix, and that requires verification of (\[eq-E\]). Before we begin our detailed analysis, we need to make a few observations.
Let $d_i$ denote the principal minor of $\mathcal{M}(2)$ for $i=1,\ldots,6$; since $\mathcal{M}(2)$ is positive and invertible, we know that these minors are all positive. Then $$\begin{aligned}
d_5=-\frac{\left(b^2 c+a^2 d-c d\right) \left(-c^3+a^2 g\right) u}{a^2} \text{ \ \ and \ \ }
d_6=\frac{ d_5\left(-d^3+b^2 h\right) }{ b^2},\end{aligned}$$ which implies $$\begin{aligned}
\label{minor}
\left(b^2 c+a^2 d-c d\right) \left(-c^3+a^2 g\right)<0 \qquad \text{ and } \qquad -d^3+b^2 h >0 .\end{aligned}$$ Next, we use *Mathematica* to solve $E_1=0$ for $\beta_{50}$ and $E_3=0$ for $\beta_{05}$, and we obtain $$\begin{aligned}
&&\beta_{50}=\frac{1}{a^2 c \left(b^2 c+a^2 d-c d\right) \left(-d^3+b^2 h\right) u}
( \alpha_{11} \beta_{41}^2 + \alpha_{12} \beta_{41} +\alpha_{13} ), \\
&&\beta_{05}=\frac{1}{b^2 d \left(b^2 c+a^2 d-c d\right) \left(c^3-a^2 g\right)} ( \alpha_{21} \beta_{41} +\alpha_{22} ),\end{aligned}$$ where the $\alpha_{ij}$’s are polynomials in $a,b,c,d,g,h,$ and $u$. Since $a,b\neq 0$, $c,d>0$, we can use (\[minor\]) to show that both $\beta_{50}$ and $\beta_{05}$ above are well defined. We now substitute these values in $E_2$ and check that $E_2$ is a quadratic polynomial in $\beta_{41}$; indeed, we can readily show that the leading coefficient of $E_2$ is nonzero if $c=a$ or $d=b$. Thus, if the discriminant $\Delta $ of this quadratic polynomial is nonnegative, then (\[eq-E\]) has at least one solution. We are now ready to deal with the two special cases: $c=a$ and $d=b$. If $c=a$, then $$\begin{aligned}
\Delta=\frac{a^2 u^2 (a-g)^2 \left(-d^3+b^2 h\right)^2 F_1(a,b,d,h) }{b^4 d^2},\end{aligned}$$ where $$F_1(a,b,d,h)= (-1+a)^2 b^2 h^2+
2 b^2 d \left(2 b^2-3 d+3 a d\right) h
-d^4 \left(3 b^2-4 d+4 a d\right)$$ is a concave upward quadratic polynomial in $h$. Notice that $\Delta \geq 0$ if and only if $F_1 \geq 0$, which means that the discriminant of $F_1$, $\Delta_1:=
16 b^2 d^2 \left(b^2-d+a d\right)^3$, needs to be zero or negative. In this case, we observe that $$\begin{aligned}
&&c=a>0,\\
&&d_3= -a b^2+a d-a^2 d+a u-a^2 u-b^2 u+d u-a d u > 0,\\
&&d_4= -d_3 (a-g)>0 \; (\Rightarrow a-g<0),\\
&&d_5= a \left(b^2-d+a d\right) (a-g) u >0,
$$ which leads to $b^2-d+a d<0$. Therefore, $\Delta_1 <0$ and $\Delta >0$.
Similarly, if $d=b$, then $$\begin{aligned}
\Delta=\frac{\left(-c^3+a^2 g\right)^2 (b-h)^2 u^2 F_2(a,b,c,h)}{a^4},\end{aligned}$$ where $$\begin{aligned}
F_2(a,b,d,h)&=&(-1+b)^2 h^2 c^2 +
2 a (-1+b) \left(-2 b^3+3 b^2 h+a h^2-b h^2\right) c \\
&&+ a^2 \left(-4 a b^3+b^4+6 a b^2 h-2 b^3 h+a^2 h^2-2 a b h^2+b^2 h^2\right)\end{aligned}$$ is a concave upward quadratic polynomial in $c$. The discriminant of $F_2$ is $\Delta_2:=16 a^2 (-1+b)^2 b^3 (b-h)^3$; we observe that $d=b>0$ and $d_6=-d_5(b-h)>0$, which leads to $b-h<0$. Therefore, $\Delta_2<0$ and $\Delta >0$, which completes the proof.
*Acknowledgments*. The authors are deeply grateful to the referee for many suggestions that led to significant improvements in the presentation. Many of the examples, and portions of the proofs of some results in this paper were obtained using calculations with the software tool *Mathematica [@Wol]*.
[CuFi1]{}
R.E. Curto and L.A. Fialkow, Flat extensions of positive moment matrices: Relations in analytic or conjugate terms, *Operator Th.: Adv. Appl*. 104(1998), 59–82.
R.E. Curto and L.A. Fialkow, Flat extensions of positive moment matrices: Recursively generated relations, *Memoirs Amer. Math. Soc*. no. 648, Amer. Math. Soc., Providence, 1998.
R.E. Curto and L.A. Fialkow, Solution of the singular quartic moment problem, J. Operator Theory 48(2002), 315–354.
R.E. Curto and L.A. Fialkow, Solution of the truncated hyperbolic moment problem, Integral Equations Operator Theory 52(2005), 181–218.
L.A. Fialkow, The truncated moment problem on parallel lines, Proc. 24th International Conference on Operator Theory, to appear.
L.A. Fialkow and J. Nie, Positivity of Riesz functionals and solutions of quadratic and quartic moment problems J. Functional Analysis 258(2010), 328–356.
J.L. Smul’jan, An operator Hellinger integral (Russian), *Mat. Sb*. 91(1959), 381–430.
Wolfram Research, Inc., *Mathematica*, Version 9.0, Wolfram Research, Inc., Champaign, IL, 2012.
[^1]: The first named author was supported by NSF Grants DMS-0801168 and DMS-1302666.
|
{
"pile_set_name": "ArXiv"
}
|
---
title: 'Constraining the Doublet Left-Right Model'
---
One of the most puzzling features of the Standard Model (SM) consists in the different treatment of left and right chiralities of fermions, as shown by the violation of parity by weak interactions. In order to restore this symmetry at high energies, Left-Right Models (LRM) have been introduced in the 70’s [@History] and they assume a symmetry between left- and right-handed fermions broken spontaneously, implying that left- and right-handed currents behave differently at low energies.
Historically, LRM have been considered with doublets in order to break the left-right symmetry spontaneously [@History]. Later the focus was on triplet models, due to their ability to generate both Dirac and Majorana masses for neutrinos. The triplet models have the advantage of potentially introducing a see-saw mechanism, though it is difficult to reconcile the very light masses of neutrinos with a TeV scale LRM without fine tuning [@Deshpande]. Moreover, combined constraints coming from meson oscillations, among other observables, tend to push the mass scale of the new scalar particles to a few TeV [@Buras] or beyond [@Mohapatra]. The new vector particles must not be far away from this scale, otherwise the couplings present in the Higgs potential would become non-perturbative. Much effort has been done to avoid these constraints, but stringent lower bounds persist [@Basecq].
Our aim is to reconsider the breaking of the left-right gauge group, via doublet rather than triplet fields, and see the constraints set on the scale and pattern of symmetry breaking. We also want to determine whether experimental data can be accommodated only through this spontaneous breakdown or if it requires also an explicit breaking of parity through different couplings in the left and right sectors.
Doublet LRM
===========
The gauge group of LRM is $ SU(3)_{c} \times SU(2)_{L} \times SU(2)_{R} \times U(1)_{B-L} $, where $ B $ is the baryon number and $ L $ is the lepton number. We call $ g_{B-L} $ the gauge coupling of $ U(1)_{B-L} $, and $ g_{L} $ and $ g_{R} $ the ones of $ SU(2)_{L} $ and $ SU(2)_{R} $, respectively – the case $ g_{L} \neq g_{R} $ explicitly violates parity. The LR symmetry is spontaneously broken into the EW symmetry, $ SU(3)_{c} \times SU(2)_{L} \times U(1)_{Y} $, where $ Y $ is the hypercharge, given by $ Y = T^{3}_{R} + \frac{B-L}{2} $. This breaking can be triggered by a scalar in any representation whose Vacuum Expectation Value (VEV) does not preserve the LR symmetry, but preserves the EW symmetry. One considers here a doublet representation $ \chi_{R} = \begin{pmatrix}
\chi^{\pm}_{R} , \chi^{0}_{R}
\end{pmatrix} $, with quantum numbers under the gauge group $ (0,0,1/2,1) $, whose VEV is $ \langle \chi_{R} \rangle = \begin{pmatrix}
0 , \frac{1}{\sqrt{2}} \kappa_{R}
\end{pmatrix} $. Since LRM is assumed to be valid at energies much higher than the scale $ \kappa $ of the EW Symmetry Breaking (EWSB), it follows that $ \kappa_{R} \gg \kappa $. The Higgs mechanism then leads to new heavy gauge bosons, called $ {W'}^{\pm} $ and $ {Z'}^{0} $, whose masses are of order $ \kappa_{R} $, coupled predominantly to right-handed fermions.
To discuss the physics occurring at the EWSB, one introduces a bidoublet $ \phi = \begin{pmatrix}
\varphi^{0}_{1} & \varphi^{+}_{2} \\
\varphi^{-}_{1} & \varphi^{0}_{2} \\
\end{pmatrix} $, $ (0,1/2,1/2,0) $, whose VEV, $ \langle \phi \rangle = \frac{\operatorname{diag} \left( \kappa_{1}, \kappa_{2} \right)}{\sqrt{2}} $, is not invariant under the SM gauge group and breaks it spontaneously into $ SU(3)_{c} \times U(1)_{\operatorname{EM}} $. Even though a second doublet $ \chi_{L} $, with quantum numbers $ (0,1/2,0,1) $, is not necessary from the point-of-view of EWSB, it is introduced in order to preserve the structural symmetry between left and right sectors before symmetry breaking. Its VEV is $ \langle \chi_{L} \rangle = \begin{pmatrix}
0 , \frac{1}{\sqrt{2}} \kappa_{L}
\end{pmatrix} $, with $ \kappa_{L} $ of the order of the EWSB scale at most, and thus it also triggers the EWSB. Since $ \chi_{L} $ is a doublet, its VEV does not contribute to the $ \rho $ parameter at tree-level, and must be constrained by other observables, in particular EW Precision Observables (EWPO). The scale of EWSB is set by $ \kappa^{2} \equiv \kappa^{2}_{1} + \kappa^{2}_{2} + \kappa^{2}_{L} $ and the SM Higgs is given by a combination of the real degrees of freedom of $ \phi $ and $ \chi_{L} $. The gauge bosons $ {W}^{\pm} $ and $ {Z}^{0} $ acquire masses of order $ \kappa $ by the Higgs mechanism, and they couple predominantly to left-handed fermions. For simplicity, we take $ \kappa_{1,2} $ and $ \kappa_{L,R} $ to be real and positive. We also consider that there is no complex phase in the Higgs potential, so that no new CP-violation terms are generated by the extended Higgs sector.
In the LRM, right-handed (left-handed) fermions come into doublets (singlets) of $ SU(2)_{R} $ and singlets (doublets) of $ SU(2)_{L} $, denoted $ Q_{R} $ ($ Q_{L} $). The mechanism responsible for giving them a mass is the Yukawa coupling $ \overline{Q}_{L} \left( Y \phi + \tilde{Y} \sigma_{2} \phi^{*} \sigma_{2} \right) Q_{R} + h.c. $, where generation indices are not shown. As in the SM, one introduces the mixing matrices $ V^{L,R} $, where $ V^{L} $ is the equivalent of the SM-CKM matrix and $ V^{R} $ is a new mixing matrix for right-handed quarks. Discrete symmetries can be imposed to relate L and R sectors, implying relations between $ V^{L} $ and $ V^{R} $ [@Maiezza]. The more general case where $ V^{L,R} $ are independent corresponds to an explicit violation of parity.
The spectrum of physical scalars of the Doublet LRM is composed of one light neutral Higgs $ h^{0} $, five heavy neutral Higgses $ H^{0}_{1,2,3} $ (CP-even) and $ A^{0}_{1,2} $ (CP-odd), and four heavy charged Higgses $ H^{\pm}_{1,2} $. All of the heavy Higgses have masses of the order of $ \kappa_{R} $. The neutral Higgses $ H^{0}_{1,2} $ and $ A^{0}_{1,2} $ couple to quarks with a strength proportional to the Yukawa couplings and VEV’s and induce Flavor Changing Neutral Currents (FCNC), whereas charged Higgses induce left- and right-handed Flavor Changing Charged Currents (FCCC).
Some differences/advantages of the doublet model compared to the triplet case are the following: (1) A VEV, whose size (of the order of the EWSB or less) is less constrained than in the triplet model, modifies the structure of FCNC couplings between Higgses and quarks; (2) This impacts the analysis of neutral-meson mixing (mainly corrected by neutral Higgs scalars in this class of models); (3) There are no doubly-charged Higgses in the theory; (4) There is no particular mechanism of mass generation for neutrinos, leaving the smallness of their masses unexplained.
The observables constraining the model will be the following: (a) EWPO (constraining VEVs and gauge couplings); (b) meson mixing and (semi-)leptonic decays (for $ V^{L} $, $ V^{R} $, Higgs masses, etc.); (c) and finally other observables as $ b \rightarrow s \gamma $ and the relation among the masses of up- and down-type quarks. The firs two categories will be discussed in the following sections.
EWPO
====
Among the observables one can use to constrain models of New Physics (NP), EWPO are of particular importance due to the accuracy reached by SM computations and experiments [@PDG]. The SM global fit of these observables shows good agreement between them, but some tension is present specially between the Forward-Backward asymmetry $ A_{FB} (b) $ and the Left-Right asymmetry $ A_{LR} (e) $. The observables we consider here are (a) Z-lineshape and asymmetries: $ A_{LR} (f) $, for $ f = e, \mu, \tau, c, b $, $ A_{FB} (f) $, for $ f = e, \mu, \tau, c, b $, the hadronic cross section at the Z-pole ($ \sigma^{0}_{\operatorname{had}} $), ratios of partial widths ($ R_{\ell} $, for $ \ell = e, \mu, \tau $, and $ R_{q} $, for $ q = c, b $) and the total width ($ \Gamma_{Z} $); (b) Mass ($ M_{W} $) and total width ($ \Gamma_{W} $) of the W; (c) Atomic parity violation of cesium and thallium.
In the SM, one usually parameterizes EWPO in terms of $ \mathcal{S} \equiv \{ m_{h}, m_{t}, \alpha_{s} (M_{Z}), \Delta \alpha, M_{Z} \} $:
$$\mathcal{X} = c_{0} + c_{1} \cdot L_{H} + c_{2} \cdot \Delta_{t} + c_{3} \cdot \Delta_{\alpha_{s}} + c_{4} \cdot \Delta^{2}_{\alpha_{s}} + c_{5} \cdot \Delta_{\alpha_{s}} \Delta_{t} + c_{6} \cdot \Delta_{\alpha} + c_{7} \cdot \Delta_{Z} ,$$
where $ \Delta_{t} = \left( \frac{m_{t}}{173.2 \, \operatorname{GeV}} \right)^{2} - 1 $, $ \Delta_{\alpha_{s}} = \frac{\alpha_{s} (M_{Z})}{0.1184} - 1 $, $ \Delta_{\alpha} = \frac{\Delta \alpha}{0.059} - 1 $, $ \Delta_{Z} = \frac{M_{Z}}{91.1876 \, \operatorname{GeV}} - 1 $ and $ {L_{H} = \log \frac{m_{h}}{125.7 \, \operatorname{GeV}}} $. The values of the coefficients $ c_{i} $ for some observables ($ \Gamma_{Z} $, $ \sigma^{0}_{\operatorname{had}} $, $ R_{b, c} $), including 2-loop fermionic EW corrections, are given by [@Freitas]. Using the numerical program [@Zfitter], one can determine the values of the coefficients $ c_{i} $ of the other observables with a level of accuracy somewhat lower. The fundamental parameters $ \mathcal{R} \equiv \{ \epsilon^2 \equiv \frac{\kappa^{2}}{\kappa^{2}_{R}}, c^{2}_R \equiv 1 - \frac{s^{2}_{W}}{1 - s^{2}_{W}} \left( \frac{g_{L}}{g_{R}} \right)^{2}, r \equiv \frac{\kappa_{2}}{\kappa_{1}}, w \equiv \frac{\kappa_{L}}{\kappa_{1}} \} $ are also present in the LRM, where $ s_{W} $ is the sine of the weak angle. The EWPO are given by $ \mathcal{X}_{LR} = \mathcal{X} + \delta \mathcal{X} $, where $ \delta \mathcal{X} $ is the Leading-Order (LO) correction to the SM and $ \frac{\delta \mathcal{X}}{\mathcal{X}} = \mathcal{O} (\epsilon^{2}) $. A similar treatment of EWPO can be found in [@Yuan].
 \[fig:fig1\]
$ w $ $ \epsilon^{2} $ $ c_{R} $ $ g_{R} $ $ g_{B-L} $ $ M_{Z'} $\[TeV\] $ \chi^{2}_{min} $
------- ------------------ ----------- ----------- ------------- ------------------- --------------------
0 0.88 0.11 0.36 3.57 13.1 26.12
1 1.04 0.40 0.39 0.90 3.8 25.14
2 1.43 0.63 0.46 0.56 2.4 24.06
\[tab:changingw\]
In order to combine the different EWPO and constrain the parameters $ \mathcal{S} \cup \mathcal{R} $, we use the [@CKMfitter] frequentist framework (with *Range fit* treatment of systematic uncertainties). The correlated constraint among the scale of LRM, $ \epsilon $, and the size of $ c_R $ is seen in Figure \[fig:fig1\]. We do not use bounds on masses coming from direct searches for the $ W' $ boson, as the latter are tied to specific assumptions on the structure of the LRM couplings [@directsearches] and the analysis should be adapted to the more general framework considered here.
The constraints are not powerful enough to constrain $ c_{R} $, $ r $ and $ w $ independently at $ 1 \sigma $. The global fit of LRM is similar to the SM one: $ \chi^{2}_{\min} \vert_{SM} = 22.24 $ and $ \chi^{2}_{\min} \vert_{LRM} = 22.19 $. The agreement is improved for some observables (e.g. $ \sigma^{0}_{\operatorname{had}} $) at the expense of others (e.g. $ \Gamma_{Z} $), [@Luiz]. Though not constrained at $ 1 \sigma $, $ w $ has an impact on the fit, as seen in Table \[tab:changingw\]. The fit prefers $ w > 0 $, though $ \chi^{2}_{\min} $ does not change by large amounts. Moreover, when $ w = 0 $, $ g_{B-L} $ reaches its perturbativity limit, $ g^{2}_{B-L} = 4 \pi $. The fact that $ w $ is pushed towards non-vanishing values is an interesting feature of EWPO, but it remains to be seen if the other sectors of the theory agree with this tendency.
Neutral-meson mixing
====================
In order to further test the LRM, in particular the scale of the masses of the Higgses and the general structure of the $ V^R $ mixing matrix, we consider meson oscillation observables. The SM calculation of these observables consists of $ {W}^{\pm} {W}^{\pm} $, $ {W}^{\pm} {G}^{\pm} $ and $ {G}^{\pm} {G}^{\pm} $ box diagrams ($ {G}^{\pm} $ is the Goldstone associated to $ {W}^{\pm} $), and it is corrected at order $ \epsilon^2 $ by (a) new boxes $ {W}^{\pm} {W'}^{\pm} $ and $ {G}^{\pm} {W'}^{\pm} $; (b) $ W $ gauge boson/charged scalar boxes, $ {W}^{\pm} {H}^{\pm}_{1,2} $ and $ {G}^{\pm} {H}^{\pm}_{1,2} $; (c) FCNC introduced by $ H^{0}_{1,2} $ and $ A^{0}_{1,2} $ at tree-level; and (d) self-energy and vertex corrections to the FCNC, necessary for gauge invariance of the $ {W}^{\pm} {W'}^{\pm} $ box [@Pal]. Usually, the tree-level Higgs exchanges dominate over the other new contributions. In the triplet case (which is similar to the limit $ \kappa_{L} = 0 $), only the pair $ H^{0}_{1} $, $ A^{0}_{1} $ contributes. In the doublet case, the presence of contributions from other Higgses, $ H^{0}_{2} $ and $ A^{0}_{2} $, and different FCNC couplings when $ \kappa_{L} \neq 0 $, as suggested by the analysis of EWPO, means that the constraint from neutral meson mixing is less stringent, in particular on the mass of FCNC Higgses.
The general structure of the neutral meson mixing observables is $ \Delta m = \sum_{i} C^{q_{1} q_{2}}_{i} \eta^{q_{1} q_{2}}_{i} \langle O_{i} \rangle $, where $ i $ runs over the number of operators, and $ q_{1}, q_{2} $ are the flavors of the up-type quarks in the box, also related to the FCCC mixing matrices arising in $ C_{i} $. The Wilson coefficients $ C_{i} $ can be computed perturbatively by matching the low energy EFT (effective Hamiltonian) and the underlying theory (SM here), whereas the matrix element $ \langle O_{i} \rangle $ can be determined from lattice QCD; $ \eta $, collecting the short-distance QCD corrections are precisely known (up to NNLO) in the SM through the use of EFT. There is also a simplified method for computing the $ \eta $’s, described by Vysotskii [@Vysotskii]. Consider the LO diagrams (for instance, the $ W W $ box) with the addition of a gluon (corresponding to a two-loop integral on a gluon momentum and a quark momentum). Vysotskii’s simplified method aims at extracting the main contributions to the short-distance coefficients by determining, at a first stage, the range of momenta of the gluon contributing to the leading-order QCD corrections. One then improves the result with the help of Renormalization Group Equations (RGE), resumming the gluon corrections thanks to the running of four-fermion operators. Finally, the quark momentum of the 2-loop integral is determined from the range of energies dominating the Inami-Lim functions (LO computation). The values of the short-distance QCD corrections in the SM for the $ K $ system using this method are given in Table \[tab:SMetas\], and they reproduce the values calculated from a systematic use of EFT [@Burasetas].
For the LR operators, only calculations of the $ \eta $ for top-top box below $ \mathcal{O} (m_{t}) $ are known [@Burasgammas]. To derive constraints from meson mixing, we compute the remaining $ \eta $’s applying the procedure described by Vysotskii, extending what was done by [@Ecker]. We give our preliminary results in Table \[tab:LRetas\]. The same approach was also employed by [@Bertolini]. More details of our calculations at LO, a possible NLO extension and the corresponding constraints on the parameters of the Doublet LRM from meson mixing will be given in ref. [@Luiz].
$ K \overline{K} $, LO $ \eta_{tt} $ $ \eta_{cc} $ $ \eta_{ct} $
----------------------------- --------------- --------------- ---------------
Vysotskii [@Vysotskii] 0.60 0.92 0.34
systematic EFT [@Burasetas] 0.612 1.12 0.35
: SM short-distance QCD corrections at LO showing a comparison between Vysotskii’s prescription and a systematic use of Effective Field Theory (EFT). Flavor thresholds are taken into account.[]{data-label="tab:SMetas"}
LO $ \overline{\eta}^{K \overline{K}}_{tt} $ $ \overline{\eta}^{K \overline{K}}_{cc} $ $ \overline{\eta}^{K \overline{K}}_{ct} $ $ \overline{\eta}^{B \overline{B}}_{tt} $
--------------------------------------------- ------------------------------------------- ------------------------------------------- ------------------------------------------- -------------------------------------------
$ W^{\pm} {W'}^{\pm} $, $ W^{\pm} H^{\pm} $ 2.89 0.78 1.50 2.19
$ G^{\pm} {W'}^{\pm} $ 2.89 0.92 1.50 2.19
: Preliminary results for the short-distance QCD corrections at LO to the LRM, using the Vysotskii’s procedure described briefly in the text. Flavor thresholds are taken into account. The $ \overline{\eta} $’s are dependent on the hadronisation scale: $ \mu_{\operatorname{had}} = 2 $ GeV is taken for the $ K $ system and $ \mu_{\operatorname{had}} = 4 $ GeV for the $ B $ systems.[]{data-label="tab:LRetas"}
LO $ \overline{\eta}^{K \overline{K}}_{tt} $ $ \overline{\eta}^{K \overline{K}}_{cc} $ $ \overline{\eta}^{K \overline{K}}_{ct} $ $ \overline{\eta}^{B \overline{B}}_{tt} $
--------------------- ------------------------------------------- ------------------------------------------- ------------------------------------------- -------------------------------------------
$ G^{\pm} H^{\pm} $ 2.89 0.31 0.41 2.18
tree-level FCNC 2.15 0.58 1.12 1.63
: Preliminary results for the short-distance QCD corrections at LO to the LRM, using the Vysotskii’s procedure described briefly in the text. Flavor thresholds are taken into account. The $ \overline{\eta} $’s are dependent on the hadronisation scale: $ \mu_{\operatorname{had}} = 2 $ GeV is taken for the $ K $ system and $ \mu_{\operatorname{had}} = 4 $ GeV for the $ B $ systems.[]{data-label="tab:LRetas"}
Outlook
=======
We reanalyze a version of LRM where the spontaneous breakdown of the LR gauge group is triggered by doublet rather than triplet representations. This changes the structure of the model and introduces a new degree of freedom, the VEV $ \kappa_{L} $ of a $ SU(2)_{L} $ doublet field. Our first concern was to set constraints in this model using EWPO. Our analysis shows that these observables impose a correlation between the scale of the left-right symmetry breaking, which occurs at the scale of several TeV scale, and the size of the couplings.
We are presently analyzing meson mixing observables in order to constrain the remaining parameters of the model. These observables are of great impact because LRM introduces FCNC through new heavy scalars. The structure of the corrections from LRM is different, in the doublet and triplet cases, and they require a good knowledge of short-distance QCD corrections, which can be large, as seen in Table \[tab:LRetas\]. We aim at investigating the consequences of this new scalar sector, especially for FCNC, which are sensitive to $ \kappa_{L} $. A joint fit of EWPO and meson mixing observables, together with leptonic and semileptonic decays, is in progress [@Luiz].
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We describe the effect of thermal motion and buffer-gas collisions on a four-level closed $N$ system interacting with strong pump(s) and a weak probe. This is the simplest system that experiences electromagnetically induced absorption (EIA) due to transfer of coherence via spontaneous emission from the excited to ground state. We investigate the influence of Doppler broadening, velocity-changing collisions (VCC), and phase-changing collisions (PCC) with a buffer gas on the EIA spectrum of optically active atoms. In addition to exact expressions, we present an approximate solution for the probe absorption spectrum, which provides physical insight into the behavior of the EIA peak due to VCC, PCC, and wave-vector difference between the pump and probe beams. VCC are shown to produce a wide pedestal at the base of the EIA peak, which is scarcely affected by the pump-probe angular deviation, whereas the sharp central EIA peak becomes weaker and broader due to the residual Doppler-Dicke effect. Using diffusion-like equations for the atomic coherences and populations, we construct a spatial-frequency filter for a spatially structured probe beam and show that Ramsey narrowing of the EIA peak is obtained for beams of finite width.'
author:
- 'E. Tilchin'
- 'O. Firstenberg'
- 'A. D. Wilson-Gordon'
title: Effects of thermal motion on electromagnetically induced absorption
---
\[sec:intro\]Introduction
=========================
The absorption spectrum of a weak probe, interacting with a pumped nearly-degenerate two-level transition, can exhibit either a sharp subnatural dip or peak at line center [@Khitrova1988JOSAB], depending on the degeneracy of the levels, the polarizations of the fields, and the absence or presence of a weak magnetic field. The phenomenon is termed electromagnetically-induced transparency (EIT) [@HarrisToday; @Fleischhauer2005RMP] when there is a dip in the probe spectrum and electromagnetically induced absorption (EIA) [@Akulshin1998PRA] when there is a peak.
In the case of orthogonal polarizations of the pump and probe, both EIT and EIA are related to the ground-level Zeeman coherence, which is induced by the simultaneous action of both fields. The simplest model system that exhibits EIT is the three-level $\Lambda$ system, where the two lower states $g_{1,2}$ are Zeeman sublevels of the ground hyperfine level $F_{g}$. In a $\Lambda$ system, quantum coherence can lead to the destructive interference between the two possible paths of excitation. As a result, if the pump field is tuned to resonance, the narrow dip in the probe absorption spectrum at the two-photon resonance can be interpreted as EIT caused by a coherent population trapping [@ArimondoCPTRev2] in the lower levels. The simplest system that exhibits EIA is the four-level $N$ system [@Taichenachev1999PRA; @GorenN2004PRA] (Fig. \[Fig. 1\], top), consisting of states $g_{1,2}$ and $e_{1,2}$ which are Zeeman sublevels of the ground ($F_{g}$) and excited ($F_{e}$) hyperfine levels, where the $g_{i}\leftrightarrow e_{i}$, $i=1,2$, transitions interact with non-saturating pump(s), and the $g_{2}\leftrightarrow e_{1}$ transition interacts with a weak probe. The $N$ system gives similar results to those obtained for a closed alkali-metal $F_{g}\rightarrow F_{e}=F_{g}+1$ transition interacting with a $\sigma_{\pm}$ polarized pump, and a weak $\pi$ polarized probe [@GorenN2004PRA; @Zigdon2007SPIE]. It has been shown [@Taichenachev1999PRA; @GorenTOP2003PRA; @GorenN2004PRA], that the EIA peak is due to transfer of coherence (TOC) from the excited state to the ground state, via spontaneous emission. The excited-state coherence only exists in systems where the coherent population trapping is incomplete so that there is some population in the excited state [@GorenTOP2003PRA; @Meshulam2007OL]. The transfer of this coherence to the ground state leads to a peak in the contribution of the ground-state two-photon coherence to the probe absorption at line center, instead of the dip that occurs in its absence (for example, in a $\Lambda$ system or a non-degenerate $N$ system) [@Zigdon2008PRA].
\[ptb\]
[Fig1.eps]{}
In this paper, we investigate the effect of the thermal motion of the alkali-metal gas on the EIA spectrum, in the presence of a buffer gas. In a previous paper [@GorenTOP2004PRA], we discussed the effect of phase-changing collisions (PCC) with the buffer gas on an $N$ system and showed that they lead to considerable narrowing of the EIA peak in both the presence and absence of Doppler broadening. These collisions increase the transverse decay rate of the optical transitions, resulting in the so-called pressure broadening of the optical spectral line, and are thus easily incorporated in the Bloch equations. However, in order to describe the overall effect of buffer-gas collisions, it is necessary to include both velocity-changing collisions (VCC) as well as PCC [@Singh1988JOSAB; @May1999PRA], which is a much greater challenge. Due to the complexity of the problem, we limit our discussion to a four-level $N$ system, and to buffer-gas pressures that are sufficiently low so that collisional decoherence of the excited state [@Failache2003PRA] can be neglected.
The Doppler effect occurs in the limit of *ballistic* atomic motion, when the mean free-path between VCC is much larger than the radiation wavelength. Due to their narrow spectral response, Raman processes such as EIT and EIA are much more sensitive to the “residual" Doppler effect, arising when there is a difference between the wavevectors of the Raman fields. In many cases however, the Raman wavelength can become much larger than the typical free-path between collisions. For example, an angular deviation of a milliradian between the two optical beams yields a superposition pattern with a wavelength in the order of a millimeter. In this limit, the atoms effectively perform a *diffusion* motion through the spatial oscillations of the superposition field, leading to the Dicke narrowing of the residual Doppler width. While the residual Doppler broadening is linearly proportional to the Raman wavevector, Dicke narrowing shows a quadratic dependence. This behavior was demonstrated in EIT with non-collimated pump and probe [@Weitz2005PRA; @Shuker2007PRA].
Recently, a model describing thermal motion and collisions for EIT was presented [@Firstenberg2007PRA; @Shuker2007PRA; @Firstenberg2008PRA], utilizing the density matrix distribution in space and velocity with a Boltzmann relaxation formalism. The model describes a range of motional phenomena, including Dicke narrowing, and diffusion in the presence of electromagnetic fields and during storage of light. This diffusion model was used to describe a spatial frequency filter for a spatially structured probe [@Firstenberg2008PRA] and also Ramsey narrowing [@Xiao2006PRL; @Xiao2008OE]. Here, we utilize a similar formalism to estimate the influence of the atomic thermal motion in a buffer-gas environment, including VCC and PCC, on the spectral shape of EIA in a four-level *N* system, with collimated or non-collimated light beams. In Sec. \[sec:DD\], the Doppler broadening and Dicke narrowing effects are studied for plane-wave fields. As the full mathematical treatment is lengthy, it appears in Appendix A. However, an approximate equation which describes the main features of the spectra is presented in Sec. \[sec:DD\]. Diffusion-like equations for the ground and excited state coherences and populations are derived in Appendix B. Two main phenomena are described using this model: (i) a spatial-frequency filter for structured probe fields which is presented in Sec. \[sec:filter\], and (ii) atomic diffusion through a finite-sized beam resulting in Ramsey narrowing of the EIA peak, which is discussed in Sec. \[sec:Ramseynarr\]. Finally, conclusions are drawn in Sec. \[sec:conc\].
\[sec:DD\]The Doppler-Dicke line shapes of EIA
==============================================
Consider the near-resonant interaction of a four-state atom in an *N* configuration, depicted in Fig. \[Fig. 1\]. The two lower states $g_{1}$ and $g_{2}$ are degenerate and belong to the ground level with zero energy, and the excited states $e_{1}$ and $e_{2}$ are degenerate with energy $\hbar
\omega_{0}$. The light field consists of three beams, each with a carrier frequency $\omega_{j}$ and wavevector $\mathbf{q}_{j},$ where $j=1,2$ denotes the two strong pump beams, and $j=p$ the weak probe, $$\mathbf{\breve{E}}(\mathbf{r},t)=\sum_{j=1,2,p}\mathbf{E}_{j}\left(
\mathbf{r},t\right) e^{-i\omega_{j}t+i\mathbf{q}_{j}\cdot\mathbf{r}}+\text{c.c.} \label{Eq. 1}$$ Here, $\mathbf{E}_{j}\left( \mathbf{r},t\right) $ are the slowly varying envelopes in space and time. The pumps drive the $g_{1}\leftrightarrow e_{1}$ and $g_{2}\leftrightarrow e_{2}$ transitions, and the probe is coupled to the $g_{2}\leftrightarrow e_{1}$ transition.
\[ptb\]
[Fig2.eps]{}
Our model will incorporate four relaxation rates: $\Gamma,$ the spontaneous emission rate from the each of the excited states to all the ground states; $\Gamma_{\text{pcc}},$ the pressure broadening of the optical transitions resulting from PCC; $\gamma_{\text{vcc}}$, the velocity autocorrelation relaxation rate ($1/\gamma_{\text{vcc}}$ is the time it takes the velocity vector to vary substantially) [@Sobelman1967SPU], which is proportional to the rate of VCC; and $\gamma$ is the homogenous decoherence rate within the ground and excited state manifolds due, for example, to spin-exchange and spin-destruction collisions [^1]. In the model, the transition $g_{1}\leftrightarrow e_{2}$ is forbidden (due to some selection rule such as angular momentum).
To focus the discussion, we assume that all three beams are continuous waves, namely $\mathbf{E}_{j}(\mathbf{r},t)=\mathbf{E}_{j}(\mathbf{r})$. We then obtain stationary Rabi frequencies, given by $V_{j}=V_{j}\left(
\mathbf{r}\right) =\mu_{j}\mathbf{E}_{j}(\mathbf{r})/\hbar$, where $\mu_{j}$ is the transition dipole moment. The complete set of Bloch equations for the four-level *N* system consists of sixteen equations [@GorenN2004PRA]. In order to simplify the application of the theory to EIA, we assume that $V_{p}\ll V_{1,2}<\Gamma$ and that the pump transitions are well below saturation, so that in the absence of the probe, the population concentrates in the $g_{2}$ state, the $g_{2} \leftrightarrow e_{2}$ dipole is excited, and the $e_{2}$ state is empty up to second order in the pump field [@GorenN2004PRA]. The equations can then be written up to the first order in the probe field $V_{p}$ [@Taichenachev1999PRA], which reduces the number of Bloch equations to five.
The complete analytical development is presented in Appendix A, and an example of the calculated probe absorption spectrum for collinear and degenerate beams ($\mathbf{q}_{1}=\mathbf{q}_{2}=\mathbf{q}$) is given in Fig. \[Fig. 2\](a) (blue line). For the numerical calculations, we have considered the $D_{2}$ line of $^{85}$Rb (wavelength 780 nm) at room temperature, with a total spontaneous emission rate $\Gamma=2\pi\times6$ MHz [@Lezama1999PRA]. Other parameters are indicated in the figure caption and described in what follows.
Four complex frequencies control the EIA dynamics, each relating to a different coherence in the process:
\[Eq. 3\]$$\begin{aligned}
\xi_{1} & =\left( \Delta_{p}-\Delta_{1}\right) -(\mathbf{q}_{p}-\mathbf{q}_{1})\cdot\mathbf{v+}i(\gamma+\gamma_{\text{vcc}}),\label{Eq. 3a}\\
\xi_{2} & =\Delta_{p}-\mathbf{q}_{p}\cdot\mathbf{v+}i(\tilde{\Gamma}+\gamma_{\text{vcc}}),\label{Eq. 3b}\\
\xi_{3} & =\left( \Delta_{p}-\Delta_{2}\right) -(\mathbf{q}_{p}-\mathbf{q}_{2})\cdot\mathbf{v+}i(\Gamma+\gamma+\gamma_{\text{vcc}}),\label{Eq. 3c}\\
\xi_{4} & =\left( \Delta_{p}-\Delta_{1}-\Delta_{2}\right) -(\mathbf{q}_{p}-\mathbf{q}_{1}-\mathbf{q}_{2})\cdot\mathbf{v+}i(\tilde{\Gamma}+\gamma_{\text{vcc}}), \label{Eq. 3d}$$ with the one-photon detunings $\Delta_{j}=\omega_{j}-\omega_{e_{j}g_{j}}$ ($j=1,2$) and $\Delta_{p}=\omega_{p}-\omega_{e_{1}g_{2}}$, and $\tilde{\Gamma
}=\Gamma/2+\Gamma_{\text{pcc}}+\gamma$. The frequency $\xi_{2}$ is related to the probe transition and includes the one-photon Doppler shift $\mathbf{q}_{p}\cdot\mathbf{v}$. $\xi_{1}$ and $\xi_{3}$ relate to the slowly varying ground and excited state coherences and include the residual Doppler shift $(\mathbf{q}_{p}-\mathbf{q}_{i})\cdot\mathbf{v}$ and the Raman (two-photon) detuning. $\xi_{4}$ relates to the three-photon transition (whose direct optical-dipole is forbidden), required for the EIA process. Note that the fast optical decay rates ($\Gamma$ or $\tilde{\Gamma}$) is absent only from $\xi_{1}$.
In EIA, in contrast to EIT, a strong optical-dipole transition ($g_{2}\leftrightarrow e_{2}$) is excited even in the absence of the probe. Its excitation depends on its resonance with the pump field, and is thus affected by Doppler broadening. This leads to velocity-dependent equations even in zero-order in the probe field, and introduces the additional complex frequency
$$\xi_{5}=-\Delta_{2}+\mathbf{q}_{2}\cdot\mathbf{v+}i(\tilde{\Gamma}+\gamma_{\text{vcc}}),$$
with the one-photon Doppler shift $\mathbf{q}_{2}\cdot\mathbf{v}$. The overall dynamics is thus governed by the five equations (\[Eq. A11a\])-(\[Eq. A11e\]).
We start by calculating the probe absorption spectrum for uniform pump and probe fields (plane waves) by solving the equations analytically. The spectrum depends on $18$ different integrals over velocity, of the form $$G_{i}=\int d^{3}v\frac{\xi_{\alpha}\cdots\xi_{\beta}}{\xi_{5}\xi_{d}}F(\mathbf{v}), \label{Gs}$$ where $F(\mathbf{v})=\left( 2\pi v_{\text{th}}^{2}\right) ^{-3/2}e^{-\mathbf{v}^{2}/2v_{\text{th}}^{2}}$ is the Boltzmann velocity distribution, and $v_{\text{th}}^{2}=k_{b}T/m$ is the mean thermal velocity. The determinant $\xi_{d}$, $$\xi_{d}=\xi_{1}\xi_{2}\xi_{3}\xi_{4}-\xi_{3}(\xi_{2}V_{2}^{2}+\xi_{4}V_{1}^{2})+iV_{1}V_{2}bA\Gamma\left( \xi_{2}+\xi_{4}\right) , \label{xie_d}$$ introduces the power broadening effect (first and second terms), *i.e.* the dependence of the Raman spectral width on the pump powers, and the spontaneous TOC from the excited state to the ground state (last term). The last term is associated with the TOC due to its dependence on the parameter $b$, which sets the amount of TOC in the original dynamic equations (\[Eq. A1\]), and can take either the value 0 (no TOC) or 1 [@Taichenachev1999PRA]. The spontaneous decay branching ratio is given by $A^{2}=\mu_{e_{1}g_{1}}^{2}/\left( \mu_{e_{1}g_{1}}^{2}+\mu_{e_{1}g_{2}}^{2}\right) $ [@GorenN2004PRA]. The TOC term in Eq. (\[xie\_d\]) depends on the complex frequency $$\begin{aligned}
\xi_{2}+\xi_{4} & =\left( 2\Delta_{p}-\Delta_{1}-\Delta_{2}\right)
-(2\mathbf{q}_{p}-\mathbf{q}_{1}-\mathbf{q}_{2})\cdot\mathbf{v}\nonumber\\
& \mathbf{+}2i(\Gamma/2+\Gamma_{\text{pcc}}+\gamma+\gamma_{\text{vcc}}).
\label{Xi2Xi4}$$ It is important to note that, although each of the individual frequencies $\xi_{2}$ and $\xi_{4}$ is affected by a Doppler shift (either one- or three-photon), the sum $\xi_{2}+\xi_{4}$ exhibits *only a residual Doppler shift* (assuming nearly collinear pumps, $\mathbf{q}_{1}\approx\mathbf{q}_{2}$). Nevertheless the relaxation rate $(\Gamma
/2+\Gamma_{\text{pcc}}+\gamma+\gamma_{\text{vcc}})$ is the same as that characterizing the decay of the optical transitions. As a consequence, even when $\Gamma_{\text{pcc}}$ is much smaller than the optical Doppler width, it plays a significant role in determining the intensity of the EIA spectrum. This is in contrast to one- and two-photon processes (such as EIT), in which $\Gamma_{\text{pcc}}$ is irrelevant when it is much smaller than the Doppler width. It can also be seen that when $\mathbf{q}_{1}\approx\mathbf{q}_{2},$ the various residual Doppler shifts are negligible compared to the relaxation rates in the determinant $\xi_{d},$ so that $\xi_{d}$ is only weakly dependent on these shifts.
Examining the absorption spectrum in Fig. \[Fig. 2\](a), we observe the narrow absorption peak on top of the broad one-photon curve. Moreover, as can be seen in the inset, the EIA resonance consists of two independent features: a “pedestal" at the base and a sharp absorption peak at the center. In order to obtain physical insight into these features, we have derived an approximate solution for the probe absorption which incorporates the main contributions to the EIA, namely the underlying EIT mechanism plus the spontaneous TOC. The approximate Fourier transform of the nondiagonal density-matrix element for the probe is $$R_{e_{1}g_{2}}=n_{0}\left[ -G_{4}+V_{2}^{2}G_{5}+iV_{1}V_{2}bA\Gamma
\frac{iG_{2}G_{3}\gamma_{\text{vcc}}}{1-iG_{1}\gamma_{\text{vcc}}}\right]
V_{p}, \label{Eq. 2}$$ where $G_{1}=\int d^{3}v\frac{\xi_{2}\xi_{3}\xi_{4}F(\mathbf{v})}{\xi_{d}}$, $G_{2}=\int d^{3}v\frac{\xi_{3}\xi_{4}F(\mathbf{v})}{\xi_{d}}$, $G_{3}=\int
d^{3}v\frac{\xi_{2}\xi_{4}F(\mathbf{v})}{\xi_{5}\xi_{d}}$, $G_{4}=\int
d^{3}v\frac{\xi_{1}\xi_{3}\xi_{4}F(\mathbf{v})}{\xi_{d}}$, $G_{5}=\int
d^{3}v\frac{\xi_{3}F(\mathbf{v})}{\xi_{d}}$, and $n_{0}$ is the number density of the active atoms. It can be shown that Eq. (\[Eq. 2\]) is valid provided $\gamma_{\text{vcc}}\ll\Gamma_{\text{pcc}}+\Gamma/2.$ For an atom at rest and in the absence of collisions, so that $\gamma_{\text{vcc}}=0,$ $v_{\text{th}}\rightarrow0,$ and $\Gamma_{\text{pcc}}=0,$ Eq. (\[Eq. 2\]) is identical to the expression obtained by Taichenachev *et al*. [@Taichenachev1999PRA] (with $b=1$), $$R_{e_{1}g_{2}}^{\text{rest}}=\frac{in_{0}V_{p}}{\Gamma/2-i\Delta_{p}}\left[
1+\frac{2A\left\vert V_{1}\right\vert ^{2}/\Gamma}{2\left( 1-A^{2}\right)
\left\vert V_{2}\right\vert ^{2}/\Gamma-i\Delta_{p}}\right] . \label{Eq. 2a}$$ The first term in the square brackets in Eqs. (\[Eq. 2\]) and (\[Eq. 2a\]) describes the one-photon (background) absorption, and the other terms are the EIA peak.
For a moving atom, the spectrum resulting from Eq. (\[Eq. 2\]) is shown in Fig. \[Fig. 2\](a) (red dashed line) and is compared with the exact solution; evidently, there is a good agreement between the spectra. Despite the small discrepancy in the intensity of the sharp peak, the approximate solution preserves the main features in the resonance. When plotted separately in Fig. \[Fig. 2\](b), the three terms in Eq. (\[Eq. 2\]) can be identified with the different spectral features: $-G_{4}$ (black dashed line) describes the background absorption; $V_{2}^{2}G_{5}$ (brown dotted line), which constitutes the total peak in the absence of VCC, describes the wide pedestal; and $iG_{2}G_{3}\gamma_{\text{vcc}}/(1-G_{1}\gamma_{\text{vcc}})$ (green dashed-dotted line) describes the sharp EIA peak, induced by VCC.
\[ptb\]
[Fig3.eps]{}
Fig. \[Fig. 3\] shows the effect of varying the VCC rate, for a fixed PCC rate ($\Gamma_{\text{pcc}}=5\Gamma$) and zero pump-probe angular deviation. The width of the pedestal feature depends on the VCC rate and is given by $\gamma_{\text{vcc}}+\gamma,$ while the width of the narrow peak shows only a very weak dependence on $\gamma_{\text{vcc}}$. Increasing the VCC rate leads to a decrease in the overall EIA intensity, but to an increase in the ratio between the amplitude of the narrow peak and the pedestal baseline.
We now turn to explore the residual (two-photon and four-photon) Doppler and Dicke effects due to wave-vector mismatch between the pump fields and the probe, introduced in principle either by a frequency detuning between the fields, $|\mathbf{q}_{p}|\neq|\mathbf{q}_{1,2}|,$ or due to an angular deviation between them, $\mathbf{q}_{p}\nparallel\mathbf{q}_{1,2}$. We mainly focus on the latter, which may be found in a nearly degenerate level scheme, and we further take the two pump fields to be the same, namely $\mathbf{q}_{1}=\mathbf{q}_{2}$. Figure \[Fig. 4\] presents the probe absorption spectrum for different values of the wave-vector difference, $\delta
\mathbf{q=}$ $\mathbf{q}_{p}-\mathbf{q}_{1,2},$ when $\gamma_{\text{vcc}}=0.1\Gamma$ and $\Gamma_{\text{pcc}}=\Gamma$. As can be seen, increasing $\delta\mathbf{q}$ broadens the EIA spectrum (see inset). This is analogous to the broadening of an EIT transmission peak in a similar configuration [@Shuker2007PRA]. However, the wide collisionally-broadened pedestal remains unaffected by the changes in $\delta\mathbf{q}$, indicating that it mostly originates from homogenous decay processes. Figure \[Fig. 5\](a) summarizes the full-width at half-maximum (FWHM) of the EIA peak for $\Gamma_{\text{pcc}}=\Gamma$ and for various values of $\gamma_{\text{vcc}}$, as a function of $\delta\mathbf{q}$. Because of the difficulty of separating the sharp peak from the background in the calculated spectra [^2], the widths of the sharp EIA peak were obtained only from the third term in Eq. (\[Eq. 2\]). In contrast to an EIT peak, which does not depend on $\gamma_{\text{vcc}}$ when $\delta\mathbf{q=0}$ (collinear degenerate beams) [@Firstenberg2008PRA], the FWHM of the EIA peak at $\delta\mathbf{q=0}$ depends weakly on the VCC rate (although barely noticeable in the figure). This difference derives from the effect of collisions on the pump absorption in the case of EIA, as described earlier.
\[ptb\]
[Fig4.eps]{}
For $\delta\mathbf{q\neq0}$ the FWHM of the peak in the Dicke limit (high $\gamma_{\text{vcc}}$) depends on $\gamma_{\text{vcc}}$ and is proportional to the residual Doppler-Dicke width, $2v_{\text{th}}\delta\mathbf{q}^{2}/\gamma_{\text{vcc}}$. In this limit, the results are well approximated by the analytic expression [@Firstenberg2008PRA] \[dotted lines in Fig. \[Fig. 5\](a)\]: $$\text{FWHM}=2\times\frac{2}{a^{2}}\gamma_{\text{vcc}}H\left( a\frac
{v_{\text{th}}\delta q}{\gamma_{\text{vcc}}}\right) , \label{Eq. 5}$$ where $H\left( x\right) =e^{-x}-1+x$ and $a^{2}=2/\ln2$. Increasing the pump-probe angular deviation reduces the efficiency of the EIA process and thus results in a decrease in the probe absorption \[Fig. \[Fig. 5\](b)\]. This is of course the opposite trend to that of EIT (blue stars), where the depth of the dip decreases (the absorption increases) with increasing $\delta
q$ [@Firstenberg2008PRA].
\[ptb\]
[Fig5.eps]{}
\[sec:filter\]Spatial-frequency filter
======================================
We now to turn to discuss the results of our model from the viewpoint of a spatial-frequency filter for a structured probe beam. When non-uniform beams are considered, the different spatial frequencies that comprise the beams result in different Doppler and Dicke widths. Consequently, the various spatial-frequency components experience different absorption and refraction in the medium. Specifically, the dependence of the absorption on the transverse wave-vectors of the probe beam manifests a filter for the probe in Fourier space.
We assume an optical configuration of two collinear uniform pumps (plane waves with $V_{1}$ and $V_{2}$ constant) and a spatially varying propagating probe, $V_{p}=V_{p}(\mathbf{r},t)$. Since the medium exhibits a non-local response due the atomic motion, the evolution of the probe is more naturally described in the Fourier space $V_{p}(\mathbf{k},\omega)$ where $\mathbf{k}$ and $\omega$ are the spatial and temporal frequencies of the envelope of the probe. Under these assumptions, the model results in a Diffusion-like equations for the populations and coherences of the atomic medium, derived in Appendix B. To simplify the general dynamics of Eqs. (\[Eq. B8a\]) and (\[Eq. B9\]), we take the stationary case \[$\omega=0,$ $V_{p}=V_{p}(\mathbf{k})$\] and assume that the carrier wave-vector of the probe is the same as that of the pumps, $\mathbf{q}_{p}=\mathbf{q}_{1}=\mathbf{q}_{2},$ so that $\delta\mathbf{q}_{1}=\delta\mathbf{q}_{2}=0$. Taking the Fourier transform \[see Eq. (\[Eq. A13\])\], we obtain a set of steady-state equations for the spatially-dependent atomic coherences, $R_{g_{1}g_{2}}(\mathbf{k})$, $R_{e_{1}e_{2}}(\mathbf{k})$, and $R_{e_{1}g_{2}}(\mathbf{k}),$
\[Eq. 6\]$$\begin{aligned}
& \left[ i\left( \Delta_{p}-\Delta_{1}\right) -\gamma-K_{\text{1p}}\left\vert V_{1}\right\vert ^{2}-K_{\text{3p}}\left\vert V_{2}\right\vert
^{2}-Dk^{2}\right] R_{g_{1}g_{2}}\nonumber\\
& =(Dk^{2}-bA\Gamma)R_{e_{1}e_{2}}+K_{\text{1p}}V_{1}^{\ast}V_{p}n_{0},\label{Eq. 6a}\\
& \left[ i\left( \Delta_{p}-\Delta_{2}\right) -\Gamma-\gamma
-Dk^{2}\right] R_{e_{1}e_{2}}\nonumber\\
& =-V_{1}V_{2}^{\ast}(K_{\text{1p}}+K_{\text{3p}})R_{g_{1}g_{2}}-V_{2}^{\ast
}(K_{\text{1p}}+K_{\text{pump}})V_{p}n_{0},\label{Eq. 6b}\\
& R_{e_{1}g_{2}}=iK_{\text{1p}}\left( V_{1}R_{g_{1}g_{2}}+V_{p}n_{0}\right)
\label{Eq. 6c}$$ where $K_{\text{1p}}=iG_{\text{1p}}/\left( 1-iG_{\text{1p}}\gamma
_{\text{vcc}}\right) $ is the one-photon absorption spectrum with $G_{\text{1p}}=\int F\left( \mathbf{v}\right) /\xi_{2}d^{3}v\mathbf{\ }$; $K_{\text{3p}}=iG_{\text{3p}}/\left( 1-iG_{\text{3p}}\gamma_{\text{vcc}}\right) $ is the three-photon absorption spectrum with $G_{\text{3p}}=\int
F\left( \mathbf{v}\right) /\xi_{4}d^{3}v;\mathbf{\ }$and $K_{\text{pump}}=iG_{\text{pump}}/\left( 1-iG_{\text{pump}}\gamma_{\text{vcc}}\right) $ is the one-photon (pump) absorption spectrum with $G_{\text{pump}}=\int F\left(
\mathbf{v}\right) /\xi_{5}d^{3}\mathbf{v}$, as described in Appendix B. Solving Eq. (\[Eq. 6\]) for $R_{e_{1}g_{2}}\left( \mathbf{k},\omega\right)
$, substituting the result into the expression for the linear-susceptibility \[Eq. (\[Eq. A16\])\], assuming that $V_{1}=\eta V_{2}$ ($0<\eta\leq1$), and neglecting all the terms proportional to $1/\Gamma$, we obtain
\[Eq. 7\]$$\begin{aligned}
& \chi_{e_{1}g_{2}}\left( \mathbf{k}\right) =\frac{g}{c}iKn_{0}\left(
1+\text{L}\right) ,\label{Eq. 7a}\\
& \text{L}=\frac{\eta\left( 2bA-\eta\right) \Gamma_{\text{p}}}{-i\Delta
_{p}+\gamma+\left( \eta^{2}+1-2bA\eta\right) \Gamma_{\text{p}}+Dk^{2}},\label{Eq. 7b}$$ where $D=v_{th}/\gamma_{\text{vcc}}$ is the diffusion coefficient, $\Gamma_{\text{p}}=K\left\vert V_{2}\right\vert ^{2}$ is the power broadening, and $K_{\text{1p}}\approx K_{\text{3p}}\approx K_{\text{pump}}=K=\int F\left(
\mathbf{v}\right) /\left[ \mathbf{q}_{p}\cdot\mathbf{v+}i\left(
\Gamma/2+\Gamma_{\text{pcc}}+\gamma+\gamma_{\text{vcc}}\right) \right]
d^{3}v$ for $\Delta_{p}\ll$ $\Gamma_{\hom}=\gamma+\Gamma_{\text{p}}$. In the case where $\eta=A,$ Eqs. (\[Eq. 7\]) is similar to Eq. (\[Eq. 2a\]) obtained by Taichenachev *et al.* [@Taichenachev1999PRA], except for the diffusion term $Dk^{2}$, which vanishes for an atom at rest.
\[ptb\]
[Fig6.eps]{}
The imaginary part of the susceptibility in Eq. (\[Eq. 7\]) yields the absorption of the probe for various values of $\mathbf{k}$. The first term in the brackets in Eq. (\[Eq. 7a\]) is the linear one-photon absorption, and the second term is the $k$-dependent EIA contribution. Thus, the real part of $L$ in Eq. (\[Eq. 7b\]) describes an absorbing spatial-frequency filter, the same way as was done for EIT [@Shuker2008PRL; @Firstenberg2009NP]. Fig. \[Fig. 6\] summarizes several examples of the EIA spatial filter behavior as a function of $k$ for $\Delta_{p}=0,$ $\Delta_{p}=\pm\Gamma_{\hom},$ and $\Delta_{p}=\pm2\Gamma_{\hom}$. At $\Delta_{p}=0,$ the curve is a Lorentzian and maximum absorption is achieved. When $\Delta_{p}\neq0$ the filter becomes more transparent.
\[sec:Ramseynarr\]Ramsey narrowing
==================================
We now consider the *N* system interacting with collinear probe and pump beams that have finite widths. Due to thermal motion, the alkali atoms spend a period of time in the interaction region and then leave the light beams, evolve ‘in the dark’, and diffuse back inside. Such a random periodic motion was described recently by Xiao *et al.* [@Xiao2006PRL; @Xiao2008OE] for an EIT system, and was shown to result in a cusp-like spectrum. Near its center, the line is much narrower than that expected from time-of-flight broadening and power broadening, and the effect, resulting from the contribution of bright-dark-bright atomic trajectories of random durations, was named Ramsey narrowing.
\[ptb\]
[Fig7.eps]{}
Ramsey-narrowed spectra can be calculated analytically from the diffusion equations of the atomic coherences when the light fields of both the probe and pump beams have finite widths [@Firstenberg2008PRA]. The EIA spectrum resulting from a one-dimensional uniform light-sheet of thickness $2a$ in the $x-$direction is derived analytically in Appendix B \[Eq. (\[B15\])\]. In Fig. \[Fig. 7\], we show the spectrum for two different thicknesses and the fitted Lorentzian curves. Near the resonance, the EIA line for the $100$ $\mu
m$ sheet is spectrally sharper than the fitted Lorentzian – the characteristic signature of Ramsey narrowing. In contrast, the EIA peak calculated for a $10$ $mm$ beam is well fitted by the Lorentzian. In addition, the EIA contrast deteriorates as the beam becomes narrower, since the interaction area decreases and fewer atoms interact with the fields.
\[sec:conc\]Conclusions
=======================
In this paper, we extended the theory that describes the effect of buffer-gas collisions on three-level $\Lambda$ systems in an EIT configuration [@Firstenberg2007PRA; @Shuker2007PRA; @Firstenberg2008PRA] to the case of a four-level closed $N$ system which is the simplest system that experiences EIA due to TOC. Using this formalism, we investigated the influence of collisions of optically active atoms with a buffer gas on the EIA peak. In addition to the exact expressions, we presented an approximate solution for the probe absorption spectrum, which provides a physical insight into the behavior of the EIA peak due to VCC, PCC, and wave-vector difference between the pump and probe beams. VCC were shown to produce a wide pedestal at the base of the EIA peak; increasing the pump-probe angular deviation scarcely affects the pedestal whereas the sharp central EIA peak becomes weaker and broader due to the residual Doppler-Dicke effect. Using diffusion-like equations for the atomic coherences and populations, the spatial-frequency filter and the Ramsey-narrowed spectrum were analytically obtained.
In extending the description from the $\Lambda$ to the $N$ schemes, we have considered several elements that are likely to be important in other four-level systems. These include the diffusion of excited-state coherences and the influence of the thermal motion on the optical dipole in the absence of the probe. The latter introduces a Doppler contribution into the pumping terms and consequently affects the power broadening of the narrow resonances.
Reduced density matrix
======================
Consider the near-resonant interaction of a light field consisting of one or two moderately strong pumps and a weak probe, as given in Eq. (\[Eq. 1\]), with the four-level degenerate $N$ ** system of Fig. \[Fig. 1\](a). We use the first-order approximation in the probe amplitude, $V_{p}$, and assume that $V_{2}<\Gamma,$ $V_{1}\leqslant V_{2},$ $V_{p}<V_{1,2}$. Since the pump transitions are assumed non-saturated, the atomic population in the absence of the probe concentrates in the $g_{2}$ state, and the population in other states can be neglected. The $g_{2} \leftrightarrow e_{2}$ dipole, excited in the absence of the probe, is of importance and is thus considered. The resulting Bloch equations are [@Taichenachev1999PRA]
\[Eq. A1\]$$\begin{aligned}
\dot{\breve{\rho}}_{g_{1}g_{2}}^{\left( 1\right) ,i} & \left( \omega
_{p}-\omega_{1}\right) =-\left[ i\left( \omega_{e_{1}g_{2}}-\omega
_{e_{1}g_{1}}\right) +\gamma\right] \breve{\rho}_{g_{1}g_{1}}^{\left(
1\right) ,i}\nonumber\\
& +i\breve{V}_{1}^{\ast}\breve{\rho}_{e_{1}g_{2}}^{\left( 1\right)
,i}-i\breve{V}_{2}\breve{\rho}_{g_{1}e_{2}}^{\left( 1\right) ,i}+bA\Gamma\breve{\rho}_{e_{1}e_{2}}^{\left( 1\right) ,i},\label{Eq. A1a}\\
\dot{\breve{\rho}}_{e_{1}g_{2}}^{\left( 1\right) ,i} & \left( \omega
_{p}\right) =-\left[ i\omega_{e_{1}g_{2}}+\Gamma/2+\Gamma_{\text{pcc}}\right] \breve{\rho}_{e_{1}g_{2}}^{\left( 1\right) ,i}\nonumber\\
& +i\breve{V}_{p}\breve{\rho}_{g_{2}g_{2}}^{\left( 0\right) ,i}+i\breve
{V}_{1}\breve{\rho}_{g_{1}g_{2}}^{\left( 1\right) ,i},\label{Eq. A1b}\\
\dot{\breve{\rho}}_{e_{1}e_{2}}^{\left( 1\right) ,i} & \left( \omega
_{p}-\omega_{2}\right) =-\left[ i\left( \omega_{e_{1}g_{2}}-\omega
_{e_{2}g_{2}}\right) +\Gamma+\gamma\right] \breve{\rho}_{e_{1}e_{2}}^{\left( 1\right) ,i}\nonumber\\
& +i\breve{V}_{p}\breve{\rho}_{g_{2}e_{2}}^{\left( 0\right) ,i}+i\breve
{V}_{1}\breve{\rho}_{g_{1}e_{2}}^{\left( 1\right) ,i}-i\breve{V}_{2}^{\ast
}\breve{\rho}_{e_{1}g_{2}}^{\left( 1\right) ,i},\label{Eq. A1c}\\
\dot{\breve{\rho}}_{g_{1}e_{2}}^{\left( 1\right) ,i} & \left( \omega
_{p}-\omega_{1}-\omega_{2}\right) =-\left[ i\left( \omega_{e_{1}g_{2}}-\omega_{e_{1}g_{1}}-\omega_{e_{2}g_{2}}\right) \right. \nonumber\\
& +\Gamma/2+\left. \Gamma_{\text{pcc}}\right] \breve{\rho}_{g_{1}g_{1}}^{\left( 1\right) ,i}-i\breve{V}_{2}^{\ast}\breve{\rho}_{g_{1}g_{2}}^{\left( 1\right) ,i},\label{Eq. A1d}\\
\dot{\breve{\rho}}_{g_{2}e_{2}}^{\left( 0\right) ,i} & \left( -\omega
_{2}\right) =-\left[ \Gamma/2+\Gamma_{\text{pcc}}-i\omega_{e_{2}g_{2}}\right] \breve{\rho}_{g_{2}e_{2}}^{\left( 0\right) ,i}\left( -\omega
_{2}\right) \nonumber\\
& +i\breve{V}_{2}^{\ast}\left( \breve{\rho}_{e_{2}e_{2}}^{\left( 0\right)
,i}-\breve{\rho}_{g_{2}g_{2}}^{\left( 0\right) ,i}\right) . \label{Eq. A1e}$$ Here, $\breve{\rho}_{ss^{\prime}}^{\left( j\right) ,i}$ is the density-matrix element of the $i-$th atom (one of many identical particles) to the $j-$th order in the probe, and apart from $\breve{\rho}_{g_{2}g_{2}}^{\left( 0\right) ,i}\approx1,$ $\breve{\rho}_{ss}^{\left( 0\right)
,i}=0$. We also consider the envelopes of the pumps to be constant in time so that $V_{1,2}$ is shorthand for $V_{1,2}\left( \mathbf{r}\right) $. The wave equation for the probe field is
$$\left( \nabla^{2}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}\right)
\mathbf{\breve{E}}_{p}\left( \mathbf{r},t\right) =\frac{4\pi}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}\mathbf{\breve{P}}_{e_{1}g_{2}}\left(
\mathbf{r},t\right) , \label{Eq. A2}$$
where $\mathbf{\breve{P}}_{e_{1}g_{2}}\left( \mathbf{r},t\right)
=\mathbf{P}_{e_{1}g_{2}}\left( \mathbf{r},t\right) e^{-i\omega_{p}t}e^{-i\mathbf{q}_{p}\cdot t}$ is the contribution of the $e_{1}\leftrightarrow g_{2}$ transition to the expectation value of the polarization, $\mathbf{P}_{e_{1}g_{2}}$ is the slowly varying polarization, and $\nabla^{2}$ is the three-dimensional Laplacian operator. With Eq. (\[Eq. 1\]), and assuming without loss of generality that $\mathbf{\hat{q}}_{p}=\mathbf{\hat{z}}q_{p}$, as shown in Fig. \[Fig. 1\](b), Eq. (\[Eq. A2\]) can be written in the paraxial approximation as $$\left( \frac{\partial}{\partial t}+c\frac{\partial}{\partial z}-i\frac
{c}{2q_{p}}\nabla_{\bot}^{2}\right) V_{p}\left( \mathbf{r},t\right)
=i\frac{g}{\mu_{e_{1}g_{2}}^{\ast}}\mathbf{P}_{e_{1}g_{2}}\left(
\mathbf{r},t\right) , \label{Eq. A3}$$ where $\nabla_{\bot}^{2}$ is the transverse Laplacian operator, and $g=2\pi\omega_{p}\left\vert \mu_{e_{1}g_{2}}\right\vert ^{2}/\hbar$ is a coupling constant .
Following [@Firstenberg2008PRA], we introduce a density-matrix distribution function in space and velocity, $$\breve{\rho}_{ss^{\prime}}^{{}}=\breve{\rho}_{ss^{\prime}}^{{}}\left(
\mathbf{r},\mathbf{v},t\right) =\underset{i}{\sum}\breve{\rho}_{ss^{\prime}}^{i}\left( t\right) \delta\left( \mathbf{r-r}_{i}\left( t\right)
\right) \delta\left( \mathbf{v-v}_{i}\left( t\right) \right) ,
\label{Eq. A4}$$ where the time dependence of $\breve{\rho}_{ss^{\prime}}^{i}\left( t\right)
$ is determined by Eqs. (\[Eq. A1\]). Differentiating Eq. (\[Eq. A4\]) with respect to time, we arrive at $$\begin{aligned}
& \frac{\partial}{\partial t}\breve{\rho}_{ss^{\prime}}^{{}}+\mathbf{v}\cdot\frac{\partial}{\partial\mathbf{r}}\breve{\rho}_{ss^{\prime}}^{{}}+\left[ \frac{\partial}{\partial t}\breve{\rho}_{ss^{\prime}}^{{}}\right]
_{\operatorname{col}}\nonumber\\
& =\underset{i}{\sum}\frac{\partial}{\partial t}\breve{\rho}_{ss^{\prime}}^{i}\left( t\right) \delta\left( \mathbf{r-r}_{i}\left( t\right)
\right) \delta\left( \mathbf{v-v}_{i}\left( t\right) \right) ,
\label{Eq. A5}$$ where the effect of velocity-changing collisions is taken in the strong collision limit in the form of a Boltzmann relaxation term [@Sobelman1967SPU], $$\left[ \frac{\partial}{\partial t}\breve{\rho}_{ss^{\prime}}^{{}}\right]
_{\operatorname{col}}=-\gamma_{\text{vcc}}\left[ \breve{\rho}_{ss^{\prime}}^{{}}\left( \mathbf{r},\mathbf{v},t\right) -\breve{R}_{ss^{\prime}}^{{}}\left( \mathbf{r},t\right) F(\mathbf{v})\right] , \label{Eq. A6}$$ with $\breve{R}_{ss^{\prime}}^{{}}=\breve{R}_{ss^{\prime}}^{{}}\left(
\mathbf{r},t\right) =\int d^{3}\mathbf{v}\breve{\rho}_{ss^{\prime}}^{{}}\left( \mathbf{r},\mathbf{v},t\right) $ being the density-number of atoms per unit volume, near $\mathbf{r}$ in space, and$$F=F(\mathbf{v})=\left( 2\pi v_{\text{th}}\right) ^{-3/2}e^{-\mathbf{v}^{2}/2v_{\text{th}}},\text{ }v_{\text{th}}=\frac{k_{b}T}{m}$$ is the Boltzmann distribution.
Before writing the coupled dynamics of the internal and motional degrees of freedom, we introduce the slowly varying envelopes of the density-matrix elements, $\rho_{ss^{\prime}}=\rho_{ss^{\prime}}\left( \mathbf{r},\mathbf{v},t\right) $, as$$\begin{aligned}
& \breve{\rho}_{g_{1}g_{2}}=\rho_{g_{1}g_{2}}e^{-i\left( \omega_{p}-\omega_{1}\right) t}e^{i\left( \mathbf{q}_{p}-\mathbf{q}_{1}\right)
\cdot\mathbf{r}},\nonumber\\
& \breve{\rho}_{e_{1}g_{2}}=\rho_{e_{1}g_{2}}e^{-i\omega_{p}t}e^{i\mathbf{q}_{p}\cdot\mathbf{r}},\nonumber\\
& \breve{\rho}_{e_{1}e_{2}}=\rho_{e_{1}e_{2}}e^{i\left( \mathbf{q}_{p}-\mathbf{q}_{2}\right) \cdot\mathbf{r}},\nonumber\\
& \breve{\rho}_{g_{1}e_{2}}=\rho_{g_{1}e_{2}}e^{-i\left( \omega_{p}-\omega_{1}-\omega_{2}\right) t}e^{i\left( \mathbf{q}_{p}-\mathbf{q}_{1}-\mathbf{q}_{2}\right) \cdot\mathbf{r}},\nonumber\\
& \breve{\rho}_{g_{2}e_{2}}=\rho_{g_{2}e_{2}}e^{i\omega_{2}t}e^{-i\mathbf{q}_{2}\cdot\mathbf{r}}, \label{Eq. A9}$$ and similarly the slowly varying densities $R_{ss^{\prime}}^{{}}=\int
d^{3}\mathbf{v}\rho_{ss^{\prime}}^{{}}$. Eqs. (\[Eq. A1\]) then become
\[Eq. A11\]$$\begin{aligned}
& \left[ \frac{\partial}{\partial t}+\mathbf{v}\cdot\frac{\partial}{\partial\mathbf{r}}-i\xi_{1}\right] \rho_{g_{1}g_{2}}-\gamma_{\text{vcc}}R_{g_{1}g_{2}}F\nonumber\\
& =i\left( V_{1}^{\ast}\rho_{e_{1}g_{2}}-V_{2}\rho_{g_{1}e_{2}}\right)
+bA\Gamma\rho_{e_{1}e_{2}},\label{Eq. A11a}\\
& \left[ \frac{\partial}{\partial t}+\mathbf{v}\cdot\frac{\partial}{\partial\mathbf{r}}-i\xi_{2}\right] \rho_{e_{1}g_{2}}-\gamma_{\text{vcc}}R_{e_{1}g_{2}}F\nonumber\\
& =i\left[ V_{p}n_{0}F+V_{1}\rho_{g_{1}g_{2}}\right] ,\label{Eq. A11b}\\
& \left[ \frac{\partial}{\partial t}+\mathbf{v}\cdot\frac{\partial}{\partial\mathbf{r}}-i\xi_{3}\right] \rho_{e_{1}e_{2}}-\gamma_{\text{vcc}}R_{e_{1}e_{2}}F\nonumber\\
& =i\left( V_{1}\rho_{g_{1}e_{2}}-V_{2}^{\ast}\rho_{e_{1}g_{2}}\right)
+iV_{p}^{{}}\rho_{g_{2}e_{2}},\label{Eq. A11c}\\
& \left[ \frac{\partial}{\partial t}+\mathbf{v}\cdot\frac{\partial}{\partial\mathbf{r}}-i\xi_{4}\right] \rho_{g_{1}e_{2}}-\gamma_{\text{vcc}}R_{g_{1}e_{2}}F\nonumber\\
& =-iV_{2}^{\ast}\rho_{g_{1}g_{2}},\label{Eq. A11d}\\
& \left[ \frac{\partial}{\partial t}+\mathbf{v}\cdot\frac{\partial}{\partial\mathbf{r}}-i\xi_{5}\right] \rho_{g_{2}e_{2}}^{{}}-\gamma
_{\text{vcc}}R_{g_{2}e_{2}}F\nonumber\\
& =-iV_{2}^{\ast}n_{0}F, \label{Eq. A11e}$$ where $\xi_{i}$ ($i=1-5$) are given in Eq. (\[Eq. 3\]). The expectation value of the polarization density $\mathbf{P}_{e_{1}g_{2}}\left(
\mathbf{r},t\right) $ in terms of the number density $R_{e_{1}g_{2}}\left(
\mathbf{r},t\right) $ is $\mathbf{P}_{e_{1}g_{2}}\left( \mathbf{r},t\right)
=\mu_{e_{1}g_{2}}^{\ast}R_{e_{1}g_{2}}\left( \mathbf{r},t\right) $, and Eq. (\[Eq. A3\]) becomes
$$\left( \frac{\partial}{\partial t}+c\frac{\partial}{\partial z}-i\frac
{c}{2q_{p}}\nabla_{\bot}^{2}\right) V_{p}\left( \mathbf{r},t\right)
=igR_{e_{1}g_{2}}\left( \mathbf{r},t\right) . \label{Eq. A12}$$
We now consider the case of stationary plane-wave pumps. For this case, it is convenient to introduce the Fourier transform $$f\left( \mathbf{r},t\right) =\underset{-\infty}{\overset{+\infty}{\int}}\frac{d^{3}k}{2\pi}e^{i\mathbf{kr}}\underset{-\infty}{\overset{+\infty}{\int
}}\frac{d\omega}{2\pi}e^{-i\omega t}f\left( \mathbf{k},\omega\right) ,
\label{Eq. A13}$$ and write Eqs. (\[Eq. A11\]) as
\[Eq. A14\]$$\begin{aligned}
& \left[ \omega-\mathbf{k\cdot v}+\xi_{1}\right] \rho_{g_{1}g_{2}}-i\gamma_{\text{vcc}}R_{g_{1}g_{2}}\left( \mathbf{k},\omega\right)
F\nonumber\\
& =\left( V_{2}\rho_{g_{1}e_{2}}-V_{1}^{\ast}\rho_{e_{1}g_{2}}\right)
+ibA\Gamma\rho_{e_{1}e_{2}},\label{Eq. A14a}\\
& \left[ \omega-\mathbf{k\cdot v}+\xi_{2}\right] \rho_{e_{1}g_{2}}-i\gamma_{\text{vcc}}R_{e_{1}g_{2}}\left( \mathbf{k},\omega\right)
F\nonumber\\
& =-\left( V_{p}n_{0}F+V_{1}\rho_{g_{1}g_{2}}\right) ,\label{Eq. A14b}\\
& \left[ \omega-\mathbf{k\cdot v}+\xi_{3}\right] \rho_{e_{1}e_{2}}-i\gamma_{\text{vcc}}R_{e_{1}e_{2}}\left( \mathbf{k},\omega\right)
F\nonumber\\
& =\left( V_{2}^{\ast}\rho_{e_{1}g_{2}}-V_{1}\rho_{g_{1}e_{2}}\right)
-V_{p}^{{}}\rho_{g_{2}e_{2}},\label{Eq. A14c}\\
& \left[ \omega-\mathbf{k\cdot v}+\xi_{4}\right] \rho_{g_{1}e_{2}}-i\gamma_{\text{vcc}}R_{g_{1}e_{2}}\left( \mathbf{k},\omega\right)
F\nonumber\\
& =V_{2}^{\ast}\rho_{g_{1}g_{2}},\label{Eq. A14d}\\
& \left[ \omega-\mathbf{k\cdot v}+\xi_{5}\right] \rho_{g_{2}e_{2}}-i\gamma_{\text{vcc}}R_{g_{2}e_{2}}\left( \mathbf{k},\omega\right)
F\nonumber\\
& =V_{2}^{\ast}n_{0}F, \label{Eq. A14e}$$ and Eq. (\[Eq. A12\]) as
$$\left( ik_{z}-i\frac{\omega}{c}+i\frac{k^{2}}{2q_{p}}\right) V_{p}\left(
\mathbf{k},\omega\right) =i\frac{g}{c}R_{e_{1}g_{2}}\left( \mathbf{k},\omega\right) . \label{Eq. A15}$$
The linear susceptibility $\chi_{e_{1}g_{2}}\left( \mathbf{k},\omega\right)
$ is defined by $$R_{e_{1}g_{2}}\left( \mathbf{k},\omega\right) =\chi_{e_{1}g_{2}}\left(
\mathbf{k},\omega\right) \frac{c}{g}V_{p}\left( \mathbf{k},\omega\right) .
\label{Eq. A16}$$
In order to find the probe absorption spectrum, we solve Eqs. (\[Eq. A14\]) analytically, obtain an expression for $\rho_{ss^{\prime}}$, and formally integrate it over velocity. This leads to an expression for $R_{ss^{\prime}}$ in terms of integrals over velocity, in the form of Eq. (\[Gs\]), such as $G_{1}=\int d^{3}v\frac{\xi_{2}\xi_{3}\xi_{4}F(\mathbf{v})}{\xi_{d}}$, which can be evaluated numerically. In the general case, the resulting expression for $R_{ss^{\prime}}$ is very complicated and is not reproduced here. In order to explore the underlying physics, we developed an approximate expression for the Fourier transform of the density-matrix element that refers to the probe transition, namely, $R_{e_{1}g_{2}}$ \[see Eq. (\[Eq. 2\])\].
One can verify that in the absence of the pumps ($V_{1}=V_{2}=0$), the resulting one-photon complex spectrum simplifies to the well known result for the strong collision regime, $K=iG/\left( 1-i\gamma_{\text{vcc}}G\right) $, where $G=\int d^{3}\mathbf{v}F /\left( \omega-\mathbf{k\cdot v}+\xi
_{2}\right) $ [@Sobelman1967SPU].
Diffusion in the presence of fields
===================================
In order to obtain diffusion-like equations for the density-matrix elements and the probe fields, we begin by integrating Eqs. (\[Eq. A11a\]) and (\[Eq. A11c\]) over velocity and obtain
\[Eq. B1\]$$\begin{aligned}
& \left[ \frac{\partial}{\partial\mathbf{r}}+i\delta\mathbf{q}_{1}\right]
\cdot\mathbf{J}_{g_{1}g_{2}}+\left[ \frac{\partial}{\partial t}-i\left(
\Delta_{p}-\Delta_{1}\right) +\gamma\right] R_{g_{1}g_{2}}\nonumber\\
& =i\left( V_{1}^{\ast}R_{e_{1}g_{2}}-V_{2}R_{g_{1}e_{2}}\right) +bA\Gamma
R_{e_{1}e_{2}},\label{Eq. B1a}\\
& \left[ \frac{\partial}{\partial\mathbf{r}}+i\delta\mathbf{q}_{2}\right]
\cdot\mathbf{J}_{e_{1}e_{2}}+\Biggl[\frac{\partial}{\partial t}-i\left(
\Delta_{p}-\Delta_{2}\right) \nonumber\\
& +\Gamma+\gamma\Biggr]R_{e_{1}e_{2}}=i\left( V_{1}R_{g_{1}e_{2}}-V_{2}^{\ast}R_{e_{1}g_{2}}+V_{p}R_{g_{2}e_{2}}\right) , \label{Eq. B1b}$$ where $\mathbf{J}_{ss^{\prime}}=\mathbf{J}_{ss^{\prime}}\left( \mathbf{r},t\right) =\int d^{3}v\mathbf{v}\rho_{ss^{\prime}} $ is the envelope of the current density. Expanding $\rho_{g_{1}g_{2}} $ and $\rho_{e_{1}e_{2}} $ in Eqs. (\[Eq. A11a\]) and (\[Eq. A11c\]) as $\rho_{ss^{\prime}}
=R_{ss^{\prime}} F +1/\gamma_{\text{vcc}}\rho_{ss^{\prime}}^{(1)} ,$ multiplying Eqs. (\[Eq. A11a\]) and (\[Eq. A11c\]) by $\mathbf{v}$, integrating the resulting equations over velocity using
$$\int d^{3}v_{j}v_{i}\frac{\partial}{\partial x_{i}}R_{ss^{\prime}} F
=\delta_{ij}v_{\text{th}}\frac{\partial}{\partial x_{i}}R_{ss^{\prime}} ,
\label{Eq. B2}$$
defining the current density of the density matrix by $$\gamma_{\text{vcc}}\mathbf{J}_{ss^{\prime}} =\int d^{3}v_{j}\rho_{ss^{\prime}}^{\left( 1\right) } , \label{Eq. B3}$$ and retaining the leading terms in $1/\gamma_{\text{vcc}}$, we obtain
\[Eq. B4\]$$\begin{aligned}
& \mathbf{J}_{g_{1}g_{2}}+D\left[ \frac{\partial}{\partial\mathbf{r}}+i\delta\mathbf{q}_{1}\right] R_{g_{1}g_{2}}\nonumber\\
& =\frac{i}{\gamma_{\text{vcc}}}\left( V_{1}^{\ast}\mathbf{J}_{e_{1}g_{2}}-V_{2}\mathbf{J}_{g_{1}g_{2}}\right) -\frac{bA\Gamma}{\gamma_{\text{vcc}}}\mathbf{J}_{e_{1}e_{2}},\label{Eq. B4a}\\
& \mathbf{J}_{e_{1}e_{2}}+D\left[ \frac{\partial}{\partial\mathbf{r}}+i\delta\mathbf{q}_{2}\right] R_{e_{1}e_{2}}\nonumber\\
& =\frac{i}{\gamma_{\text{vcc}}}\left( V_{1}\mathbf{J}_{g_{1}e_{2}}-V_{2}^{\ast}\mathbf{J}_{e_{1}g_{2}}+\tilde{V}_{p}\mathbf{J}_{g_{2}e_{2}}\right) , \label{Eq. B4b}$$ where $D=v_{\text{th}}/\gamma_{\text{vcc}}$. Substituting $\mathbf{J}_{g_{1}g_{2}}$, $\mathbf{J}_{e_{1}e_{2}}$ from Eq. (\[Eq. B4\]) into Eq. (\[Eq. B1\]), we get
\[Eq. B5\]$$\begin{aligned}
& \left[ \frac{\partial}{\partial t}-i\left( \Delta_{p}-\Delta_{1}\right)
+\gamma-D\left( \frac{\partial}{\partial\mathbf{r}}+i\delta\mathbf{q}_{1}\right) ^{2}\right] R_{g_{1}g_{2}}\nonumber\\
& =i\left( V_{1}^{\ast}R_{e_{1}g_{2}}-V_{2}R_{g_{1}e_{2}}\right) +bA\Gamma
R_{e_{1}e_{2}}-D\left( \frac{\partial}{\partial\mathbf{r}}+i\delta
\mathbf{q}_{1}\right) \nonumber\\
& \times\left[ \frac{i}{\gamma_{\text{vcc}}}\left( V_{1}^{\ast}\mathbf{J}_{e_{1}g_{2}}-V_{2}\mathbf{J}_{g_{1}e_{2}}\right) -\frac{bA\Gamma
}{\gamma_{\text{vcc}}}\mathbf{J}_{e_{1}e_{2}}\right] ,\label{Eq. B5a}\\
& \left[ \frac{\partial}{\partial t}-i\left( \Delta_{p}-\Delta_{2}\right)
+\Gamma+\gamma-D\left( \frac{\partial}{\partial\mathbf{r}}+i\delta
\mathbf{q}_{2}\right) ^{2}\right] R_{e_{1}e_{2}}\nonumber\\
& =i\left( V_{1}R_{g_{1}e_{2}}-V_{2}^{\ast}R_{e_{1}g_{2}}+V_{p}R_{g_{2}e_{2}}\right) -D\left( \frac{\partial}{\partial\mathbf{r}}+i\delta
\mathbf{q}_{2}\right) \nonumber\\
& \times\left[ \frac{i}{\gamma_{\text{vcc}}}\left( V_{1}\mathbf{J}_{g_{1}e_{2}}-V_{2}^{\ast}\mathbf{J}_{e_{1}g_{2}}+V_{p}\mathbf{J}_{g_{2}e_{2}}\right) \right] . \label{Eq. B5b}$$ In order to calculate $R_{e_{1}g_{2}}$, $R_{g_{1}e_{2}}$, $R_{g_{2}e_{2}},$ and $\mathbf{J}_{e_{1}g_{2}}$, $\mathbf{J}_{g_{1}e_{2}}$, $\mathbf{J}_{g_{2}e_{2}}$, we assume in Eqs. (\[Eq. A11b\]), (\[Eq. A11d\]), and (\[Eq. A11e\]) that the envelopes change slowly enough such that $\left\vert
\partial/\partial t+\mathbf{v}\cdot\partial/\partial\mathbf{r}\right\vert $ $\ll\left\vert \xi_{2,4,5}\right\vert ,$ and get
\[Eq. B6\]$$\begin{aligned}
-i\xi_{2}\rho_{e_{1}g_{2}}= & \gamma_{\text{vcc}}R_{e_{1}g_{2}} F+i\left(
V_{p}n_{0}F+V_{1}\rho_{g_{1}g_{2}}\right) ,\label{Eq. B6a}\\
-i\xi_{4}\rho_{g_{1}e_{2}}= & \gamma_{\text{vcc}}R_{g_{1}e_{2}}
F-iV_{2}^{\ast}\rho_{g_{1}g_{2}},\label{Eq. B6b}\\
-i\xi_{5}\rho_{g_{2}e_{2}}= & \gamma_{\text{vcc}}R_{g_{2}e_{2}}
F-iV_{2}^{\ast}n_{0}F. \label{Eq. B6c}$$ Solving Eq. (\[Eq. B6\]) formally for $\rho_{e_{1}g_{2}}$, $\rho_{g_{1}e_{2}}$, $\rho_{g_{2}e_{2}}$ and substituting only their leading parts, *i.e.* $\rho_{ss^{\prime}}=$ $R_{ss^{\prime}} F $, we find
\[Eq. B7\]$$\begin{aligned}
\rho_{e_{1}g_{2}} & =\left[ \gamma_{\text{vcc}}R_{e_{1}g_{2}} \right.
-V_{1}R_{g_{1}g_{2}}) - \left. V_{p} n_{0}\right] F/\xi_{2}, \label{Eq. B7a}\\
\rho_{g_{1}e_{2}} & =\left[ \gamma_{\text{vcc}}R_{g_{1}e_{2}} +V_{2}^{\ast
}R_{g_{1}g_{2}} \right] F/\xi_{4},\label{Eq. B7b}\\
\rho_{g_{2}e_{2}} & =\left[ \gamma_{\text{vcc}}R_{g_{2}e_{2}} +V_{2}^{\ast
}n_{0}\right] F/\xi_{5}. \label{Eq. B7c}$$ Integrating Eqs. (\[Eq. B7\]) over velocity we get
\[Eq. B8\]$$\begin{aligned}
& R_{e_{1}g_{2}} =iK_{\text{1p}}\left[ V_{1}R_{g_{1}g_{2}} +V_{p}
n_{0}\right] ,\label{Eq. B8a}\\
& R_{g_{1}e_{2}} =-iK_{\text{3p}}^{{}}V_{2}^{\ast}R_{g_{1}g_{2}}
,\label{Eq. B8b}\\
& R_{g_{2}e_{2}} =-iK_{\text{pump}}V_{2}^{\ast}n_{0}, \label{Eq. B8c}$$ where $K_{\text{1p}}=iG_{\text{1p}}/\left( 1-G_{\text{1p}}\gamma_{\text{vcc}}\right) $ is the one-photon absorption spectrum with $G_{\text{1p}}=\int F
/\xi_{2}d^{3}v\mathbf{\ }$, $K_{\text{3p}}=iG_{\text{3p}}/\left(
1-G_{\text{3p}}\gamma_{\text{vcc}}\right) $ is the three-photon absorption spectrum with $G_{\text{3p}}=\int F/\xi_{4}d^{3}v\mathbf{\ }$and $K_{\text{pump}}=iG_{\text{pump}}/\left( 1-G_{\text{pump}}\gamma_{\text{vcc}}\right) $ is the one-photon (pump) absorption spectrum with $G_{\text{pump}}=\int F/\xi_{5}d^{3}\mathbf{v}$. In the case of collinear pump and probe beams $\delta\mathbf{q=}\delta\mathbf{q}_{1,2}=\mathbf{q}_{p}-\mathbf{q}_{1,2}=\delta q\widehat{\mathbf{z}}$, Eqs. (\[Eq. B5\]) and (\[Eq. B8\]) form a closed set when
$$\begin{aligned}
& \left( \frac{\partial}{\partial\mathbf{r}}+i\delta\mathbf{q}_{1,2}\right)
\cdot\frac{iV_{1,2}\left( \mathbf{r}\right) }{\gamma_{\text{vcc}}}\mathbf{J}_{e_{1}g_{2}},\\
& \left( \frac{\partial}{\partial\mathbf{r}}+i\delta\mathbf{q}_{1,2}\right)
\cdot\frac{iV_{1,2}\left( \mathbf{r}\right) }{\gamma_{\text{vcc}}}\mathbf{J}_{g_{1}e_{2}},\text{ }\\
& \left( \frac{\partial}{\partial\mathbf{r}}+i\delta\mathbf{q}_{2}\right)
\cdot\frac{iV_{p}\left( \mathbf{r,}t\right) }{\gamma_{\text{vcc}}}\mathbf{J}_{g_{2}e_{2}}$$
can be neglected in Eq. (\[Eq. B5\]). These terms vanish completely in the special case of pump and probe which are plane waves $\left( \partial
/\partial\mathbf{r=0}\right) $, and also collinear and degenerate $\left(
\delta\mathbf{q=0}\right) $. They can also be neglected whenever $\left\vert
V_{1,2,p}\right\vert \ll\gamma_{\text{vcc}}$ as is the case in many realistic situations. However, the term $\left( \partial/\partial\mathbf{r}+i\delta\mathbf{q}_{1}\right) \cdot bA\Gamma/\gamma_{\text{vcc}}\mathbf{J}_{e_{1}e_{2}}$ in Eq. (\[Eq. B5a\]) cannot be neglected in the case of collinear pump and probe beams since $bA\Gamma/\gamma_{\text{vcc}}$ does not go to zero.
Substituting Eq. (\[Eq. B4b\]) into Eq. (\[Eq. B5a\]), and Eq. (\[Eq. B8\]) into Eq. (\[Eq. B5\]), we find:
\[Eq. B9\]$$\begin{aligned}
& \left\{ \frac{\partial}{\partial t}-i\left( \Delta_{p}-\Delta_{1}\right)
+\gamma+K_{\text{1p}}\left\vert V_{1}\right\vert ^{2}+K_{\text{3p}}\left\vert
V_{2}\right\vert ^{2}\right\} R_{g_{1}g_{2}}\nonumber\\
& =D\left( \frac{\partial}{\partial\mathbf{r}}+i\delta\mathbf{q}_{1}\right)
^{2}R_{g_{1}g_{2}}+D\left( \frac{\partial}{\partial\mathbf{r}}+i\delta
\mathbf{q}_{2}\right) ^{2}R_{e_{1}e_{2}}\nonumber\\
& +bA\Gamma R_{e_{1}e_{2}}-K_{\text{1p}}V_{1}^{\ast}V_{p}n_{0},\label{Eq. B9a}\\
& \left\{ \frac{\partial}{\partial t}-i\left( \Delta_{p}-\Delta_{2}\right)
+\Gamma+\gamma\right\} R_{e_{1}e_{2}}\nonumber\\
& =D\left( \frac{\partial}{\partial\mathbf{r}}+i\delta\mathbf{q}_{2}\right)
^{2}R_{e_{1}e_{2}}+V_{1}V_{2}^{\ast}\left( K_{\text{1p}}+K_{\text{3p}}\right) R_{g_{1}g_{2}}\nonumber\\
& +V_{2}^{\ast}\left( K_{\text{1p}}+K_{\text{pump}}\right) V_{p}n_{0}.
\label{Eq. B9b}$$ These are the final diffusion-like coupled equations for the ground- and excited-state coherences.
In order to investigate the Ramsey narrowing of the EIA peak, we consider finite probe and pump beams and restrict the discussion to collinear EIA. We assume that the fields are stationary and overlap in their cross sections with negligible variation along the $z-$direction, $V_{p}\left( \mathbf{r},t\right) =V_{p}w\left( \mathbf{r}_{\bot}\right) $, $V_{1}\left(
\mathbf{r}\right) =V_{1}w\left( \mathbf{r}_{\bot}\right) $, $V_{2}\left(
\mathbf{r},t\right) =V_{2}w\left( \mathbf{r}_{\bot}\right) ,$ where $w\left( \mathbf{r}_{\bot}\right) $ is the transverse profile of the fields. We further take $\delta q=0$ and $\Delta_{1}=\Delta_{2}=0$ for brevity. In the diffusion regime, we rewrite Eqs. (\[Eq. B8\]) and (\[Eq. B9\]) as
\[Eq. B10\]$$\begin{aligned}
& \left[ i\Delta_{p}+\gamma+\left( K_{\text{1p}}\left\vert V_{1}\right\vert
^{2}+K_{\text{3p}}\left\vert V_{2}\right\vert ^{2}\right) w\left(
\mathbf{r}_{\bot}\right) ^{2}\right] R_{g_{1}g_{2}}=\nonumber\\
& bA\Gamma\left( 1+\frac{D}{\gamma_{\text{vcc}}}\nabla_{\bot}^{2}\right)
R_{e_{1}e_{2}}-K_{\text{1p}}V_{1}^{\ast}V_{p}n_{0}w\left( \mathbf{r}_{\bot
}\right) ^{2},\label{Eq. B10a}\\
& R_{e_{1}g_{2}}=iK_{\text{1p}}\left( V_{1}R_{g_{1}g_{2}}+V_{p}n_{0}\right)
w\left( \mathbf{r}_{\bot}\right) ,\label{Eq. B10b}\\
& \left( i\Delta_{p}+\Gamma+\gamma-D\nabla_{\bot}^{2}\right) R_{e_{1}e_{2}}\nonumber\\
& =V_{1}\left( K_{\text{1p}}+K_{\text{3p}}\right) R_{g_{1}g_{2}}V_{2}^{\ast}w\left( \mathbf{r}_{\bot}\right) ^{2}\nonumber\\
& +V_{p}\left( K_{\text{1p}}+K_{\text{pump}}\right) n_{0}V_{2}^{\ast
}w\left( \mathbf{r}_{\bot}\right) ^{2},\label{Eq. B10c}\\
& R_{g_{1}e_{2}}=-iK_{\text{3p}}V_{2}^{\ast}R_{g_{1}g_{2}}w\left(
\mathbf{r}_{\bot}\right) ,\label{Eq. B10d}\\
& R_{g_{1}e_{2}}=-iK_{\text{pump}}V_{2}^{\ast}n_{0}w\left( \mathbf{r}_{\bot
}\right) . \label{Eq. B10e}$$ We further consider a probe and pump beams with a uniform intensity and phase within a sheet of thickness $2a$ in the $x-$direction (one-dimensional stepwise beams):
$$w\left( x,y\right) =\left\{
\genfrac{}{}{0pt}{}{1\text{ for }\left\vert x\right\vert \leq a}{0\text{ for
}\left\vert x\right\vert >a}\right. .$$ The solution for $R_{g_{1}g_{2}}$, symmetric in $x$ and decaying as $\left\vert x\right\vert \rightarrow\infty$, is given by
\[Eq. B13\]$$\begin{aligned}
& R_{g_{1}g_{2}}\left( \left\vert x\right\vert \leq a\right) =\nonumber\\
& C_{2}\cosh\left( k_{1}x\right) +C_{1}\cosh\left( k_{2}x\right)
+\frac{bA\Gamma\beta_{2}+D\alpha_{2}^{2}\beta_{1}}{\left( D\alpha_{1}\alpha_{2}\right) ^{2}+bA\Gamma\beta_{3}},\label{Eq. B13a}\\
& R_{e_{1}e_{2}}\left( \left\vert x\right\vert \leq a\right) =\frac
{C_{1}\left( k_{2}^{2}-\alpha_{2}^{2}\right) D\gamma_{\text{vcc}}}{bA\Gamma\left( D\alpha_{2}^{2}+\gamma_{\text{vcc}}\right) }\cosh\left(
k_{2}x\right) \nonumber\\
& +\frac{C_{2}\left( k_{1}^{2}-\alpha_{1}^{2}\right) D\gamma_{\text{vcc}}}{bA\Gamma\left( D\alpha_{2}^{2}+\gamma_{\text{vcc}}\right) }\cosh\left(
k_{1}x\right) +\frac{\beta_{1}\beta_{3}-\beta_{2}\alpha_{1}^{2}D}{\beta
_{3}bA\Gamma-\left( D\alpha_{1}\alpha_{2}\right) ^{2}},\\
& R_{g_{1}g_{2}}\left( \left\vert x\right\vert >a\right) =\nonumber\\
& \frac{C_{3}bA\Gamma\left( D\alpha_{2}^{2}+\gamma_{\text{vcc}}\right)
}{\left( \alpha_{3}^{2}-\alpha_{2}^{2}\right) D\gamma_{\text{vcc}}}e^{-\alpha_{2}\left( \left\vert x\right\vert -a\right) }+C_{4}e^{-\alpha_{3}\left( \left\vert x\right\vert -a\right) },\\
& R_{e_{1}e_{2}}\left( \left\vert x\right\vert >a\right) =C_{3}e^{-\alpha_{2}\left( \left\vert x\right\vert -a\right) },\end{aligned}$$ where $\alpha_{1}^{2}=(-i\Delta_{p}+\gamma+K_{\text{1p}}\left\vert
V_{1}\right\vert ^{2}+K_{\text{3p}}\left\vert V_{2}\right\vert ^{2})/D,$ $\alpha_{2}^{2}=\alpha_{3}^{2}+\Gamma/D,$ $\alpha_{3}^{2}=\left( -i\Delta
_{p}+\gamma\right) /D,$ and $\beta_{1}=V_{1}^{\ast}V_{p}K_{\text{1p}}n_{0},$ $\beta_{2}=V_{1}V_{2}^{\ast}(K_{\text{1p}}+K_{\text{3p}}),\beta_{3}=V_{2}^{\ast}V_{p}(K_{\text{1p}}+K_{\text{pump}})n_{0}.$ The complex diffusion wave-numbers are obtained from
$$\begin{aligned}
& 2D\gamma_{\text{vcc}}k_{1,2}^{2}=D\gamma_{\text{vcc}}\alpha_{+}^{2}+\beta_{3}bA\Gamma\\
& \mp\left[ (D\gamma_{\text{vcc}})^{2}\alpha_{-}^{4}+\beta_{3}bA\Gamma\left( 4+2D\gamma_{\text{vcc}}\alpha_{+}^{2}+\beta_{3}bA\Gamma
\right) \right] ^{1/2},\end{aligned}$$
with $\alpha_{\pm}^{2}=\alpha_{2}^{2}\pm\alpha_{1}^{2}.$ The coefficients $C_{i}$ ($i=1-4$) are obtained from the continuity conditions of $R_{ss^{\prime}}$ and $\left( \partial/\partial x\right) R_{ss^{\prime}}$at $\left\vert x\right\vert =a$. From Eq. (\[Eq. B10b\]) one finds $$\begin{aligned}
& R_{e_{1}g_{2}}\left( \left\vert x\right\vert \leq a\right) =iK_{2}\Biggl[V_{1}\left( C_{2}\cosh\left( k_{1}x\right) \right. \nonumber\\
& +C_{1}\cosh\left( k_{2}x\right) +\left. \frac{bA\Gamma\beta_{2}+D\alpha_{2}^{2}\beta_{1}}{\left( D\alpha_{1}\alpha_{2}\right) ^{2}+bA\Gamma\beta_{3}}\right) +V_{p}n_{0}\Biggr] , \label{B15}$$ and the energy absorption at frequency $\omega_{p}$ is finally calculated from $P\left( \Delta\right) =(\hbar\omega_{p}/a)\text{Im}\int_{-a}^{a}dxR_{e_{1}g_{2}}\left( x\right) .$ Two examples for the resulting spectrum are given in Fig. \[Fig. 7\].
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[^1]: The fact that the ‘inner’ decoherence rate $\gamma$ is shared by both the ground and the excited manifolds, does not imply that their total decoherence rate is the same; the coherence between the two excited states decays via the $e\rightarrow g$ relaxation channels and therefore decays much faster than the ground-state coherence. Incorporating different values of $\gamma$ for the ground and excited states does not lead to substantial changes in the collision-induced phenomena explored here.
[^2]: When $\gamma_{\text{vcc}}\rightarrow0$, the third term in Eq. (\[Eq. 2\]) gradually vanishes, and the second term ($V_{2}^{2}G_{5}$) is responsible for the EIA peak, as indicated by the brown-dotted line in Fig. \[Fig. 2\](b). Its width is limited by homogenous broadening mechanisms and determined by $\gamma$ and $\Gamma_{\text{pcc}}$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Extensions of the standard model of particle physics, in particular those based on string theory, often predict a new $U(1)$ gauge symmetry in a hidden sector. The corresponding gauge boson, called hidden photon, naturally interacts with the ordinary photon via gauge kinetic mixing, leading to photon - hidden photon oscillations. In this framework, one expects photon disappearance as a function of the mass of the hidden photon and the mixing angle, loosely constrained from theory. Several experiments have been carried out or are planned to constrain the mass-mixing plane.
In this contribution we derive new constraints on the hidden photon parameters, using very high energy $\gamma$-rays detected from the Crab Nebula, whose broad-band spectral characteristics are well understood. The very high energy $\gamma$-ray observations offer the possibility to provide bounds in a broad mass range at a previously unexplored energy and distance scale. Using existing data that were taken with several Cherenkov telescopes, we discuss our results in the context of current constraints and consider the possibilities of using astrophysical data to search for hidden photon signatures.
author:
- 'Hannes-Sebastian Zechlin'
- Dieter Horns
- Javier Redondo
bibliography:
- 'biblio.bib'
title: 'New Constraints on Hidden Photons using Very High Energy Gamma-Rays from the Crab Nebula'
---
[ address=[University of Hamburg, Institut für Experimental Physik, Luruper Chaussee 149, D-22761 Hamburg, Germany]{} ]{}
[ address=[University of Hamburg, Institut für Experimental Physik, Luruper Chaussee 149, D-22761 Hamburg, Germany]{} ]{}
[ address=[Deutsches Elektronen Synchrotron (DESY), Notkestrasse 85, D-22607 Hamburg, Germany]{} ]{}
Introduction
============
Typical extensions of the current standard model of particle physics often contain extra $U(1)$ gauge degrees of freedom. Many models based on string compactifications show that standard model particles are uncharged under the additional $U(1)$ symmetries. Therefore these degrees of freedom belong to a “hidden sector”, i.e., an experimentally so far unobserved set of fields uncharged[^1] under the standard model gauge group. The existence of high mass particles charged under both the visible and hidden sectors (mediators) or gravity can produce nevertheless small interactions between the two sectors. Assuming that hidden sector particles are light[^2], note that current accelerator based experiments can be largely insensitive to the subtle effects of their existence. Therefore in the recent past, many high-precision experiments have been dedicated to search for new light particles (e.g. [@Ahlers:2007rd] or [@DESY08] for an overview) and many ideas for new experiments are under consideration [@Patras:2008].
Natural models contain at least one hidden photon [@Holdom:1986] (or sometimes called paraphoton), i.e., the corresponding gauge boson to the new $U(1)_h$ gauge group. Here we consider a minimal theory with just *one*[^3] $U(1)_h$ gauge group ([@Okun:1982], [@Holdom:1986], [@Ahlers:2007rd], [@Jaeckel:2008fi]), in addition to the normal electromagnetic gauge. The most general low energy Lagrangian allowed by the symmetries is $$\begin{aligned}
\label{lagrangian}
\mathcal{L} & = & -\frac{1}{4} F^{\mu \nu} F_{\mu \nu} - \frac{1}{4} B^{\mu \nu} B_{\mu \nu} - \frac{\sin \chi}{2} F^{\mu \nu} B_{\mu \nu} + \nonumber \\
& & + \frac{\cos^2 \chi}{2} \mu^2 B^{\mu} B_{\mu}\end{aligned}$$ where $F_{\mu \nu}$ is the field strength tensor for the ordinary photon gauge field $A_{\mu}$, defined by , and $B_{\mu \nu}$ the same tensor for the hidden photon field $B_{\mu}$. The third term, also allowed by gauge invariance, corresponds to a non-diagonal kinetic term, the so-called kinetic mixing, where $\chi$ is the mixing angle between photons and hidden photons. We assume $\chi$ to be small. The last term describes a possible mass $\mu$ of the hidden photon. It can arise either from Higgs or Stückelberg mechanisms, but in the former case the model suffers from additional constraints [@Ahlers:2008].
The non-zero kinetic mixing states that the $A_\mu$ and $B_\mu$ fields are non-orthogonal. The transformation to an orthogonal basis, with canonical kinetic term, is done by the redefinition $$B_{\mu} \rightarrow S_{\mu} - \sin \chi A_{\mu}$$ Moreover, one observes that now contains a mass term that mixes photons with hidden photons (find the relevant part of the redefined $\mathcal{L}$ below, expanded for $\chi \ll 1$) $$\mathcal{L} = \dots + \frac{1}{2} \mu^2 \left( S^\mu S_\mu - 2 \chi S^\mu A_\mu + \chi^2 A^\mu A_\mu \right)$$ leading to vacuum $\gamma$-$\gamma_s$ oscillations (if hidden photons are not massless). Here $\gamma_s$ is the quantum of the field $S_\mu$, that being orthogonal to the photon is completely sterile with respect to electromagnetic interactions. Note also that the diagonalization causes a multiplicative renormalization of the electric charge.
It is interesting to note that the oscillation effect is completely analogous to the phenomenology of neutrino oscillations. Thus the oscillations of photons open the possibility to search for hidden photons via, e.g., ”light shining through a wall“ (LSW) experiments ([@Okun:1982], [@Cavity:2008], [@Ahlers:2007rd]).
In order to compute the oscillation probability at a distance $L$, one has to solve the equations of motion for the Lagragian to find the propagation eigenstates (see e.g. [@Ahlers:2007rd] for a short review). One obtains the following result for the oscillation probability (in natural units): $$\label{oscprob}
P_{\gamma \rightarrow \gamma_s}(L) = \sin^{2} (2\chi) \sin^{2} \left( \frac{\mu^2}{4E} L \right),$$ where $E$ stands for the energy. Hence the oscillation length is observed to $$L_{osc} = \frac{4 \pi E}{\mu^2} \simeq 8 \left( \frac{E}{\textnormal{TeV}} \right) \left( \frac{\mu}{10^{-7}\textnormal{eV}} \right)^{-2} \textnormal{kpc}.$$
The values of the mixing parameters of hidden photons $\chi$ and $\mu$ are unspecified from theory and experimentally unknown because these particles have not been detected until today. However, there already exists a broad range of restrictions on these parameters provided by several experiments. The stronger constraints originate from tests of the Coulomb law, CMB measurements, LSW experiments, and searches of $\gamma_s$’s radiated from the Sun (see e.g. [@Ahlers:2008] and references therein). Find a composite plot of the current bounds in [@Ahlers:2008], Fig. 1.
Considering the experiments, it turns out that the bounds were obtained using energies up to $\mathcal{O}(100\, \textnormal{GeV})$ (LEP) and distances up to $\mathcal{O}(1\,\textnormal{AU})$ (solar searches), so it’s worth mentioning that no constraints exist using the very high energy range ($\mathcal{O}(\textnormal{TeV})$) and distances with $\mathcal{O}(\textnormal{kpc})$. In this work we will provide such (astronomical) bounds using very high energy $\gamma$-rays from the Crab Nebula.
The Crab Nebula (Messier 1) is a remnant of a supernova explosion that occurred in the year AD 1054. A pulsar (PSR B0531+21) is located near the geometrical center of the nebula. Today, the remnant is observed to be of the plerionic type at a distance of from Earth [@Trimble:1973], with a bright continuum emission from radio to very high energy $\gamma$-rays, peaked in the near-infrared and optical range. The whole emission is predominantly produced by non-thermal processes, mainly by synchrotron and inverse Compton interactions of energetic electrons. The mechanism producing the VHE spectrum is inverse Compton scattering of accelerated electrons (up to PeV energies) on different low energy seed photon fields, dominated by the synchroton field (see e.g. Figure 10 in [@Aharonian:2004]).
The broad-band VHE spectrum can be parametrised by (see [@Aharonian:2004] for the coefficients $p_i$) $$\label{invCom}
\log \left\{ \frac{\nu f_{\nu}}{\textnormal{erg }(\textnormal{cm}^2 \textnormal{ s})^{-1}} \right\} = \sum_{i=0}^{5} p_i \log^{i} \left( \frac{E}{\textnormal{TeV}} \right),$$ where $\nu f_\nu = E^2 \frac{\mathrm{d}N}{\mathrm{d}E}$ is the differential energy-flux of the Crab nebula.
Upper limits measured for the diameter of the nebula in the VHE regime are $\alpha_{\textnormal{c}} < 2'$ at and $\alpha_{\textnormal{c}} < 3'$ at $E > 10\,\textnormal{TeV}$ [@Aharonian:2004]. However, considering models of the VHE emission of the nebula (e.g. [@Aharonian:2004]) leads to diameters of $\mathcal{O}(\textnormal{arcsec})$, which we assume here.
Before we explain the method of giving new constraints, there remains one subtlety we have to mention briefly. The oscillation probability is calculated under the assumption that the photons can be represented as plane waves. Considering production and detection processes, this assumption does not hold under normal circumstances. Rather, must be calculated using quantum mechanical wave packets having a coherence length $D$. When working with wave packets of different masses $m_1 \neq m_2$, the velocities would differ by a factor , $\Delta m^2 = m_1^2 - m_2^2$, so they separate by $L \Delta \beta$ after traveling a distance L. Thus oscillations freeze out, if $$L \geqslant L_{coh} = \frac{D}{\Delta \beta}$$ (see [@Nussinov:1976]). In our case, $\Delta m^2 = \mu^2$. Detailed quantum mechanical calculations with wave packets reveal (see [@Giunti:1998]), that for relativistic particles $$\begin{aligned}
L_{coh} & = & \frac{4 \sqrt{2} \sigma_x E^2}{\mu^2} \\
& \simeq & 1.8 \times 10^{22} \left( \frac{E}{\textnormal{TeV}} \right)^2 \left( \frac{\mu}{10^{-7} \textnormal{eV} }\right)^{-2} \left( \frac{\sigma_x}{\textnormal{m}} \right) \textnormal{pc}, \nonumber\end{aligned}$$ where $\sigma_x \equiv \sqrt{\sigma_{xP}^2 + \sigma_{xD}^2}$, with $\sigma_{xP}, \sigma_{xD}$ the spatial uncertainties of the production and the detection process, respectively. To get oscillations, three conditions have to be satisfied: (a) $L_{coh} > d_{\textnormal{c}} \approx 2 \, \textnormal{kpc}$, (b) $L_{osc} < L_{coh}$, and (c) $\sigma_x \ll L_{osc}$.
For the production process one finds $\sigma_{xP}$ calculating the interaction length for electrons in the nebula, which interact via inverse Compton scattering with various seed photons. A lower limit is found assuming the Thomson regime, $\gamma \epsilon \ll m_e c^2$, with $\gamma$ the Lorentz-factor of the electron, $\epsilon$ the energy of the seed photon, $m_e$ the electron rest mass, and $c$ the speed of light in vacuum. In this case the interaction length is given by $$\lambda^{-1} = \sigma_{\textnormal{T}} \int_{\epsilon} \mathrm{d}\epsilon' n_{\textnormal{b}}(\epsilon'),$$ where $\sigma_\textnormal{T}$ is the Thomson cross section and $n_{\textnormal{b}}(\epsilon) =
\frac{\mathrm{d}n}{\mathrm{d}\epsilon}$ is the differential (seed-) photon density. For a rough lower limit, approximating the main seed field (synchrotron) in [@Aharonian:2004] and using the extension of the nebula it reveals $\lambda \gtrsim 1 \,\textnormal{kpc}$.
Comparing $\lambda$ to the radius of the nebula in the VHE regime $r_{\textnormal{c}} \ll 1 \,\textnormal{pc}$, it should be clear that $r_{\textnormal{c}}$ is the right estimation for $\sigma_{xP}$, because $r_{\textnormal{c}} \ll \lambda$.
The spatial uncertainty of the detection process, $\sigma_{xD}$, does not contribute. Considering the detection process in detail one finds that $\sigma_{xD} \ll \sigma_{xP}$ (Zechlin, et al., in prep.).
Taking these results into account, one observes that the conditions given above hold for masses $\mu \lesssim 10^{-5} \,\textnormal{eV}$.
Method
======
As mentioned above, this work investigates the very high energy data available from the Crab nebula. The VHE regime is mainly covered by ground-based detection techniques, due to the fact that non-thermal sources typically produce power-law type spectra in which the flux drops with increasing energy. As a consequence large effective detection areas of $\mathcal{O}(10^{5}\,\textnormal{m}^2)$ are required, sufficient to compensate for small photon fluxes. The data used here were taken with (stereoscopic) imaging air Cherenkov telescopes (IACTs), especially by HEGRA [@Aharonian:2004], H.E.S.S. [@HessICRC:2007], MAGIC [@Albert:2008], and the Whipple 10m telescope [@Grube:2007].
If hidden photons of mass $\mu$ exist, the energy dependent oscillation probability influences the observable spectrum. Some photons will convert to hidden photons during their propagation to Earth which are not detectable because they initialize no air-shower. Due to the broad-band VHE spectrum that shows no intrinsic absorption lines in this region these spectral signatures should be measurable or could be used to constrain $\chi$ and $\mu$. The data and the expected spectral signature are shown in Fig.\[fig:sig\].
The differential spectra are measured as functions of energy-bins (see e.g. [@Aharonian:2004]), where $\overline{E}$ denotes the geometric mean energy of a bin with width $\Delta E$. Thus, one has to average (at the Crab distance $d_{\textnormal{c}}$) over the bin-size of the energy-bin centered on $\overline{E}$, giving $$\overline{P}_{\gamma \rightarrow \gamma_s}(\overline{E}, \Delta E) = \frac{1}{\Delta E} \int\limits_{\Delta E(\overline{E})} \mathrm{d}E \, P_{\gamma \rightarrow \gamma_s}(E, d_{\textnormal{c}})$$ Therefore, the predicted energy-flux observed is $$\label{signature}
\left. y(\overline{E}, \Delta E) = (\nu f_{\nu})\right\vert_{E=\overline{E}} \cdot (1-\overline{P}_{\gamma \rightarrow \gamma_s}(\overline{E}, \Delta E))$$ for the parameters $\chi$ and $\mu$, where the inverse Compton flux $\nu f_{\nu}$ from was used.
Hence one has to fit the spectral signature to the data. This can be done applying a goodness-of-fit test, here the method of least squares is used (see e.g. [@Yao:2006]). In our case, $\chi^2_{\textnormal{lsq}}$ is given by the expression $$\label{chisq}
\chi^2_{\textnormal{lsq}} = \sum\limits_{i=1}^N \left( \frac{y_i - y(E_i, \Delta E_i, \chi, \mu)}{\sigma_i} \right)^2$$ where the sum runs over all data points, given by $(E_i,y_i,\sigma_i)$. $y_i$ stands for the measured energy-flux at energy $E_i$ with a statistical error $\sigma_i$. The method of least squares can be applied considering arbitrary confidence levels. The following results are calculated using the $68.3 \% $ confidence level which is constrained by (for one fit parameter).
Solving this numerically under the conditions explained above for every allowed mass $\mu$ ($\lesssim 10^{-5} \, \textnormal{eV}$, see above) one gets a fit value $\chi_{\textnormal{fit}}$ for $\chi$ for every $\mu$ ($\chi_{\textnormal{fit}}$ depends on the desired confidence level). Interpreting this in the disappearance approach it is clear that one can exclude all values $\chi \geq \chi_{\textnormal{fit}}$.
Results
=======
Clearly, comparing the data of the different telescopes (see Fig.\[fig:sig\]) it is worth mentioning that the results of the experiments differ among each other. Physically and within the experimental errors, all experiments must measure the same flux at a constant energy $E$. We choose to renormalize the energy scale of the instruments within their respective systematic uncertainties to avoid smearing of signatures. The scaling factor can be found by a fit of on every data set.
---------- ------- -------- ----------
HEGRA 1.000 0.0346 1.43(15)
H.E.S.S. 0.921 0.0800 4.05(7)
MAGIC 0.968 0.1165 0.82(9)
Whipple 0.944 0.1036 1.21(7)
combined - 0.0343 1.60(41)
---------- ------- -------- ----------
\[tab:res\]
For energies above $10\,\textnormal{TeV}$, the re-scaled spectra of H.E.S.S. and HEGRA still deviate from each other. Since the H.E.S.S. data are not consistent with the model considered (see ), we choose to ignore the data for energies above 10TeV.
Applying the method described above and using the HEGRA, H.E.S.S., MAGIC, and Whipple data we can conclude with a value $\chi_{\textnormal{fit}} \simeq 0.0343$ (the result converges to this value with increasing mass). The result is shown in Fig.\[fig:res\]. Results for separate experiments can be found in . To test the goodness of the fit, the minimum of the reduced chi-squared value, defined by , is given in the table for a specific mass $\mu$, where $n$ is the number of degrees of freedom.
Due to experimental errors on the distance $d_{\textnormal{c}}$ and the energy $E_i$ of each data point, the error on the mass $\mu$ of every point excluded from the mass-mixing plane can be approximated to be 9%.
Conclusions
===========
Comparing our results to the constraints given in [@Ahlers:2008], Fig.1, the limits obtained from measuring deviations from the Coulomb law are better. But the bounds given here are the best constraints on the hidden photon parameters using oscillation effects of photons directly. For the first time, we got astronomical limits considering new energy and distance ranges.
We would like to acknowledge J.Grube for sending us the data, published in their proceedings ([@Grube:2007]). would like to acknowledge the ”Bundesministerium für Bildung und Forschung“ (BMBF) for making the participation on the conference possible.
[^1]: In case of direct renormalizable couplings of the corresponding gauge boson to standard model matter precise measurements of the electroweak theory have shown that their masses must exceed a few hundred GeV *[@Cavity:2008]*.
[^2]: The masses typically considered belong to the sub-eV range.
[^3]: Of course theories with more than one additional $U(1)$ gauge symmetry exist, but they would be more cumbersome to handle [@Okun:1982].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'For a model 1d asymmetric double-well potential we calculated so-called survival probability (i.e. the probability for a particle initially localized in one well to remain there). We use a semiclassical (WKB) solution of the Schroedinger equation. It is shown that behavior essentially depends on transition probability, and on one dimensionless parameter $\Lambda $ which is a ratio of characteristic frequencies for low energy non-linear in-well oscillations and inter wells tunneling. For the potential describing a finite motion (double-well) one has always a regular behavior. For $\Lambda \ll 1$ there is well defined resonance pairs of levels and the survival probability has coherent oscillations related to resonance splitting. However for $\Lambda \gg 1$ no oscillations at all for the survival probability, and there is almost an exponential decay with the characteristic time determined by Fermi golden rule. In this case one may not restrict oneself to only resonance pair levels. The number of perturbed by tunneling levels grows proportionally to $\sqrt \Lambda $ (by other words instead of isolated pairs there appear the resonance regions containing the sets of strongly coupled levels). In the region of intermediate values of $\Lambda $ one has a crossover between both limiting cases, namely the exponential decay with subsequent long period recurrent behavior.'
address: |
$^1$ Institute of Problems of Chemical Physics, RAS\
142432 Moscow Region, Chernogolovka, Russia;\
$^2$ Lab. Spectrometrie Physique, UJF\
BP 87, St. Martin d’Heres, Cedex, France;\
$^3$ Laue-Langevin Institute, F-38042, Grenoble, France.
author:
- 'V.A.Benderskii $^{1 , 2}$ , E. I. Kats $^{3}$ [^1]'
title: 'Coherent oscillations and incoherent tunneling in one - dimensional asymmetric double - well potential'
---
Introduction
============
Double level systems and models appear in various contexts in physics, chemistry and biology. The recurrent interest to the topic is related mainly with fairly rich and interesting physics of the systems, and with the experimental activity on several classes of systems which can be viewed as good physical realization of double level models (including fashionable quantum dots, see e.g. [@YF99]). Among the possible types of behavior we will particularly be concerned with coherent oscillations and incoherent (dissipative like) tunneling. Our goal is to propose a simple mathematical model to illustrate crossover from coherent oscillations to dissipative tunneling (decay or relaxation), which are also related to incoherent transitions in multidimensional oscillator systems. In a certain sense this crossover reveals many features of chaotic behavior. It is a common wisdom now that classical chaos is defined as extreme complexity of the trajectories in phase space, with the trajectories being very sensitive to small changes in the initial conditions [@ZC79], [@LL83]. As well evidently that the state vector (wave function) of a closed quantum system strictly speaking does not exhibit chaotic motion, as a consequence of the unitary nature of time evolution. But in fact since in quantum mechanics trajectories in the phase space can not be introduced due to Heisenberg uncertainty principle, the standard classical concept of the stability becomes ambiguous (see e.g.[@WJ82], [@BE1], [@BE2], [@NA], [@KH00]).
We put forward a simple (but yet non-trivial) model of 1d asymmetric double-well potential which can be used to describe under relatively weak assumptions a crossover from coherent oscillations (say mechanical behavior) to incoherent decay or dissipative tunneling (say ergodic behavior). The essential part of the model we will present is to illustrate this semiclassical quasi-chaotic behavior. In fact the illustration was made long ago by Fermi, Pasta, and Ulam [@FPU]. They performed computer studies of energy sharing and ergodicity for weakly coupled systems of $N$ oscillators. Later on, the results of [@FPU] were confirmed and refined (see e.g. [@JA63], [@FW63]). But all these papers were devoted to systems with many degrees of freedom ($N \gg 1$ dimensional phase space) for the cases where the motion is nearly integrable and irregular in different energy regions. Level statistics for such kind of mixed systems (i.e. when behavior is regular and chaotic in different phase space regions) changes gradually from Poisson to Wigner type of distributions [@BR84], [@CG86], [@ZM86]. Thus these systems become non-integrable when the energy exceeds a certain critical value. Just on the contrary we will propose and investigate in 1d a conservative system with time independent Hamiltonian which is evidently always integrable, and it does not generate classical chaos.
For the sake of completeness let us note that the tunneling in the mixed (i.e. regular-chaotic) systems has been studied as well for two level systems when one of the levels interacts with a chaotic state [@TU94], [@SW01] (see also review [@BT93] and references therein). In the case of a resonance between the tunneling doublet and suitable chaotic states, the tunneling is enhanced (so-called chaos assisted tunneling) and has very strong resonance dependence on quantum numbers. Similar effects due to transverse vibrations take place for isolated Fermi resonances in tunneling systems [@BV99].
Our paper has the following structure. Section II contains basic equations necessary for our investigation. Section III is devoted to the calculation of so-called survival probability. We use the semiclassical approach [@LL] (see also [@b1] and references herein). The last section IV contains the summary. The appendix to our paper is devoted to the technical and methodical details of the calculations.
Asymmetric 1d double-well potential
===================================
The simple model studied in this paper consists of a quantum particle in one dimensional asymmetric double-well potential $U(X)$ with one-parameter dependent shape. Using the tunneling distance $a_0$ and the characteristic frequency of the oscillations around the left minimum $\Omega _0$, we can introduced the so-called semiclassical parameter $\gamma \equiv m\Omega _0 a_0^2/\hbar \gg 1 $ ($m$ is a mass of a particle, and further we will set $\hbar = 1$ measuring energies in the units of frequency), which is assumed to be sufficiently large, i.e. the tunneling matrix element should be small in $\Omega _0$ scale. The choice of the model potential is dictated by the principle of minimal requirements. Our aim is to describe in the frame work of one universal model crossover from symmetric double-well potential to so-called decay potential, and to do it we need a parameter to make the right well ($R$-well) deeper and wider than the left well ($L$-well).
Using $\Omega _0$ and $a_0$ to set corresponding scales, the model potential satisfying these minimal requirements can be written in the following dimensionless form $$\begin{aligned}
\label{b1}
V(x) = \frac{1}{2}x^2(1-x)\left [1 + \frac{1}{b^2}x\right ] \, ,\end{aligned}$$ where $V \equiv U/(\Omega _0 \gamma )$, and $x \equiv X/a_0$. The dimensionless parameter $b$ allows us to change the shape of the right well ($R$-well), and to consider both limiting cases, namely a traditional symmetric double-well potential (for $b =1$), and for $b \to \infty $ a decay potential (or by other words to change the level spacings from $\Omega _0^{-1}$ scale to zero ). In fact it can be shown (see below and the Appendix to the paper) that qualitatively all our results do not depend on the concrete form of the one parametric potential satisfying these requirements (only on the density of $R$-states). Behavior in both limiting cases are well known, and for $b=1$ one has coherent quantum oscillations, typical for any two-level systems, while for $b \to \infty $ there is a continuum spectrum of eigen states for $x \to +\infty $ and one can find an ergodic behavior (incoherent decay). Our main goal in this section is to study crossover between both limits at variations of $b$.
The general procedure for searching semiclassical solutions of the Schroedinger equation with the model potential (\[b1\]) has a tricky point. The fact is that in the $L$-well we have a discrete eigenvalue spectrum (stationary states) while for the $R$-well in the case $b \gg 1$ we have quasi-stationary states, which are characterized by wave functions $\Psi _n(X)$ exponentially increased in the region of $\varepsilon \gg V(X)$. Both kind of states are defined on different sheets of complex energetic surfaces [@LL], and to treat both kind a states one should use different tools, namely, the standard quantization of the stationary states from the discrete part of the spectra [@LL], and proposed long ago by Zeldovich [@zeld] for quasi-stationary states the flux probability conservation law, which leads to the Lorentzian envelope for spectral distribution functions. Unlike [@zeld] in our case we get the Lorentzian envelope filled by $\delta $-peaks of the final states.
The procedure is described in the Appendix, and it includes three steps (see [@LL], [@zeld], and we will use notations from [@b1]):
- First one should find the action $W_L$ in the classically allowed region (i.e.$W_L$ between turning points) in the left well ($L$-well), and apply the semiclassical quantization. For the low energy states in the $L$-well it leads to the following relation $$\begin{aligned}
\label{b2}
\gamma W_L = \pi \left [n + \frac{1}{2} + \chi _n \right ] \equiv \pi \varepsilon _n \, ,\end{aligned}$$ where, integer numbers $n$ numerate eigenvalues, $\chi _n$ is determined by an exponentially small phase shift, and the last r.h.s. of (\[b2\]) is in fact the definition for eigenvalues $\varepsilon _n$.
- Second, the same should be done for the right well ($R$-well). The calculation is almost trivial in the limit $b \gg 1$ (when the potential (\[b1\]) becomes strongly asymmetric) $$\begin{aligned}
\label{b3}
\gamma W_R = \gamma W_R^{(0)} + \pi \beta \varepsilon
\, ,\end{aligned}$$ where the dimensionless energy $\varepsilon $ is counted from the bottom of the $L$-well, the action $W_R^{(0)}$ is $$\begin{aligned}
\label{b4}
\gamma W_R^{(0)} = \frac{\pi }{16 b}(b^2 - 1)^2(b^2 + 1)
\, ,\end{aligned}$$ and $$\begin{aligned}
\label{b5}
\beta = \frac{b^2 +1}{b} \simeq b \, , \, {\rm for} \quad b \gg 1
\, ,\end{aligned}$$ Note that the parameter $\beta = 2\Omega _0/\omega _R $ is proportional to the density of states in the $R$-well ($\omega _R$ is the frequency of non-linear oscillations in $R$-well at $\varepsilon = 0$), and therefore knowing the magnitude $\beta $ one can compute the density of states in the $R$-well, which grows proportional to $b$ for $b \gg 1$. It is convenient to rewrite (\[b3\]) - (\[b4\]) in the same form as (\[b1\]) $$\begin{aligned}
\label{b6}
\gamma W_R = \pi \left [n_R + \frac{1}{2} + \alpha _n + \beta \chi \right ]
\, ,\end{aligned}$$ where $n_R$ and $\alpha _n$ are integer and correspondingly fractional parts of the quantity $$\begin{aligned}
\label{b7}
\frac{\gamma W_R^{(0)}}{\pi} + \beta \left (n + \frac{1}{2} \right ) - \frac{1}{2}
\, .\end{aligned}$$ The physical meaning of $\alpha _n$ is the deviation from a resonance between the $n$-th level in the $L$-well and the nearest level in the $R$-well. By the definition of a fractional part $|\alpha _n| < 1/2\beta $.
- And as the last step, again using the quantization rule, one can find the spectrum.
It turns out (see Appendix) that the spectrum and the behavior of the system depends crucially on the parameter $\Lambda \equiv \beta R_n$, where $$\begin{aligned}
\label{b8}
R_n = \frac{2^{n+2} \gamma ^{n +1/2}}{\pi ^{1/2} n!} \exp (- 2 \gamma W_B)\end{aligned}$$ is the $\beta $ independent decay rate of the $n$-th metastable state of the $L$-well at $b \to \infty $ ($W_B$ is the action in the classically forbidden (between turning points) region).
For $\Lambda \ll 1$ solving the quantization relation (\[A1\]), one can easily find $$\begin{aligned}
\label{b9}
\varepsilon _{n \pm} = n + \frac{1}{2} \pm \frac{1}{2 \beta } \left [\sqrt {\alpha _n^2 + \frac{4}{\pi ^2}
\Lambda } - \alpha _n \right ] \,
.\end{aligned}$$ This expression (\[b9\]) determines the resonance pairs of the levels, so-called two-level systems.
Besides from the same quantization rule (\[A1\]) we get analytically (i.e. for arbitrary values of $\Lambda $) eigenvalues for the $R$-well in the vicinity of the resonance doublet $$\begin{aligned}
\label{b10}
\varepsilon _{n m} = n + \frac{1}{2} + \frac{1}{2 \beta } \left [\sqrt {(m - \alpha _n)^2 + \frac{4}{\pi ^2}
\Lambda } - (m - \alpha _n)\right ] \quad m = \pm 1 , \pm 2 , .... \end{aligned}$$ These levels are numerated by the quantum number $m$.
For $\Lambda \ll 1$ all displacements of the levels due to tunneling are small, and two-level system approximation is valid (i.e. there is well defined isolated resonance pairs of levels with splitting $\propto (R_n
/\beta )^{1/2}$). The situation becomes completely different for $\Lambda \geq 1$. In the limit $\Lambda \gg 1$ we get almost equidistant spectrum of mixed $L - R$ levels in the vicinity of the following values of $\chi $ (see Appendix for the details) $$\begin{aligned}
\label{b11}
\chi \equiv \chi _{n m} = \pm \frac{m + 1/2 - \alpha _n}{\beta }\left
[1 + \frac{1}{\pi \Lambda }\right ]
\, .\end{aligned}$$
The given above expressions (\[b10\]) - (\[b11\]) show that the number of the perturbed by tunneling levels grows proportionally to $\sqrt \Lambda $. In Fig. 1 we have shown the displacements of the levels perturbed by tunneling. These displacements are decreased very rapidly for the levels with quantum numbers larger than $\sqrt \Lambda $. The scales in this figure are given by the semiclassical parameter $\gamma $ which relates to the $L$-well and the barrier. Once the scales are fixed the $R$-well is characterized by the eigenfrequency $\propto 1/b$ at $\varepsilon = 0$ (or what is the same by the density of states or by the action $W_R$ in the $R$-well).
Summarizing the results of this section, thus we have shown that instead of isolated two level systems taking place for $\Lambda \ll 1$, in the opposite limit $\Lambda \gg 1$ there appear the resonance regions containing the sets of strongly coupled levels. The resonance widths are determined by tunneling matrix elements ($H_{1 2}^2 = \omega _L \omega _R \exp (- 2\gamma
W_B)/4 \pi ^2
= R_n/\beta $). In spite of the fact that for any finite values of $\Lambda $ (and $b$) we have only the discrete spectrum of real eigenstates, found above mixing of $L - R$ states very closely resembles the representation of quasi-stationary states in terms of eigenstates of a continuous spectrum. This behavior can be formulated by other words in terms of the so-called recurrence time, i.e. the characteristic time when the system is returned to the initial state. For a finite motion (i.e. for a finite value of $b$) the behavior of the system remains regular. The recurrence time (i.e. in the case merely coherent oscillation period) is proportional to $1/H_{1 2}$ for $\Lambda \ll 1$, while for $\Lambda \gg 1$ this time scales as $1/\omega _R$ (as a long-period time scale).
Survival probability
====================
The tunneling dynamics can be characterized by the time evolution of the initially prepared localized state $\Psi (0)$, and by the definition the survival probability of the state is $$\begin{aligned}
\label{b12}
P(t) \equiv |\langle \Psi (0) | \Psi (t) \rangle |^2
\, .\end{aligned}$$ For the stationary states evidently $P(t) = 1$, while for quasi-stationary (decaying states), the survival probability reads $$\begin{aligned}
\label{b13}
P(t) = \exp (- \Gamma t)
\, ,\end{aligned}$$ where $\Gamma $ is the decay rate which should be found, and we use $\omega _R^{-1}$ for the time scale.
The simplest case is the coherent tunneling dynamics of two-level states. Let us consider the $ n - n^\prime $ resonance region. The eigenfunctions of isolated $R$ and $L$ wells, $\Psi _n^L$, and $\Psi _{n^\prime }^R$. If one has the initial state $$\Psi (0) = \Psi _n^L \, ,$$ the survival probability can be easily calculated $$\begin{aligned}
\label{b14}
P(t) = \frac{1}{2}\left [1 + cos\left (2t\sqrt {\frac{R_n}{\beta }}\right )\right ]
\, .\end{aligned}$$
Normalized wave functions in the $L$-well can be calculated trivially, and using standard semiclassical wave functions for the $R$-well, we are in a position to compute the survival probability for a general case as a function of $\Lambda $. The results are shown in Fig. 2.
For $\Lambda \ll 1$, $P(t)$ oscillates with characteristic time scales proportional to $H_{12}^{-1} = \sqrt {\beta /R_n}$. In the region $\Lambda \simeq 1$ these oscillations are strongly suppressed. The reason for the suppression of oscillations is related to interference of the states with energies in the resonance region. As a result of the interference the total probability for the system to return back from the $R$-well is decreased, and low-frequency modulation of coherent tunneling is raised. The period of the modulation grows with $\beta $, and in the limit $\Lambda \gg 1$ we get the dense spectrum of states in the $R$-well, and almost exponential decay for $P(t)$ with $\beta $-independent relaxational time $\tau \propto R_n^{-1}$. In this case the survival probability (i.e. the probability to keep the system in its initial state) for the time interval $ \ll 1/\omega _R$ decay almost exponentially with time, and the characteristic relaxation time $\tau $ is determined by Fermi golden rule, i.e. $\tau ^{-1} \propto H_{1 2}^2/\omega _R$. This result is also conformed to van Hove statement [@HO55] concerning quasi-chaotic behavior of semi-classical systems at time scales of the order of $\omega _R/H_{1 2}^2$.
We can relate the phenomenom described above (i.e. almost vanishing probability for back-flow from the $R$ to $L$ well) to the Fermi golden rule for a transition probability $$\begin{aligned}
\label{b15}
W_{fi} = 2\pi |H_{f i}|^2 \rho _f
\, ,\end{aligned}$$ where $H_{i f}$ is the matrix element between the initial state $E_i$ and the final state $E_f$, and $\rho _f$ is the density of final states. For our case ($H_{i f} \equiv H_{12} = \sqrt {R_n/\beta }$, and $\rho _f = \beta /2$) we get easily $$W_{i f} = \pi R_n \, ,$$ which does not depend on $\rho _f$. Therefore the Fermi golden rule corresponds to the limit when the back flow from the $R$-well is totally suppressed due to the interference.
The survival probability can be related also to spectral distribution of the initially localized in the $L$-well states. Indeed, by the definition of the spectral distribution $S(E)$ of the initially prepared localized state is determined by the transition amplitudes in expansion over the eigenstates $(\Psi _n , E_n)$: $$\begin{aligned}
\label{b16}
S(E) = \sum _{n}|\langle \Psi (0) | \Psi _n \rangle |^2 \delta (E - E_n)
\, ,\end{aligned}$$ and therefore $$\begin{aligned}
\label{b17}
\langle \Psi (0) | \Psi (t) \rangle = \int _{-\infty }^{+\infty } S(E) \exp (- i E t) dE
\, .\end{aligned}$$ For $\Psi (0) \equiv \Psi _i^L$ the spectral distribution is a set of $\delta $-peaks with Lorentzian envelope $$\begin{aligned}
\label{b18}
S(E) = \frac{2}{\pi } \frac{\sqrt {R_i\beta }}{\beta (E - E_i)^2 + R_i}\delta (E - E_i)
\, .\end{aligned}$$ Crossover from the coherent oscillations to exponential decay occurs when the Lorentzian envelope begins to fill up by $\delta $-peaks of the final states. Note that the width of the Lorentzian envelope (\[b18\]) does not depend on the final state density (see Appendix and also [@zeld]). We have shown the results of the calculation of the spectral distribution in Fig. 3.
Conclusion
==========
Let us sum up the results of our paper. We investigated the behavior of a quantum particle in 1d asymmetric double-well potential with one parameter dependent shape, which allows us to consider in the frame work of one universal model the crossover from the traditional symmetric double well potential to the decay one. We have shown that behavior essentially depends on transition probability, and on dimensionless parameter $\Lambda $ which is a ratio of characteristic frequencies for low energy non-linear in-well oscillations and inter wells tunneling. For the potential describing a finite motion (double-well) strictly speaking one has always a regular behavior. For $\Lambda \ll 1$ there is well defined resonance pairs of levels and the survival probability has coherent oscillations related to resonance splitting. However for $\Lambda \gg 1$ there are no oscillations at all for the survival probability, and there is almost an exponential decay with the characteristic time determined by Fermi golden rule. In this case one may not restrict oneself to only resonance pair levels. The number of perturbed by tunneling levels grows proportionally to $\sqrt \Lambda $ (by other words instead of isolated pairs there appear the resonance regions containing the sets of strongly coupled levels). In the region of intermediate values of $\Lambda $ one has a crossover between both limiting cases, namely the exponential decay with subsequent long period recurrent behavior.
However a number of remarks related to our results are in order. Many features often classified as evidences of quantum chaos in fact as we have illustrated in our model can occur for well defined states possessing only discrete energy levels. The deviation from two level system behavior, taking place for $\Lambda \gg 1$ has nothing to do with random or chaotic properties of the system. It means only that due to well known phenomenom of level repulsion the two level approximation is not adequate. Lorentzian envelope (see Fig. 3) we found arises from the interaction of a single level in $L$-well with a set of levels in the $R$-well and not with appearance of level widths (imaginary self-energy contributions).
One should distinguish between short-time and long-time behavior, and the boundary between them depends on the parameter $\Lambda $. Short-time returns ($\propto \beta $) are governed by one or a small number of semiclassical paths, while long-time returns ($\propto R_n^{-1}$) arise from interference between many paths. In the limit $\Lambda \ll 1$, exponential decay occurs for short-time dynamics, while the system remains regular for long-time scales, in contrast with chaotic models we discussed in the Introduction. Nevertheless the tunneling in the limit of $\Lambda \gg 1$ can induce vibrational relaxation for localized $R$-levels. The relaxation appears due to tunneling recurrences, and results in redistribution of initial energy over all levels coupled with a single $L$-level.
Main physical idea of our paper, namely that specific quasi-chaotic behavior is associated with the fact that one level in $L$- well in a certain condition ($\Lambda \gg 1$) is coupled to a set of almost dense levels in the $R$-well, was discussed in the literature long ago [@HO55] (see also [@zeld]), mainly qualitatively. Our achievement is that we alone seem to have propose the concrete and tractable analytical model to illustrate and to investigate explicitely and quantitatively this statement.
In this respect our results are quite different from numerical investigations of billiard-type systems (see e.g. review article [@BT93]), showing universal behavior of level spacings in finite chaotic systems. Our results (for the totally integrable 1d model) demonstrate that level spacing distribution is not a specific feature of quantum systems with chaotic classical counterpart limit. Our finding of the equidistant regular level distribution is a result of the interaction of the single $L$-level with several (of the order of 10 for our particular choice of the parameters) $R$-levels (which in own turn are regular ones). As well we should distinguish our model and dynamic tunneling ones [@WI88], [@FD98]. The latter assumed strong coupling of the tunneling system with an environment which destroys the coherence, whereas in our model the coherence is destroyed by the tunneling itself due to the high density of $R$-states, breaking two level approximation.
Note also at the very end of the paper that results presented here not only interested in their own right (at least in our opinion) but they might be directly tested experimentally since there are many molecular systems where investigated in the paper 1d asymmetric potential is a reasonable model for the reality. And not only molecular systems, for instance recently as a controllable two-level system, double quantum dots are proposed for realizing a single quantum bit in solid state systems. Experimentally [@YF99] in these systems there observed two distinct regimes characterizing the nature of low-energy dynamics:
\(i) relaxational regime, when an excited-state electron population decays exponentially in time with a rate correctly given by Fermi golden rule;
\(ii) vibronic regime, when the population oscillates for some number of cycles before decaying.
And what’s more, at short times the averaged excited-state populations oscillates but has a decaying envelope. The similarity with the behavior we found in the paper is evident.[^2]
The research described in this publication was made possible in part by RFFR Grants 97-03-33687a and 00-02-11785. The numerical results origin from a collaboration with E.V.Vetoshkin whose contribution is greatly acknowledged. Also we express our sincere gratitude to the referee who read our manuscript and made a number of valuable comments.
{#section .unnumbered}
The semi-classical wave function is represented in the well known WKB form [^3] $$\Psi = \exp \left (i W\right ) \, ,$$ The action $W$ should satisfy to WKB equation $$\begin{aligned}
\label{a1}
\frac{1}{2} \left (\frac{d W}{d X}\right )^2 = \frac{\varepsilon }{\gamma } - V(X) \, ,\end{aligned}$$ and two turning points, which are boundaries of classically allowed regions, are situated near zeros of $V(X) - \varepsilon /\gamma $.
For the asymmetric double-well potential (\[b1\]) the Bohr - Sommerfeld [@LL] quantization equations read $$\begin{aligned}
\label{A1}
tg (\gamma W_L)tg (\gamma W_R) = 4 \exp(2\gamma W_B) \, ,\end{aligned}$$ where $W_B$ is the action in the classically forbidden region in between the turning points $X_1 , X_2$ in the left and right wells, and $W_{L , R}$ are the coordinate independent actions in the classically allowed regions inside of the $L$ (respectively $R$) well. Using the following expansion $$\tan z = \sum _{m=0}^{\infty } 2 z\left [z^2 - \pi ^2\left (m +\frac{1}{2}^2\right )\right ]^{-1} \, ,$$ one gets the almost equidistant spectrum of the mixed $L-R$ levels, and in this condition the solution of (\[A1\]) leads to the expressions (\[b9\]), (\[b10\]) presented in the main text of the paper.
The time evolution of any initially prepared state can be described by a superposition of the eigenfunctions of the discrete and continuous spectra with time dependent phases. For the potential (\[b1\]) with $b \gg 1$ the initial finite motion, i.e. the initial density distribution $$\begin{aligned}
\label{n1}
\rho (t) = \int _{X_1}^{X_2} |\Psi (X , t)|^2 d X \,\end{aligned}$$ concentrated in the $L$-well at $t=0$ decreases exponentially with time $$\begin{aligned}
\label{n2}
\rho (t) = \rho (0) \exp \left (-\eta t\right ) \, .\end{aligned}$$ Eq. (\[n2\]) signifies that the wave functions of quasi-stationary states have the form $$\begin{aligned}
\label{n3}
\Psi _n(X ,t) = \Psi _n(X) \exp \left (\left (-i \varepsilon _n - \eta _n/2 \right )t\right ) \, ,\end{aligned}$$ and the eigenvalues are complex and lies on the lower half-space of $(\varepsilon , \eta )$ plane. The quantization of the stationary states of a discrete spectrum is performed by the requirement [@LL] $$|\Psi (X , t)|^2 \to 0 \, , \, {\rm at} \, |X| \to \infty \, .$$ This condition is impossible to impose to quasi-stationary states, since the wave functions $\Psi _n(X)$ exponentially is increased in the region of $\varepsilon \gg V(X)$. The physically meaningful boundary condition as was noted first by Zeldovich [@zeld] for quasi-stationary states can be written as a conservation law for the flux probability from the $L$-well through the barrier. The difference between stationary and quasi-stationary states disappears as it should be at $\eta \to 0$.
The expansion of the initially quasi-stationary state is dominated by the continuum spectrum eigenfunctions with the energies close to the real parts of the eigenvalues $\varepsilon _n$. These eigenfunctions have the form $$\begin{aligned}
\label{n4}
\Psi _k(X) = \left (
\begin{array}{cc}
A(k)\phi _k^0 (X) \, , \quad X < X_m \\
\sqrt {\frac{2}{\pi }} \sin(k X + \delta (k)) \, , \quad X > X_m
\end{array}
\right ) \, ,\end{aligned}$$ where $X_m$ is the left turning point of the $R$-well, the localized wave function $\phi _k^0$ is normalized to unity, and the phase is given $$\begin{aligned}
\label{n5}
\delta (k) = \delta _0 - \arctan \frac{k_2}{k-k_1} \, ,\end{aligned}$$ and $\delta _0$ is $k$-independent component, $k_1 = \sqrt {2 m \varepsilon _n} $, $k_2 = k_1\eta _n/4\varepsilon _n$. For the eigenfunctions with the energies $\varepsilon $ and $\varepsilon ^\prime $ close to $\varepsilon _n$ we get $$\begin{aligned}
\label{n6}
\int _{-\infty }^{X} \phi _k(X^\prime )\phi _{k^\prime }(X^\prime ) d X^\prime =
\frac{1}{2 m}\left (\frac{1}{\varepsilon - \varepsilon ^\prime }\right )\left (
\phi _k^\prime \frac{d\phi _k}{dX} - \phi _k^\prime \frac{d \phi _k^\prime }{d X}\right )
\, .\end{aligned}$$ From (\[n4\]), (\[n5\]), and (\[n6\]) in the limit $\varepsilon - \varepsilon ^\prime \to 0$ we get $$\begin{aligned}
\label{n7}
A^2(k) = \frac{2}{\pi } \sqrt {\frac{2 \varepsilon _n}{m}} \frac{\eta _n}{4(\varepsilon - \varepsilon
_n)^2 + \eta _n^2} \, .\end{aligned}$$ Expressions (\[n5\]), (\[n7\]) are valid for a continuous spectrum, for discrete levels the phase shift as well is governed by the probability flux from the $R$-well into classically forbidden region, and instead of (\[n5\]) it leads $$\begin{aligned}
\label{n9}
\delta = \arctan \sqrt{{R_n}{\beta }}\frac{1}{\varepsilon _n - \varepsilon
_{nm}} \, ,\end{aligned}$$ and instead of (\[n7\]) one can easily find $$\begin{aligned}
\label{n10}
A^2(\varepsilon _{nm}) = \frac{2}{\pi }\frac{\sqrt
{R_n}}{\beta (\varepsilon _n - \varepsilon _{nm})^2 + R_n}
\, , \end{aligned}$$ Note that (\[n10\]) has almost the same form as (\[n7\]), although it depends on discrete energy levels, and besides it has a different coefficient due to different normalization condition.
The relation (\[n7\]) shows that the probability density of the continuous spectrum eigenstates exhibits the Lorentzian distribution around the real part of the quasi-stationary eigenvalues $\varepsilon _n$. Expressions (\[n7\]) -(\[n10\]) are equivalent to the spectral distribution (\[b18\]) presented in the main body of the paper.
Few words concerning numerical results presented in the main text in the figures 1 - 3. The calculations have been performed to check:
\(i) semiclassical approximation for the model potential (\[b1\]);
\(ii) the spectral distribution (\[b18\]).
We used the numerical diagonalization of the Hamiltonian matrix in the basis set of trial functions, which includes: so-called instanton wave functions of the $L$-well (see [@b1]), and the WKB functions of $R$-well. This basis was orthonormalized by using standard Schmidt method [@morse]. For the $L$-well highly excited states near the barrier top have been also included. In all numerical calculations we set the value of $\alpha _0$ (so-called defect of a resonance) as zero. All results presented on the figures do not depend on this particular choice.
The numerical results confirm that Eq. (\[b18\]) is quiet accurate in the whole range of $\Lambda $ where the transition from coherent oscillations to exponential decay occurs. Note that since $R$-levels with the negative energy are not mixed with $L$-levels, and besides the resonance region is sufficiently narrow ($R_n =0.01$), we need not diagonalize huge matrices. For our purposes the diagonalization of the matrix $3000 \times 3000$ is more than sufficient to find eigenvalues in the resonance region around the $n=0$ $L$-level.
H.S.Yang, S.P.Feofilov, D.K.Williams, et al., Physica B, [**263**]{}, 476 (1999). G.M.Zaslavsky, Phys. Repts., [**80**]{}, 157 (1980). A.J.Lichtenberg, M.A.Lieberman, Regular and Stochastic Motion, Springer, New York (1983). Y.Weissman, J.Jortner, J. Chem. Phys., [**77**]{}, 1469 (1982); ibid., 1486. M.V.Berry, in Chaotic Behavior of Deterministic Systems, ed. by G.Iooss, R.Helleman, R.Stora, North-Holland, Amsterdam (1983). M.V.Berry, Proc. Royal .Soc., A, [**423**]{}, 219 (1989). K.Nakamura, Quantum Chaos, Kluver, Dordrecht (1997). L.Kaplan, E.J.Heller, Phys. Rev. E, [**62**]{}, 409 (2000). E.Fermi, J.Pasta, S.Ulam, Los Alamos Scientific Laboratory Report, LA-1940 (1955). E.A.Jackson, J. of Math. Phys., [**4**]{}, 686 (1963). J.Ford, J.Waters, J. of Math. Phys., [**4**]{}, 1293 (1963). M.V.Berry, M.Robnik, J. Phys. A, [**17**]{}, 2413 (1984). E.Caurier, B.Grammaticos, Europhys. Lett., [**2**]{}, 417 (1986). T.Zimmermann, H.D.Meyer, H.Koppel, L.S.Cederbaum, Phys. Rev. A, [**33**]{}, 4334 (1986). S.Tomsovic, D.Ullmo, Phys. Rev. E., [**50**]{}, 145 (1994). D.A.Steck, W.H.Windell, M.G.Raizen, Science, [**293**]{}, 274 (2001). O.Bohigas, S.Tomsovich, D.Ullmo, Phys. Repts., [**223**]{}, 43 (1993). V.A.Benderskii, E.V.Vetoshkin, H.P.Trommsdorf, Chem. Phys., [**244**]{}, 273 (1999). L.D.Landau, E.M.Lifshits, Quantum Mechanics (non-relativistic theory), Pergamon Press, New York (1965). V.A.Benderskii, E.V.Vetoshkin, Chem. Phys., [**257**]{}, 203 (2000). Ya.B.Zeldovich, JETP, [**12**]{}, 542 (1961). L.van Hove, Physica, [**21**]{}, 517 (1955). M.Wilkinson, J. Phys. A., [**21**]{}, 1173 (1988). S.D.Frischat, E.Doron, Phys. Rev. E., [**57**]{}, 1421 (1998). A.M.Polyakov, Nucl.Phys. B, [**129**]{}, 429 (1977). P.M.Morse, M.Feshbach, Methods of Theoretical Physics, Mc.Graw Hill, N.Y. (1953).
Figure Captions.
Fig. 1
The eigenvalues as functions of $\Lambda $ for the zero-point level ($n=0$) of the $L$-well. Dashed lines indicate the limits of $\Lambda \ll 1 $, and $\Lambda \gg
1$; $\gamma = 10$ , $\alpha _0 =0$.
Fig. 2
The survival probability for different values of $\Lambda $ and $\gamma = 10$ :
\(a) $\Lambda = 0.02$ , $b=5$ (solid line) ; $\Lambda = 0.5$ , $b=116$ (dashed line) ;
\(b) $\Lambda = 0.5$ , $b=116$ (solid line) ; $\Lambda = 4.0$ , $b=929$ (dashed line) ;
\(c) $\Lambda = 4.0$ , $b=929$ (solid line) ; $\Lambda = 16.0$ , $b=3715$ (dashed line) .
Fig. 3
The spectral distribution for different values of $\Lambda $ and $\gamma = 10 $:
\(a) $\Lambda = 0.02$ , $b=5$ ;
\(b) $\Lambda = 4.0$ , $b=929$ ;
\(c) $\Lambda = 20.0$ , $b=4644$ .
[^1]: Also, from L. D. Landau Institute for Theoretical Physics, RAS, 117940, Kosygina 2, Moscow, Russia.
[^2]: All characteristics of our model are not specific only for 1d case. For $\Lambda \gg 1$ one can expect similar behavior and for multidimensional systems.
[^3]: Equivalently it can be represented in the so-called instanton or minimum action tunneling path formalism [@pol] (see also [@b1]) in the form of $\Psi =
\exp (-\gamma W_E)$, which is more efficient for classically inaccessible parts of phase space.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We propose a cipher similar to the One Time Pad and McEliece cipher based on a subband coding scheme. The encoding process is an approximation to the One Time Pad encryption scheme. We present results of numerical experiments which suggest that a brute force attack to the proposed scheme does not result in all possible plaintexts, as the One Time Pad does, but still the brute force attack does not compromise the system. However, we demonstrate that the cipher is vulnerable to a chosen-plaintext attack.'
address:
- 'Department of Mathematics, University of Wyoming, Laramie, WY 82071-3036'
- 'Department of Mathematics, University of Wyoming, Laramie, WY 82071-3036'
- 'Department of Mathematics, University of Wyoming, Laramie, WY 82071-3036'
author:
- Ryan Harkins
- Eric Weber
- Andrew Westmeyer
title: Encryption Schemes using Finite Frames and Hadamard Arrays
---
Introduction
============
In this paper, we propose a private key cipher, the idea for which comes from frame theory and multiple access communications. The cipher has similarities to the Hill cipher, the One Time Pad, and the McEliece cipher [@HAC; @Cha]. Indeed, one of the design goals for our cipher is to approximate the One Time Pad.
Our design goals include the following:
1. Include randomness in the encryption process;
2. Require the key be shared only once;
3. Use a relatively small key size;
4. Computationally fast;
5. Robust to brute force attacks.
Our proposed cipher implements items 1-4 above; the purpose of the present paper is to give some demonstration of item 5. We remark here that 5 is not sufficient for the cipher to be a good one, but certainly is necessary. We will demonstrate that this cipher is vulnerable to a chosen-plaintext attack. It is unknown if this cipher is robust against a known-plaintext attack.
Our cipher can be described as follows: consider a communications channel; we divide the channel into two subbands, one which will carry the message, and the other which will carry noise, or as we call it in this paper, garbage. The message, along with the garbage is transmitted over the channel; the recipient then filters out the garbage, leaving only the message. This procedure is carried out using orthogonal frames. The procedure requires the construction of orthogonal frames; the easiest way to do this is using Fourier frames (also called harmonic frames). However, as will be described, these frames are not good for our purposes here, and so we present several alternative methods for constructing orthogonal frames.
The paper is organized as follows. In Section \[S:frames\], we give a short introduction to frames, and in particular orthogonal frames. In Section \[S:schemes\], we give an account of several methods for constructing orthogonal frames, with remarks regarding our design goals. In Section \[S:results\], we present the results and conclusions of our numerical experiments and the chosen-plaintext attack. In the Appendix, we provide psuedocode to describe the experiments.
Introduction to Frames {#S:frames}
======================
Frames for Hilbert spaces are being used in many signal processing applications such as sampling theory, multiple access communications, etc. Frames provide redundancy via overcompleteness, where bases do not, and it is this redundancy that makes them advantageous to use in these settings. In this paper, we will utilize this redundancy of frames for the purpose of encryption.
Let $H$ be a Hilbert space over the field ${\mathbb}{F}$ with scalar product $\langle \cdot , \cdot \rangle$ and norm $\| \cdot \|$, where ${\mathbb}{F}$ denotes either ${\mathbb}{R}$ or ${\mathbb}{C}$. A frame for $H$ is a sequence ${\mathbb}{X} := \{x_n\}_{n \in {\mathbb}{Z}}$ such that there exist constants $0 < A \leq B < \infty$ such that for all $v \in H$, $$\label{E:frame}
A \|v \|^2 \leq \sum_{n \in {\mathbb}{Z}} | \langle v, x_n \rangle |^2 \leq B \|v \|^2.$$ Clearly, a frame spans the Hilbert space. Moreover, $\{x_n\}$ defines the following *frame operator* $$S_{{\mathbb}{X}}: H \to H: v \mapsto \sum_{n \in {\mathbb}{Z}} \langle v , x_n \rangle x_n$$ which is positive and invertible. Define $\{\tilde{x}_n\} \subset H$, the *standard dual* of $\{x_n\}$ by $\tilde{x}_n := S_{{\mathbb}{X}}^{-1} x_n$, then for all $v \in H$, $$v = \sum_{n \in {\mathbb}{Z}} \langle v , x_n \rangle \tilde{x}_n = \sum_{n \in {\mathbb}{Z}} \langle v , \tilde{x}_n \rangle x_n.$$ If $A = B = 1$, the frame is said to be *Parseval*, and then for all $v \in H$, $$v = \sum_{n \in {\mathbb}{Z}} \langle v , x_n \rangle x_n.$$ For elementary frame theory, see [@HL; @Casazza].
If $H$ is finite dimensional ($H$ will always be assumed to be so from here on, unless specifically stated), then a frame sequence (possibly finite) is any spanning set $\{x_n\}$ such that $\sum_{n \in {\mathbb}{Z}} \| x_n \|^2 < \infty$. If only a finite number of $x_n$’s are non-zero, then $\{x_n\}$ is a finite frame, and we will discard those that are zero. See [@Casazza2; @Dykema; @Fickus] for more on finite frames.
For convenience of notation, we make the following definition.
An $n\times n$ real matrix, $M$, is an orthogonal matrix if $M^TM=kI_n$ for some constant $k$.
The (finite) Parseval frames in $H$ are characterized by the following proposition.
\[P:PF\] Let $\{x_n\}_{n=1}^{M} \subset H$, where $H$ has dimension $N$. The following are equivalent:
1. $\{x_n\}$ is a Parseval frame for $H$;
2. the $M \times N$ matrix whose $i$th row is $x_i$ (as a row vector) has columns which are orthonormal;
3. there exists a Hilbert space $K$ of dimension $M-N$ and vectors $\{y_n\}_{n =1}^{M} \subset K$ such that the $M \times M$ matrix formed by $$\left(
\begin{aligned}
&x_1 & &| & &y_1 \\
& \vdots & &| & & \vdots \\
&x_M & &| & &y_M
\end{aligned}
\right)$$ is a unitary matrix.
Here we write the vectors $x_i$ and $y_i$ as row vectors with respect to any orthonormal bases for $H$ and $K$, respectively.
The proof of the equivalence of 1 and 2 is in [@Fickus]. The proof of the equivalence of 1 and 2 is, for infinite frames, contained in [@HL Corollary 1.3, Theorem 1.7]. The case for finite frames is analogous.
Another way to view Proposition \[P:PF\] is that $\{x_n\}$ is a Parseval frame for $H$ if and only if $\{x_n\}$ is the inner direct summand of an orthonormal basis $\{x_n \oplus y_n\}$ for some superspace $H \oplus K$ of $H$.
Two frames $\{x_n\}_{n=1}^{M} \subset H$ and $\{y_n\}_{n=1}^{M} \subset K$ are *orthogonal* if for all $v \in H$, $\sum_{n=1}^{M} \langle v , x_n \rangle y_n = 0$.
\[P:ortho\] Suppose $\{x_n\}_{n=1}^{M} \subset H$ and $\{y_n\}_{n=1}^{M} \subset K$ are Parseval frames; they are orthogonal if and only if $$\left(
\begin{aligned}
&x_1 & &| & &y_1 \\
& \vdots & &| & & \vdots \\
&x_M & &| & &y_M
\end{aligned}
\right)
:= \left(P | Q \right)$$ has columns which form an orthonormal set.
($\Leftarrow$) Consider the two matrices $P$ and $Q$ whose rows are $\{x_n\}$ and $\{y_n\}$, respectively. A straight forward computation demonstrates that for $v \in H$, $$\label{E:ortho}
\sum_{n =1}^{M} \langle v , x_n \rangle y_n = Q^{*} P v,$$ where $Q^{*}$ is the conjugate transpose of $Q$. It follows that if the above matrix has orthonormal columns, then $Q^{*} P = 0$, and thus the frames $\{x_n\}$ and $\{y_n\}$ are orthogonal.
($\Rightarrow$) Conversely, suppose the Parseval frames are orthogonal. Note that by Proposition 1, the left part $P$ of the above matrix has orthonormal columns; likewise the right part of the matrix $Q$ also has orthonormal columns. By equation (\[E:ortho\]), we must have that the columns of the left part of the matrix are orthogonal to the columns of the right part of the matrix. Hence, the columns of the matrix form an orthonormal set.
Note that if $\{x_n\}$ is orthogonal to $\{y_n\}$, then $\{y_n\}$ is orthogonal to $\{x_n\}$.
Let ${\mathbb}{X} := \{x_n\}_{n=1}^{M} \subset H$; the analysis operator $\Theta_{{\mathbb}{X}}$ of $\{x_n\}$ is given by: $$\Theta_{{\mathbb}{X}}: H \to {\mathbb}{F}^M : v \mapsto (\langle v, x_1 \rangle, \langle v, x_2 \rangle, \dots , \langle v, x_M \rangle ).$$ The matrix representation of $\Theta_{{\mathbb}{X}}$ is given as the matrix $P$ in Proposition \[P:ortho\]. The proof of Proposition \[P:ortho\] shows that two frames $\{x_n\}$ and $\{y_n\}$ are orthogonal if and only if their analysis operators $\Theta_{{\mathbb}{X}}$ and $\Theta_{{\mathbb}{Y}}$ have orthogonal ranges in ${\mathbb}{F}^M$.
Encryption Using Orthogonal Frames
----------------------------------
We present here an overview of our proposed private key encryption scheme using orthogonal frames. For motivation, consider that the One-Time Pad is an unconditionally secure cipher, which is optimal of all unconditionally secure ciphers in terms of key length [@HAC]. Our encryption scheme, which is similar to a subband coding scheme, is an effort to approximate the One-Time Pad. The (private) key for this encryption scheme is two orthogonal Parseval frames $\{x_n\}_{n=1}^{M} \subset H$ and $\{y_n\}_{n=1}^{M} \subset K$. Let $\Theta_{{\mathbb}{X}}$ and $\Theta_{{\mathbb}{Y}}$ respectively denote their analysis operators. Suppose $m \in H$ is a message; let $g \in K$ be a non-zero vector chosen at random. The ciphertext $c \in {\mathbb}{F}^{M}$ is given as follows: $$c := \Theta_{{\mathbb}{X}} m + \Theta_{{\mathbb}{Y}} g.$$ To recover the message, we apply $\Theta_{{\mathbb}{X}}^{*}$: $$\begin{aligned}
\Theta_{{\mathbb}{X}}^{*} c &= \Theta_{{\mathbb}{X}}^{*} \Theta_{{\mathbb}{X}} m + \Theta_{{\mathbb}{X}}^{*} \Theta_{{\mathbb}{Y}} g \\
&= \sum_{n = 1}^{M} \langle m, x_n \rangle x_n + \sum_{n = 1}^{M} \langle m, y_n \rangle x_n \\
&= m + 0
= m.\end{aligned}$$
There are several things to note about our scheme:
1. The frame $\{x_n\}$ need not be Parseval, but Parseval frames are in general easier to work with. Since the Parseval frames form only a small subset of all possible frames, using general frames would allow a much greater choice of specific encryption keys.
2. The frame $\{y_n\}$ need not be Parseval; it need not even be a frame, though again Parseval frames simplify matters. If $\{y_n\}$ is not a frame, then $\Theta_{{\mathbb}{Y}}$ has non-trivial kernel, and $\Theta_{{\mathbb}{Y}}g$ could be 0 if g is chosen to be in the kernel. (Below we will actually use scalar multiples of Parseval frames for both $\{x_n\}$ and $\{y_n\}$.)
3. Just as with the One-Time Pad, when done properly, encoding a message twice results in two different ciphertexts.
4. Unlike the One-Time Pad, in which a brute force attack results in all possible plaintexts, it appears unlikely that a brute force attack on our system would result in the same. Our simulations indicate that an attack produces either a text which is very close to the original plaintext or is gibberish (see graphs below for more.) However, at this time, we cannot prove why this is so.
If $\{x_n\}_{n=1}^{M} \subset H$ and $\{y_n\}_{n=1}^{M} \subset K$ are orthogonal frames, then $M \geq dim(H) + dim(K)$.
Let $\Theta_{{\mathbb}{X}}$ and $\Theta_{{\mathbb}{Y}}$ be the respective analysis operators. Note that by the (lower) frame inequality in equation \[E:frame\], both $\Theta_{{\mathbb}{X}}$ and $\Theta_{{\mathbb}{Y}}$ are one-to-one. Moreover, the orthogonality of the frames is equivalent to the orthogonality of the ranges of $\Theta_{{\mathbb}{X}}$ and $\Theta_{{\mathbb}{Y}}$. Combining these two observations establishes the proposition.
For convenience, we will assume that $M = dim(H) + dim(K)$. The ciphertext is $$c = \Theta_{{\mathbb}{X}} m + \Theta_{{\mathbb}{Y}} g$$ where $\{x_n\}$ and $\{y_n\}$ are orthogonal Parseval frames. Since they are orthogonal, we write $$c = \left( \Theta_{{\mathbb}{X}} | \Theta_{{\mathbb}{Y}} \right) m \oplus g$$ where the matrix $\left( \Theta_{{\mathbb}{X}} | \Theta_{{\mathbb}{Y}} \right)$ is an isometry. Therefore, our encryption procedure involves generating a large orthogonal matrix.
The next section discusses several ways of constructing such matrices. Since the encryption scheme is a private key system, we wish to have a relatively small key size; that is to say that the entire matrix is too much information to be used as the key. We discuss below some of the strengths and weaknesses of the various construction techniques.
Five Encryption Schemes {#S:schemes}
=======================
The cipher algorithm depends upon generating a pair of random orthogonal frames, each of which is the size of the message. This is equivalent to producing a random orthogonal matrix of twice the size of the message. We investigate here several methods for doing so. The first method takes the view of producing orthogonal frames using Fourier frames. The remaining methods take the view of producing orthogonal matrices.
Once the orthogonal frames, or orthogonal matrix, is determined, the encryption and decryption process is the same. If the frames are given by ${\mathbb}{X}$ and ${\mathbb}{Y}$, then we write the matrix $(\Theta_{{\mathbb}{X}} | \Theta_{{\mathbb}{Y}} )$; if on the other hand the matrix is $A$, we think of $A = (\Theta_{{\mathbb}{X}} | \Theta_{{\mathbb}{Y}})$. Given a message $m$, choose at random a vector $g$, called the “garbage” or “noise”, and compute $(\Theta_{{\mathbb}{X}} | \Theta_{{\mathbb}{Y}} ) m \oplus g = c$ to yield the cipher text $c$. The recipient computes $$(\Theta_{{\mathbb}{X}} | \Theta_{{\mathbb}{Y}} )^{T} c = (\Theta_{{\mathbb}{X}} | \Theta_{{\mathbb}{Y}} )^{T} (\Theta_{{\mathbb}{X}} | \Theta_{{\mathbb}{Y}} ) m \oplus g = K m \oplus 0 = K m,$$ where $K$ is the square of the norm of any column of the matrix $\Theta_{{\mathbb}{X}}$. Dividing by $K$ then reproduces the message.
Scheme \#1
----------
The first algorithm utilizes the Discrete Cosine Transform. The original idea came from using the Discrete Fourier transform, which involves complex exponentials. The Discrete Cosine Transform, in matrix form, is given by: $$C=[c_{kn}]=\left[ \lambda_k\sqrt{\frac{2}{M}} \cos \left\{ \frac{k\pi}{M}(n+1/2) \right\} \right],$$ where $n=1,\dots, M$, $\lambda_1=1/\sqrt{2}$ and $\lambda_k=1$ for all $k=2,\dots, N$. Note that this is normalized to be a unitary matrix. Assuming that $M=2N$, one can permute the columns of $C$ to yield $C'$, and divide the resulting matrix in half vertically: $$C' = (\Theta_{{\mathbb}{X}} | \Theta_{{\mathbb}{Y}} ).$$ The resulting divided matrix can then be viewed as the analysis operators for two orthogonal frames, each for ${\mathbb}{R}^N$, consisting of cosine bases projected onto smaller subspaces, (Proposition \[P:ortho\], see also [@ALTW02]). Moreover, the frame vectors can be weighted, which is accomplished by a diagonal, invertible matrix $D$. Let $P$ denote a permutation matrix.
The (private) key for the cipher then consists of the matrix $D$ (or simply its diagonal entries), and the permutation corresponding to $P$. The encryption algorithm of a message $m$ of length $N$ then consists of randomly generating a garbage vector $g \in {\mathbb}{R}^N$ and computing the ciphertext $c$: $$c = C D P (m \oplus g).$$ To decrypt the message, we apply the matrix $Q P^{T}D^{-1}C^T$ to the ciphertext, where $Q$ is the projection of ${\mathbb}{R}^M$ onto the first $N$ co-ordinates: $$Q P^{T}D^{-1}C^T C D P(m \oplus g) = Q (m \oplus g) = m.$$
We note that the only knowledge unknown to an adversary is $D$ and $P$; the adversary will know $C$. Hence, $C$ is irrelevant to the cipher algorithm. Because of this, the algorithm reduces to rearrangement followed by weighting of the entries of the message and the garbage. We conclude that our first algorithm is a poor one.
Scheme \#2
----------
The second scheme involves using Hadamard arrays to generate orthogonal matrices. We first start with the definition of Hadamard arrays. We remark here that this scheme is related to linear codes [@delsarte].
[@Wallis] A Hadamard array $H[h,k,\lambda ]$ based on the indeterminates $x_1,~x_2,\ldots,x_k$, with $k\leq h$, is an $h\times h$ matrix with entries chosen from $\{\pm x_1,~\pm x_2,\ldots,\pm x_k\}$ in such a way that:
1. In any row there are $\lambda$ entries $\pm x_1$, $\lambda$ entries $\pm x_2$, $\ldots,$ $\lambda$ entries $\pm x_k$, and similarly for the columns.
2. The rows and columns are (formally) pairwise orthogonal, respectively.
The matrices we use for our encryption scheme are of $h=k$, $\lambda=1$. The only possible Hadamard arrays of this type are for $h=1,2,4,8$ [@ag]. For indeterminants $A$ through $H$, we have the Hadamard array $$H[8,8,1]=\begin{bmatrix}A&B&C&D&|&E&F&G&H\\-B&A&D&-C&|&F&-E&-H&G\\-C&-D&A&B&|&G&H&-E&-F\\-D&C&-B&A&|&H&-G&F&-E\\
-E&-F&-G&-H&|&A&B&C&D\\-F&E&-H&G&|&-B&A&-D&C\\-G&H&E&-F&|&-C&D&A&-B\\-H&-G&F&E&|&-D&-C&B&A\\
\end{bmatrix}.$$
For $\Theta=H[8,8,1]$, $\Theta^T \Theta=K I_8$ where $K=A^2+B^2+\dots + H^2$.
The Hadamard arrays allow easy construction of matrices (and hence tight frames) needed in our encryption schemes. For the encryption process, we now have only $\Theta$ to construct instead of computing the matrices $C$, $D$, and $P$.
The encryption process starts with a message $m$ of arbitrary length, and dividing $m$ into blocks $m_1,\dots , m_q$ of length 4 (padding the last block with $0$’s if necessary). Then random vectors $g_1, \dots , g_q$ of length 4 are chosen, and the matrix $N$ is applied successively to $m_i \oplus g_i$. The ciphertext is then $$c = \Theta(m_1 \oplus g_1) \oplus \dots \oplus \Theta(m_q \oplus g_q) .$$
The message is then decrypted by dividing $c$ into blocks $c_1, \dots, c_q$ of size 8, computing $K \Theta^{T} c_i$ for $i=1, \dots ,q$, and reconstructing the message using the first four entries of these resulting blocks.
Because of the ease of construction of the Hadamard arrays, the system is quite easy to implement. Unlike the first scheme, the key for the recipient has now been reduced to knowing the chosen entries for $\Theta$, hence in this case the key is the entries $A,B, \dots, H$ of the matrix $\Theta$. Since Hadamard arrays are small, however, we wish to find an algorithm to generate larger orthogonal matrices.
Scheme \# 3
-----------
Our next scheme is an attempt to produce larger orthogonal matrices. Starting with Hadamard arrays $A$ and $M$ with $A^TA=kI_8$ and $M^TM=pI_8$ for constants $k$ and $p$, we construct a new $16 \times 16$ orthogonal matrix $$S = \begin{bmatrix}A&MA\\-M^TA&A\end{bmatrix}.$$ Repeat this procedure with Hadamard arrays $B$ and $N$ to get $$T = \begin{bmatrix}B&NB\\-N^T B&B\end{bmatrix}.$$ The matrices $S$ and $T$ are then used to construct a $32 \times 32$ orthogonal matrix: $$U = \begin{bmatrix}S&TS\\-T^T S&S\end{bmatrix}.$$ This “blow up” construction is iterated to get the appropriate size matrix for our plain text.
In this encryption scheme, the key is the entries of the matrices $A$, $B$, $M$, $N$, etc., and their positions in the construction. This method, however is computationally inefficient.
Scheme \#4
----------
We first define the tensor product, $\otimes$, of two matrices, $A$ and $B$. The sizes of the matrices is irrelevant.
[@comb] Let $$A=\begin{bmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\
\vdots&~ &\ddots&~ \\
a_{m1}&a_{m2}&\cdots&a_{mn}
\end{bmatrix}.$$ Then $$A\otimes B:=\begin{bmatrix}a_{11}B&a_{12}B&\cdots&a_{1n}B\\
\vdots&~ &\ddots&~ \\
a_{m1}B&a_{m2}B&\cdots&a_{mn}B
\end{bmatrix}.$$
If $A$ is an $m\times n$ and $B$ is a $p\times q$, then $A\otimes B$ is an $mp\times nq$ matrix. The tensor product will be the critical element of construction in this and the next scheme. Note that if $A$ and $B$ are orthogonal matrices, then $A \otimes B$ is also an orthogonal matrix.
A Hadamard matrix is a square orthogonal matrix with entries consisting of $\pm 1$’s.
We start with an Hadamard *matrix* (not an array), $H$, of a chosen size $2^p$, and then two Hadamard *arrays*, $A$ and $B$ of choice sizes 2,4, or 8. We then construct the new matrix via the tensor products: $$C=\begin{bmatrix}H\otimes A & (H\otimes B)(H\otimes A)\\ -(H\otimes B)^T(H\otimes A)& H\otimes A\end{bmatrix}.$$ $C$ is now an orthogonal matrix. This matrix is size adaptive with respect to powers of 2 since each matrix is of some order of 2, and the size of $H$ can be chosen.
However, the Hadamard matrix property that $H^TH=I_n$ is actually a disadvantage. Let $$H=\begin{bmatrix}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\end{bmatrix}.$$ Then our matrix is $$C=\begin{bmatrix}H\otimes A & (H\otimes B)(H\otimes A)\\ -(H\otimes B)^T(H\otimes A)& H\otimes A\end{bmatrix}=$$ $$\begin{bmatrix} A&A&A&A &|& 4BA&0&0&0 \\
A&-A&A&-A &|& 0&4BA&0&0 \\
A&A&-A&-A &|& 0&0&4BA&0 \\
A&-A&-A&A &|& 0&0&0&4BA \\
-&-&-&- &|& -&-&-&- \\
-4B^{T}A&0&0&0 &|& A&A&A&A \\
0&-4B^{T}A&0&0 &|& A&-A&A&-A \\
0&0&-4B^{T}A&0 &|& A&A&-A&-A \\
0&0&0&-4B^{T}A &|& A&-A&-A&A \end{bmatrix}.$$
The resulting matrix is relatively sparse, which is undesirable for maintaining secrecy.
Scheme \#5
----------
We choose $p$ Hadamard arrays $H_1,H_2,\ldots,H_p$. Each array can have its own size, say $e_i\times e_i$ for $1\leq i\leq p$, where each $e_i$ is either 2,4, or 8. We then construct our $e_1e_2\cdots e_p$-sized matrix $M$ by the tensor product of these $p$ matrices: $$M = \bigotimes^{p}_{i=1}H_i:=H_1\otimes H_2\otimes\cdots\otimes H_p.$$ The ciphertext then is $c = M(m\oplus g)$. With this construction, we eliminate the sparsity that was shown in scheme \#4. Note that the key in this case is the entries of the first rows of $H_1$ to $H_p$, hence is an array of numbers of size $e_1 + e_2 + \cdots + e_p$, and hence is relatively small.
We ran some numerical experiments, using scheme \#5 to obtain information regarding several things:
1. We wanted to see if a brute force attack would be a feasible way of defeating the cipher. The results of the experiments and also the computations below suggest that the answer is no.
2. One advantage of the One Time Pad is that a brute force attack results in all possible plaintext messages, forcing an adversary to choose which was the original message. We wanted to determine if this was also true of our proposed cipher. The results of our experiments indicate that the answer to this is also no.
3. Finally, we wanted to determine if the size of the entries of the garbage vector $g$ mattered. The experiments and the computations below suggest that the answer is yes.
The results of our experiments, in the form of graphs, are given below.
Experimental Results and Conclusions {#S:results}
====================================
We want to know how accurate a guess has to be in order to break the cipher. We suppose that an adversary knows that we are using scheme \#5, that is the adversary knows the structure of the matrix $M$, but not the entries. We let $M$ be the original matrix of size $n$, $\tilde{M}$ be the adversary’s guess, and $w$ be the original plaintext $m$ concatenated with the garbage $g$ (i.e. $w=m \oplus g$). Then we consider $\tilde{w} := (1/\tilde{k})\tilde{M}^TMw$ where $\tilde{k} = \|\tilde{M}\|^2$. Since we assume that the structure of $M$ is known by the adversary, we consider $\tilde{M}=M+P$, where $P$ is a matrix with the same structure as $M$. For simplicity, we let $M_i$ denote the $i$th row of the matrix $M$ and likewise for $P$. Note that $k=\langle M_i,M_i\rangle$ since $(1/k)M^TM=I_n$, and $\tilde{k}=\langle\tilde{M}_i,\tilde{M}_i\rangle=\langle M_i+P_i,M_i+P_i\rangle=||M_i||^2+2\langle M_i,P_i\rangle+||P_i||^2$.
We rewrite to get the following: $$\begin{gathered}
(1/\tilde{k})\tilde{M}^TM=(k/\tilde{k})I+(1/\tilde{k})P^TM= \\
{\displaystyle\frac}{\langle M_i,M_i\rangle}{\langle M_i+P_i,M_i+P_i\rangle}I+{\displaystyle\frac}{1}{\langle M_i+P_i,M_i+P_i\rangle}\begin{bmatrix}\langle P_1^T,M^T_1\rangle & \cdots & \langle P_1^T,M^T_n\rangle \\ \vdots & \ddots & ~ \\ \langle P_n^T,M^T_1\rangle & \cdots & \langle P_n^T,M^T_n\rangle \end{bmatrix}.\end{gathered}$$
Let $\tilde{w}=\left(\tilde{w}_1,\tilde{w}_2,\ldots,\tilde{w}_n\right).$ Then we have that for $1 \leq j \leq n$: $$\tilde{w}_j = \left({\displaystyle\frac}{\langle P_j^T,M_j^T\rangle+\langle M_j,M_j\rangle}{\langle M_j+P_j,M_j+P_j\rangle}w_j+\sum^n_{\begin{array}{c}i=1 \\ i\not=j\end{array}}{\displaystyle\frac}{\langle P_j^T,M_i^T\rangle}{\langle M_i+P_i,M_i+P_i\rangle}w_i\right)$$
For an adversary’s guess to be close, $${\displaystyle\frac}{\langle P_j^T,M_j^T\rangle+\langle M_j,M_j\rangle}{\langle M_j+P_j,M_j+P_j\rangle} \approx 1$$ and $$\sum^n_{\begin{array}{c}i=1 \\ i\not=j\end{array}}{\displaystyle\frac}{\langle P_j^T,M_i^T\rangle}{\langle M_i+P_i,M_i+P_i\rangle} \approx 0.$$ We break this up into cases.
- Assume $||P||$ is relatively large compared to $\|M\|$; that is, the guess is far from the actual matrix. We have $$\begin{gathered}
\left|{\displaystyle\frac}{\langle P_i^T,M_i^T\rangle}{\langle M_i+P_i,M_i+P_i\rangle}\right|=\left|{\displaystyle\frac}{\langle P_i^T,M_i^T\rangle}{||M_i||^2+2\langle P_i,M_i\rangle+||P_i||^2}\right|= \\
\left|{\displaystyle\frac}{\langle P_i^T,M_i^T\rangle/||P_i||^2}{(||M_i||^2/||P_i||^2)+(2\langle P_i,M_i\rangle/||P_i||^2)+1}\right|\to 0 \mbox{ as } ||P||\to\infty.\end{gathered}$$ However, when we look at the $\tilde{w}_i$ coefficients, we see the following: $$\left|{\displaystyle\frac}{\langle M_j,M_j\rangle}{\langle M_j+P_j,M_j+P_j\rangle}\right|= \left|{\displaystyle\frac}{(\langle M_j,M_j\rangle/||P_j||^2)}{(||M_i||^2/||P_i||^2)+(2\langle P_i,M_i\rangle/
||P_i||^2)+1}\right|\to 0 \mbox{ as } ||P_j||\to\infty.$$
- We assume $||P||$ is small relative to $||M||$; that is, the guess is close. Then we have using the same arguments: $$\begin{gathered}
\left|{\displaystyle\frac}{\langle P_i^T,M_i\rangle}{\langle
M_i+P_i,M_i+P_i\rangle}\right|=\left|{\displaystyle\frac}{\langle
P_i^T,M_i\rangle}{||M_i||^2+2\langle M_i,P_i\rangle
+||P_i||^2}\right|= \\
\left|{\displaystyle\frac}{\langle P_i^T,M_i\rangle/||M_i||^2}{1+(2\langle
M_i,P_i\rangle/||M_i||^2)+(||P_i||^2/||M_i||^2)}\right|\to 0 \mbox{
as }||P_i||\to 0.\end{gathered}$$ So, the better the guess, the smaller the ‘extra’ coefficients will be. Likewise, for the $\tilde{w}_j$ coefficients, $$\begin{gathered}
\left|{\displaystyle\frac}{\langle M_j,M_j\rangle}{\langle
M_j,P_j\rangle}\right|=\left|{\displaystyle\frac}{||M_j||^2}{||M_j||^2+2\langle
M_j,P_j\rangle +||P_j||^2}\right|= \\
\left|{\displaystyle\frac}{1}{1+(2\langle
M_j,P_j\rangle/||M_j||^2)+(||P_j||^2/||M_j||^2)}\right|\to 1
\mbox{ as }||P_j||\to 0.\end{gathered}$$
Our first question is whether an adversary can figure out how small the perturbation $P$ must be in order to get a “good guess”. The adversary knows the size of $M$ and $||Mw||$; we assume additionally that the adversary knows the structure of $M$. For convenience, assume that the encryption matrix $M=A\otimes B\otimes C$ for 3 Hadamard arrays, $A,B$, and $C$. We then let $\tilde{M}=(A+a)\otimes(B+b)\otimes(C+c)$ for (small norm) perturbation matrices $a,b$ and $c$. We reformulate our question: How big can $||a||,||b||$, and $||c||$ be such that $||M^TMw-\tilde{M}^TMw||<{\varepsilon}$, where ${\varepsilon}$ is some acceptable tolerance for error? (Here, for a matrix $A$, $\| A \|$ denotes the operator norm of $A$. Below, $\| \cdot \|$ shall denote both Hilbert space norm for vectors and operator norm for matrices.)
We let $||a||\approx||b||\approx||c||\approx\beta$ and $||A||\approx||B||\approx||C||\approx\gamma$. We may assume that $\gamma\gg\beta$. If we write out $\tilde{M}$ in terms of the tensor products, we get $$\tilde{M}=A\otimes B\otimes C+A\otimes B\otimes c +\cdots+
a\otimes b\otimes c \mbox{ and } ||\tilde{M}||\leq\gamma^3+3\gamma^2\beta+3\gamma\beta^2+\beta^3.$$ Given any ${\varepsilon}>0$, we choose $\delta={\varepsilon}/||Mw||$. If $|3\gamma^2\beta|<\delta$, then $$\begin{aligned}
||M^TMw-\tilde{M}^TMw|| &\leq||M^T-\tilde{M}^T|| ||Mw|| \leq (3\gamma^2\beta+3\gamma\beta^2+\beta^3) ||Mw|| \\
& \approx 3(\gamma^2\beta) ||Mw|| <\delta ||Mw||={\varepsilon}.\end{aligned}$$
These computations suggest that the larger the entries of the garbage vector $g$ are, the closer a guess must be in order to reasonably recover the message. This is corroborated by the experiments we ran (see the graphs below). Thus, we can control the accuracy an adversary would need in order to break the cipher.
Chosen-Plaintext Attack
-----------------------
We will demonstrate here a chosen-plaintext attack on the cipher which will break the system. A chosen-plaintext attack is an attack mounted by an adversary which chooses a plaintext and is then given the corresponding ciphertext.
The encryption algorithm proposed above is vulnerable to a chosen-plaintext attack.
We assume the adversary knows the length of the message band and subsequently the length of the noise band. Let the length of the message band be $N_m$ and the length of the noise band be $N_n$. The attack is as follows:
1. Determine the range of the noise band $K$ of $\Theta$. That is, determine $(\Theta_{{\mathbb}{X}} | \Theta_{{\mathbb}{Y}}) (0 \oplus {\mathbb}{R}^{N_n})$. Choose any plaintext $m$ of size $N_m$. Encode the plaintext twice, with output, say, $e_0$ and $e_1$. Compute $e_1 - e_0 = \Theta (m \oplus g_1) - \Theta (m \oplus g_0) = \Theta (0 \oplus g_1 - 0 \oplus g_0)$. Notice that this yields a vector $f_1 = \Theta (0 \oplus g_1 - 0 \oplus g_0)$ in the range of the noise band of $\Theta$. Encode the plaintext a third time, with output $e_2$, and compute $f_2 = e_2 - e_0$. Compute $f_3, \dots , f_m$ until the collection $\{f_1, \dots , f_m \}$ contain a linearly independent subset of size $N_n$. This determines the range of the noise band $K$ of $\Theta$.
2. Determine the range of the message band $T$ of $\Theta$. That is, determine what is $(\Theta_{{\mathbb}{X}} | \Theta_{{\mathbb}{Y}}) ({\mathbb}{R}^{N_m} \oplus 0)$. Choose any plaintext $m_1$ of size $N_m$; encode the plaintext, with output $e_1$; then project $e_1$ onto the orthogonal complement of $K$. This yields a vector $x_1$ in $T$. Choose another plaintext $m_2$ and repeat, yielding vector $x_2 \in T$. Repeat until the collection $\{x_1, \dots , x_q\}$ contains a linearly independent subset of size $N_m$. This set determines $T$.
3. Determine the message part of $\Theta$. That is, determine $\Theta_{{\mathbb}{X}}$. Suppose in Step 2, $\{m_1, \dots , m_{N_m} \}$ is such that $\{x_1, \dots , x_{N_m} \}$ is linearly independent. If we write $\Theta = ( \Theta_{{\mathbb}{X}} | \Theta_{{\mathbb}{Y}} )$, then we now have the following system of equations: $$( \Theta_{{\mathbb}{X}} | \Theta_{{\mathbb}{Y}} ) m_k \oplus 0 = x_k \text{ for $k = 1, \dots N_m$.}$$ Given this system of equations, now solve for $\Theta_{{\mathbb}{X}}$.
4. Unencode ciphertexts. Given any ciphertext $e$, the adversary computes the following: $$\begin{aligned}
K^{-1} (\Theta_{{\mathbb}{X}} | 0 )^T e &= K^{-1} (\Theta_{{\mathbb}{X}} | 0)^T (\Theta_{{\mathbb}{X}} | \Theta_{{\mathbb}{Y}}) m \oplus g \\
&= K^{-1} \Theta_{{\mathbb}{X}}^T \Theta_{{\mathbb}{X}} m \\
&= m\end{aligned}$$ where $K$ is the square of the norm of any column of $\Theta_{{\mathbb}{X}}$.
Concluding Remarks
------------------
The proposed cipher appears to be robust to brute force attacks, but is not robust against a chosen-plaintext attack. We mention, however, that we do not know if the scheme is robust to a known-plaintext attack. Moreover, this is a private symmetric key cipher; it would be desirable if this method could be altered to be used as a public key cipher. We reiterate that the McEliece cipher is a public key system and is similar in flavor to the cipher presented here.
The ultimate downfall of the cipher is the linearity. We suggest that perhaps there is possibly a way of introducing non-linearity into the algorithm to defeat a chosen-plaintext attack. However, at this point, we know of no methods to accomplish this.
Pseudo-Code
===========
Encoder.cpp
-----------
1. Calculate Matrix
1. Input the possible range of entries for A, B, C
2. Make A, B, C either 4x4 or 8x8 Hadamard arrays with entries chosen randomly from the range (for simplicity, we are using the 4x4 Hadamard array)
3. Compute tensor product $A \oplus B \oplus C$
2. Encode Message
1. Compute $m \oplus g$ by converting the message to ASCII and filling $g$ with random numbers
2. Compute $(A \otimes B \otimes C) (m \oplus g)$
Hacker.cpp
----------
Hacker.cpp–this code attempts a brute force method on a cypher text.
1. Input min, max, range of key guesses
2. Input ciphertext
3. For all possible values of the twelve variables in use
1. Fill the matrices with the possible values
2. Tensor matrices together
3. Calculate possible text messages
4. Output text to file for later examination
Analyzer.cpp
------------
This code takes the output of Hacker.cpp and calculates the frequency of occurrence of every ASCII symbol.
1. For each line of text, count number of appearances of each ASCII value
2. Output information to text file
Acknowledgements {#acknowledgements .unnumbered}
================
This work was done while all three authors were at the University of Wyoming, at which time the first author was an undergraduate student, and the third author was a graduate student. The first and second authors were supported by NSF grant DMS-0308634. The third author was supported by a Basic Research Grant from the University of Wyoming.
We thank Bryan Shader, Eric Moorhouse, and Cliff Bergman for helpful discussions.
Graphs
======
How to read the following graphs. We carried out the following computations to simulate a brute force attack on the cipher:
1. for a sample plaintext, encode the plaintext using scheme \#5 making the following choices: approximate entry size for the matrices and approximate size for the garbage entries;
2. decode the ciphertext using every combination of key entry and key entry $\pm 1$;
3. converted the decoded ciphertext in the previous step to ASCII values;
4. counted the appearance of each value in the resulting combinations.
The graphs represent the number of appearances within all possible key guesses from step 2 above. The plaintext is given in the title of the graph; the ASCII values are the $x$-axis of the graph, and the approximate key sizes and garbage sizes are given in the graph captions.
Note that in figures 3 and 7, the key size and garbage size are the same. The graphs show that most of the characters that appear in the simulated brute force attack are those that are in the original message.
[References]{} Agaian, S.S.: *Hadamard Matrices and Their Applications*. New York: Springer-Verlag, 1985. Aldroubi, A., D. Larson, W.S. Tang, and E. Weber, *Geometric Aspects of Frame Representations of Abelian Groups*, preprint (2002) (available on ArXiv.org; math.FA/0308250). Benedetto, J. and Fickus, M.: *Finite Normalized Tight Frames*, *Adv. Comput. Math.* [**18**]{}, (2003) no. 2-4, 357–385. Casazza, P.: *The Art of Frame Theory*, *Taiwanese Math. J.* [**4**]{} (2000) no. 2, 129–201. Casazza, P., Kovacević, J.: *Uniform Tight Frames for Signal Processing and Communications*, SPIE Proc. vol. 4478 (2001), 129–135. Chabaud, F., *On the security of some cryptosystems based on error-correcting codes* *Advances in Cryptology – EUROCRYPT ’94*, Lecture Notes in Computer Science 950, Springer-Verlag, 1995, pp. 131–139. Delsarte, P. and Goethals, J. M.: *Tri-weight Codes and Generalized Hadamard Matrices*, Information and Control [**15**]{} (1969), p. 196-206. Dykema, K., Freeman, D., Kornelson, K., Larson, D., Ordower, M., Weber, E.: *Ellipsoidal Tight Frames and Projection Decompositions of Operators*, preprint (2003). Han, D. and Larson, D.: *Frames, Bases, and Group Representations* Mem. Amer. Math. Soc. [**147**]{}, (2000) no. 697. van Lint, J.H.: *A Course in Combinatorics*, Cambridge: Cambridge University Press, 1992. Menezes, A., van Oorschot, P. and Vanstone, S.: *Handbook of Applied Cryptography* CRC Press, 1997. Wallis, W.D.: *Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices*, Berlin: Springer-Verlag, 1972.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Non-uniform sampling arises when an experimenter does not have full control over the sampling characteristics of the process under investigation. Moreover, it is introduced intentionally in algorithms such as Bayesian optimization and compressive sensing. We argue that Stochastic Differential Equations (SDEs) are especially well-suited for characterizing second order moments of such time series. We introduce new initial estimates for the numerical optimization of the likelihood, based on incremental estimation and initialization from autoregressive models. Furthermore, we introduce model truncation as a purely data-driven method to reduce the order of the estimated model based on the SDE likelihood. We show the increased accuracy achieved with the new estimator in simulation experiments, covering all challenging circumstances that may be encountered in characterizing a non-uniformly sampled time series. Finally, we apply the new estimator to experimental rainfall variability data.'
author:
- 'ExxonMobil Research and Engineering Company, Annandale, New Jersey, USA.'
title: 'Accurate Characterization of Non-Uniformly Sampled Time Series using Stochastic Differential Equations'
---
Introduction
============
Non-uniformly sampled time series are found in a wide range of applications. They typically occur when the experimenter has limited control over signal sampling. Sampling times may be determined by a natural process, for example in turbulent flow characterization using laser-Doppler anemometry [@damaschke2018ldaspectra], analysis of climate time series [@thirumalai2020climate] and astronomy [@khan2017exoplanets]. Alternatively, non-uniform sampling can occur because of an irregular human sampling, e.g. in power systems sensor data [@stankovic2017power], oil production surveillance [@tewari201production] and vital signs measurements in medical applications [@tan2020medical; @barbieri2020icureadmission]. Finally, non-uniform sampling is a key attribute of certain algorithms, including Bayesian optimization [@ghahramani2015bayesianopt] and compressive sensing [@turunctur2019compressive].
It has been shown theoretically that alias-free spectral estimates can be obtained using non-uniform sampling[@shapiro1960aliasfree]. As most theoretical results, this result is asymptotic, i.e., it holds in the limit where the number of observations $N$ tends to infinity. The result can easily be understood intuitively, as follows: As a signal is sampled at random times for a long enough time, any desired number of observations is a available at an arbitrarily low sampling interval. Hence, the spectrum can be estimated alias-free up to an arbitrary high frequency. Following the same intuition, it is clear that this asymptotic result breaks down for finite $N$, because the shortest time interval will be finite in that case.
The objective of this work is to accurately characterize the second order moments of a non-uniformly sampled stationary stochastic process, which can be expressed in terms of the power spectral density in the frequency domain. Given the close correspondence between the error in the log power spectrum and the Kullback-Leibler discrepancy [@broersen2006automatic], we will use “spectral estimation” as a shorthand for the aforementioned objective. In the time domain, the second order moments are expressed using the covariance function, or, equivalently, the kernel of a Gaussian processes [@rasmussen2003gaussian].
Existing approaches include non-parametric methods, such as the Lomb-Scargle spectral estimate [@springford2020lombscargle], and the slotting technique for estimation of the covariance function [@rehfeld2011comparison]. Parametric techniques include discrete-time autoregressive models [@broersen2007beyondrate] and stochastic differential equations with random initial estimates [@kelly2014astronomy]. While parametric techniques show the most promising results, previous work also report inaccurate spectral spectral estimates at higher frequencies for higher-order models, and sensitivity to initial conditions of the maximum likelihood (ML) fitting procedure [@jones1984sensitive-init; @broersen2006sensitive-init], which limit the practical use of these algorithms.
Our main contribution is to propose a new algorithm for spectral estimation using Stochastic Differential Equations (SDEs) based on incremental parameter estimation and data-driven model truncation, that has been validated in simulation experiments. We explain how model truncation in SDEs results greater accuracy than discrete-time models through analyzing the SDE(1) case. For the problem of spectral estimation from non-uniformly sampled time series, asymptotic theory provides a poor description of actual behavior. Hence, simulation experiments are indispensable to establish the accuracy of an estimator, and are therefore a key component of this paper.
In section \[sec:SDE-motivation\], we motivate the usage of SDEs for spectral estimation and provide a number of basic definitions and results. Section \[sec:MLE\] describes the new SDE parameter estimation procedure. Section \[sec:simulations\] contains the design of the simulation experiments, which is used in section \[sec:truncation\] to quantify the error reduction achieved through model truncation, and in section \[sec:performance\] for the overall evaluation of the proposed estimator. Finally, we apply the estimator to experimental rainfall variability data in section \[sec:rainfall\].
\[sec:SDE-motivation\]Stochastic Differential Equations: Motivation and definitions
===================================================================================
We use stochastic differential equations (SDEs) to estimate the power spectral density of a time series. We motivate this choice over available alternatives as follows:
- By using *a parametric model* we can formulate of a maximum likelihood (ML) estimator. Unlike many non-parametric estimators, ML estimators have desirable theoretical properties such as asymptotic efficiency. Parametric spectral estimators have been most successful at achieving the benefit of high-frequency spectra in finite samples [@broersen2007beyondrate], whereas non-parametric methods such as the Lomb-Scargle estimator [@springford2020lombscargle] have very high variance, which mostly limits their use to finding a single peak in the spectrum under high signal-to-noise conditions. Similarly, in the time domain, correlation estimated with the non-parametric slotting technique violate the positive-definiteness property that is required for a valid autocorrelation function [@rehfeld2011comparison]. Parametric models can achieve the same flexibility as non-parametric estimates by increasing the model order.
- The SDE model is a *continuous time* model, which can operate directly on non-uniform samples without the need to introduce a regular grid as is required for discrete-time estimates. Moreover, estimated discrete time time series models may have poles that do not have a continuous-time counterpart [@soderstrom1991contarma].
The stochastic differential equation of order $p$ for the process $y$ as a function of time $t$ is given by: $$\frac{d^{p}y}{dt^{p}}+\sum_{i=1}^{p}a_{i}\frac{d^{i-1}y}{dt}=\epsilon\label{eq:sde}$$ where $a_{i}$ are the SDE coefficients, collectively denoted $\mathbf{a}$, and $\epsilon$ is continuous Gaussian white noise with standard deviation $\sigma_{\epsilon}$.
The SDE model can be rewritten as an equivalent state-space model: $$\begin{aligned}
\text{\ensuremath{\frac{d\mathbb{\mathbf{z}}}{dt}}} & =\mathbf{A}\mathbf{z}+\mathbf{\epsilon}\label{eq:state-space}\\
y & =C\text{\ensuremath{\mathbf{z}}}\nonumber \end{aligned}$$ where the state $\mathbf{z}$ and $\mathbb{\epsilon}$ are $p$-dimensional time series and $C\mathbf{z}=z_{1}$. The matrix $\mathbf{A}$ and the covariance matrix $\mathbf{Q}$ of $\mathbb{\epsilon}$ can be computed from $\left\{ \mathbf{a},\sigma_{\epsilon}\right\} $, see [@jones1981fitcar].
The SDE process has a power spectral density given by [@sarkka2019sde]: $$h_{y}\left(f\right)=\mathbf{C}\left(\mathbf{A}-i2\pi f\mathbf{I}\right)^{-1}\mathbf{Q}\left(\mathbf{A}+i2\pi f\mathbf{I}\right)^{-T}\mathbf{C}^{T}$$ and the covariance function at lag $\tau>0$ of: $$R\left(\tau\right)=\mathbf{\mathbf{P}}_{s}\exp\left(\mathbf{A}t\right)^{T}\label{eq:autocov}$$ where $\mathbf{\mathbf{P}}_{s}$ is the stationary covariance matrix.
The state space equation may be diagonalized to represent the SDE as: $$\begin{aligned}
\text{\ensuremath{\frac{d\mathbb{\mathbf{z}}'}{dt}}} & =\Lambda\mathbb{\mathbf{z}}'+\mathbf{\epsilon'}\\
y= & C'\text{\ensuremath{\mathbf{z}}}'\end{aligned}$$ where the eigenvalues are equal to the the roots $r_{i}$ (collectively denoted $\mathbf{r}$) of the characteristic equation of (\[eq:sde\]) for $\epsilon=0$, and $C'\text{\ensuremath{\mathbf{z}}}'=\sum z'_{i}$. We refer to this parameterization as the **roots parameterization**, while we refer to the $\left\{ \mathbf{a},\sigma_{\epsilon}\right\} $ parameterization as the **coefficients** **parameterization**.
The key advantage of the roots parameterization is that stationarity can be expressed as the requirement that the real part of the roots is negative : $\textrm{Re}\left(r_{i}\right)<0$. Furthermore, the computational complexity of the likelihood computation is more efficient for large model order $p$.
Despite the advantages, many researchers use the coefficient representation. This may be motivated by the reduced computational complexity for lower-order models. In addition, SDEs can be used to parameterize a kernel as part of a larger machine learning model, e.g. in deep learning models [@chen2018neuralode] or for posterior sampling using a probabilistic program [@carpenter2017stan]. Currently many major automatic differentiation packages such as PyTorch [@paszke2017pytorch-autodiff] and the Stan autodiff library [@carpenter2015stan-autodiff] do not support complex-valued parameters, therefore necessitating the use of the coefficient representation. Therefore, we consider both representations in this work.
\[sec:MLE\]Maximum likelihood estimation
========================================
Likelihood computation
----------------------
The exact log likelihood $L$ is computed recursively using the Kalman filtering equations. Process stationarity is exploited for the first observation, i.e., it has the stationary covariance matrix $\mathbf{P}_{s}$. Given a value for the SDE parameters $\mathbf{a}$ or $\mathbf{r}$, the analytical expression for the Maximum Likelihood standard deviation $\sigma_{\epsilon}$ is used [@jones1981fitcar]. For unstable models, or errors due to the finite machine precision, a log likelihood of $L=-\infty$ is produced.
The computation largely follows cited algorithms, with the following improvements to increase computational efficiency:
1. For the coefficient representation, we follow [@sarkka2019sde] for the measurement and likelihood **** steps. The prediction step computes the conditional mean $\mathbf{\mu}_{n|n-1}$ and covariance matrix $\mathbf{P}_{n|n-1}$ of the state $\mathbf{z}_{n}$: $$\begin{aligned}
\mathbf{\mu}_{n|n-1} & = & \mathbf{F}\mathbf{\mu}_{n-1}\\
\mathbf{P}_{n|n-1} & = & \mathbf{F}\mathbf{P}_{n-1}\mathbf{F}^{T}+\int_{\tau=0}^{\Delta t}\exp(\mathbf{A}\tau)\mathbf{Q}\exp(\mathbf{A}\tau)^{T}d\tau\end{aligned}$$
where $\mathbf{F}=\exp\left(\mathbf{A}t\right)$. To compute the integral, instead of using the Matrix Fraction Decomposition proposed in [@sarkka2019sde], we eliminate the covariance matrix $\mathbf{Q}$ to yield: $$\begin{aligned}
\mathbf{P}_{n|n-1} & =\mathbf{P}_{s}-\mathbf{F}(\mathbf{P}_{s}-\mathbf{P}_{n-1})\mathbf{F}^{T}\end{aligned}$$
1. For the root representation, we use the algorithm in [@jones1981fitcar], with the following alternative computation for the stationary covariance matrix $\mathbf{P}'_{s}$: $$\left[\mathbf{P}'_{s}\right]_{ij}=-\frac{\left[\mathbf{Q}'_{s}\right]_{ij}}{\left(r_{i}+\bar{r_{j}}\right)},$$ which is a corollary of eq. 30 in [@jones1981fitcar] for $t_{k}-t_{k-1}\rightarrow\infty$.
Given the extensive literature on Kalman filtering, these improvements may have been previously reported in the literature. We still report them here to accurately represent to algorithms used in this work.
Optimization
------------
Optimization of the likelihood is performed using the Limited-Memory BFGS algorithm [@nocedal2006optimization], with derivatives obtained through automatic differentiation. For the coefficient representation, a necessary condition for stability is that all coefficients $a_{i}$ are positive; it is also sufficient for $p\le2$ [@mattuck2011diffeq-stability]. To improve optimization results, parameter values are constrained to a configurable interval: $a_{i}\in\left\langle a_{l},a_{h}\right\rangle $. Wide limits should be set to allow for a wide range of models. As an indication, $a_{l}$ is related to the duration $D$ of the time series $a_{l}\lesssim1/D$, and $a_{h}$ is related to the shortest sampling interval $\Delta t_{m}$, $a_{h}\gtrsim1/\Delta t_{m}$. In the presented simulation results, we use $a_{i}\in\left\langle 10^{-3},10^{3}\right\rangle $. In the roots representation, the real and imaginary part of the roots are similarly constrained.
Initialization\[subsec:Initialization\]
---------------------------------------
Accurate initialization of the optimization is key to achieving high-quality estimates. The sensitivity of SDE parameter estimation to initial conditions has been acknowledged in the literature [@jones1984sensitive-init], in particular for higher-order models [@broersen2006sensitive-init].
The first element in the algorithm is **incremental estimation**: The estimate for the SDE($p$) model is initiated from a lower order SDE($p'$) model ($p'<p$). The motivation for incremental estimation is the observation that per parameter, lower order models often have the largest contribution to the model fit. The lower order models are expanded by adding a single random real root (for $p'=p-1$) or a conjugate pair of random complex roots (for $p'=p-2$) to the lower order model. An alternative for a this root initialization would be to use a large initial value, as this corresponds to a model that is most similar to the lower-order model. However, this extreme initialization does not result in successful convergence to a finite value during the numerical optimization.
The second element is **initiation from autoregressive** (AR) **models** estimated from resampled data using the Burg estimator [@percival2020spectral]. Resampling is performed using nearest neighbor interpolation and linear interpolation. Intentionally, basic interpolation methods are used here, because more advanced interpolation methods tend to produce artificially smooth signals, or, in the frequency domain, a power spectrum with a very large dynamic range. This results in less accurate models, since the estimators attempts to fit the artificially introduced low power spectral density at high frequencies, at the expense of modeling the actual process dynamics [@broersen2006automatic]. AR roots that do not have a corresponding continuous-time root [@soderstrom1991contarma] are replaced by randomly generated roots.
The usage of autoregressive models is similar to the methodology proposed in [@broersen2009spurious] for reducing the variance in spectral estimates based on AR models. However, our work is distinct in the following aspects: (i) we do not need to introduce an arbitrary criterion to remove roots in the upper half of the spectrum, which could eliminate true spectral peaks, and (ii) we only use the AR model as an initial estimate, allowing further optimization of the likelihood during numerical optimization.
These main components are supplemented with purely random initialization [@kelly2014astronomy; @goodfellow2016deep] and a truncation phase that occurs after incremental estimation, where successively lower model orders are initiated from the most significant roots of higher order models.
Implementation
--------------
The estimator is implemented in Julia 1.4, using [Optim.jl](https://julianlsolvers.github.io/Optim.jl/stable/) [@mogensen2018optim] for L-BFGS optimization. Gradients are computed with automatic differentiation using [Zygote.jl](https://fluxml.ai/Zygote.jl/latest/) [@innes2019zygote]. Julia was used because it combines an expressive syntax with high execution speed. Zygote is a package for automatic differentiation that supports all functions used in the likelihood computation, notably including the matrix exponential [@branvcik2008expadjoint], and supports complex-valued parameters.
\[sec:simulations\]Design of experiments
========================================
\[subsec:Test-processes\]Test processes
---------------------------------------
The experiment is designed to cover the following process characteristics that are challenging for parameter estimation from non-uniformly sampled data:
1. **Overfit**: When the order of the estimated model matches the order that of the generating process, accurate models can be estimated [@broersen2007beyondrate]. However, the additional flexibility of a higher order parameters can lead to large errors, more so than the small statistical error that is observed in parameter estimation from regularly sampled data.
2. **High dynamic range**: Estimation of the spectral density at frequencies where the true density is low is challenging for many estimators, due to a phenomenon similar to spectral leakage [@bos2002autoregressive].
3. **Spectral details beyond average sampling rate**: While asymptotic theory predicts alias-free estimates, capturing spectral details at higher frequency remains challenging in practice, because limited information high-frequency information is available in finite samples.
4. **Model misspecification**: Any estimation procedure should continue to work well when the actual process cannot be described exactly using the estimated model structure.
To cover these characteristics, we use the following test processes:
- **Case A**: SDE(1) process with parameter $a_{1}=-1/200$. Covariance function: $R\left(\tau\right)=\exp\left(a_{1}\tau\right)$. Addresses characteristics \#1 and \#2.
- **Case B**: Squared exponential covariance with scaling parameter $l=0.3$ with added random noise with $\sigma_{w}=0.01$. Covariance: $R\left(\tau\right)=\exp\left(-\tau^{2}/l^{2}\right)+\sigma_{w}^{2}\delta\left(\tau\right)$. Addresses \#2, \#4.
- **Case C**: SDE(4) process with roots $\text{\ensuremath{\left\{ -0.10\pm2\pi\cdot0.25i,-0.5\pm2\pi\cdot1.5im\right\} } }$. Addresses \#1, \#3.
- **White noise**: A temporally uncorrelated process. Covariance function $R\left(\tau\right)=\delta\left(\tau\right)$, where $\delta\left(0\right)=1$, and $0$ elsewhere. Addresses: \#1.
Non-uniform sampling times are generated by drawing $N=200$ time intervals from a Poisson distribution with average sampling interval $T_{av}=1$. Samples $\mathbf{y}$ from a process $P$ are drawn for the resulting sampling times $\mathbf{t}$. In this way, $S=50$ time series $\left\{ \mathbf{t},\mathbf{y}\right\} $ are generated for each process. For the simulated data, SDE(8) models are estimated. The results of the simulation experiments are discussed in the subsequent sections.
Kullback-Leibler Discrepancy
----------------------------
Our objective is to accurately characterize the second-order moments of a random process, which we quantify using the The Kullback-Leibler Discrepancy (KLD). The KLD has a number of desirable properties. First, it has units that are statistically meaningful. For an unbiased estimate of a $d$-dimensional parameter $\theta$ that achieves the Cramér-Rao lower bound, the expected value of the KLD is asymptotically equal to $d/2$: $\textrm{E}\left[D\left(\hat{\mathbf{\theta}}\mathbf{\Vert\mathbf{\theta}}\right)\right]=d/2$. For the SDE($p$) model we estimate $p+1$ parameters (adding 1 to $p$ for estimation of $\sigma_{\epsilon}$), yielding: $$\textrm{E}\left[D\left(\hat{\mathbf{\theta}}\mathbf{\Vert\mathbf{\theta}}\right)\right]=\left(p+1\right)/2.\label{eq:expect-kld}$$ A second desirable property is that, for time series models, the KLD is asymptotically equivalent to the spectral distortion (Root Mean Square Error (RMSE) of the log power spectrum) and the normalized one-step ahead prediction error. See e.g. [@broersen2006automatic] for further background on these properties.
The KLD for a zero-mean multivariate Gaussian model for a random vector $\mathbf{y}$ with covariance matrix $\hat{\Sigma}$ with respect to the true zero-mean distribution with covariance matrix $\Sigma$ is given by: $$D\left(\Sigma\Vert\hat{\Sigma}\right)=\frac{1}{2}\left(\mathrm{tr}\left(\hat{\Sigma}^{-1}\Sigma-I\right)-\log\text{\ensuremath{\left(\left|\Sigma\right|/\hat{\left|\Sigma\right|}\right)}}\right)\label{eq:kld}$$ For SDE models, $\mathbf{y}$ is the vector of time series observations at times $\mathbf{t}$. The covariance matrix is computed using the covariance function from eq. \[eq:autocov\]. The choice of time steps $\mathbf{t}$ determines the time scale at which we evaluate the process.
One value for $\mathbf{t}$ is the original time points of the dataset to which the estimated model is fitted. The resulting KLD is referred to as $D_{o}$. A second value for $\mathbf{t}$ is a regularly spaced grid at interval $T$, referred to as $D_{T}.$ The value of $T$ corresponds to the time scale of interest at which we evaluate the second order moments. In the frequency domain, the corresponds to evaluating the power spectrum for frequencies up to the corresponding Nyquist frequency, $f=1/2T$.
The KLD does not suffer from some problems associated with some alternative ways to evaluate estimates:
- **Look for the “correct” or “actual” order**: Practical processes typically cannot be described exactly by a finite order model. Even if such a finite order model would exist, estimating a model of this order may not result in the most accurate estimate due to estimation errors.
- **RMSE of estimated SDE coefficients or autocovariance**: **** A small change coefficients or autocovariance values can result in a completely different process, e.g. changing from stable to unstable (coefficients), or positive-definite (valid) to not positive-definite (invalid) for autocovariances [@rehfeld2011comparison].
The base implementation of the KLD is computationally expensive. If required, a more efficient asymptotic expression can be derived specifically for SDE models. We do not elaborate on this here, because this computation is only used to evaluate model performance in simulations. It not part of the estimation algorithm, and so it will not increase computation times for the end user. Furthermore, the generic expression allows evaluation of non-SDE processes such as the squared exponential covariance function.
\[sec:truncation\]Data-driven model truncation
==============================================
In this section we describe the phenomenon of parameter divergence in SDE parameter estimation, and how it can be exploited to reduce the cost of model overfit.
Model truncation for the SDE(1) model
-------------------------------------
As parameters are optimized to maximize the likelihood, parameter estimates can diverge to infinity, resulting in numerical problems in the likelihood computation [@jones1981fitcar]. Also, it has been reported that not all AR models have a continuous-time counterpart [@soderstrom1991contarma]. In this section, we show how these phenomena are closely related through a theoretical analysis of the the SDE(1) model. Furthermore, we show how this phenomenon ultimately results in more accurate estimates.
The SDE(1) model is an important model for many applications. It is also known in the literature as the Ornstein-Uhlenbeck process and is a special case of the Matérn kernel [@rasmussen2003gaussian]. For data regularly sampled at interval $T$, the maximum likelihood estimate of the SDE(1) parameter $\hat{a}$ can be computed analytically [@sarkka2019sde]:
$$\hat{a}=\frac{1}{T}\log\left[\hat{\alpha}\right]$$ where $\hat{\alpha}$ is the estimated AR(1) parameter. For a white noise process, the estimated AR(1) parameter is distributed symmetrically around $\hat{\alpha}=0$ with standard deviation $1/\sqrt{N}$, where $N$ is the number of observations [@priestley1981spectral]. Hence, $\hat{\alpha}$ is negative for 50% of signals. In this case, the AR(1) process has no continuous-time counterpart.
It can be shown that, under these conditions, the likelihood monotonically increases for $a\rightarrow\infty$. In this limit, the SDE(1) model is equivalent to the white noise model. In practice, the true model order is unknown, and so it is critical that an estimator returns an accurate estimate under these circumstances. This is achieved using data-driven model truncation. With model trunction, the estimation algorithm for an SDE($p$) model can return a lower SDE($p'$) model if the likelihood indicates that the lower order model fits better to the data. For the SDE(1) case, this amounts in returning an SDE(0) or white noise model when $\hat{\alpha}<0$. We discuss model truncation for higher order models in the next section.
Model truncation for higher order models
----------------------------------------
As described in section \[subsec:Initialization\], SDE($p$) models are estimated incrementally. If parameter divergence occurs, the model returned by the optimization procedure has large but finite values because of the limits introduced on parameter values. In this case, the lower order model will have a larger likelihood, and is consequently the final maximum likelihood estimate. While this phenomena can be easily analyzed for the SDE(1) model estimated from white noise, this is an important phenomena more generally, as it occurs whenever the order of the estimated model is greater than the true model order. Since the true model order is unknown for experimental data, it is desirable to estimate a high order model, so that a wide range of processes can be represented. Model truncation reduces the cost of overfit, i.e., the statistical estimation error induced by estimation SDE($p$) models where $p$ exceeds the true model order.
KLD reduction achieved in white noise
-------------------------------------
Adding to the theoretical analysis of the SDE(1) case, we quantify the improvement that is achieved with data-driven model truncation in a simulation experiment where SDE models are estimated from a non-uniformly sampled white noise process. The Kullback-Leibler discrepancy $D_{o}$ as a function of model order is given in figure \[fig:whitenoise\], along with the theoretical expected value for the KLD $D_{o}$ from eq \[eq:expect-kld\]. We use $D_{o}$ here, because it uses the same time vector $\mathbf{t}$ as the likelihood. Therefore, we can use the theoretical expectation (\[eq:expect-kld\]).
As expected, we observe a significant reduction in the error for higher order models due to model truncation. Because the theoretical expression is accurate for the discrete-time AR models, these results also quantify the error reduction compared to AR models.
![\[fig:whitenoise\]Kullback-Leibler Discrepancy (KLD) $D_{o}$ as a function of SDE model order $p$ estimated from non-uniformly sampled white noise. Because of data-driven model truncation, the KLD observed in simulation experiments for estimated SDE($p$) models is below the theoretical prediction for the KLD.](figures/whitenoise){width="1\columnwidth"}
\[sec:performance\]Estimator performance
========================================
In this section, we discuss estimator performance across the test cases A, B and C introduced in section \[sec:simulations\]. Figure \[fig:mle-accuracy\] shows the average KLD at interval $T=0.2$, $D_{0.2}$, as well as sample spectral estimates compared to the true spectrum up to the Nyquist frequency corresponding to the KLD interval: $f=1/2T=2.5$. Note that we use a KLD at a time interval $T$ that is considerably shorter than the average sampling interval $T_{av}=1$. This allows us to quantify the estimator capability to characterize the process at short time scales, or, in the frequency domain, up to frequencies beyond the Nyquist rate corresponding the the average sampling time: $f_{N,Tav}=0.5$. Estimates are shown for the SDE(8) ML estimate obtained using the roots parameterization ($\mathbf{r}$), referred to as the “MLE root” estimate.
We conclude that this estimator reliably estimates the power spectrum for the test processes, which cover all of the challenging conditions listed in section \[subsec:Test-processes\]. Accurate estimates can be obtained well above $f_{N,Tav}$. The squared exponential case is the most challenging case, because of the joint occurrence of model misfit and a high dynamic range. For this case, estimates are less accurate across the entire frequency range because of statistical estimation errors.
We specifically draw attention to the absence of large erroneous peaks in the spectral estimates, as they have been reported previously in the literature, including a study of the autoregressive (AR) ML estimator for similar test cases [@dewaele2018kernel]. The absence of these peaks in the SDE estimate is a consequence of the data-driven model truncation introduced in section \[sec:truncation\], and results in substantially more accurate spectral estimates. Compared to [@dewaele2018kernel], the model accuracy is much improved for cases A and B. Only in case C, the AR ML estimate in [@dewaele2018kernel] is more accurate. However, for this case, a model of the true order (AR(4)) was estimated instead of an AR(8) model, explaining the absence of erroneous peaks. In general, we cannot assume knowledge of the true model order of the generating process for a given experimental dataset.
Given the remarkable success in suppressing erroneous peaks compared to previous work, the relevance of case C is to show the capability of the estimator to accurately estimate true spectral peaks at a high frequency. Here, our work has a key advantage over the algorithm proposed in [@broersen2009spurious], in which *all* high frequency AR roots are eliminated.
![\[fig:mle-accuracy\]Accuracy of SDE(8) models estimated from simulated data, sampled non-uniformly at an average sampling time of $T_{av}=1$ ($S=50$ simulation runs for each case A, B and C). **Left top:** Kullback-Leibler Discrepancy at time interval $T=0.2$ ($D_{0.2}$), averaged over all simulation runs. **Remaining graphs:** Representative sample spectral estimates computed from the SDE(8) estimates, compared to the true power spectrum up a frequency of $1/2T=2.5$. Accurate estimates are achieved across the entire frequency range, going well beyond the Nyquist frequency for the average sampling interval $f_{N,Tav}=0.5$, which is indicated by the black dashed line.](figures/mle-accuracy){width="1\columnwidth"}
Figure \[fig:cmp-estimators\] compares the accuracy of the “MLE root” estimator to two alternative estimators: the coefficient parameterization “MLE coef” and random root initiation (“random init root”). While all estimators perform well for the basic SDE(1) case A, the “MLE coef” and “random init root” estimators have reduced quality for the more complex cases B and C, that require accurate higher order SDE estimates for accurate results.
For the “MLE coef” estimator, the reduced quality can be explained from the fact that the likelihood computation is less numerically stable. Also, for models of order $p>3$, the search space of positive coefficients also contains non-stationary models. The performance degradation is greatest for the “random init root” estimator, with an increase of a factor of 4 case C. While random initiation converges to a good solution for many simulations, it occasionally fails to find a good optimum resulting in a very large error. This performance degradation is expected to increase further with increasing model order.
![\[fig:cmp-estimators\]Comparison of the estimation accuracy of 3 alternative SDE parameter estimators. The reported values are the Kullback-Leibler Discrepancy $D_{0.2}$ relative to the “MLE root” estimate.](figures/cmp-estimators){width="1\columnwidth"}
\[sec:rainfall\]Analysis of monsoon rainfall variability
========================================================
Long-term variability in monsoon rainfall is studied using radiometric-dated, speleothem oxygen isotope $\delta^{18}O$ records [@sinha2015trends]. The data is intrinsically irregularly sampled, because it is formed by natural deposition rather than experimenter controlled sampling. For the same reason, irregular sampling occurs for many other long-term climate records as well, e.g. ice core data [@petit1999climate].
The average sampling rate of speleothem data depends on the measurement location. The current dataset is suitable for algorithm benchmarking because it has a higher average sampling rate than datasets collected from other locations. This allows us to study algorithm performance for different sampling rates, by subsampling the original data, and comparing the results to estimates obtained from the full dataset. Also, the dataset is publicly available as supplementary material to [@sinha2015trends] for reproducibility of results. The oxygen isotope data consists of $N=1848$ irregularly sampled observations of $\delta^{18}O$ anomalies over a time span of $2147$ years, resulting in average sampling interval of $T_{0}=1.16$ years.
We estimate SDE(8) models from detrended $\delta^{18}O$ data. A reference estimate is obtained using the complete dataset. To emulate a lower sampling rate, we then estimate models from $5$ random subsets of the original data, at an average sampling interval of $T_{av}=5$ years. The resulting estimated spectra and model fit are given in figure \[fig:monsoon-psd\]. The reference SDE(8) model is truncated to an SDE(1) model. Models estimated from the subsampled data similarly exploit model truncation to achieve accurate spectral estimates, avoiding erroneous peaks in the estimate, despite a much lower sampling frequency compared to the full dataset. The maximum likelihood is achieved at either order $p=1$ (4 out of 5 subsets) or $p=5$ (1/5) (see figure \[fig:monsoon-psd\], bottom).
![\[fig:monsoon-psd\]**Top:** Power spectra estimated from detrended speleothem oxygen isotope $\delta^{18}O$ data using SDE(8) estimates. Estimates are based either on all data (black dashed line), or on subsets of data (colored lines) that are randomly subsampled at an average sampling interval $T=5$ years. While based on a much smaller dataset, the estimates from subsampled data remain accurate. **Bottom**: Model fit (negative log likelihood $L$) as a function of model order for the models estimated from subsampled data, with colors corresponding to the same subsets as in the top spectrum plots. The yellow diamond indicates the order of the ML estimate after data-driven model truncation.](figures/rainfall-results){width="1\columnwidth"}
\[sec:conclusions\]Concluding remarks
=====================================
The proposed SDE-based method for spectral estimation from non-uniformly sampled data provides a more accurate estimate than existing methods. This is achieved using more accurate initialization combined with data-driven model truncation. We have shown the performance of this estimator in simulation experiments, and by application of the algorithm to experimental rainfall variability data.
Further advances can be achieved through model regularization. One way to achieve regularization is by means of order selection. Novel order selection criteria can be developed based on the reported behavior of SDE estimators in figure \[fig:whitenoise\]. This behavior deviates significantly from the theoretical behavior on which criteria such as the Akaike Information Criterion are based.
Alternatively, regularization can be achieved by introducing informative priors in a Bayesian approach. This has shown promise in discrete time estimation, albeit at the cost of a considerably higher computational load [@dewaele2018kernel]. Given the results in ML estimation, we expect that a continuous-time model will also produce superior results for Bayesian estimation.
An additional benefit of a Bayesian approach is that it has the flexibility to produce a posterior summary that is relevant to a particular quantity of interest. This may be exploited to produce an accurate spectral estimate for a frequency range of interest, independent of the (average) sampling frequency. To accommodate Bayesian inference, the code accompanying this paper can compute posterior samples for the SDE model parameters using Hamiltonian Monte Carlo sampling.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'In this paper we introduce the notion of Hom-pre-Lie bialgebra in the general framework of the cohomology theory for Hom-Lie algebras. We show that Hom-pre-Lie bialgebras, standard Manin triples for Hom-pre-Lie algebras and certain matched pairs of Hom-pre-Lie algebras are equivalent. Due to the usage of the cohomology theory, it makes us successfully study the coboundary Hom-pre-Lie bialgebras. The notion of Hom-${\mathfrak s}$-matrix is introduced, by which we can construct Hom-pre-Lie bialgebras naturally. Finally we introduce the notions of Hom-${{\mathcal{O}}}$-operators on Hom-pre-Lie algebras and Hom-L-dendriform algebras, by which we construct Hom-${\mathfrak s}$-matrices.'
address:
- 'Department of Mathematics, Jilin University, Changchun 130012, Jilin, China'
- 'University of Haute Alsace, IRIMAS- Département de Mathématiques, Université de Haute Alsace, France'
- 'Department of Mathematics, Jilin University, Changchun 130012, Jilin, China'
author:
- Shanshan Liu
- Abdenacer Makhlouf
- Lina Song
title: 'On Hom-pre-Lie bialgebras '
---
Introduction
============
For a given algebraic structure determined by a set of multiplications of various arities and a set of relations among the operations, a bialgebra structure on this algebra is obtained by a corresponding set of comultiplications together with a set of compatibility conditions between the multiplications and comultiplications. A good compatibility condition is prescribed by a rich structure theory and effective constructions. The most famous examples of bialgebras are associative bialgebras and Lie bialgebras, which have important applications in both mathematics and mathematical physics.
The notion of a Hom-Lie algebra was introduced by Hartwig, Larsson and Silvestrov in [@HLS] as part of a study of deformations of the Witt and the Virasoro algebras. In a Hom-Lie algebra, the Jacobi identity is twisted by a linear map (homomorphism), and it is called Hom-Jacobi identity. Different types of Hom-algebras were introduced and widely studied. Recently, in [@ELMS], Elchinger, Lundengard, Makhlouf and Silvestrov extend the result in [@HLS] to the case of $(\sigma,\tau)$-derivations. Due to the importance of the aforementioned bialgebra theory, the bialgebra theory for Hom-algebras was deeply studied in [@Cai-Sheng; @MS3; @sheng1; @TY; @Yao1; @Yao3]
Pre-Lie algebras (also called left-symmetric algebras, quasi-associative algebras, Vinberg algebras and so on) are a class of nonassociative algebras that appeared in many fields in mathematics and mathematical physics. See the survey [@Pre-lie; @algebra; @in; @geometry] and the references therein for more details. The notion of left-symmetric bialgebra was introduced in [@Bai]. The author also introduced the notion of ${\mathfrak s}$-matrices in [@Bai] to produce left-symmetric bialgebras. The notion of a Hom-pre-Lie algebra was introduced in [@MS2] and play important roles in the study of Hom-Lie bialgebras and Hom-Lie 2-algebras [@sheng1; @SC]. Recently, Hom-pre-Lie algebras were studied from several aspects. The geometrization of Hom-pre-Lie algebras was studied in [@Qing]; universal $\alpha$-central extensions of Hom-pre-Lie algebras were studied in [@sunbing].
The purpose of this paper is to give a systematic study of the bialgebra theory for Hom-pre-Lie algebras. Note that the notion of a Hom-pre-Lie bialgebra was already introduced in [@QH] under the terminology of Hom-left-symmetric algebra. However the bialgebra structure given in [@QH] does not enjoy a coboundary theory. This is also one of our motivation to study Hom-pre-Lie bialgebras that enjoy a rich structure theory. Our Hom-pre-Lie bialgebra structure enjoy the following properties:
- equivalent to a Manin triple for Hom-pre-Lie algebras as well as certain matched pair of Hom-pre-Lie algebras;
- Hom-${\mathfrak s}$-matrices can be defined to produce Hom-pre-Lie bialgebras;
- ${{\mathcal{O}}}$-operators on Hom-pre-Lie algebras can be defined to give Hom-${\mathfrak s}$-matrices in the semidirect product Hom-pre-Lie algebras.
The paper is organized as follows. In Section 2, we recall relevant definitions and results about matched pairs of Hom-Lie algebras and matched pairs of Hom-pre-Lie algebras. In Section 3, we introduce the notion of Hom-pre-Lie bialgebra and show that it is equivalent to Manin triples as well as matched pairs. In Section 4, we study coboundary Hom-pre-Lie bialgebras and introduce the notion of a Hom-${\mathfrak s}$-matrix, by which we can construct a Hom-pre-Lie bialgebra naturally. In Section 5, we introduce the notion of an ${{\mathcal{O}}}$-operator and the notion of a Hom-L-dendriform algebra, by which we can construct Hom-${\mathfrak s}$-matrices.
[**Acknowledgements.** ]{} We give warmest thanks to Yunhe Sheng for helpful comments.
Preliminaries
=============
In this section, we briefly recall matched pairs of Hom-Lie algebras and matched pairs of Hom-pre-Lie algebras, as preparation for our later study of Hom-pre-Lie bialgebras.
[([@HLS])]{} A [**Hom-Lie algebra**]{} is a triple $({\mathfrak g},[\cdot,\cdot]_{\mathfrak g},\phi_{\mathfrak g})$ consisting of a linear space ${\mathfrak g}$, a skew-symmetric bilinear map $[\cdot,\cdot]_{\mathfrak g}:\wedge^2{\mathfrak g}\longrightarrow {\mathfrak g}$ and an algebra morphism $\phi_{\mathfrak g}:{\mathfrak g}\longrightarrow {\mathfrak g}$, satisfying: $$[\phi_{\mathfrak g}(x),[y,z]_{\mathfrak g}]_{\mathfrak g}+[\phi_{\mathfrak g}(y),[z,x]_{\mathfrak g}]_{\mathfrak g}+[\phi_{\mathfrak g}(z),[x,y]_{\mathfrak g}]_{\mathfrak g}=0,\quad
\forall~x,y,z\in {\mathfrak g}.$$ A Hom-Lie algebra $({\mathfrak g},[\cdot,\cdot]_{\mathfrak g},\phi_{\mathfrak g})$ is said to be regular if $\phi_{\mathfrak g}$ is invertible.
[([@AEM; @sheng3])]{}\[defi:hom-lie representation\] A [**representation**]{} of a Hom-Lie algebra $({\mathfrak g},[\cdot,\cdot]_{\mathfrak g},\phi_{\mathfrak g})$ on a vector space $V$ with respect to $\beta\in{\mathfrak {gl}}(V)$ is a linear map $\rho:{\mathfrak g}\longrightarrow {\mathfrak {gl}}(V)$, such that for all $x,y\in {\mathfrak g}$, the following equalities are satisfied: $$\begin{aligned}
\label{hom-lie-rep-1}\rho(\phi_{\mathfrak g}(x))\circ \beta&=&\beta\circ \rho(x),\\
\label{hom-lie-rep-2}\rho([x,y]_{\mathfrak g})\circ \beta&=&\rho(\phi_{\mathfrak g}(x))\circ\rho(y)-\rho(\phi_{\mathfrak g}(y))\circ\rho(x).\end{aligned}$$
We denote a representation by $(V,\beta,\rho)$. For all $x\in\mathfrak{g}$, we define ${\mathrm{ad}}_{x}:\mathfrak{g}{\,\rightarrow\,}\mathfrak{g}$ by $$\begin{aligned}
{\mathrm{ad}}_{x}(y)=[x,y]_{\mathfrak g},\quad\forall y \in \mathfrak{g}.\end{aligned}$$ Then ${\mathrm{ad}}:{\mathfrak g}\longrightarrow{\mathfrak {gl}}(\frak g)$ is a representation of the Hom-Lie algebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak g},\phi_{\mathfrak g})$ on ${\mathfrak g}$ with respect to $\phi_{\mathfrak g}$, which is called the adjoint representation.
Let $({\mathfrak g},[\cdot,\cdot]_{\mathfrak g},\phi_{\mathfrak g})$ be a Hom-Lie algebra, $(V,\beta_V,\rho_V)$ and $(W,\beta_W,\rho_W,)$ its representations. Then $(V\otimes W,\beta_V\otimes \beta_W,\rho_V\otimes \beta_W+\beta_V\otimes \rho_W)$ is a representation of $({\mathfrak g},[\cdot,\cdot]_{\mathfrak g},\phi_{\mathfrak g})$.
[([@sheng1])]{} A [**matched pair of Hom-Lie algebras**]{}, which is denoted by $({\mathfrak g},{\mathfrak g}',\rho,\rho')$, consists of two Hom-Lie algebras $({\mathfrak g},[\cdot,\cdot]_{\mathfrak g},\phi_{\mathfrak g})$ and $({\mathfrak g}',[\cdot,\cdot]_{{\mathfrak g}'},\phi_{{\mathfrak g}'})$, together with representations $\rho:{\mathfrak g}\longrightarrow {\mathfrak {gl}}(g')$ and $\rho':{\mathfrak g}'\longrightarrow {\mathfrak {gl}}(g)$ with respect to $\phi_{{\mathfrak g}'}$ and $\phi_{\mathfrak g}$ respectively, such that for all $ x,y \in {\mathfrak g}, x',y'\in {\mathfrak g}'$ the following conditions are satisfied: $$\begin{aligned}
\label{matche-pair-1}\rho'(\phi_{\mathfrak g}'(x'))[x,y]_{\mathfrak g}&=&[\rho'(x')(x),\phi_{\mathfrak g}(y)]_{\mathfrak g}+[\phi_{\mathfrak g}(x),\rho'(x')(y)]_{\mathfrak g}\\
\nonumber &&+\rho'(\rho(y)(x'))(\phi_{\mathfrak g}(x))-\rho'(\rho(x)(x'))(\phi_{\mathfrak g}(y)),\\
\label{matche-pair-2}\rho(\phi_{\mathfrak g}(x))[x',y']_{{\mathfrak g}'}&=&[\rho(x)(x'),\phi_{{\mathfrak g}'}(y')]_{{\mathfrak g}'}+[\phi_{{\mathfrak g}'}(x'),\rho(x)(y')]_{{\mathfrak g}'}\\
\nonumber &&+\rho(\rho'(y')(x))(\phi_{{\mathfrak g}'}(x'))-\rho(\rho'(x')(x))(\phi_{{\mathfrak g}'}(y')).\end{aligned}$$
We define $\phi_d:{\mathfrak g}\oplus {\mathfrak g}'\longrightarrow {\mathfrak g}\oplus {\mathfrak g}'$ by $$\phi_d(x,x')=(\phi_{\mathfrak g}(x),\phi_{{\mathfrak g}'}(x')),$$ and define a skew-symmetric bilinear map $[\cdot,\cdot]_d:\wedge^2({\mathfrak g}\oplus {\mathfrak g}')\longrightarrow{\mathfrak g}\oplus {\mathfrak g}'$ by $$[(x,x'),(y,y')]_d=([x,y]_{\mathfrak g}+\rho'(x')(y)-\rho'(y')(x),[x',y']_{{\mathfrak g}'}+\rho(x)(y')-\rho(y)(x')).$$
[([@sheng1])]{} With the above notations, $({\mathfrak g}\oplus {\mathfrak g}',[\cdot,\cdot]_d,\phi_d)$ is a Hom-Lie algebra if and only if $({\mathfrak g},{\mathfrak g}',\rho,\rho')$ is a matched pair of Hom-Lie algebras.
[([@MS2])]{} A [**Hom-pre-Lie algebra**]{} $(A,\cdot,\alpha)$ is a vector space $A$ equipped with a bilinear product $\cdot:A\otimes A\longrightarrow A$, and $\alpha\in {\mathfrak {gl}}(A)$, such that for all $x,y,z\in A$, $\alpha(x \cdot y)=\alpha(x)\cdot \alpha(y)$ and the following equality is satisfied: $$\begin{aligned}
(x\cdot y)\cdot \alpha(z)-\alpha(x)\cdot (y\cdot z)=(y\cdot x)\cdot \alpha(z)-\alpha(y)\cdot (x\cdot z).\end{aligned}$$ A Hom-pre-Lie algebra $(A,\cdot,\alpha)$ is said to be regular if $\alpha$ is invertible.
Let $(A,\cdot,\alpha)$ be a Hom-pre-Lie algebra. We always assume that it is regular, i.e. $\alpha$ is invertible. The commutator $[x,y]=x\cdot y-y\cdot x$ gives a Hom-Lie algebra $(A,[\cdot,\cdot],\alpha)$, which is denoted by $A^C$ and called the sub-adjacent Hom-Lie algebra of $(A,\cdot,\alpha)$.
[([@LSS])]{} A [**morphism**]{} from a Hom-pre-Lie algebra $(A,\cdot,\alpha)$ to a Hom-pre-Lie algebra $(A',\cdot',\alpha')$ is a linear map $f:A\longrightarrow A'$ such that for all $x,y\in A$, the following equalities are satisfied: $$\begin{aligned}
\label{homo-1}f(x\cdot y)&=&f(x)\cdot' f(y),\hspace{3mm}\forall x,y\in A,\\
\label{homo-2}f\circ \alpha&=&\alpha'\circ f.\end{aligned}$$
[([@QH])]{}\[defi:hom-pre representation\] A [**representation**]{} of a Hom-pre-Lie algebra $(A,\cdot,\alpha)$ on a vector space $V$ with respect to $\beta\in{\mathfrak {gl}}(V)$ consists of a pair $(\rho,\mu)$, where $\rho:A\longrightarrow {\mathfrak {gl}}(V)$ is a representation of the sub-adjacent Hom-Lie algebra $A^C$ on $V$ with respect to $\beta\in{\mathfrak {gl}}(V)$, and $\mu:A\longrightarrow {\mathfrak {gl}}(V)$ is a linear map, for all $x,y\in A$, satisfying: $$\begin{aligned}
\label{rep-1}\beta\circ \mu(x)&=&\mu(\alpha(x))\circ \beta,\\
\label{rep-2}\mu(\alpha(y))\circ\mu(x)-\mu(x\cdot y)\circ \beta&=&\mu(\alpha(y))\circ\rho(x)-\rho(\alpha(x))\circ\mu(y).\end{aligned}$$
We denote a representation of a Hom-pre-Lie algebra $(A,\cdot,\alpha)$ by $(V,\beta,\rho,\mu)$. Furthermore, Let $L,R:A\longrightarrow {\mathfrak {gl}}(A)$ be linear maps, where $L_xy=x\cdot y,\quad R_xy=y\cdot x$. Then $(A,\alpha,L,R)$ is also a representation, which we call the regular representation.
Let $(A,\cdot,\alpha)$ be a Hom-pre-Lie algebra. For any integer $s$, define $L^s,R^s:A\longrightarrow {\mathfrak {gl}}(A)$ by $$L^s_xy=\alpha^s(x)\cdot y,\quad R^s_xy=y\cdot \alpha^s(x), \quad \forall x,y \in A.$$ Then $(A,\alpha,L^s,R^s)$ is a representation of a Hom-Lie algebra $(A,\cdot,\alpha)$.
For all $x,y,z\in A$, we have $$\begin{aligned}
L^s_{\alpha(x)}\alpha(y)=\alpha^{s+1}(x)\cdot \alpha(y)=\alpha(\alpha^s(x)\cdot y)=\alpha(L^s_xy),\end{aligned}$$ which implies that $L^s_{\alpha(x)}\circ \alpha=\alpha\circ L^s_x$. Similarly, we have $R^s_{\alpha(x)}\circ \alpha=\alpha\circ R^s_x$.
By the definition of a Hom-pre-Lie algebra, we have $$\begin{aligned}
\nonumber L^s_{\alpha(x)}L^s_y(z)-L^s_{\alpha(y)}L^s_x(z)
\nonumber&=&\alpha^{s+1}(x)\cdot(\alpha^s(y)\cdot z)-\alpha^{s+1}(y)\cdot(\alpha^s(x)\cdot z)\\
\nonumber&=&(\alpha^s(x)\cdot \alpha^s(y))\cdot\alpha(z)-(\alpha^s(y)\cdot \alpha^s(x))\cdot\alpha(z)\\
\nonumber &=&\alpha^s([x,y])\cdot\alpha(z)\\
\nonumber &=&L^s_{[x,y]}\alpha(z).\end{aligned}$$ Similarly, we have $$R^s_{\alpha(y)}\circ R^s_x-R^s_{x\cdot y}\circ \alpha=R^s_{\alpha(y)}\circ L^s_x-L^s_{\alpha(x)}\circ R^s_y.$$ This finishes the proof.
The notion of matched pair of Hom-pre-Lie algebras was given in [@QH].
[([@QH])]{} A [**matched pair of Hom-pre-Lie algebras**]{} $(A,B,l_A,r_A,l_B,r_B)$ consists of two Hom-pre-Lie algebras $(A,\cdot,\alpha_A)$ and $(B,\circ,\alpha_B)$, together with linear maps $l_A,r_A:A\longrightarrow {\mathfrak {gl}}(B)$ and $l_B,r_B:B\longrightarrow {\mathfrak {gl}}(A)$ such that $(B,\alpha_B,l_A,r_A)$ and $(A,\alpha_A,l_B,r_B)$ are representations and for all $x,y\in A, a,b\in B$, satisfying the following conditions: $$\begin{aligned}
\label{pre-matche-pair-1}r_A(\alpha_A(x))\{a,b\}&=&r_A(l_B(b)x)\alpha_B(a)-r_A(l_B(a)x)\alpha_B(b)+\alpha_B(a)\circ(r_A(x)b)\\
\nonumber &&-\alpha_B(b)\circ(r_A(x)a),\\
\label{pre-matche-pair-2}l_A(\alpha_A(x))(a\circ b)&=&-l_A(l_B(a)x-r_B(a)x)\alpha_B(b)+(l_A(x)a-r_A(x)a)\circ \alpha_B(b)\\
\nonumber &&+r_A(r_B(b)x)\alpha_B(a)+\alpha_B(a)\circ(l_A(x)b),\\
\label{pre-matche-pair-3}r_B(\alpha_B(a))[x,y]&=&r_B(l_A(y)a)\alpha_A(x)-r_B(l_A(x)a)\alpha_A(y)+\alpha_A(x)\cdot(r_B(a)y)\\
\nonumber &&-\alpha_A(y)\cdot(r_B(a)x),\\
\label{pre-matche-pair-4}l_B(\alpha_B(a))(x\cdot y)&=&-l_B(l_A(x)a-r_A(x)a)\alpha_A(y)+(l_B(a)x-r_B(a)x)\cdot \alpha_A(y)\\
\nonumber &&+r_B(r_A(y)a)\alpha_A(x)+\alpha_A(x)\cdot(l_B(a)y),\end{aligned}$$ where $[\cdot,\cdot]$ is the Lie bracket of the sub-adjacent Hom-Lie algebra $A^C$ and $\{\cdot,\cdot\}$ is the Lie bracket of the sub-adjacent Hom-Lie algebra $B^C.$
We define a bilinear operation $\diamond:\otimes^2(A\oplus B)\lon(A\oplus B)$ by $$\label{Hom-pre-lie matched pair}
(x+a)\diamond(y+b):=x\cdot y+l_B(a)y+r_B(b)x+a\circ b+l_A(x)b+r_A(y)a,$$ and a linear map $\alpha_A\oplus \alpha_B:A\oplus B\longrightarrow A\oplus B$ by $$(\alpha_A\oplus \alpha_B)(x+a):=\alpha_A(x)+\alpha_B(a).$$ The following is proved in [@QH].
[([@QH])]{}\[matched-pair-Hom-pre-Lie algebra\] With the above notations, $(A\oplus B,\diamond,\alpha_A\oplus \alpha_B)$ is a Hom-pre-Lie algebra if and only if $(A,B,l_A,r_A,l_B,r_B)$ is a matched pair of Hom-pre-Lie algebras.
Hom-pre-Lie bialgebras
======================
In this section, we introduce the notions of Manin triple for Hom-pre-Lie algebras and Hom-pre-Lie bialgebra. We show that Hom-pre-Lie bialgebras, standard Manin triples for Hom-pre-Lie algebras and certain matched pairs of Hom-pre-Lie algebras are equivalent.
Let $(V,\beta,\rho,\mu)$ be a representation of a Hom-pre-Lie algebra $(A,\cdot,\alpha)$. In the sequel, we always assume that $\beta$ is invertible. For all $x\in A,u\in V,\xi\in V^*$, define $\rho^*:A\longrightarrow{\mathfrak {gl}}(V^*)$ and $\mu^*:A\longrightarrow{\mathfrak {gl}}(V^*)$ as usual by $$\langle \rho^*(x)(\xi),u\rangle=-\langle\xi,\rho(x)(u)\rangle,\quad \langle \mu^*(x)(\xi),u\rangle=-\langle\xi,\mu(x)(u)\rangle,\quad\forall x\in A,~\xi\in V^*,~u\in V.$$ Then define $\rho^\star:A\longrightarrow{\mathfrak {gl}}(V^*)$ and $\mu^\star:A\longrightarrow{\mathfrak {gl}}(V^*)$ by $$\begin{aligned}
\label{eq:1.3}\rho^\star(x)(\xi):=\rho^*(\alpha(x))\big{(}(\beta^{-2})^*(\xi)\big{)},\\
\label{eq:1.4}\mu^\star(x)(\xi):=\mu^*(\alpha(x))\big{(}(\beta^{-2})^*(\xi)\big{)}.\end{aligned}$$
[([@LSS])]{}\[dual-rep\] Let $(V,\beta,\rho,\mu)$ be a representation of a Hom-pre-Lie algebra $(A,\cdot,\alpha)$. Then $(V^*,(\beta^{-1})^*,\rho^\star-\mu^\star,-\mu^\star)$ is a representation of $(A,\cdot,\alpha)$, which is called the dual representation of $(V,\beta,\rho,\mu)$.
Let $(A,\cdot,\alpha)$ be a Hom-pre-Lie algebra. Then $(A^*,(\alpha^{-1})^*,{\mathrm{ad}}^\star,-R^\star)$ is a representation of $(A,\cdot,\alpha)$, where ${\mathrm{ad}}=L-R$ is the adjoint representation of the sub-adjacent Hom-Lie algebra $A^C.$
In the sequel, when there is a Hom-pre-Lie algebra structure on $A^*$, we will use ${\mathcal{L}},~{\mathcal{R}}$ and $\mathfrak{ad}:={\mathcal{L}}-{\mathcal{R}}$ to denote the corresponding operations.
Before we introduce the notions of Manin triple and Hom-pre-Lie bialgebra, we give an important relation between matched pairs of Hom-pre-Lie algebras and matched pairs of the associated sub-adjacent Hom-Lie algebras.
\[matched-pair-equivalent\] Let $(A,\cdot,\alpha)$ and $(A^\ast,\circ,(\alpha^{-1})^\ast)$ be Hom-pre-Lie algebras. Then $(A^C,(A^\ast)^C,L^\star,{\mathcal{L}}^\star)$ is a matched pair of Hom-Lie algebras if and only if $(A,A^\ast,{\mathrm{ad}}^\star,-R^\star,\mathfrak{ad}^\star,-{\mathcal{R}}^\star)$ is a matched pair of Hom-pre-Lie algebras.
We denote the bracket of the sub-adjacent Hom-pre-Lie algebra $(A^*)^C$ by $\{\cdot,\cdot\}$. For all $x,y\in A, \xi,\eta\in A^\ast$, we have $$\begin{aligned}
\label{matched-1}
\nonumber &&\langle R^\star_{\alpha(\alpha^{-2}(x))}\{\xi,\eta\}-R^\star_{\mathfrak{ad}^\star_\eta\alpha^{-2}(x)}(\alpha^{-1})^\ast(\xi)+R^\star_{\mathfrak{ad}^\star_\xi\alpha^{-2}(x)}(\alpha^{-1})^\ast(\eta)-(\alpha^{-1})^\ast(\xi)\circ R^\star_{\alpha^{-2}(x)}\eta\\
\nonumber &&+(\alpha^{-1})^\ast(\eta)\circ R^\star_{\alpha^{-2}(x)}\xi,\alpha^2(y)\rangle\\
\nonumber&=&-\langle\{\xi,\eta\},R_{\alpha^{-2}(x)}y\rangle+\langle (\alpha^{-1})^\ast(\xi),R_{\alpha^{-1}(\mathfrak{ad}^\star_\eta\alpha^{-2}(x))}(y)\rangle-\langle (\alpha^{-1})^\ast(\eta),R_{\alpha^{-1}(\mathfrak{ad}^\star_\xi\alpha^{-2}(x))}(y)\rangle\\
\nonumber &&-\langle {\mathcal{L}}_{(\alpha^{-1})^\ast(\xi)}R^\star_{\alpha^{-2}(x)}\eta,\alpha^2(y)\rangle+\langle {\mathcal{L}}_{(\alpha^{-1})^\ast(\eta)}R^\star_{\alpha^{-2}(x)}\xi,\alpha^2(y)\rangle\\
\nonumber &=&\langle-\{\xi,\eta\},L_y\alpha^{-2}(x)\rangle+\langle(\alpha^{-1})^\ast(\xi),L_y\alpha^{-1}(\mathfrak{ad}^\star_\eta\alpha^{-2}(x))\rangle-\langle(\alpha^{-1})^\ast(\eta),L_y\alpha^{-1}(\mathfrak{ad}^\star_\xi\alpha^{-2}(x))\rangle\\
\nonumber &&+\langle R^\star_{\alpha^{-2}(x)}\eta,{\mathcal{L}}^\ast_{(\alpha^{-1})^\ast(\xi)}\alpha^2(y)\rangle-\langle R^\star_{\alpha^{-2}(x)}\xi,{\mathcal{L}}^\ast_{(\alpha^{-1})^\ast(\eta)}\alpha^2(y)\rangle\\
\nonumber &=&\langle L^\star_{\alpha(y)}\{\xi,\eta\},x\rangle-\langle(\alpha^{-1})^\ast L^\ast_y(\alpha^{-1})^\ast(\xi),\mathfrak{ad}^\star_\eta\alpha^{-2}(x)\rangle+\langle(\alpha^{-1})^\ast L^\ast_y(\alpha^{-1})^\ast(\eta),\mathfrak{ad}^\star_\xi\alpha^{-2}(x)\rangle\\
\nonumber &&-\langle\eta,R_{\alpha^{-3}(x)}\alpha^{-2}({\mathcal{L}}^\ast_{(\alpha^{-1})^\ast(\xi)}\alpha^2(y))\rangle+\langle\xi,R_{\alpha^{-3}(x)}\alpha^{-2}({\mathcal{L}}^\ast_{(\alpha^{-1})^\ast(\eta)}\alpha^2(y))\rangle\\
\nonumber&=&\langle L^\star_{\alpha(y)}\{\xi,\eta\},x\rangle+\langle \mathfrak{ad}_{\alpha^\ast(\eta)}\alpha^\ast (L^\ast_y(\alpha^{-1})^\ast(\xi)),\alpha^{-2}(x)\rangle
-\langle \mathfrak{ad}_{\alpha^\ast(\xi)}\alpha^\ast (L^\ast_y(\alpha^{-1})^\ast(\eta)),\alpha^{-2}(x)\rangle\\
\nonumber &&+\langle L^\ast_{\alpha^{-2}({\mathcal{L}}^\star_\xi y)}\eta,\alpha^{-3}(x)\rangle-\langle L^\ast_{\alpha^{-2}({\mathcal{L}}^\star_\eta y)}\xi,\alpha^{-3}(x)\rangle\\
\nonumber &=&\langle L^\star_{\alpha(y)}\{\xi,\eta\},x\rangle+\langle \{\eta,L^\star_{\alpha^{-1}(y)}\alpha^\ast(\xi)\},\alpha^{-1}(x)\rangle-\langle \{\xi,L^\star_{\alpha^{-1}(y)}\alpha^\ast(\eta)\},\alpha^{-1}(x)\rangle\\
\nonumber &&+\langle L^\star_{\alpha^{-3}({\mathcal{L}}^\star_\xi y)}(\alpha^2)^\ast(\eta),\alpha^{-3}(x)\rangle-\langle L^\star_{\alpha^{-3}({\mathcal{L}}^\star_\eta y)}(\alpha^2)^\ast(\xi),\alpha^{-3}(x)\rangle\\
\nonumber &=&\langle L^\star_{\alpha(y)}\{\xi,\eta\},x\rangle+\langle \{\eta,\alpha^\ast (L^\star_y \xi)\},\alpha^{-1}(x)\rangle-\langle \{\xi,\alpha^\ast (L^\star_y \eta)\},\alpha^{-1}(x)\rangle+\langle L^\star_{{\mathcal{L}}^\star_\xi y}(\alpha^{-1})^\ast(\eta),x\rangle\\
\nonumber &&-\langle L^\star_{{\mathcal{L}}^\star_\eta y}(\alpha^{-1})^\ast(\xi),x\rangle\\
\nonumber&=&\langle L^\star_{\alpha(y)}\{\xi,\eta\}+\{(\alpha^{-1})^\ast(\eta),L^\star_y \xi\}-\{(\alpha^{-1})^\ast(\xi),L^\star_y \eta)\}+L^\star_{{\mathcal{L}}^\star_\xi y}(\alpha^{-1})^\ast(\eta)-L^\star_{{\mathcal{L}}^\star_\eta y}(\alpha^{-1})^\ast(\xi),x\rangle,\end{aligned}$$ which implies that $\eqref{matche-pair-2}\Longleftrightarrow \eqref{pre-matche-pair-1}$. We also have $$\begin{aligned}
\nonumber &&\langle-\mathfrak{ad}^\star_{(\alpha^{-1})^\ast(\xi)}(x\cdot y)-\mathfrak{ad}^\star_{L^\star_x \xi}\alpha(y)+{\mathcal{L}}^\star_{\xi}x \cdot\alpha(y)+{\mathcal{R}}^\star_{R^\star_y\xi}\alpha(x)+\alpha(x)\cdot \mathfrak{ad}^\star_\xi y,(\alpha^{-2})^\ast(\eta)\rangle\\
\nonumber&=&\langle x\cdot y,\mathfrak{ad}_\xi \eta\rangle+\langle \alpha(y),\mathfrak{ad}_{\alpha^\ast(L^\star_x\xi)}\eta\rangle-\langle\alpha(y),L^\ast_{{\mathcal{L}}^\star_\xi x}(\alpha^{-2})^\ast(\eta)\rangle-\langle \alpha(x),{\mathcal{R}}_{\alpha^\ast(R^\star_y\xi)}\eta\rangle\\
\nonumber &&-\langle \mathfrak{ad}^\star_\xi y,L^\ast_{\alpha(x)}(\alpha^{-2})^\ast(\eta)\rangle\\
\nonumber&=&\langle L_x y,\{\xi,\eta\}\rangle+\langle \alpha(y),\{\alpha^\ast(L^\star_x \xi),\eta\}\rangle-\langle \alpha(y),L^\star_{\alpha^{-1}({\mathcal{L}}^\star_\xi x)}\eta\rangle-\langle\alpha(x),{\mathcal{L}}_\eta\alpha^\ast(R^\star_y \xi) \rangle-\langle \mathfrak{ad}^\star_\xi y,L^\star_x \eta\rangle\\
\nonumber &=&-\langle \alpha^2(y),L^\star _{\alpha(x)}\{\xi,\eta\}\rangle+\langle \alpha^2(y),\{L^\star_x \xi,(\alpha^{-1})^\ast(\eta)\}\rangle-\langle \alpha^2(y),(\alpha^{-1})^\ast( L^\star_{\alpha^{-1}({\mathcal{L}}^\star_\xi x)}\eta)\rangle\\
\nonumber &&-\langle R_{\alpha^{-1}(y)}\alpha^{-1}({\mathcal{L}}^\ast_\eta \alpha(x)),\xi\rangle+\langle y,\mathfrak{ad}_{\alpha^\ast(\xi)}(\alpha^2)^\ast(L^\star_x \eta)\rangle\\
\nonumber &=&-\langle \alpha^2(y),L^\star _{\alpha(x)}\{\xi,\eta\}\rangle+\langle \alpha^2(y),\{L^\star_x \xi,(\alpha^{-1})^\ast(\eta)\}\rangle-\langle \alpha^2(y),L^\star_{{\mathcal{L}}^\star_\xi x}(\alpha^{-1})^\ast(\eta)\rangle\\
\nonumber&& -\langle \alpha^2({\mathcal{L}}^\ast_\eta \alpha(x))\cdot\alpha^2(y),(\alpha^{-3})^\ast(\xi)\rangle+\langle \alpha^2(y),(\alpha^{-2})^\ast \mathfrak{ad}_{\alpha^\ast(\xi)}(\alpha^2)^\ast(L^\star_x \eta)\rangle\\
\nonumber &=&-\langle \alpha^2(y),L^\star_{\alpha(x)}\{\xi,\eta\}\rangle+\langle \alpha^2(y),\{L^\star_x \xi,(\alpha^{-1})^\ast(\eta)\}\rangle-\langle \alpha^2(y),L^\star_{{\mathcal{L}}^\star_\xi x}(\alpha^{-1})^\ast(\eta)\rangle\\
\nonumber &&-\langle L_{\alpha({\mathcal{L}}^\star_\eta x)}\alpha^2(y),(\alpha^{-3})^\ast(\xi)\rangle+\langle \alpha^2(y),\{(\alpha^{-1})^\ast(\xi),L^\star_x \eta\}\rangle\\
\nonumber&=&\langle \alpha^2(y),-L^\star _{\alpha(x)}\{\xi,\eta\}+\{L^\star_x \xi,(\alpha^{-1})^\ast(\eta)\}-L^\star_{{\mathcal{L}}^\star_\xi x}(\alpha^{-1})^\ast(\eta)+L^\star_{{\mathcal{L}}^\star_\eta x}(\alpha^{-1})^\ast(\xi)+\{(\alpha^{-1})^\ast(\xi),L^\star_x \eta\}\rangle,\end{aligned}$$ which implies that $\eqref{matche-pair-2}\Longleftrightarrow \eqref{pre-matche-pair-4}$.
Thus, we have $\eqref{matche-pair-2}\Longleftrightarrow\eqref{pre-matche-pair-1} \Longleftrightarrow \eqref{pre-matche-pair-4}$. Similarly, we have $\eqref{matche-pair-1}\Longleftrightarrow\eqref{pre-matche-pair-2} \Longleftrightarrow \eqref{pre-matche-pair-3}$. This finishes the proof.
Now we introduce the notion of quadratic Hom-pre-Lie algebra and the notion of Manin triple.
\[invariant\] A [**quadratic Hom-pre-Lie algebra**]{} is a Hom-pre-Lie algebra $(A,\cdot,\alpha)$ equipped with a nondegenerate skew-symmetric bilinear form $\omega\in \wedge^2 A^\ast$ such that for all $x,y,z\in A$, the following invariant conditions hold: $$\begin{aligned}
\label{invariant-1}\omega(\alpha(x),\alpha(y))&=&\omega(x,y),\\
\label{invariant-2}\omega(x\cdot y,\alpha(z))&=&-\omega(\alpha(y),[x,z]).\end{aligned}$$
We denote a quadratic Hom-pre-Lie algebra by $(A,\cdot,\alpha,\omega)$.
A [**Manin triple**]{} for Hom-pre-Lie algebras is a triple $({\mathcal{A}},A_1,A_2)$ in which $({\mathcal{A}},\cdot,\alpha,\omega)$ is a quadratic Hom-pre-Lie algebra, $(A_1,\cdot_1,\alpha_1)$ and $(A_2,\cdot_2,\alpha_2)$ are isotropic Hom-pre-Lie sub-algebras of ${\mathcal{A}}$ such that
- ${\mathcal{A}}=A_1\oplus A_2$ as vector spaces,
- $\alpha=\alpha_1\oplus \alpha_2.$
Two Manin triples $({\mathcal{A}},A_1,A_2)$ and $({\mathcal{B}},B_1,B_2)$ with the bilinear forms $\omega_1$ and $\omega_2$ respectively are isomorphic if there exists an isomorphism of Hom-pre-Lie algebras $f:{\mathcal{A}}\longrightarrow {\mathcal{B}}$ such that $$f(A_1)=B_1, \quad f(A_2)=B_2,\quad \omega_1(x,y)=\omega_2(f(x),f(y)),\quad\forall x,y \in {\mathcal{A}}.$$ Let $(A,\cdot,\alpha)$ and $(A^\ast,\circ,(\alpha^{-1})^\ast)$ be two Hom-pre-Lie algebras, for all $x,y\in A, \xi,\eta\in A^*,$ define a bilinear operator $\diamond:\otimes^2(A\oplus A^*)\lon(A\oplus A^*)$ by $$\label{Hom-pre-lie standard-matched pair}
(x+\xi)\diamond(y+\eta):=x\cdot y+\mathfrak{ad}^\star(\xi)y-{\mathcal{R}}^\star(\eta)x+\xi\circ \eta+{\mathrm{ad}}^\star(x)\eta-R^\star(y)\xi.$$ If $(A\oplus A^*,\diamond,\alpha\oplus (\alpha^{-1})^*)$ is a Hom-pre-Lie algebra, such that $(A,\cdot,\alpha)$ and $(A^\ast,\circ,(\alpha^{-1})^\ast)$ are Hom-pre-Lie subalgebras, by computation, the natural nondegenerate skew-symmetric bilinear form $\bar{\omega}$ on $A\oplus A^\ast$ given by $$\label{standard manin triple}
\bar{\omega}(x+\xi,y+\eta)=\langle \xi,y \rangle-\langle \eta,x \rangle,$$ is invariant. Consequently, $(A\oplus A^\ast,A,A^\ast)$ is a Manin triple, which is called the [**standard Manin triple**]{}.
\[Manin triple isomorphic\] Every Manin triple is isomorphic to the standard Manin triple.
Let $({\mathcal{A}},A_1,A_2)$ be a Manin triple with a nondegenerate skew-symmetric invariant bilinear form $\omega$. For all $x\in A_1, u\in A_2$, define a linear map $f:{\mathcal{A}}\longrightarrow A_1\oplus A_1^*$ by $f(x,u)=(x,\omega(u,\cdot))$. Since $\omega$ is nondegenerate, $f$ is an isomorphism between vector spaces. Thus, $f$ induces a Manin triples structure on $(A_1\oplus A_1^*,A_1,A_1^*)$.
First for all $x,y\in A, u,v\in A_2$, we have $$\begin{aligned}
\nonumber \bar{\omega}( f(x+u),f(y+v))
=\bar{\omega}(x+\omega(u,\cdot),y+\omega(v,\cdot))=
\omega(u,y)-\omega(v,x)=\omega(x+u,y+v),\end{aligned}$$ which implies that the induced bilinear form on $A_1\oplus A_1^* $ is exactly $\bar{\omega}$ given by .
Then we assume the induced Hom-pre-Lie algebra structure on $A_1\oplus A_1^\ast$ is given by $(A_1\oplus A_1^*,\cdot',\alpha_{A_1}\oplus \alpha_{A_1^*})$. By , we obtain $\alpha_{A_1^\ast}=(\alpha_{A_1}^{-1})^\ast$. For all $x,y\in A, \xi,\eta\in A^\ast$, we have $$\begin{aligned}
\label{equivalent-3-4}
\nonumber \bar{\omega}(x \cdot' \xi,\alpha(y))&=&-\bar{\omega}((\alpha^{-1})^\ast(\xi),[x,y])\\
\nonumber &=&-\langle (\alpha^{-2})^\ast(\xi),L_\alpha(x) \alpha(y)-R_\alpha(x) \alpha(y)\rangle\\
\nonumber &=&\langle L^\ast_{\alpha(x)} (\alpha^{-2})^\ast(\xi)-R^\ast_{\alpha(x)} (\alpha^{-2})^\ast(\xi),\alpha(y)\rangle\\
\nonumber &=& \langle L^\star_x \xi-R^\star_x \xi,\alpha(y)\rangle\\
&=&\bar{\omega}({\mathrm{ad}}^\star_x \xi,\alpha(y)),\end{aligned}$$ and $$\begin{aligned}
\label{equivalent-3-5}
\nonumber \bar{\omega}(x\cdot'\xi,(\alpha^{-1})^\ast(\eta))&=&\bar{\omega}((\alpha^{-1})^\ast(\xi),[\eta,x])\\
\nonumber &=&-\bar{\omega}(\eta\circ \xi,\alpha(x))\\
\nonumber &=&-\langle {\mathcal{R}}_{(\alpha^{-1})^\ast(\xi)} (\alpha^{-1})^\ast(\eta),\alpha^2(x)\rangle\\
\nonumber &=&\langle (\alpha^{-1})^\ast(\eta),{\mathcal{R}}^\ast_{(\alpha^{-1})^\ast(\xi)} \alpha^2(x)\rangle\\
\nonumber &=& \langle (\alpha^{-1})^\ast(\eta),{\mathcal{R}}^\star_\xi x\rangle\\
&=&-\bar{\omega}({\mathcal{R}}^\star_\xi x,(\alpha^{-1})^\ast(\eta)).\end{aligned}$$ Thus, we have $x\cdot'\xi={\mathrm{ad}}^\star_x \xi-{\mathcal{R}}^\star_\xi x$. Similarly, we have $\xi\cdot' x=\mathfrak{ad}^\star_\xi x-R^\star_x \xi$. Thus, we deduce that $(x+\xi)\cdot'(y+\eta)=(x+\xi)\diamond(y+\eta)$, which implies that $(A_1\oplus A_1^\ast,A_1,A_1^\ast)$ is the standard Manin triple.
For a Hom-Lie algebra $({\mathfrak g},[\cdot,\cdot]_{\mathfrak g},\phi_{\mathfrak g})$ and a representation $(V,\beta,\rho)$, recall that a $1$-cocycle $\delta$ associated to $(V,\beta,\rho)$ is a linear map form ${\mathfrak g}$ to $V$ satisfying: $$\delta([x,y]_{\mathfrak g})=\rho(\phi_{\mathfrak g}(x))\delta(y)-\rho(\phi_{\mathfrak g}(y))\delta(x).$$
A pair of Hom-pre-Lie algebras $(A,\cdot,\alpha)$ and $(A^\ast,\circ,(\alpha^{-1})^\ast)$ is called a [**Hom-pre-Lie bialgebra**]{} if the following conditions hold:
- $\varphi^\ast$ is a $1$-cocycle of the sub-adjacent Hom-Lie algebra $A^C$ associated to the representation $(A\otimes A,L^{-2}\otimes\alpha+\alpha \otimes {\mathrm{ad}}^{-2})$, where $\varphi^\ast:A\longrightarrow A\otimes A$ is the dual of $\circ:A^\ast\otimes A^\ast\longrightarrow A^\ast$, i.e. $\langle \varphi^\ast(x),\xi\otimes\eta\rangle=\langle x,\xi\circ \eta\rangle$.
- $\psi^\ast$ is a $1$-cocycle of the sub-adjacent Hom-Lie algebra $(A^\ast)^C$ associated to the representation $(A^*\otimes A^*,{\mathcal{L}}^{-2}\otimes(\alpha^{-1})^\ast+(\alpha^{-1})^\ast\otimes \mathfrak{ad}^{-2})$, where $\psi^\ast:A^\ast\longrightarrow A^\ast\otimes A^\ast$ is the dual of $\cdot:A\otimes A\longrightarrow A^\ast$, i.e. $\langle \psi^\ast(\xi),x\otimes y\rangle=\langle \xi,x\cdot y\rangle$.
We denote a Hom-pre-Lie bialgebra by $(A,A^\ast,\varphi^*,\psi^*)$ or simply $(A,A^\ast)$.
Now we are ready to give the main result of this section.
\[equivalent\] Let $(A,\cdot,\alpha)$ and $(A^\ast,\circ,(\alpha^{-1})^\ast)$ be two Hom-pre-Lie algebras. Then the following conditions are equivalent:
- $(A,A^\ast)$ is a Hom-pre-Lie bialgebra,
- $(A,A^\ast,{\mathrm{ad}}^\star,-R^\star,\mathfrak{ad}^\star,-{\mathcal{R}}^\star)$ is a matched pair of Hom-pre-Lie algebras,
- $(A\oplus A^\ast,A,A^\ast)$ is the standard Manin triple for Hom-pre-Lie algebras.
First, we prove that $\rm(i)$ is equivalent to $\rm(ii)$. We have $$\begin{aligned}
\label{cocycle-1}
\nonumber && \langle-{\mathrm{ad}}^\star_{\alpha(x)}(\xi\circ\eta)-{\mathrm{ad}}^\star_{{\mathcal{L}}^\star_\xi x}(\alpha^{-1})^\ast(\eta)+L^\star_x\xi\circ (\alpha^{-1})^\ast(\eta)+R^\star({\mathcal{R}}^\star_\eta x)(\alpha^{-1})^\ast(\xi)\\
\nonumber&&+(\alpha^{-1})^\ast(\xi)\circ{\mathrm{ad}}^\star_x \eta,\alpha^2(y)\rangle\\
\nonumber &=&\langle[x,y],\xi\circ\eta \rangle+\langle {\mathrm{ad}}_{\alpha^{-1}({\mathcal{L}}^\star_\xi x)}y,(\alpha^{-1})^\ast(\eta)\rangle+\langle (\alpha^2)^*(L^\star_x \xi)\circ \alpha^*(\eta),y\rangle\\
\nonumber&&-\langle R_{\alpha^{-1}({\mathcal{R}}^\star_\eta x)}y,(\alpha^{-1})^\ast(\xi)\rangle+\langle \alpha^\ast(\xi)\circ (\alpha^2)^*({\mathrm{ad}}^\star_x \eta),y\rangle\\
\nonumber &=&\langle\varphi^\ast[x,y],\xi\otimes \eta\rangle-\langle{\mathrm{ad}}_y \alpha^{-1}({\mathcal{L}}^\star_\xi x),(\alpha^{-1})^\ast(\eta)\rangle+\langle L^\star_{\alpha^{-2}(x)}(\alpha^2)^*(\xi)\circ \alpha^*(\eta),y \rangle\\
\nonumber&&-\langle L_y\alpha^{-1}({\mathcal{R}}^\star_\eta x),(\alpha^{-1})^\ast(\xi)\rangle+\langle \alpha^\ast(\xi)\circ {\mathrm{ad}}^\star_{\alpha^{-2}(x)}(\alpha^2)^*(\eta),y \rangle\\
\nonumber &=&\langle \varphi^\ast[x,y],\xi\otimes \eta\rangle+\langle\alpha^{-1}({\mathcal{L}}^\star_\xi x),{\mathrm{ad}}^*_y (\alpha^{-1})^\ast(\eta)\rangle+\langle L^*_{\alpha^{-1}(x)}\xi\circ \alpha^*(\eta),y \rangle\\
\nonumber&&+\langle \alpha^{-1}({\mathcal{R}}^\star_\eta x),L^*_y(\alpha^{-1})^\ast(\xi) \rangle +\langle \alpha^\ast(\xi)\circ {\mathrm{ad}}^*_{\alpha^{-1}(x)}\eta,y \rangle\\
\nonumber &=&\langle \varphi^\ast[x,y],\xi\otimes \eta \rangle+\langle {\mathcal{L}}^\star_\xi x,(\alpha^{-1})^\ast{\mathrm{ad}}^\star_{\alpha^{-1}(y)}\alpha^\ast(\eta) \rangle+\langle (L^{-2})^*_{\alpha(x)}\xi\circ \alpha^*(\eta),y\rangle\\
\nonumber&&+\langle {\mathcal{R}}^\star_\eta x,(\alpha^{-1})^\ast L^\star_{\alpha^{-1}(y)}\alpha^\ast(\xi)\rangle+\langle \alpha^\ast(\xi)\circ ({\mathrm{ad}}^{-2})^*_{\alpha(x)}\eta,y \rangle\\
\nonumber &=&\langle \varphi^\ast[x,y],\xi\otimes \eta \rangle+\langle {\mathcal{L}}^\star_\xi x,{\mathrm{ad}}^\star_y \eta \rangle-\langle(L_{\alpha(x)}^{-2}\otimes \alpha)\varphi^*(y),\xi\otimes \eta \rangle+\langle{\mathcal{R}}^\star_\eta x, L^\star_y \xi \rangle\\
\nonumber&&-\langle (\alpha\otimes{\mathrm{ad}}_{\alpha(x)}^{-2})\varphi^*(y),\xi\otimes \eta\rangle\\
\nonumber &=&\langle \varphi^\ast[x,y],\xi\otimes \eta \rangle-\langle x,\alpha^*(\xi)\circ (\alpha^2)^*({\mathrm{ad}}^\star_y \eta) \rangle-\langle(L_{\alpha(x)}^{-2}\otimes \alpha)\varphi^*(y),\xi\otimes \eta \rangle\\
\nonumber&&-\langle x, (\alpha^2)^*(L^\star_y \xi)\circ \alpha^*(\eta)\rangle-\langle (\alpha\otimes{\mathrm{ad}}_{\alpha(x)}^{-2})\varphi^*(y),\xi\otimes \eta\rangle\\
\nonumber &=&\langle \varphi^\ast[x,y],\xi\otimes \eta \rangle-\langle x,\alpha^*(\xi)\circ {\mathrm{ad}}^\star_{\alpha^{-2}(y)}(\alpha^2)^*(\eta) \rangle-\langle(L_{\alpha(x)}^{-2}\otimes \alpha)\varphi^*(y),\xi\otimes \eta \rangle\\
\nonumber&&-\langle x, L^\star_{\alpha^{-2}(y)}(\alpha^2)^*(\xi)\circ \alpha^*(\eta)\rangle-\langle (\alpha\otimes{\mathrm{ad}}_{\alpha(x)}^{-2})\varphi^*(y),\xi\otimes \eta\rangle\\
\nonumber &=&\langle \varphi^\ast[x,y],\xi\otimes \eta \rangle-\langle x,\alpha^*(\xi)\circ {\mathrm{ad}}^*_{\alpha^{-1}(y)}\eta \rangle-\langle(L_{\alpha(x)}^{-2}\otimes \alpha)\varphi^*(y),\xi\otimes \eta \rangle\\
\nonumber&&-\langle x, L^*_{\alpha^{-1}(y)}\xi\circ \alpha^*(\eta)\rangle-\langle (\alpha\otimes{\mathrm{ad}}_{\alpha(x)}^{-2})\varphi^*(y),\xi\otimes \eta\rangle\\
\nonumber &=&\langle \varphi^\ast[x,y],\xi\otimes \eta \rangle-\langle x,\alpha^*(\xi)\circ ({\mathrm{ad}}^{-2})_{\alpha(y)}^*\eta \rangle-\langle(L_{\alpha(x)}^{-2}\otimes \alpha)\varphi^*(y),\xi\otimes \eta \rangle\\
\nonumber&&-\langle x, (L^{-2})_{\alpha(y)}^*\xi\circ \alpha^*(\eta)\rangle-\langle (\alpha\otimes{\mathrm{ad}}_{\alpha(x)}^{-2})\varphi^*(y),\xi\otimes \eta\rangle\\
\nonumber &=&\langle \varphi^\ast[x,y],\xi\otimes \eta \rangle+\langle (\alpha\otimes {\mathrm{ad}}_{\alpha(y)}^{-2}) \varphi^*(x),\xi\otimes \eta\rangle-\langle(L_{\alpha(x)}^{-2}\otimes \alpha)\varphi^*(y),\xi\otimes \eta \rangle\\
\nonumber&&+\langle(L_{\alpha(y)}^{-2})\otimes\alpha)\varphi^* x, \xi\otimes \eta\rangle-\langle (\alpha\otimes{\mathrm{ad}}_{\alpha(x)}^{-2})\varphi^*(y),\xi\otimes \eta\rangle\\
&=& \langle \varphi^\ast[x,y]-(L^{-2}_{\alpha(x)}\otimes\alpha+\alpha\otimes {\mathrm{ad}}^{-2}_{\alpha(x)})\varphi^\ast(y)+(L^{-2}_{\alpha(y)}\otimes\alpha+\alpha\otimes {\mathrm{ad}}^{-2}_{\alpha(y)})\varphi^\ast(x),\xi\otimes\eta\rangle,\end{aligned}$$ which means that holds if and only if $\varphi^\ast$ is a $1$-cocycle of the sub-adjacent Hom-Lie algebra $A^C$ associated to the representation $(A\otimes A,L^{-2}\otimes\alpha+\alpha \otimes {\mathrm{ad}}^{-2})$. By Proposition \[matched-pair-equivalent\], we can obtain that $\eqref{pre-matche-pair-2} \Longleftrightarrow \eqref{pre-matche-pair-3}$. Therefore, $\varphi^\ast$ is a $1$-cocycle of the sub-adjacent Hom-Lie algebra $A^C$ associated to the representation $(A\otimes A,L^{-2}\otimes\alpha+\alpha \otimes {\mathrm{ad}}^{-2})$ if and only if $\eqref{pre-matche-pair-2}$ and $\eqref{pre-matche-pair-3}$ hold. Similarly, we can prove that $\psi^\ast$ is a 1-cocycle of the sub-adjacent Hom-Lie algebra $(A^\ast)^C$ associated to the representation $(A^*\otimes A^*,{\mathcal{L}}^{-2}\otimes(\alpha^{-1})^\ast+(\alpha^{-1})^\ast\otimes \mathfrak{ad}^{-2})$ if and only if $\eqref{pre-matche-pair-1}$ and $\eqref{pre-matche-pair-4}$ hold. Thus, we obtain that condition $\rm(i)$ is equivalent to condition $\rm(ii)$.
Next, we prove that $\rm(ii)$ is equivalent to $\rm(iii)$. Let $(A,A^\ast,{\mathrm{ad}}^\star,-R^\star,\mathfrak{ad}^\star,-{\mathcal{R}}^\star)$ be a matched pair of Hom-pre-Lie algebras. By Theorem \[matched-pair-Hom-pre-Lie algebra\], $(A\oplus A^\ast,\diamond,\alpha\oplus (\alpha^{-1})^*)$ is a Hom-pre-Lie algebra, where $``\diamond$” is given by . Let $\omega$ be the natural nondegenerate skew-symmetric bilinear form on $A\oplus A^*$ given by . We only need to prove that $\omega$ satisfies the invariant conditions. For all $x,y,z\in A, \xi,\eta,\gamma\in A^\ast$, we have $$\begin{aligned}
\label{equivalent-3-1}
\nonumber \omega(\alpha(x)+(\alpha^{-1})^\ast(\xi),\alpha(y)+(\alpha^{-1})^\ast(\eta))
\nonumber&=&\langle (\alpha^{-1})^\ast(\xi),\alpha(y)\rangle-\langle (\alpha^{-1})^\ast(\eta),\alpha(x)\rangle\\
\nonumber&=&\langle\xi,y\rangle-\langle\eta,x\rangle\\
&=&\omega(x+\xi,y+\eta),\end{aligned}$$ which implies that holds. Moreover, we have $$\begin{aligned}
\label{equivalent-3-2}
\nonumber && \omega((x+\xi)\diamond(y+\eta),\alpha(z)+(\alpha^{-1})^\ast(\gamma))\\
\nonumber &=&\omega(x\cdot y+\mathfrak{ad}^\star_\xi y-{\mathcal{R}}^\star_\eta x+\xi\circ \eta+{\mathrm{ad}}^\star_x \eta-R^\star_y \xi,\alpha(z)+(\alpha^{-1})^\ast(\gamma))\\
\nonumber &=&\langle \xi\circ \eta+{\mathrm{ad}}^\star_x \eta-R^\star_y \xi,\alpha(z)\rangle-\langle(\alpha^{-1})^\ast(\gamma), x\cdot y+\mathfrak{ad}^\star_\xi y-{\mathcal{R}}^\star_\eta x\rangle\\
\nonumber &=&\langle\alpha^\ast(\xi)\circ \alpha^\ast(\eta),z \rangle-\langle \eta,\alpha^{-1}(x)\cdot \alpha^{-1}(z)-\alpha^{-1}(z)\cdot \alpha^{-1}(x)\rangle+\langle \xi,\alpha^{-1}(z)\cdot \alpha^{-1}(y)\rangle\\
&&-\langle \gamma,\alpha^{-1}(x)\cdot \alpha^{-1}(y)\rangle+\langle y,\alpha^\ast(\xi)\circ \alpha^\ast(\gamma)\rangle-\alpha^\ast(\gamma)\circ \alpha^\ast(\xi)\rangle-\langle x,\alpha^\ast(\gamma)\circ \alpha^\ast(\eta)\rangle,\end{aligned}$$ and $$\begin{aligned}
\label{equivalent-3-3}
\nonumber && -\omega(\alpha(y)+(\alpha^{-1})^\ast(\eta),[x+\xi,z+\gamma])\\
\nonumber &=&-\omega(\alpha(y)+(\alpha^{-1})^\ast(\eta),x\cdot z+\mathfrak{ad}^\star_\xi z-{\mathcal{R}}^\star_\gamma x+\xi\circ \gamma+{\mathrm{ad}}^\star_x \gamma-R^\star_z \xi)\\
\nonumber&&-\omega(\alpha(y)+(\alpha^{-1})^\ast(\eta),-z\cdot x-\mathfrak{ad}^\star_\gamma x+{\mathcal{R}}^\star_\xi z-\gamma\circ \xi-{\mathrm{ad}}^\star_z \xi+R^\star_x \gamma)\\
\nonumber &=&-\langle (\alpha^{-1})^\ast(\eta),x\cdot z+\mathfrak{ad}^\star_\xi z-{\mathcal{R}}^\star_\gamma x-z\cdot x-\mathfrak{ad}^\star_\gamma x+{\mathcal{R}}^\star_\xi z\rangle\\
\nonumber&&+\langle\xi\circ \gamma+{\mathrm{ad}}^\star_x \gamma-R^\star_z \xi-\gamma\circ \xi-{\mathrm{ad}}^\star_z \xi+R^\star_x \gamma,\alpha(y)\rangle\\
\nonumber &=&-\langle\eta,\alpha^{-1}(x)\cdot \alpha^{-1}(z) \rangle+\langle z,\alpha^*(\xi)\circ\alpha^*(\eta)-\alpha^*(\eta)\circ\alpha^*(\xi)\rangle-\langle x,\alpha^*(\eta)\circ\alpha^*(\gamma)\rangle\\
\nonumber&& +\langle\eta,\alpha^{-1}(z)\cdot \alpha^{-1}(x) \rangle-\langle x,\alpha^*(\gamma)\circ\alpha^*(\eta)-\alpha^*(\eta)\circ\alpha^*(\gamma)\rangle+\langle z,\alpha^*(\eta)\circ\alpha^*(\xi)\rangle \\
\nonumber && +\langle \alpha^\ast(\xi)\circ \alpha^\ast(\gamma),y\rangle-\langle \gamma,\alpha^{-1}(x)\cdot\alpha^{-1}(y)-\alpha^{-1}(y)\cdot\alpha^{-1}(x)\rangle+\langle \xi, \alpha^{-1}(y)\cdot \alpha^{-1}(z)\rangle \\
\nonumber&& -\langle \alpha^\ast(\gamma)\circ \alpha^\ast(\xi),y\rangle+\langle \xi,\alpha^{-1}(z)\cdot\alpha^{-1}(y)-\alpha^{-1}(y)\cdot\alpha^{-1}(z)\rangle-\langle \gamma,\alpha^{-1}(y)\cdot \alpha^{-1}(x)\rangle \\
\nonumber &=& \langle\alpha^\ast(\xi)\circ \alpha^\ast(\eta),z \rangle-\langle \eta,\alpha^{-1}(x)\cdot \alpha^{-1}(z)-\alpha^{-1}(z)\cdot \alpha^{-1}(x)\rangle+\langle \xi,\alpha^{-1}(z)\cdot \alpha^{-1}(y)\rangle\\
&&-\langle \gamma,\alpha^{-1}(x)\cdot \alpha^{-1}(y)\rangle+\langle y,\alpha^\ast(\xi)\circ \alpha^\ast(\gamma)\rangle-\alpha^\ast(\gamma)\circ \alpha^\ast(\xi)\rangle-\langle x,\alpha^\ast(\gamma)\circ \alpha^\ast(\eta)\rangle,\end{aligned}$$ which implies that holds. Thus, $(A\oplus A^\ast,A,A^\ast)$ is a standard Manin triple. Conversely, if $(A\oplus A^\ast,A,A^\ast)$ is the standard Manin triple, then it is obvious that $(A,A^\ast,{\mathrm{ad}}^\star,-R^\star,\mathfrak{ad}^\star,-{\mathcal{R}}^\star)$ is a matched pair of Hom-pre-Lie algebras.
Coboundary Hom-pre-Lie bialgebras
=================================
In this section, we study coboundary Hom-pre-Lie bialgebras and introduce the notion of a Hom-${\mathfrak s}$-matrix, which gives rise to a Hom-pre-Lie bialgebra naturally.
A Hom-pre-Lie bialgebra $(A,A^\ast,\varphi^\ast,\psi^\ast)$ is called [**coboundary**]{} if $\varphi^\ast$ is a $1$-coboundary of the sub-adjacent Hom-Lie algebra $A^C$ associated to the representation $(A\otimes A,L^{-2}\otimes\alpha+\alpha \otimes {\mathrm{ad}}^{-2})$, that is, there exists an $r\in A\otimes A$ such that $$\label{coboundary}
\varphi^\ast(x)=(L_x^{-2}\otimes\alpha+\alpha \otimes {\mathrm{ad}}_x^{-2})r,\quad \forall x \in A.$$
Let $(A,\cdot,\alpha)$ be a Hom-pre-Lie algebra, and $r\in A\otimes A$, suppose that $\varphi^\ast$ is a $1$-coboundary of the sub-adjacent Hom-Lie algebra $A^C$ associated to the representation $(A\otimes A,L^{-2}\otimes\alpha+\alpha \otimes {\mathrm{ad}}^{-2})$. Then it is obvious that $\varphi^\ast$ is a $1$-cocycle of the sub-adjacent Hom-Lie algebra $A^C$ associated to the representation $(A\otimes A,L^{-2}\otimes\alpha+\alpha \otimes {\mathrm{ad}}^{-2})$.
\[pro:condition\] With the above notations, $(A,A^\ast,\varphi^*,\psi^*)$ is a Hom-pre-Lie bialgebra if and only if the following two conditions are satisfied:
- $\circ:A^\ast\otimes A^\ast\longrightarrow A^\ast$ define a Hom-pre-Lie algebra structure on $A^\ast$, where $``\circ$” is given by $\langle \varphi^\ast(x),\xi\otimes\eta\rangle=\langle x,\xi\circ \eta\rangle$.
- $\psi^\ast$ is a $1$-cocycle of the sub-adjacent Hom-Lie algebra $(A^\ast)^C$ associated to the representation $(A^*\otimes A^*,{\mathcal{L}}^{-2}\otimes(\alpha^{-1})^\ast+(\alpha^{-1})^\ast\otimes \mathfrak{ad}^{-2})$, where $\psi^\ast:A^\ast\longrightarrow A^\ast\otimes A^\ast$ is given by $\langle \psi^\ast(\xi),x\otimes y\rangle=\langle \xi,x\cdot y\rangle$.
For all $r\in A\otimes A$, the linear map $r^\sharp:A^*\longrightarrow A$ is defined by $$\langle r^\sharp(\xi),\eta\rangle=\langle r,\xi\otimes \eta\rangle,\quad \forall \xi,\eta \in A^*.$$
\[O-Operator\] Let $(A,\cdot,\alpha)$ be a Hom-pre-Lie algebra and $\varphi^*:A\longrightarrow A\otimes A$ defined by . If $r\in A\otimes A$ satisfies $$\label{pro-1}
r^\sharp\circ (\alpha^{-1})^*=\alpha\circ r^\sharp.$$ Then, for all $\xi, \eta\in A^*$, we have $$\label{pro-2}
\xi\circ\eta={\mathrm{ad}}_{r^\sharp(\xi)}^\star\eta-R_{{\sigma(r)}^\sharp(\eta)}^\star\xi={\mathrm{ad}}_{\alpha({r^\sharp}(\xi))}^*((\alpha^{-2})^*(\eta))-R_{\alpha({{\sigma(r)}^\sharp(\eta)})}^*((\alpha^{-2})^*(\xi)),$$ where $\sigma:A\otimes A\longrightarrow A\otimes A$ is the flip operator defined by $\sigma(x\otimes y)=y\otimes x$ for all $x,y\in A$. Furthermore, we have $$\label{pro-3}
r^\sharp(\alpha^*(\xi))\cdot r^\sharp(\alpha^*(\eta))-r^\sharp(\alpha^*(\xi\circ \eta))=[[r,r]](\xi,\eta),$$ where $[[r,r]]=[r_{12},r_{23}]-r_{13}\cdot r_{12}+r_{13}\cdot r_{23}$.
Before proving this result, let us explain the notations. Let $(A,\cdot,\alpha)$ be a Hom-pre-Lie algebra and $r=\sum_i x_i\otimes y_i \in A\otimes A$. Set $$r_{12}=\sum_ix_i\otimes y_i\otimes \alpha, \quad r_{13}=\sum_ix_i\otimes \alpha \otimes y_i, \quad r_{23}=\sum_i \alpha\otimes x_i \otimes y_i.$$
Let $r=\sum_i x_i\otimes y_i$. Here the Einstein summation convention is used. By and , for all $z\in A,~\xi,~\eta\in A^*$, we have $$\begin{aligned}
\nonumber &&\langle z,\xi\circ \eta\rangle = \langle\varphi^*(z),\xi\otimes \eta\rangle \\
\nonumber&=&\langle (L_z^{-2}\otimes\alpha+\alpha \otimes {\mathrm{ad}}_z^{-2})(x_i\otimes y_i),\xi\otimes \eta\rangle\\
\nonumber &=& \langle \alpha^{-2}(z)\cdot x_i \otimes\alpha(y_i),\xi\otimes \eta\rangle+\langle\alpha(x_i)\otimes[\alpha^{-2}(z),y_i],\xi\otimes \eta\rangle\\
\nonumber&=& \langle\alpha^{-2}(z)\cdot x_i ,\xi\rangle\langle\alpha(y_i),\eta\rangle+\langle\alpha(x_i),\xi\rangle\langle[\alpha^{-2}(z),y_i],\eta\rangle\\
\nonumber &=& \langle \alpha^{-2}(z)\cdot \langle\alpha(y_i),\eta\rangle x_i ,\xi\rangle+\langle[\alpha^{-2}(z),\langle\alpha(x_i),\xi\rangle y_i],\eta\rangle\\
\nonumber&=& \langle \alpha^{-2}(z)\cdot {\sigma(r)}^\sharp(\alpha^*(\eta)) ,\xi\rangle+\langle[\alpha^{-2}(z),r^\sharp(\alpha^*(\xi))],\eta\rangle\\
\nonumber &=& \langle R_{{\sigma(r)}^\sharp(\alpha^*(\eta))}\alpha^{-2}(z),\xi\rangle-\langle {\mathrm{ad}}_{r^\sharp(\alpha^*(\xi))}\alpha^{-2}(z),\eta\rangle \\
\nonumber&=& -\langle z,(\alpha^{-2})^* R_{{\sigma(r)}^\sharp(\alpha^*(\eta))}^*\xi\rangle+\langle z,(\alpha^{-2})^*{\mathrm{ad}}_{r^\sharp(\alpha^*(\xi))}^*\eta\rangle\\
\nonumber &=& -\langle z,(\alpha^{-2})^* R_{\alpha^{-1}({\sigma(r)}^\sharp(\alpha^*(\eta)))}^\star(\alpha^2)^*(\xi)\rangle+\langle z,(\alpha^{-2})^*{\mathrm{ad}}_{\alpha^{-1}(r^\sharp(\alpha^*(\xi)))}^\star(\alpha^2)^*(\eta)\rangle\\
\nonumber&=& -\langle z, R_{\alpha({\sigma(r)}^\sharp(\alpha^*(\eta)))}^\star\xi\rangle+\langle z,{\mathrm{ad}}_{\alpha(r^\sharp(\alpha^*(\xi)))}^\star\eta\rangle\\
\nonumber&=&-\langle z, R_{{\sigma(r)}^\sharp(\eta)}^\star\xi\rangle+\langle z,{\mathrm{ad}}_{r^\sharp(\xi)}^\star\eta\rangle,\end{aligned}$$ which implies that holds.
Moreover, for all $\theta\in A^*$, we have $$\begin{aligned}
\nonumber &&\langle\alpha(r^\sharp(\xi))\cdot \alpha(r^\sharp(\eta))-\alpha(r^\sharp(\xi\circ \eta)),\theta\rangle \\
\nonumber &=& \langle r^\sharp((\alpha^{-1})^*(\xi))\cdot r^\sharp((\alpha^{-1})^*(\eta)),\theta\rangle-\langle \xi\circ \eta,{\sigma(r)}^\sharp(\alpha^*(\theta))\rangle\\
\nonumber&=& \langle r^\sharp((\alpha^{-1})^*(\xi))\cdot r^\sharp((\alpha^{-1})^*(\eta)),\theta\rangle-\langle {\mathrm{ad}}_{\alpha({r^\sharp}(\xi))}^*((\alpha^{-2})^*(\eta))-R_{\alpha({{\sigma(r)}^\sharp(\eta)})}^*((\alpha^{-2})^*(\xi)),{\sigma(r)}^\sharp(\alpha^*(\theta))\rangle\\
\nonumber &=& \langle r^\sharp((\alpha^{-1})^*(\xi))\cdot r^\sharp((\alpha^{-1})^*(\eta)),\theta\rangle+\langle (\alpha^{-2})^*(\eta),r^\sharp((\alpha^{-1})^*(\xi))\cdot {\sigma(r)}^\sharp(\alpha^*(\theta))\rangle\\
\nonumber&&-\langle(\alpha^{-2})^*(\eta),{\sigma(r)}^\sharp(\alpha^*(\theta))\cdot r^\sharp((\alpha^{-1})^*(\xi)) \rangle-\langle(\alpha^{-2})^*(\xi),{\sigma(r)}^\sharp(\alpha^*(\theta))\cdot {\sigma(r)}^\sharp((\alpha^{-1})^*(\eta))\rangle\\
\nonumber&=& \langle \langle x_i,(\alpha^{-1})^*(\xi)\rangle y_i\cdot \langle x_j,(\alpha^{-1})^*(\eta)\rangle y_j,\theta\rangle+\langle \langle x_i,(\alpha^{-1})^*(\xi)\rangle y_i\cdot \langle y_j,\alpha^*(\theta)\rangle x_j,(\alpha^{-2})^*(\eta)\rangle\\
\nonumber && -\langle \langle y_i,\alpha^*(\theta)\rangle x_i\cdot \langle x_j,(\alpha^{-1})^*(\xi)\rangle y_j,(\alpha^{-2})^*(\eta)\rangle-\langle \langle y_i,\alpha^*(\theta)\rangle x_i\cdot \langle y_j,(\alpha^{-1})^*(\eta)\rangle x_j,(\alpha^{-2})^*(\xi)\rangle\\
\nonumber&=& \langle x_i,(\alpha^{-1})^*(\xi)\rangle\langle x_j,(\alpha^{-1})^*(\eta)\rangle\langle y_i\cdot y_j,\theta\rangle+\langle x_i,(\alpha^{-1})^*(\xi)\rangle\langle y_j,\alpha^*(\theta)\rangle\langle y_i\cdot x_j,(\alpha^{-2})^*(\eta)\rangle\\
\nonumber && -\langle y_i,\alpha^*(\theta)\rangle\langle x_j,(\alpha^{-1})^*(\xi)\rangle\langle x_i\cdot y_j,(\alpha^{-2})^*(\eta)\rangle-\langle y_i,\alpha^*(\theta)\rangle\langle y_j,(\alpha^{-1})^*(\eta)\rangle\langle x_i\cdot x_j,(\alpha^{-2})^*(\xi)\rangle\\
\nonumber&=& \langle \alpha(x_i)\otimes \alpha(x_j)\otimes y_i\cdot y_j-\alpha(x_i)\otimes[x_j,y_i]\otimes \alpha(y_j)-x_i\cdot x_j\otimes \alpha(y_j)\otimes\alpha(y_i),(\alpha^{-2})^*(\xi)\otimes(\alpha^{-2})^*(\eta)\otimes\theta\rangle \\
\nonumber &=& [[r,r]]((\alpha^{-2})^*(\xi),(\alpha^{-2})^*(\eta),\theta) \\
&=&\nonumber\langle [[r,r]]((\alpha^{-2})^*(\xi),(\alpha^{-2})^*(\eta)),\theta\rangle,\end{aligned}$$ which implies that holds. This finishes the proof.
\[cor:dualalg\] Let $(A,\cdot,\alpha)$ be a Hom-pre-Lie algebra. Let $r\in Sym^2(A)$ satisfying and $[[r,r]]=0$, then $(A^\ast,\circ,(\alpha^{-1})^\ast)$ is a Hom-pre-Lie algebra, where $\circ$ is given by .
By Proposition \[O-Operator\] and $(A^*,(\alpha^{-1})^*,{\mathrm{ad}}^\star,-R^\star)$ a representation of a Hom-pre-Lie algebra $(A,\cdot,\alpha)$. For all $\xi,\eta,\delta\in A^*,$ we have $$\begin{aligned}
&&(\xi\circ \eta)\circ (\alpha^{-1})^\ast(\delta)-(\alpha^{-1})^\ast(\xi)\circ(\eta\circ \delta)-(\eta\circ \xi)\circ (\alpha^{-1})^\ast(\delta)+(\alpha^{-1})^\ast(\eta)\circ(\xi\circ \delta)\\
&=&{\mathrm{ad}}_{r^\sharp(\xi\circ\eta)}^\star(\alpha^{-1})^\ast(\delta)-R_{r^\sharp((\alpha^{-1})^\ast(\delta))}^\star (\xi\circ\eta)-{\mathrm{ad}}_{r^\sharp((\alpha^{-1})^\ast(\xi))}^\star (\eta\circ\delta)+R_{r^\sharp(\eta\circ\delta)}^\star(\alpha^{-1})^\ast(\xi)\\
&&-{\mathrm{ad}}_{r^\sharp(\eta\circ\xi)}^\star(\alpha^{-1})^\ast(\delta)+R_{r^\sharp((\alpha^{-1})^\ast(\delta))}^\star (\eta\circ \xi)+{\mathrm{ad}}_{r^\sharp((\alpha^{-1})^\ast(\eta))}^\star (\xi\circ\delta)-R_{r^\sharp(\xi\circ\delta)}^\star(\alpha^{-1})^\ast(\eta)\\
&=&{\mathrm{ad}}_{r^\sharp(\xi\circ\eta)}^\star(\alpha^{-1})^\ast(\delta)-R_{r^\sharp((\alpha^{-1})^\ast(\delta))}^\star ({\mathrm{ad}}_{r^\sharp(\xi)}^\star\eta-R_{r^\sharp(\eta)}^\star\xi)\\
&&-{\mathrm{ad}}_{r^\sharp((\alpha^{-1})^\ast(\xi))}^\star ({\mathrm{ad}}_{r^\sharp(\eta)}^\star\delta-R_{r^\sharp(\delta)}^\star\eta)+R_{r^\sharp(\eta\circ\delta)}^\star(\alpha^{-1})^\ast(\xi)\\
&&-{\mathrm{ad}}_{r^\sharp(\eta\circ\xi)}^\star(\alpha^{-1})^\ast(\delta)+R_{r^\sharp((\alpha^{-1})^\ast(\delta))}^\star ({\mathrm{ad}}_{r^\sharp(\eta)}^\star\xi-R_{r^\sharp(\xi)}^\star\eta)\\
&&+{\mathrm{ad}}_{r^\sharp((\alpha^{-1})^\ast(\eta))}^\star ({\mathrm{ad}}_{r^\sharp(\xi)}^\star\delta-R_{r^\sharp(\delta)}^\star\xi)-R_{r^\sharp(\xi\circ\delta)}^\star(\alpha^{-1})^\ast(\eta)\\
&=&{\mathrm{ad}}_{r^\sharp(\xi\circ\eta)}^\star(\alpha^{-1})^\ast(\delta)-{\mathrm{ad}}_{r^\sharp(\eta\circ\xi)}^\star(\alpha^{-1})^\ast(\delta)-{\mathrm{ad}}_{\alpha(r^\sharp(\xi))}^\star{\mathrm{ad}}_{r^\sharp(\eta)}^\star\delta+{\mathrm{ad}}_{\alpha(r^\sharp(\eta))}^\star{\mathrm{ad}}_{r^\sharp(\xi)}^\star\delta\\
&&-R_{\alpha(r^\sharp(\delta))}^\star L_{r^\sharp(\xi)}^\star \eta+L_{\alpha(r^\sharp(\xi))}^\star R_{r^\sharp(\delta)}^\star \eta-R_{\alpha(r^\sharp(\xi))}^\star R_{r^\sharp(\delta)}^\star \eta-R_{r^\sharp(\xi\circ\delta)}^\star (\alpha^{-1})^*(\eta)\\
&&+R_{\alpha(r^\sharp(\delta))}^\star L_{r^\sharp(\eta)}^\star \xi-L_{\alpha(r^\sharp(\eta))}^\star R_{r^\sharp(\delta)}^\star \xi+R_{\alpha(r^\sharp(\eta))}^\star R_{r^\sharp(\delta)}^\star \xi+R_{r^\sharp(\eta\circ\delta)}^\star (\alpha^{-1})^*(\xi)\\
&=&({\mathrm{ad}}_{r^\sharp(\xi\circ\eta)}^\star-{\mathrm{ad}}_{r^\sharp(\eta\circ\xi)}^\star-{\mathrm{ad}}_{r^\sharp(\xi)\cdot r^\sharp(\eta)-r^\sharp(\eta)\cdot r^\sharp(\xi)}^\star)((\alpha^{-1})^\ast(\delta))\\
&&+(R_{r^\sharp(\xi)\cdot r^\sharp(\delta)}^\star-R_{r^\sharp(\xi\circ\delta)}^\star)((\alpha^{-1})^\ast(\eta))-(R_{r^\sharp(\eta)\cdot r^\sharp(\delta)}^\star-R_{r^\sharp(\eta\circ\delta)}^\star)((\alpha^{-1})^\ast(\xi)).\end{aligned}$$ For all $ x\in A$, we have $$\begin{aligned}
&&\langle(\xi\circ \eta)\circ (\alpha^{-1})^\ast(\delta)-(\alpha^{-1})^\ast(\xi)\circ(\eta\circ \delta)-(\eta\circ \xi)\circ (\alpha^{-1})^\ast(\delta)+(\alpha^{-1})^\ast(\eta)\circ(\xi\circ \delta),x\rangle\\
&=&-\langle {\mathrm{ad}}_{{r^\sharp(\xi)\cdot r^\sharp(\eta)}-{r^\sharp(\xi\circ\eta)}}^\star(\alpha^{-1})^\ast(\delta),x\rangle+\langle {\mathrm{ad}}_{{r^\sharp(\eta)\cdot r^\sharp(\xi)}-{r^\sharp(\eta\circ\xi)})}^\star(\alpha^{-1})^\ast(\delta),x\rangle\\
&&+\langle R_{{r^\sharp(\xi)\cdot r^\sharp(\delta)}-{r^\sharp(\xi\circ\delta)})}^\star(\alpha^{-1})^\ast(\eta),x\rangle-\langle R_{{r^\sharp(\eta)\cdot r^\sharp(\delta)}-{r^\sharp(\eta\circ\delta)})}^\star(\alpha^{-1})^\ast(\xi),x\rangle\\
&=&-\langle {\mathrm{ad}}_{\alpha({r^\sharp(\xi))\cdot \alpha(r^\sharp(\eta))}-{\alpha(r^\sharp(\xi\circ\eta))}}^*(\alpha^{-3})^\ast(\delta),x\rangle+\langle {\mathrm{ad}}_{\alpha({r^\sharp(\eta))\cdot \alpha(r^\sharp(\xi))}-{\alpha(r^\sharp(\eta\circ\xi))}}^*(\alpha^{-3})^\ast(\delta),x\rangle\\
&&+\langle R_{\alpha({r^\sharp(\xi))\cdot \alpha(r^\sharp(\delta))}-{\alpha(r^\sharp(\xi\circ\delta))}}^*(\alpha^{-3})^\ast(\eta),x\rangle-\langle R_{\alpha({r^\sharp(\eta))\cdot \alpha(r^\sharp(\delta))}-{\alpha(r^\sharp(\eta\circ\delta))}}^*(\alpha^{-3})^\ast(\xi),x\rangle\\
&=&-\langle {\mathrm{ad}}_{[[r,r]]((\alpha^{-2})^*(\xi),(\alpha^{-2})^*(\eta))}^*(\alpha^{-3})^\ast(\delta),x\rangle+\langle {\mathrm{ad}}_{[[r,r]]((\alpha^{-2})^*(\eta),(\alpha^{-2})^*(\xi))}^*(\alpha^{-3})^\ast(\delta),x\rangle\\
&&+\langle R_{[[r,r]]((\alpha^{-2})^*(\xi),(\alpha^{-2})^*(\delta))}^*(\alpha^{-3})^\ast(\eta),x\rangle-\langle R_{[[r,r]]((\alpha^{-2})^*(\eta),(\alpha^{-2})^*(\delta))}^*(\alpha^{-3})^\ast(\xi),x\rangle\\
&=&0.\end{aligned}$$ Thus, $(A^\ast,\circ,(\alpha^{-1})^\ast)$ is a Hom-pre-Lie algebra.
\[pro:cocycle\] Let $(A,\cdot,\alpha)$ be a Hom-pre-Lie algebra and $\varphi^*:A\longrightarrow A\otimes A$ defined by . If $r\in A\otimes A$ satisfies and $(A^*,\circ,(\alpha^{-1})^*)$ is a Hom-pre-Lie algebra, where $\circ$ is given by . Then $\psi^\ast$ is a $1$-cocycle of the sub-adjacent Hom-Lie algebra $(A^\ast)^C$ associated to the representation $(A^*\otimes A^*,{\mathcal{L}}^{-2}\otimes(\alpha^{-1})^\ast+(\alpha^{-1})^\ast\otimes \mathfrak{ad}^{-2})$ if and only if the following equation holds: $$\label{cocycle}
(P(x\cdot y)-P(\alpha(x))P(y))(r-\sigma(r))=0,$$ where $P(x)=L^{-2}_x\otimes \alpha+\alpha\otimes L^{-2}_x$.
By , for all $\xi,\eta\in A^*$, we have $$\begin{aligned}
\nonumber \langle r^\sharp((\alpha^{-1})^*(\xi))-\alpha(r^\sharp(\xi)),\eta\rangle&=&\langle r,(\alpha^{-1})^*(\xi)\otimes \eta\rangle-\langle r^\sharp(\xi),\alpha^*(\eta)\rangle \\
\nonumber&=&\langle (\alpha^{-1}\otimes {{\rm{Id}}})r,\xi\otimes \eta\rangle-\langle r,\xi\otimes \alpha^*(\eta)\rangle\\
\nonumber&=&\langle (\alpha^{-1}\otimes {{\rm{Id}}}-{{\rm{Id}}} \otimes \alpha)r,\xi\otimes \eta\rangle\\
\nonumber&=&0,\end{aligned}$$ which implies that $$\label{1-cocycle-1}
(\alpha^{-1}\otimes {{\rm{Id}}})r=({{\rm{Id}}} \otimes \alpha)r.$$ Let $r=\sum_i x_i\otimes y_i$. Here the Einstein summation convention is used. For all $x,y\in A, \xi,\eta\in A^*$, we have $$\begin{aligned}
&&\langle-L^\star_{\alpha(x)}\{\xi,\eta\}+\{L_x^\star \xi,(\alpha^{-1})^*(\eta)\}+\{(\alpha^{-1})^*(\xi),L_x^\star \eta\}+L^\star_{{\mathcal{L}}^\star_\eta x}(\alpha^{-1})^*(\xi)-L^\star_{{\mathcal{L}}^\star_\xi x}(\alpha^{-1})^*(\eta),\alpha^2(y)\rangle \\
&=&\langle \{\xi,\eta\},x\cdot y\rangle+\langle L^\star_x\xi\circ (\alpha^{-1})^*(\eta),\alpha^2(y) \rangle-\langle(\alpha^{-1})^*(\eta)\circ L^\star_x\xi,\alpha^2(y)\rangle+\langle(\alpha^{-1})^*(\xi)\circ L^\star_x\eta,\alpha^2(y)\rangle\\
&&- \langle L^\star_x\eta\circ (\alpha^{-1})^*(\xi),\alpha^2(y)\rangle-\langle(\alpha^{-1})^*(\xi),\alpha^{-1}({\mathcal{L}}^\star_\eta x)\cdot y)\rangle +\langle(\alpha^{-1})^*(\eta),\alpha^{-1}({\mathcal{L}}^\star_\xi x)\cdot y)\rangle\\
&=& \langle\xi\circ \eta,x\cdot y\rangle-\langle\eta\circ \xi,x\cdot y\rangle+\langle (\alpha^2)^*(L^\star_x\xi)\circ \alpha^*(\eta),y\rangle-\langle\alpha^*(\eta)\circ (\alpha^2)^*(L^\star_x\xi),y \rangle \\
&&+\langle\alpha^*(\xi)\circ (\alpha^2)^*(L^\star_x\eta),y\rangle -\langle (\alpha^2)^*(L^\star_x\eta)\circ \alpha^*(\xi),y\rangle-\langle(\alpha^{-1})^*(\xi),R_y \alpha^{-1}({\mathcal{L}}^\star_\eta x) \rangle\\
&&+\langle(\alpha^{-1})^*(\eta),R_y \alpha^{-1}({\mathcal{L}}^\star_\xi x) \rangle\\
&=& \langle\xi\otimes\eta,\varphi^*(x\cdot y)\rangle-\langle\eta\otimes \xi,\varphi^*(x\cdot y)\rangle+\langle L_{\alpha^{-2}(x)}^\star(\alpha^2)^*(\xi)\circ \alpha^*(\eta),y\rangle\\
&& -\langle \alpha^*(\eta)\circ L_{\alpha^{-2}(x)}^\star(\alpha^2)^*(\xi),y \rangle+ \langle \alpha^*(\xi)\circ L_{\alpha^{-2}(x)}^\star(\alpha^2)^*(\eta),y \rangle-\langle L_{\alpha^{-2}(x)}^\star(\alpha^2)^*(\eta)\circ \alpha^*(\xi),y\rangle\\
&&+\langle (\alpha^{-1})^*R_y^*(\alpha^{-1})^*(\xi),{\mathcal{L}}^\star_\eta x\rangle-\langle (\alpha^{-1})^*R_y^*(\alpha^{-1})^*(\eta),{\mathcal{L}}^\star_\xi x\rangle \\
&=& \langle\xi\otimes\eta,\varphi^*(x\cdot y)\rangle-\langle\eta\otimes \xi,\varphi^*(x\cdot y)\rangle+\langle L_{\alpha^{-1}(x)}^*\xi\circ \alpha^*(\eta),y\rangle-\langle \alpha^*(\eta)\circ L_{\alpha^{-1}(x)}^*\xi,y\rangle\\
&& +\langle \alpha^*(\xi)\circ L_{\alpha^{-1}(x)}^*\eta,y\rangle-\langle L_{\alpha^{-1}(x)}^*\eta\circ \alpha^*(\xi),y\rangle+\langle (\alpha^{-1})^*R_{\alpha^{-1}(y)}^\star\alpha^*(\xi),{\mathcal{L}}^\star_\eta x \rangle\\
&&-\langle (\alpha^{-1})^*R_{\alpha^{-1}(y)}^\star\alpha^*(\eta),{\mathcal{L}}^\star_\xi x \rangle\\
&=& \langle\xi\otimes\eta,\varphi^*(x\cdot y)\rangle-\langle\eta\otimes \xi,\varphi^*(x\cdot y)\rangle+\langle (L^{-2})^*_{\alpha(x)}\xi\circ \alpha^*(\eta),y\rangle-\langle \alpha^*(\eta)\circ (L^{-2})^*_{\alpha(x)}\xi,y\rangle\\
&& +\langle \alpha^*(\xi)\circ (L^{-2})^*_{\alpha(x)}\eta,y\rangle-\langle (L^{-2})^*_{\alpha(x)}\eta\circ \alpha^*(\xi),y\rangle+\langle R^\star_y\xi,{\mathcal{L}}^\star_\eta x \rangle-\langle R^\star_y\eta,{\mathcal{L}}^\star_\xi x \rangle\\
&=& \langle\xi\otimes\eta,\varphi^*(x\cdot y)\rangle-\langle\eta\otimes \xi,\varphi^*(x\cdot y)\rangle+\langle (L^{-2})^*_{\alpha(x)}\xi\otimes \alpha^*(\eta),\varphi^*(y)\rangle-\langle \alpha^*(\eta)\otimes(L^{-2})^*_{\alpha(x)}\xi,\varphi^*(y)\rangle\\
&& +\langle \alpha^*(\xi)\otimes (L^{-2})^*_{\alpha(x)}\eta,\varphi^*(y)\rangle-\langle (L^{-2})^*_{\alpha(x)}\eta\otimes \alpha^*(\xi),\varphi^*(y)\rangle-\langle \alpha^*(\eta)\circ (\alpha^2)^*(R^\star_y\xi), x \rangle\\
&&+\langle \alpha^*(\xi)\circ (\alpha^2)^*(R^\star_y\eta), x \rangle\\
&=& \langle\xi\otimes\eta,\varphi^*(x\cdot y)\rangle-\langle\eta\otimes \xi,\varphi^*(x\cdot y)\rangle+\langle (L^{-2})^*_{\alpha(x)}\xi\otimes \alpha^*(\eta),\varphi^*(y)\rangle-\langle \alpha^*(\eta)\otimes(L^{-2})^*_{\alpha(x)}\xi,\varphi^*(y)\rangle\\
&& +\langle \alpha^*(\xi)\otimes (L^{-2})^*_{\alpha(x)}\eta,\varphi^*(y)\rangle-\langle (L^{-2})^*_{\alpha(x)}\eta\otimes \alpha^*(\xi),\varphi^*(y)\rangle-\langle \alpha^*(\eta)\circ R^\star_{\alpha^{-2}(y)}(\alpha^2)^*(\xi), x \rangle\\
&&+\langle \alpha^*(\xi)\circ R^\star_{\alpha^{-2}(y)}(\alpha^2)^*(\eta), x \rangle\\
&=& \langle\xi\otimes\eta,\varphi^*(x\cdot y)\rangle-\langle\eta\otimes \xi,\varphi^*(x\cdot y)\rangle+\langle (L^{-2})^*_{\alpha(x)}\xi\otimes \alpha^*(\eta),\varphi^*(y)\rangle-\langle \alpha^*(\eta)\otimes(L^{-2})^*_{\alpha(x)}\xi,\varphi^*(y)\rangle\\
&& +\langle \alpha^*(\xi)\otimes (L^{-2})^*_{\alpha(x)}\eta,\varphi^*(y)\rangle-\langle (L^{-2})^*_{\alpha(x)}\eta\otimes \alpha^*(\xi),\varphi^*(y)\rangle-\langle \alpha^*(\eta)\circ R^*_{\alpha^{-1}(y)}\xi, x \rangle\\
&&+\langle \alpha^*(\xi)\circ R^*_{\alpha^{-1}(y)}\eta, x \rangle\\
&=&\langle\xi\otimes\eta,\varphi^*(x\cdot y)\rangle-\langle\eta\otimes \xi,\varphi^*(x\cdot y)\rangle+\langle (L^{-2})^*_{\alpha(x)}\xi\otimes \alpha^*(\eta),\varphi^*(y)\rangle-\langle \alpha^*(\eta)\otimes(L^{-2})^*_{\alpha(x)}\xi,\varphi^*(y)\rangle\\
&& +\langle \alpha^*(\xi)\otimes (L^{-2})^*_{\alpha(x)}\eta,\varphi^*(y)\rangle-\langle (L^{-2})^*_{\alpha(x)}\eta\otimes \alpha^*(\xi),\varphi^*(y)\rangle-\langle \alpha^*(\eta)\circ (R^{-2})^*_{\alpha(y)}\xi, x \rangle\\
&&+\langle \alpha^*(\xi)\circ (R^{-2})^*_{\alpha(y)}\eta, x \rangle\\
&=&\langle\xi\otimes\eta,\varphi^*(x\cdot y)\rangle-\langle\xi\otimes\eta,(L^{-2}_{\alpha(x)}\otimes \alpha)\varphi^*(y) \rangle-\langle\xi\otimes\eta,(\alpha \otimes L^{-2}_{\alpha(x)})\varphi^*(y)\rangle-\langle\xi\otimes\eta,(\alpha \otimes R^{-2}_{\alpha(y)})\varphi^*(x)\rangle\\
&&-\langle\eta\otimes\xi,\varphi^*(x\cdot y)\rangle+\langle\eta\otimes\xi,(L^{-2}_{\alpha(x)}\otimes \alpha)\varphi^*(y) \rangle+\langle\eta\otimes\xi,(\alpha \otimes L^{-2}_{\alpha(x)})\varphi^*(y)\rangle+\langle\eta\otimes\xi,(\alpha \otimes R^{-2}_{\alpha(y)})\varphi^*(x)\rangle\end{aligned}$$ Moreover, we have $$\begin{aligned}
\label{1-cocycle-2}
\nonumber \langle\eta\otimes\xi,\varphi^*(x\cdot y)\rangle
\nonumber &=& \langle \eta\otimes\xi,(L^{-2}_{x\cdot y}\otimes \alpha+\alpha\otimes {\mathrm{ad}}^{-2}_{x\cdot y})r\rangle \\
\nonumber &=& \langle \eta\otimes\xi,(L^{-2}_{x\cdot y}\otimes \alpha+\alpha\otimes {\mathrm{ad}}^{-2}_{x\cdot y})(x_i \otimes y_i)\rangle\\
\nonumber &=&\langle \eta\otimes\xi,L^{-2}_{x\cdot y} x_i\otimes \alpha(y_i)+\alpha(x_i)\otimes {\mathrm{ad}}^{-2}_{x\cdot y}y_i\rangle\\
\nonumber &=&\langle \xi\otimes \eta,\alpha(y_i)\otimes L^{-2}_{x\cdot y} x_i+{\mathrm{ad}}^{-2}_{x\cdot y}y_i \otimes \alpha(x_i)\rangle\\
&=&\langle \xi\otimes \eta,(\alpha \otimes L^{-2}_{x\cdot y}+{\mathrm{ad}}^{-2}_{x\cdot y}\otimes \alpha)\sigma(r)\rangle.\end{aligned}$$ Similarly, we have $$\begin{aligned}
\label{1-cocycle-3}\langle\eta\otimes\xi,(L^{-2}_{\alpha(x)}\otimes \alpha)\varphi^*(y) \rangle=\langle\xi\otimes \eta,(\alpha \otimes L^{-2}_{\alpha(x)})(\alpha\otimes L^{-2}_y+{\mathrm{ad}}^{-2}_y \otimes \alpha)\sigma(r) \rangle,\\
\label{1-cocycle-4}\langle\eta\otimes\xi,(\alpha\otimes L^{-2}_{\alpha(x)})\varphi^*(y) \rangle=\langle\xi\otimes \eta,( L^{-2}_{\alpha(x)}\otimes \alpha)(\alpha\otimes L^{-2}_y+{\mathrm{ad}}^{-2}_y \otimes \alpha)\sigma(r) \rangle,\\
\label{1-cocycle-5}\langle\eta\otimes\xi,(\alpha\otimes R^{-2}_{\alpha(y)})\varphi^*(x) \rangle=\langle\xi\otimes \eta,( R^{-2}_{\alpha(y)}\otimes \alpha)(\alpha\otimes L^{-2}_x+{\mathrm{ad}}^{-2}_x \otimes \alpha)\sigma(r) \rangle.\end{aligned}$$ Thus, by - and the definition of a Hom-pre-Lie algebra, we have $$\begin{aligned}
&&\langle-L^\star_{\alpha(x)}\{\xi,\eta\}+\{L_x^\star \xi,(\alpha^{-1})^*(\eta)\}+\{(\alpha^{-1})^*(\xi),L_x^\star \eta\}+L^\star_{{\mathcal{L}}^\star_\eta x}(\alpha^{-1})^*(\xi)-L^\star_{{\mathcal{L}}^\star_\xi x}(\alpha^{-1})^*(\eta),\alpha^2(y)\rangle \\
&=&\langle\xi\otimes\eta,(L^{-2}_{x\cdot y}\otimes \alpha+\alpha\otimes {\mathrm{ad}}^{-2}_{x\cdot y})r\rangle-\langle\xi\otimes\eta,(L^{-2}_{\alpha(x)}\otimes \alpha)(L^{-2}_ y\otimes \alpha+\alpha\otimes {\mathrm{ad}}^{-2}_y)r \rangle\\
&&-\langle\xi\otimes\eta,(\alpha \otimes L^{-2}_{\alpha(x)})(L^{-2}_ y\otimes \alpha+\alpha\otimes {\mathrm{ad}}^{-2}_y)r\rangle-\langle\xi\otimes\eta,(\alpha \otimes R^{-2}_{\alpha(y)})(L^{-2}_ x\otimes \alpha+\alpha\otimes {\mathrm{ad}}^{-2}_x)r\rangle\\
&&-\langle\xi\otimes\eta,(\alpha \otimes L^{-2}_{x\cdot y}+{\mathrm{ad}}^{-2}_{x\cdot y} \otimes\alpha)\sigma(r)\rangle+\langle\xi\otimes\eta,(\alpha \otimes L^{-2}_{\alpha(x)})(\alpha \otimes L^{-2}_ y+ {\mathrm{ad}}^{-2}_y \otimes \alpha)\sigma(r)\rangle\\
&&+\langle\xi\otimes\eta,(L^{-2}_{\alpha(x)}\otimes \alpha)(\alpha \otimes L^{-2}_ y+ {\mathrm{ad}}^{-2}_y \otimes \alpha)\sigma(r)\rangle+\langle\xi\otimes\eta,(R^{-2}_{\alpha(y)}\otimes \alpha)(\alpha \otimes L^{-2}_ x+ {\mathrm{ad}}^{-2}_x \otimes \alpha)\sigma(r)\rangle\\
&=&\langle \xi\otimes \eta,\alpha^{-2}(x\cdot y)\cdot x_i\otimes \alpha(y_i)\rangle+\alpha(x_i)\otimes \alpha^{-2}(x\cdot y)\cdot y_i-\alpha^{-1}(x)\cdot(\alpha^{-2}(y)\cdot x_i)\otimes\alpha^2(y_i)\\
&&-\alpha^{-1}(x)\cdot \alpha(x_i)\otimes \alpha^{-1}(y)\cdot \alpha(y_i)-\alpha^{-1}(y)\cdot \alpha(x_i)\otimes \alpha^{-1}(x)\cdot \alpha(y_i)-\alpha^2(x_i)\otimes \alpha^{-1}(x)\cdot(\alpha^{-2}(y)\cdot y_i)\rangle\\
&&+\langle \xi\otimes \eta, -\alpha(y_i)\otimes \alpha^{-2}(x\cdot y)\cdot x_i-\alpha^{-2}(x\cdot y)\cdot y_i\otimes\alpha(x_i)+\alpha^2(y_i)\otimes \alpha^{-1}(x)\cdot(\alpha^{-2}(y)\cdot x_i)\rangle\\
&&+\alpha^{-1}(y)\cdot \alpha(y_i)\otimes \alpha^{-1}(x)\cdot \alpha(x_i)+\alpha^{-1}(x)\cdot \alpha(y_i)\otimes\alpha^{-1}(y)\cdot \alpha(x_i)+\alpha^{-1}(x)\cdot(\alpha^{-2}(y)\cdot y_i)\otimes \alpha^2(x_i)\rangle\\
&=&\langle \xi\otimes \eta,(L^{-2}_{x\cdot y}\otimes \alpha+\alpha\otimes L^{-2}_{x\cdot y}-(L^{-2}_{\alpha(x)}\otimes \alpha+\alpha\otimes L^{-2}_{\alpha(x)})(L^{-2}_y\otimes \alpha+\alpha\otimes L^{-2}_y))(x_i\otimes y_i)\rangle\\
&&-\langle \xi\otimes \eta,(L^{-2}_{x\cdot y}\otimes \alpha+\alpha\otimes L^{-2}_{x\cdot y}-(L^{-2}_{\alpha(x)}\otimes \alpha+\alpha\otimes L^{-2}_{\alpha(x)})(L^{-2}_y\otimes \alpha+\alpha\otimes L^{-2}_y))(y_i\otimes x_i)\rangle\\
&=&\langle \xi\otimes \eta, (P(x\cdot y)-P(\alpha(x))P(y))(r-\sigma(r)) \rangle,\end{aligned}$$ which implies that holds if and only if holds. By Proposition \[matched-pair-equivalent\], $\eqref{matche-pair-2}\Longleftrightarrow\eqref{pre-matche-pair-1} \Longleftrightarrow \eqref{pre-matche-pair-4}$, and by Theorem \[equivalent\], $\psi^\ast$ is a 1-cocycle of the sub-adjacent Hom-Lie algebra $(A^\ast)^C$ associated to the representation $(A^*\otimes A^*,{\mathcal{L}}^{-2}\otimes(\alpha^{-1})^\ast+(\alpha^{-1})^\ast\otimes \mathfrak{ad}^{-2})$ if and only if $\eqref{pre-matche-pair-1}$ and $\eqref{pre-matche-pair-4}$ hold. Thus, $\psi^\ast$ is a $1$-cocycle of the sub-adjacent Hom-Lie algebra $(A^\ast)^C$ associated to the representation $(A^*\otimes A^*,{\mathcal{L}}^{-2}\otimes(\alpha^{-1})^\ast+(\alpha^{-1})^\ast\otimes \mathfrak{ad}^{-2})$ if and only if holds.
Let $(A,\cdot,\alpha)$ be a Hom-pre-Lie algebra. Assume that $r\in A\otimes A$ is symmetric and satisfies . Then the equation $[[r,r]]=0$ is called the [**Hom-${\mathfrak s}$-equation**]{} in $(A,\cdot,\alpha)$ and $r$ is called a [**Hom-${\mathfrak s}$-matrix**]{}. A [**triangular Hom-pre-Lie bialgebra**]{} is a coboundary Hom-pre-Lie bialgebra, in which $r$ is a Hom-${\mathfrak s}$-matrix.
Let $(A,\cdot,\alpha)$ be a Hom-pre-Lie algebra and $r$ a Hom-${\mathfrak s}$-matrix. Then $(A,A^\ast,\varphi^*,\psi^*)$ is a Hom-pre-Lie bialgebra, where $\varphi^*:A\longrightarrow A\otimes A$ defined by and $\psi^*$ is the dual of the multiplication $\cdot$ in $A$.
By Corollary \[cor:dualalg\], $(A^\ast,\circ,(\alpha^{-1})^\ast)$ is a Hom-pre-Lie algebra, where $\circ$ is given by . By Proposition \[pro:cocycle\], $\psi^\ast$ is a $1$-cocycle of the sub-adjacent Hom-Lie algebra $(A^\ast)^C$ associated to the representation $(A^*\otimes A^*,{\mathcal{L}}^{-2}\otimes(\alpha^{-1})^\ast+(\alpha^{-1})^\ast\otimes \mathfrak{ad}^{-2})$. By Proposition \[pro:condition\] $(A,A^\ast,\varphi^*,\psi^*)$ is a Hom-pre-Lie bialgebra.
Hom-${{\mathcal{O}}}$-operators, Hom-L-dendriform algebras and Hom-${\mathfrak s}$-matrices
===========================================================================================
In this section, we introduce the notions of Hom-${{\mathcal{O}}}$-operator on Hom-pre-Lie algebra and Hom-L-dendriform algebra, by which we construct Hom-${\mathfrak s}$-matrices.
Let $(A,\cdot,\alpha)$ be a Hom-pre-Lie algebra and $(V,\beta,\rho,\mu)$ be a representation of $(A,\cdot,\alpha)$. A linear map $T:V\longrightarrow A$ is called [**Hom-${{\mathcal{O}}}$-operator**]{} if for all $u,v\in V$, the following equalities are satisfied $$\begin{aligned}
\label{eq:opreator-1}T\circ \beta&=&\alpha\circ T,\\
\label{eq:operator-2}T(u)\cdot T(v)&=&T(\rho(T(\beta^{-1}(u)))(v)+\mu(T(\beta^{-1}(v)))(u)).\end{aligned}$$
Let $(A,\cdot,\alpha)$ be a Hom-pre-Lie algebra and $r\in A\otimes A$ is symmetric. Then $r$ satisfies and $[[r,r]]=0$ if and only if $r^{\sharp}\circ (\alpha^{-1})^*$ is a Hom-${{\mathcal{O}}}$-operator associated to the representation $(A^*,(\alpha^{-1})^*,{\mathrm{ad}}^\star,-R^\star)$.
By Proposition \[O-Operator\], it is straightforward.
A [**Hom-L-dendriform algebra**]{} $(A,\triangleright,\triangleleft,\alpha)$ is a vector space $A$ equipped with two bilinear products $\triangleright,\triangleleft:A\otimes A\longrightarrow A$ and $\alpha\in {\mathfrak {gl}}(V)$, such that for all $x,y,z \in A$, $\alpha(x\triangleright y)=\alpha(x)\triangleright \alpha(y)$, $\alpha(x\triangleleft y)=\alpha(x)\triangleleft \alpha(y)$, and the following equalities are satisfied $$\begin{aligned}
\label{eq:L-1}\\ \nonumber (x\triangleright y)\triangleright \alpha(z)+(x\triangleleft y)\triangleright \alpha(z)+\alpha(y)\triangleright(x\triangleright z)-(y\triangleleft x)\triangleright\alpha(z)-(y\triangleright x)\triangleright\alpha(z)-\alpha(x)\triangleright(y\triangleright z)=0,\\
\label{eq:L-2}(x\triangleright y)\triangleleft\alpha(z)+\alpha(y)\triangleleft(x\triangleright z)+\alpha(y)\triangleleft(x\triangleleft z)-(y\triangleleft x)\triangleleft \alpha(z)-\alpha(x)\triangleright(y\triangleleft z)=0.\end{aligned}$$
Let $(A,\triangleright,\triangleleft,\alpha)$ be a Hom-L-dendriform algebra.
- The bilinear product $\bullet:A\otimes A\longrightarrow A$ given by $$x\bullet y=x\triangleright y+x\triangleleft y, \quad \forall x,y \in A,$$ define a Hom-pre-Lie algebra. $(A,\bullet,\alpha)$ is called the associated horizontal Hom-pre-Lie algebra of $(A,\triangleright,\triangleleft,\alpha)$ and $(A,\triangleright,\triangleleft,\alpha)$ is called a compatible Hom-L-dendriform algebra structure on the Hom-pre-Lie algebra $(A,\bullet,\alpha)$.
- The bilinear product $\cdot:A\otimes A\longrightarrow A$ given by $$x\cdot y=x\triangleright y-y\triangleleft x, \quad \forall x,y \in A,$$ defines a Hom-pre-Lie algebra. $(A,\cdot,\alpha)$ is called the associated vertical Hom-pre-Lie algebra of $(A,\triangleright,\triangleleft,\alpha)$ and $(A,\triangleright,\triangleleft,\alpha)$ is called a compatible Hom-L-dendriform algebra structure on the Hom-pre-Lie algebra $(A,\cdot,\alpha)$.
- Both $(A,\bullet,\alpha)$ and $(A,\cdot,\alpha)$ have the same sub-adjacent Hom-Lie algebra $A^C$ defined by $$[x,y]=x\triangleright y+x\triangleleft y-y\triangleleft x-y\triangleright x, \quad \forall x,y \in A.$$
It is straightforward.
Let $(A,\triangleright,\triangleleft,\alpha)$ be a Hom-L-dendriform algebra. Then and can be rewritten as $$\begin{aligned}
\ \alpha(x)\triangleright(y\triangleright z)-(x\bullet y)\triangleright \alpha(z)&=&\alpha(y)\triangleright (x\triangleright z)-(y\bullet x)\triangleright \alpha(z),\\
\ \alpha(x)\triangleright(y\triangleleft z)-(x\triangleright y)\triangleleft \alpha(z)&=&\alpha(y)\triangleleft (x\bullet z)-(y\triangleleft x)\triangleleft\alpha(z).\end{aligned}$$
\[rep\] Let $A$ be a vector space with two bilinear products $\triangleright,\triangleleft:A\otimes A\longrightarrow A$.
- $(A,\triangleright,\triangleleft,\alpha)$ is a Hom-L-dendriform algebra if and only if $(A,\bullet,\alpha)$ is a Hom-pre-Lie algebra and $(A,\alpha,L_\triangleright,R_\triangleleft)$ is a representation of $(A,\bullet,\alpha)$.
- $(A,\triangleright,\triangleleft,\alpha)$ is a Hom-L-dendriform algebra if and only if $(A,\cdot,\alpha)$ is a Hom-pre-Lie algebra and $(A,\alpha,L_\triangleright,-L_\triangleleft)$ is a representation of $(A,\cdot,\alpha)$.
We only prove the condition $\rm(1)$. If $(A,\triangleright,\triangleleft,\alpha)$ is a Hom-L-dendriform algebra, then for all $x,y\in A$, we have $$L_\triangleright(\alpha(x))\alpha(y)=\alpha(x)\triangleright\alpha(y)=\alpha(x\triangleright y)=\alpha(L_\triangleright (x)y),$$ which implies that $L_\triangleright(\alpha(x))\circ \alpha=\alpha\circ L_\triangleright(x)$. Similarly, we have $R_\triangleleft(\alpha(x))\circ \alpha=\alpha\circ R_\triangleleft(x)$.
For all $x,y,z\in A$, by , we have $$\begin{aligned}
&&L_\triangleright([x,y])\alpha(z)-L_\triangleright(\alpha(x))L_\triangleright(y)z+L_\triangleright(\alpha(y))L_\triangleright(x)z\\
&=& [x,y]\triangleright \alpha(z)-\alpha(x)\triangleright(y\triangleright z)+\alpha(y)\triangleright(x\triangleright z) \\
&=& (x\triangleright y)\triangleright\alpha(z)+(x\triangleleft y)\triangleright\alpha(z)-(y\triangleright x)\triangleright\alpha(z)-(y\triangleleft x)\triangleright\alpha(z)-\alpha(x)\triangleright(y\triangleright z)+\alpha(y)\triangleright(x\triangleright z) \\
&=&0,\end{aligned}$$ which implies that $$L_\triangleright([x,y])\circ\alpha=L_\triangleright(\alpha(x))L_\triangleright(y)-L_\triangleright(\alpha(y))L_\triangleright(x).$$ Similarly, we have $$R_\triangleleft(\alpha(y))\circ R_\triangleleft(x)-R_\triangleleft(x\bullet y)\circ\alpha=R_\triangleleft(\alpha(y))\circ L_\triangleright(x)-L_\triangleright(\alpha(x))\circ R_\triangleleft(y).$$ Thus $(A,\alpha,L_\triangleright,R_\triangleleft)$ is a representation of the Hom-pre-Lie algebra $(A,\bullet,\alpha)$. The converse part can be proved similarly. We omit details. The proof is finished.
Let $(A,\triangleright,\triangleleft,\alpha)$ be a Hom-L-dendriform algebra. Define two bilinear products $\triangleright^t,\triangleleft^t:A\otimes A\longrightarrow A$ by $$\label{transpose}
x\triangleright^t y=x\triangleright y, \quad x\triangleleft^ty=-y\triangleleft x, \quad \forall x,y \in A.$$ Then $(A,\triangleright^t,\triangleleft^t,\alpha)$ is a Hom-L-dendriform algebra. The associated horizontal Hom-pre-Lie algebra of $(A,\triangleright^t,\triangleleft^t,\alpha)$ is the associated vertical Hom-pre-Lie algebra $(A,\cdot,\alpha)$ of $(A,\triangleright,\triangleleft,\alpha)$ and the associated vertical Hom-pre-Lie algebra of $(A,\triangleright^t,\triangleleft^t,\alpha)$ is the associated horizontal Hom-pre-Lie algebra $(A,\bullet,\alpha)$ of $(A,\triangleright,\triangleleft,\alpha)$, that is, $$\bullet^t=\cdot,\quad \cdot^t=\bullet.$$
It is straightforward.
Let $(A,\triangleright,\triangleleft,\alpha)$ be a Hom-L-dendriform algebra. The Hom-L-dendriform algebra $(A,\triangleright^t,\triangleleft^t,\alpha)$ given by is called the transpose of $(A,\triangleright,\triangleleft,\alpha)$.
For brevity, we only give the study of the vertical Hom-pre-Lie algebra.
\[O-operator-L\] Let $(A,\cdot,\alpha)$ be a Hom-pre-Lie algebra and $(V,\beta,\rho,\mu)$ be a representation of $(A,\cdot,\alpha)$. Suppose that $T:V\longrightarrow A$ is a Hom-${{\mathcal{O}}}$-operator. Then there exists a Hom-L-dendriform algebra structure on $V$ defined by $$u\triangleright v=\rho(T(\beta^{-1}(u)))v, \quad u\triangleleft v=-\mu(T(\beta^{-1}(u)))v, \quad \forall u,v \in V.$$
First by $T\circ \beta=\alpha\circ T$ and $(V,\beta,\rho,\mu)$ is a representation of $(A,\cdot,\alpha)$, for all $u,v\in V$ we have $$\beta(u\triangleright v)=\beta(\rho(T(\beta^{-1}(u)))v)=\rho(\alpha(T(\beta^{-1}(u))))\beta(v)=\rho(T(u))\beta(v)=\beta(u)\triangleright\beta(v).$$ Similarly, we have $\beta(u\triangleleft v)=\beta(u)\triangleleft\beta(v).$ Furthermore, by and $(V,\beta,\rho,\mu)$ being a representation of $(A,\cdot,\alpha)$, for all $u,v,w\in V$, we have $$\begin{aligned}
&&(u\triangleright v)\triangleright\beta(w)+(u\triangleleft v)\triangleright\beta(w)+\beta(v)\triangleright(u\triangleright w)-(v\triangleleft u)\triangleright\beta(w)-(v\triangleright u)\triangleright\beta(w)-\beta(u)\triangleright(v\triangleright w)\\
&=& \rho(T(\beta^{-1}(u)))v\triangleright\beta(w)-\mu(T(\beta^{-1}(u)))v\triangleright\beta(w)+\beta(v)\triangleright \rho(T(\beta^{-1}(u)))w\\
&&+\mu(T(\beta^{-1}(v)))u\triangleright\beta(w)-\rho(T(\beta^{-1}(v)))u\triangleright\beta(w)-\beta(u)\triangleright\rho(T(\beta^{-1}(v)))w\\
&=& \rho(T(\beta^{-1}(\rho(T(\beta^{-1}(u)))v)))\beta(w)-\rho(T(\beta^{-1}(\mu(T(\beta^{-1}(u)))v)))\beta(w)+\rho(T(v))\rho(T(\beta^{-1}(u)))w \\
&&+\rho(T(\beta^{-1}(\mu(T(\beta^{-1}(v)))u)))\beta(w)-\rho(T(\beta^{-1}(\rho(T(\beta^{-1}(v)))u)))\beta(w)-\rho(T(u))\rho(T(\beta^{-1}(v)))w\\
&=&\rho(\alpha^{-1}(T(u)\cdot T(v)))\beta(w)-\rho(\alpha^{-1}(T(v)\cdot T(u)))\beta(w)+\rho(T(v))\rho(T(\beta^{-1}(u)))w-\rho(T(u))\rho(T(\beta^{-1}(v)))w\\
&=&0,\end{aligned}$$ which implies that holds.
Similarly, we have $$(u\triangleright v)\triangleleft\beta(w)+\beta(v)\triangleleft(u\triangleright w)+\beta(v)\triangleleft(u\triangleleft w)-(v\triangleleft u)\triangleleft\beta(w)-\beta(u)\triangleright(v\triangleleft w)=0,$$ which implies that holds. This finishes the proof.
\[O-operator-L1\] With the above conditions. $T$ is a homomorphism from the associated vertical Hom-pre-Lie algebra of $(V,\triangleright,\triangleleft,\beta)$ to Hom-pre-Lie algebra $(A,\cdot,\alpha)$. Moreover, $T(V)=\{T(u)|u\in V\}\subset A$ is a Hom-pre-Lie subalgebra of $(A,\cdot,\alpha)$ and there is an induced Hom-L-dendriform algebra structure on $T(V)$ given by $$T(u)\triangleright T(v)=T(u\triangleright v), \quad T(u)\triangleleft T(v)=T(u\triangleleft v), \quad \forall u,v \in V.$$
\[Operator-L\] Let $(A,\cdot,\alpha)$ be a Hom-pre-Lie algebra. Then there exists a compatible Hom-L-dendriform algebra structure on $(A,\cdot,\alpha)$ such that $(A,\cdot,\alpha)$ is the associated vertical Hom-pre-Lie algebra if and only if there exists an invertible Hom-${{\mathcal{O}}}$-operator $T$ associated to a representation $(V,\beta,\rho,\mu)$.
Let $T$ be an invertible Hom-${{\mathcal{O}}}$-operator $T$ associated to a representation $(V,\beta,\rho,\mu)$. By Theorem \[O-operator-L\], Corollary \[O-operator-L1\] and , there exists a Hom-L-dendriform algebra on $T(V)$ given by $$x\triangleright y=T(\rho(T(\beta^{-1}(u)))T^{-1}(y))=T(\rho(\alpha^{-1}(x))T^{-1}(y)).$$ Similarly, we have $x\triangleleft y=-T(\mu(\alpha^{-1}(x))T^{-1}(y)).$
Moreover, by , we have $$x\triangleright y-y\triangleleft x=T(\rho(\alpha^{-1}(x))T^{-1}(y)+\mu(\alpha^{-1}(y))T^{-1}(x))=x\cdot y.$$ Conversely, Let $(A,\triangleright,\triangleleft,\alpha)$ be a Hom-L-dendriform algebra and $(A,\cdot,\alpha)$ be the associated vertical Hom-pre-Lie algebra. By Theorem \[rep\], $(A,\alpha,L_\triangleright,-L_\triangleleft)$ is a representation of $(A,\cdot,\alpha)$ and $\alpha:A\longrightarrow A$ is a Hom-${{\mathcal{O}}}$-operator of $(A,\cdot,\alpha)$ associated to $(A,\alpha,L_\triangleright,-L_\triangleleft)$.
In the sequel, we give the relation between Hessian structure and Hom-L-dendriform algebras.
[([@LSS])]{} A Hessian structure on a regular Hom-pre-Lie algebra $(A,\cdot,\alpha)$ is a symmetric nondegenerate $2$-cocycle ${\mathcal{B}}\in Sym^2(A^\ast)$, i.e. $\partial_T{\mathcal{B}}=0$, satisfying ${\mathcal{B}}\circ (\alpha\otimes\alpha)={\mathcal{B}}$. More precisely, $$\begin{aligned}
\label{hom-hessian-1}{\mathcal{B}}(\alpha(x),\alpha(y))&=&{\mathcal{B}}(x,y),\\
\label{hom-hessian-2}{\mathcal{B}}(x\cdot y,\alpha(z))-{\mathcal{B}}(\alpha(x),y\cdot z)&=&{\mathcal{B}}(y\cdot x,\alpha(z))-{\mathcal{B}}(\alpha(y),x\cdot z),\quad \forall x,y,z\in A.\end{aligned}$$
Let $A$ be a vector space, for all ${\mathcal{B}}\in Sym^2(A^\ast)$, the linear map ${\mathcal{B}}^\sharp:A \longrightarrow A^\ast$ is given by $$\label{hessian}
\langle {\mathcal{B}}^\sharp(x),y\rangle={\mathcal{B}}(x,y),\quad \forall x,y\in A.$$
Let $(A,\cdot,\alpha)$ be a Hom-pre-Lie algebra with a Hessian structure ${\mathcal{B}}$. Then there exists a compatible Hom-L-dendriform algebra structure on $(A,\cdot,\alpha)$ given by $${\mathcal{B}}(x\triangleright y,z)=-{\mathcal{B}}(y,[\alpha^{-1}(x),\alpha^{-2}(z)]), \quad {\mathcal{B}}(x\triangleleft y,z)=-{\mathcal{B}}(y,\alpha^{-2}(z)\cdot\alpha^{-1}(x)), \quad \forall x,y,z\in A.$$
By and , we obtain that $({\mathcal{B}}^\sharp)^{-1}\circ (\alpha^{-1})^\ast=\alpha\circ ({\mathcal{B}}^\sharp)^{-1}.$ Thus, we have $$\label{hessian-operator1}
({\mathcal{B}}^\sharp)^{-1}\circ (\alpha^{-1})^\ast\circ (\alpha^{-1})^\ast=\alpha\circ ({\mathcal{B}}^\sharp)^{-1}\circ (\alpha^{-1})^\ast.$$ For all $x,y,z\in A, \xi,\eta,\gamma\in A^*$, set $x=\alpha(({\mathcal{B}}^\sharp)^{-1}(\xi)), y=\alpha(({\mathcal{B}}^\sharp)^{-1}(\eta)), z=\alpha(({\mathcal{B}}^\sharp)^{-1}(\gamma))$, we have $$\begin{aligned}
&&\langle ({\mathcal{B}}^\sharp)^{-1}((\alpha^{-1})^*(\xi))\cdot ({\mathcal{B}}^\sharp)^{-1}((\alpha^{-1})^*(\eta))-({\mathcal{B}}^\sharp)^{-1}(\alpha^{-1})^*({\mathrm{ad}}^\star(({\mathcal{B}}^\sharp)^{-1}(\alpha^{-1})^*(\alpha^*(\xi)))\eta\\
&&-R^\star(({\mathcal{B}}^\sharp)^{-1}(\alpha^{-1})^*(\alpha^*(\eta)))\xi),(\alpha^{-2})^*(\gamma)\rangle\\
&=&\langle x\cdot y-({\mathcal{B}}^\sharp)^{-1}(\alpha^{-1})^*({\mathrm{ad}}^\star_{({\mathcal{B}}^\sharp)^{-1}(\xi)}\eta-R^\star_{({\mathcal{B}}^\sharp)^{-1}(\eta)}\xi),(\alpha^{-2})^*(\gamma)\rangle \\
&=&\langle x \cdot y,{\mathcal{B}}^\sharp(\alpha(z))\rangle-\langle{\mathrm{ad}}^\star_{\alpha^{-1}(x)}{\mathcal{B}}^\sharp(\alpha^{-1}(y))-R^\star_{\alpha^{-1}(y)}{\mathcal{B}}^\sharp(\alpha^{-1}(x)),z \rangle \\
&=&{\mathcal{B}}(\alpha(z),x\cdot y)+\langle {\mathcal{B}}^\sharp(\alpha^{-1}(y)),[\alpha^{-2}(x),\alpha^{-2}(z)]\rangle-\langle {\mathcal{B}}^\sharp(\alpha^{-1}(x)),\alpha^{-2}(z)\cdot \alpha^{-2}(y)\rangle\\
&=&{\mathcal{B}}(\alpha(z),x\cdot y)+{\mathcal{B}}([\alpha^{-2}(x),\alpha^{-2}(z)],\alpha^{-1}(y))-{\mathcal{B}}(\alpha^{-1}(x),\alpha^{-2}(z)\cdot \alpha^{-2}(y))\\
&=&{\mathcal{B}}(\alpha(z),x\cdot y)+{\mathcal{B}}([x,z],\alpha(y))-{\mathcal{B}}(\alpha(x),z\cdot y)\\
&=&0,\end{aligned}$$ which implies that $$\begin{aligned}
\label{Hessian-o-operator-2}
\nonumber &&({\mathcal{B}}^\sharp)^{-1}((\alpha^{-1})^*(\xi))\cdot ({\mathcal{B}}^\sharp)^{-1}((\alpha^{-1})^*(\eta))\\
&=&({\mathcal{B}}^\sharp)^{-1}(\alpha^{-1})^*({\mathrm{ad}}^\star(({\mathcal{B}}^\sharp)^{-1}(\alpha^{-1})^*(\alpha^*(\xi)))\eta-R^\star(({\mathcal{B}}^\sharp)^{-1}(\alpha^{-1})^*(\alpha^*(\eta)))\xi).\end{aligned}$$ By and , we deduce that $({\mathcal{B}}^\sharp)^{-1}\circ (\alpha^{-1})^*$ is a Hom-${{\mathcal{O}}}$-operator associated to the representation $(A^*,(\alpha^{-1})^*,{\mathrm{ad}}^\star,-R^\star)$. By Theorem \[Operator-L\], there is a compatible Hom-L-dendriform algebra structure on $A$ defined by $$\begin{aligned}
{\mathcal{B}}(x\triangleright y,z)
&=&{\mathcal{B}}(({\mathcal{B}}^\sharp)^{-1}(\alpha^{-1})^*({\mathrm{ad}}^\star_{\alpha^{-1}(x)}\alpha^*({\mathcal{B}}^\sharp(y))),z)\\
&=&\langle {\mathrm{ad}}^\star_{\alpha^{-1}(x)}\alpha^*({\mathcal{B}}^\sharp(y)),\alpha^{-1}(z)\rangle\\
&=&-\langle {\mathcal{B}}^\sharp(y),[\alpha^{-1}(x),\alpha^{-2}(z)]\rangle\\
&=&-{\mathcal{B}}(y,[\alpha^{-1}(x),\alpha^{-2}(z)]).\end{aligned}$$ Similarly, we have ${\mathcal{B}}(x\triangleleft y,z)=-{\mathcal{B}}(y,\alpha^{-2}(z)\cdot\alpha^{-1}(x))$. The proof is finished.
Next we consider the semi-direct product Hom-pre-Lie algebra $A\ltimes_{(\rho^{\star}-\mu^{\star},-\rho^{\star})}V^*$. Any linear map $T:V\longrightarrow A$ can be view as an element $\bar{T}\in \otimes^2(A\oplus V^*)$ via $$\bar{T}(\xi+u,\eta+v)=\langle T(u),\eta\rangle, \quad \forall \xi+u,\eta+v\in A^*\oplus V,$$ then $r=\bar{T}+\sigma(\bar{T})$ is symmetric.
\[semi-product\] Let $(A,\cdot,\alpha)$ be a Hom-pre-Lie algebra, $(V,\beta,\rho,\mu)$ be a representation of $(A,\cdot,\alpha)$ and $T:V\longrightarrow A$ be a linear map satisfying $T\circ \beta=\alpha\circ T$. Then $r=\bar{T}+\sigma(\bar{T})$ is a Hom-${\mathfrak s}$-matrix in the Hom-pre-Lie algebra $A\ltimes_{(\rho^{\star}-\mu^{\star},-\rho^{\star})}V^*$ if and only if $T\circ \beta$ is a Hom-${{\mathcal{O}}}$-operator.
Let $\{v_1,\dots,v_n\}$ be a basis of $V$ and $\{v_1^*,\dots,v_n^*\}$ be its dual basis. It is obvious that $\bar{T}$ can be expressed by $\bar{T}=T(v_i)\otimes v_i^*$. Here the Einstein summation convention is used. Therefore, we can write $r=T(v_i)\otimes v_i^*+v_i^*\otimes T(v_i)$. Then we have $$\begin{aligned}
r_{12} \cdot r_{23}
&=&-\alpha(T(v_i))\otimes \mu^\star(T(v_j))v_i^*\otimes(\beta^{-1})^*(v_j^*)+(\beta^{-1})^*(v_i^*)\otimes T(v_i)\cdot T(v_j)\otimes(\beta^{-1})^*(v_j^*)\\
&&+(\beta^{-1})^*(v_i^*)\otimes \rho^\star(T(v_i))v_j^*\otimes \alpha(T(v_j))-(\beta^{-1})^*(v_i^*)\otimes \mu^\star(T(v_i))v_j^*\otimes \alpha(T(v_j)),\\
-r_{23} \cdot r_{12}
&=&-\alpha(T(v_j))\otimes \rho^\star(T(v_i))v_j^*\otimes(\beta^{-1})^*(v_i^*)+\alpha(T(v_j))\otimes \mu^\star(T(v_i))v_j^*\otimes(\beta^{-1})^*(v_i^*)\\
&&-(\beta^{-1})^*(v_j^*)\otimes T(v_i)\cdot T(v_j) \otimes(\beta^{-1})^*(v_i^*)+(\beta^{-1})^*(v_j^*)\otimes \mu^\star(T(v_j))v_i^*\otimes \alpha(T(v_i)),\\
-r_{13} \cdot r_{12}
&=&-T(v_i)\cdot T(v_j)\otimes (\beta^{-1})^*(v_j^*)\otimes(\beta^{-1})^*(v_i^*)-\rho^\star(T(v_i))v_j^* \otimes\alpha(T(v_j))\otimes (\beta^{-1})^*(v_i^*)\\
&&+\mu^\star(T(v_i))v_j^* \otimes\alpha(T(v_j))\otimes (\beta^{-1})^*(v_i^*)+ \mu^\star(T(v_j))v_i^*\otimes(\beta^{-1})^*(v_j^*)\otimes \alpha(T(v_i)),\\r_{13} \cdot r_{23}
&=&-\alpha(T(v_i))\otimes (\beta^{-1})^*(v_j^*)\otimes\mu^\star(T(v_j))v_i^*+(\beta^{-1})^*(v_i^*)\otimes\alpha(T(v_j))\otimes \rho^\star(T(v_i))v_j^* \\
&&-(\beta^{-1})^*(v_i^*)\otimes\alpha(T(v_j))\otimes \mu^\star(T(v_i))v_j^*+(\beta^{-1})^*(v_i^*)\otimes(\beta^{-1})^*(v_j^*)\otimes-T(v_i)\cdot T(v_j).\end{aligned}$$ By direct computations, we have $$\begin{aligned}
&=&\langle(\beta^{-1})^*(v_i^*),v_m\rangle v_m^*\otimes T(v_i)\cdot T(v_j)\otimes \langle(\beta^{-1})^*(v_j^*),v_n\rangle v_n^*\\
&&+\langle(\beta^{-1})^*(v_i^*),v_m\rangle v_m^*\otimes \langle \rho^\star(T(v_i))v_j^*,v_n\rangle v_n^*\otimes \alpha(T(v_j))\\
&&-\alpha(T(v_j))\otimes \langle \rho^\star(T(v_i))v_j^*,v_m\rangle v_m^*\otimes \langle(\beta^{-1})^*(v_i^*),v_n\rangle v_n^*\\
&&-\langle(\beta^{-1})^*(v_j^*),v_m\rangle v_m^*\otimes T(v_i)\cdot T(v_j)\otimes \langle(\beta^{-1})^*(v_i^*),v_n\rangle v_n^*\\
&&-T(v_i)\cdot T(v_j)\otimes \langle(\beta^{-1})^*(v_j^*),v_m\rangle v_m^*\otimes\langle(\beta^{-1})^*(v_i^*),v_n\rangle v_n^*\\
&&-\langle \rho^\star(T(v_i))v_j^*,v_m\rangle v_m^*\otimes \alpha(T(v_j))\otimes\langle(\beta^{-1})^*(v_i^*),v_n\rangle v_n^*\\
&&+\langle \mu^\star(T(v_i))v_j^*,v_m\rangle v_m^*\otimes \alpha(T(v_j))\otimes\langle(\beta^{-1})^*(v_i^*),v_n\rangle v_n^*\\
&&+\langle \mu^\star(T(v_j))v_i^*,v_m\rangle v_m^*\otimes\langle(\beta^{-1})^*(v_j^*),v_n\rangle v_n^*\otimes\alpha(T(v_i))\\
&&-\alpha(T(v_i))\otimes\langle(\beta^{-1})^*(v_j^*),v_m\rangle v_m^*\otimes \langle \mu^\star(T(v_j))v_i^*,v_n\rangle v_n^*\\
&&+\langle(\beta^{-1})^*(v_i^*),v_m\rangle v_m^*\otimes \alpha(T(v_j))\otimes\langle \rho^\star(T(v_i))v_j^*,v_n\rangle v_n^*\\
&&-\langle(\beta^{-1})^*(v_i^*),v_m\rangle v_m^*\otimes \alpha(T(v_j))\otimes\langle \mu^\star(T(v_i))v_j^*,v_n\rangle v_n^*\\
&&+\langle(\beta^{-1})^*(v_i^*),v_m\rangle v_m^*\otimes \langle(\beta^{-1})^*(v_j^*),v_n\rangle v_n^*\otimes T(v_i)\cdot T(v_j)\\
&=&v_m^*\otimes\langle v_i^*,\beta^{-1}(v_m)\rangle\langle v_j^*,\beta^{-1}(v_n)\rangle T(v_i)\cdot T(v_j)\otimes v_n^*\\
&&-v_m^*\otimes v_n^*\otimes \langle v_i^*,\beta^{-1}(v_m)\rangle\langle v_j^*,\rho(T(\beta^{-1}(v_i)))\beta^{-2}(v_n)\rangle\alpha(T(v_j))\\
&&+\langle v_j^*,\rho(T(\beta^{-1}(v_i)))\beta^{-2}(v_m)\rangle\langle v_i^*,\beta^{-1}(v_n)\rangle \alpha(T(v_j))\otimes v_m^*\otimes v_n^*\\
&&-v_m^*\otimes\langle v_j^*,\beta^{-1}(v_m)\rangle\langle v_i^*,\beta^{-1}(v_n)\rangle T(v_i)\cdot T(v_j)\otimes v_n^*\\
&&-\langle v_j^*,\beta^{-1}(v_m)\rangle\langle v_i^*,\beta^{-1}(v_n)\rangle T(v_i)\cdot T(v_j)\otimes v_m^*\otimes v_n^*\\
&&+v_m^*\otimes\langle v_j^*,\rho(T(\beta^{-1}(v_i)))\beta^{-2}(v_m)\rangle\langle v_i^*,\beta^{-1}(v_n)\rangle\alpha(T(v_j))\otimes v_n^*\\
&&-v_m^*\otimes\langle v_j^*,\mu(T(\beta^{-1}(v_i)))\beta^{-2}(v_m)\rangle\langle v_i^*,\beta^{-1}(v_n)\rangle\alpha(T(v_j))\otimes v_n^*\\
&&-v_m^*\otimes v_n^*\otimes\langle v_i^*,\mu(T(\beta^{-1}(v_j)))\beta^{-2}(v_m)\rangle\langle v_j^*,\beta^{-1}(v_n)\rangle\alpha(T(v_i))\\
&&+\langle v_j^*,\beta^{-1}(v_m)\rangle\langle v_i^*,\mu(T(\beta^{-1}(v_j)))\beta^{-2}(v_n)\rangle\alpha(T(v_i))\otimes v_m^*\otimes v_n^*\\
&&-v_m^*\otimes\langle v_i^*,\beta^{-1}(v_m)\rangle\langle v_j^*,\rho(T(\beta^{-1}(v_i)))\beta^{-2}(v_n)\rangle\alpha(T(v_j))\otimes v_n^*\\
&&+v_m^*\otimes\langle v_i^*,\beta^{-1}(v_m)\rangle\langle v_j^*,\mu(T(\beta^{-1}(v_i)))\beta^{-2}(v_n)\rangle\alpha(T(v_j))\otimes v_n^*\\
&&+v_m^*\otimes v_n^*\otimes\langle v_i^*,\beta^{-1}(v_m)\rangle\langle v_j^*,\beta^{-1}(v_n)\rangle T(v_i)\cdot T(v_j)\\
&=&v_m^*\otimes T(\beta^{-1}(v_m))\cdot T(\beta^{-1}(v_n))\otimes v_n^*-v_m^*\otimes v_n^*\otimes T\beta(\rho(T(\beta^{-2}(v_m)))\beta^{-2}(v_n))\\
&&+T\beta(\rho(T(\beta^{-2}(v_n)))\beta^{-2}(v_m))\otimes v_m^*\otimes v_n^*- v_m^*\otimes T(\beta^{-1}(v_n))\cdot T(\beta^{-1}(v_m))\otimes v_n^*\\
&&-T(\beta^{-1}(v_n))\cdot T(\beta^{-1}(v_m))\otimes v_m^*\otimes v_n^*+v_m^*\otimes T\beta(\rho(T(\beta^{-2}(v_n)))\beta^{-2}(v_m))\otimes v_n^*\\
&&-v_m^*\otimes T\beta(\mu(T(\beta^{-2}(v_n)))\beta^{-2}(v_m))\otimes v_n^*-v_m^*\otimes v_n^*\otimes T\beta(\mu(T(\beta^{-2}(v_n)))\beta^{-2}(v_m))\\
&&+T\beta(\mu(T(\beta^{-2}(v_m)))\beta^{-2}(v_n))\otimes v_m^*\otimes v_n^*-v_m^*\otimes T\beta(\rho(T(\beta^{-2}(v_m)))\beta^{-2}(v_n))\otimes v_n^*\\
&&+v_m^*\otimes T\beta(\mu(T(\beta^{-2}(v_m)))\beta^{-2}(v_n))\otimes v_n^*+v_m^*\otimes v_n^*\otimes T(\beta^{-1}(v_m))\cdot T(\beta^{-1}(v_n))\\
&=&v_m^*\otimes (T(\beta^{-1}(v_m))\cdot T(\beta^{-1}(v_n))-T\beta(\rho(T(\beta^{-2}(v_m)))\beta^{-2}(v_n)
-\mu(T(\beta^{-2}(v_n)))\beta^{-2}(v_m))\otimes v_n^*\\
&&+v_m^*\otimes v_n^*\otimes (T(\beta^{-1}(v_m))\cdot T(\beta^{-1}(v_n))-T\beta(\rho(T(\beta^{-2}(v_m)))\beta^{-2}(v_n)
-\mu(T(\beta^{-2}(v_n)))\beta^{-2}(v_m))\\
&&-(T(\beta^{-1}(v_n))\cdot T(\beta^{-1}(v_m))-T\beta(\rho(T(\beta^{-2}(v_n)))\beta^{-2}(v_m)
-\mu(T(\beta^{-2}(v_m)))\beta^{-2}(v_n))\otimes v_m^*\otimes v_n^*\\
&&-v_m^*\otimes (T(\beta^{-1}(v_n))\cdot T(\beta^{-1}(v_m))-T\beta(\rho(T(\beta^{-2}(v_n)))\beta^{-2}(v_m)
-\mu(T(\beta^{-2}(v_m)))\beta^{-2}(v_n))\otimes v_n^*,\end{aligned}$$ which implies that $[[r,r]]=0$ if and only if for all $u,v\in V$, $$T\beta(\beta^{-2}(u))\cdot T\beta(\beta^{-2}(v))=T\beta(\rho(T\beta(\beta^{-1}(\beta^{-2}(u))))\beta^{-2}(v)+\mu(T\beta(\beta^{-1}(\beta^{-2}(v))))\beta^{-2}(u)).$$ Furthermore, since $T\circ \beta=\alpha\circ T$, it is obvious that $T\circ \beta$ satisfies $(T\circ \beta)\circ \beta=\alpha \circ(T\circ \beta)$. Thus, $r$ is a Hom-${\mathfrak s}$-matrix in the Hom-pre-Lie algebra $A\ltimes_{(\rho^{\star}-\mu^{\star},-\rho^{\star})}V^*$ if and only if $T\circ \beta$ is a Hom-${{\mathcal{O}}}$-operator.
Let $(A,\triangleright,\triangleleft,\alpha)$ be a Hom-L-dendriform algebra. Then $r=v_i\otimes v_i^*+v_i^*\otimes v_i$ is a Hom-${\mathfrak s}$-matrix in the associated vertical Hom-pre-Lie algebra $A\ltimes_{(L_\triangleright^*+L_\triangleleft^*,L_\triangleleft^*)}A^*.$
By Proposition \[rep\] and Proposition \[dual-rep\], $(A^*,(\alpha^{-1})^*,L_\triangleright^*+L_\triangleleft^*,L_\triangleleft^*)$ is a dual representation of the associated vertical Hom-pre-Lie algebra $(A,\cdot,\alpha)$. Moreover, $\alpha={\rm{Id}}\circ \alpha:A\longrightarrow A$ is a Hom-${{\mathcal{O}}}$-operator associated to the representation $(A,\alpha,L_\triangleright,-L_\triangleleft)$. By Theorem \[semi-product\], $r=v_i\otimes v_i^*+v_i^*\otimes v_i$ is a Hom-${\mathfrak s}$-matrix.
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{
"pile_set_name": "ArXiv"
}
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[**Asymmetric and Moving-Frame Approaches** ]{}
[**to Navier-Stokes Equations**]{}[^1]
[Xiaoping Xu]{}
[Institute of Mathematics, Academy of Mathematics & System Sciences]{}
[Chinese Academy of Sciences, Beijing 100080, P.R. China]{} [^2]
[**Abstract**]{}
Introduction
============
The most fundamental differential equations in the motion of incompressible viscous fluid are Navier-Stokes equations: $$u_t+uu_x+vu_y+wu_z+\frac{1}{\rho}p_x=\nu (u_{xx}+u_{yy}+u_{zz})
,\eqno(1.1)$$ $$v_t+uv_x+vv_y+wv_z+\frac{1}{\rho}p_y=\nu (v_{xx}+v_{yy}+v_{zz})
,\eqno(1.2)$$ $$w_t+uw_x+vw_y+ww_z+\frac{1}{\rho}p_z=\nu (w_{xx}+w_{yy}+w_{zz})
,\eqno(1.3)$$ $$u_x+v_y+w_z=0,\eqno(1.4)$$ where $(u,v,w)$ stands for the velocity vector of the fluid, $p$ stands for the pressure of the fluid, $\rho$ is the density constant and $\nu$ is the coefficient constant of the kinematic viscosity.
The Lie point symmetries of the two-dimensional special case of the above equations ($u_z=v_z=w=0$) were obtained by Pukhnachev \[P1\] and Buchnev \[Ba\]. Moreover, certain group-invariant solutions were found in the works of Pukhnachev \[Pv1\], Kochin-Kibel’-Roze \[KKR\] and Bytev \[Bv1\], \[Bv2\]. Futhermore, Gryn \[G\] obtained certain exact solution describing flows between porous walls in the presence of injection and suction at identical rates, and Polyanin \[Pa\] used the method of generalized separation of variables to find certain exact solutions.
Assuming nullity of certain components of the tensor of momentum flow density, Landau \[Ll\] found a exact solution of Navier-Stokes equations (1.1)-(1.4), which describes axially symmetrical jet discharging from a thin pipe into unbounded space. The Lie point symmetries of the above three-dimensional equations were obtained by Buchnev \[Ba\] and Pukhnachev \[Pv2\]. Moreover, Kapitanskii \[K\] found certain cylindrical invariant solutions of the equations and Yakimov \[Y\] obtained exact solutions with a singularity of the type of a vortex filament situated on a half line. Shen \[S1, S2\] rewrote Navier-Stokes equations in terms of complex variables and found certain exact solutions. Brutyan and Karapivskii \[BK\] got exact solutions describing the evolution of a vortex structure in a generalized shear flow. Furthermore, Leipnik \[Lr\] obtained exact solutions by recursive series of diffusive quotients, and Vyskrebtsov \[V\] studied self-similar solutions for an axisymmetric flow of a viscous incompressible flow.
From algebraic point of view, it seems to us that there are not enough exact solutions that fully reflect the fundamental natures of Navier-Stokes equations. In this paper, we introduce a method of imposing asymmetric conditions on the velocity vector with respect to independent variables and obtain two families of non-steady asymmetric solutions with rotation. One of the families contains two arbitrary parameter functions of $t$ and an arbitrary number of parameter constants. Using Fourier expansion and this family of solutions, one can obtain discontinuous solutions that may be useful in study of shock waves. Another family of solutions are partially cylindrical invariant, contain two parameter functions of $t$ and structurally depend on two arbitrary polynomials, which may be used to describe incompressible fluid in a nozzle. In order to better reflect the rotating nature of flow, we also give a method of moving frame and find five families of non-steady rotating solutions with various parameters. In particular, one family of solutions blow up at any point on a moving plane with a line deleted, which may be used to study turbulence. Another family can be used to obtain discontinuous rotating solutions. Most of our solutions are globally analytic with respect to spacial variables. Below we give a more detailed introduction.
The equations (1.1)-(1.4) are invariant under orthogonal transformations $\{T_A\mid A\in O(n,{\mathbb}{R})\}$ with $$T_A\left[\left(\begin{array}{c}x\\ y\\ z\end{array}\right)\right]=
\left(\begin{array}{c}x\\ y\\ z\end{array}\right)A,\qquad
T_A\left[\left(\begin{array}{c}u\\ v\\ w\end{array}\right)\right]=
\left(\begin{array}{c}u\\ v\\
w\end{array}\right)A,\qquad T_A(p)=p.\eqno(1.5)$$ Moreover, they are invariant under the time translation $t\mapsto t+a$ with $a\in{\mathbb}{R}$ and the scaling $T_b$ with $0\neq b\in{\mathbb}{R}$: $$T_b(u)=b^{-1}u(b^2t,bx,by,bz),\qquad T_b(v)=
b^{-1}v(b^2t,bx,by,bz),\eqno(1.6)$$ $$T_b(w)=b^{-1}w(b^2t,bx,by,bz),\qquad T_b(p)=
b^{-2}p(b^2t,bx,by,bz).\eqno(1.7)$$ The most interesting symmetries of Navier-Stokes equations are the following time-dependent translations: $$T_{1{\alpha}}(u)=u(t,x+{\alpha},y,z)-{\alpha}',\qquad
T_{1{\alpha}}(v)=v(t,x+{\alpha},y,z),\eqno(1.8)$$ $$T_{1{\alpha}}(w)=w(t,x+{\alpha},y,z),\qquad T_{1{\alpha}}(p)=
p(t,x+{\alpha},z)+\rho{{\alpha}'}'x\eqno(1.9)$$ and its permutations on $(u,x),\;(v,y),\;(w,z)$, and $$T_{2{\alpha}}(u)=u,\qquad T_{2{\alpha}}(v)=v,\qquad T_{2{\alpha}}(w)=w,\qquad
T_{2{\alpha}}(p)= p+{\alpha},\eqno(1.10)$$ where ${\alpha}$ is an arbitrary function of $t$. The above transformations transform solutions of Navier-Stokes equations into their solutions. Our goal in this paper is to find exact solutions of Navier-Stokes equations modulo the above symmetries. In other words, the above symmetries will be used to simplify our ansatzes for exact solutions and related arguments.
For convenience, we always assume that all the involved partial derivatives of related functions always exist and we can change orders of taking partial derivatives. In fluid dynamics, rotation-free solutions of Navier-Stokes equations, namely, $$u_y-v_x=0,\qquad v_z-w_y=0,\qquad w_x-u_z=0,\eqno(1.11)$$ are not so interesting. From pure mathematical point of view, a rotation-free solution is equivalent to a time-dependent harmonic function $f(t,x,y,x)$ (i.e., $f_{xx}+f_{yy}+f_{zz}=0$), where $$u=f_x,\qquad v=f_y,\qquad w=f_z.\eqno(1.12)$$ Practically, steady solutions (or time-independent) are not very important. In general, it is difficult to find exact non-steady rotating solutions for Navier-Stokes equations (1.1)-(1.4) due to their nonlinearity.
Using certain finite-dimensional stable range of the nonlinear term, we found in \[X1\] a family of exact solutions with seven parameter functions for the equation of nonstationary transonic gas flows found by Lin, Reisner and Tsien \[LRT\], which blow up on a moving line. These solutions may reflect partial phenomena of gust. In \[X2\], we use various ansatzes with undermined functions and the technique of moving frame to find basic solutions modulo the Lie point symmetries with parameter functions for the classical non-steady boundary layer problems. These two works motivated us to solve Navier-Stokes equations by algebraic methods.
Our first idea is to impose suitable asymmetric conditions on the velocity vector with respect to independent variables. For instance, assuming $$u={\gamma}(t)x+y\phi(t,x^2+y^2),\;\;v={\gamma}(t)y-x\phi(t,x^2+y^2),\;\;w=
\psi(t,x^2+y^2)-2{\gamma}(t)z, \eqno(1.13)$$ we obtain the following solution of Navier-Stokes equations (see Theorem 2.4): $$u=\frac{{\alpha}'}{2{\alpha}} x+\frac{{\beta}y}{x^2+y^2}+y\sum_{i=0}^\infty\frac{({\alpha}{\partial}_t)^i(\Im)}{i!(i+1)!{\alpha}}
\left(\frac{x^2+y^2}{4\nu{\alpha}}\right)^i,\eqno(1.14)$$ $$v=\frac{{\alpha}'}{2{\alpha}}y-\frac{{\beta}x}{x^2+y^2}-x\sum_{i=0}^\infty\frac{({\alpha}{\partial}_t)^i(\Im)}{i!(i+1)!{\alpha}}
\left(\frac{x^2+y^2}{4\nu{\alpha}}\right)^i,\eqno(1.15)$$ $$w={\alpha}\sum_{s=0}^\infty\frac{({\alpha}{\partial}_t)^s({\varphi})}{(s!)^2}
\left(\frac{x^2+y^2}{4\nu{\alpha}}\right)^s-\frac{{\alpha}'}{{\alpha}}z,\eqno(1.16)$$ $$\begin{aligned}
\hspace{2cm}p&=&\frac{\rho(({\alpha}')^2-2{\alpha}{{\alpha}'}')(x^2+y^2)}{8{\alpha}^2}
+\frac{\rho({\alpha}{{\alpha}'}'-2({\alpha}')^2)z^2}{{\alpha}^2}+{\beta}'\arctan\frac{y}{x}
\\ & &+2\nu\rho{\alpha}\sum_{i,s=0}^\infty\frac{[({\alpha}{\partial}_t)^i(\Im)]
[({\alpha}{\partial}_t)^s({\varphi})]}{i!(i+1)!s!(s+1)!({\alpha})^2}
\left(\frac{x^2+y^2}{4\nu{\alpha}}\right)^{i+s+1},\hspace{2.4cm}(1.17)\end{aligned}$$ where ${\alpha},{\beta}$ are any functions in $t$ and $\Im,{\varphi}$ are arbitrary polynomials in $t$. The above solution can be used to describe incompressible fluid in a nozzle. The polynomials $\Im$ and ${\varphi}$ can be replaced by the other functions as long as the related power series converge.
As we emphasized earlier, people are interested in solutions that are not rotation free. To better capture the rotating nature of fluid, we introduce the following moving frames: $${{\cal X}}=x\cos{\alpha}+(y\cos{\beta}+z\sin{\beta})\sin{\alpha},\;\;
{{\cal Y}}=-x\sin{\alpha}+(y\cos{\beta}+z\sin{\beta})\cos{\alpha},\eqno(1.18)$$ $${{\cal Z}}=-y\sin{\beta}+z\cos{\beta},\qquad{{\cal U}}=u\cos{\alpha}+(v\cos{\beta}+w\sin{\beta})
\sin{\alpha},\eqno(1.19)$$ $${{\cal V}}=-u\sin{\alpha}+(v\cos{\beta}+w\sin{\beta})\cos{\alpha},\;\;
{{\cal W}}=-v\sin{\beta}+w\cos{\beta},\eqno(1.20)$$ where ${\alpha}$ and ${\beta}$ are functions in $t$. Here we exclude the translation components because Navier-Stokes equations are invariant under the transformations of the type $T_{1{\alpha}}$ in (1.8) and (1.9), and we want to consider solutions modulo these transformations. With respect to the above rotating frames, Navier-Stokes equations change to more complicated system of partial differential equations. Imposing asymmetric conditions on the moving frames, we find another five families of non-steady rotating solutions with various parameters. For instance, we have the following solution of Navier-Stokes equations (see Theorem 3.3): $$u=\left(\frac{{{\alpha}'}'}{2{\alpha}'}+6\nu{{\cal Y}}{{\cal X}}^{-2}\right)({{\cal Y}}\sin{\alpha}-{{\cal X}}\cos{\alpha})
-{\alpha}'({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha}),\eqno(1.21)$$ $$\begin{aligned}
\hspace{0.6cm}v&=&-\left(\frac{{{\alpha}'}'}
{2{\alpha}'}+6\nu{{\cal Y}}{{\cal X}}^{-2}\right)({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha})\cos{\beta}+{\alpha}'({{\cal X}}\cos{\alpha}-{{\cal Y}}\sin{\alpha})\cos{\beta}\\ &&-{\beta}'{{\cal Z}}\cos{\beta}+\left({\beta}'{{\cal X}}\sin{\alpha}+{\beta}'{{\cal Y}}\cos{\alpha}-\frac{{{\alpha}'}'}{{\alpha}'}{{\cal Z}}\right)\sin{\beta},
\hspace{3.7cm}(1.22)\end{aligned}$$ $$\begin{aligned}
\hspace{0.6cm}w&=&-\left(\frac{{{\alpha}'}'}
{2{\alpha}'}+6\nu{{\cal Y}}{{\cal X}}^{-2}\right)({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha})\sin{\beta}+{\alpha}'({{\cal X}}\cos{\alpha}-{{\cal Y}}\sin{\alpha})\sin{\beta}\\ &&-{\beta}'{{\cal Z}}\sin{\beta}+\left(\frac{{{\alpha}'}'}{{\alpha}'}{{\cal Z}}-{\beta}'{{\cal X}}\sin{\alpha}-{\beta}'{{\cal Y}}\cos{\alpha}\right)\cos{\beta},
\hspace{3.6cm}(1.23)\end{aligned}$$ $$\begin{aligned}
p&=&\rho\{
\frac{(2{\alpha}'{{{\alpha}'}'}'+4({\alpha}')^4-3({{\alpha}'}')^2)({{\cal X}}^2+{{\cal Y}}^2)}{8({\alpha}')^2}-
\frac{3({\beta}')^2({{\cal X}}^2\sin^2{\alpha}+{{\cal Y}}^2\cos^2{\alpha})}{2}
\\ & &+12\nu({\alpha}'{{\cal Y}}{{\cal X}}^{-1}-\nu{{\cal X}}^{-2})
+({{\beta}'}'-4{\beta}'{\gamma}){{\cal Z}}({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha})\\ &
&-3({\beta}')^2{{\cal X}}{{\cal Y}}\sin{\alpha}\;\cos {\alpha}+
\frac{({\alpha}'{{{\alpha}'}'}'+({\alpha}')^2({\beta}')^2-2({{\alpha}'}')^2){{\cal Z}}^2}{2({\alpha}')^2}
\}.\hspace{3cm}(1.24)\end{aligned}$$ The above solution blows up at any point on the following rotating plane with a line deleted: $$\begin{aligned}
\hspace{2cm}& &\{(x,y,z)\in{\mathbb}{R}^3\mid x\cos{\alpha}+y\sin{\alpha}\;\cos{\beta}+z\sin{\alpha}\;\sin{\beta}=0,\\ & &-x\sin{\alpha}+y\cos{\alpha}\;\cos{\beta}+z\cos{\alpha}\;\sin{\beta}\neq
0\}.\hspace{4.3cm}(1.25)\end{aligned}$$ This type of solutions may be applied in studying turbulence. Since all of our solutions in this paper only involve elementary functions and integrations, they may be applied to engineering problems with the help of computer, although they appear sophisticated in format. They can also be used to solve certain initial value problems for Navier-Stokes equations because they contain parameter functions.
As we all know that in general, it is impossible to find all the solutions of nonlinear partial differential equations both analytically and algebraically. In our arguments throughout this paper, we always search for reasonable sufficient conditions of obtaining exact solutions. For instance, we treat nonzero functions like “nonzero constants” for this purpose because our approaches in this paper are completely algebraic. Of course, one can use our methods in this paper to get more solutions, in particular, by considering the support and discontinuity of the related functions. We want to remind the reader that we always put arguments (proofs) before our theorems (conclusions) due to our purpose of finding exact solutions.
The paper is organized as follows. Section 2 is devoted to our asymmetric approaches. We present the general settings for the moving-frame approach in Section 3 and find two families of exact solutions. In Section 4, we use the moving frames and certain ansatzes involving given irrational functions to find another three families of exact solutions.
Asymmetric Approaches
=====================
In this section, we will solve incompressible Navier-Stokes equations (1.1)-(1.4) by imposing asymmetric assumptions on $u,\;v$ and $w$.
For convenience of computation, we denote $$\Phi_1=u_t+uu_x+vu_y+wu_z-\nu
(u_{xx}+u_{yy}+u_{zz}),\eqno(2.1)$$ $$\Phi_2=v_t+uv_x+vv_y+wv_z-\nu
(v_{xx}+v_{yy}+v_{zz}),\eqno(2.2)$$ $$\Phi_3=w_t+uw_x+vw_y+ww_z-\nu (w_{xx}+w_{yy}+w_{zz})
.\eqno(2.3)$$ Then Navier-Stokes equations become $$\Phi_1+\frac{1}{\rho}p_x=0,\qquad
\Phi_2+\frac{1}{\rho}p_y=0,\qquad
\Phi_3+\frac{1}{\rho}p_z=0
\eqno(2.4)$$ and $u_x+v_y+w_z=0.$ Our strategy is first to solve the following compatibility conditions: $${\partial}_y(\Phi_1)={\partial}_x(\Phi_2),\qquad
{\partial}_z(\Phi_1)={\partial}_x(\Phi_3),\qquad{\partial}_z(\Phi_2)={\partial}_y(\Phi_3)
\eqno(2.5)$$ and then find $p$ via (2.4).
Let us first look for simplest non-steady solutions of Navier-Stokes equations (indeed, the corresponding Euler equations) that are not rotation free. This will help the reader to better understand our later approaches. Assume $$u={\gamma}_1x-{\alpha}_1y-{\alpha}_2z,\;\;v={\alpha}_1x+{\gamma}_2y-{\alpha}_3z,\;\;w={\alpha}_2x+{\alpha}_3y
+{\gamma}_3z,\eqno(2.6)$$ where ${\alpha}_i$ and ${\gamma}_i$ are functions in $t$ such that ${\gamma}_1+{\gamma}_2+{\gamma}_3=0$. Then $$\Phi_1=({\gamma}_1'+{\gamma}_1^2-{\alpha}_1^2-{\alpha}_2^2)x-({\alpha}_1'-{\alpha}_1{\gamma}_3
+{\alpha}_2{\alpha}_3)y+({\alpha}_1{\alpha}_3-{\alpha}_2'+{\alpha}_2{\gamma}_2)z,\eqno(2.7)$$ $$\Phi_2=({\alpha}_1'-{\alpha}_1{\gamma}_3-{\alpha}_2{\alpha}_3)x+({\gamma}_2'+{\gamma}_2^2-{\alpha}_1^2
-{\alpha}_3^2)y-({\alpha}_3'+{\alpha}_1{\alpha}_2-{\alpha}_3{\gamma}_1)z,\eqno(2.8)$$ $$\Phi_3=({\alpha}_2'+{\alpha}_1{\alpha}_3-{\alpha}_2{\gamma}_2)x
+({\alpha}_3'-{\alpha}_1{\alpha}_2-{\alpha}_3{\gamma}_1)y+({\gamma}_3'+{\gamma}_3^2-{\alpha}_2^2
-{\alpha}_3^2)z.\eqno(2.9)$$ Furthermore, $${\partial}_y(\Phi_1)={\partial}_x(\Phi_2){\Longrightarrow}{\gamma}_3=\frac{{\alpha}_1'}{{\alpha}_1},\eqno(2.10)$$ $${\partial}_z(\Phi_1)={\partial}_x(\Phi_3){\Longrightarrow}{\gamma}_2=\frac{{\alpha}_2'}{{\alpha}_2},\eqno(2.11)$$ $${\partial}_z(\Phi_2)={\partial}_y(\Phi_3){\Longrightarrow}{\gamma}_1=\frac{{\alpha}_3'}{{\alpha}_3}.\eqno(2.12)$$
Note $${\gamma}_1+{\gamma}_2+{\gamma}_3=0\sim \frac{{\alpha}_1'}{{\alpha}_1}+
\frac{{\alpha}_2'}{{\alpha}_2}+\frac{{\alpha}_3'}{{\alpha}_3}=0\sim
{\alpha}_1{\alpha}_2{\alpha}_3=c\eqno(2.13)$$ for some real constant. Moreover, $$\Phi_1=({{\alpha}_3'}'{\alpha}_3^{-1}
-{\alpha}_1^2-{\alpha}_2^2)x-{\alpha}_2{\alpha}_3y+{\alpha}_1{\alpha}_3z,\eqno(2.14)$$ $$\Phi_2=-{\alpha}_2{\alpha}_3x+({{\alpha}_2'}'{\alpha}_2^{-1}-{\alpha}_1^2
-{\alpha}_3^2)y-{\alpha}_1{\alpha}_2z,\eqno(2.15)$$ $$\Phi_3={\alpha}_1{\alpha}_3x
-{\alpha}_1{\alpha}_2y+({{\alpha}_1'}'{\alpha}^{-1}_1-{\alpha}_2^2 -{\alpha}_3^2)z.\eqno(2.16)$$ By (2.4), $$\begin{aligned}
p&=&\frac{\rho}{2}[
({\alpha}_1^2+{\alpha}_2^2-{{\alpha}_3'}'{\alpha}_3^{-1})x^2+
({\alpha}_1^2+{\alpha}_3^2-{{\alpha}_2'}'{\alpha}_2^{-1})y^2+
({\alpha}_2^2+{\alpha}_3^2-{{\alpha}_1'}'{\alpha}_1^{-1})z^2]\\ &
&+\rho({\alpha}_2{\alpha}_3xy-{\alpha}_1{\alpha}_3xz+{\alpha}_1{\alpha}_2yz)\hspace{7.9cm}(2.17)\end{aligned}$$ modulo the transformation in (1.10).
[**Proposition 2.1**]{}. [*Let ${\alpha}_1,\;{\alpha}_2$ and ${\alpha}_3$ be functions in $t$ such that ${\alpha}_1{\alpha}_2{\alpha}_3=c$ for some real constant $c$. Then we have the following solution of Navier-Stokes equations (1.1)-(1.4): $$u=\frac{{{\alpha}_3}'}{{\alpha}_3}x-{\alpha}_1y-{\alpha}_2z,\;\;v={\alpha}_1x
+\frac{{{\alpha}_2}'}{{\alpha}_2}y -{\alpha}_3z,\;\;w={\alpha}_2x+{\alpha}_3y
+\frac{{{\alpha}_1}'}{{\alpha}_1}z\eqno(2.18)$$ and $p$ is given in (2.17).*]{}
Next we assume $$v=-\frac{{{\beta}'}'}{2{\beta}'} y,\qquad w=\psi(t,z),\eqno(2.19)$$ where ${\beta}$ is a function in $t$, $\psi$ is a function of $t,z$ and $v$ is so written just for computational convenience by our earlier experience in \[X1, X2\]. According to (1.4), $$u=f(t,y,z)+\left(\frac{{{\beta}'}'}{2{\beta}'}-\psi_z\right)x\eqno(2.20)$$ for some function $f$ of $t,y,z$. Then $$\begin{aligned}
\hspace{1cm}\Phi_1&=&
f_t+f\left(\frac{{{\beta}'}'}{2{\beta}'}-\psi_z\right)-\frac{{{\beta}'}'}{2{\beta}'}
y f_y+\psi f_z-\nu(f_{yy}+f_{zz})
\\ & &+\left[\left(\frac{{{\beta}'}'}{2{\beta}'}-\psi_z\right)^2
+\frac{{\beta}'{{{\beta}'}'}'-({{\beta}'}')^2}{2({\beta}')^2}
-\psi_{zt}-\psi\psi_{zz}+\nu\psi_{zzz}\right]x,
\hspace{1.8cm}(2.21)\end{aligned}$$ $$\Phi_2=\frac{(3({{\beta}'}')^2-2{\beta}'{{{\beta}'}'}')y}{4({\beta}')^2},\qquad
\Phi_3=\psi_t+\psi\psi_z-\nu\psi_{zz}.\eqno(2.22)$$ Thus (2.5) is equivalent to the following equations: $${\cal T}\left[f_t+f\left(\frac{{{\beta}'}'}{2{\beta}'}-\psi_z\right)-\frac{{{\beta}'}'}{2{\beta}'}
y f_y+\psi f_z-\nu(f_{yy}+f_{zz})\right]=0,\eqno(2.23)$$ $${\cal T}\left[\psi_z^2-\frac{{{\beta}'}'}{{\beta}'}\psi_z
-\psi_{zt}-\psi\psi_{zz}+\nu\psi_{zzz}\right]=0\eqno(2.24)$$ with ${\cal T}={\partial}_y,\;{\partial}_z$.
Given $b,c\in{\mathbb}{R}$ and a function ${\gamma}$ of $t$, we set $$\xi_0=be^{\sqrt{{\gamma}'}z+\nu{\gamma}}-ce^{-\sqrt{{\gamma}'}z-\nu{\gamma}},\qquad\xi_1=
b\sin(\sqrt{{\gamma}'}z-\nu{\gamma}),\eqno(2.25)$$ $$\zeta_0=be^{\sqrt{{\gamma}'}z+\nu{\gamma}}+ce^{-\sqrt{{\gamma}'}z-\nu{\gamma}},\qquad\zeta_1=
b\cos(\sqrt{{\gamma}'}z-\nu{\gamma}).\eqno(2.26)$$ Then we have the following solution of (2.24): $$\psi=\frac{\xi_r}{{\beta}'\sqrt{({\gamma}')^3}}-\frac{{{\gamma}'}'}{2{\gamma}'}z\eqno(2.27)$$ with $r=0,1$. Write $$f=\frac{\hat f}{\sqrt{{\beta}'{\gamma}'}}.\eqno(2.28)$$ The equation (2.23) is implied by the following equation: $$\hat f_t-\frac{\hat f\zeta_r}{{\beta}'{\gamma}'}-\frac{{{\beta}'}'}{2{\beta}'}
y \hat f_y-\frac{{{\gamma}'}'}{2{\gamma}'}z\hat f_z+ \frac{\hat
f_z\xi_r}{{\beta}'\sqrt{({\gamma}')^3}}-\nu(\hat f_{yy}+\hat
f_{zz})=0.\eqno(2.29)$$
To solve the above equation, we assume $$\hat f=h(t,y)+g(t,y)\zeta_r,\eqno(2.30)$$ where $h$ and $g$ are functions in $t,y$. Then (2.29) is implied by the following two equations: $$g_t-\frac{{{\beta}'}'}{2{\beta}'}yg_y-\nu g_{yy}-\frac{h}{{\beta}'{\gamma}'}=0,\eqno(2.31)$$ $$h_t-\frac{{{\beta}'}'}{2{\beta}'}yh_y-\nu h_{yy}
-\frac{(4{\delta}_{0,r}bc+{\delta}_{1,r}b^2)g}{{\beta}'{\gamma}'}=0.\eqno(2.32)$$ We will solve the above system of partial differential equations according to the following two cases:
[*Case 1*]{}. $r=0,\;bc=0$ or $r=1,\;b=0$
In this case, we have the following solutions: $$g=\sum_{i=1}^md_{1,i}e^{\nu(a_{1,i}^2-b_{1,i}^2){\beta}+a_{1,i}
\sqrt{{\beta}'}y}\sin(b_{1,i}(2\nu
a_{1,i}{\beta}+\sqrt{{\beta}'}y)+c_{1,i})+h\int\frac{dt}{{\beta}'{\gamma}'},
\eqno(2.33)$$ $$h=
\sum_{s=1}^nd_{2,s}e^{\nu(a_{2,s}^2-b_{2,s}^2){\beta}+a_{2,s}
\sqrt{{\beta}'}y}\sin(b_{2,s}(2\nu
a_{2,s}{\beta}+\sqrt{{\beta}'}y)+c_{2,s}),\eqno(2.34)$$ where $a_{1,i},b_{1,i},c_{1,i},d_{1,i}$ and $a_{2,s},b_{2,s},c_{2,s},d_{2,s}$ are real constants.
[*Case 2*]{}. $r=0,\;bc\neq 0$ or $r=1,\;b\neq 0$ if $r=1$.
For convenience, we denote $$a=\sqrt{4{\delta}_{0,r}bc+{\delta}_{1,r}b^2}.\eqno(2.35)$$ Then we have the following solution of the system (2.31) and (2.32): $$\begin{aligned}
& &g=\cosh\left(a\int\frac{dt}{{\beta}'{\gamma}'}\right)
\sum_{i=1}^md_{1,i}e^{\nu(a_{1,i}^2-b_{1,i}^2){\beta}+a_{1,i}
\sqrt{{\beta}'}y}\sin(b_{1,i}(2\nu a_{1,i}{\beta}+\sqrt{{\beta}'}y)+c_{1,i})\\
& &+\frac{
\sinh\left(a\int\frac{dt}{{\beta}'{\gamma}'}\right)}{a}\sum_{s=1}^nd_{2,s}e^{\nu(a_{2,s}^2-b_{2,s}^2){\beta}+a_{2,s} \sqrt{{\beta}'}y}\sin(b_{2,s}(2\nu
a_{2,s}{\beta}+\sqrt{{\beta}'}y)+c_{2,s}),
\hspace{0.5cm}(2.36)\end{aligned}$$ $$\begin{aligned}
& &h=a\sinh\left(a\int\frac{dt}{{\beta}'{\gamma}'}\right)
\sum_{i=1}^md_{1,i}e^{\nu(a_{1,i}^2-b_{1,i}^2){\beta}+a_{1,i}
\sqrt{{\beta}'}y}\sin(b_{1,i}(2\nu a_{1,i}{\beta}+\sqrt{{\beta}'}y)+c_{1,i})\\
& &+ \cosh\left(\int\frac{adt}{{\beta}'{\gamma}'}\right)
\sum_{s=1}^nd_{2,s}e^{\nu(a_{2,s}^2-b_{2,s}^2){\beta}+a_{2,s}
\sqrt{{\beta}'}y}\sin(b_{2,s}(2\nu a_{2,s}{\beta}+\sqrt{{\beta}'}y)+c_{2,s}).
\hspace{0.4cm}(2.37)\end{aligned}$$
In any case, $$\Phi_1=\left(\frac{2{\beta}'{{{\beta}'}'}'-({{\beta}'}')^2}{4({\beta}')^2}
+\frac{2{\gamma}'{{{\gamma}'}'}'-({{\gamma}'}')^2}{4({\gamma}')^2}+\frac{{{\beta}'}'{{\gamma}'}'}
{2{\beta}'{\gamma}'}+\frac{{\delta}_{0,r}4bc+{\delta}_{1,r}b^2}{({\beta}'{\gamma}')^2}\right)x
,\eqno(2.38)$$ $$\Phi_3=\left[\frac{(3({{\gamma}'}')^2-2{\gamma}'{{{\gamma}'}'}')z^2}
{8({\gamma}')^2}+\frac{\xi_r^2-(-1)^r(2{{\beta}'}'{\gamma}'+5{\beta}'{{\gamma}'}')
\zeta_r}{2({\beta}')^3({\gamma}')^3}\right]_z. \eqno(2.39)$$ Thus (2.4), (2.38), (3.39) and the first equation in (2.22) give $$\begin{aligned}
p&=&
\rho\left[\frac{(2{\beta}'{{{\beta}'}'}'-3({{\beta}'}')^2)y^2}{8({\beta}')^2}
+\frac{2{\gamma}'{{{\gamma}'}'}'-3({{\gamma}'}')^2}
{8({\gamma}')^2}z^2+\frac{(-1)^r(2{{\beta}'}'{\gamma}'+5{\beta}'{{\gamma}'}')
\zeta_r-\xi_r^2}{2({\beta}')^3({\gamma}')^3}\right]\\ & &+\frac{\rho
x^2}{2} \left(\frac{({{\beta}'}')^2-2{\beta}'{{{\beta}'}'}'}{4({\beta}')^2}
+\frac{({{\gamma}'}')^2-2{\gamma}'{{{\gamma}'}'}'}{4({\gamma}')^2}-\frac{{{\beta}'}'{{\gamma}'}'}
{2{\beta}'{\gamma}'}-\frac{{\delta}_{0,r}4bc+{\delta}_{1,r}b^2}{({\beta}'{\gamma}')^2}\right)
\hspace{1.3cm}(2.40)\end{aligned}$$ modulo the transformation in (1.10).
By (2.20), (2.27), (2.28), (2.33), (2.34), (2.36) and (2.37), we obtain:
[**Theorem 2.2**]{}. [*Let ${\beta},{\gamma}$ be functions in $t$, and let $b,c,a_{1,i},b_{1,i},c_{1,i},d_{1,i}$ and $a_{2,s},b_{2,s},c_{2,s},\\ d_{2,s}$ be real constants. Define $\xi_r$ and $\zeta_r$ in (2.25) and (2.26). For $r=0,1$, we have the following solution of Navier-Stokes equations (1.1)-(1.4): $$v=-\frac{{{\beta}'}'}{2{\beta}'} y,\qquad w=
\frac{\xi_r}{{\beta}'\sqrt{({\gamma}')^3}}-\frac{{{\gamma}'}'}{2{\gamma}'}z,\eqno(2.41)$$ $p$ is given (2.40), and $$\begin{aligned}
\hspace{1cm}u&=&\frac{\zeta_r}{\sqrt{{\beta}'{\gamma}'}}
\sum_{i=1}^md_{1,i}e^{\nu(a_{1,i}^2-b_{1,i}^2){\beta}+a_{1,i}
\sqrt{{\beta}'}y}\sin(b_{1,i}(2\nu a_{1,i}{\beta}+\sqrt{{\beta}'}y)+c_{1,i})\\
& &+\left(\frac{{{\beta}'}'}{2{\beta}'}
+\frac{{{\gamma}'}'}{2{\gamma}'}-\frac{\zeta_r}{{\beta}'{\gamma}'}\right)x
+\frac{1+\zeta_r\int\frac{dt}{{\beta}'{\gamma}'}}{\sqrt{{\beta}'{\gamma}'}}
\sum_{s=1}^nd_{2,s}e^{\nu(a_{2,s}^2-b_{2,s}^2){\beta}+a_{2,s}
\sqrt{{\beta}'}y}\\ & &\times\sin(b_{2,s}(2\nu
a_{2,s}{\beta}+\sqrt{{\beta}'}y)+c_{2,s})\hspace{6.6cm}
(2.42)\end{aligned}$$ if $r=0,\;bc=0$, and $$\begin{aligned}
u&=&\left(\frac{{{\beta}'}'}{2{\beta}'}+\frac{{{\gamma}'}'}{2{\gamma}'}-
\frac{\zeta_r}{{\beta}'{\gamma}'}\right)x+
\frac{1}{\sqrt{{\beta}'{\gamma}'}}\left(\zeta_r\cosh
\left(\int\frac{adt}{{\beta}'{\gamma}'}\right)
+a\sinh\left(\int\frac{adt}{{\beta}'{\gamma}'}\right)\right)
\\ & &\times\sum_{i=1}^md_{1,i}e^{\nu(a_{1,i}^2-b_{1,i}^2){\beta}+a_{1,i}
\sqrt{{\beta}'}y}\sin(b_{1,i}(2\nu a_{1,i}{\beta}+\sqrt{{\beta}'}y)+c_{1,i})\\
& & + \frac{1}{\sqrt{{\beta}'{\gamma}'}}
\left(\cosh\left(\int\frac{adt}{{\beta}'{\gamma}'}\right)
+\frac{\zeta_r}{a}\sinh\left(\int\frac{adt}{{\beta}'{\gamma}'}\right)\right)
\sum_{s=1}^nd_{2,s}e^{\nu(a_{2,s}^2-b_{2,s}^2){\beta}+a_{2,s}
\sqrt{{\beta}'}y}\\ & &\times\sin(b_{2,s}(2\nu
a_{2,s}{\beta}+\sqrt{{\beta}'}y)+c_{2,s})\hspace{7.6cm}
(2.43)\end{aligned}$$ if $r=0,\;bc\neq 0$ or $r=1,\;b\neq 0$, where $a$ is defined in (2.35).*]{}
[**Remark 2.3**]{}. We can use Fourier expansion to solve the system (2.31) and (2.32) for $g({\beta},\sqrt{{\beta}'}y)$ and $h({\beta},\sqrt{{\beta}'}y)$ with given $g(0,\sqrt{{\beta}'}y)$ and $h(0,\sqrt{{\beta}'}y)$. In this way, we can obtain discontinuous solutions of Navier-Stokes equations (1.1)-(1.4), which may be useful in studying shock waves.
Set $$\varpi=x^2+y^2.\eqno(2.44)$$ Consider $$u=\frac{{\alpha}'}{2{\alpha}} x+y\phi(t,\varpi),\qquad v=\frac{{\alpha}'}{2{\alpha}}y
-x\phi(t,\varpi),\qquad
w=\psi(t,\varpi)-\frac{{\alpha}'}{{\alpha}}z,\eqno(2.45)$$ where ${\alpha}$ is a function in $t$ and $\phi,\psi$ are functions in $t,\varpi$. Then (2.1)-(2.3) give $$\Phi_1=\frac{2{\alpha}{{\alpha}'}'-({\alpha}')^2}{4{\alpha}^2} x+y\phi_t
+\frac{{\alpha}'y}{{\alpha}}(\varpi\phi)_\varpi-x\phi^2-
4y\nu(\varpi\phi)_{\varpi\varpi},\eqno(2.46)$$ $$\Phi_2=\frac{2{\alpha}{{\alpha}'}'-({\alpha}')^2}{4{\alpha}^2}y-x\phi_t
-\frac{{\alpha}'x}{{\alpha}}(\varpi\phi)_\varpi-y\phi^2+
4x\nu(\varpi\phi)_{\varpi\varpi},\eqno(2.47)$$ $$\Phi_3=\frac{2({\alpha}')^2-{\alpha}{{\alpha}'}'}{{\alpha}^2}z+\psi_t
-\frac{{\alpha}'}{{\alpha}}\psi+\frac{{\alpha}'}{{\alpha}}\varpi
\psi_\varpi-4\nu(\psi_\varpi+\varpi\psi_{\varpi\varpi}).\eqno(2.48)$$
Note that ${\partial}_y(\Phi_1)={\partial}_x(\Phi_2)$ becomes $$(\varpi\phi)_{\varpi t}+\frac{{\alpha}'}{{\alpha}}
((\varpi\phi)_\varpi+\varpi(\varpi\phi)_{\varpi\varpi})
-4\nu((\varpi\phi)_{\varpi\varpi}+
\varpi(\varpi\phi)_{\varpi\varpi\varpi})=0.\eqno(2.49)$$ Set $$\hat\phi={\alpha}(\varpi\phi)_\varpi.\eqno(2.50)$$ Then (2.49) becomes $$\hat\phi_t+\frac{{\alpha}'}{{\alpha}}\varpi\hat\phi_\varpi-
4\nu(\hat\phi_\varpi+\varpi\hat\phi_{\varpi\varpi})=0.\eqno(2.51)$$ So we have the solution $$\hat\phi=\sum_{i=0}^\infty\frac{({\alpha}{\partial}_t)^i(\Im)}{(i!)^2}
\left(\frac{\varpi}{4\nu{\alpha}}\right)^i\eqno(2.52)$$ for a polynomial $\Im$ in $t$. By (2.50)-(2.52), we have $$\phi={\beta}\varpi^{-1}+\sum_{i=0}^\infty\frac{({\alpha}{\partial}_t)^i(\Im)}{i!(i+1)!{\alpha}}
\left(\frac{\varpi}{4\nu{\alpha}}\right)^i\eqno(2.53)$$ for a function ${\beta}$ of $t$.
Note $$\phi_t={\beta}'\varpi^{-1}+\sum_{i=0}^\infty\left[\frac{({\alpha}{\partial}_t)^{i+1}(\Im)}{i!(i+1)!{\alpha}^2}
-\frac{{\alpha}'({\alpha}{\partial}_t)^i(\Im)}{(i!)^2!{\alpha}^2}\right]
\left(\frac{\varpi}{4\nu{\alpha}}\right)^i,\eqno(2.54)$$ $$\frac{{\alpha}'}{{\alpha}}
((\varpi\phi)_\varpi=\frac{{\alpha}'}{{\alpha}^2}\hat\phi=
\frac{{\alpha}'}{{\alpha}^2}\sum_{i=0}^\infty\frac{({\alpha}{\partial}_t)^i(\Im)}{(i!)^2}
\left(\frac{\varpi}{4\nu{\alpha}}\right)^i,\eqno(2.55)$$ $$4\nu(\varpi\phi)_{\varpi\varpi}=\frac{4\nu\hat\phi_\varpi}{{\alpha}}=
\sum_{i=1}^\infty\frac{({\alpha}{\partial}_t)^i(\Im)}{(i-1)!i!{\alpha}^2}
\left(\frac{\varpi}{4\nu{\alpha}}\right)^{i-1}=\sum_{i=0}^\infty\frac{({\alpha}{\partial}_t)^{i+1}
(\Im)}{i!(i+1)!{\alpha}^2}
\left(\frac{\varpi}{4\nu{\alpha}}\right)^i.\eqno(2.56)$$ Thus $$\phi_t+\frac{{\alpha}'}{{\alpha}}(\varpi\phi)_\varpi-
4\nu(\varpi\phi)_{\varpi\varpi}={\beta}'\varpi^{-1}.\eqno(2.57)$$ Therefore, $$\Phi_1=
\frac{2{\alpha}{{\alpha}'}'-({\alpha}')^2}{4{\alpha}^2}
x+\frac{{\beta}'y}{x^2+y^2}-x\phi^2\eqno(2.58)$$ and $$\Phi_2=
\frac{2{\alpha}{{\alpha}'}'-({\alpha}')^2}{4{\alpha}^2}
y-\frac{{\beta}'x}{x^2+y^2}-y\phi^2.\eqno(2.59)$$
On the other hand, Equations ${\partial}_z(R_1)={\partial}_x(R_3)$ and ${\partial}_z(R_2)={\partial}_y(R_3)$ are implied by the following differential equation: $$\psi_t
-\frac{{\alpha}'}{{\alpha}}\psi+\frac{{\alpha}'}{{\alpha}}\varpi
\psi_\varpi-4\nu(\psi_\varpi+\varpi\psi_{\varpi\varpi})=0\eqno(2.60)$$ (cf. (2.48)). Similarly, we have the solution: $$\psi={\alpha}\sum_{s=0}^\infty\frac{({\alpha}{\partial}_t)^s({\varphi})}{(s!)^2}
\left(\frac{\varpi}{4\nu{\alpha}}\right)^s,\eqno(2.61)$$ where ${\varphi}$ is a polynomial in $t$. With this $\psi$, $$\Phi_3=\frac{2({\alpha}')^2-{\alpha}{{\alpha}'}'}{{\alpha}^2}z.\eqno(2.62)$$ By (2.4), (2.44), (2.45), (2.53), (2.58), (2.59), (2.61) and (2.62), we obtain:
[**Theorem 2.4**]{}. [*Let ${\alpha},{\beta}$ be any functions in $t$ and let $\Im,{\varphi}$ be polynomials in $t$. We have the following solution of Navier-Stokes equations (1.1)-(1.4): $$u=\frac{{\alpha}'}{2{\alpha}} x+\frac{{\beta}y}{x^2+y^2}+y\sum_{i=0}^\infty\frac{({\alpha}{\partial}_t)^i(\Im)}{i!(i+1)!{\alpha}}
\left(\frac{x^2+y^2}{4\nu{\alpha}}\right)^i,\eqno(2.63)$$ $$v=\frac{{\alpha}'}{2{\alpha}}y-\frac{{\beta}x}{x^2+y^2}-x\sum_{i=0}^\infty\frac{({\alpha}{\partial}_t)^i(\Im)}{i!(i+1)!{\alpha}}
\left(\frac{x^2+y^2}{4\nu{\alpha}}\right)^i,\eqno(2.64)$$ $$w={\alpha}\sum_{s=0}^\infty\frac{({\alpha}{\partial}_t)^s({\varphi})}{(s!)^2}
\left(\frac{x^2+y^2}{4\nu{\alpha}}\right)^s-\frac{{\alpha}'}{{\alpha}}z,\eqno(2.65)$$ $$\begin{aligned}
\hspace{2cm}p&=&\frac{\rho(({\alpha}')^2-2{\alpha}{{\alpha}'}')(x^2+y^2)}{8{\alpha}^2}
+\frac{\rho({\alpha}{{\alpha}'}'-2({\alpha}')^2)z^2}{{\alpha}^2}+{\beta}'\arctan\frac{y}{x}
\\ & &+2\nu\rho{\alpha}\sum_{i,s=0}^\infty\frac{[({\alpha}{\partial}_t)^i(\Im)]
[({\alpha}{\partial}_t)^s({\varphi})]}{i!(i+1)!s!(s+1)!({\alpha})^2}
\left(\frac{x^2+y^2}{4\nu{\alpha}}\right)^{i+s+1}.\hspace{2.4cm}(2.66)\end{aligned}$$*]{}
[**Remark 2.5**]{}. The above solution can be used to describe incompressible fluid in a nozzle. The polynomials $\Im$ and ${\varphi}$ can be replaced by the other functions as long as the related power series converge.
Moving-Frame Approach I
=======================
In this section, we will present the general settings for the moving-frame approach and find two families of exact solutions.
Let ${\alpha},{\beta}$ be given functions of $t$. Denote $$T=\left(\begin{array}{ccc}\cos{\alpha}&\sin{\alpha}\;\cos{\beta}&\sin{\alpha}\;\sin{\beta}\\
-\sin{\alpha}&\cos{\alpha}\;\cos{\beta}&\cos{\alpha}\;\sin{\beta}\\ 0&
-\sin{\beta}&\cos{\beta}\end{array}\right)\eqno(3.1)$$ and $$Q=\left(\begin{array}{ccc}0&{\alpha}'&{\beta}'\sin{\alpha}\\
-{\alpha}'&0&{\beta}'\cos{\alpha}\\-{\beta}'\sin{\alpha}&-{\beta}'\cos{\alpha}&0
\end{array}\right).\eqno(3.2)$$ Then $$T^{-1}=T^t=\left(\begin{array}{ccc}\cos{\alpha}&-\sin{\alpha}&0\\
\sin{\alpha}\;\cos{\beta}&\cos{\alpha}\;\cos{\beta}&-\sin{\beta}\\\sin{\alpha}\;\sin{\beta}&\cos{\alpha}\;\sin{\beta}&\cos{\beta}\end{array}\right)\eqno(3.3)$$ and $$\frac{d}{dt}(T)=QT.\eqno(3.4)$$ Define the moving frames: $$\left(\begin{array}{c}{\cal U}\\{\cal V}\\ {\cal
W}\end{array}\right)=T\left(\begin{array}{c}u\\ v\\
w\end{array}\right),\qquad \left(\begin{array}{c}{\cal X}\\{\cal
Y}\\ {\cal
Z}\end{array}\right)=T\left(\begin{array}{c}x\\ y\\
z\end{array}\right).\eqno(3.5)$$ Note $${\Delta}={\partial}_x^2+{\partial}_y^2+{\partial}_z^2={\partial}_{\cal X}^2+{\partial}_{\cal Y}^2+
{\partial}_{\cal Z}^2,\qquad
u{\partial}_x+v{\partial}_y+z{\partial}_z={{\cal U}}{\partial}_{{\cal X}}+{{\cal V}}{\partial}_{{\cal Y}}+{{\cal W}}{\partial}_{{\cal Z}},\eqno(3.6)$$ $$u_x+v_y+w_z={{\cal U}}_{{\cal X}}+{{\cal V}}_{{\cal Y}}+{{\cal W}}_{{\cal Z}},\qquad
\left(\begin{array}{c}{{\cal X}}_t\\ {{\cal Y}}_t\\ Z_t\end{array}\right)=
Q\left(\begin{array}{c}{{\cal X}}\\{{\cal Y}}\\ {{\cal Z}}\end{array}\right),\eqno(3.7)$$ $$\left(\begin{array}{c}{\partial}_{{\cal X}}\\
{\partial}_{{\cal Y}}\\ {\partial}_{{\cal Z}}\end{array}\right)=T\left(\begin{array}{c}{\partial}_x\\
{\partial}_y\\ {\partial}_z\end{array}\right),\qquad
\left(\begin{array}{c}{\partial}_t({{\cal U}})\\ {\partial}_t({{\cal V}})\\
{\partial}_t({{\cal W}})\end{array}\right)=Q\left(\begin{array}{c}{{\cal U}}\\ {{\cal V}}\\
{{\cal W}}\end{array}\right)+T\left(\begin{array}{c}u_t\\ v_t\\
w_t\end{array}\right).\eqno(3.8)$$
Write ${{\cal U}},{{\cal V}},{{\cal W}},p$ as functions in $t,{{\cal X}},{{\cal Y}},{{\cal Z}}$. Set $$\begin{aligned}
& &R_1={{\cal U}}_t+{\alpha}'({{\cal Y}}{{\cal U}}_{{\cal X}}-{{\cal X}}{{\cal U}}_{{\cal Y}}-{{\cal V}})+{\beta}'({{\cal Z}}{{\cal U}}_{{\cal X}}-{{\cal X}}{{\cal U}}_{{\cal Z}}-{{\cal W}})\sin{\alpha}\\
&
&+{\beta}'({{\cal Z}}{{\cal U}}_{{\cal Y}}-{{\cal Y}}{{\cal U}}_{{\cal Z}})\cos{\alpha}+{{\cal U}}{{\cal U}}_{{\cal X}}+{{\cal V}}{{\cal U}}_{{\cal Y}}+{{\cal W}}{{\cal U}}_{{\cal Z}}-\nu{\Delta}({{\cal U}}),
\hspace{4cm}(3.9)\end{aligned}$$ $$\begin{aligned}
& &R_2={{\cal V}}_t+{\alpha}'({{\cal Y}}{{\cal V}}_{{\cal X}}-{{\cal X}}{{\cal V}}_{{\cal Y}}+{{\cal U}})+{\beta}'({{\cal Z}}{{\cal V}}_{{\cal X}}-{{\cal X}}{{\cal V}}_{{\cal Z}})\sin{\alpha}\\
&&+{\beta}'({{\cal Z}}{{\cal V}}_{{\cal Y}}-{{\cal Y}}{{\cal V}}_{{\cal Z}}-{{\cal W}})\cos{\alpha}+{{\cal U}}{{\cal V}}_{{\cal X}}+{{\cal V}}{{\cal V}}_{{\cal Y}}+{{\cal W}}{{\cal V}}_{{\cal Z}}-\nu{\Delta}({{\cal V}}),
\hspace{3cm}(3.10)\end{aligned}$$ $$\begin{aligned}
& &R_3={{\cal W}}_t+{\alpha}'({{\cal Y}}{{\cal W}}_{{\cal X}}-{{\cal X}}{{\cal W}}_{{\cal Y}})+{\beta}'({{\cal Z}}{{\cal W}}_{{\cal X}}-{{\cal X}}{{\cal W}}_{{\cal Z}}+{{\cal U}})\sin{\alpha}\\&&+{\beta}'({{\cal Z}}{{\cal W}}_{{\cal Y}}-{{\cal Y}}{{\cal W}}_{{\cal Z}}+{{\cal V}})\cos{\alpha}+{{\cal U}}{{\cal W}}_{{\cal X}}+{{\cal V}}{{\cal W}}_{{\cal Y}}+{{\cal W}}{{\cal W}}_{{\cal Z}}-\nu{\Delta}({{\cal W}}),\hspace{2.3cm}(3.11)\end{aligned}$$ Then Navier-Stokes equations (1.1)-(1.4) become $$R_1+\frac{1}{\rho}p_{_{{\cal X}}}=0,\qquad R_2+
\frac{1}{\rho}p_{_{{\cal Y}}}=0,\qquad R_3+
\frac{1}{\rho}p_{_{{\cal Z}}}=0,\eqno(3.12)$$ $${{\cal U}}_{{\cal X}}+{{\cal V}}_{{\cal Y}}+{{\cal W}}_{{\cal Z}}=0.\eqno(3.13)$$ Instead of solving the equations in (3.12), we will first solve the following compatibility equations: $${\partial}_{{\cal Y}}(R_1)={\partial}_{{\cal X}}(R_2),\qquad{\partial}_{{\cal Z}}(R_1)={\partial}_{{\cal X}}(R_3),
\qquad {\partial}_{{\cal Z}}(R_2)={\partial}_{{\cal Y}}(R_3)\eqno(3.14)$$ for ${{\cal U}},{{\cal V}},{{\cal W}}$, and then find $p$ from the equations in (3.12).
Let $f$ be a function in $t,{{\cal Y}},{{\cal Z}}$ such that ${\partial}_{{\cal Y}}^2(f)={\partial}_{{\cal Z}}^2(f)=0$, and let $\phi,\psi$ be functions in $t,{{\cal X}}$. Suppose that ${\gamma}$ is a function of $t$. Assume $${{\cal U}}=f-2{\gamma}'{{\cal X}},\qquad{{\cal V}}=\phi+{\gamma}'{{\cal Y}},\qquad{{\cal W}}=\psi+{\gamma}'{{\cal Z}}.\eqno(3.15)$$ Then $$\begin{aligned}
\hspace{0.5cm}& &R_1=f_t-2{{\gamma}'}'{{\cal X}}-{\alpha}'(3{\gamma}'{{\cal Y}}+{{\cal X}}f_{{\cal Y}}+\phi
)-{\beta}'(3{\gamma}'{{\cal Z}}+{{\cal X}}f_{{\cal Z}}+\psi)\sin{\alpha}\\ & &+{\beta}'({{\cal Z}}f_{{\cal Y}}-{{\cal Y}}f_{{\cal Z}})\cos{\alpha}-2{\gamma}'(f-2{\gamma}'{{\cal X}})+f_{{\cal Y}}(\phi+{\gamma}'{{\cal Y}})+f_{{\cal Z}}(\psi+{\gamma}'{{\cal Z}})
,\hspace{1cm}(3.16)\end{aligned}$$ $$\begin{aligned}
\hspace{1.3cm}R_2&=&\phi_t+{{\gamma}'}'{{\cal Y}}+{\alpha}'({{\cal Y}}\phi_{{\cal X}}-3{\gamma}'{{\cal X}}+f)
+{\beta}'{{\cal Z}}\phi_{{\cal X}}\sin{\alpha}-{\beta}'\psi\cos{\alpha}\\
&&+(f-2{\gamma}'{{\cal X}})\phi_{{\cal X}}+{\gamma}'\phi+({\gamma}')^2{{\cal Y}}-\nu\phi_{{{\cal X}}{{\cal X}}},\hspace{4.8cm}(3.17)\end{aligned}$$ $$\begin{aligned}
\hspace{1cm}R_3&=&\psi_t+{{\gamma}'}'{{\cal Z}}+{\alpha}'{{\cal Y}}\psi_{{\cal X}}+{\beta}'({{\cal Z}}\psi_{{\cal X}}-3{\gamma}'{{\cal X}}+f)\sin{\alpha}-\nu\psi_{{{\cal X}}{{\cal X}}}\\&&
+{\beta}'\phi\cos{\alpha}+(f-2{\gamma}'{{\cal X}})\psi_{{\cal X}}+{\gamma}'(\psi+{\gamma}'{{\cal Z}}).\hspace{4.7cm}(3.18)\end{aligned}$$ Now (3.14) becomes $$\begin{aligned}
\hspace{1.5cm}& &
\phi_{t{{\cal X}}}+({\alpha}'{{\cal Y}}+{\beta}'{{\cal Z}}\sin{\alpha}+f)\phi_{{{\cal X}}{{\cal X}}}-{\beta}'\psi_{{\cal X}}\cos{\alpha}-2{\gamma}'({{\cal X}}\phi_{{\cal X}})_{{\cal X}}\\ & &+{\gamma}'\phi_{{\cal X}}-\nu\phi_{{{\cal X}}{{\cal X}}{{\cal X}}}=f_{t{{\cal Y}}}-{\beta}'
f_{{\cal Z}}\cos{\alpha}-{\gamma}'f_{{\cal Y}},\hspace{4.8cm}(3.19)\end{aligned}$$ $$\begin{aligned}
\hspace{1.5cm}& &\psi_{t{{\cal X}}}+({\alpha}'{{\cal Y}}+{\beta}'{{\cal Z}}\sin{\alpha}+f)\psi_{{{\cal X}}{{\cal X}}}-\nu\psi_{{{\cal X}}{{\cal X}}{{\cal X}}} +{\beta}'\phi_{{\cal X}}\cos{\alpha}\\&&-2({\gamma}'{{\cal X}}\psi_{{\cal X}})_{{\cal X}}+{\gamma}'\psi_{{\cal X}}=
f_{t{{\cal Z}}}+{\beta}'
f_{{\cal Y}}\cos{\alpha}-{\gamma}'f_{{\cal Z}},\hspace{4cm}(3.20)\end{aligned}$$ $${\alpha}'f_{{\cal Z}}+({\beta}'\sin{\alpha}+f_{{\cal Z}})\phi_{{\cal X}}=
({\alpha}'+f_{{\cal Y}})\psi_{{\cal X}}+{\beta}'f_{{\cal Y}}\sin{\alpha}.\eqno(3.21)$$
By (3.19) and (3.20), we take $$f=-{\alpha}'{{\cal Y}}-{\beta}'{{\cal Z}}\sin{\alpha}\eqno(3.22)$$ modulo the transformations of type $T_{1{\alpha}}$ in (1.8) and (1.9). Note that (3.21) is implied by (3.22). Integrating (3.19) and (3.20), we obtain $$\phi_t-2{\gamma}'{{\cal X}}\phi_{{\cal X}}+{\gamma}'\phi -\nu\phi_{{{\cal X}}{{\cal X}}}-{\beta}'\psi\cos{\alpha}=[({\beta}')^2
\sin{\alpha}\;\cos{\alpha}+{\alpha}'{\gamma}'-{{\alpha}'}']{{\cal X}}+{\beta}_1,\eqno(3.23)$$ $$\begin{aligned}
\hspace{2cm}& &\psi_t-2{\gamma}'{{\cal X}}\psi_{{\cal X}}+{\gamma}'\psi-\nu\psi_{{{\cal X}}{{\cal X}}} +{\beta}'\phi\cos{\alpha}\\ &=& -[({\beta}'\sin{\alpha})'
+{\alpha}'{\beta}'\cos{\alpha}-{\gamma}'{\beta}'\sin{\alpha}]{{\cal X}}+{\beta}_2,
\hspace{3.9cm}(3.24)\end{aligned}$$ where ${\beta}_1$ and ${\beta}_2$ are arbitrary functions of $t$. To solve the above problem, we write $${\beta}'=\frac{{\varphi}'}{\cos{\alpha}},\qquad {\gamma}=\frac{1}{4}\ln\mu'\eqno(3.25)$$ and set $$\left(\begin{array}{c}\hat\phi\\ \hat\psi\end{array}\right)
=\sqrt[4]{\mu'}\left(\begin{array}{cc}\cos{\varphi}&-\sin{\varphi}\\
\sin{\varphi}&\cos{\varphi}\end{array}\right) \left(\begin{array}{c}\phi\\
\psi\end{array}\right),\eqno(3.26)$$ $$\left(\begin{array}{c}{\gamma}_1\\ {\gamma}_2\end{array}\right)
=\int\frac{1}{\sqrt[4]{\mu'}} \left(\begin{array}{cc}\cos{\varphi}&
\sin{\varphi}\\
-\sin{\varphi}&\cos{\varphi}\end{array}\right) \left(\begin{array}{c}({\varphi}')^2
\tan{\alpha}+\frac{{\alpha}'{\mu'}'}{4\mu'}-{{\alpha}'}'
\\ -({\varphi}'\tan{\alpha})'-{\alpha}'{\varphi}'+\frac{{\mu'}'{\varphi}'}{4\mu'}\tan{\alpha}\end{array}\right)dt.\eqno(3.27)$$ Then (3.23) and (3.24) are equivalent to: $$\hat\phi_t-\frac{{\mu'}'}{2\mu'}{{\cal X}}\hat\phi_{{\cal X}}-\nu\phi_{{{\cal X}}{{\cal X}}}={\gamma}_1'\sqrt{\mu'}{{\cal X}}+{\varphi}_1',\eqno(3.28)$$ $$\hat\psi_t-\frac{{\mu'}'}{2\mu'}{{\cal X}}\hat\psi_{{\cal X}}-\nu\psi_{{{\cal X}}{{\cal X}}}={\gamma}_2'\sqrt{\mu'}{{\cal X}}+{\varphi}_2',\eqno(3.29)$$ where ${\varphi}_1$ and ${\varphi}_2$ are arbitrary functions of $t$. Thus $$\hat\phi={\gamma}_1\sqrt{\mu'}{{\cal X}}+{\varphi}_1+\sum_{i=1}^md_{1,i}e^{\nu
(a_{1,i}^2-b_{1,i}^2)\mu+a_{1,i}\sqrt{\mu'}{{\cal X}}}\sin(b_{1,i}(2a_{1,i}
\mu+\sqrt{\mu'}{{\cal X}})+c_{i,1}),\eqno(3.30)$$ $$\hat\psi={\gamma}_2\sqrt{\mu'}{{\cal X}}+{\varphi}_2+ \sum_{s=1}^nd_{1,s}e^{\nu
(a_{2,s}^2-b_{1,s}^2)\mu+a_{2,s}\sqrt{\mu'}{{\cal X}}}\sin(b_{2,s}(2a_{2,s}
\mu+\sqrt{\mu'}{{\cal X}})+c_{2,s}),\eqno(3.31)$$ where $a_{1,i},a_{2,s},b_{1,i},b_{2,s},c_{1,i},c_{2,s},d_{1,i}$ and $d_{2,s}$ are real constants. According (3.26), we have $$\begin{aligned}
\hspace{1cm}\phi&=&\sqrt[4]{\mu'}({\gamma}_1\cos{\varphi}+{\gamma}_2\sin{\varphi})
{{\cal X}}+{\sigma}_1+\frac{\cos{\varphi}}{\sqrt[4]{\mu'}}\sum_{i=1}^md_{1,i}e^{\nu
(a_{1,i}^2-b_{1,i}^2)\mu+a_{1,i}\sqrt{\mu'}{{\cal X}}}\\
& &\times\sin(b_{1,i}(2a_{1,i}+\sqrt{\mu'}{{\cal X}})+c_{i,1})+
\frac{\sin{\varphi}}{\sqrt[4]{\mu'}}\sum_{s=1}^nd_{1,s}e^{\nu
(a_{2,s}^2-b_{1,s}^2)\mu+a_{2,s}\sqrt{\mu'}{{\cal X}}}\\
& &\times\sin(b_{2,s}(2a_{2,s}+\sqrt{\mu'}{{\cal X}})+c_{2,s}),
\hspace{6.9cm}(3.32)\end{aligned}$$ $$\begin{aligned}
\hspace{1cm}\psi&=&\sqrt[4]{\mu'}({\gamma}_2\cos{\varphi}-{\gamma}_1\sin{\varphi})
{{\cal X}}+{\sigma}_2-\frac{\sin{\varphi}}{\sqrt[4]{\mu'}}\sum_{i=1}^md_{1,i}e^{\nu
(a_{1,i}^2-b_{1,i}^2)\mu+a_{1,i}\sqrt{\mu'}{{\cal X}}}\\
& &\times\sin( b_{1,i}(2a_{1,i}+\sqrt{\mu'}{{\cal X}})+c_{i,1})+
\frac{\cos{\varphi}}{\sqrt[4]{\mu'}}\sum_{s=1}^nd_{1,s}e^{\nu
(a_{2,s}^2-b_{1,s}^2)\mu+a_{2,s}\sqrt{\mu'}{{\cal X}}}\\
& &\times\sin(b_{2,s}(2a_{2,s} +\sqrt{\mu'}{{\cal X}})+c_{2,s}),
\hspace{6.9cm}(3.33)\end{aligned}$$ where ${\sigma}_1$ and ${\sigma}_2$ are arbitrary functions of $t$.
To find the pressure $p$, we recalculate $$\begin{aligned}
R_1&=&(({\varphi}')^2{{\cal Y}}-2{\varphi}'\psi)\tan{\alpha}-2{\alpha}'\phi
-{{\alpha}'}'{{\cal Y}}-({{\varphi}'}'+{\alpha}'{\varphi}'(1+\sec^2{\alpha})){{\cal Z}}\\ & &-\frac{{\mu'}'({\alpha}'{{\cal Y}}+{\varphi}'{{\cal Z}}\tan{\alpha})}{\mu'}+(({\alpha}')^2+({\varphi}')^2
\tan^2{\alpha}) {{\cal X}}+\frac{(3({\mu'}')^2-2{{\mu'}'}'){{\cal X}}}{4(\mu')^2}
,\hspace{0.9cm}(3.34)\end{aligned}$$ $$R_2=\frac{(4{{\mu'}'}'-3({\mu'}')^2){{\cal Y}}}{16(\mu')^2}
-({\alpha}')^2{{\cal Y}}+ {\sigma}_1'+({\varphi}'{{\cal X}}-{\alpha}'{{\cal Z}}){\varphi}'\tan{\alpha}-
\frac{({\alpha}'{\mu'}'+{{\alpha}'}'\mu'){{\cal X}}}{\mu'},\eqno(3.35)$$ $$\begin{aligned}
\hspace{2cm}R_3&=&\frac{(4{{\mu'}'}'-3({\mu'}')^2){{\cal Z}}}{16(\mu')^2}
-\frac{{\mu'}'{{\cal X}}+{\alpha}'\mu'{{\cal Y}}}{\mu'}{\varphi}'\tan{\alpha}+ {\sigma}_2'\\&
&-({\varphi}')^2{{\cal Z}}\tan^2{\alpha}-({{\varphi}'}'+{\alpha}'{\varphi}'(1+\sec^2{\alpha})){{\cal X}}.
\hspace{3.7cm}(3.36)\end{aligned}$$ Thus $$\begin{aligned}
p&=&\rho\{
({{\alpha}'}'{{\cal Y}}+({{\varphi}'}'\tan{\alpha}+{\alpha}'{\varphi}'(1+\sec^2{\alpha})){{\cal Z}}){{\cal X}}+\frac{{\mu'}'({\alpha}'{{\cal Y}}+{\varphi}'{{\cal Z}}\tan{\alpha}){{\cal X}}}{\mu'}\\
& &-{\sigma}_1'{{\cal Y}}-{\sigma}_2'{{\cal Z}}+\frac{{{\cal X}}^2}{2}
\left(\frac{({{\mu'}'}'-(3{\mu'}')^2)}{4(\mu')^2}-({\alpha}')^2-({\varphi}')^2\tan^2{\varphi}\right)
\\ &
&+\frac{(3({\mu'}')^2-4{{\mu'}'}')({{\cal Y}}^2+{{\cal Z}}^2)}{32(\mu')^2}+({\alpha}'{{\cal Z}}-{\varphi}'{{\cal X}}){\varphi}'{{\cal Y}}\tan{\alpha}\\ & &+\frac{({\alpha}')^2{{\cal Y}}^2+({\gamma}')^2{{\cal Z}}^2\tan^2{\alpha}}{2}
+2\int({\alpha}'\phi+{\varphi}'\psi\tan{\alpha})d{{\cal X}}\}
\hspace{3.9cm}(3.37)\end{aligned}$$ modulo the transformation in (1.10). In summary, we have:
[**Theorem 3.1**]{}. [*Let ${\alpha},{\varphi},\mu,{\sigma}_1,{\sigma}_2$ be functions of $t$ with $\mu'> 0$. Take real constants $\{a_{1,i},a_{2,s},b_{1,i},b_{2,s},c_{1,i},c_{2,s},d_{1,i},d_{2,s}\mid
i=1,...,m;s=1,...,n\}$. Denote $${\beta}=\int\frac{{\varphi}'dt}{\cos{\alpha}}\eqno(3.38)$$ and define ${\gamma}_1,{\gamma}_2$ by (3.27). Take the notations ${{\cal X}},{{\cal Y}},{{\cal Z}}$ given in (3.1) and (3.5). In terms of the functions $\phi$ in (3.32) and $\psi$ in (3.33), we have the following solution of the Navier-Stokes equations (1.1)-(1.4): $$u=-\left(\frac{{\mu'}'{{\cal X}}}{2\mu'}
+{\alpha}'{{\cal Y}}+{\varphi}'{{\cal Z}}\tan{\alpha}\right)\cos{\alpha}-\left(\phi+\frac{{\mu'}'{{\cal Y}}}{4\mu'}\right)\sin{\alpha},\eqno(3.39)$$ $$\begin{aligned}
\hspace{2cm}v&=&\left(\frac{{\mu'}'{{\cal X}}}{2\mu'}
-{\alpha}'{{\cal Y}}-{\varphi}'{{\cal Z}}\tan{\alpha}\right)\sin{\alpha}\;\cos{\beta}\\ &&
+\left(\phi+\frac{{\mu'}'{{\cal Y}}}{4\mu'}\right)\cos{\alpha}\;\cos{\beta}-
\left(\psi+\frac{{\mu'}'{{\cal Z}}}{4\mu'}\right)\sin{\beta},
\hspace{3.2cm}(3.40)\end{aligned}$$ $$\begin{aligned}
\hspace{2cm}w&=&\left(\frac{{\mu'}'{{\cal X}}}{2\mu'}
-{\alpha}'{{\cal Y}}-{\varphi}'{{\cal Z}}\tan{\alpha}\right)\sin{\alpha}\;\sin{\beta}\\ &&
+\left(\phi+\frac{{\mu'}'{{\cal Y}}}{4\mu'}\right)\cos{\alpha}\;\sin{\beta}+
\left(\psi+\frac{{\mu'}'{{\cal Z}}}{4\mu'}\right)\cos{\beta}\hspace{3.3cm}(3.41)\end{aligned}$$ and $p$ is given in (3.37).*]{}
[**Remark 3.2**]{}. We can use Fourier expansion to solve the system (3.28) and (3.29) for $\hat\phi(\mu,\sqrt{\mu'}{{\cal X}})$ and $\hat\psi(\mu,\sqrt{\mu'}{{\cal X}})$ with given $\hat\phi(0,\sqrt{\mu'}{{\cal X}})$ and $\hat\psi(0,\sqrt{\mu'}{{\cal X}})$. In this way, we can obtain discontinuous solutions of Navier-Stokes equations (1.1)-(1.4), which may be useful in studying shock waves.
Let $f,g,h$ be functions of $t,{{\cal X}},{{\cal Y}},{{\cal Z}}$ that are linear in ${{\cal X}},{{\cal Y}},{{\cal Z}}$ and $f_{{\cal X}}+g_{_{{\cal Y}}}+h_{{\cal Z}}=0$. Based on our experience in \[X2\], we assume $${{\cal U}}=f-6\nu{{\cal X}}^{-1},\qquad{{\cal V}}=g-6\nu{{\cal Y}}{{\cal X}}^{-2},\qquad{{\cal W}}=h.\eqno(3.42)$$ Then $$\begin{aligned}
\hspace{1.1cm}R_1&=&f_t+ff_{{\cal X}}+f_{{\cal Y}}g+f_{{\cal Z}}h
-6\nu f_{{\cal X}}{{\cal X}}^{-1}+{\alpha}'({{\cal Y}}f_{{\cal X}}-{{\cal X}}f_{{\cal Y}}-g)
\\
& &+{\beta}'({{\cal Z}}f_{{\cal X}}-{{\cal X}}f_{{\cal Z}}-h)\sin{\alpha}+{\beta}'({{\cal Z}}f_{{\cal Y}}-{{\cal Y}}f_{{\cal Z}})\cos{\alpha}\\
& &+6\nu (f-{{\cal Y}}f_{{\cal Y}}+2{\alpha}'{{\cal Y}}+{\beta}'{{\cal Z}}\sin{\alpha}){{\cal X}}^{-2}-24\nu^2{{\cal X}}^{-3},\hspace{3.2cm}(3.43)\end{aligned}$$ $$\begin{aligned}
\hspace{0.5cm}R_2&=&g_t+fg_{_{{\cal X}}}+gg_{_{{\cal Y}}}+g_{_{{\cal Z}}}h+
{\alpha}'({{\cal Y}}g_{_{{\cal X}}}-{{\cal X}}g_{_{{\cal Y}}} +f)+{\beta}'({{\cal Z}}g_{_{{\cal X}}}-{{\cal X}}g_{_{{\cal Z}}})\sin{\alpha}\\
& &-6\nu ({\alpha}'+g_{_{{\cal X}}}){{\cal X}}^{-1} -6\nu(g+{\beta}'{{\cal Z}}\cos{\alpha}-{\alpha}'{{\cal X}}+{{\cal Y}}g_{_{{\cal Y}}}){{\cal X}}^{-2}\\ & &+{\beta}'({{\cal Z}}g_{_{{\cal Y}}}-g_{_{{\cal Z}}}{{\cal Y}}-h)\cos{\alpha}+12\nu{{\cal Y}}(f+{\alpha}'{{\cal Y}}+{\beta}'{{\cal Z}}\sin{\alpha}){{\cal X}}^{-3},
\hspace{1.8cm}(3.44)\end{aligned}$$ $$\begin{aligned}
& &R_3=h_t+fh_{{\cal X}}+gh_{{\cal Y}}+hh_{{\cal Z}}+{\alpha}'({{\cal Y}}h_{{\cal X}}-{{\cal X}}h_{{\cal Y}})+{\beta}'({{\cal Z}}h_{{\cal X}}-{{\cal X}}h_{{\cal Z}}+f)\sin{\alpha}\\&&+{\beta}'({{\cal Z}}h_{{\cal Y}}-{{\cal Y}}h_{{\cal Z}}+g)\cos{\alpha}-6\nu(h_x+{\beta}'\sin{\alpha}){{\cal X}}^{-1}
-6\nu(h_{{\cal Y}}+{\beta}'\cos{\alpha}) {{\cal Y}}{{\cal X}}^{-2}.
\hspace{0.3cm}(3.45)\end{aligned}$$ According the negative powers of ${{\cal X}}$ in (3.14), we take $$f={\gamma}{{\cal X}}-{\alpha}'{{\cal Y}}-{\beta}'{{\cal Z}}\sin{\alpha},\qquad
g={\alpha}'{{\cal X}}+{\gamma}{{\cal Y}}-{\beta}'{{\cal Z}}\cos{\alpha},\eqno(3.46)$$ $$h=-({\beta}'{{\cal X}}\sin{\alpha}+{\beta}'{{\cal Y}}\cos{\alpha}+2{\gamma}{{\cal Z}})\eqno(3.47)$$ modulo the transformations of type in (1.8) and (1.9) for some function ${\gamma}$ of $t$. With the above data, we have: $$\begin{aligned}
\hspace{1.1cm}R_1&=&({\gamma}'+{\gamma}^2-({\alpha}')^2+
3({\beta}')^2\sin^2{\alpha}){{\cal X}}+12\nu {\alpha}'{{\cal Y}}{{\cal X}}^{-2}-24\nu^2{{\cal X}}^{-3}
\\ & &+(3({\beta}')^2\sin{\alpha}\;\cos {\alpha}-{{\alpha}'}'-2{\alpha}'{\gamma}){{\cal Y}}+(4{\beta}'{\gamma}-{{\beta}'}'){{\cal Z}}\sin{\alpha}, \hspace{2cm}(3.48)\end{aligned}$$ $$\begin{aligned}
\hspace{1.1cm}R_2&=&({\gamma}'+{\gamma}^2-({\alpha}')^2+
3({\beta}')^2\cos^2{\alpha}){{\cal Y}}-12\nu {\alpha}'{{\cal X}}^{-1}\\&
&+({{\alpha}'}'+2{\alpha}'{\gamma}+3({\beta}')^2\sin{\alpha}\;\cos {\alpha}){{\cal X}}+(4{\beta}'{\gamma}-{{\beta}'}'){{\cal Z}}\cos{\alpha},\hspace{1.9cm}(3.49)\end{aligned}$$ $$R_3=(4{\gamma}^2-2{\gamma}'-({\beta}')^2){{\cal Z}}+(4{\beta}'{\gamma}-{{\beta}'}')
({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha}).\eqno(3.50)$$
By (3.48)-(3.50), (3.14) is now equivalent to $$-{{\alpha}'}'-2{\alpha}'{\gamma}={{\alpha}'}'+2{\alpha}'{\gamma}{\Longrightarrow}{\gamma}=-\frac{{{\alpha}'}'}{2{\alpha}'}.\eqno(3.51)$$ Thus $${{\cal U}}=-\frac{{{\alpha}'}'}{2{\alpha}'}{{\cal X}}-{\alpha}'{{\cal Y}}-{\beta}'{{\cal Z}}\sin{\alpha}-6\nu{{\cal X}}^{-1},\qquad
{{\cal V}}={\alpha}'{{\cal X}}-\frac{{{\alpha}'}'}{2{\alpha}'}{{\cal Y}}-{\beta}'{{\cal Z}}\cos{\alpha}-6\nu{{\cal Y}}{{\cal X}}^{-2},\eqno(3.52)$$ $${{\cal W}}=\frac{{{\alpha}'}'}{{\alpha}'}{{\cal Z}}-{\beta}'{{\cal X}}\sin{\alpha}-{\beta}'{{\cal Y}}\cos{\alpha}\eqno(3.53)$$ by (3.42), (3.46), (3.47) and (3.51). Moreover, (3.12) and (3.48)-(3.50) imply $$\begin{aligned}
p&=&\rho\{
\frac{(2{\alpha}'{{{\alpha}'}'}'+4({\alpha}')^4-3({{\alpha}'}')^2)({{\cal X}}^2+{{\cal Y}}^2)}{8({\alpha}')^2}-
\frac{3({\beta}')^2({{\cal X}}^2\sin^2{\alpha}+{{\cal Y}}^2\cos^2{\alpha})}{2}
\\ & &+12\nu({\alpha}'{{\cal Y}}{{\cal X}}^{-1}-\nu{{\cal X}}^{-2})
+({{\beta}'}'-4{\beta}'{\gamma}){{\cal Z}}({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha})\\ &
&-3({\beta}')^2{{\cal X}}{{\cal Y}}\sin{\alpha}\;\cos {\alpha}+
\frac{({\alpha}'{{{\alpha}'}'}'+({\alpha}')^2({\beta}')^2-2({{\alpha}'}')^2){{\cal Z}}^2}{2({\alpha}')^2}
\}.\hspace{3cm}(3.54)\end{aligned}$$ By (3.3) and (3.5), we have the following theorem:
[**Theorem 3.3**]{}. [*Let ${\alpha}$ and ${\beta}$ be functions of $t$ with ${\alpha}'\neq 0$. In terms of the notations ${{\cal X}},{{\cal Y}},{{\cal Z}}$ given in (3.1) and (3.5), we have the following solution of Navier-Stokes equations (1.1)-(1.4): $$u=\left(\frac{{{\alpha}'}'}{2{\alpha}'}+6\nu{{\cal Y}}{{\cal X}}^{-2}\right)({{\cal Y}}\sin{\alpha}-{{\cal X}}\cos{\alpha})
-{\alpha}'({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha}),\eqno(3.55)$$ $$\begin{aligned}
\hspace{0.6cm}v&=&-\left(\frac{{{\alpha}'}'}
{2{\alpha}'}+6\nu{{\cal Y}}{{\cal X}}^{-2}\right)({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha})\cos{\beta}+{\alpha}'({{\cal X}}\cos{\alpha}-{{\cal Y}}\sin{\alpha})\cos{\beta}\\ &&-{\beta}'{{\cal Z}}\cos{\beta}+\left({\beta}'{{\cal X}}\sin{\alpha}+{\beta}'{{\cal Y}}\cos{\alpha}-\frac{{{\alpha}'}'}{{\alpha}'}{{\cal Z}}\right)\sin{\beta},
\hspace{3.7cm}(3.56)\end{aligned}$$ $$\begin{aligned}
\hspace{0.6cm}w&=&-\left(\frac{{{\alpha}'}'}
{2{\alpha}'}+6\nu{{\cal Y}}{{\cal X}}^{-2}\right)({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha})\sin{\beta}+{\alpha}'({{\cal X}}\cos{\alpha}-{{\cal Y}}\sin{\alpha})\sin{\beta}\\ &&-{\beta}'{{\cal Z}}\sin{\beta}+\left(\frac{{{\alpha}'}'}{{\alpha}'}{{\cal Z}}-{\beta}'{{\cal X}}\sin{\alpha}-{\beta}'{{\cal Y}}\cos{\alpha}\right)\cos{\beta}\hspace{3.8cm}(3.57)\end{aligned}$$ and $p$ is given in (3.54). The above solution blows up at any point on the following rotating plane with a line deleted: $$\begin{aligned}
\hspace{2cm}& &\{(x,y,z)\in{\mathbb}{R}^3\mid x\cos{\alpha}+y\sin{\alpha}\;\cos{\beta}+z\sin{\alpha}\;\sin{\beta}=0,\\ & &-x\sin{\alpha}+y\cos{\alpha}\;\cos{\beta}+z\cos{\alpha}\;\sin{\beta}\neq
0\}.\hspace{4.2cm}(3.58)\end{aligned}$$*]{}
We remark that the above solution may be applied to study turbulence.
Moving-Frame Approach II
========================
Motivated from the solution $\psi$ in (2.27) of the equation (2.24), we will solve Navier-Stokes equations by ansatzes with given irrational functions under the moving frames in (3.5).
First we rewrite (3.9)-(3.11): $$\begin{aligned}
& &R_1={{\cal U}}_t+({\alpha}'{{\cal Y}}+{\beta}'{{\cal Z}}\sin{\alpha}+{{\cal U}}){{\cal U}}_{{\cal X}}+({{\cal V}}-{\alpha}'{{\cal X}}+{\beta}'{{\cal Z}}\cos{\alpha}){{\cal U}}_{{\cal Y}}\\
&&+({{\cal W}}-{\beta}'({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha})){{\cal U}}_{{\cal Z}}-{\alpha}'{{\cal V}}-{\beta}'{{\cal W}}\sin{\alpha}-\nu{\Delta}({{\cal U}}),
\hspace{3.2cm}(4.1)\end{aligned}$$ $$\begin{aligned}
& &R_2={{\cal V}}_t+({\alpha}'{{\cal Y}}+{\beta}'{{\cal Z}}\sin{\alpha}+{{\cal U}}){{\cal V}}_{{\cal X}}+({{\cal V}}-{\alpha}'{{\cal X}}+{\beta}'{{\cal Z}}\cos{\alpha}){{\cal V}}_{{\cal Y}}\\
&&+({{\cal W}}-{\beta}'({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha})){{\cal V}}_{{\cal Z}}+{\alpha}'{{\cal U}}-{\beta}'{{\cal W}}\cos{\alpha}-\nu{\Delta}({{\cal V}}),
\hspace{3.2cm}(4.2)\end{aligned}$$ $$\begin{aligned}
& &R_3={{\cal W}}_t+({\alpha}'{{\cal Y}}+{\beta}'{{\cal Z}}\sin{\alpha}+{{\cal U}}){{\cal W}}_{{\cal X}}+({{\cal V}}-{\alpha}'{{\cal X}}+{\beta}'{{\cal Z}}\cos{\alpha}){{\cal W}}_{{\cal Y}}\\
&&+({{\cal W}}-{\beta}'({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha})){{\cal W}}_{{\cal Z}}+
{\beta}'({{\cal U}}\sin{\alpha}+{{\cal V}}\cos{\alpha})-\nu{\Delta}({{\cal W}}).
\hspace{2.3cm}(4.3)\end{aligned}$$ Let ${\alpha}_1,{\beta}_1,{\gamma}$ be functions of $t$, and let $a,b$ be real numbers. Set $$\xi_0=e^{{\alpha}_1{{\cal Y}}+{\beta}_1{{\cal Z}}}-ae^{-{\alpha}_1{{\cal Y}}-{\beta}_1{{\cal Z}}},\qquad\zeta_0=
e^{{\alpha}_1{{\cal Y}}+{\beta}_1{{\cal Z}}}+ae^{-{\alpha}_1{{\cal Y}}-{\beta}_1{{\cal Z}}},\eqno(4.4)$$ $$\xi_1=\sin({\alpha}_1{{\cal Y}}+{\beta}_1{{\cal Z}}),\qquad\zeta_1=\cos({\alpha}_1{{\cal Y}}+{\beta}_1{{\cal Z}}),
\eqno(4.5)$$ $$\phi_0=e^{{\gamma}{{\cal X}}}-be^{-{\gamma}{{\cal X}}},\qquad\zeta_0=
e^{{\gamma}{{\cal X}}}+be^{-{\gamma}{{\cal X}}},\eqno(4.6)$$ $$\phi_1=\sin({\gamma}{{\cal X}}),\qquad
\psi_1=\cos({\gamma}{{\cal X}}),\qquad{\Delta}_1={\partial}_{{\cal Y}}^2+{\partial}_{{\cal Z}}^2.\eqno(4.7)$$
Suppose that $f$ and $h$ are functions in $t,{{\cal Y}},{{\cal Z}}$. According to (3.46) and (3.47), we assume $${{\cal U}}=-{\alpha}'{{\cal Y}}-{\beta}'{{\cal Z}}\sin{\alpha}-(f_{{\cal Y}}+h_{{\cal Z}}){{\cal X}}-({\alpha}_1{\sigma}+{\beta}_1\tau)\zeta_r\phi_s,\eqno(4.8)$$ $${{\cal V}}={\alpha}'{{\cal X}}-{\beta}'{{\cal Z}}\cos{\alpha}+f+{\sigma}{\gamma}\xi_r\psi_s,\qquad
{{\cal W}}={\beta}'({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha})+h+\tau{\gamma}\xi_r\psi_s.\eqno(4.9)$$ By (4.1)-(4.3), we have $$\begin{aligned}
&
&R_1= -({\alpha}_1{\sigma}+{\beta}_1\tau)'\zeta_r\phi_s-
({\alpha}_1{\sigma}+{\beta}_1\tau)[(-1)^r({\alpha}_1'{{\cal Y}}+{\beta}_1'{{\cal Z}})\xi_r\phi_s+{\gamma}'{{\cal X}}\zeta_r\psi_s]
\\ & &-(f_{{{\cal Y}}t}+h_{{{\cal Z}}t}){{\cal X}}+((f_{{\cal Y}}+h_{{\cal Z}}){{\cal X}}+({\alpha}_1{\sigma}+{\beta}_1\tau)\zeta_r\phi_s)(
f_{{\cal Y}}+h_{{\cal Z}}+{\gamma}({\alpha}_1{\sigma}+{\beta}_1\tau)\zeta_r\psi_s)
\\ & &-{\alpha}'(f+{\alpha}'{{\cal X}}-{\beta}'{{\cal Z}}\cos{\alpha}+{\gamma}{\sigma}\xi_r\psi_s)
-{\beta}'({\beta}'({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha})+h+{\gamma}\tau\xi_r\psi_s)\sin{\alpha}\\ &&-(f+{\gamma}{\sigma}\xi_r\psi_s)({\alpha}'+(f_{{{\cal Y}}{{\cal Y}}}+h_{{{\cal Y}}{{\cal Z}}}){{\cal X}}+(-1)^r{\alpha}_1({\alpha}_1{\sigma}+{\beta}_1\tau)\xi_r\phi_s)
-(h+{\gamma}\tau\xi_r\psi_s)\\ & &\times
({\beta}'\sin{\alpha}+(f_{{{\cal Y}}{{\cal Z}}}+h_{{{\cal Z}}{{\cal Z}}}){{\cal X}}+(-1)^r{\beta}_1({\alpha}_1{\sigma}+{\beta}_1\tau)\xi_r\phi_s)
+\nu\{{\Delta}_1(f_{{\cal Y}}+h_{{\cal Z}}){{\cal X}}\\ & &+({\alpha}_1{\sigma}+{\beta}_1\tau)[(-1)^r({\alpha}_1^2+{\beta}_1^2)+(-1)^s{\gamma}^2]
\zeta_r\phi_s\}-{{\alpha}'}'{{\cal Y}}-({\beta}'\sin{\alpha})'{{\cal Z}}\\&
&=\{({\gamma}(f_{{\cal Y}}+h_{{\cal Z}})-{\gamma}'){{\cal X}}\zeta_r\psi_s-
(-1)^r({\alpha}_1'{{\cal Y}}+{\beta}_1'{{\cal Z}}+{\alpha}_1f+{\beta}_1h
)\xi_r\phi_s\}({\alpha}_1{\sigma}+{\beta}_1\tau)
\\ & &+\{({\alpha}_1{\sigma}+{\beta}_1\tau)[(-1)^s\nu{\gamma}^2
+(-1)^r\nu({\alpha}_1^2+{\beta}_1^2)+f_{{\cal Y}}+h_{{\cal Z}}]-({\alpha}_1{\sigma}+\tau{\beta}_1)'\}\zeta_r\phi_s
\\ &&-{\gamma}\{2({\sigma}{\alpha}'+\tau{\beta}'\sin{\alpha})
+[{\sigma}(f_{{{\cal Y}}{{\cal Y}}}+h_{{{\cal Y}}{{\cal Z}}})+\tau(f_{{{\cal Y}}{{\cal Z}}}+h_{{{\cal Z}}{{\cal Z}}})]{{\cal X}}\} \xi_r\psi_s
-(f_{{{\cal Y}}t}+h_{{{\cal Z}}t}){{\cal X}}\\
&&+(f_{{\cal Y}}+h_{{\cal Z}})^2{{\cal X}}-f({\alpha}'+(f_{{{\cal Y}}{{\cal Y}}}+h_{{{\cal Y}}{{\cal Z}}}){{\cal X}})
-h({\beta}'\sin{\alpha}+(f_{{{\cal Y}}{{\cal Z}}}+h_{{{\cal Z}}{{\cal Z}}}){{\cal X}})
\\ & &-{\alpha}'(f+{\alpha}'{{\cal X}}-{\beta}'{{\cal Z}}\cos{\alpha})
-{\beta}'({\beta}'({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha})+h)\sin{\alpha}-{{\alpha}'}'{{\cal Y}}\\ &
&-({\beta}'\sin{\alpha})'{{\cal Z}}+\nu{\Delta}_1(f_{{\cal Y}}+h_{{\cal Z}}){{\cal X}}+{\gamma}({\alpha}_1{\sigma}+{\beta}_1\tau)^2({\delta}_{r,1}+4a{\delta}_{r,0})\phi_s\psi_s
, \hspace{2cm}(4.10)\end{aligned}$$ $$\begin{aligned}
&
&R_2={{\alpha}'}'{{\cal X}}-({\beta}'\cos{\alpha})'{{\cal Z}}+
f_t+({\gamma}{\sigma})'\xi_r\psi_s+{\gamma}{\sigma}(({\alpha}_1'{{\cal Y}}+{\beta}_1'{{\cal Z}})\zeta_r\psi_s
+(-1)^s{\gamma}'{{\cal X}}\xi_r\phi_s)\\ &
&-{\alpha}'[{\alpha}'{{\cal Y}}+{\beta}'{{\cal Z}}\sin{\alpha}+(f_{{\cal Y}}+h_{{\cal Z}}){{\cal X}}+({\alpha}_1{\sigma}+{\beta}_1\tau)\zeta_r\phi_s]
-{\beta}'[{\beta}'({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha})\\ & &+
h+{\gamma}\tau\xi_r\psi_s]\cos{\alpha}-[(f_{{\cal Y}}+h_{{\cal Z}}){{\cal X}}+({\alpha}_1{\sigma}+{\beta}_1\tau)\zeta_r\phi_s]
({\alpha}'+(-1)^s{\gamma}^2{\sigma}\xi_r\phi_s)
\\ & &+(f+{\gamma}{\sigma}\xi_r\psi_s)(f_{{\cal Y}}+
{\alpha}_1{\gamma}{\sigma}\zeta_r\psi_s)+(h+{\gamma}\tau\xi_r\psi_s)(f_{{\cal Z}}-{\beta}'\cos{\alpha}+{\beta}_1{\gamma}{\sigma}\zeta_r\psi_s)\\& &
-\nu[{\Delta}_1(f)+{\gamma}{\sigma}((-1)^r({\alpha}_1^2+{\beta}_1^2)+(-1)^s{\gamma}^2)]\xi_r\psi_s
\\ &&={{\alpha}'}'{{\cal X}}+f_t+{\gamma}{\sigma}({\alpha}_1'{{\cal Y}}+{\beta}_1'{{\cal Z}}+{\alpha}_1f+{\beta}_1h) \zeta_r\psi_s
+(-1)^s{\gamma}{\sigma}({\gamma}'-{\gamma}(f_{{\cal Y}}+h_{{\cal Z}})){{\cal X}}\xi_r\phi_s \\ &
&+\{{\gamma}[{\sigma}f_{{\cal Y}}+\tau f_{{\cal Z}}-\nu{\sigma}[(-1)^r({\alpha}_1^2+{\beta}_1^2)+(-1)^s{\gamma}^2]
-2\tau{\beta}'\cos{\alpha}]+({\gamma}{\sigma})'\}\xi_r\psi_s
\\ & &-2{\alpha}'({\alpha}_1{\sigma}+{\beta}_1\tau)\zeta_r\phi_s-({\beta}'\cos{\alpha})'{{\cal Z}}-{\alpha}'[{\alpha}'{{\cal Y}}+{\beta}'{{\cal Z}}\sin{\alpha}+2(f_{{\cal Y}}+h_{{\cal Z}}){{\cal X}}]
\\ &
&-{\beta}'[{\beta}'({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha})+h]\cos{\alpha}+ff_{{\cal Y}}+h(f_{{\cal Z}}-{\beta}'\cos{\alpha})
-\nu{\Delta}_1(f)\\ & &
+{\gamma}^2{\sigma}({\alpha}_1{\sigma}+{\beta}_1\tau)(4b{\delta}_{s,0}+{\delta}_{s,1})\xi_r\zeta_r
, \hspace{7.9cm}(4.11)\end{aligned}$$ $$\begin{aligned}
& &R_3=({\beta}'\sin{\alpha})'{{\cal X}}+({\beta}'\cos{\alpha})'{{\cal Y}}+h_t
+(\tau{\gamma})'\xi_r\psi_s+ \tau{\gamma}[({\alpha}_1'{{\cal Y}}+{\beta}_1'{{\cal Z}})
\zeta_r\psi_s\\ & &+(-1)^s{\gamma}'{{\cal X}}\xi_r\phi_s] -[(f_{{\cal Y}}+h_{{\cal Z}}){{\cal X}}+
({\alpha}_1{\sigma}+{\beta}_1\tau)\zeta_r\phi_s]({\beta}'\sin{\alpha}+(-1)^s
\tau{\gamma}^2\xi_r\phi_s)\\ & &+(f+{\sigma}{\gamma}\xi_r\psi_s)
({\beta}'\cos{\alpha}+h_{{\cal Y}}+{\alpha}_1\tau{\gamma}\zeta_r\psi_s)+(h+\tau{\gamma}\xi_r\psi_s)
(h_{{\cal Z}}+{\beta}_1\tau{\gamma}\zeta_r\psi_s)\hspace{4cm}\end{aligned}$$$$\begin{aligned}
& &-{\beta}'({\alpha}'{{\cal Y}}+{\beta}'{{\cal Z}}\sin{\alpha}+(f_{{\cal Y}}+h_{{\cal Z}}){{\cal X}}+({\alpha}_1{\sigma}+{\beta}_1\tau)
\zeta_r\phi_s)\sin{\alpha}+{\beta}'({\alpha}'{{\cal X}}-{\beta}'{{\cal Z}}\cos{\alpha}\\ &
&+f+{\sigma}{\gamma}\xi_r\psi_s)\cos{\alpha}-\nu[{\Delta}_1(h)+{\gamma}\tau((-1)^r({\alpha}_1^2+{\beta}_1^2)
+(-1)^s{\gamma}^2)]\xi_r\psi_s
\\ &&={\gamma}\tau({\alpha}_1'{{\cal Y}}+{\beta}_1'{{\cal Z}}+{\alpha}_1f+{\beta}_1h)\zeta_r\psi_s
+\{(\tau{\gamma})'-\nu{\gamma}\tau[(-1)^r({\alpha}_1^2+{\beta}_1^2) +(-1)^s{\gamma}^2]\\
&&+{\gamma}(2{\beta}'{\sigma}\cos{\alpha}+{\sigma}h_{{\cal Y}}+\tau h_{{\cal Z}}) \}\xi_r\psi_s
-2{\beta}'({\alpha}_1{\sigma}+{\beta}_1\tau)\zeta_r\phi_s\sin{\alpha}+
(-1)^s{\gamma}\tau({\gamma}'\\ & &-{\gamma}(f_{{\cal Y}}+h_{{\cal Z}})){{\cal X}}\xi_r\phi_s
+({\beta}'\sin{\alpha})'{{\cal X}}+({\beta}'\cos{\alpha})'{{\cal Y}}+h_t-{\beta}'(f_{{\cal Y}}+h_{{\cal Z}}){{\cal X}}\sin{\alpha}\\ &&+f({\beta}'\cos{\alpha}+h_{{\cal Y}})+hh_{{\cal Z}}-
{\beta}'({\alpha}'{{\cal Y}}+{\beta}'{{\cal Z}}\sin{\alpha}+(f_{{\cal Y}}+h_{{\cal Z}}){{\cal X}})\sin{\alpha}+{\beta}'({\alpha}'{{\cal X}}\\ &
&-{\beta}'{{\cal Z}}\cos{\alpha}+f)\cos{\alpha}-\nu{\Delta}_1(h)
+{\gamma}^2\tau({\alpha}_1{\sigma}+{\beta}_1\tau)(4b{\delta}_{s,0}+{\delta}_{s,1})\xi_r\zeta_r.
\hspace{2.1cm}(4.12)\end{aligned}$$
By the coefficients of $\xi_r\psi_s$ in the equation ${\partial}_{{\cal Y}}(R_1)={\partial}_{{\cal X}}(R_2)$, we have $${\gamma}^2{\sigma}=(-1)^{r+s+1}{\alpha}_1({\alpha}_1{\sigma}+{\beta}_1\tau),\qquad
[{\sigma}(f_{{{\cal Y}}{{\cal Y}}}+h_{{{\cal Y}}{{\cal Z}}})+\tau(f_{{{\cal Y}}{{\cal Z}}}+h_{{{\cal Z}}{{\cal Z}}})]_{{\cal Y}}=0.\eqno(4.13)$$ Moreover, the coefficients of $\zeta_r\phi_s$ in the equation ${\partial}_{{\cal Y}}(R_1)={\partial}_{{\cal X}}(R_2)$ suggest $$(f_{{\cal Y}}+h_{{\cal Z}})_{{\cal Y}}=0,\eqno(4.14)$$ which implies the second equation in (4.13). According the coefficients of $\xi_r\phi_s$ in the equation ${\partial}_{{\cal Y}}(R_1)={\partial}_{{\cal X}}(R_2)$, we get $${\sigma}{\beta}_1h_{{\cal Y}}={\alpha}_1(\tau f_{{\cal Z}}-2\tau{\beta}'\cos{\alpha}).\eqno(4.15)$$ Furthermore, the coefficients of $\zeta_r\psi_s$ in the equation ${\partial}_{{\cal Y}}(R_1)={\partial}_{{\cal X}}(R_2)$ yield $${\alpha}_1{\beta}'\sin{\alpha}={\alpha}'{\beta}_1.\eqno(4.16)$$ Symmetrically, we have (4.16), $${\gamma}^2\tau=(-1)^{r+s+1}{\beta}_1({\alpha}_1{\sigma}+{\beta}_1\tau),\qquad
(f_{{\cal Y}}+h_{{\cal Z}})_{{\cal Z}}=0\eqno(4.17)$$ and $$f_{{\cal Z}}= h_{{\cal Y}}+2{\beta}'\cos{\alpha}.\eqno(4.18)$$ By the first equation in (4.13) and (4.17), we have $${\sigma}{\beta}_1=\tau{\alpha}_1.\eqno(4.19)$$ Then (4.15) is implied by (4.18) and (4.19). Note that the equations of the coefficients $\xi_r\psi_s,\;\zeta_r\psi_s,\;\xi_r\phi_s$ and $\zeta_r\phi_s$ in ${\partial}_{{\cal Z}}(R_2)={\partial}_{{\cal Y}}(R_3)$ are implied by (4.16), (4.18) and (4.19).
According to (4.14) and the second equation in (4.17), $$f_{{\cal Y}}+h_{{\cal Z}}={\gamma}_1,\eqno(4.20)$$ a function of $t$. Under the conditions in (4.16), the first equation in (4.17), and (4.18)-(4.20), ${\partial}_{{\cal Y}}(R_1)={\partial}_{{\cal X}}(R_2)$ becomes $${\alpha}'h_{{\cal Z}}-{\beta}'h_{{\cal Y}}\sin{\alpha}={{\alpha}'}',\eqno(4.21)$$ ${\partial}_{{\cal Z}}(R_1)={\partial}_{{\cal X}}(R_3)$ is equivalent to $${\beta}'h_{{\cal Z}}\sin{\alpha}+{\alpha}'
h_y={\beta}'{\gamma}_1\sin{\alpha}-({\beta}'\sin{\alpha})'-2{\alpha}'{\beta}'\cos{\alpha}\eqno(4.22)$$ and ${\partial}_{{\cal Z}}(R_2)={\partial}_{{\cal Y}}(R_3)$ says $$(ff_{{\cal Y}}+hf_{{\cal Z}})_{{\cal Z}}=(fh_{{\cal Y}}+hh_{{\cal Z}})_{{\cal Y}}+2{\beta}'{\gamma}_1\cos{\alpha}.\eqno(4.23)$$ By (4.18) and (4.20)-(4.22), we assume $f_{{\cal Y}},\;f_{{\cal Z}},\;h_{{\cal Y}}$ and $h_{{\cal Z}}$ are functions of $t$. Then (4.23) can be written as $$(f_{{\cal Y}}+h_{{\cal Z}})f_{{\cal Z}}=(f_{{\cal Y}}+h_{{\cal Z}})h_{{\cal Y}}+2{\beta}'{\gamma}_1\cos{\alpha},\eqno(4.24)$$ which is implied by (4.18) and (4.20). Solving (4.21) and (4.22), we get $$h_{{\cal Y}}=\frac{{\alpha}'{\beta}'{\gamma}_1\sin{\alpha}-({\alpha}'{\beta}'\sin{\alpha})'
-2({\alpha}')^2{\beta}'\cos{\alpha}}{({\alpha}')^2+({\beta}')^2\sin^2{\alpha}},\eqno(4.25)$$ $$h_{{\cal Z}}= \frac{{\alpha}'{{\alpha}'}'+({\beta}')^2{\gamma}_1\sin^2{\alpha}-
({\beta}'\sin{\alpha})({\beta}'\sin{\alpha})'-{\alpha}'({\beta}')^2\sin
2{\alpha}}{({\alpha}')^2+({\beta}')^2\sin^2{\alpha}}.\eqno(4.26)$$ Moreover, $$f_{{\cal Y}}=\frac{{\gamma}_1({\alpha}')^2-{\alpha}'{{\alpha}'}'+
({\beta}'\sin{\alpha})({\beta}'\sin{\alpha})'+{\alpha}'({\beta}')^2\sin
2{\alpha}}{({\alpha}')^2+({\beta}')^2\sin^2{\alpha}}\eqno(4.27)$$ by (4.20) and (4.26), and $$f_{{\cal Z}}=\frac{{\alpha}'{\beta}'{\gamma}_1\sin{\alpha}-({\alpha}'{\beta}'\sin{\alpha})'
+2({\beta}')^2\sin^2{\alpha}\;\cos{\alpha}}{({\alpha}')^2+({\beta}')^2\sin^2{\alpha}}
\eqno(4.28)$$ by (4.18) and (4.25). With above data, we take $$f=f_{{\cal Y}}{{\cal Y}}+f_{{\cal Z}}{{\cal Z}},\qquad h=h_{{\cal Y}}{{\cal Y}}+h_{{\cal Z}}{{\cal Z}}\eqno(4.29)$$ by the transformations of the type in (1.8) and (1.9). Furthermore, (4.17), (4.19) and the first equation in (4.17) yield $r+s+1\in 2{\mathbb}{Z}$, $${\alpha}_1={\varphi}{\alpha}',\qquad {\gamma}=\pm{\varphi}\sqrt{({\alpha}')^2+({\beta}')^2\sin^2{\alpha}},
\eqno(4.30)$$ $${\beta}_1={\varphi}{\beta}'\sin{\alpha},\qquad{\sigma}=\mu{\alpha}',\qquad
\tau=\mu{\beta}'\sin{\alpha}.\eqno(4.31)$$ In particular, ${\alpha},{\beta},{\gamma}_1,{\varphi}$ and $\mu$ are arbitrary functions of $t$. According (4.9)-(4.11), the pressure $$\begin{aligned}
&
&p=\rho \{{\gamma}\mu{\varphi}^{-1}[({\gamma}'-{\gamma}{\gamma}_1){{\cal X}}\zeta_r\phi_s
-(({\varphi}{\alpha}')'{{\cal Y}}+({\varphi}{\beta}'\sin{\alpha})'{{\cal Z}}+{\varphi}({\alpha}'f+{\beta}'h\sin{\alpha}) )\xi_r\psi_s] \\
& & +(-1)^s{\varphi}^{-1}[({\gamma}\mu)'-{\gamma}\mu{\varphi}'{\varphi}^{-1}]\zeta_r\psi_s
+2\mu(({\alpha}')^2+({\beta}')^2\sin^2{\alpha})\xi_r\phi_s+2{\alpha}'f{{\cal X}}\\
& & +\frac{({\alpha}')^2+({\beta}')^2\sin^2{\alpha}+{\gamma}_1'-{\gamma}_1^2}{2}{{\cal X}}^2
+2{\beta}'h{{\cal X}}\sin{\alpha}+[({\beta}'\sin{\alpha})'-{\alpha}'{\beta}'\cos{\alpha}]{{\cal X}}{{\cal Z}}\\ & & +\left(\frac{({\beta}')^2}{2}\sin2{\alpha}+{{\alpha}'}'\right){{\cal X}}{{\cal Y}}-
\frac{1}{2}{\gamma}^4\mu^2{\varphi}^{-2}[({\delta}_{r,1}+4a{\delta}_{r,0})\phi_s^2
+(4b{\delta}_{s,0}+{\delta}_{s,1})\xi_r^2]
\\ & &+[({\beta}'\cos{\alpha})'+{\alpha}'{\beta}'\sin{\alpha}-f_{{{\cal Z}}t}-f_{{\cal Y}}f_{{\cal Z}}-h_{{\cal Y}}h_{{\cal Z}}]{{\cal Y}}{{\cal Z}}+\frac{({\beta}')^2-h_{{{\cal Z}}t}-f_{{\cal Z}}^2-h_{{\cal Z}}^2}{2}{{\cal Z}}^2
\\ & &+ \frac{({\alpha}')^2+({\beta}')^2\cos{\alpha}-f_{{{\cal Y}}t}-f_{{\cal Y}}^2-h_{{\cal Y}}^2}{2}{{\cal Y}}^2
\}\hspace{6.9cm}(4.32)\end{aligned}$$ modulo the transformation in (1.10). By (3.3) and (3.5), we have the following theorem:
[**Theorem 4.1**]{}. [*Let ${\alpha},{\beta},{\gamma}_1,{\varphi}$ and $\mu$ be arbitrary functions of $t$ such that ${\varphi}\neq 0$ and $({\alpha}')^2+({\beta}')^2\sin^2{\alpha}\neq 0$. Take any real constants $a$ and $b$. The notations ${{\cal X}},{{\cal Y}}$ and ${{\cal Z}}$ are defined in (3.5) via (3.1), and the notations $\xi_r,\zeta_r,\phi_r$ and $\psi_r$ are defined in (4.4)-(4.7) with ${\alpha}_1,{\beta}_1$ and ${\gamma}$ given in (4.30) and (4.31). Moreover, $f_{{\cal Y}},f_{{\cal Z}},h_{{\cal Y}},h_{{\cal Z}}$ and $f, h$ are given in (4.25)-(4.29). Assume $(r,s)\in\{(0,1),(1,0)\}$. We have the following solution of the Navier-Stokes equations (1.1)-(1.4): $$\begin{aligned}
\hspace{1.7cm} u&=&-{\alpha}'({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha})
-(f+\mu{\alpha}'{\gamma}\xi_r\psi_s)\sin{\alpha}\\ & & -({\gamma}_1{{\cal X}}+ {\varphi}\mu(({\alpha}')^2+({\beta}')^2\sin^2{\alpha})\zeta_r\phi_s)\cos{\alpha},\hspace{4cm}(4.33)\end{aligned}$$ $$\begin{aligned}
\hspace{1.2cm}v&=&(f\cos{\alpha}-{\beta}'{{\cal Z}})\cos{\beta}-({\alpha}'\sin{\alpha}\;\cos{\beta}+{\beta}'\cos{\alpha}\;\sin{\beta}){{\cal Y}}\\ & &-({\gamma}_1{{\cal X}}+ {\varphi}\mu(({\alpha}')^2+({\beta}')^2\sin^2{\alpha})\zeta_r\phi_s)\sin{\alpha}\;\cos{\beta}-h\sin{\beta}\\ & &+({\alpha}'\cos{\alpha}\;\cos{\beta}-
{\beta}'\sin{\alpha}\;\sin{\beta})({{\cal X}}+{\gamma}\mu\xi_r\psi_s),\hspace{4cm}(4.34)\end{aligned}$$ $$\begin{aligned}
\hspace{1.2cm}w&=&({\beta}'\cos{\alpha}\;\cos{\beta}-
{\alpha}'\sin{\alpha}\;\sin{\beta}){{\cal Y}}+(f\cos{\alpha}-{\beta}'{{\cal Z}})\sin{\beta}\\& & -({\gamma}_1{{\cal X}}+
{\varphi}\mu(({\alpha}')^2+({\beta}')^2\sin^2{\alpha})\zeta_r\phi_s)\sin{\alpha}\;\sin{\beta}+h\cos{\beta}\\&&+({\alpha}'\cos{\alpha}\;\sin{\beta}+{\beta}'\sin{\alpha}\;\cos{\beta})({{\cal X}}+{\gamma}\mu\xi_r\psi_s)
\hspace{4cm}(4.35)\end{aligned}$$ and $p$ is given in (4.32). The above solution is globally analytic in $x,y,z$.*]{}
Let ${\gamma}_1,{\gamma}_2$ be functions of $t$ and let $a,b,c$ be real numbers. Denote $$\phi_0=
e^{{\gamma}_1{{\cal Y}}+{\gamma}_2{{\cal Z}}}-ae^{-{\gamma}_1{{\cal Y}}-{\gamma}_2{{\cal Z}}},
\qquad\phi_1=\sin({\gamma}_1{{\cal Y}}+{\gamma}_2{{\cal Z}}),\eqno(4.36)$$ $$\psi_0=
e^{{\gamma}_1{{\cal Y}}+{\gamma}_2{{\cal Z}}}+ae^{-{\gamma}_1{{\cal Y}}-{\gamma}_2{{\cal Z}}},
\qquad\psi_1=\cos({\gamma}_1{{\cal Y}}+{\gamma}_2{{\cal Z}}),\eqno(4.37)$$ $$\xi_0=
be^{{\gamma}_1{{\cal Y}}+{\gamma}_2{{\cal Z}}}-ce^{-{\gamma}_1{{\cal Y}}-{\gamma}_2{{\cal Z}}},
\qquad\xi_1=c\sin({\gamma}_1{{\cal Y}}+{\gamma}_2{{\cal Z}}+b),\eqno(4.38)$$ $$\zeta_0=
be^{{\gamma}_1{{\cal Y}}+{\gamma}_2{{\cal Z}}}+ce^{-{\gamma}_1{{\cal Y}}-{\gamma}_2{{\cal Z}}},
\qquad\zeta_1=c\cos({\gamma}_1{{\cal Y}}+{\gamma}_2{{\cal Z}}+b).\eqno(4.39)$$ Suppose that ${\sigma},\tau$ are functions of $t$ and $f,k,h$ are functions in $t,{{\cal X}},{{\cal Y}},{{\cal Z}}$ such that $h$ and $g$ are linear in ${{\cal X}},Y,{{\cal Z}}$ and $$f_{{\cal X}}+k_{{\cal Y}}+h_{{\cal Z}}=0.\eqno(4.40)$$ Motivated from the above solution, we consider the solution of the form: $${{\cal U}}=-{\alpha}'{{\cal Y}}-{\beta}'{{\cal Z}}\sin{\alpha}+f-({\gamma}_1^2+{\gamma}_2^2)
(\tau\zeta_r{{\cal X}}+{\sigma}\psi_r{{\cal X}}^2),\eqno(4.41)$$ $${{\cal V}}={\alpha}'{{\cal X}}-{\beta}'{{\cal Z}}\cos{\alpha}+k+{\gamma}_1(\tau\xi_r+2{\sigma}\phi_r{{\cal X}}),\eqno(4.42)$$ $${{\cal W}}={\beta}'({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha})+h+{\gamma}_2(\tau\xi_r+2{\sigma}\phi_r{{\cal X}})
.\eqno(4.43)$$
For convenience of computation, we denote $${\gamma}={\gamma}_1^2+{\gamma}_2^2,\qquad f^\ast=f-f_x{{\cal X}}\qquad{\Delta}_1={\partial}_{{\cal Y}}^2+{\partial}_{{\cal Z}}^2.\eqno(4.44)$$ Now (4.1) becomes $$\begin{aligned}
&
&R_1=-{{\alpha}'}'{{\cal Y}}-({\beta}'\sin{\alpha})'{{\cal Z}}+f_t-(-1)^r{\gamma}({\gamma}_1'{{\cal Y}}+{\gamma}_2'{{\cal Z}})(\tau\xi_r{{\cal X}}+{\sigma}\phi_r{{\cal X}}^2)\\
& & +((-1)^r\nu {\gamma}^2\tau-({\gamma}\tau)')\zeta_r{{\cal X}}+
(f-{\gamma}(\tau\zeta_r{{\cal X}}+{\sigma}\psi_r{{\cal X}}^2))(f_{{\cal X}}-{\gamma}(\tau\zeta_r+2{\sigma}\psi_r{{\cal X}}))
\\ & &+(k+{\gamma}_1(\tau\xi_r+2{\sigma}\phi_r{{\cal X}})) [f_{{\cal Y}}-2{\alpha}'-(-1)^r{\gamma}{\gamma}_1
(\tau\xi_r{{\cal X}}+{\sigma}\phi_r{{\cal X}}^2)]-\nu{\Delta}_1(f)\\ & &
+(h+{\gamma}_2(\tau\xi_r+2{\sigma}\phi_r{{\cal X}}))[f_{{\cal Z}}-2{\beta}'\sin{\alpha}-(-1)^r{\gamma}{\gamma}_2
(\tau\xi_r{{\cal X}}+{\sigma}\phi_r{{\cal X}}^2)]+2\nu{\gamma}{\sigma}\psi_r\\&&-{\alpha}'({\alpha}'{{\cal X}}-{\beta}'{{\cal Z}}\cos{\alpha})
-({\beta}')^2({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha})\sin{\alpha}+((-1)^r\nu
{\gamma}^2{\sigma}-({\gamma}{\sigma})')\psi_r{{\cal X}}^2
\\&&=-(({\alpha}')^2+({\beta}')^2\sin^2{\alpha}){{\cal X}}-({{\alpha}'}'+2^{-1}({\beta}')^2\sin2{\alpha}){{\cal Y}}+({\alpha}'{\beta}'\cos{\alpha}-({\beta}'\sin{\alpha})'){{\cal Z}}\\ &
&+{\gamma}^2[\tau^2(4b{\delta}_{0,r}+c{\delta}_{1,r})c{{\cal X}}+3{\sigma}\tau(2{\delta}_{0,r}(ab+c)+
{\delta}_{1,r}c\cos b){{\cal X}}^2+2{\sigma}^2 (4a{\delta}_{0,r}+{\delta}_{1,r}){{\cal X}}^3]
\\ & &-(-1)^r{\gamma}({\gamma}_1'{{\cal Y}}+{\gamma}_2'{{\cal Z}}+k{\gamma}_1+h{\gamma}_2)
(\tau\xi_r{{\cal X}}+{\sigma}\phi_r{{\cal X}}^2)+ff_{{\cal X}}+kf_{{\cal Y}}+hf_{{\cal Z}}\\ & &+((-1)^r\nu
{\gamma}^2{\sigma}-({\gamma}{\sigma})'-3{\gamma}{\sigma}f_{{\cal X}})\psi_r{{\cal X}}^2+\nu(2{\sigma}\psi_r-{\Delta}_1(f))-{\gamma}\tau f^\ast\zeta_r\\
& &-[(({\gamma}\tau)'+2{\gamma}\tau f_{{\cal X}}-(-1)^r\nu
{\gamma}^2\tau)\zeta_r+2{\gamma}{\sigma}f^\ast\psi_r]{{\cal X}}+f_t\\
&& +({\gamma}_1 (f_{{\cal Y}}-2{\alpha}') +{\gamma}_2(f_{{\cal Z}}-2{\beta}'\sin{\alpha}))
(\tau\xi_r+2{\sigma}\phi_r{{\cal X}}).\hspace{4.8cm}(4.45)\end{aligned}$$ To solve (3.14), we assume $${\gamma}_1'{{\cal Y}}+{\gamma}_2'{{\cal Z}}+k{\gamma}_1+h{\gamma}_2=0\eqno(4.46)$$ and $$(-1)^r\nu{\gamma}^2{\sigma}-({\gamma}{\sigma})'-3{\gamma}{\sigma}f_{{\cal X}}=0,\eqno(4.47)$$ Moreover, (4.2) and (4.3) become $$\begin{aligned}
& &R_2={{\alpha}'}'{{\cal X}}-({\beta}'\cos{\alpha})'{{\cal Z}}+
(({\gamma}_1\tau)'-(-1)^r\nu{\gamma}{\gamma}_1\tau)\xi_r+2(({\gamma}_1{\sigma})'-
(-1)^r\nu{\gamma}{\gamma}_1{\sigma})\phi_r{{\cal X}}\\ &
&+k_t+({\gamma}_1'{{\cal Y}}+{\gamma}_2'{{\cal Z}}){\gamma}_1(\tau\zeta_r+2{\sigma}\psi_r{{\cal X}})
+(f-{\gamma}(\tau\zeta_r{{\cal X}}+{\sigma}\psi_r{{\cal X}}^2))(2{\alpha}'+k_{{\cal X}}+2{\gamma}_1{\sigma}\phi_r)
\\ & &+(k+{\gamma}_1(\tau\xi_r+2{\sigma}\phi_r{{\cal X}}))(k_{{\cal Y}}+{\gamma}_1^2(\tau\zeta_r
+2{\sigma}\psi_r{{\cal X}}))-({\beta}')^2({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha})\cos{\alpha}\\ & &
-{\alpha}'({\alpha}'{{\cal Y}}+{\beta}'{{\cal Z}}\sin{\alpha})+(h+{\gamma}_2(\tau\xi_r+2{\sigma}\phi_r{{\cal X}}))(k_{{\cal Z}}-2{\beta}'\cos{\alpha}+{\gamma}_1{\gamma}_2
(\tau\zeta_r+2{\sigma}\psi_r{{\cal X}}))\\
&&=({{\alpha}'}'-2^{-1}({\beta}')^2\sin2{\alpha}+f_{{\cal X}}(2{\alpha}'+k_{{\cal X}})){{\cal X}}-(({\alpha}')^2+({\beta}')^2\cos^2{\alpha}){{\cal Y}}+k_t+kk_{{\cal Y}}\\ & & +
[\tau({\gamma}_1k_{{\cal Y}}+{\gamma}_2(k_{{\cal Z}}-2{\beta}'\cos{\alpha}))+({\gamma}_1\tau)'-(-1)^r\nu{\gamma}{\gamma}_1\tau]\xi_r
-(({\beta}'\cos{\alpha})'+{\alpha}'{\beta}'\sin{\alpha}){{\cal Z}}\\&&+{\gamma}{\sigma}(2{\gamma}_1{\sigma}\phi_r-2{\alpha}'
-k_{{\cal X}})\psi_r{{\cal X}}^2+f^\ast(2{\alpha}'+k_{{\cal X}}+2{\gamma}_1{\sigma}\phi_r)+h(k_{{\cal Z}}-2{\beta}'\cos{\alpha})
\\ &&+{\gamma}{\gamma}_1\tau^2\xi_r\zeta_r
+ \{2{\gamma}{\gamma}_1{\sigma}\tau\xi_r\psi_r +2[({\gamma}_1{\sigma})'-{\sigma}({\gamma}_1(h_{{\cal Z}}+
(-1)^r\nu{\gamma})\\ & &+{\gamma}_2(2{\beta}'\cos{\alpha}-k_{{\cal Z}}))]\phi_r
-{\gamma}\tau(2{\alpha}'+k_{{\cal X}})\zeta_r\}{{\cal X}}\hspace{6.2cm}(4.48)\end{aligned}$$ $$\begin{aligned}
& &R_3=({\beta}'\sin{\alpha})'{{\cal X}}+({\beta}'\cos{\alpha})'{{\cal Y}}+
({\gamma}_2\tau)'\xi_r+2({\gamma}_2{\sigma})'\phi_r{{\cal X}}-(-1)^r\nu{\gamma}_2^3(\tau\xi_r+{\sigma}\phi_r{{\cal X}})\\ &
&+({\gamma}_1'{{\cal Y}}+{\gamma}_2'{{\cal Z}}){\gamma}_2(\tau\zeta_r+2{\sigma}\psi_r{{\cal X}})
+(f-{\gamma}(\tau\zeta_r{{\cal X}}+{\sigma}\psi_r{{\cal X}}^2))(2{\beta}'\sin{\alpha}+h_{{\cal X}}+2{\gamma}_2{\sigma}\phi_r)
\\ & &+(k+{\gamma}_1(\tau\xi_r+2{\sigma}\phi_r{{\cal X}}))(2{\beta}'\cos{\alpha}+
h_{{\cal Y}}+{\gamma}_1{\gamma}_2(\tau\zeta_r +2{\sigma}\psi_r{{\cal X}})-({\beta}')^2{{\cal Z}}+h_t\\ &
&+{\alpha}'{\beta}'({{\cal X}}\cos{\alpha}-{{\cal Y}}\sin{\alpha})
+(h+{\gamma}_2(\tau\xi_r+2{\sigma}\phi_r{{\cal X}}))(h_{{\cal Z}}+{\gamma}_2^2(\tau\zeta_r+2{\sigma}\psi_r{{\cal X}}))\\ &
&=[({\beta}'\sin{\alpha})'+{\alpha}'{\beta}'\cos{\alpha}+f_{{\cal X}}(2{\beta}'\sin{\alpha}+h_{{\cal X}})]{{\cal X}}+
[({\beta}'\cos{\alpha})'-{\alpha}'{\beta}'\sin{\alpha}]{{\cal Y}}\\ & & +[({\gamma}_2\tau)'+({\gamma}_1(2{\beta}'\cos{\alpha}+
h_{{\cal Y}})+{\gamma}_2h_{{\cal Z}}-(-1)^r\nu{\gamma}{\gamma}_2)\tau]\xi_r+\{
2{\gamma}{\gamma}_2\tau{\sigma}\xi_r\psi_r +2[({\gamma}_2{\sigma})'\\ &
&-{\gamma}_2{\sigma}(k_{{\cal Y}}+(-1)^r\nu{\gamma})+{\gamma}_1{\sigma}(2{\beta}'\cos{\alpha}+h_{{\cal Y}})]\phi_r-{\gamma}\tau(2{\beta}'\sin{\alpha}+ h_{{\cal X}})\zeta_r\}{{\cal X}}\\
& &+f^\ast(2{\beta}'\sin{\alpha}+h_{{\cal X}}+2{\gamma}_2{\sigma}\phi_r) +k(2{\beta}'\cos{\alpha}+
h_{{\cal Y}})+h_t+hh_{{\cal Z}}+{\gamma}{\gamma}_2\tau^2\xi_r\zeta_r\\ &
&+{\gamma}{\sigma}\psi_r(2{\gamma}_2{\sigma}\phi_r-2{\beta}'\sin{\alpha}-h_{{\cal X}}){{\cal X}}^2-({\beta}')^2{{\cal Z}}.
\hspace{6cm}(4.49)\end{aligned}$$
By the coefficients of ${{\cal X}}^2$ in ${\partial}_{{\cal Z}}(R_2)={\partial}_{{\cal Y}}(R_3)$, we have: $${\gamma}_2(2{\alpha}'+k_{{\cal X}})={\gamma}_1(2{\beta}'\sin{\alpha}+h_{{\cal X}}).\eqno(4.50)$$ According to (4.46), $$k_{{\cal X}}{\gamma}_1+h_{{\cal X}}{\gamma}_2=0,\;\;{\gamma}_1'+{\gamma}_1k_{{\cal Y}}+{\gamma}_2h_{{\cal Y}}=0,\;\;
{\gamma}_2'+{\gamma}_1k_{{\cal Z}}+{\gamma}_2h_{{\cal Z}}=0.\eqno(4.51)$$ Solving (4.50) and the first equation in (4.51), we obtain $$k_{{\cal X}}=2{\gamma}^{-1}{\gamma}_2({\beta}'{\gamma}_1\sin{\alpha}-{\alpha}'{\gamma}_2),\qquad h_{{\cal X}}=
-2{\gamma}^{-1}{\gamma}_1({\beta}'{\gamma}_1\sin{\alpha}-{\alpha}'{\gamma}_2).\eqno(4.52)$$ Moreover, the coefficients of ${{\cal X}}$ in ${\partial}_{{\cal Z}}(R_2)={\partial}_{{\cal Y}}(R_3)$ give $${\gamma}_1'{\gamma}_2-{\gamma}_1{\gamma}_2'+{\gamma}_1{\gamma}_2(k_{{\cal Y}}-h_{{\cal Z}})+{\gamma}_2^2k_{{\cal Z}}-{\gamma}_1^2h_{{\cal Y}}-2{\gamma}{\beta}'\cos{\alpha}=0. \eqno(4.53)$$ By (4.51), the above equation can be rewritten as $$k_{{\cal Z}}-h_{{\cal Y}}=2{\beta}'\cos{\alpha}.
\eqno(4.54)$$ Furthermore, (4.50) and the coefficients of ${{\cal X}}^0$ in ${\partial}_{{\cal Z}}(R_2)={\partial}_{{\cal Y}}(R_3)$ show that $f$ is a function of $t$ and ${\gamma}_1{{\cal Y}}+{\gamma}_2{{\cal Z}}$. According to the coefficients of ${{\cal X}}$ in ${\partial}_{{\cal Y}}(R_1)={\partial}_{{\cal X}}(R_2)$ and ${\partial}_{{\cal Z}}(R_1)={\partial}_{{\cal X}}(R_3)$, we take $$f^\ast={\varphi}{\vartheta}_r+{\sigma}{\tilde}\varpi\phi_r+{\alpha}_1
,\eqno(4.55)$$ where ${\varphi}$ and ${\alpha}_1$ are functions of $t$, and $${\tilde}\varpi={\gamma}_1{{\cal Y}}+{\gamma}_2{{\cal Z}},\qquad
{\vartheta}_0=b_1e^{{\tilde}\varpi}-c_1e^{-{\tilde}\varpi},\qquad
{\vartheta}_1=c_1\sin({\tilde}\varpi+b_1)\eqno(4.56)$$ for $b_1,c_1\in{\mathbb}{R}$.
Now the coefficients of ${{\cal X}}$ in ${\partial}_{{\cal Y}}(R_1)={\partial}_{{\cal X}}(R_2)$ and ${\partial}_{{\cal X}}(R_1)={\partial}_{{\cal X}}(R_3)$ give $${\cal T}[2{\alpha}_1{\gamma}{\sigma}\psi_r+(({\gamma}\tau)'+2{\gamma}\tau f_{{\cal X}}-(-1)^r\nu
{\gamma}^2\tau)\zeta_r]=0,\qquad{\cal T}={\partial}_{{\cal Y}},{\partial}_{{\cal Z}}.\eqno(4.57)$$ Moreover, the coefficients of ${{\cal X}}^0$ in ${\partial}_{{\cal Y}}(R_1)={\partial}_{{\cal X}}(R_2)$ and ${\partial}_{{\cal X}}(R_1)={\partial}_{{\cal X}}(R_3)$ yield $$\begin{aligned}
&&[(f_{{\cal X}}-(-1)^r{\gamma}\nu){\varphi}+{\varphi}'){\vartheta}_r
+((f_{{\cal X}}-(-1)^r\nu{\gamma}){\sigma}+{\sigma}'){\tilde}\varpi\phi_r
-{\alpha}_1{\gamma}\tau\zeta_r]_{{\cal Y}}\\
&=&2[({\gamma}_1{\sigma})'-{\sigma}({\gamma}_1(h_{{\cal Z}}+
(-1)^r\nu{\gamma})-{\gamma}_2h_{{\cal Y}})]\phi_r\\
& &+2{{\alpha}'}'+2{\alpha}'f_{{\cal X}}+k_{{{\cal X}}t}+h_{{\cal X}}h_{{\cal Y}}-k_{{\cal X}}h_{{\cal Z}},\hspace{7.1cm}(4.58)\end{aligned}$$ $$\begin{aligned}
&&
[(f_{{\cal X}}-(-1)^r{\gamma}\nu){\varphi}+{\varphi}'){\vartheta}_r
+((f_{{\cal X}}-(-1)^r\nu{\gamma}){\sigma}+{\sigma}'){\tilde}\varpi\phi_r
-{\alpha}_1{\gamma}\tau\zeta_r]_{{\cal Z}}\\ &=&2({\beta}'\sin{\alpha})'+2{\beta}'f_{{\cal X}}\sin{\alpha}+h_{{{\cal X}}t}-k_{{\cal Y}}h_{{\cal X}}+k_{{\cal X}}k_{{\cal Z}}\\
& &+2[({\gamma}_2{\sigma})'-{\gamma}_2{\sigma}(k_{{\cal Y}}+(-1)^r\nu{\gamma})+{\gamma}_1{\sigma}k_{{\cal Z}}]\phi_r. \hspace{6cm}(4.59)\end{aligned}$$ Thus we have: $$2{{\alpha}'}'+2{\alpha}'f_{{\cal X}}+k_{{{\cal X}}t}+h_{{\cal X}}h_{{\cal Y}}-k_{{\cal X}}h_{{\cal Z}}=0,\eqno(4.60)$$ $$2({\beta}'\sin{\alpha})'+2{\beta}'f_{{\cal X}}\sin{\alpha}+h_{{{\cal X}}t}-k_{{\cal Y}}h_{{\cal X}}+k_{{\cal X}}k_{{\cal Z}}=0.\eqno(4.61)$$
For simplicity, we only consider two special cases a follows.
[*Case 1*]{}. ${\vartheta}_r=\zeta_r,\;{\sigma}=0,\;{\gamma}_1={\alpha}'\mu$ and ${\gamma}_2={\beta}'\mu\sin{\alpha}$.
In this case, $$k_{{\cal X}}=h_{{\cal X}}=0\eqno(4.62)$$ by (4.52). According to (4.60) and (4.61), we have $${\beta}'\sin{\alpha}=d{\alpha}',\qquad
f_{{\cal X}}=-\frac{{{\alpha}'}'}{{\alpha}'}.\eqno(4.63)$$ Moreover, (4.51) becomes $$k_{{\cal Y}}+dh_{{\cal Y}}=-\frac{(\mu{\alpha}')'}{\mu{\alpha}'},\qquad
k_{{\cal Z}}+dh_{{\cal Z}}=-d\frac{(\mu{\alpha}')'}{\mu{\alpha}'}.\eqno(4.64)$$ According to (4.40) and (4.54), $$h_{{\cal Z}}=\frac{{{\alpha}'}'}{{\alpha}'}-k_{{\cal Y}},\qquad
h_{{\cal Y}}=k_{{\cal Z}}-2{\beta}'\cos{\alpha}.\eqno(4.65)$$ Substituting (4.65) into (4.64), we obtain $$k_{{\cal Y}}+dk_{{\cal Z}}=2d{\beta}'\cos{\alpha}-\frac{(\mu{\alpha}')'}{\mu{\alpha}'},\qquad
k_{{\cal Z}}-dk_{{\cal Y}}=-d\frac{(\mu{\alpha}')'+{{\alpha}'}'\mu}{\mu{\alpha}'}.\eqno(4.66)$$
For convenience of computation, we write $$\mu=\frac{\sqrt{{\beta}_1'}}{{\alpha}'}{\Longrightarrow}{\gamma}=(1+d^2){\beta}_1',\;\;{\gamma}_1=
\sqrt{{\beta}_1},\;\;{\gamma}_2=d\sqrt{{\beta}_1}.\eqno(4.67)$$ From (4.66), $$k_{{\cal Y}}=\frac{1}{1+d^2}\left(2d^2{\alpha}'\cot{\alpha}+
(d^2-1)\frac{{{\beta}_1'}'}{2{\beta}_1'}+d^2\frac{{{\alpha}'}'}{{\alpha}'}\right),
\eqno(4.68)$$ $$k_{{\cal Z}}=\frac{d}{1+d^2}\left(2d^2{\alpha}'\cot{\alpha}-\frac{{{\beta}_1'}'}{{\beta}_1'}-\frac{{{\alpha}'}'}{{\alpha}'}\right).
\eqno(4.69)$$ By (4.65), $$h_{{\cal Z}}=\frac{1}{1+d^2}\left((1-d^2)\frac{{{\beta}_1'}'}{2{\beta}_1'}
+\frac{{{\alpha}'}'}{{\alpha}'}-2d^2{\alpha}'\cot{\alpha}\right),\eqno(4.70)$$ $$h_{{\cal Y}}=-\frac{d}{1+d^2}\left(2{\alpha}'\cot{\alpha}+\frac{{{\beta}_1'}'}{{\beta}_1'}+\frac{{{\alpha}'}'}{{\alpha}'}\right).
\eqno(4.71)$$ Furthermore, (4.57) becomes $$({\gamma}\tau)'+2{\gamma}\tau f_{{\cal X}}-(-1)^r\nu
{\gamma}^2\tau=0{\Longrightarrow}{\gamma}\tau=({\alpha}')^2e^{(-1)^r\nu(1+d^2){\beta}_1}.\eqno(4.72)$$ So $$\tau=\frac{({\alpha}')^2e^{(-1)^r\nu(1+d^2){\beta}_1}}
{(1+d^2){\beta}_1'}.\eqno(4.73)$$ Note that (4.58) and (4.59) are implied by $$(f_{{\cal X}}-(-1)^r{\gamma}\nu){\varphi}+{\varphi}'-{\alpha}_1{\gamma}\tau=0{\Longrightarrow}{\alpha}_1=\frac{{\alpha}'{\varphi}'-({{\alpha}'}'+(-1)^r\nu(1+d^2){\alpha}'{\beta}_1'){\varphi}}
{({\alpha}')^3e^{(-1)^r\nu(1+d^2){\beta}_1}}.\eqno(4.74)$$
Observe $$\begin{aligned}
\hspace{2cm}{{\cal U}}&=&-\frac{{{\alpha}'}'}{{\alpha}'}{{\cal X}}-{\alpha}'({{\cal Y}}+d{{\cal Z}})
+\frac{{\alpha}'{\varphi}'-({{\alpha}'}'+(-1)^r\nu(1+d^2){\alpha}'{\beta}_1'){\varphi}}
{({\alpha}')^3e^{(-1)^r\nu(1+d^2){\beta}_1}}\\
& &+({\varphi}-({\alpha}')^2e^{(-1)^r\nu(1+d^2){\beta}_1}
{{\cal X}})\zeta_r,\hspace{6.1cm}(4.75)\end{aligned}$$ $$\begin{aligned}
{{\cal V}}&=&
\frac{1}{1+d^2}\left(2d^2{\alpha}'({{\cal Y}}+d{{\cal Z}})\cot{\alpha}+
((d^2-1){{\cal Y}}-2d{{\cal Z}})\frac{{{\beta}_1'}'}{2{\beta}_1'}+(d^2{{\cal Y}}-d{{\cal Z}})\frac{{{\alpha}'}'}
{{\alpha}'}\right)\\
& &+{\alpha}'({{\cal X}}-d{{\cal Z}}\cot{\alpha})+\frac{({\alpha}')^2e^{(-1)^r\nu(1+d^2){\beta}_1}}
{(1+d^2)\sqrt{{\beta}_1'}}\xi_r,\hspace{5.8cm}(4.76)\end{aligned}$$ $$\begin{aligned}
{{\cal W}}&=&\frac{1}{1+d^2}\left(((1-d^2){{\cal Z}}-2d{{\cal Y}})\frac{{{\beta}_1'}'}{2{\beta}_1'}
+({{\cal Z}}-d{{\cal Y}})\frac{{{\alpha}'}'}{{\alpha}'}-2d({{\cal Y}}+d{{\cal Z}}){\alpha}'\cot{\alpha}\right)
\\ & &+d{\alpha}'({{\cal X}}+{{\cal Y}}\cot{\alpha})+\frac{d({\alpha}')^2e^{(-1)^r\nu(1+d^2){\beta}_1}}
{(1+d^2)\sqrt{{\beta}_1'}}\xi_r.\hspace{5.5cm}(4.77)\end{aligned}$$ Moreover, $$\begin{aligned}
\hspace{1cm}R_1&=&(4b{\delta}_{0,r}+c{\delta}_{1,r})c({\alpha}')^2e^{(-1)^r\nu(1+d^2){\beta}_1}[
({\alpha}')^2e^{(-1)^r\nu(1+d^2){\beta}_1}{{\cal X}}-{\varphi}]\\
&&-\frac{(2({{\alpha}'}')^2-{{{\alpha}'}'}'){{\cal X}}}{({\alpha}')^2}-\frac{2({\alpha}')^3e^{(-1)^r\nu(1+d^2){\beta}_1}}
{\sqrt{{\beta}_1'}}\xi_r-({\alpha}')^2(1+d^2){{\cal X}}\\ & &
+d(({\alpha}')^2\cot{\alpha}-{{\alpha}'}'){{\cal Z}}-({{\alpha}'}'+(d{\alpha}')^2\cot{\alpha}){{\cal Y}},\hspace{4cm}(4.78)\end{aligned}$$ $$\begin{aligned}
R_2&=&{\gamma}_1\left[\frac{{\alpha}'(2{{\alpha}'}'{\beta}_1'-{\alpha}'{{\beta}_1'}')e^{(-1)^r\nu(1+d^2){\beta}_1}}
{(1+d^2)({\beta}_1')^2}\xi_r+\frac{({\alpha}')^4e^{(-1)^r2\nu(1+d^2){\beta}_1}}
{(1+d^2){\beta}_1'}\xi_r\zeta_r\right]\\ &&
-({{\alpha}'}'+(d{\alpha}')^2\cot{\alpha}){{\cal X}}+[k_{{{\cal Y}}t}-({\alpha}')^2(1+d^2\csc^2{\alpha})]{{\cal Y}}+\frac{1}{2}(k^2+h^2)_{{\cal Y}}\\ & &+[(k_{{\cal Z}}-{\beta}'\cos{\alpha})'-d({\alpha}')^2]{{\cal Z}}+2{\alpha}'f^\ast
-2({\alpha}')^3e^{(-1)^r\nu(1+d^2){\beta}_1}\zeta_r{{\cal X}},
\hspace{1.8cm}(4.79)\end{aligned}$$ $$\begin{aligned}
R_3&=&{\gamma}_2\left[\frac{{\alpha}'(2{{\alpha}'}'{\beta}_1'-{\alpha}'{{\beta}_1'}')e^{(-1)^r\nu(1+d^2){\beta}_1}}
{(1+d^2)({\beta}_1')^2}\xi_r+\frac{({\alpha}')^4e^{(-1)^r2\nu(1+d^2){\beta}_1}}
{(1+d^2){\beta}_1'}\xi_r\zeta_r\right]\\ && +
d(({\alpha}')^2\csc{\alpha}-{{\alpha}'}'){{\cal X}}+((h_{{\cal Y}}+{\beta}'\cos{\alpha})'-d({\alpha}')^2){{\cal Y}}+\frac{1}{2}(k^2+h^2)_{{\cal Z}}\\
& &+(h_{{{\cal Z}}t}-(d{\alpha}')^2\csc^2{\alpha}){{\cal Z}}+2d{\alpha}'f^\ast
-2d({\alpha}')^3e^{(-1)^r\nu(1+d^2){\beta}_1}\zeta_r{{\cal X}}.
\hspace{2.4cm}(4.80)\end{aligned}$$ By (3.12), $$\begin{aligned}
& &p
=\rho\{\frac{2({\alpha}')^3e^{(-1)^r\nu(1+d^2){\beta}_1}\xi_r{{\cal X}}}
{\sqrt{{\beta}_1'}}+\frac{({\alpha}')^2}{2}({{\cal Y}}^2+d^2\csc^2{\alpha}({{\cal Y}}^2+{{\cal Z}}^2)+2d{{\cal Y}}{{\cal Z}})
-\frac{2{\alpha}'{\varphi}\zeta_r}{\sqrt{{\beta}_1'}}\\
& &+\frac{(2({{\alpha}'}')^2-{{{\alpha}'}'}'){{\cal X}}^2} {2({\alpha}')^2}
+\frac{{\alpha}'{{{\alpha}'}'}'-({{\alpha}'}')^2}{2(1+d^2){\alpha}'}
(2d{{\cal Y}}{{\cal Z}}-d^2{{\cal Y}}^2-{{\cal Z}}^2)
-\frac{({\alpha}')^4e^{(-1)^r2\nu(1+d^2){\beta}_1}\xi_r^2} {2(1+d^2){\beta}_1'}
\\ & &
+\frac{(-1)^r{\alpha}'({\alpha}'{{\beta}_1'}'-2{{\alpha}'}'{\beta}_1') \zeta_r}
{(1+d^2)({\beta}_1')^2e^{(-1)^{r+1}\nu(1+d^2){\beta}_1}} +
\frac{2{\alpha}'[({{\alpha}'}'+(-1)^r\nu(1+d^2){\alpha}'{\beta}_1'){\varphi}-{\alpha}'{\varphi}']({{\cal Y}}+d{{\cal Z}})}
{({\alpha}')^3e^{(-1)^r\nu(1+d^2){\beta}_1}}
\\& &-\frac{1}{2(1+d^2)^2}[\left(2d^2{\alpha}'({{\cal Y}}+d{{\cal Z}}\cot{\alpha})+
\frac{{{\beta}_1'}'((d^2-1){{\cal Y}}-2d{{\cal Z}})}{2{\beta}_1'}+\frac{{{\alpha}'}'(d^2{{\cal Y}}-d{{\cal Z}})}
{{\alpha}'}\right)^2\\& &+
\left(\frac{{{\beta}_1'}'((1-d^2){{\cal Z}}-2d{{\cal Y}})}{2{\beta}_1'}
+\frac{{{\alpha}'}'({{\cal Z}}-d{{\cal Y}})}{{\alpha}'}-2d({{\cal Y}}+d{{\cal Z}}){\alpha}'\cot{\alpha}\right)^2]-\frac{({\alpha}')^2(1+d^2){{\cal X}}^2}{2}
\\ &&+\frac{d}{1+d^2}({{\alpha}'}'\cot{\alpha}-({\alpha}')^2\csc^2{\alpha})((1-d^2){{\cal Y}}{{\cal Z}}+2d({{\cal Z}}^2-
{{\cal Y}}^2))
+\frac{{\beta}_1'{{{\beta}_1'}'}'-({{\beta}_1'}')^2} {4(1+d^2)({\beta}_1')^2}\\
& &\times(4d{{\cal Y}}{{\cal Z}}+(1-d^2)({{\cal Y}}^2-{{\cal Z}}^2))
+({{\alpha}'}'+(d{\alpha}')^2\cot{\alpha}){{\cal X}}{{\cal Y}}+d({{\alpha}'}'-({\alpha}')^2\cot{\alpha}){{\cal X}}{{\cal Z}}\hspace{4cm}\end{aligned}$$ $$\begin{aligned}
&
&+(4b{\delta}_{0,r}+c{\delta}_{1,r})c({\alpha}')^2e^{(-1)^r\nu(1+d^2){\beta}_1}[{\varphi}{{\cal X}}-
2^{-1}({\alpha}')^2e^{(-1)^r\nu(1+d^2){\beta}_1}{{\cal X}}^2 ]\} \hspace{2cm}(4.81)\end{aligned}$$ modulo the transformation in (1.10).
By (3.3) and (3.5), we have the following theorem:
[**Theorem 4.2**]{}. [*Let ${\alpha},{\beta}_1,{\varphi}$ be functions of $t$ and let $b,c,d$ be real constants. Denote $${\beta}=d\ln|\csc{\alpha}-\cot{\alpha}|\eqno(4.82)$$ (so the first equation in (4.63) holds). Define the moving frame ${{\cal X}},\;{{\cal Y}}$ and ${{\cal Z}}$ by (3.1) and (3.5), and $$\xi_0=be^{\sqrt{{\beta}_1}({{\cal Y}}+d{{\cal Z}})}-ce^{-\sqrt{{\beta}_1}({{\cal Y}}+d{{\cal Z}})},\qquad
\xi_1=c\sin[\sqrt{{\beta}_1}({{\cal Y}}+d{{\cal Z}})+b],\eqno(4.83)$$ $$\zeta_0=be^{\sqrt{{\beta}_1}({{\cal Y}}+d{{\cal Z}})}+ce^{-\sqrt{{\beta}_1}({{\cal Y}}+d{{\cal Z}})},\qquad
\zeta_1=c\cos[\sqrt{{\beta}_1}({{\cal Y}}+d{{\cal Z}})+b].\eqno(4.84)$$ For $r=0,1$, we have the following solution of the Navier-Stokes equations (1.1)-(1.4): $$\begin{aligned}
u&=&\left[\frac{{\alpha}'{\varphi}'-({{\alpha}'}'+(-1)^r\nu(1+d^2){\alpha}'{\beta}_1'){\varphi}}
{({\alpha}')^3e^{(-1)^r\nu(1+d^2){\beta}_1}}+({\varphi}-({\alpha}')^2e^{(-1)^r\nu(1+d^2){\beta}_1}{{\cal X}})
\zeta_r-\frac{{{\alpha}'}'}{{\alpha}'}{{\cal X}}\right]\cos{\alpha}\\& &-\frac{1}{1+d^2}\left(2d^2{\alpha}'({{\cal Y}}+d{{\cal Z}})\cot{\alpha}+
((d^2-1){{\cal Y}}-2d{{\cal Z}})\frac{{{\beta}_1'}'}{2{\beta}_1'}+(d^2{{\cal Y}}-d{{\cal Z}})\frac{{{\alpha}'}'}
{{\alpha}'}\right)\sin{\alpha}\\
& &+{\alpha}'({{\cal X}}\sin{\alpha}-({{\cal Y}}+2d{{\cal Z}})\cos{\alpha})
+\frac{({\alpha}')^2e^{(-1)^r\nu(1+d^2){\beta}_1}}
{(1+d^2)\sqrt{{\beta}_1'}}\xi_r\sin{\alpha},\hspace{2.7cm}(4.85)\end{aligned}$$ $$\begin{aligned}
v&=&\left[\frac{{\alpha}'{\varphi}'-({{\alpha}'}'+(-1)^r\nu(1+d^2){\alpha}'{\beta}_1'){\varphi}}
{({\alpha}')^3e^{(-1)^r\nu(1+d^2){\beta}_1}}+({\varphi}-({\alpha}')^2e^{(-1)^r\nu(1+d^2){\beta}_1}{{\cal X}})
\zeta_r\right]\sin{\alpha}\;\cos{\beta}\\& &+\frac{2d{\alpha}'(d\cos{\alpha}\;\cos{\beta}-\sin{\beta})({{\cal Y}}+d{{\cal Z}})\cot{\alpha}}
{1+d^2}+\frac{{{\alpha}'}'(d\cos{\alpha}\;\cos{\beta}+\sin{\beta})(d{{\cal Y}}-{{\cal Z}})}
{(1+d^2){\alpha}'}\\ & &
+\frac{[((d^2-1){{\cal Y}}-2d{{\cal Z}})\cos{\alpha}\;\cos{\beta}-((1-d^2){{\cal Z}}-2d{{\cal Y}})\sin{\beta}]{{\beta}_1'}'}
{2(1+d^2){\beta}_1'}-\frac{{{\alpha}'}'}{{\alpha}'}{{\cal X}}\sin{\alpha}\;\cos{\beta}\\ &
&+{\alpha}'{{\cal X}}(\cos{\alpha}\;\cos{\beta}-d\sin{\beta})-{\alpha}'{{\cal Y}}(\cos{\alpha}\;\cos{\beta}-d\cos{\alpha}\;\sin{\beta})\\
& &-d{\alpha}'{{\cal Z}}\csc{\alpha}\;\cos{\beta}+\frac{({\alpha}')^2e^{(-1)^r\nu(1+d^2){\beta}_1}(\cos{\alpha}\;\cos{\beta}-d\sin{\beta})}
{(1+d^2)\sqrt{{\beta}_1'}}\xi_r,\hspace{2.1cm}(4.86)\end{aligned}$$ $$\begin{aligned}
w&=&\left[\frac{{\alpha}'{\varphi}'-({{\alpha}'}'+(-1)^r\nu(1+d^2){\alpha}'{\beta}_1'){\varphi}}
{({\alpha}')^3e^{(-1)^r\nu(1+d^2){\beta}_1}}+({\varphi}-({\alpha}')^2e^{(-1)^r\nu(1+d^2){\beta}_1}{{\cal X}})
\zeta_r\right]\sin{\alpha}\;\sin{\beta}\\& &+\frac{2d{\alpha}'(d\cos{\alpha}\;\sin{\beta}+\cos{\beta})({{\cal Y}}+d{{\cal Z}})\cot{\alpha}}
{1+d^2}+\frac{{{\alpha}'}'(d\cos{\alpha}\;\sin{\beta}-\cos{\beta})(d{{\cal Y}}-{{\cal Z}})}
{(1+d^2){\alpha}'}\\ & &
+\frac{[((d^2-1){{\cal Y}}-2d{{\cal Z}})\cos{\alpha}\;\sin{\beta}+((1-d^2){{\cal Z}}-2d{{\cal Y}})\cos{\beta}]{{\beta}_1'}'}
{2(1+d^2){\beta}_1'}-\frac{{{\alpha}'}'}{{\alpha}'}{{\cal X}}\sin{\alpha}\;\sin{\beta}\\ &
&+{\alpha}'{{\cal X}}(\cos{\alpha}\;\sin{\beta}+d\cos{\beta})-{\alpha}'{{\cal Y}}(\cos{\alpha}\;\sin{\beta}+d\cos{\alpha}\;\cos{\beta})\\
& &-d{\alpha}'{{\cal Z}}\csc{\alpha}\;\sin{\beta}+\frac{({\alpha}')^2e^{(-1)^r\nu(1+d^2){\beta}_1}(\cos{\alpha}\;\sin{\beta}+d\cos{\beta})}
{(1+d^2)\sqrt{{\beta}_1'}}\xi_r\hspace{2.2cm}(4.87)\end{aligned}$$ and $p$ is given in (4.81).*]{}
[*Case 2*]{}. ${\gamma}_2={\alpha}_1=0,\;({\gamma}\tau)'+2{\gamma}\tau f_{{\cal X}}-(-1)^r\nu
{\gamma}^2\tau=0$ and ${\gamma}_1\neq 0$.
According to (4.51) and (4.54), $$k_{{\cal Y}}=-\frac{{\gamma}_1'}{{\gamma}_1},\qquad k_{{\cal Z}}=0,\qquad
h_{{\cal Y}}=-2{\beta}'\cos{\alpha}.\eqno(4.88)$$ Note ${\gamma}={\gamma}_1^2$. Moreover, (4.52) says $$k_{{\cal X}}=0,\qquad h_{{\cal X}}=-2{\beta}'\sin{\alpha}.\eqno(4.89)$$ Furthermore, (4.61) yields $$f_{{\cal X}}=\frac{{\gamma}_1'}{{\gamma}_1}.\eqno(4.90)$$ Besides, (4.40) implies $$h_{{\cal Z}}=-(f_{{\cal X}}+k_{{\cal Y}})=0.\eqno(4.91)$$ Under the condition (4.60) and (4.61), (4.58) and (4.59) are equivalent to $$f_{{\cal X}}=k_{{\cal Y}}={\gamma}_1'=0,\qquad -(-1)^r{\gamma}\nu{\varphi}+{\varphi}'=0.\eqno(4.92)$$
Write ${\gamma}_1=a_1$ as a real constant. We have: $${\sigma}=a_2e^{(-1)^r\nu a_1^2t},\qquad
{\varphi}=a_1e^{(-1)^r\nu a_1^2t},\qquad\tau =a_1^{-1}e^{(-1)^r\nu
a_1^2t} \eqno(4.93)$$ for $a_2\in{\mathbb}{R}$ (cf. (4.47)). By (4.60) $${\beta}'=\pm
\sqrt{\frac{-{{\alpha}'}'}{\sin2{\alpha}}},\eqno(4.94)$$ that is, $${\beta}=\pm\int
\sqrt{\frac{-{{\alpha}'}'}{\sin2{\alpha}}}\;dt.\eqno(4.95)$$ Thus $${{\cal U}}=-{\alpha}'{{\cal Y}}-{\beta}'{{\cal Z}}\sin{\alpha}+a_1e^{(-1)^r\nu
a_1^2t}({\vartheta}_r+a_2{{\cal Y}}\phi_r-\zeta_r{{\cal X}}-a_1a_2\psi_r{{\cal X}}^2),\eqno(4.96)$$ $${{\cal V}}={\alpha}'{{\cal X}}-{\beta}'{{\cal Z}}\cos{\alpha}+e^{(-1)^r\nu
a_1^2t}(\xi_r+2a_1a_2\phi_r{{\cal X}}),\qquad{{\cal W}}=-{\beta}'({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha}).
\eqno(4.97)$$ Moreover, $$\begin{aligned}
&&
R_1=-\frac{{{\alpha}'}'}{2}{{\cal Y}}+a_1^2e^{(-1)^r2\nu
a^2_1t}[(4b{\delta}_{0,r}+c{\delta}_{1,r})c{{\cal X}}+3a_2(2{\delta}_{0,r}(ab+c)+
{\delta}_{1,r}c\cos b){{\cal X}}^2\\ & &+2a_1^2a_2^2
(4a{\delta}_{0,r}+{\delta}_{1,r}){{\cal X}}^3
+a_2(2{\delta}_{0,r}(ab-c)+{\delta}_{1,r}c\sin b){{\cal Y}}+
{\delta}_{1,r}cc_1\sin(b-b_1)\\
& &+2{\delta}_{0,r}(bc_1-b_1c)+a_1a_2(2{\delta}_{0,r}(c_1-ab_1)-
{\delta}_{1,r}c_1\sin b_1){{\cal X}}] +({\alpha}'{\beta}'\cos{\alpha}-({\beta}'\sin{\alpha})'){{\cal Z}}\\ & &-(({\alpha}')^2+({\beta}'\sin{\alpha})^2){{\cal X}}+ e^{(-1)^r\nu
a^2_1t}(a_1a_2\phi-2{\alpha}') ) (\xi_r+2a_1a_2e^{(-1)^r\nu
a^2_1t}\phi_r{{\cal X}}),\hspace{0.8cm}(4.98)\end{aligned}$$ $$\begin{aligned}
R_2&=&2a_1e^{(-1)^r\nu
a_1^2t}[({\vartheta}_r+a_2{{\cal Y}}\phi_r)({\alpha}'+a_1a_2e^{(-1)^r\nu
a_1^2t}\phi_r)-{\alpha}'(\zeta_r{{\cal X}}+a_1a_2\psi_r{{\cal X}}^2)]\\ & &
-\frac{{{\alpha}'}'}{2}{{\cal X}}+(3({\beta}')^2\cos^2{\alpha}-({\alpha}')^2){{\cal Y}}-(({\beta}'\cos{\alpha})'+{\alpha}'{\beta}'\sin{\alpha}){{\cal Z}}\\&& +a_1e^{(-1)^r2\nu a_1^2t}(
\xi_r\zeta_r+2a_1a_2\xi_r\psi_r{{\cal X}}+2a_1^2a_2^2\phi_r\psi_r{{\cal X}}^2)
,\hspace{4.2cm}(4.99)\end{aligned}$$ $$R_3=({\alpha}'{\beta}'\cos{\alpha}-({\beta}'\sin{\alpha})'){{\cal X}}-(({\beta}'\cos{\alpha})'+{\alpha}'{\beta}'\sin{\alpha}){{\cal Y}}+{{\alpha}'}'{{\cal Z}}\csc2{\alpha}.\eqno(4.100)$$ By (3.12), $$\begin{aligned}
&&p
=\rho\{\frac{{{\alpha}'}'{{\cal X}}{{\cal Y}}}{2}-a_1^2e^{(-1)^r2\nu
a^2_1t}[\frac{(4b{\delta}_{0,r}+c{\delta}_{1,r})c{{\cal X}}^2}{2}
+a_2(2{\delta}_{0,r}(ab+c)+ {\delta}_{1,r}c\cos b){{\cal X}}^3\\ & &+a_1^2a_2^2
(4a{\delta}_{0,r}+{\delta}_{1,r})\frac{{{\cal X}}^4}{2}
+a_2(2{\delta}_{0,r}(ab-c)+{\delta}_{1,r}c\sin b){{\cal X}}{{\cal Y}}+
({\delta}_{1,r}cc_1\sin(b-b_1)\\
& &+2{\delta}_{0,r}(bc_1-b_1c)){{\cal X}}+\frac{a_1a_2(2{\delta}_{0,r}(c_1-ab_1)-
{\delta}_{1,r}c_1\sin b_1){{\cal X}}^2}{2}]
+\frac{(({\alpha}')^2+({\beta}'\sin{\alpha})^2){{\cal X}}^2}{2}\\
& &+(({\beta}'\sin{\alpha})'-{\alpha}'{\beta}'\cos{\alpha}){{\cal X}}{{\cal Z}}- e^{(-1)^r\nu
a^2_1t}(a_1a_2\phi-2{\alpha}')(\xi_r{{\cal X}}+a_1a_2e^{(-1)^r\nu
a^2_1t}\phi_r{{\cal X}}^2)\\& & - (-1)^re^{(-1)^r\nu a_1^2t}[
a_2e^{(-1)^r\nu
a_1^2t}[a_1a_2{{\cal Y}}\phi_r\psi_r-2^{-1}a_2(a_1(4a{\delta}_{0,1}+{\delta}_{1,r}){{\cal Y}}^2+
\phi_r^2)\\ &
&+{\vartheta}_r\zeta_r-({\delta}_{0,1}2(ab_1+c_1)+{\delta}_{1,r}c_1){{\cal Y}}]+{\alpha}'({\varepsilon}_r+a_2{{\cal Y}}\psi_r-a_1^{-1}a_2\phi_r)]-\frac{e^{(-1)^r2\nu
a_1^2t} \xi_r^2}{2}\\ & &
-\frac{(3({\beta}')^2\cos^2{\alpha}-({\alpha}')^2){{\cal Y}}^2}{2}
+(({\beta}'\cos{\alpha})'+{\alpha}'{\beta}'\sin{\alpha}){{\cal Y}}{{\cal Z}}-\frac{{{\alpha}'}'{{\cal Z}}^2\csc2{\alpha}}{2}
\}\hspace{0.9cm}(4.101)\end{aligned}$$ where modulo the transformation in (1.10), where $${\varepsilon}_0=b_1e^{{\tilde}\varpi}+c_1e^{-{\tilde}\varpi},\qquad
{\varepsilon}_1=c_1\cos({\tilde}\varpi+b_1)\eqno(4.102)$$ in connection with (4.56) and ${\tilde}\varpi=a_1{{\cal Y}}$.
By (3.3) and (3.5), we have the following theorem:
[**Theorem 4.3**]{}. [*Let ${\alpha}$ be a function of $t$ and let $a,a_1,a_2,b,b_1,c,c_1$ be real constants. Denote ${\beta}$ as in (4.91). Define the moving frame ${{\cal X}},\;{{\cal Y}}$ and ${{\cal Z}}$ by (3.1) and (3.5), and $$\phi_0=
e^{a_1{{\cal Y}}}-ae^{-a_1{{\cal Y}}},\;\;\phi_1=\sin(a_1{{\cal Y}}),\;\;\psi_0=
e^{a_1{{\cal Y}}}+ae^{-a_1{{\cal Y}}},\;\; \psi_1=\cos(a_1{{\cal Y}}),\eqno(4.103)$$ $$\xi_0=
be^{a_1{{\cal Y}}}-ce^{-a_1{{\cal Y}}},\;\;\xi_1=c\sin(a_1{{\cal Y}}+b),\;\;\zeta_0=
be^{a_1{{\cal Y}}}+ce^{-a_1{{\cal Y}}},\eqno(4.104)$$ $$\zeta_1=c\cos(a_1{{\cal Y}}+b),
\;\;{\vartheta}_0=
b_1e^{a_1{{\cal Y}}}-c_1e^{-a_1{{\cal Y}}},\;\;{\vartheta}_1=c_1\sin(a_1{{\cal Y}}+b_1),\eqno(4.105)$$ $${\varepsilon}_0=
b_1e^{a_1{{\cal Y}}}+c_1e^{-a_1{{\cal Y}}},\qquad{\varepsilon}_1=c_1\cos(a_1{{\cal Y}}+b_1).\eqno(4.106)$$ For $r=0,1$, we have the following solution of Navier-Stokes equations (1.1)-(1.4): $$\begin{aligned}
\hspace{1cm}u&=&
[-{\alpha}'{{\cal Y}}+a_1e^{(-1)^r\nu
a_1^2t}({\vartheta}_r+a_2{{\cal Y}}\phi_r-\zeta_r{{\cal X}}-a_1a_2\psi_r{{\cal X}}^2)]\cos{\alpha}\\ & &-[{\alpha}'{{\cal X}}+e^{(-1)^r\nu
a_1^2t}(\xi_r+2a_1a_2\phi_r{{\cal X}})]\sin{\alpha},\hspace{5cm}(4.107)\end{aligned}$$ $$\begin{aligned}
& &v=
[-{\alpha}'{{\cal Y}}+a_1e^{(-1)^r\nu
a_1^2t}({\vartheta}_r+a_2{{\cal Y}}\phi_r-\zeta_r{{\cal X}}-a_1a_2\psi_r{{\cal X}}^2)]\sin{\alpha}\;\cos{\beta}-{\beta}'{{\cal Z}}\cos{\beta}\\ & &-[{\alpha}'{{\cal X}}+e^{(-1)^r\nu
a_1^2t}(\xi_r+2a_1a_2\phi_r{{\cal X}})]\cos{\alpha}\;\cos{\beta}+{\beta}'({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha})\sin{\beta},\hspace{0.4cm}(4.108)\end{aligned}$$ $$\begin{aligned}
& &w=
[-{\alpha}'{{\cal Y}}+a_1e^{(-1)^r\nu
a_1^2t}({\vartheta}_r+a_2{{\cal Y}}\phi_r-\zeta_r{{\cal X}}-a_1a_2\psi_r{{\cal X}}^2)]\sin{\alpha}\;\sin{\beta}-{\beta}'{{\cal Z}}\sin{\beta}\\ & &-[{\alpha}'{{\cal X}}+e^{(-1)^r\nu
a_1^2t}(\xi_r+2a_1a_2\phi_r{{\cal X}})]\cos{\alpha}\;\sin{\beta}-{\beta}'({{\cal X}}\sin{\alpha}+{{\cal Y}}\cos{\alpha})\cos{\beta}\hspace{0.6cm}(4.109)\end{aligned}$$ and $p$ is given in (4.101).*]{}
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[^1]: 2000 Mathematical Subject Classification. Primary 35C05, 35Q35; Secondary 35C10, 35C15.
[^2]: Research supported by China NSF 10431040
|
{
"pile_set_name": "ArXiv"
}
|
[**SU(N) gauge theories in 2+1 dimensions:\
glueball spectra and k-string tensions**]{}\
[**Abstract**]{}
We calculate the low-lying glueball spectrum and various string tensions in $SU(N)$ lattice gauge theories in $2+1$ dimensions, and extrapolate the results to the continuum limit. We do so for for the range $N\in[2,16]$ so as to control the $N$-dependence with a useful precision. We observe a number of striking near-degeneracies in the various $J^{PC}$ sectors of the glueball spectrum, in particular between $C=+$ and $C=-$ states. We calculate the string tensions of flux tubes in a number of representations, and provide evidence that the leading correction to the $N$-dependence of the $k$-string tensions is $\propto 1/N$ rather than $\propto 1/N^2$, and that the dominant binding of $k$ fundamental flux tubes into a $k$-string is via pairwise interactions. We comment on the possible implications of our results for the dynamics of these gauge theories.
Introduction {#section_intro}
============
In this paper on $SU(N)$ gauge theories in $2+1$ dimensions we provide a calculation of the light glueball spectrum and of the confining string tension for fluxes in various representations. This work improves significantly upon earlier work [@MT98_SUN; @HMMT_spin; @BLMT02_glue]; in particular, we cover a wider range of $N$, from $N=2$ to $N=16$, and we calculate the continuum limit of more excited states, by using a much larger basis of operators. And we do all this with greater accuracy than before. Since one of the motivations of such calculations is to look for interesting regularities in the spectrum, one needs accuracy and many states. Another motivation is to provide a testbed for analytic methods or models, e.g. closed flux tube [@NIJP] or constituent gluon [@FB13] models, and this makes similar demands.
In Section \[section\_lattice\_energies\] we begin with a brief description of the lattice setup and outline how we calculate the masses of glueballs as well as the energies of flux tubes from which we extract string tensions. We also list the main systematic errors one needs to be aware of. We then move on, in Section \[section\_tensions\], to the results of our string tension calculations and, in Section \[section\_glueballs\], to our results for the glueball spectrum. Our calculations point to a number of interesting regularities which we discuss in the concluding section of the paper.
Our results for the glueball spectrum in this paper are largely consistent with the earlier work in [@MT98_SUN; @HMMT_spin; @BLMT02_glue] but are much more precise and extensive. Some of our preliminary results appeared in [@AARLMT_SONSUN] where it was shown how a comparison of $SO(N)$ and $SU(N)$ gauge theories can provide an explanation for the very weak $N$-dependence of a number of glueball mass ratios. While the spectrum in this paper supersedes that quoted in [@AARLMT_SONSUN], none of the conclusions of that paper are affected by the changes.
Calculating energies on a lattice {#section_lattice_energies}
=================================
Here we outline how our lattice calculations are performed. The methods we use are standard and so we shall be brief.
lattice setup {#section_latticesetup}
-------------
Our lattice field variables are $SU(N)$ matrices, $U_l$, residing on the links $l$ of the periodic $L^2_s L_t$ lattice, with lattice spacing $a$. The Euclidean path integral is $$Z=\int {\cal{D}}U \exp\{- \beta S[U]\}\,,
\label{eqn_Z}$$ where ${\cal{D}}U$ is the Haar measure over all the lattice link matrices and we use the standard plaquette action, $$\beta S = \beta \sum_p \left\{1-\frac{1}{N} {\text{ReTr}} U_p\right\}
\quad ; \quad \beta=\frac{2N}{ag^2}.
\label{eqn_S}$$ Here $U_p$ is the ordered product of link matrices around the plaquette $p$. We write $\beta=2N/ag^2$, where $g^2$ has dimensions of mass and this becomes the continuum coupling when $a\to 0$. Monte Carlo calculations are performed using a standard Cabibbo-Marinari heat bath plus over-relaxation algorithm. (See e.g. [@MT98_SUN] for more details.)
calculating glueball masses {#subsection_glue_calc}
---------------------------
The quantum numbers of our particle (‘glueball’) states are as follows [@MT98_SUN; @HMMT_spin] (We ignore momentum since we always take $p=0$). In continuum $D=2+1$ the rotation group is Abelian and the spins are $J=0, \pm1,\pm2,...$. (More exotic values — anyons – are possible in two spatial dimensions, but we shall assume that they do not occur in $SU(N)$ gauge theories.) Parity, $P$, does not commute with rotations: $J \stackrel{P}{\to} -J$. So particle (‘glueball’) states can be labelled by parity $P=\pm$ and $|J|$ as well as charge conjugation $C$. (For notational simplicity we shall use the label $J^{PC}$ rather than the more correct $|J|^{PC}$ in this paper.) Now, for $J\neq 0$ the state $|J\rangle$ will have an orthogonal partner $|-J\rangle$ that is degenerate with it. So the states $\propto \{|J\rangle \pm |-J\rangle|\}$, which are degenerate and non-null, have parities $P=\pm$ respectively. That is to say we have the well known parity doubling for states with $J\neq 0$ in 2 space dimensions $$M_{JC}^{P=+} = M_{JC}^{P=-}\,, \qquad J\neq 0\,.
\label{eqn_Pdoubling}$$ For $J=0$ the argument breaks down because the $P=-$ state can clearly be null. We also recall [@MT98_SUN; @HMMT_spin]. that on a square lattice, where the full rotation group is broken down to the group of rotations by $\pi/2$, states with $J^P= 1^+, 1^-$ (and odd spins in general) will still be degenerate, while states with $J^P= 2^+, 2^-$ (and even non-zero spins in general) need not be – although any non-degeneracy is due to lattice spacing correction and so will usually be very small. Note also that the finite volume periodic boundary conditions may also break the $2^{P=\pm}$ degeneracy, even in the continuum limit, but not that of odd spins.
The fact that on a square lattice we have exact rotational invariance only under rotations of $\pi/2$ implies that states that tend to continuum states with $J=0$ or $J=4$ or $J=8$,... will all fall into the same representation of the lattice rotation group, and similarly for $J=2,6,...$ and for $J=1,3,...$. For simplicity, we shall label states belonging to these representations as $J=0$, $J=2$ and $J=1$. What are the real spins of these states has been investigated in [@HMMT_spin] and we shall refer to the detailed implications of that work for our results later on. For now we merely remark that in some channels the actual ground states are not those with the lowest spin $J$.
Ground state masses $M$ are calculated from the asymptotic time dependence of correlators, i.e. $$\langle \phi^\dagger(t) \phi(0) \rangle
= \sum_n |\langle vac|\phi^\dagger|n \rangle|^2 e^{-E_n t}
\stackrel{t\to\infty}{\propto} e^{-Mt}\,,
\label{eqn_M}$$ where $M$ is the mass of the lightest state with the quantum numbers of the operator $\phi$. (If $\phi$ has the quantum numbers of the vacuum then we use vacuum-subtracted operator $\phi - \langle \phi \rangle$, so as to remove the contribution of the vacuum in eqn(\[eqn\_M\]).) The operator $\phi$ will be the product of $SU(N)$ link matrices around some contractible closed path, with the trace then taken. We will use zero momentum operators, $p=0$, so that there is no momentum integral on the right side of eqn(\[eqn\_M\]). To calculate the excited states $E_n$ in eqn(\[eqn\_M\]), one starts with a number of operators, $\phi_i$, and calculates all their (cross-)correlators $\langle \phi_i^\dagger(t) \phi_j(0) \rangle$. One uses this as the basis for a systematic variational calculation in $e^{-Ht_0}$ where $H$ is the Hamiltonian (corresponding to our lattice transfer matrix) and $t_0$ is some convenient distance. Typically we choose $t_0=a$. The procedure produces a set of orthonormal operators $\psi_i$ that are our best variational estimates for the true eigenstates. We then improve the energy estimate by using eqn(\[eqn\_M\]) with the correlators $C_i(t)=\langle \psi_i^\dagger(t) \psi_i(0) \rangle$. With these operators we can hope to have good overlaps onto the desired states, so that one can evaluate their masses at values of $t$ where the signal has not yet disappeared into the statistical noise. To achieve this in practice, we use blocked and smeared operators (for details see e.g. [@MT98_SUN; @BLMTUW_ops]) and a large number of such operators – in our case $O(100)$ for each set of quantum numbers. Some examples of the basic loops we employ are displayed in Table \[table\_glueloops\].
An important aside. The accuracy of such a calculation is constrained by the fact that the statistical errors are roughly independent of $t$ (for pure gauge theories) while the desired ‘signal’ is decreasing exponentially with $t$. So if the overlap of the desired state onto our basis is small, the relevant correlator will disappear into the statistical ‘noise’ before we get to large enough $t$ for the correlator to be dominated by a simple exponential. Similarly the correlator of a more massive state will disappear into the ‘noise’ at smaller $t=an_t$ and this may make ambiguous the judgement of whether it is dominated by a single exponential. All this may provide an important source of systematic error.
We extract the energy of the state by going to vale of $t=an_t$ that is large enough that $C_i(t=an_t) \propto \exp\{-aE_in_t\}$ within our errors. For illustrative purposes it is convenient to define the effective energy $$E_{eff,i}(n_t) = -\ln \frac{C_i(n_t)}{C_i(n_t-1)}.
\label{eqn_Eeff}$$ If $\tilde{t}=a\tilde{n}_t$ is the lowest value of $t$ for which $E_{eff,i}(n_t)$ is independent of $t\geq\tilde{t}$ within our errors, then we can say that a single exponential dominates and $E_{eff,i}(\tilde{t})$ provides an estimate of the energy $E_i$ of the state. (In practice we extract our $E_i$ from a fit over a range of $t$.) That is to say, we search for a ‘plateau’ in the values of $E_{eff}(n_t)$ against $n_t$. As a concrete example we show in Fig.\[fig\_meff\_JPC\_su6\] the values of $E_{eff,i}(n_t)$ for our six lightest $0^{++}$ glueballs in a calculation in $SU(6)$ on a $60^3$ lattice at $\beta=206.84$, which corresponds to our smallest value of $a$. As we can see, the overlaps are not far from $\sim 100\%$ so that it is easy to identify a plateau in $E_{eff}$ even for the highly excited states. In Fig.\[fig\_meff\_JPC\_su8\] we show the ground states for various quantum numbers, as obtained in $SU(8)$ at a comparably small value of $a$. Here the overlaps of some higher excited states are not quite so good, but an $E_{eff}$ plateau can still be plausibly identified. All this to indicate that while the mass estimates given in this paper should be largely reliable, this is less certain for the most massive states.
calculating flux tube energies {#subsection_flux_calc}
------------------------------
We calculate the string tension by calculating the energy of a flux tube of length $l$ that closes on itself by winding once around a spatial torus of size $l$. We use exactly the same technique as for the glueballs, except that the operator $\phi$ is now the product of $SU(N)$ link matrices taken around a non-contractible closed path that winds once around the spatial torus. The simplest such operator is the spatial Polyakov loop $$\phi(n_t) = l_p(n_t) = \sum_{n_y} \mathrm{Tr}
\left\{\prod^{L_x}_{n_x=1} U_x(n_x,n_y,n_t)\right\} \,,
\label{eqn_poly}$$ where we take the product of the link matrices in the $x$-direction around the $x$-torus of length $l=aL_x$. Here $(x,y,t)=(an_x,an_y,an_t)$, and we sum over $n_y$ to produce an operator with zero transverse momentum, $p_\perp = p_y = 0$. (We only consider flux tubes with zero transverse momentum, since we do not expect to learn anything qualitatively new from $p_\perp \neq 0$.) In addition to this simple straight-line operator we construct other winding operators with a variety of kinks and loops extending from the original straight line and we also use smeared and blocked operators just as for the glueballs. Using this large basis of operators we apply a variational procedure as for the glueballs, and this allows us to calculate not only the ground state energy but also excitation energies of the flux tube. We can label the flux tubes states by their transverse parities and their longitudinal momenta, and by their longitudinal parities if the momentum is zero. We can do all this for flux tubes carrying flux in any representation ${\cal{R}}$ by taking the trace in eqn(\[eqn\_poly\]) in that representation, i.e. by using $\mathrm{Tr}_{\cal{R}}$. All this is described in detail in for example [@BLMT01_strings; @AABBMT11_strings].
Since we will focus in this paper on the string tension rather than on the full flux tube spectrum [@AABBMT11_strings] we only need to calculate the ground state energy, $E_{gs}(l)$, of the flux tube. We extract the string tension $\sigma$ from $E_{gs}(l)$ using the ‘Nambu-Goto’ formula $$E_{gs}(l)
\stackrel{NG}{=}
\sigma l \left(1-\frac{\pi}{3\sigma l^2}\right)^{1/2}\,,
\label{eqn_gsNG}$$ which arises from the light-cone quantisation of the bosonic string and is known to provide an excellent approximation to the lattice calculations [@AABBMT11_strings; @AAMT16_strings] for reasons that have now become well understood [@OA] (see also [@SD_long]). We calculate in this way not only the fundamental string tension, $\sigma_f$, but also string tensions, $\sigma_{\cal{R}}$, for the flux in a number of other representations ${\cal{R}}$ as discussed in Section \[section\_tensions\].
The reliability of such calculations depends on how well we can calculate $E_{gs}(l)$. As ${\cal{R}}$ becomes larger, so does the corresponding ground state energy $E_{gs,{\cal{R}}}(l)$ which makes the calculation less reliable. To give some indication of the uncertainties involved we show in Fig.\[fig\_Ekeff\_su8\] the effective energies, as defined in eqn(\[eqn\_Eeff\]), for the representations of interest in this paper. We do this for the case of $SU(8)$ at our smallest lattice spacing. We also show the estimates of $E_{gs,{\cal{R}}}(l)$ that we obtain from these effective energies. We see that the identification of the effective energy ‘plateau’ appears to be unambiguous in all cases.
ambiguities and systematic errors {#subsection_sys_errors}
---------------------------------
A calculation of the glueball mass spectrum is subject to a number of systematic errors that one needs to control. (For flux tubes this is also the case, but because we are only interested in their ground states here, they are less important.) We discuss in this section our control over some of these systematic errors.
### missing states {#subsubsection_missing_states}
Our variational bases are, of course, incomplete and as we calculate ever higher excited states we will, at some point, begin to miss states. As it happens we have two sets of operators that were independently produced and which differ substantially, and comparing the spectra obtained using these bases provides some check on whether we are in fact missing any states in the range of interest to us here. One set of operators was used for our $N=3,6$ calculations, while the other was used for $N=2,4,8,12,16$. However we have also performed some extra calculations to compare the results obtained using the two bases. One comparison was in $SU(4)$ at $\beta=51$ on a $40^248$ lattice. Another was in $SU(6)$, comparing the spectrum on a $54^260$ lattice at $\beta=206.0$ with that from the second basis on a $60^3$ lattice at $\beta=206.84$. This latter comparison has a much smaller lattice spacing and hence a more reliable estimate of the masses of the heavier states, although the differing lattice sizes and values of $\beta$ make it less than ideal. (But since to leading order $aM \propto 1/\beta$, the expected shift in the masses should be a negligible $\approx 0.4\%$.) We compare the spectra obtained with the two bases of operators in Tables \[table\_msu4\_opcomp\], \[table\_msu6\_opcomp\]. We include for each set of quantum numbers as many states as we will eventually include in our continuum extrapolations. We see that the spectra are broadly consistent within errors. (This provides us with some confidence that we are not missing any states in the range of energies being considered. (The consistency of the spectra obtained with the two bases is of course a prerequisite for the large $N$ extrapolations we perform later in this paper.)
### multi-glueball and unstable states {#subsubsection_multiglueballs}
The full glueball spectrum will contain multi-glueball states. Such states are represented by multi-trace operators and so as $N$ increases the overlap of such a state onto our basis of single trace operators will vanish by standard large-$N$ counting. However at modest values of $N$ such states may be present. Here two $J^{PC}=0^{++}$ glueballs with angular momentum $J$ can produce $|J|^{\pm +}$ states (with $P=-$ for $J\neq 0$). Similarly $|J|^{\pm -}$ states can arise from a $0^{++}$ and $0^{--}$ glueball with angular momentum $J$. Some of our more massive states certainly fall into this ‘dangerous’ mass range. However the overlap suppression tells us that such a state will manifest itself through an effective energy that is much larger at small $t$ and then decreases, asymptoting to $2m_{0^{++}}$ or $m_{0^{++}}+m_{0^{--}}$ at large $t$. Since we order our states by the value of $E_{eff}(n_t)$ at $t=a$, it is unlikely that any such states would appear in our energy range, and indeed we see no obvious sign of any such states.
A closely related issue concerns genuine single glueball states that are heavy enough to decay into lighter glueballs. Again the decay width will vanish as $N\to\infty$ and at moderate $N$ we expect the state to resemble a narrow resonance. So the effective energy should have something like an approximate plateau at small $t$, but at large $t$ it will drift down to the threshold value of its decay products. However because the error/signal ratio grows exponentially with $t$, this large-$t$ behaviour is unlikely to be visible and the state will appear just like any stable state. Of course things may be different at small $N$. To see what happens at smaller $N$ we turn to our $SU(2)$ calculation, where the decay widths should be largest, and to our smallest lattice spacing, which corresponds to $\beta=30.0$, performed on a $120^290$ lattice, We show in Fig. \[fig\_meff\_2pp\_su2\] the effective energies of the lightest few $2^{++}$ states. The horizontal line corresponds to the threshold energy of the potential decay products, i.e. $aE_{th}=2am_{0^{++}} \simeq 0.43$. We see that while several of the states are heavier than this theshold energy, there is no sign, within our errors, of a drift in the effective mass towards this threshold energy as $t$ increases. There is of course a decrease at small $t$, as the higher excited state contributions to the correlator die away, but that does not persist to larger $t$, as one would expect it to do if the decay width was substantial.
All this encourages us to conclude that any systemtic errors due to multiglueball states or the instability of the states we consider are unlikely to be substantial in the calculations of this paper.
### finite V corrections {#subsubsection_finiteV}
The leading finite volume correction, at large volumes, to the mass of a glueball comes from the emission of the lightest glueball, of mass $m$, which then winds around the spatial torus of length $l$, before being reabsorbed by the glueball. So these contributions are typically $\propto \exp(-ml)$ and they are suppressed at large $N$ because the triple glueball coupling is $g^2_{GGG} \sim 1/N$. Now for our calculations in this paper $ml$ varies between $\sim 25$ at small $N$ and $\sim 12$ at our largest values of $N$, so we can assume that these leading finite volume effects are completely insignificant.
In practice, as is well known, the important finite volume corrections are quite different and arise from the presence of finite volume states composed of, for example, a winding flux tube together with a conjugate winding flux tube. These states are described by double-trace operators so their overlap onto our basis of single-trace glueball operators will vanish at large $N$ but is known to be substantial at small values of $N$. Now, when both flux tubes are in their ground states, the energy of such a ‘winding’ state is $E_{T} \sim 2\sigma_f l$, and this can contribute to both $0^{++}$ and $2^{++}$. For $J=1$ or $P=-$ or $C=-$, one needs to have one or both of the flux tubes in a suitable excited state and this has a much larger energy. Our strategy to control these finite volume corrections is therefore the very simple one of making $l$ large enough at small $N$ that all the $0^{++}$ and $2^{++}$ glueball states that we consider have energies no greater than $E_T$. We then rely on the large-$N$ suppression of the overlaps to allow ourselves smaller values of $l$ at larger $N$. (A decreasing overlap means higher effective masses at small $t$, pushing the state out of our range of masses, which is determined by the effective mass at $t=a$.) Our specific choices are listed in Table \[table\_sizeN\]. Of course one needs to ask whether the large-$N$ suppression is actually sufficient to eliminate such extra finite volume states at our ‘large’ values of $N$. This can only be answered convincingly by explicit finite volume studies. So in Table \[table\_msu8\_Vcomp\] we compare the spectrum obtained at $\beta=306.25$ on a $44^248$ lattice (our standard size) with the spectrum on a much larger $60^248$ lattice. Any finite volume state should be apparent from a large upward shift in its mass as we go to the larger lattice. As we see in Table \[table\_msu8\_Vcomp\] the spectra are consistent, with no sign of any finite volume states. The smallest volumes we use are in $SU(16)$ so we perform another comparison in that case. We list in Table \[table\_msu16\_Vcomp\] the spectrum on a $22^230$ lattice (our standard volume) and that on a larger $26^230$ lattice, both at $\beta=800$. We see that these two spectra are entirely consistent at the level of 2 standard deviations. All this strongly suggests that these finite volume corrections are insignificant in the calculations presented in this paper.
### nearly degenerate states {#subsubsection_degeneracy}
One of the interesting features of our results is that there are a number of nearly degenerate states. When these have the same quantum numbers and therefore arise from a single variational calculation (as described above) the variational procedure can induce an extra splitting, which is driven by statistical fluctuations and is therefore on the order of the errors. Moreover this may also lead to the two states being ordered differently in different bins, leading possibly to biases in the estimated errors and eventually to unsatisfactory fits when performing continuum or large $N$ extrapolations. Any ambiguities here are small as long as the errors are small, but some of the most interesting near-degeneracies occur in the $J=1$ sector where all the states are massive and the statistical errors are substantial.
String tensions {#section_tensions}
===============
fundamental string tension and $N$-dependence {#section_ftension}
---------------------------------------------
The ground state of the fundamental flux tube of length $l$ that winds once around our spatial torus is usually the lightest of all the states (on the volumes we typically use) and hence it is the state whose energy, $E_{gs,f}(l)$, we can calculate most accurately and most reliably. We extract the string tension $\sigma_f$ from the energy using the ‘Nambu-Goto’ formula in eqn(\[eqn\_gsNG\]).
We list in Tables \[table\_param\_su2\]-\[table\_param\_su16\] the resulting values of $a\surd\sigma_f$ for our various lattice calculations. To obtain values in the continuum limit we need to express $\surd\sigma_f$ in units of a quantity with dimensions of mass, and an obvious choice is the coupling $g^2$ which in $2+1$ dimensions has dimensions $[m^1]$. The lattice coupling scheme that we choose to use is the ‘mean-field improved’ coupling $g^2_I$ defined by [@b_imp] $$\beta_I \equiv \beta \times \langle \frac{1}{N} {\text{Tr}}U_p \rangle
=
\frac{2N}{ag^2_I}\,.
\label{eqn_gMFI}$$ Since different choices of lattice coupling differ at $O(g^2)$, the leading correction to the continuum limit will be $O(a)$ rather than $O(a^2)$. We therefore extrapolate our string tensions to the continuum limit, at each value of $N$, using $$\frac{\beta_I}{2N^2}a\surd\sigma_f
=
\left.\frac{\surd\sigma_f}{g^2_IN}\right|_a
=
\left.\frac{\surd\sigma_f}{g^2N}\right|_{a=0} + c_1 ag^2_IN + c_2 (ag^2_IN)^2 + \ldots \,.
\label{eqn_sigf_cont}$$ We have used the ’t Hooft coupling, $g^2N$, since that is what needs to keep fixed for a smooth $N\to\infty$ limit. We provide some examples of such continuum extrapolations in Fig.\[fig\_k1g\_cont\]. As is apparent from the figure, we can get good fits with just the leading $O(a)$ correction. We list in Table \[table\_sigf\_cont\] the continuum limit for each $N$, together with the coefficients of the linear correction, and the goodness of fit as measured by the total $\chi^2$ and the number of degrees of freedom, $n_{dof}$, of the fit.
We can now extrapolate our results to $N=\infty$. (For $N=2$ and $N=4$ we use the values on lattices of medium size, where the flux tube energies are small enough that one expects to avoid the systematic error associated with the large energies one obtains on large lattices.) We expect the leading correction to be $O(1/N^2)$ [@tHooft_N] and indeed we find that we get a marginally acceptable fit to all our calculated values with just this correction, $$\frac{\surd\sigma_f}{g^2N}
=
0.196573(81) -\frac{0.1162(9)}{N^2} \qquad N\geq 2 \quad \chi^2/n_{dof}=11.6/5\,,
\label{eqn_sigf_N}$$ which is displayed in Fig.\[fig\_k1g\_m02g\_N\]. In order to see how robust this result is to the inclusion of higher order corrections in $1/N^2$, or to dropping the lowest $N$ data point, we perform the corresponding fits, giving $$\frac{\surd\sigma_f}{g^2N}
= 0.19636(12) -\frac{0.1085(32)}{N^2}-\frac{0.029(12)}{N^4} \qquad N\geq 2 \quad \chi^2/n_{dof}=5.0/4\,,
\label{eqn_sigf_Nb}$$ $$\frac{\surd\sigma_f}{g^2N}
=
0.19642(11) -\frac{0.1118(26)}{N^2} \qquad \qquad \qquad N\geq 3 \quad \chi^2/n_{dof}=6.0/4\,.
\label{eqn_sigf_Nc}$$ In the rest of the paper we shall use the result obtained in eqn(\[eqn\_sigf\_Nb\]).
$k$-string tensions and $N$-dependence {#section_ktensions}
--------------------------------------
We calculate string tensions in other representations in the same way as for the fundamental. A flux tube carrying a flux that transforms under the $Z_N$ centre like the product of $k$ fundamental fluxes is called a $k$-string. For $N\geq 2k$ the lightest flux tube in each $k$-sector is stable against decay as one can infer for $k=2,3,4$ from the energies of the various ground states listed in Tables \[table\_Egs\_su4\]-\[table\_Egs\_su16\] and from the continuum string tension ratios listed in Table \[table\_sigksigf\_cont\]. So there is no ambiguity in extracting the flux tube energy, and hence the string tension using eqn(\[eqn\_gsNG\]). The ground states of these flux tubes fall into the totally antisymmetric representation to an excellent approximation [@BLMT01_strings; @AAMT16_strings] and so we label them as $2A,3A,4A$ respectively. One might ask how this feature can survive screening by gluons, but it turns out that for long flux tubes the effects of screening are extremely weak [@AAMT16_strings]. The flux tubes in the other representations listed in Table \[table\_sigksigf\_cont\] are in principle not stable; for example $\sigma_{2S} > 2\sigma_f$ and $\sigma_{3M} > \sigma_{2A} + \sigma_f$ (albeit not by much in this case). Since the least ambiguous string tensions are those that are associated with stable flux tubes, we will focus here on the $2A,3A,4A$ string tensions.
We obtain the continuum limit of $\surd\sigma_k/\surd\sigma_f$ from our calculated values using the standard fit $$\left.\frac{\surd\sigma_k}{\surd\sigma_f}\right|_{a}
=
\left.\frac{\surd\sigma_k}{\surd\sigma_f}\right|_{a=0}
+ c_1 a^2\sigma_f\,,
\label{eqn_sigksigfcont}$$ where we include only an $O(a^2)$ correction term since that suffices to obtain an acceptable fit in all cases. In Fig. \[fig\_kf\_cont\_su8\] we show the fits for $k=2,3,4$ in the case of $SU(8)$. We see that the $a$-dependence is so small as to be consistent with zero. Our continuum extrapolations are listed in Table \[table\_sigksigf\_cont\].
We begin by fitting $\sigma_{2A}/\sigma_f$ for $N\geq 4$. In addition to the values in Table \[table\_sigksigf\_cont\] we also add the constraint $\lim_{N\to\infty}\sigma_{2A} = 2 \sigma_f$ from the theoretical expectation that $\lim_{N\to\infty}\sigma_{k} = k \sigma_f$. We attempt separate fits: one that is in powers of $1/N$, and one in powers of $1/N^2$. These give the best fits $$\frac{\sigma_{2A}}{\sigma_f}
\stackrel{N\geq4}{=}
2 - \frac{1.406(49)}{N} - \frac{4.68(21)}{N^2} \qquad \chi^2/n_{df}=1.5\,,
\label{eqn_sigk2A_Ndep}$$ $$\frac{\sigma_{2A}}{\sigma_f}
\stackrel{N\geq4}{=}
2 - \frac{17.52(26)}{N^2} - \frac{5.70(75)}{N^4} \qquad \chi^2/n_{df}=27.6\,.
\label{eqn_sigk2A_NNdep}$$ Since $n_{df}=3$, the first fit is entirely acceptable, but the second is not. The second fit is not much improved if we restrict ourselves to $N\geq 6$. It is clear that our calculations imply that the leading correction to $\sigma_{2A}/\sigma_f$ is $\propto 1/N$ rather than $\propto 1/N^2$.
We repeat the exercise for $\sigma_{3A}/\sigma_f$, this time adding the constraint $\lim_{N\to\infty}\sigma_{3A} = 3 \sigma_f$. Fitting to $N\geq 6$ we obtain $$\frac{\sigma_{3A}}{\sigma_f}
\stackrel{N\geq6}{=}
3 - \frac{4.24(17)}{N} - \frac{16.3(1.2)}{N^2} \qquad \chi^2/n_{df}=0.13\,,
\label{eqn_sigk3A_Ndep}$$ $$\frac{\sigma_{3A}}{\sigma_f}
\stackrel{N\geq6}{=}
3 - \frac{68.0(1.0)}{N^2} - \frac{963(43}{N^4} \qquad \chi^2/n_{df}=38.6\,.
\label{eqn_sigk3A_NNdep}$$ Again this clearly points to the leading correction being $\propto 1/N$ and not $\propto 1/N^2$.
Finally we look at $\sigma_{4A}/\sigma_f$ for $N\geq 8$, where we obtain $$\frac{\sigma_{4A}}{\sigma_f}
\stackrel{N\geq8}{=}
4 - \frac{9.08(46)}{N} - \frac{31.1(3.9)}{N^2} \qquad \chi^2/n_{df}=0.14
\label{eqn_sigk4A_Ndep}$$ while the best fit with $N\to N^2$ has an unacceptable $\chi^2/n_{df}=12.1$.
In the above we have restricted our fits to $N\geq 2k$ since the center symmetry allows a $k$-string to mix with a $(N-k)$-string, and the latter will have a lower string tension if $N < 2k$, and so it will then provide the lightest flux tube in the $k$-sector. So, for example, if we extrapolate eqn(\[eqn\_sigk2A\_Ndep\]) to $SU(3)$, we expect to find ${\sigma_{2A}}\stackrel{su3}{=}{\sigma_f}$. Interestingly enough, substituting $N=3$ in eqn(\[eqn\_sigk2A\_Ndep\]) does indeed give a value very close to unity. We can extend the argument to $SU(2)$, where $k=2$ can mix with the $k=0$ vacuum, and indeed the $O(1/N)$ fit in eqn(\[eqn\_sigk2A\_Ndep\]) gives us ${\sigma_{2A}}\stackrel{su2}{\sim} 0 $. Similarly we find ${\sigma_{3A}}\stackrel{su4}{\sim}{\sigma_f}$ and ${\sigma_{3A}}\stackrel{su5}{\sim}{\sigma_{k=2}}$ (where we estimate the value of $\sigma_{k=2}$ in $SU(5)$ using eqn(\[eqn\_sigk2A\_Ndep\])). Of course this rough agreement is only indicative: as we go to smaller $N$ we must expect that the omitted higher order terms in $1/N$ will become significant. Indeed a constructive way to look at this is to use the constraints from $N < 2k$ to fix, with no extra work, the next higher order terms in the expansion in powers of $N$, which our fitting for $N\geq 2k$ is not sensitive to.
Another interesting feature becomes apparent when we look at the coefficient $c_1(k)$ of the leading $1/N$ correction in the above equations. We observe that $$c_1(k=4):c_1(k=3):c_1(k=2) \approx 6:3:1
= \frac{4(4-1)}{2}:\frac{3(3-1)}{2}:\frac{2(2-1)}{2}
\label{eqn_c1k}$$ i.e. $c_1(k) \propto k(k-1)/2$, which is just the number of pairwise interactions amongst the $k$ fundamental strings which are bound into the $k$-string. A similar rough proportionality appears to hold when we look at the coefficient $c_2(k)$ of the second, $1/N^2$, correction term. This appears to suggest that the interactions binding the $k$ fundamental strings into the $k$-string are predominantly pairwise.
It is also interesting to compare our results to the Karabali-Kim-Nair (KKN) analysis [@Nair] which provides a prediction not only for the $k$ and $N$ dependence of $\sigma_k$, but also predicts its absolute magnitude: $$\frac{\surd\sigma_k}{g^2N} \stackrel{KKN}{=}\sqrt{\frac{k(N-k)}{N-1}}
\times \sqrt{\frac{1-\frac{1}{N^2}}{8\pi}}.
\label{eqn_KKN}$$ Although we know from earlier studies [@BBMT06_string] that eqn(\[eqn\_KKN\]) does not fit the calculated values of $\sigma_f/g^2N$ perfectly, it does come within $\sim 2\%$ for $N\geq 3$, and within $3\%$ even if we include $SU(2)$. The prefactor in eqn(\[eqn\_KKN\]) is simply the quadratic Casimir of the represention which, for reasons obvious from the above discussion, we take to be the totally antisymmetric one. From the values of $\surd\sigma_f/g^2N$ in Table \[table\_sigf\_cont\] and $\surd\sigma_k/\surd\sigma_f$ in Table \[table\_sigksigf\_cont\] we form the ratio $\surd\sigma_k/g^2N$ which we plot in Fig.\[fig\_ksig\_Nair\]. We observe very close agreement between the string tensions predicted by eqn(\[eqn\_KKN\]) and our calculated values. Indeed the agreement is clearly better for $\sigma_{k\geq 2}$ than for $\sigma_f$. Of course, since $\lim_{N\to\infty}\sigma_k = k\sigma_f$, any disagreement will be the same for $\sigma_k$ and for $\sigma_f$ in the large $N$ limit. Nonetheless it is clear that the simple formula in eqn(\[eqn\_KKN\]) provides a remarkably good approximation to all our $k$-string tensions for all values of $N$.
Glueball spectra {#section_glueballs}
================
We calculate glueball masses as described in Section \[subsection\_glue\_calc\]. A particular concern is the finite volume states which might appear in our spectrum of excited glueball states, as discussed in Section \[subsection\_sys\_errors\]. The lightest such state is composed of a ground state flux loop and its conjugate, so it will have a mass $m_T\sim 2\sigma_f l$, and where it might appear is in the $0^{++}$ and $2^{++}$ spectra. Since our glueball operators are single trace while these finite volume states are double trace, large-$N$ counting tells us that such states will decouple from our correlators at larger $N$. So our strategy is to have a large enough volume $l^2$ at small $N$ so that $m_T$ is heavier than the states we calculate, and then to relax the volume to a smaller value at larger $N$ where we expect such states to become invisible. A summary of the ‘standard’ volumes we have chosen to use, expressed in the relevant physical units, is presented in Table \[table\_sizeN\]. We check these choices with some explicit comparisons between smaller and larger volumes. For example in $SU(8)$ at $\beta=306.25$ we compare the spectrum on our ‘standard’ $44^2 48$ volume to the one on a larger $60^2 48$ volume. The resulting glueball masses are given in Table \[table\_msu8\_Vcomp\], where we include all the excitations for which we will attempt to obtain continuum limits. We see that the spectrum on the $l=44a$ volume matches that on the $l=60a$ volume within say $2\sigma$, suggesting that for this moderately large value of $N$ the choice of $l\surd\sigma_f \sim 3.8$ is sufficient to exclude finite volume corrections (of whatever source) at the level of our statistical errors. In Table \[table\_msu16\_Vcomp\] we perform a similar comparison in $SU(16)$ at $\beta=800$ where our ‘standard’ volume is $l=22a$, corresponding to to $l\surd\sigma_f \sim 3.1$. Again we see that the glueball spectra agree, within say $2\sigma$, indicating that our reduction in $l\surd\sigma_f$ with increasing $N$ (based on the idea of the large-$N$ suppression of finite volume corrections) is indeed appropriate. The specific lattices and $\beta$-values used are listed in Tables \[table\_param\_su2\]-\[table\_param\_su16\], along with the values of the string tension and the mass gap. For $N\in[2,6]$ there are two sets of volumes listed: the larger are used for glueball calculations while the smaller are used for string tension calculations. The reason for doing this is that the energy of the flux loop grows with $l$, so we can perform a (statistically) more precise calculation with smaller $l$. This is only possible, of course, because we have a very precise theoretical control of finite volume corrections in this case.
Having obtained our ‘infinite’ volume glueball masses in this way, for various lattice spacings, we extrapolate the results to the continuum limit. Mostly we do so for the dimensionless ratio $am/a\surd\sigma_f = m/\surd\sigma_f $ since in general $a\surd\sigma_f$ is our most accurately calculated physical quantity on the lattice. We perform the continuum extrapolation in the standard way $$\left.\frac{M}{\surd\sigma_f}\right|_{a}
=
\left.\frac{M}{\surd\sigma_f}\right|_{a=0}
+ c_1 a^2\sigma_f + c_2 (a^2\sigma_f)^2+ ... \,,
\label{eqn_MKcont}$$ and we find in nearly all cases that the first $O(a^2)$ correction suffices for a good fit. The results of these continuum extrapolations are listed in Tables \[table\_mksu2\_cont\]-\[table\_mksu16\_cont\], together with the $\chi^2$ and number of degrees of freedom $n_{dof}$ of the fit. In Table \[table\_mksu8\_cont\] we also show the values one finds for the coefficient $c_1$ in eqn(\[eqn\_MKcont\]); they are very similar for other values of $N$. We see that the lattice corrections are very modest even at our coarsest lattice spacing, where $a^2\sigma_f \sim 0.1$. Note that at the coarser values of $a$, the masses of higher excited states can be too large for them to be identified, and then the value of $n_{dof}$ is smaller than for the ground states.
In addition to these continuum fits we also calculate in one or two cases the continuum limit of the ratio $m/g^2N$ just as we did for the string tension in eqn(\[eqn\_sigf\_cont\]). This might seem an attractive way to calculate continuum physics because the error on the denominator $g^2_I N$ comes from the average plaquette and is negligible. In practice, however, this advantage is more than outweighed by the fact that the leading correction is $O(a)$ rather than $O(a^2)$, so we generally focus on the extrapolations using eqn(\[eqn\_MKcont\]). We will also look at some of the masses in units of the mass gap, which is of interest for the reasons discussed in [@AARLMT_SONSUN].
Finally we extrapolate our continuum mass ratios to $N=\infty$ using $$\left.\frac{M}{\surd\sigma_f}\right|_N
=
\left.\frac{M}{\surd\sigma_f}\right|_\infty
+ \frac{c_1}{N^2} + \frac{c_2}{N^4} + ...
\label{eqn_MKN}$$ with the results listed in Tables \[table\_mksuN\_J0\]-\[table\_mksuN\_J1\]. We note that in general fits with just the leading $O(1/N^2)$ correction are acceptable, although for our lightest and most accurately calculated masses, it helps to include a further $O(1/N^4)$ correction if one wishes to include the $SU(2)$ values in the fits.
$|J|=0, 4, ...$ glueballs {#subsection_J0glueballs}
-------------------------
Since the argument for parity doubling breaks down for $J=0$ glueballs, the $0^{++}$, $0^{--}$, $0^{-+}$, and $0^{+-}$ glueball masses are, in principle, all unrelated. The lightest glueball state is the $0^{++}$ ground state and, as we have seen in Fig.\[fig\_meff\_JPC\_su6\], its mass and that of the lowest few $0^{++}$ excitations can be calculated very accurately. The $0^{--}$ is the next lightest ground state state and the lowest few $0^{--}$ masses can also be calculated accurately. The lightest $0^{-+}$ and $0^{+-}$ are quite heavy and the mass estimates for them are correspondingly less reliable.
In Fig.\[fig\_mJ0\_cont\_su6\] we plot the lattice values of $m/\surd\sigma_f$, calculated in $SU(6)$, against the value of $a^2\sigma_f$ for the lightest six $0^{++}$ states, the lightest three $0^{--}$ states, and for the $0^{-+}$ ground state. We also show the continuum extrapolations, and note that in all cases a leading $O(a^2)$ correction suffices to give an acceptable fit.
We observe some striking regularities in Fig.\[fig\_mJ0\_cont\_su6\]. Firstly, each of the three $0^{--}$ states is nearly (but not exactly) degenerate with an excited $0^{++}$ state: to be specific, the first, second and fifth excited $0^{++}$ states. Secondly, the ground state $0^{-+}$ is degenerate, within errors, with the fourth excited $0^{++}$. In fact earlier analyses [@HMMT_spin] have revealed that the lightest ‘$0^{-+}$’ state is in fact a $4^{-+}$ state so, by parity doubling, there should be a degenerate $4^{++}$ state which will appear in what we label as the set of ‘$0^{++}$’ states. In other words, we believe that what we have called the fourth $0^{++}$ excitation is in fact a $4^{++}$ state. Moreover this $4^{++}$ state is nearly degenerate with the fifth $0^{++}$ excitation (and also with the second excited $0^{--}$). We also note that the ground state $0^{+-}$ appears to be nearly degenerate with the first excited $0^{-+}$, and that in those cases where we have calculations (i.e. for $N=4,12,16$) the first excited $0^{+-}$ appears to be nearly degenerate with the second excited $0^{-+}$. (With the caveat that for these very massive states the errors are large.)
As we can see in Fig.\[fig\_mJ0\_cont\_su6\] the gaps between the states that are ‘nearly degenerate’ are much smaller than the typical gaps between other states. That is to say, these regularities appear to be real rather than statistical coincidences.
While we have chosen to use $SU(6)$ in Fig. \[fig\_mJ0\_cont\_su6\] to display these features they are in fact common to all $SU(N)$ as we see from the continuum limits listed in Tables \[table\_mksu2\_cont\]-\[table\_mksu16\_cont\] and the large-$N$ extrapolations in Table \[table\_mksuN\_J0\], together with Fig.\[fig\_m0K\_N\] where we show the continuum values of $m/\surd\sigma_f$ for these states for all our values of $N$.
$|J|=2, 6, ...$ glueballs {#subsection_J2glueballs}
-------------------------
For $J=2$ glueballs we have parity doubling so we expect to have $m_{2^{++}} = m_{2^{-+}}$ and $m_{2^{--}} = m_{2^{+-}}$ up to lattice spacing and finite volume corrections. We list the $J=2$ glueball masses in units of the string tension, and after an extrapolation to the continuum limit, in Tables \[table\_mksu2\_cont\]-\[table\_mksu16\_cont\] and we see that the parity doubling expectations are broadly satisfied, albeit with an apparent exception in the case of the third state in the $2^{\pm +}$ sectors in both $SU(3)$ and $SU(8)$, as well as the fourth excited state in $SU(3)$. (For $SU(3)$ these may be finite volume effects, while in the case of $SU(8)$, where finite volume corrections should be large-$N$ suppressed, it may just be an unlikely statistical fluctuation.)
As an example we plot in Fig.\[fig\_mJ2\_cont\_su12\] our values in $SU(12)$ of the lightest few $2^{++}$ and $2^{--}$ glueball masses as a function of $a^2\sigma_f$ together with the corresponding linear continuum extrapolations. Just as for $J=0$ we observe a striking pattern of near-degeneracies: the first, second and fourth $2^{++}$ excited states appear to be nearly degenerate with the lightest three $2^{--}$ states. We also see from Tables \[table\_mksu2\_cont\]-\[table\_mksu16\_cont\] that this near-degeneracy holds for all $N$ except in $SU(3)$ and $SU(8)$ for those excited states where the parity doubling is poor, and, in any case, in those cases the near-degeneracy holds between the $2^{-+}$ and $2^{\pm -}$.
The large-$N$ extrapolations of our $J=2$ spectra are listed in Table \[table\_mksuN\_J2\]. In the $2^{++}$ sector the second and fourth excited states have such poor fits that we do not attempt to provide any estimate based on a large-$N$ extrapolation. Instead we provide an estimate based on the average of the $SU(12)$ and $SU(16)$ values (with errors enhanced to encompass the two values.) Since the fits to the supposedly degenerate $2^{-+}$ parity partners are mostly acceptable, it is these that we plot in Fig.\[fig\_m2K\_N\] against $1/N^2$, together with the lightest $2^{--}$ states. We clearly see the near-degeneracies discussed above. We also see that the large-$N$ limits of the second and third excited $2^{-+}$ states are very similar. All this is very similar to what we observed in the $J=0$ sector, displayed in Fig.\[fig\_m0K\_N\].
One might ask if there is a $J=6$ state located in the $J=2$ sector, just like the $J=4$ state in the $J=0$ sector. The answer given in [@HMMT_spin] for $SU(2)$, suggests that $m_{gs}^{J=6} \sim 1.5 m_{gs}^{J=2}$ which means that it will be heavier than the states we consider here.
$|J|=1, 3, ...$ glueballs {#subsection_J1glueballs}
-------------------------
For $J=1$ (and indeed any odd $J$) we expect to have exact parity doubling, not only in the continuum limit but even on a square lattice in a finite volume as long as the latter respects $\pi/2$ rotational invariance. Our calculated masses are consistent with parity doubling, as we see for example in Tables \[table\_msu4\_opcomp\], \[table\_msu6\_opcomp\] and Tables-\[table\_msu8\_Vcomp\],-\[table\_msu16\_Vcomp\].
We list in Tables \[table\_mksu2\_cont\]-\[table\_mksu16\_cont\] the values that we obtain for the $J=1$ glueball masses expressed in units of the string tension, after an extrapolation to the continuum limit. For simplicity we refer to the $|J|=odd$ states as $J=1$ both here and in the Tables, but will shortly return to the question of their true spin.) A striking feature of the $J=1$ spectrum is that for $N\geq 3$ the lightest two $1^{++}$ states are nearly degenerate, within our errors, and that the next two states are also nearly degenerate. (By parity doubling the same is true for the $1^{-+}$ states.) The gap between these two pairs of states is much larger than any gap within each pair, making it plausible that the near-degeneracy is not just a statistical accident but has a dynamical origin. Moreover the fact that all these states are very massive, with correspondingly large statistical errors, creates some technical problems for the variational procedure, where the ordering of the states may differ in different (jack-knife) data bins so that the observed small splittings between the states in each pair may be enhanced, or even entirely driven, by the statistical fluctuations. This may be the reason for the fact that many of the continuum fits are quite poor, and that we occasionally see discrepancies between the $P=\pm$ $J=1$ masses.
Another feature of the $J=1$ spectrum, that differs from the $J=0,2$ spectra, is that the lightest state is the $C=-$ ground state. Moreover the first excited $C=-$ state appears to be roughly degenerate with the pair of nearly degenerate $C=+$ ground states and, albeit now within larger errors, the third excited $C=-$ state is nearly degenerate with the next pair of nearly degenerate $C=+$ states. All this is very reminiscent of what we saw for $J=0,2$, but with $C=-$ and $C=+$ interchanged.
To illustrate these features of the $J=1^{\pm +}$ and $J=1^{\pm -}$ spectra, we plot in Fig. \[fig\_mJ1Pav\_cont\_su8\] the masses of the $J=1$ states against $a^2\sigma_f$ for our $SU(8)$ calculation. We have averaged the $P=\pm$ masses since they should be degenerate, and this should help to suppress fluctuations. The fact that the lightest state is $C=-$ is unambiguous. We also see good evidence that the lightest two $C=+$ states are (nearly) degenerate, and that the first excited $C=-$ state is (nearly) degenerate with them. The next two $C=+$ states have similar masses to each other and may be degenerate although the errors on these massive states are so large that this is something of a conjecture. Equally, the next two $C=-$ states have similar masses to each other and also, interestingly, to the second pair of $C=+$ states.
These features are characteristic of all our $SU(N\geq 3)$ spectra and hence also of the $N\to\infty$ extrapolations listed in Table \[table\_mksuN\_J1\]. These extrapolations have reasonably small errors, and the (very near) degeneracy of the lightest two $C=+$ states is convincing as is the approximate degeneracy of the next two excited $C=+$ states. It is also clear that the $C=-$ ground state is the lightest $J=1$ state and that it is not degenerate with any other state. However the first excited $C=-$ state appears to be (nearly) degenerate with the lightest pair of $C=+$ states. The second $C=-$ excitation does not appear to be very close to any other states, but the third $C=-$ excitation appears to be nearly degenerate with the second pair of $C=+$ states. One can, as for $SU(8)$, reduce the fluctuations a little by averaging the $P=\pm$ values. We show the result of doing so in Fig. \[fig\_m1K\_N\] where we plot these averaged values for all our $SU(N)$ continuum limits against $1/N^2$. Apart from $SU(2)$ where there are no $C=-$ states, and where the lightest four $C=+$ states definitely do not form nearly degenerate pairs, all the other $SU(N)$ spectra display the features emphasised above.
Given these striking regularities, it is interesting to ask whether there is any evidence that some of these states might not be $|J|=3,5,...$ rather than $J=1$. Now the analysis for $SU(2)$ in [@HMMT_spin] and the extension to $SU(3)$ and $SU(5)$ in Section 6.4 of the thesis referred to there does in fact provide evidence that the lightest two states in the ‘$1^{++}$’ sector are $J=3$ and not $J=1$, while in the ‘$1^{--}$’ sector the ground state is $J=1$ but the first excited state is $J=3$. If we take these assignments as correct then this will alter our conclusions as follows: the lightest $J=1^{P=\pm}$ state has $C=-$ and is lighter than the lightest $J=3$ state and is much lighter than the lightest $C=+$ $J=1$ state. The lightest $J=3^{P=\pm}$ states consist of three nearly degenerate states: two with $C=+$ and one with $C=-$. For the higher excited states we have no evidence concerning their spin, other than that it is odd.
Discussion {#section_conclusion}
==========
A striking feature of the fundamental string tension expressed in units of the coupling, is how small are the lattice corrections, as we can see in Fig. \[fig\_k1g\_cont\]. Even at the coarsest lattice spacings, where $a\surd\sigma_f \sim 0.3$, the correction is less than $10\%$. Indeed a simple $O(ag^2_IN)$ correction term is all that is needed despite the precision of our lattice calculations of $a^2\sigma_f$ and the substantial range of $a$ being fitted. Moreover the same is true of the mass gap, as we see in Table \[table\_sigf\_cont\]. The lattice corrections are even smaller when we consider the masses of the lightest glueballs expressed in units of the string tension, $m_G/\surd\sigma_f$, as we see in Figs. \[fig\_mJ0\_cont\_su6\], \[fig\_mJ2\_cont\_su12\] and \[fig\_mJ1Pav\_cont\_su8\]. The same is true for the tensions of the stable higher representation flux tubes, as we see in Fig. \[fig\_kf\_cont\_su8\] in the case of $SU(8)$.
This remarkably precocious scaling may have to do with the fact that the theory is super-renormalisable: the dimensionless running coupling decreases linearly as the distance scale is reduced, $\tilde{g}^2(l) = lg^2$, so that corrections that are higher powers of $g^2$ are necessarily accompanied by higher powers in $a$. (In contrast to 4 dimensions where the leading $O(a^2)$ lattice correction is a power series in $g^2$.)
A striking feature of our (continuum extrapolated) values of $m_G/\surd\sigma_f$, $m_G/g^2N$ and $\surd\sigma_f/g^2N$, is how weak is their variation with $N$, over the whole range of $N\geq 2$, as we see in Figs. \[fig\_m0K\_N\], \[fig\_m2K\_N\] and \[fig\_m1K\_N\]. A possible explanation for this has been given in our earlier paper [@AARLMT_SONSUN] where we discuss the constraints on the $N$-dependence that arise when we consider $SO(N)$ as well as $SU(N)$ gauge theories.
In any case all this suggests that there may be some underlying simplicity in the dynamics of $D=2+1$ $SU(N)$ gauge theories, and this motivates a close examination of our results for unexpected regularities. We have indeed found evidence for a number of these (some already remarked upon in earlier less precise calculations) which we shall now summarise, beginning with the string tensions.
In addition to the stable fundamental flux tube, it is known that the ground states of $k$-strings (i.e. flux tubes which carry the flux of local static sources consisting of $k$ fundamental charges) are also stable, and that these flux tubes are in the totally anti-symmetric representation (when not very short). We confirm all this with our more accurate calculations. Furthermore, our large range of $N$ allows us to make an unambiguous statement that the leading corrections to the $N=\infty$ limit of $\sqrt{\sigma_k/\sigma_f}$ are $O(1/N)$ rather than $O(1/N^2)$. Moreover the $O(1/N)$ binding energy is consistent with being produced by a pairwise interaction between the $k$ fundamental strings that are bound into a $k$-string. And we confirm previous observations that to a very good approximation the string tensions are proportional to the quadratic Casimir of the representation. It is intriguing that the absolute value of $\surd\sigma_k/g^2N$ is very close to that predicted in [@Nair] as we see in Fig.\[fig\_ksig\_Nair\].
Turning now to our results for glueballs, here the striking feature is a number of unexpected near-degeneracies amongst glueballs with different $J^{PC}$ quantum numbers. (This is in addition to the expected parity doubling for $J\neq 0$ which we also observe.)
For $J=0$ we saw that the first, second and fifth excited $0^{++}$ glueballs are nearly degenerate with the lightest three $0^{--}$ glueball states. The fourth excited ‘$0^{++}$’ state is presumably the $4^{++}$ ground state given its near degeneracy with the ‘$0^{-+}$’ ground state which is believed to be, in reality, the $4^{-+}$ ground state [@HMMT_spin]. The only $0^{++}$ states that are not nearly degenerate with another $J=0$ state (amongst the lightest 6 states) are the ground state and the third excited state. We note that the mass of the latter is very nearly twice the mass of the former. (Although this may well be accidental.)
In the $J=2$ glueball sector we observe a nearly identical pattern of degeneracies. The first, second and fourth excited $2^{++}$ glueballs are nearly degenerate with the lightest three $2^{--}$ glueball states. Only the ground and third excited of these lightest $2^{++}$ states are not nearly degenerate with some other $J=2$ state..
In the $J=1$ sector the lightest glueball is a $1^{--}$. There are two very nearly degenerate ‘$1^{++}$’ ground states which are nearly degenerate with the first excited ‘$1^{--}$’ state. But there is in fact good evidence [@HMMT_spin] that these three nearly degenerate states are $J=3$ rather than $J=1$. It also appears that the next two pairs of excited states in the $1^{++}$ and $1^{--}$ sectors are nearly degenerate and nearly degenerate with each other, although their large masses mean that this observation is more speculative.
We finish by recalling that in the simplest closed flux tube model of glueballs [@NIJP], the $C=+$ and $C=-$ glueball states in two spatial dimensions are degenerate. We also note that while we have a number of near-degeneracies, they do not appear to be exact. That is to say, even if they point to some kind of relatively simple dynamics, this is likely to be the property of a field theory from which the $SU(N)$ gauge theory is – at the very least – a small perturbation. Helping to identify such a neighbouring field theory which one may hope to be analytically tractable, is a motivation for the present study.
Acknowledgements {#acknowledgements .unnumbered}
================
AA has been partially supported by an internal program of the University of Cyprus under the name of BARYONS. In addition, AA acknowledges the hospitality of the Cyprus Institute where part of this work was carried out. MT acknowledges partial support under STFC grant ST/L000474/1. The numerical computations were carried out on the computing cluster in Oxford Theoretical Physics.
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Tables: some lattice data and operators {#section_appendix_results}
=======================================
[cccccc]{} $\parbox{0.65cm}{\rotatebox{0}{\includegraphics[height=0.65cm]{path1.eps}}}$ & $\parbox{1.3cm}{\rotatebox{0}{\includegraphics[height=1.3cm]{path2.eps}}}$ & $\parbox{1.3cm}{\rotatebox{0}{\includegraphics[height=1.3cm]{path3.eps}}}$ & $\parbox{1.3cm}{\rotatebox{0}{\includegraphics[height=1.3cm]{path4.eps}}}$ & $\parbox{1.3cm}{\rotatebox{0}{\includegraphics[height=0.65cm]{path5.eps}}}$ & $\parbox{1.95cm}{\rotatebox{0}{\includegraphics[height=0.65cm]{path6.eps}}}$\
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$\parbox{1.3cm}{\rotatebox{0}{\includegraphics[height=0.65cm]{path7.eps}}}$ & $\parbox{1.3cm}{\rotatebox{0}{\includegraphics[height=1.3cm]{path8.eps}}}$ & $\parbox{1.3cm}{\rotatebox{0}{\includegraphics[height=1.3cm]{path9.eps}}}$ & $\parbox{1.95cm}{\rotatebox{0}{\includegraphics[height=1.95cm]{path10.eps}}}$ & $\parbox{1.95cm}{\rotatebox{0}{\includegraphics[height=1.95cm]{path11.eps}}}$ & $\parbox{1.95cm}{\rotatebox{0}{\includegraphics[height=1.95cm]{path12.eps}}}$\
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$\parbox{1.95cm}{\rotatebox{0}{\includegraphics[height=1.3cm]{path13.eps}}}$ & $\parbox{1.95cm}{\rotatebox{0}{\includegraphics[height=1.3cm]{path14.eps}}}$ & $\parbox{1.3cm}{\rotatebox{0}{\includegraphics[height=1.3cm]{path15.eps}}}$ & $\parbox{1.3cm}{\rotatebox{0}{\includegraphics[height=1.3cm]{path16.eps}}}$ & $\parbox{1.95cm}{\rotatebox{0}{\includegraphics[height=0.65cm]{path17.eps}}}$ & $\parbox{1.95cm}{\rotatebox{0}{\includegraphics[height=0.65cm]{path18.eps}}}$\
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$\parbox{1.95cm}{\rotatebox{0}{\includegraphics[height=1.95cm]{path19.eps}}}$ & $\parbox{1.95cm}{\rotatebox{0}{\includegraphics[height=1.95cm]{path20.eps}}}$ & $\parbox{1.95cm}{\rotatebox{0}{\includegraphics[height=1.95cm]{path21.eps}}}$ & $\parbox{1.95cm}{\rotatebox{0}{\includegraphics[height=1.95cm]{path22.eps}}}$ & $\parbox{1.95cm}{\rotatebox{0}{\includegraphics[height=1.95cm]{path23.eps}}}$ & $\parbox{1.95cm}{\rotatebox{0}{\includegraphics[height=1.95cm]{path24.eps}}}$\
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$\parbox{1.3cm}{\rotatebox{0}{\includegraphics[height=1.3cm]{path25.eps}}}$ & $\parbox{1.95cm}{\rotatebox{90}{\includegraphics[height=1.3cm]{path26.eps}}}$ & $\parbox{1.95cm}{\rotatebox{90}{\includegraphics[height=1.3cm]{path27.eps}}}$ & $\parbox{1.95cm}{\rotatebox{90}{\includegraphics[height=1.3cm]{path28.eps}}}$ & $\parbox{1.95cm}{\rotatebox{90}{\includegraphics[height=1.3cm]{path29.eps}}}$ & $\parbox{1.95cm}{\rotatebox{90}{\includegraphics[height=1.95cm]{path30.eps}}}$\
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$\parbox{1.95cm}{\rotatebox{0}{\includegraphics[height=1.3cm]{path31.eps}}}$ & $\parbox{1.95cm}{\rotatebox{0}{\includegraphics[height=1.3cm]{path32.eps}}}$ & $\parbox{1.95cm}{\rotatebox{0}{\includegraphics[height=1.3cm]{path33.eps}}}$ & $\parbox{1.95cm}{\rotatebox{90}{\includegraphics[height=0.65cm]{path34.eps}}}$ & $\parbox{1.95cm}{\rotatebox{90}{\includegraphics[height=0.65cm]{path35.eps}}}$ & $\parbox{1.95cm}{\rotatebox{90}{\includegraphics[height=1.95cm]{path36.eps}}}$\
\
[|cll|cll|]{}\
$J^{PC}$ & I & II & $J^{PC}$ & I & II\
$0^{++}$ & 0.5471(20) & 0.5441(20) & $0^{--}$ & 0.8049(23) & 0.7996(34)\
& 0.8197(29) & 0.8247(27) & & 0.995(12) & 1.0129(30)\
& 1.009(10) & 1.023(4) & & 1.188(6) & 1.1869(40)\
& 1.027(32) & 1.075(6) & & 1.262(6) & 1.260(6)\
& 1.141(16) & 1.139(17) & & 1.302(8) & 1.299(8)\
& 1.198(6) & 1.182(4) & & 1.322(19) & 1.337(6)\
& 1.229(18) & 1.235(8) & & &\
& 1.253(6) & 1.259(7) & & &\
$0^{-+}$ & 1.128(14) & 1.167(15) & $0^{+-}$ & 1.252(22) & 1.310(6)\
& 1.317(8) & 1.317(6) & & 1.487(11) & 1.478(11)\
& 1.394(37) & 1.455(8) & & 1.596(11) & 1.575(11)\
$2^{++}$ & 0.9048(30) & 0.9047(27) & $2^{-+}$ & 0.9054(44) & 0.9105(39)\
& 1.092(5) & 1.082(16) & & 1.071(12) & 1.091(12)\
& 1.241(9) & 1.242(7) & & 1.248(6) & 1.251(7)\
& 1.253(23) & 1.281(6) & & 1.210(22) & 1.279(7)\
& 1.274(22) & 1.318(24) & & 1.399(9) & 1.388(9)\
$2^{--}$ & 1.067(9) & 1.077(6) & $2^{+-}$ & 1.052(14) & 1.067(12)\
& 1.260(7) & 1.233(6) & & 1.255(9) & 1.256(6)\
& 1.421(8) & 1.364(34) & & 1.414(12) & 1.343(32)\
& 1.48(3) & 1.42(3) & & 1.44(5) & 1.38(3)\
& 1.541(10) & 1.497(7) & & 1.509(8) & 1.545(11)\
$1^{++}$ & 1.292(7) & 1.276(7) & $1^{-+}$ & 1.290(8) & 1.240(32)\
& 1.300(8) & 1.305(8) & & 1.297(7) & 1.311(9)\
& 1.471(8) & 1.404(41) & & 1.468(8) & 1.447(10)\
& 1.475(6) & 1.458(10) & & 1.490(8) & 1.473(13)\
& 1.461(25) & 1.426(34) & & 1.497(8) & 1.505(12)\
$1^{--}$ & 1.243(6) & 1.247(7) & $1^{+-}$ & 1.241(7) & 1.242(5)\
& 1.292(7) & 1.280(8) & & 1.232(22) & 1.276(7)\
& 1.338(26) & 1.40(1) & & 1.403(8) & 1.421(8)\
& 1.455(7) & 1.411(9) & & 1.455(7) & 1.403(9)\
& 1.463(10) & 1.446(9) & & 1.462(7) & 1.453(8)\
[|ccc|ccc|]{}\
$J^{PC}$ & I & II & $J^{PC}$ & I & II\
$0^{++}$ & 0.2922(7) & 0.2948(10) & $0^{--}$ & 0.4326(11) & 0.4331(10)\
& 0.4467(9) & 0.4466(17) & & 0.5481(14) & 0.5493(14)\
& 0.5554(19) & 0.5502(21) & & 0.6453(39) & 0.6496(19)\
& 0.5793(14) & 0.5795(26) & & 0.6826(41) & 0.6861(18)\
& 0.6289(38) & 0.6331(27) & & 0.7154(19) & 0.7077(45)\
& 0.6426(28) & 0.6337(33) & & 0.7452(43) & 0.7111(64)\
& 0.6433(61) & 0.652(13) & & 0.7808(50) & 0.7631(51)\
$0^{-+}$ & 0.6346(34) & 0.6296(50) & $0^{+-}$ & 0.7121(36) & 0.7202(43)\
& 0.7112(36) & 0.7109(66) & & 0.8100(47) & 0.7949(45)\
& 0.8107(52) & 0.789(9) & & 0.8557(58) & 0.8611(52)\
& 0.804(13) & 0.812(14) & & 0.8821(79) & 0.8847(77)\
$2^{++}$ & 0.4914(16) & 0.4932(16) & $2^{-+}$ & 0.4926(14) & 0.4933(19)\
& 0.5970(16) & 0.5992(16) & & 0.5964(18) & 0.5980(32)\
& 0.6846(23) & 0.6752(13) & & 0.6766(31) & 0.6707(74)\
& 0.6882(82) & 0.7029(48) & & 0.7013(34) & 0.6963(44)\
& 0.7728(49) & 0.7237(86) & & 0.7801(46) & 0.7661(52)\
$2^{--}$ & 0.5773(42) & 0.5871(17) & $2^{+-}$ & 0.5830(26) & 0.5869(34)\
& 0.6766(37) & 0.6787(21) & & 0.6861(26) & 0.6793(34)\
& 0.7622(47) & 0.7477(61) & & 0.7782(19) & 0.7681(46)\
& 0.8276(28) & 0.7942(51) & & 0.7880(49) & 0.7759(96)\
& 0.8361(55) & 0.781(12) & & 0.8438(63) & 0.8573(26)\
$1^{++}$ & 0.7074(42) & 0.7077(33) & $1^{-+}$ & 0.7086(32) & 0.6924(69)\
& 0.7068(39) & 0.7082(33) & & 0.7124(35) & 0.7119(86)\
& 0.7783(98) & 0.7794(54) & & 0.7771(75) & 0.7884(27)\
& 0.7739(73) & 0.7818(65) & & 0.7806(96) & 0.776(12)\
$1^{--}$ & 0.6685(40) & 0.6716(41) & $1^{+-}$ & 0.6672(29) & 0.6695(36)\
& 0.6896(44) & 0.6966(36) & & 0.6989(46) & 0.6959(43)\
& 0.7637(46) & 0.7563(47) & & 0.7603(34) & 0.7645(47)\
& 0.7771(47) & 0.7725(40) & & 0.7839(44) & 0.7832(24)\
$N$ $l\surd\sigma_f\approx$ $2\sigma l/m_{0^{++}} \approx$
----- ------------------------- --------------------------------
2 5.5 2.4
3 5.2 2.4
4 5.1 2.4
6 4.2 2.0
8 3.8 1.8
12 3.6 1.7
16 3.1 1.5
: Lattice sizes used in our $SU(N)$ calculations, expressed in physical units[]{data-label="table_sizeN"}
[|cll|cll|]{}\
$J^{PC}$ & $60^2 48$ & $44^2 48$ & $J^{PC}$ & $60^2 48$ & $44^2 48$\
$0^{++}$ & 0.3569(13) & 0.3552(13) & $0^{--}$ & 0.5248(14) & 0.5227(13)\
& 0.5428(18) & 0.5436(16) & & 0.6630(44) & 0.6669(32)\
& 0.6747(20) & 0.6745(20) & & 0.7848(31) & 0.785(3)\
& 0.6953(42) & 0.697(5) & & 0.822(5) & 0.835(3)\
& 0.7652(59) & 0.762(6) & & 0.860(7) & 0.864(7)\
& 0.7801(54) & 0.7805(26) & & &\
& 0.789(11) & 0.8172(22) & & &\
$0^{-+}$ & 0.7678(47) & 0.7705(43) & $0^{+-}$ & 0.876(8) & 0.885(3)\
& 0.882(9) & 0.877(7) & & 0.974(5) & 0.942(9)\
$2^{++}$ & 0.5965(22) & 0.5972(46) & $2^{-+}$ & 0.5973(22) & 0.5934(32)\
& 0.7276(23) & 0.7218(41) & & 0.7177(51) & 0.7233(31)\
& 0.8204(59) & 0.8318(69) & & 0.8303(54) & 0.806(13)\
& 0.8542(23) & 0.8506(76) & & 0.8543(29) & 0.838(6)\
& 0.9188(85) & 0.901(18) & & 0.928(10) & 0.920(7)\
$2^{--}$ & 0.7022(50) & 0.7055(43) & $2^{+-}$ & 0.7110(41) & 0.7087(49)\
& 0.8221(26) & 0.8253(30) & & 0.8289(27) & 0.8282(23)\
& 0.9281(39) & 0.9237(33) & & 0.9097(83) & 0.903(14)\
& 0.9761(91) & 0.939(27) & & 0.959(10) & 0.942(26)\
$1^{++}$ & 0.8516(65) & 0.8504(58) & $1^{-+}$ & 0.8540(62) & 0.8550(62)\
& 0.8719(65) & 0.8680(57) & & 0.8639(59) & 0.8646(65)\
& 0.934(9) & 0.941(8) & & 0.954(10) & 0.959(8)\
& 0.963(10) & 0.957(8) & & 0.952(9) & 0.952(9)\
$1^{--}$ & 0.798(14) & 0.8131(54) & $1^{+-}$ & 0.8185(21) & 0.800(12)\
& 0.850(6) & 0.8414(56) & & 0.840(6) & 0.8500(30)\
& 0.896(25) & 0.884(17) & & 0.878(20) & 0.898(21)\
& 0.945(10) & 0.9521(32) & & 0.944(4) & 0.9539(34)\
[|cll|cll|]{}\
$J^{PC}$ & $26^2 30$ & $22^2 30$ & $J^{PC}$ & $26^2 30$ & $22^2 30$\
$0^{++}$ & 0.5640(29) & 0.5609(22) & $0^{--}$ & 0.8270(45) & 0.8311(35)\
& 0.8643(56) & 0.8532(38) & & 1.057(8) & 1.027(16)\
& 1.056(7) & 1.060(6) & & 1.227(9) & 1.237(8)\
& 1.118(7) & 1.103(8) & & 1.310(11) & 1.291(10)\
& 1.195(10) & 1.204(10) & & 1.346(14) & 1.375(13)\
& 1.235(11) & 1.236(7) & & 1.398(17) & 1.409(9)\
$0^{-+}$ & 1.232(12) & 1.211(10) & $0^{+-}$ & 1.290(64) & 1.325(46)\
& 1.367(15) & 1.367(10) & & 1.48(9) & 1.36(5)\
& 1.556(22) & 1.540(16) & & 1.647(21) & 1.638(16)\
$2^{++}$ & 0.9380(58) & 0.9357(39) & $2^{-+}$ & 0.9404(66) & 0.9494(43)\
& 1.157(8) & 1.149(7) & & 1.152(10) & 1.150(6)\
& 1.297(10) & 1.294(8) & & 1.313(11) & 1.297(9)\
& 1.343(17) & 1.353(8) & & 1.339(16) & 1.281(33)\
& 1.444(23) & 1.439(11) & & 1.385(71) & 1.479(13)\
$2^{--}$ & 1.119(9) & 1.111(5) & $2^{+-}$ & 1.127(7) & 1.122(6)\
& 1.299(17) & 1.298(8) & & 1.301(11) & 1.310(9)\
& 1.435(15) & 1.464(11) & & 1.35(8) & 1.466(12)\
& 1.557(18) & 1.517(12) & & 1.501(16) & 1.522(12)\
$1^{++}$ & 1.353(15) & 1.344(10) & $1^{-+}$ & 1.331(13) & 1.323(10)\
& 1.393(15) & 1.400(9) & & 1.370(10) & 1.366(12)\
& 1.499(17) & 1.494(11) & & 1.41(7) & 1.514(11)\
& 1.526(16) & 1.518(11) & & 1.45(9) & 1.504(13)\
$1^{--}$ & 1.271(12) & 1.282(9) & $1^{+-}$ & 1.270(11) & 1.287(8)\
& 1.317(19) & 1.335(11) & & 1.330(14) & 1.330(9)\
& 1.490(17) & 1.443(12) & & 1.488(11) & 1.475(10)\
& 1.478(20) & 1.485(14) & & 1.475(15) & 1.449(56)\
[|cl|cll|cl|]{}\
$\beta$ & $\tfrac{1}{N}\text{Tr}\langle U_p\rangle$ & $L_s^2L_t$ & $a\surd\sigma_f$ & $am_G$ & $L_s^2L_t$ & $a\surd\sigma_f$\
30.0 & 0.96639434(4) & $120^290$ & 0.04579(7) & 0.2167(5) & $84^2100$ & 0.04573(6)\
26.5 & 0.96191273(4) & $104^280$ & 0.05194(7) & 0.2459(6) & $72^290$ & 0.05186(7)\
23.5 & 0.95699656(4) & $96^264$ & 0.05869(8) & 0.2775(6) & $68^280$ & 0.05869(7)\
20.0 & 0.94937113(8) & $80^264$ & 0.06929(7) & 0.3272(10) & $56^272$ & 0.06926(7)\
16.0 & 0.93649723(10) & $68^248$ & 0.08760(11) & 0.4112(12) & $44^254$ & 0.08743(11)\
12.0 & 0.91482126(21) & $50^240$ & 0.11847(14) & 0.5594(14) & $34^240$ & 0.11832(17)\
9.0 & 0.88544949(27) & $36^232$ & 0.16075(16) & 0.7624(15) & $22^232$ & 0.16082(16)\
7.0 & 0.85112757(60) & $28^224$ & 0.21217(45) & 1.0012(22) & $18^232$ & 0.21214(25)\
6.0 & 0.82477909(68) & $25^220$ & 0.25098(50) & 1.1908(26) & $18^220$ & 0.25260(25)\
5.0 & 0.7868676(11) & $20^220$ & 0.3134(11) & 1.4689(60) & $14^220$ & 0.3113(8)\
4.5 & 0.7608386(13) & $18^220$ & 0.3557(17) & 1.665(6) & $14^220$ & 0.3524(13)\
[|c|cll|cl|]{}\
$\beta$ & lattice & $\tfrac{1}{N}\text{Tr}\langle U_p\rangle$ & $a\surd\sigma_f$ & lattice & $am_G$\
49.450 & $52\times 74^2$ & 0.9452122(3) & 0.069694(69)& $74^3$ & 0.3046(11)\
40.400 & $44\times 62^2$ & 0.9326799(3) & 0.086106(92)& $62^3$ & 0.3759(6)\
33.130 & $34\times 48^2$ & 0.917513(2) & 0.10611(10)& $48^3$ & 0.4625(15)\
26.788 & $27\times 40^2$ & 0.897308(2) & 0.13326(15)& $38^240$ & 0.5799(18)\
19.154 & $18\times 40^2$ & 0.854215(2) & 0.19292(22)& $26^240$ & 0.8337(33)\
13.621 & $13\times 36^2$ & 0.789856(2) & 0.28954(38)& $18^236$ & 1.236(6)\
[|cl|cll|cl|]{}\
$\beta$ & $\tfrac{1}{N}\text{Tr}\langle U_p\rangle$ & $L_s^2L_t$ & $a\surd\sigma_f$ & $am_G$ & $L_s^2L_t$ & $a\surd\sigma_f$\
86.0 & 0.94081019(5) & $70^280$ & 0.07378(8) & 0.3115(11) & $50^280$ & 0.073711(46)\
74.0 & 0.93099416(7) & $60^268$ & 0.08637(10) & 0.3650(12) & $44^268$ & 0.08634(8)\
63.0 & 0.91861509(13) & $50^256$ & 0.10252(11) & 0.4350(15) & $36^256$ & 0.10228(10)\
51.0 & 0.89878825(22) & $40^248$ & 0.12838(17) & 0.5441(20) & $30^248$ & 0.12814(15)\
40.0 & 0.86961166(24) & $30^236$ & 0.16758(20) & 0.7096(18) & $22^232$ & 0.16745(16)\
28.0 & 0.8093380(7) & $20^224$ & 0.25205(33) & 1.059(4) & $14^224$ & 0.25147(26)\
[|c|cll|cl|]{}\
$\beta$ & lattice & $\tfrac{1}{N}\text{Tr}\langle U_p\rangle$ & $a\surd\sigma_f$ & lattice & $am_G$\
206.84 & $52.56^2$ & 0.9425827(1) & 0.07031(4) & $60^3$ & 0.2922(7)\
206.0 & $54^260$ & 0.9423440(1) & 0.07060(7) & $54^260$ & 0.2948(10)\
171.0 & — & 0.9302656(2) & 0.085827(21) & $50^3$ & 0.3583(12)\
139.870 & $34.48^2$ & 0.9142825(5) & 0.10630(12) & $40^3$ & 0.4434(18)\
113.176 & $27.40^2$ & 0.893280(2) & 0.13360(17) & $32^240$ & 0.5574(27)\
81.019 & $18.40^2$ & 0.848402(2) & 0.19334(23) & $22^240$ & 0.8025(27)\
58.906 & $13.36^2$ & 0.786018(2) & 0.2816(5) & $16^236$ & 1.1731(66)\
[|cclll|]{}\
$\beta$ & lattice & $\tfrac{1}{N}\text{Tr}\langle U_p\rangle$ & $a\surd\sigma_f$ & $am_G$\
370.0 & $54^260$ & 0.94220766(4) & 0.07041(8) & 0.2926(9)\
306.25 & — & — & 0.085880(24) & —\
306.25 & $44^248$ & 0.92989047(5) & 0.08597(10) & 0.3552(13)\
250.0 & $36^240$ & 0.9136369(2) & 0.10624(11) & 0.4408(17)\
200.0 & $28^232$ & 0.8911821(2) & 0.13590(23) & 0.5634(24)\
145.0 & $20^224$ & 0.8473983(4) & 0.19408(35) & 0.8053(31)\
106.0 & $14^216$ & 0.7857764(9) & 0.27911(67) & 1.1620(76)\
[|cclll|]{}\
$\beta$ & $L_s^2L_t$ & $\tfrac{1}{N}\text{Tr}\langle U_p\rangle$ & $a\surd\sigma_f$ & $am_G$\
830.0 & $50^260$ & 0.94150228(3) & 0.07105(8) & 0.2940(9)\
700.0 & $42^250$ & 0.93037943(4) & 0.08485(8) & 0.3508(9)\
565.0 & $34^240$ & 0.9132337(1) & 0.10647(15) & 0.4418(17)\
450.0 & $28^230$ & 0.8901571(2) & 0.13649(19) & 0.5636(22)\
355.0 & $20^224$ & 0.8591249(3) & 0.17767(27) & 0.7286(30)\
260.0 & $14^216$ & 0.8031790(5) & 0.25405(50) & 1.045(5)\
[|cclll|]{}\
$\beta$ & $L_s^2L_t$ & $\tfrac{1}{N}\text{Tr}\langle U_p\rangle$ & $a\surd\sigma_f$ & $am_G$\
1180.0 & $34^248$ & 0.92624511(4) & 0.08984(10) & 0.3716(10)\
980.0 & $28^240$ & 0.91071306(7) & 0.10976(15) & 0.4514(18)\
800.0 & $20^2,22^2,26^230$ & 0.8897983(2) & 0.13664(14) & 0.5609(22)\
560.0 & $15^224$ & 0.8394872(4) & 0.20369(31) & 0.8356(39)\
430.0 & $11^216$ & 0.7861389(5) & 0.27859(40) & 1.1461(84)\
[|c|cccc|ccc|]{}\
$\beta$ & $l/a$ & $aE_{k=0}$ & $aE_{k=2A}$ & $aE_{k=2S}$ & $l/a$ & $aE_{k=0}$ & $aE_{k=2A}$\
28.0 & 14 & 1.794(33) & 1.1627(66) & 1.99(2) & 20 & 2.73(12) & 1.672(9)\
40.0 & 22 & 1.271(7) & 0.8101(24) & 1.4244(74) & 30 & 1.797(14) & 1.126(5)\
51.0 & 30 & 1.046(6) & 0.6519(28) & 1.150(20) & 40 & 1.418(12) & 0.8846(28)\
63.0 & 36 & 0.7749(58) & 0.4941(14) & 0.853(10) & 50 & 1.086(15) & 0.6998(26)\
74.0 & 44 & 0.6870(56) & 0.4346(10) & 0.7623(72) & 60 & 0.952(7) & 0.5973(17)\
86.0 & 50 & 0.5669(35) & 0.3578(7) & 0.6306(37) & 70 & 0.800(13) & 0.5067(18)\
[|cc|cccccc|]{}\
$\beta$ & $l/a$ & $aE_{k=0}$ & $aE_{k=2A}$ & $aE_{k=2S}$ & $aE_{k=3A}$ & $aE_{k=3M}$ & $aE_{k=4}$\
106.0 & 14 & 2.25(11) & 1.881(32) & 2.47(9) & 2.42(8) & 3.31(54) & 2.50(8)\
145.0 & 20 & 1.537(13) & 1.294(9) & 1.654(15) & 1.629(14) & 2.204(27) & 1.760(18)\
200.0 & 28 & 1.034(8) & 0.8867(26) & 1.1299(68) & 1.1221(54) & 1.457(30) & 1.2011(35)\
250.0 & 36 & 0.8260(39) & 0.6992(12) & 0.8883(56) & 0.8864(29) & 1.193(9) & 0.9520(24)\
306.25 & 44 & 0.6606(25) & 0.5582(17) & 0.7137(41) & 0.7056(43) & 0.9545(57) & 0.7593(17)\
370.0 & 54 & 0.5466(26) & 0.4576(13) & 0.5933(23) & 0.5818(23) & 0.7919(23) & 0.6267(27)\
[|cc|cccccc|]{}\
$\beta$ & $l/a$ & $aE_{k=0}$ & $aE_{k=2A}$ & $aE_{k=2S}$ & $aE_{k=3A}$ & $aE_{k=3M}$ & $aE_{k=4}$\
260.0 & 14 & 1.78(3) & 1.658(14) & 1.904(25) & 2.23(5) & 2.67(8) & 2.66(8)\
355.0 & 20 & 1.258(11) & 1.1449(51) & 1.3283(77) & 1.5787(73) & 1.878(12) & 1.76(8)\
450.0 & 28 & 1.0505(65) & 0.9480(39) & 1.106(10) & 1.3027(72) & 1.5891(58) & 1.518(33)\
565.0 & 34 & 0.7721(43) & 0.7007(19) & 0.8177(48) & 0.9663(35) & 1.1667(43) & 1.1403(68)\
700.0 & 42 & 0.6068(24) & 0.5512(15) & 0.6481(26) & 0.7562(40) & 0.9216(54) & 0.894(10)\
830.0 & 50 & 0.5047(16) & 0.4569(16) & 0.5483(23) & 0.6261(25) & 0.7567(23) & 0.7536(44)\
[|cc|cccccc|]{}\
$\beta$ & $l/a$ & $aE_{k=0}$ & $aE_{k=2A}$ & $aE_{k=2S}$ & $aE_{k=3A}$ & $aE_{k=3M}$ & $aE_{k=4}$\
430.0 & 11 & 1.652(24) & 1.553(14) & 1.769(21) & 2.253(43) & 2.346(47) & 2.796(84)\
560.0 & 15 & 1.198(11) & 1.1552(73) & 1.2900(89) & 1.641(15) & 1.767(13) & 2.022(25)\
800.0 & 22 & 0.7938(50) & 0.7537(28) & 0.8506(31) & 1.0775(50) & 1.2250(40) & 1.320(20)\
980.0 & 28 & 0.6579(39) & 0.6196(15) & 0.7003(26) & 0.8846(30) & 1.0131(31) & 1.0836(74)\
1180.0 & 34 & 0.5355(27) & 0.5025(23) & 0.5659(22) & 0.7162(45) & 0.8043(51) & 0.9010(47)\
Tables: continuum and large-N limits {#section_appendix_continuum}
====================================
group $\surd\sigma_f/g^2N$ $\chi^2/n_{dof}$ $m_G/g^2N$ $c_1$ $\chi^2/n_{dof}$
-------------- ---------------------- ------------------ ------------ ----------- ------------------
$SU(2)^a$ 0.16745(11) 4.7/9 0.7930(11) -0.036(2) 16.5/9
$SU(2)^b$ 0.16780(15) 7.4/6 0.7930(11) -0.035(2) 7.0/9
$SU(3)$ 0.18389(17) 1.0/4 0.8066(20) -0.042(3) 1.5/4
$SU(4)^a$ 0.18957(16) 2.8/4 0.8057(22) -0.038(4) 2.8/4
$SU(4)^b$ 0.18968(21) 2.0/4 0.8048(27) -0.038(4) 2.8/4
$SU(6)$ 0.19329(11) 3.3/5 0.8060(22) -0.035(4) 4.0/5
$SU(8)$ 0.19486(16) 11.4/4 0.8072(28) -0.033(5) 1.7/4
$SU(12)$ 0.19557(24) 3.6/4 0.8120(26) -0.040(4) 1.5/4
$SU(16)$ 0.19549(27) 1.4/3 0.8093(35) -0.037(6) 2.0/3
$SU(\infty)$ 0.19636(12) 5.0/4 0.8102(12) 6.6/7
: Continuum limit of $\surd\sigma_f/g^2N$ and the mass gap, $m_G/g^2N$, with the total $\chi^2$ of the fit to $n_{dof}$ degrees of freedom, and with the coefficient $c_1$ of the $ag^2_IN$ correction term. Also the extrapolation to $N=\infty$. Superscripts $a,b$ indicate medium and large lattice volumes respectively.[]{data-label="table_sigf_cont"}
[|c|cccccc|]{}\
group & $k=0$ & $k=2A$ & $k=2S$ & $k=3A$ & $k=3M$ & $k=4$\
SU(2) & 1.497(10) & – & – & – & – & –\
SU(4$)^a$ & 1.4617(37) & 1.1649(11) & 1.5377(38) & – & – & –\
SU(4)$^b$ & 1.4600(60) & 1.1619(21) & – & – & – & –\
SU(6) & 1.4297(42) & 1.2753(22) & 1.4961(41) & 1.3573(33) & 1.6861(67) & –\
SU(8) & 1.4389(28) & 1.3234(18) & 1.4971(30) & 1.4875(28) & 1.7306(37) & 1.5424(25)\
SU(12) & 1.4311(24) & 1.3626(19) & 1.4921(47) & 1.5920(30) & 1.7428(43) & 1.7385(52)\
SU(16) & 1.4178(34) & 1.3771(22) & 1.4590(26) & 1.6356(37) & 1.7605(37) & 1.8210(54)\
[|clc|clc|]{}\
$J^{PC}$ & $M/\sqrt{\sigma}$ & $\chi^2/n_{df}$ & $J^{PC}$ & $M/\sqrt{\sigma}$ & $\chi^2/n_{df}$\
$0^{++}$ & 4.7367(55) & 6.6/9 & $0^{-+}$ & 9.884(25) & 12.5/9\
& 6.861(14) & 8.3/5 & & 11.079(30) & 16.4/9\
& 8.382(14) & 8.1/9 & & 11.50(5) & 19.8/7\
& 9.278(16) & 7.7/9 & & 12.39(7) & 12.4/7\
& 9.708(21) & 7.2/8 & & &\
& 9.910(32) & 7.0/5 & & &\
& 10.05(7) & 1.1/4 & & &\
$2^{++}$ & 7.762(10) & 12.6/9 & $2^{-+}$ & 7.795(12) & 11.5/9\
& 9.107(20) & 11.1/9 & & 9.123(24) & 18.7/9\
& 10.138(35) & 5.7/9 & & 10.220(33) & 7.0/8\
& 10.624(23) & 18.3/7 & & 10.617(30) & 8.4/6\
& 10.823(40) & 6.2/5 & & 10.978(32) & 10.6/7\
$1^{++}$ & 10.553(31) & 18.9/9 & $1^{-+}$ & 10.538(28) & 22.7/9\
& 11.049(37) & 11.9/9 & & 11.020(32) & 19.4/9\
& 11.994(45) & 28.5/7 & & 11.840(51) & 17.9/7\
& 12.273(50) & 19.7/7 & & 12.037(65) & 2.2/7\
[|clc|clc|]{}\
$J^{PC}$ & $M/\sqrt{\sigma}$ & $\chi^2/n_{df}$ & $J^{PC}$ & $M/\sqrt{\sigma}$ & $\chi^2/n_{df}$\
$0^{++}$ & 4.3683(73) & 2.0/4 & $0^{--}$ & 6.391(14) & 2.1/4\
& 6.486(13) & 5.7/4 & & 7.983(26) & 3.8/3\
& 8.012(27) & 5.3/4 & & 9.338(38) & 0.2/3\
& 8.632(29) & 3.6/4 & & 10.145(52) & 2.4/3\
& 9.353(34) & 1.3/3 & & &\
& 9.414(37) & 3.2/3 & & &\
$0^{-+}$ & 9.234(43) & 2.0/4 & $0^{+-}$ & 10.55(6) & 1.1/4\
& 10.47(8) & 1.1/2 & & 11.55(12) & 1.4/3\
& 11.44(10) & 15.5/2 & & &\
$2^{++}$ & 7.241(17) & 1.5/4 & $2^{-+}$ & 7.261(16) & 6.8/4\
& 8.730(22) & 2.5/4 & & 8.760(23) & 16.6/4\
& 9.589(46) & 0.9/3 & & 9.992(33) & 1.6/3\
& 10.082(54) & 4.6/3 & & 10.05(9) & 7.6/3\
& 9.99(19) & 9.6/3 & & 10.86(9) & 6.9/3\
$2^{--}$ & 8.599(30) & 2.8/4 & $2^{+-}$ & 8.665(24) & 8.3/4\
& 9.940(44) & 3.5/3 & & 9.871(63) & 1.1/3\
& 11.285(54) & 7.2/3 & & 11.17(9) & 2.5/2\
& 12.09(12) & 5.7/3 & & &\
& 12.09(12) & 5.7/3 & & &\
$1^{++}$ & 10.221(50) & 11.0/4 & $1^{-+}$ & 10.316(46) & 2.8/4\
& 10.443(44) & 2.3/4 & & 10.294(55) & 15.3/4\
& 11.415(87) & 2.6/3 & & 11.50(5) & 5.1/3\
$1^{--}$ & 10.039(44) & 7.0/4 & $1^{+-}$ & 9.964(58) & 4.8/4\
& 10.341(60) & 2.1/4 & & 10.260(86) & 3.7/3\
& 11.25(8) & 3.7/2 & & 11.27(8) & 5.0/2\
[|clc|clc|]{}\
$J^{PC}$ & $M/\sqrt{\sigma}$ & $\chi^2/n_{df}$ & $J^{PC}$ & $M/\sqrt{\sigma}$ & $\chi^2/n_{df}$\
$0^{++}$ & 4.242(9) & 3.3/4 & $0^{--}$ & 6.216(15) & 6.3/4\
& 6.440(13) & 8.6/4 & & 7.867(22) & 9.0/4\
& 7.930(24) & 3.1/3 & & 9.248(30) & 5.8/4\
& 8.319(32) & 2.6/3 & & 9.795(52) & 0.3/3\
& 9.040(51) & 4.3/3 & & 10.44(5) & 20.5/3\
& 9.297(31) & 4.8/3 & & 10.36(8) & 10.5/3\
& 9.560(74) & 5.3/3 & & &\
& 9.988(88) & 0.5/2 & & &\
$0^{-+}$ & 9.204(30) & 4.6/4 & $0^{+-}$ & 10.29(7) & 5.4/4\
& 10.327(60) & 5.8/4 & & 11.39(10) & 7.6/2\
& 11.47(10) & 0.2/3 & & 12.45(11) & 12.3/3\
$2^{++}$ & 7.091(17) & 2.7/4 & $2^{-+}$ & 7.096(20) & 2.9/4\
& 8.597(27) & 7.3/4 & & 8.531(32) & 13.7/4\
& 9.785(32) & 3.5/4 & & 9.736(45) & 3.6/4\
& 9.981(55) & 5.4/3 & & 10.117(65) & 6.7/3\
& 10.85(7) & 8.9/3 & & 11.01(6) & 1.6/3\
$2^{--}$ & 8.368(40) & 2.1/4 & $2^{+-}$ & 8.475(30) & 5.7/4\
& 9.737(24) & 3.8/4 & & 9.823(27) & 4.1/4\
& 10.889(47) & 6.3/4 & & 10.923(45) & 10.7/4\
& 11.62(9) & 12.9/3 & & 11.39(8) & 9.0/3\
& 12.06(5) & 2.6/3 & & 12.01(11) & 0.8/2\
$1^{++}$ & 9.952(62) & 5.0/3 & $1^{-+}$ & 9.997(51) & 6.4/4\
& 10.09(6) & 0.7/3 & & 10.14(6) & 8.0/4\
& 11.20(7) & 18.5/3 & & 11.12(10) & 9.2/4\
& 11.27(8) & 0.2/3 & & 11.58(6) & 15.7/3\
& 11.59(8) & 2.3/3 & & 11.49(9) & 3.9/3\
$1^{--}$ & 9.797(39) & 2.4/4 & $1^{+-}$ & 9.668(41) & 3.3/4\
& 10.071(37) & 9.1/3 & & 10.043(42) & 9.1/4\
& 10.88(10) & 2.5/4 & & 10.81(12) & 4.4/3\
& 11.26(5) & 11.5/3 & & 11.21(7) & 17.0/3\
& 11.41(7) & 2.5/3 & & 11.43(6) & 3.6/3\
[|clc|clc|]{}\
$J^{PC}$ & $M/\sqrt{\sigma}$ & $\chi^2/n_{df}$ & $J^{PC}$ & $M/\sqrt{\sigma}$ & $\chi^2/n_{df}$\
$0^{++}$ & 4.164(8) & 2.3/4 & $0^{--}$ & 6.148(13) & 2.7/4\
& 6.357(13) & 1.2/4 & & 7.804(21) & 1.0/4\
& 7.927(18) & 1.5/4 & & 9.155(39) & 5.0/3\
& 8.232(32) & 3.3/3 & & 9.777(74) & 4.6/2\
& 9.003(56) & 2.4/3 & & &\
& 9.106(54) & 2.2/2 & & &\
& 9.24(13) & 7.2/2 & & &\
$0^{-+}$ & 9.030(44) & 3.6/4 & $0^{+-}$ & 10.089(53) & 3.0/4\
& 10.175(56) & 5.3/4 & & 11.48(7) & 17.6/3\
$2^{++}$ & 6.983(19) & 2.9/4 & $2^{-+}$ & 6.996(16) & 6.7/4\
& 8.513(21) & 4.5/4 & & 8.472(26) & 4.7/4\
& 9.728(31) & 1.6/4 & & 9.744(36) & 11.5/4\
& 10.02(10) & 4.2/2 & & 10.024(56) & 9.6/2\
& 11.30(14) & 0.5/2 & & 10.97(9) & 27.6/2\
$2^{--}$ & 8.318(34) & 7.8/4 & $2^{+-}$ & 8.275(34) & 2.3/4\
& 9.626(46) & 10.3/4 & & 9.700(57) & 6.2/4\
& 10.911(57) & 1.6/3 & & 10.94(9) & 11.3/2\
& 11.811(67) & 2.3/2 & & 11.20(10) & 2.5/2\
$1^{++}$ & 10.163(45) & 9.0/4 & $1^{-+}$ & 9.863(57) & 3.0/4\
& 10.084(44) & 9.0/4 & & 10.154(46) & 3.1/4\
& 11.194(89) & 3.4/4 & & 11.15(10) & 6.2/4\
& 11.21(11) & 9.1/2 & & &\
$1^{--}$ & 9.568(43) & 1.8/4 & $1^{+-}$ & 9.469(43) & 1.8/4\
& 9.872(57) & 6.6/4 & & 10.040(50) & 6.9/4\
& 10.874(57) & 1.4/3 & & 10.838(61) & 8.3/3\
& 11.11(9) & 1.5/2 & & 11.10(9) & 1.1/2\
[|cllc|cllc|]{}\
$J^{PC}$ & $M/\sqrt{\sigma}$ & $c_1$ & $\chi^2/n_{df}$ & $J^{PC}$ & $M/\sqrt{\sigma}$ & $c_1$ & $\chi^2/n_{df}$\
$0^{++}$ & 4.144(10) & 0.2(6) & 1.5/4 & $0^{--}$ & 6.102(14) & 0.3(1.1) & 5.3/4\
& 6.332(15) & -3.1(1.3) & 3.7/4 & & 7.803(26) & -2.8(3.3) & 6.0/4\
& 7.855(22) & -0.9(2.3) & 3.2/4 & & 9.177(38) & -1.9(4.8) & 2.2/3\
& 8.090(36) & -1.6(2.9) & 3.3/3 & & 9.803(38) & -16.3(4.6) & 1.3/2\
& 8.969(78) & -13.1(11.8) & 0.5/2 & & 10.00(11) & -1.5(8.8) & 1.9/2\
& 9.152(40) & -9.0(4.6) & 0.2/2 & & & &\
& 9.58(6) & -14.8(5.9) & 6.4/2 & & & &\
$0^{-+}$ & 9.035(47) & -7.1(4.2) & 3.0/4 & $0^{+-}$ & 10.35(6) & -14.9(5.3) & 11.2/4\
& 10.14(7) & 2.3(7.7) & 4.7/2 & & 11.31(10) & -14.6(8.3) & 8.1/2\
$2^{++}$ & 6.952(18) & -0.5(1.7) & 6.7/4 & $2^{-+}$ & 6.942(21) & -1.2(1.7) & 4.8/4\
& 8.412(43) & -1.7(3.7) & 4.2/4 & & 8.486(38) & -3.9(3.3) & 4.8/4\
& 9.392(73) & 1.4(5.3) & 1.5/3 & & 9.760(37) & -5.8(3.9) & 0.3/3\
& 9.811(60) & -3.2(6.0) & 7.9/3 & & 9.982(66) & -12.2(6.0) & 3.9/3\
& 10.724(70) & -0.7(6.3) & 4.0/3 & & 10.98(9) & -11.2(9.6) & 5.0/3\
$2^{--}$ & 8.273(22) & -2.6(2.8) & 5.3/4 & $2^{+-}$ & 8.273(34) & -0.8(2.7) & 2.1/4\
& 9.667(35) & -7.4(4.0) & 0.5/3 & & 9.637(42) & -1.3(4.5) & 1.7/3\
& 10.758(62) & -6.1(7.0) & 4.1/3 & & 10.940(54) & -14.7(7.1) & 4.3/3\
& 11.29(11) & -9.4(12.7) & 1.0/2 & & 11.06(14) & -11.4(22.4) & 2.3/2\
$1^{++}$ & 9.941(60) & -6.6(5.3) & 2.2/4 & $1^{-+}$ & 9.959(50) & -7.5(4.5) & 7.8/4\
& 10.074(70) & -5.1(5.1) & 5.8/4 & & 10.054(61) & -0.2(5.6) & 3.0/4\
& 11.12(9) & 1.3(7.6) & 11.3/3 & & 11.05(12) & -11.2(9.4) & 1.5/3\
& 10.94(16) & 30.9(15.0) & 11.6/3 & & & &\
$1^{--}$ & 9.449(50) & -1.2(5.0) & 2.8/4 & $1^{+-}$ & 9.505(47) & -4.2(4.7) & 4.0/4\
& 9.822(52) & -5.5(4.1) & 1.3/4 & & 9.859(43) & -2.8(5.3) & 5.1/4\
& 10.71(7) & 7.4(6.6) & 9.1/3 & & 10.76(8) & 1.0(7.4) & 3.8/3\
& 11.06(7) & -3.7(9.2) & 3.4/2 & & 11.07(6) & -4.1(6.9) & 6.1/2\
[|clc|clc|]{}\
$J^{PC}$ & $M/\sqrt{\sigma}$ & $\chi^2/n_{df}$ & $J^{PC}$ & $M/\sqrt{\sigma}$ & $\chi^2/n_{df}$\
$0^{++}$ & 4.140(9) & 2.5/4 & $0^{--}$ & 6.070(15) & 3.3/4\
& 6.320(14) & 7.7/4 & & 7.787(25) & 1.6/4\
& 7.852(20) & 5.9/4 & & 9.132(40) & 7.5/3\
& 8.188(20) & 4.2/4 & & 9.656(40) & 0.4/3\
& 8.955(44) & 4.9/3 & & 9.987(71) & 0.7/2\
& 9.173(40) & 2.5/2 & & 10.263(61) & 2.3/2\
$0^{-+}$ & 8.991(50) & 3.9/4 & $0^{+-}$ & 10.21(6) & 7.2/4\
& 10.16(7) & 4.3/4 & & 11.29(9) & 4.5/4\
& 11.23(8) & 0.55/3 & & 12.10(11) & 4.4/3\
$2^{++}$ & 6.938(18) & 3.4/4 & $2^{-+}$ & 6.947(16) & 0.3/4\
& 8.384(50) & 0.6/4 & & 8.506(25) & 6.2/4\
& 9.505(45) & 1.5/4 & & 9.611(54) & 6.5/4\
& 9.862(60) & 2.2/3 & & 9.802(60) & 2.8/3\
& 10.727(76) & 2.4/2 & & 10.826(70) & 5.8/2\
$2^{--}$ & 8.285(31) & 5.4/4 & $2^{+-}$ & 8.247(35) & 2.7/4\
& 9.619(44) & 4.0/4 & & 9.656(30) & 1.3/4\
& 10.692(53) & 3.9/4 & & 10.86(6) & 5.8/4\
& 11.16(11) & 0.6/2 & & 11.21(7) & 2.1/2\
& 12.04(8) & 9.6/2 & & 12.02(10) & 3.5/2\
$1^{++}$ & 9.876(50) & 5.4/4 & $1^{-+}$ & 9.924(52) & 1.0/4\
& 10.08(8) & 17.1/4 & & 10.07(7) & 3.0/4\
& 11.08(8) & 15.2/4 & & 11.24(6) & 19.4/4\
& 11.21(10) & 20.8/2 & & 11.16(11) & 0.7/2\
$1^{--}$ & 9.388(47) & 2.5/4 & $1^{+-}$ & 9.448(50) & 3.8/4\
& 9.724(51) & 3.5/4 & & 9.805(75) & 1.6/4\
& 10.70(7) & 7.8/4 & & 10.80(7) & 2.9/4\
& 10.98(7) & 3.1/2 & & 10.95(9) & 3.7/2\
[|clc|clc|]{}\
$J^{PC}$ & $M/\sqrt{\sigma}$ & $\chi^2/n_{df}$ & $J^{PC}$ & $M/\sqrt{\sigma}$ & $\chi^2/n_{df}$\
$0^{++}$ & 4.129(11) & 1.9/3 & $0^{--}$ & 6.098(19) & 0.4/3\
& 6.295(18) & 0.7/3 & & 7.751(32) & 6.6/3\
& 7.746(45) & 2.9/3 & & 9.022(66) & 3.8/3\
& 8.157(41) & 2.3/2 & & 9.716(50) & 5.6/2\
& 9.037(61) & 0.2/2 & & 9.935(86) & 3.9/2\
& 9.047(54) & 2.1/2 & & 10.40(10) & 1.1/2\
$0^{-+}$ & 9.012(54) & 1.2/3 & $0^{+-}$ & 10.07(8) & 0.5/3\
& 10.28(10) & 5.6/3 & & 11.44(14) & 1.0/3\
& 11.63(9) & 1.6/3 & & 12.63(17) & 0.1/2\
$2^{++}$ & 6.937(30) & 3.4/3 & $2^{-+}$ & 6.991(30) & 3.8/3\
& 8.458(38) & 5.1/3 & & 8.498(35) & 0.3/3\
& 9.721(56) & 4.5/3 & & 9.81(6) & 10.8/3\
& 9.923(52) & 3.6/2 & & 9.99(6) & 3.9/2\
& 10.93(9) & 0.5/2 & & 10.85(11) & 5.0/2\
$2^{--}$ & 8.164(54) & 6.6/3 & $2^{+-}$ & 8.305(37) & 2.4/3\
& 9.614(46) & 1.8/3 & & 9.610(73) & 2.6/3\
& 10.71(8) & 0.1/2 & & 10.92(11) & 6.1/2\
& 11.19(13) & 0.6/2 & & 11.53(10) & 7.4/2\
$1^{++}$ & 9.90(7) & 1.0/3 & $1^{-+}$ & 9.80(9) & 0.4/3\
& 9.88(13) & 24.5/3 & & 9.94(8) & 3.2/3\
& 11.19(17) & 0.5/2 & & 11.43(9) & 0.5/2\
& 11.23(14) & 8.1/2 & & 11.23(17) & 9.2/2\
$1^{--}$ & 9.53(6) & 6.1/3 & $1^{+-}$ & 9.36(8) & 9.4/3\
& 9.85(6) & 4.0/3 & & 9.82(8) & 5.5/3\
& 10.76(11) & 2.1/2 & & 10.71(12) & 1.8/3\
& 11.20(19) & 1.1/2 & & 11.07(16) & 2.0/2\
[|c|lcccc|]{}\
$J^{PC}$ & $M/\sqrt{\sigma}$ & $c_1$ & $c_2$ & $N\geq $ & $\chi^2/n_{df}$\
$0^{++}$ & 4.116(6) & 2.00(15) & 1.94(53) & 2 & 4.1/4\
& 6.308(10) & 1.72(20) & & 3 & 5.8/4\
& 7.844(14) & 1.59(39) & & 3 & 10.8/4\
& 8.147(19) & 2.63(91) & & 4 & 8.9/3\
& 8.950(25) & 3.06(18) & & 2 & 9.6/5\
& 9.087(20) & 3.25(23) & & 2 & 6.3/5\
$0^{-+}$ & 8.998(28) & 2.53(21) & & 3 & 4.7/4\
& 10.101(33) & 3.89(24) & & 2 & 4.1/5\
$0^{--}$ & 6.060(9) & 2.88(21) & & 3 & 6.0/4\
& 7.759(15) & 1.92(37) & & 3 & 1.5/4\
& 9.110(35) & 2.11(55) & & 3 & 3.2/4\
& 9.709(30) & 1.7(1.5) & & 4 & 6.0/3\
$0^{+-}$ & 10.133(37) & 3.38(94) & & 3 & 17.0/4\
& 11.350(57) & 1.70(3) & & 3 & 3.3/4\
[|c|lcccc|]{}\
$J^{PC}$ & $M/\sqrt{\sigma}$ & $c_1$ & $c_2$ & $N\geq$ & $\chi^2/n_{df}$\
$2^{++}$ & 6.914(13) & 2.62(32) & 3.1(1.2) & 2 & 0.5/4\
& 8.423(15) & 2.74(14) & & 2 & 3.8/5\
& \[9.59(9)\] & & & &\
& 9.866(34) & 1.91(84) & & 3 & 4.0/4\
& \[10.81(10)\] & & & &\
$2^{-+}$ & 6.930(13) & 2.42(34) & 4.20(1.25) & 2 & 5.1/4\
& 8.488(21) & 0.41(88) & & 4 & 1.9/3\
& 9.732(32) & 0.19(1.25) & & 4 & 7.3/3\
& 9.905(30) & 2.83(24) & & 2 & 10.3/5\
& 10.914(39) & 0.26(26) & & 2 & 5.4/5\
$2^{--}$ & 8.223(18) & 3.24(48) & & 3 & 5.8/4\
& 9.592(24) & 2.72(56) & & 3 & 3.9/4\
& 10.715(41) & 3.2(1.4) & & 4 & 4.6/3\
& 11.34(6) & 7.0(1.6) & & 3 & 28.4/4\
$2^{+-}$ & 8.224(20) & 3.92(40) & & 3 & 6.1/4\
& 9.629(25) & 2.74(68) & & 3 & 2.6/4\
& 10.870(41) & 1.74(1.06) & & 3 & 3.9/4\
& 11.26(6) & 1.37(2.38) & & 3 & 11.5/4\
[|c|lccc|]{}\
$J^{PC}$ & $M/\sqrt{\sigma}$ & $c_1$ & $N\geq$ & $\chi^2/n_{df}$\
$1^{++}$ & 9.912(26) & 2.57(24) & 2 & 10.5/5\
& 9.984(32) & 4.2(28) & 2 & 10.6/5\
& 11.040(43) & 3.77(35) & 2 & 2.5/5\
& 11.157(80) & 1.65(2.48) & 4 & 2.7/3\
$1^{-+}$ & 9.886(27) & 2.67(21) & 2 & 14.5/5\
& 9.969(30) & 4.12(25) & 2 & 12.2/5\
& 11.195(40) & 2.55(37) & 2 & 15.9/5\
$1^{--}$ & 9.401(29) & 5.88(65) & 3 & 5.87/4\
& 9.740(32) & 5.33(80) & 3 & 3.39/4\
& 10.684(44) & 4.9(1.2) & 3 & 3.04/4\
& 10.98(6) & 4.4(1.6) & 4 & 1.35/3\
$1^{+-}$ & 9.380(31) & 4.95(80) & 3 & 3.61/4\
& 9.828(36) & 3.85(98) & 3 & 5.77/4\
& 10.712(46) & 4.6(1.3) & 3 & 3.60/4\
& 10.99(6) & 3.6(2.1) & 4 & 0.86/3\
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Information geometry, that is a differential geometric method of information theory, gives a natural definition of informational quantity from the projection theorem. In this letter, we report that the entropy production and recent results in information thermodynamics can be obtained from this projection in a unified way. This result implies that a calculation of the entropy production can be regarded as an optimization problem to minimize the length. Moreover, we geometrically discuss the hierarchy of thermodynamic inequalities and the additivity of the partial entropy productions. The violation of this additivity gives a measure of the integrated information theory.'
author:
- 'Sosuke Ito$^{1,2}$'
title: Unified framework for the second law of thermodynamics and information thermodynamics based on information geometry
---
Information geometry [@amari2007methods; @amari2016information] is a theory of differential geometry for organizing various results in information theory, probability theory and statistics. The application of information geometry has been discussed in a variety of fields including the machine learning [@amari1992boltzmann], neuroscience [@amari1995neural], statistical physics [@tanaka2000meanfield; @Brody2008eqilibrium] and thermodynamics [@uffink1999uncertainty; @crooks2007measuring; @ito2018infogeo; @dechant2018infogeo]. In an application of information geometry, the projection theorem [@cover2012elements; @amari2001hierarchy] has a crucial role. For example, the projection theorem provides the conventional definitions of information quantities such as the mutual information, the transfer entropy and information integration [@amari2016information; @oizumi2016unified; @amari2017integration].
In last two decades, the second law of thermodynamics has been discussed in the field of stochastic thermodynamics [@sekimoto2010stochastic; @seifert2012stochastic]. In stochastic thermodynamics, the fluctuation theorem [@Jarzynski1997relation; @Crooks1999relation] leads to an expression of the second law by the Kullback-Leibler divergence [@kawai2007dissipation]. Recent results in information thermodynamics [@parrondo2015thermodynamics] such as the second law of information thermodynamics [@sagawa2010generalized; @still2012thermodynamics; @sagawa2012fluctuation; @ito2013information; @hartich2014stochastic; @horowitz2014thermodynamics; @ito2015maxwell; @spinney2016transfer; @ito2016backward; @crooks2016marginal; @Auconi2018backward], are also based on the Kullback-Leibler divergence [@ito2016flow]. Although the relationship between information geometry and thermodynamics has been discussed [@weinhold1975metric; @ruppeiner1979thermodynamics; @salamon1983thermodynamic; @feng2008length; @sivak2012thermodynamic; @polettini2013nonconvexity; @machta2015dissipation; @lahiri2016universal; @tajima2017efficiency; @rotskoff2017geometric; @takahashi2017shortcuts; @shimazaki2018neurons] and the Kullback-Leibler divergence has a crucial role in information geometry [@amari2007methods; @amari2016information], an information geometric interpretation of the entropy production has been elusive.
In this letter, we show that the entropy production can be derived from the projection theorem. To introduce a manifold related to reversible dynamics, the entropy production can be considered as the minimum length from this manifold. This fact leads to a novel interpretation of the entropy production as an optimization problem to minimize the length.
In addition, we show that the partial entropy productions in information thermodynamics can also be derived from the projection onto other manifolds. From the inclusion property of manifolds, we obtain a hierarchy that the bound by information thermodynamics is always tighter than the bound by thermodynamics. Moreover, we show that non-additivity of the partial entropy productions gives a measure of the integrated information theory [@Oizumi2014integrated; @Tononi2016integrated] known as the stochastic interaction [@Barrett2011practical; @Ay2015information]. The integrated information theory has been intensively discussed to seek a measure of dividing complex neural networks into several parts. This result provides a novel quantity of the integrated information theory form a view point of thermodynamics. If the stochastic interaction vanishes, the additivity of the partial entropy production holds, and its condition gives a nontrivial quadrangle in information geometry. We analytically illustrate these results by the two spins model.
*The projection theorem.–* We first introduce the projection theorem in information geometry [@cover2012elements; @amari2001hierarchy]. We consider a geometry of the joint probability $p_{\boldsymbol{S}}(\boldsymbol{s})$, where ${\boldsymbol{S}}=\{S_1, ..., S_N \}$ is the set of random variables and ${\boldsymbol{s}}=\{s_1, ..., s_N \}$ is the set of events. The set of probabilities gives a manifold, and the probability $p_{\boldsymbol{S}}(\boldsymbol{s})$ corresponds to a point on this manifold. If we discuss the projection theorem in information geometry, the metric is given by the Fisher information, and the connection is given by the dual affine connections [@amari2007methods].
![Schematic of the projection theorem. The set of probabilities gives a manifold $\mathcal{M}$, and the probability $p$ corresponds to a point. If $\mathcal{M}$ is flat, we have a unique solution $q^*$ of the optimization problem to minimize the Kullback-Leibler divergence between the probability $p$ and the probability $q \in \mathcal{M}$.[]{data-label="fig0"}](2ndlawgeofig0.eps){width="7cm"}
We here consider the set of probabilities $\mathcal{M}$, and an optimization problem to minimize the Kullback-Leibler divergence between $p_{\boldsymbol{S}}(\boldsymbol{s})$ and a probability $q_{\boldsymbol{S}}(\boldsymbol{s})$ on the manifold $\mathcal{M}$, $$\begin{aligned}
{\rm min}_{q_{\boldsymbol{S}} \in \mathcal{M}} D(p_{\boldsymbol{S}}||q_{\boldsymbol{S}}),\end{aligned}$$ where $D(p_{\boldsymbol{S}}||q_{\boldsymbol{S}}) = \sum_{\boldsymbol{s}} p_{\boldsymbol{S}}(\boldsymbol{s}) \ln [p_{\boldsymbol{S}}(\boldsymbol{s}) /q_{\boldsymbol{S}}(\boldsymbol{s})]$ is the Kullback-Leibler divergence between two probabilities $p_{\boldsymbol{S}}$ and $q_{\boldsymbol{S}}$. If the manifold $\mathcal{M}$ is flat, $q^*_{\boldsymbol{S}} \in \mathcal{M}$ exists for any probabilities $q_{\boldsymbol{S}} \in \mathcal{M}$ such that $$\begin{aligned}
D(p_{\boldsymbol{S}}||q_{\boldsymbol{S}})=D(p_{\boldsymbol{S}}||q^*_{\boldsymbol{S}})+D(q^*_{\boldsymbol{S}}||q_{\boldsymbol{S}}).\end{aligned}$$ This fact is known as the Pythagorean theorem in information geometry [@amari2007methods; @amari2016information], and it indicates that the geodesic connecting $p_{\boldsymbol{S}}$ and $q^*_{\boldsymbol{S}}$ is orthogonal to the dual geodesic connecting $q^*_{\boldsymbol{S}}$ and $q_{\boldsymbol{S}}$. In this case, the optimization problem has a unique solution $q^*_{\boldsymbol{S}}$, $$\begin{aligned}
{\rm min}_{q_{\boldsymbol{S}} \in \mathcal{M}} D(p_{\boldsymbol{S}}||q_{\boldsymbol{S}}) = D(p_{\boldsymbol{S}}||q^*_{\boldsymbol{S}}).\end{aligned}$$ It means that the Kullback-Leibler divergence $D(p_{\boldsymbol{S}}||q^*_{\boldsymbol{S}})$ gives the minimum length from the manifold $\mathcal{M}$ (see also Fig. 1).
This projection theorem is useful to define informational quantities. We here show an example to define the mutual information between random variables $\boldsymbol{X}$ and $\boldsymbol{Y}$. At first, we consider the set of the probabilities, $$\begin{aligned}
\mathcal{M}_{\rm I} = \left\{ q_{\boldsymbol{X}, \boldsymbol{Y}} \left| q_{\boldsymbol{X}, \boldsymbol{Y}}(\boldsymbol{x}, \boldsymbol{y})= q_{\boldsymbol{X}}(\boldsymbol{x})q_{\boldsymbol{Y}}(\boldsymbol{y})\right. \right\},\end{aligned}$$ where $q_{\boldsymbol{X}}(\boldsymbol{x})=\sum_{\boldsymbol{y}} q_{\boldsymbol{X}, \boldsymbol{Y}}(\boldsymbol{x}, \boldsymbol{y})$ and $q_{\boldsymbol{Y}}(\boldsymbol{y})=\sum_{\boldsymbol{x}} q_{\boldsymbol{X}, \boldsymbol{Y}}(\boldsymbol{x}, \boldsymbol{y})$. Two random variables $\boldsymbol{X}$ and $\boldsymbol{Y}$ are statistically independent for the probability in this manifold $\mathcal{M}_{\rm I}$. Because the Pytagorean theorem $$\begin{aligned}
D(p_{\boldsymbol{X}, \boldsymbol{Y}}||q_{\boldsymbol{X}, \boldsymbol{Y}}) = D(p_{\boldsymbol{X}, \boldsymbol{Y}}||p_{\boldsymbol{X}}p_{ \boldsymbol{Y}} )+ D(p_{\boldsymbol{X}}p_{ \boldsymbol{Y}} ||q_{\boldsymbol{X}, \boldsymbol{Y}})\end{aligned}$$ holds for any $q_{\boldsymbol{X}, \boldsymbol{Y}} \in \mathcal{M}_{\rm I}$, the mutual information $I(\boldsymbol{X};\boldsymbol{Y})$ for the joint probability $p_{\boldsymbol{X}, \boldsymbol{Y}}$ can be obtained from the projection onto this flat manifold $\mathcal{M}_{\rm I}$ [@amari2016information], $$\begin{aligned}
{\rm min}_{q_{\boldsymbol{X}, \boldsymbol{Y}} \in \mathcal{M}_{\rm I}} D(p_{\boldsymbol{X}, \boldsymbol{Y}}||q_{\boldsymbol{X}, \boldsymbol{Y}} ) = I(\boldsymbol{X};\boldsymbol{Y}), \end{aligned}$$ $$\begin{aligned}
I(\boldsymbol{X};\boldsymbol{Y}) &=D(p_{\boldsymbol{X}, \boldsymbol{Y}}||p_{\boldsymbol{X}}p_{ \boldsymbol{Y}} ) \nonumber \\
&= H(\boldsymbol{X})+ H(\boldsymbol{Y}) - H(\boldsymbol{X}, \boldsymbol{Y}),\end{aligned}$$ where $H(\boldsymbol{X})$ is the Shannon entropy defined as $H(\boldsymbol{X}) = - \sum_{\boldsymbol{x}} p_{\boldsymbol{X}} (\boldsymbol{x}) \ln p_{\boldsymbol{X}} (\boldsymbol{x})$. If the joint probability $p_{\boldsymbol{X}, \boldsymbol{Y}}$ is on the manifold $\mathcal{M}_{\rm I}$, we obtain $p_{\boldsymbol{X}, \boldsymbol{Y}}= p_{\boldsymbol{X}}p_{\boldsymbol{Y}}$ and $I(\boldsymbol{X};\boldsymbol{Y})=0$.
*The second law of thermodynamics.–* We next show that the entropy production can be obtained from the projection theorem. We consider the Markov process to define the entropy production. This process can be described by the path probability $p_{\boldsymbol{Z}, \boldsymbol{Z'}}(\boldsymbol{z}, \boldsymbol{z'})$, where $\boldsymbol{Z}$ and $\boldsymbol{Z}'$ are random variables of the state of the system $\mathcal{Z}$ at time $t$ and $t+dt$, respectively. The transition probability is given by the conditional probability $T(\boldsymbol{z'}|\boldsymbol{z}) = p_{ \boldsymbol{Z'}|\boldsymbol{Z}}(\boldsymbol{z'}|\boldsymbol{z})$.
We now introduce the following set of path probabilities $$\begin{aligned}
\mathcal{M}_{\rm R} = \{ q_{\boldsymbol{Z},\boldsymbol{Z'}} | q_{\boldsymbol{Z},\boldsymbol{Z'}}(\boldsymbol{z}, \boldsymbol{z'}) = q_{\boldsymbol{Z'}}(\boldsymbol{z'})T(\boldsymbol{z}|\boldsymbol{z'}) \},\end{aligned}$$ where $q_{\boldsymbol{Z'}}(\boldsymbol{z'}) = \sum_{\boldsymbol{z}} q_{\boldsymbol{Z},\boldsymbol{Z'}}(\boldsymbol{z}, \boldsymbol{z'})$. We call $\mathcal{M}_{\rm R}$ as *the reversible manifold*, because backward dynamics $q_{\boldsymbol{Z}|\boldsymbol{Z'}}= q_{\boldsymbol{Z},\boldsymbol{Z'}} /q_{\boldsymbol{Z'}}$ in this manifold are same as the forward dynamics $p_{ \boldsymbol{Z'}|\boldsymbol{Z}}$. If the path probability $p_{ \boldsymbol{Z},\boldsymbol{Z'}}$ is on this manifold, the detailed balance [@sm] satisfies, $T(\boldsymbol{z'}|\boldsymbol{z})p_{\boldsymbol{Z}}(\boldsymbol{z})=T(\boldsymbol{z}|\boldsymbol{z'})p_{\boldsymbol{Z'}}(\boldsymbol{z'})$.
We here consider the projection onto the manifold $\mathcal{M}_{\rm R}$. This manifold is flat, because the following Pythagorean theorem holds for any $q_{\boldsymbol{Z}, \boldsymbol{Z'}} \in \mathcal{M}_{\rm R}$, $$\begin{aligned}
D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||q_{\boldsymbol{Z}, \boldsymbol{Z'}}) &= D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||q^*_{\boldsymbol{Z}, \boldsymbol{Z'}}) + D(q^*_{\boldsymbol{Z}, \boldsymbol{Z'}}||q_{\boldsymbol{Z}, \boldsymbol{Z'}}), \\
q^*_{\boldsymbol{Z}, \boldsymbol{Z'}}(\boldsymbol{z}, \boldsymbol{z'})&=p_{\boldsymbol{Z'}}( \boldsymbol{z'}) T(\boldsymbol{z}|\boldsymbol{z}').\end{aligned}$$ The second term $D(q^*_{\boldsymbol{Z}, \boldsymbol{Z'}}||q_{\boldsymbol{Z}, \boldsymbol{Z'}})=D(p_{ \boldsymbol{Z'}}||q_{\boldsymbol{Z'}})$ can be interpreted as the degree of freedom in the probability distribution of $\boldsymbol{Z'}$. From this Pythagorean theorem, we obtain the projection theorem $$\begin{aligned}
{\rm min}_{q_{\boldsymbol{Z}, \boldsymbol{Z'}} \in \mathcal{M}_{\rm R}} D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||q_{\boldsymbol{Z}, \boldsymbol{Z'}}) = D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||q^*_{\boldsymbol{Z}, \boldsymbol{Z'}}).
\label{projection}\end{aligned}$$
In stochastic thermodynamics, the Kullback-Leibler divergence $D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||q^*_{\boldsymbol{Z}, \boldsymbol{Z'}})$ is equal to the entropy production $\sigma^{\mathcal{Z}}_{\rm tot}$ [@kawai2007dissipation; @sm], $$\begin{aligned}
D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||q^*_{\boldsymbol{Z}, \boldsymbol{Z'}}) &=\sigma^{\mathcal{Z}}_{\rm tot}=\sigma^{\mathcal{Z}}_{\rm sys}+ \sigma^{\mathcal{Z}}_{\rm bath}, \\
\sigma^{\mathcal{Z}}_{\rm sys}&= H(\boldsymbol{Z}') - H(\boldsymbol{Z}), \\
\sigma^{\mathcal{Z}}_{\rm bath}&= \sum_{\boldsymbol{z}, \boldsymbol{z'}} p_{\boldsymbol{Z}, \boldsymbol{Z'}} (\boldsymbol{z}, \boldsymbol{z'})\ln \frac{T(\boldsymbol{z}|\boldsymbol{z'})}{T(\boldsymbol{z'}|\boldsymbol{z})},\end{aligned}$$ where $\sigma^{\mathcal{Z}}_{\rm sys}$ is the entropy change of the system $\mathcal{Z}$, and $\sigma^{\mathcal{Z}}_{\rm bath}$ is the entropy change of the heat bath attached to the system $\mathcal{Z}$. From the projection theorem Eq. (\[projection\]), a calculation of the entropy production can be regarded as an optimization problem (see also Fig. 2), $$\begin{aligned}
\sigma^{\mathcal{Z}}_{\rm tot} = {\rm min}_{q_{\boldsymbol{Z}, \boldsymbol{Z'}} \in \mathcal{M}_{\rm R}} D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||q_{\boldsymbol{Z}, \boldsymbol{Z'}}).
\label{opt1}\end{aligned}$$ The entropy production is nonnegative, because the Kullback-Leibler divergence is nonnegative $D(p||q) \geq 0$. This nonnegativity is regarded as the second law of thermodynamics. If and only if the path probability is on the reversible manifold $p_{ \boldsymbol{Z},\boldsymbol{Z'}} \in \mathcal{M}_{\rm R}$, the entropy production vanishes $\sigma^{\mathcal{Z}}_{\rm tot} =0$.
![Schematic of the entropy production and the projection onto the reversible manifold. The entropy production is given by the minimum length from the reversible manifold $\mathcal{M}_{\rm R}$. This fact can be derived from the Pythagorean theorem.[]{data-label="fig1"}](2ndlawgeofig1.eps){width="8cm"}
Our formalization would be useful to detect the entropy production from the experimental data. Based on our framework, we can use optimization tool to calculate the entropy production in parallel with an estimation of informational quantities such as the mutual information and the transfer entropy [@schreiber2000transfer; @barnett2014transfer].
In addition, the result (\[opt1\]) gives a novel interpretation of the relaxation process to the equilibrium state from a view point of the learning process. In information geometry, the learning process is formalized as the reduction process of the minimum length between the distribution $p_t$ in each iteration $t$ and the manifold of the statistical model $\mathcal{M}_t$, i.e., $\lim_{t \to \infty } [{\rm min}_{q \in \mathcal{M}_t}D(p_t||q) ] \to 0$. In the same way, the relaxation process to the equilibrium state $\lim_{t \to \infty} [ \sigma_{\rm tot}^{\mathcal{Z}}] \to 0$ can be interpreted as the learning process of reversibility $\lim_{t \to \infty}[ {\rm min}_{q_{\boldsymbol{Z}, \boldsymbol{Z'}} \in \mathcal{M}_{\rm R}} D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||q_{\boldsymbol{Z}, \boldsymbol{Z'}})] \to 0$.
*The second law of information thermodynamics.–* Recent studies of stochastic thermodynamics reveal a connection between thermodynamics and information theory. If we consider the subsystem interacting other systems, informational quantities such as mutual information and the transfer entropy between them appear in a generalization of the second law [@sagawa2010generalized; @ito2013information]. This generalization is called as the second law of information thermodynamics. We show that the second law of information thermodynamics can also be derived from the projection theorem in the unified way [@sm0].
We consider the situation that $\mathcal{Z}$ is given by random variables of two systems $\mathcal{X}$ and $\mathcal{Y}$, i.e., $\boldsymbol{Z}= \{\boldsymbol{X},\boldsymbol{Y}\}$ and $\boldsymbol{Z'}= \{\boldsymbol{X'},\boldsymbol{Y'}\}$. Here we introduce two transition probabilities $p_{\boldsymbol{X}' |\boldsymbol{Y}',\boldsymbol{Z}} (\boldsymbol{x}' | \boldsymbol{y}',\boldsymbol{z})= T^{\mathcal{X}}_{\boldsymbol{y};\boldsymbol{y}'}(\boldsymbol{x'}|\boldsymbol{x})$ and $p_{\boldsymbol{Y}' | \boldsymbol{X}',\boldsymbol{Z}} (\boldsymbol{y}' | \boldsymbol{x}',\boldsymbol{z})= T^{\mathcal{Y}}_{\boldsymbol{x}; \boldsymbol{x}'}(\boldsymbol{y'}|\boldsymbol{y})$. The second law of information thermodynamics for the subsystem $\mathcal{X}$ is an inequality of the partial entropy changes $\sigma^{\mathcal{X}}_{\rm sys}+ \sigma^{\mathcal{X}}_{\rm bath}$ and information flow $\Theta^{\mathcal{X} \to \mathcal{Y}}$ from $\mathcal{X}$ to $\mathcal{Y}$, $$\begin{aligned}
&\sigma^{\mathcal{X}}_{\rm sys}+ \sigma^{\mathcal{X}}_{\rm bath} \geq \Theta^{\mathcal{X} \to \mathcal{Y}}, \\
&\sigma^{\mathcal{X}}_{\rm sys}= H(\boldsymbol{X'})-H(\boldsymbol{X}),\\
&\sigma^{\mathcal{X}}_{\rm bath}= \sum_{\boldsymbol{z}, \boldsymbol{z'}} p_{\boldsymbol{Z}, \boldsymbol{Z'}} (\boldsymbol{z}, \boldsymbol{z'})\ln \frac{T^{\mathcal{X}}_{\boldsymbol{y};\boldsymbol{y}'}(\boldsymbol{x'}|\boldsymbol{x})}{T^{\mathcal{X}}_{\boldsymbol{y'};\boldsymbol{y}}(\boldsymbol{x}|\boldsymbol{x'})}, \\
&\Theta^{\mathcal{X} \to \mathcal{Y}}= I(\boldsymbol{X'};\{\boldsymbol{Y},\boldsymbol{Y'}\}) -I(\boldsymbol{X};\{\boldsymbol{Y},\boldsymbol{Y'} \}).\end{aligned}$$ The term of information flow includes the (backward) directed information, which is given by the sum of mutual information at time $t$ ($t+dt$) and the (backward) transfer entropy [@schreiber2000transfer; @barnett2014transfer; @ito2016backward]. If we assume the bipartite condition $\mathcal{C}_{\rm BI}: p_{\boldsymbol{Z'}|\boldsymbol{Z}}(\boldsymbol{z'}|\boldsymbol{z}) = p_{\boldsymbol{X'}|\boldsymbol{Z}}(\boldsymbol{x'}|\boldsymbol{z})p_{\boldsymbol{Y'}|\boldsymbol{Z}}(\boldsymbol{y'}|\boldsymbol{z})$, information flow is equivalent to the learning rate [@hartich2014stochastic] up to order $O(dt^2)$ [@ito2016backward]. The bipartite condition $\mathcal{C}_{\rm BI}$ means that the transition probability $T^{\mathcal{X}}_{\boldsymbol{y};\boldsymbol{y}'}(\boldsymbol{x'}|\boldsymbol{x}) = p_{\boldsymbol{X'}|\boldsymbol{Z}}(\boldsymbol{x'}|\boldsymbol{z})$ ($T^{\mathcal{Y}}_{\boldsymbol{x};\boldsymbol{x}'}(\boldsymbol{y'}|\boldsymbol{y}) = p_{\boldsymbol{Y'}|\boldsymbol{Z}}(\boldsymbol{y'}|\boldsymbol{z})$) does not depend on $\boldsymbol{y}'$ ($\boldsymbol{x}'$).
We here consider the following set of path probabilities $$\begin{aligned}
\mathcal{M}_{\rm LR}^{\mathcal{X}} = \left\{ q_{\boldsymbol{Z}, \boldsymbol{Z'}} \left| q_{\boldsymbol{Z}, \boldsymbol{Z'}}(\boldsymbol{z},\boldsymbol{z'})=T^{\mathcal{X}}_{\boldsymbol{y'}; \boldsymbol{y}}(\boldsymbol{x}|\boldsymbol{x'})q_{\boldsymbol{Y}, \boldsymbol{Z'}}(\boldsymbol{y}, \boldsymbol{z'}) \right. \right\},\end{aligned}$$ where $q_{\boldsymbol{Y}, \boldsymbol{Z'}}(\boldsymbol{y}, \boldsymbol{z'}) = \sum_{\boldsymbol{x}} q_{\boldsymbol{Z},\boldsymbol{Z'}}(\boldsymbol{z}, \boldsymbol{z'})$. We call $\mathcal{M}_{\rm LR}^{\mathcal{X}} $ as [*the local reversible manifold of $\mathcal{X}$*]{}, because backward dynamics of the local system $q_{\boldsymbol{X}|\boldsymbol{Y}, \boldsymbol{Z'}}$ in this manifold are given by the forward dynamics of the local system $p_{ \boldsymbol{X'}|\boldsymbol{Y}', \boldsymbol{Z}}$. If the joint probability $p_{\boldsymbol{Z}, \boldsymbol{Z'}}$ is on this manifold, dynamics of $\mathcal{X}$ are locally reversible in time $T^{\mathcal{X}}_{\boldsymbol{y};\boldsymbol{y}'}(\boldsymbol{x'}|\boldsymbol{x}) p_{\boldsymbol{Y}' , \boldsymbol{Z}}(\boldsymbol{y}', \boldsymbol{z})=T^{\mathcal{X}}_{\boldsymbol{y'};\boldsymbol{y}}(\boldsymbol{x}|\boldsymbol{x'}) p_{\boldsymbol{Y}, \boldsymbol{Z'}}(\boldsymbol{y}, \boldsymbol{z'})$. This manifold is flat, because the following Pythagorean theorem holds for any $q_{\boldsymbol{Z}, \boldsymbol{Z'}} \in \mathcal{M}_{\rm LR}^{\mathcal{X}}$, $$\begin{aligned}
D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||q_{\boldsymbol{Z}, \boldsymbol{Z'}}) &= D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||q^{\mathcal{X}*}_{\boldsymbol{Z}, \boldsymbol{Z'}}) + D(q^{\mathcal{X}*}_{\boldsymbol{Z}, \boldsymbol{Z'}}||q_{\boldsymbol{Z}, \boldsymbol{Z'}}), \\
q^{\mathcal{X}*}_{\boldsymbol{Z}, \boldsymbol{Z'}}(\boldsymbol{z}, \boldsymbol{z'})&=T^{\mathcal{X}}_{\boldsymbol{y'};\boldsymbol{y}}(\boldsymbol{x}|\boldsymbol{x'})p_{\boldsymbol{Y}, \boldsymbol{Z'}}(\boldsymbol{y}, \boldsymbol{z'}).
\label{Pythagorean}\end{aligned}$$ The second term $D(q^*_{\boldsymbol{Z}, \boldsymbol{Z'}}||q_{\boldsymbol{Z}, \boldsymbol{Z'}})=D(p_{ \boldsymbol{Y}, \boldsymbol{Z'}}||q_{\boldsymbol{Y}, \boldsymbol{Z'}})$ can be interpreted as the degree of freedom in the probability distribution of $\boldsymbol{Y}$ and $\boldsymbol{Z'}$.
The partial entropy production can be obtained from the projection onto the local reversible manifold, because the Kullback-Leibler divergence $ D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||q^{\mathcal{X}*}_{\boldsymbol{Z}, \boldsymbol{Z'}})$ is equal to the partial entropy production, $$\begin{aligned}
D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||q^{\mathcal{X}*}_{\boldsymbol{Z}, \boldsymbol{Z'}})= \sigma^{\mathcal{X}}_{\rm partial} = \sigma^{\mathcal{X}}_{\rm sys}+ \sigma^{\mathcal{X}}_{\rm bath}- \Theta^{\mathcal{X} \to \mathcal{Y}}.\end{aligned}$$ Then, the partial entropy production can also be calculated from the following optimization problem $$\begin{aligned}
\sigma^{\mathcal{X}}_{\rm partial} = {\rm min}_{q_{\boldsymbol{Z}, \boldsymbol{Z'}} \in \mathcal{M}^{\mathcal{X}}_{\rm LR}} D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||q_{\boldsymbol{Z}, \boldsymbol{Z'}}).
\label{opt2}\end{aligned}$$ Its nonnegativity is the second law of information thermodynamics $\sigma^{\mathcal{X}}_{\rm partial} \geq 0$. If and only if the path probability is on the local reversible manifold, the partial entropy production vanishes $\sigma^{\mathcal{X}}_{\rm partial} =0$.
[*Hierarchy of the second laws.–*]{} We here show that our geometric interpretation of the second laws provides the hierarchy of the second laws. By definition, the reversible manifold is the submanifold of the local reversible manifold $\mathcal{M}_{\rm R} \subset \mathcal{M}_{\rm LR}^{\mathcal{X}}$. From this inclusion property of manifolds, we obtain ${\rm min}_{p_{\boldsymbol{Z}, \boldsymbol{Z'}} \in \mathcal{M}_{\rm R}} D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||q_{\boldsymbol{Z}, \boldsymbol{Z'}}) \geq {\rm min}_{p_{\boldsymbol{Z}, \boldsymbol{Z'}} \in \mathcal{M}^{\mathcal{X}}_{\rm LR}} D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||q_{\boldsymbol{Z}, \boldsymbol{Z'}})$, or equivalently $$\begin{aligned}
\sigma^{\mathcal{Z}}_{\rm tot} \geq \sigma^{\mathcal{X}}_{\rm partial}.
\label{hierarchy}\end{aligned}$$ This result gives the hierarchy of the second laws such that the second law of information thermodynamics always gives a tighter bound than the second law of thermodynamics (see also Fig.3).
![Schematic of the second law of information thermodynamics and its hierarchy. Because the local reversible manifold includes the reversible manifold, the second law of information thermodynamics always gives a tighter bound compared to the second law of thermodynamics.[]{data-label="fig2"}](2ndlawgeofig2.eps){width="8cm"}
Moreover, if the total system is consist of multiple systems $\mathcal{Z}_1, \mathcal{Z}_2, \dots, \mathcal{Z}_N$, we obtain the hierarchy for the two subsets $\{ \mathcal{Z}_{n_1} \dots, \mathcal{Z}_{n_k} \} \subset \{ \mathcal{Z}_{m_1} \dots, \mathcal{Z}_{m_l} \}$ as $$\begin{aligned}
\sigma^{\mathcal{Z}_{m_1} \dots, \mathcal{Z}_{m_l}}_{\rm partial} \geq \sigma^{\mathcal{Z}_{n_1} \dots, \mathcal{Z}_{n_k}}_{\rm partial},\end{aligned}$$ because of the inclusion property $\mathcal{M}_{\rm LR}^{\mathcal{Z}_{m_1} \dots, \mathcal{Z}_{m_l}} \subset \mathcal{M}_{\rm LR}^{\mathcal{Z}_{n_1} \dots, \mathcal{Z}_{n_k}}$. This hierarchy of the second laws is useful to apply information thermodynamics to complex systems. We have a lot of the second laws for complex systems, because the second law of information thermodynamics can be derived for any partition of the systems. This hierarchy indicates that we only need to investigate the inclusion property of manifolds if we want to grasp the relationship between the second laws.
[*Additivity and information integration.–*]{} We next discuss the additivity of the partial entropy productions, and the relationship between the additivity and the integrated information theory. The violation of the additivity of the partial entropy productions is given by its violation of the entropy changes in heat bathes $\Phi_{\rm bath}$ and the stochastic interaction $\Phi_{\rm SI}$ ($\Phi^{\dagger}_{\rm SI}$) [@Barrett2011practical; @Ay2015information] that is known as a measure of the integrated information theory [@sm] $$\begin{aligned}
&\sigma_{\rm tot}^{\mathcal{Z}} -\sigma_{\rm partial}^{\mathcal{X}} - \sigma_{\rm partial}^{\mathcal{Y}} =\Phi_{\rm bath} +\Phi_{\rm SI} - \Phi_{\rm SI}^{\dagger}, \\
&\Phi_{\rm bath} = \sigma^{\mathcal{Z}}_{\rm bath} - \sigma^{\mathcal{X}}_{\rm bath} - \sigma^{\mathcal{Y}}_{\rm bath}, \\
&\Phi_{\rm SI} = D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||p_{\boldsymbol{X'}| \boldsymbol{Z}} p_{\boldsymbol{Y}'| \boldsymbol{Z}}p_{\boldsymbol{Z}} ), \\
&\Phi_{\rm SI}^{\dagger} = D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||p_{\boldsymbol{X}| \boldsymbol{Z'}} p_{\boldsymbol{Y}| \boldsymbol{Z'}}p_{\boldsymbol{Z'}} ).
\label{additivity}\end{aligned}$$ Under the bipartite condition $\mathcal{C}_{\rm BI}$, two terms $\Phi_{\rm SI}$ and $\Phi_{\rm bath}$ vanish and $\Phi_{\rm SI}^{\dagger}$ is small enough $O(dt^2)$. But it is not necessary small if time evolution of two systems are strongly correlated. This result gives a novel interpretation of the integrated information theory in the context of thermodynamics. The violation of the additivity of the partial entropy productions can be interpreted as information integration, and its violation of the entropy changes in the heat bathes $\Phi_{\rm bath}$ can be a novel measure of information integration.
Under the bipartite condition $\mathcal{C}_{\rm BI}$ and the bipartite condition for time-reversal trajectories $\mathcal{C}_{\rm BI}^{*} : p_{\boldsymbol{Z}| \boldsymbol{Z'}}(\boldsymbol{z}| \boldsymbol{z'}) =p_{\boldsymbol{X}| \boldsymbol{Z'}}(\boldsymbol{x}|\boldsymbol{z'}) p_{\boldsymbol{Y}| \boldsymbol{Z'}}(\boldsymbol{y}|\boldsymbol{z'})$, we exactly obtain the additivity of the partial entropy productions $$\begin{aligned}
\sigma_{\rm tot}^{\mathcal{Z}} =\sigma_{\rm partial}^{\mathcal{X}}+\sigma_{\rm partial}^{\mathcal{Y}}.\end{aligned}$$ The hierarchy Eq. (\[hierarchy\]) is equivalent to the second law of information thermodynamics for the subsystem $\mathcal{Y}$, $\sigma_{\rm partial}^{\mathcal{Y}} \geq 0$. Under the conditions of the additivity $\mathcal{C}_{\rm BI}$ and $\mathcal{C
}^*_{\rm BI}$, we obtain the relationship between manifolds $$\begin{aligned}
\mathcal{M}_{\rm R} = \mathcal{M}^{\mathcal{X}}_{\rm LR} \cap \mathcal{M}^{\mathcal{Y}}_{\rm LR}. \end{aligned}$$
From the view point of information geometry, the additivity (\[additivity\]) gives a nontrivial quadrangle (see also Fig. \[fig3\]). The additivity can be written as $$\begin{aligned}
D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||q^{*}_{\boldsymbol{Z}, \boldsymbol{Z'}})=D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}|| q^{\mathcal{X}*}_{\boldsymbol{Z}, \boldsymbol{Z'}})+D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}|| q^{\mathcal{Y}*}_{\boldsymbol{Z}, \boldsymbol{Z'}}).\end{aligned}$$ From the Pythagorean theorem (\[Pythagorean\]), we have the following relationship $$\begin{aligned}
D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}|| q^{\mathcal{X}*}_{\boldsymbol{Z}, \boldsymbol{Z'}}) &=D(q^{\mathcal{Y}*}_{\boldsymbol{Z}, \boldsymbol{Z'}}|| q^{*}_{\boldsymbol{Z}, \boldsymbol{Z'}}),\label{additivity2} \\
D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}|| q^{\mathcal{Y}*}_{\boldsymbol{Z}, \boldsymbol{Z'}}) &=D(q^{\mathcal{X}*}_{\boldsymbol{Z}, \boldsymbol{Z'}}|| q^{*}_{\boldsymbol{Z}, \boldsymbol{Z'}})
\label{additivity2-1}\end{aligned}$$ which means that the parallel sides of a quadrangle have the same length. We call these conditions (\[additivity2\]) and (\[additivity2-1\]) as [*the rectangle*]{} in information geometry. This rectangle is not so trivial because information geometry is non-Euclidean. In information geometry, a measure of information integration $\Phi_{\rm bath} +\Phi_{\rm SI} - \Phi_{\rm SI}^{\dagger}$ quantifies a distortion of this rectangle.
![Schematic of the additivity and the rectangle. Under the both bipartite conditions $\mathcal{C}_{\rm BI}$ and $\mathcal{C}_{\rm BI}^{*}$, the reversible manifold is equal to the intersection of the local reversible manifolds. The additivity of the entropy production indicates that the parallel sides of a quadrangle $(p,{q}^{\mathcal{X}*}, {q}^{*}, {q}^{\mathcal{Y}*})$ have the same length.[]{data-label="fig3"}](2ndlawgeofig3.eps){width="7cm"}
[*Example.–*]{} We illustrate our results by the two spins model [@sm1]. Let ${\boldsymbol Z}=\{S_1, S_2\}$ and ${\boldsymbol Z'}= \{S_3, S_4\}$ be random variables of two spins at time $t$ and $t+dt$, respectively. The spin has the binary state $s_i \in \{0, 1 \}$. The path probability of the spin state is generally given by the exponential family even in nonequilibrium dynamics, $$\begin{aligned}
p_{{\boldsymbol Z},{\boldsymbol Z'}}^{\hat{\boldsymbol{\theta}}}(\boldsymbol{s})=& \exp \left[ \sum_{i} s_i \hat{\theta}^i +\sum_{i<j} s_i s_j \hat{\theta}^{ij} + \sum_{i<j<k} s_i s_j s_k \hat{\theta}^{ijk} \right. \nonumber\\
&\left. +\sum_{i<j<k<l} s_i s_j s_ks_l \hat{\theta}^{ijkl} - \phi_{{\boldsymbol Z},{\boldsymbol Z'}} (\boldsymbol{\hat{\theta}}) \right],\end{aligned}$$ where $\hat{\boldsymbol{\theta}}$ is the set of parameters, and $\phi_{{\boldsymbol Z},{\boldsymbol Z'}}(\boldsymbol{\hat{\theta}})$ is the normalization factor that satisfies $\sum_{{\boldsymbol s}} p_{{\boldsymbol Z},{\boldsymbol Z'}}^{\hat{\boldsymbol{\theta}}}(\boldsymbol{s}) =1$. The parameter $\hat{\boldsymbol{\theta}}$ in $ p_{{\boldsymbol Z},{\boldsymbol Z'}}^{\hat{\boldsymbol{\theta}}}$ gives a coordinate called as theta coordinate. The number of the elements in ${\hat{\boldsymbol{\theta}}}$ is $(2^4 -1) =15$, then the probability $ p_{{\boldsymbol Z},{\boldsymbol Z'}}^{\hat{\boldsymbol{\theta}}}$ can be represented by theta coordinate in $15$-dimensional manifold.
The both bipartite conditions $\mathcal{C}_{\rm BI}$ and $\mathcal{C}_{\rm BI}^{*}$ gives a constraint in $15$-dimensional manifold $$\begin{aligned}
\mathcal{C}_{\rm BI} : \:\hat{\theta}^{34} \: =\hat{\theta}^{134} \:= \hat{\theta}^{234} \:=\hat{\theta}^{1234} \: =0, \\
\mathcal{C}_{\rm BI}^{*} : \hat{\theta}^{12}=\hat{\theta}^{123}=\hat{\theta}^{124}=\hat{\theta}^{1234}=0,\end{aligned}$$ which is a $7$-dimensional constraint. We here discuss the case under the both bipartite conditions.
To calculate the entropy production, we consider the reversible manifold $\mathcal{M}_{\rm R}$. Under the both bipartite conditions $\mathcal{C}_{\rm BI}$ and $\mathcal{C}_{\rm BI}^{*}$, the condition of the reversible manifold is given by the following set of probabilities [@sm], $$\begin{aligned}
&\mathcal{M}_{\rm R} = \{ p_{{\boldsymbol Z},{\boldsymbol Z'}}^{{\boldsymbol{\theta}}} |{\boldsymbol{\theta}}^{\mathcal X} = \hat{\boldsymbol{\theta}}^{\mathcal X}, {\boldsymbol{\theta}}^{\mathcal Y} = \hat{\boldsymbol{\theta}}^{\mathcal Y}\} \\
&{\boldsymbol{\theta}}^{\mathcal X} = ({\theta}^1, {\theta}^{13}, {\theta}^{14} ), \: \: \: \: \hat{\boldsymbol{\theta}}^{\mathcal X} = (\hat{\theta}^3, \hat{\theta}^{13}, \hat{\theta}^{23} ), \\
&{\boldsymbol{\theta}}^{\mathcal Y} = ({\theta}^2, {\theta}^{24}, {\theta}^{23} ), \: \: \: \: \hat{\boldsymbol{\theta}}^{\mathcal Y} = (\hat{\theta}^4, \hat{\theta}^{24}, \hat{\theta}^{14} ),
\label{mani1}\end{aligned}$$ where a coordinate ${\boldsymbol{\theta}}$ represents a probability on the reversible manifold. The reversible manifold is flat, because the condition of the flatness is given by the linear constraints for the exponential family [@amari2016information]. The condition of the local reversible manifolds are also given by the linear constraint of ${\boldsymbol{\theta}}$ [@sm], $$\begin{aligned}
\mathcal{M}_{\rm LR}^{\mathcal X} = \{ p_{{\boldsymbol Z},{\boldsymbol Z'}}^{{\boldsymbol{\theta}}} |{\boldsymbol{\theta}}^{\mathcal X} = \hat{\boldsymbol{\theta}}^{\mathcal X}\}, \: \: \:
\mathcal{M}_{\rm LR}^{\mathcal Y} = \{ p_{{\boldsymbol Z},{\boldsymbol Z'}}^{{\boldsymbol{\theta}}} | {\boldsymbol{\theta}}^{\mathcal Y} = \hat{\boldsymbol{\theta}}^{\mathcal Y}\}.
\label{mani2}\end{aligned}$$ Under the both bipartite conditions, the intersection of these two manifolds is the reversible manifold $\mathcal{M}_{\rm R} = \mathcal{M}^{\mathcal{X}}_{\rm LR} \cap \mathcal{M}^{\mathcal{Y}}_{\rm LR}$.
To estimate the (partial) entropy production from the observation of the path probability $p_{{\boldsymbol Z},{\boldsymbol Z'}}^{\hat{\boldsymbol{\theta}}}$, we can calculate the following optimization problems $$\begin{aligned}
\sigma_{\rm partial}^{\mathcal{X}} &= { \rm min}_{{\boldsymbol{\theta}} |{\boldsymbol{\theta}}^{\mathcal X} = \hat{\boldsymbol{\theta}}^{\mathcal X}} D(p_{{\boldsymbol Z},{\boldsymbol Z'}}^{\hat{\boldsymbol{\theta}}} ||p_{{\boldsymbol Z},{\boldsymbol Z'}}^{{\boldsymbol{\theta}}}), \\
\sigma_{\rm partial}^{\mathcal{Y}} &= { \rm min}_{{\boldsymbol{\theta}}| {\boldsymbol{\theta}}^{\mathcal Y} = \hat{\boldsymbol{\theta}}^{\mathcal Y}} D(p_{{\boldsymbol Z},{\boldsymbol Z'}}^{\hat{\boldsymbol{\theta}}} ||p_{{\boldsymbol Z},{\boldsymbol Z'}}^{{\boldsymbol{\theta}}}), \\
\sigma_{\rm tot}^{\mathcal{Z}} &= { \rm min}_{{\boldsymbol{\theta}}| {\boldsymbol{\theta}}^{\mathcal X} = \hat{\boldsymbol{\theta}}^{\mathcal X}, {\boldsymbol{\theta}}^{\mathcal Y} = \hat{\boldsymbol{\theta}}^{\mathcal Y}} D(p_{{\boldsymbol Z},{\boldsymbol Z'}}^{\hat{\boldsymbol{\theta}}} ||p_{{\boldsymbol Z},{\boldsymbol Z'}}^{{\boldsymbol{\theta}}}).
\label{mani2}\end{aligned}$$ This problem can be numerically solved by using a conventional optimization tool. Such an optimization problem of the Kullback-Leibler divergence is well studied as a statistical inference [@Kass2011inference], a hypothesis testing [@Csiszar2004statistics], and an expectation-maximization algorithm [@Amari1995em] in the context of information geometry.
*Conclusion and discussion.–*By applying information-geometric framework, we clarify the relationship between the second law of thermodynamics and information thermodynamics. This result is complement to other geometric expressions of the second law, such as the principle of Carathèodory [@Caratheodory1976principle] and the maximum entropy thermodynamics [@jaynes1957info; @jaynes1957info2], while our result is based on the manifold of reversibility unlike the other.
Variants of the second laws could be derived from the selection of $T$ that gives another manifold. For example, if we consider $T$ of the dual dynamics [@seifert2012stochastic], we would obtain the generalized second law for non-equilibrium steady state [@Hatano2001sst]. The hierarchy does not apply only to the second laws of information thermodynamics. Our framework gives the hierarchy for variants of the second laws by using the inclusion property of manifolds corresponding to selections of $T$.
Because the second law of information thermodynamics would be essential for biochemical information processing [@ito2015maxwell; @barato2014efficiency; @sartori2014thermodynamic; @bo2015thermodynamic; @ouldridge2017thermodynamics; @mcgrath2017biochemical; @Matsumoto2018implication], this work would give a geometric insight into biochemical information processing. This work provides a physical validity of the integrated information theory [@Oizumi2014integrated; @Tononi2016integrated; @oizumi2016unified; @amari2017integration] for the biochemical information processing.
acknowledgement
===============
I am strongly grateful to Masafumi Oizumi, and Shun-ichi Amari for valuable discussions on results of this manuscript. I am grateful to Hideaki Shimazaki for critical reading of this manuscript. I am also grateful to Andreas Dechant for the discussion of information geometry and the thermodynamic uncertainty. I thank Kunihiko Kaneko, Takahiro Sagawa and Tetsuhiro Hatakeyama for valuable comments. This research is supported by JSPS KAKENHI Grant No. JP16K17780, JP19H05796 and JST Presto Grant No. JP18070368, Japan.
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Supplementary information
=========================
I. Review of the second law of thermodynamics
---------------------------------------------
We here review the second law of thermodynamics in stochastic thermodynamics. We start with the master equation $$\begin{aligned}
\frac{d}{dt} p (\boldsymbol{z}';t) = \sum_{\boldsymbol{z}} \left[ W(\boldsymbol{z} \to \boldsymbol{z}';t) p(\boldsymbol{z};t) - W(\boldsymbol{z}' \to \boldsymbol{z};t) p(\boldsymbol{z}';t) \right],
\label{mastereqsup}\end{aligned}$$ where $p(\boldsymbol{z};t)$ is the probability of the state $\boldsymbol{z}$ at time $t$, and $W(\boldsymbol{z} \to \boldsymbol{z}';t)$ is the transition rate from the state $\boldsymbol{z}$ to the state $\boldsymbol{z}'$ at time $t$. In the notation of this paper, the probability of $\boldsymbol{z}$ is given by $p_{\boldsymbol{Z}}(\boldsymbol{z}) =p (\boldsymbol{z};t)$. From the master equation (\[mastereqsup\]), we obtain the probability at time $t+dt$, $$\begin{aligned}
p(\boldsymbol{z}';t+dt) = \sum_{\boldsymbol{z}} \left[ W(\boldsymbol{z} \to \boldsymbol{z}';t) p(\boldsymbol{z};t) dt +(1- W(\boldsymbol{z}' \to \boldsymbol{z};t) dt)p(\boldsymbol{z}';t) \right].
\label{mastereqsup2}\end{aligned}$$ In the notation of the main text, $p_{\boldsymbol{Z}}(\boldsymbol{z})$ and $p_{\boldsymbol{Z}'}(\boldsymbol{z}')$ are given by $p_{\boldsymbol{Z}}(\boldsymbol{z}) = p({\boldsymbol{z}};t)$ and $p_{\boldsymbol{Z}'}(\boldsymbol{z}') = p({\boldsymbol{z}'};t+dt)$, respectively. We also obtain the relationship between $p_{\boldsymbol{Z}}$ and $p_{\boldsymbol{Z}'}$ as $$\begin{aligned}
p_{\boldsymbol{Z}'}(\boldsymbol{z}') = p(\boldsymbol{z}';t) +O(dt) =p_{\boldsymbol{Z}}(\boldsymbol{z}')+O(dt).\end{aligned}$$ Substituting $p(\boldsymbol{z};t) =1$ into Eq. (\[mastereqsup2\]), the transition probability $T(\boldsymbol{z}'|\boldsymbol{z})$ is given by $$\begin{aligned}
T(\boldsymbol{z}'|\boldsymbol{z}) =
\begin{cases}
W(\boldsymbol{z} \to \boldsymbol{z}';t) dt & (\boldsymbol{z} \neq \boldsymbol{z}'),\\
(1- \sum_{\boldsymbol{z} \neq \boldsymbol{z}'} W(\boldsymbol{z}' \to \boldsymbol{z};t) dt) & (\boldsymbol{z} = \boldsymbol{z}').
\end{cases}\label{transitionsup}\end{aligned}$$
Here, we consider the detailed balance. The condition of the detailed balance is given by $$\begin{aligned}
W(\boldsymbol{z} \to \boldsymbol{z}';t) p(\boldsymbol{z};t) = W(\boldsymbol{z}' \to \boldsymbol{z};t) p(\boldsymbol{z}';t)
\label{detailedbalance}\end{aligned}$$ for any $\boldsymbol{z}$ and $\boldsymbol{z}'$. This condition is valid if the system is in equilibrium. By using the transition probability, we obtain another expression of the detailed balance condition $$\begin{aligned}
T(\boldsymbol{z}'|\boldsymbol{z})p_{\boldsymbol{Z}}(\boldsymbol{z}) = T(\boldsymbol{z}|\boldsymbol{z}') p_{\boldsymbol{Z}'}(\boldsymbol{z}'),
\label{reversiblitysup}\end{aligned}$$ where we used $W(\boldsymbol{z}' \to \boldsymbol{z};t) p(\boldsymbol{z}';t) dt = T(\boldsymbol{z}|\boldsymbol{z}') p_{\boldsymbol{Z}}(\boldsymbol{z}') = T(\boldsymbol{z}|\boldsymbol{z}') p_{\boldsymbol{Z}'}(\boldsymbol{z}') +O(dt^2)$. Therefore, the detailed balance condition implies the reversibility of dynamics in the transition from $t$ to $t+dt$. From the identity by the Bayes’ rule $$\begin{aligned}
p_{\boldsymbol{Z}|\boldsymbol{Z}'}(\boldsymbol{z}|\boldsymbol{z}') =T(\boldsymbol{z}'|\boldsymbol{z}) \frac{ p_{\boldsymbol{Z}}(\boldsymbol{z}) }{p_{\boldsymbol{Z}'}(\boldsymbol{z}')},\end{aligned}$$ the detailed balance condition can be rewritten as $$\begin{aligned}
T(\boldsymbol{z}|\boldsymbol{z}') =p_{\boldsymbol{Z}|\boldsymbol{Z}'}(\boldsymbol{z}|\boldsymbol{z}').\end{aligned}$$
Next, we discuss the second law of thermodynamics. For the master equation, the entropy production ratio $\sigma^{\mathcal{Z}}_{\rm tot}/dt$ is defined as $$\begin{aligned}
\frac{\sigma^{\mathcal{Z}}_{\rm tot}}{dt} &= \sum_{\boldsymbol{z}, \boldsymbol{z}'} W(\boldsymbol{z} \to \boldsymbol{z}';t) p(\boldsymbol{z};t) \ln \frac{W(\boldsymbol{z} \to \boldsymbol{z}';t) p(\boldsymbol{z};t)}{W(\boldsymbol{z}' \to \boldsymbol{z};t) p(\boldsymbol{z}';t)}.\end{aligned}$$ If the detailed balance condition is valid, the entropy production vanishes $\sigma^{\mathcal{Z}}_{\rm tot}= 0$. By using the transition probability $T(\boldsymbol{z}'|\boldsymbol{z})$, we obtain another expression of the entropy production $$\begin{aligned}
\sigma^{\mathcal{Z}}_{\rm tot}
&= \sum_{\boldsymbol{z}, \boldsymbol{z}'|\boldsymbol{z} \neq \boldsymbol{z}' } W(\boldsymbol{z} \to \boldsymbol{z}';t) dt p(\boldsymbol{z};t)\ln \frac{W(\boldsymbol{z} \to \boldsymbol{z}';t) dt p(\boldsymbol{z};t)}{W(\boldsymbol{z}' \to \boldsymbol{z};t)dt p(\boldsymbol{z}';t)}\\
&= \sum_{\boldsymbol{z}, \boldsymbol{z}'| \boldsymbol{z} \neq \boldsymbol{z}'} T(\boldsymbol{z}'|\boldsymbol{z}) p_{\boldsymbol{Z}}(\boldsymbol{z}) \ln \frac{T(\boldsymbol{z}'|\boldsymbol{z}) p_{\boldsymbol{Z}}(\boldsymbol{z})}{T(\boldsymbol{z}|\boldsymbol{z}') p_{\boldsymbol{Z}'}(\boldsymbol{z}')} +O(dt^2) \\
&= \sum_{\boldsymbol{z}, \boldsymbol{z}'} T(\boldsymbol{z}'|\boldsymbol{z}) p_{\boldsymbol{Z}}(\boldsymbol{z}) \ln \frac{T(\boldsymbol{z}'|\boldsymbol{z}) p_{\boldsymbol{Z}}(\boldsymbol{z})}{T(\boldsymbol{z}|\boldsymbol{z}') p_{\boldsymbol{Z}'}(\boldsymbol{z}')}.\end{aligned}$$ To introduce two probabilities $p_{\boldsymbol{Z},\boldsymbol{Z}'}(\boldsymbol{z},\boldsymbol{z}')= T(\boldsymbol{z}'|\boldsymbol{z}) p_{\boldsymbol{Z}}(\boldsymbol{z})$ and $q^*_{\boldsymbol{Z},\boldsymbol{Z}'}(\boldsymbol{z},\boldsymbol{z}')= T(\boldsymbol{z}|\boldsymbol{z}') p_{\boldsymbol{Z}'}(\boldsymbol{z}')$, this expression can be regarded as the Kullback-Leibler divergence between two probabilities $$\begin{aligned}
\sigma^{\mathcal{Z}}_{\rm tot} &= \sum_{\boldsymbol{z}, \boldsymbol{z}'| \boldsymbol{z} \neq \boldsymbol{z}'} p_{\boldsymbol{Z},\boldsymbol{Z}'}(\boldsymbol{z},\boldsymbol{z}') \ln \frac{p_{\boldsymbol{Z},\boldsymbol{Z}'}(\boldsymbol{z},\boldsymbol{z}') }{q^*_{\boldsymbol{Z},\boldsymbol{Z}'}(\boldsymbol{z},\boldsymbol{z}')} \\
&= D(p_{\boldsymbol{Z},\boldsymbol{Z}'}||q^*_{\boldsymbol{Z},\boldsymbol{Z}'}).\end{aligned}$$
II. The second law of thermodynamics under feedback control and the projection theorem
--------------------------------------------------------------------------------------
We here consider the situation that the time evolution of the system $\mathcal{X}$ depends on the memory $\mathcal{M}$. Let $\mathcal{X}$ and $\mathcal{X'}$ be random variables of the system $\mathcal{X}$ at time $t$ and $t+dt$, respectively. Let $\boldsymbol{M}$ be a random variable of the memory $\mathcal{M}$. The transition probability of $\mathcal{X}$ depend on the state of memory, $T^{\mathcal{X}}_{\boldsymbol{m}}(\boldsymbol{x'}|\boldsymbol{x}) =p_{\boldsymbol{X'}|\boldsymbol{X},\boldsymbol{M} }(\boldsymbol{x'}|\boldsymbol{x}, \boldsymbol{m})$. The condition of feedback reversibility in the system $\mathcal{X}$ is given by $p_{\boldsymbol{X}|\boldsymbol{X'}, \boldsymbol{M}}(\boldsymbol{x}|\boldsymbol{x'}, \boldsymbol{m}) =T^{\mathcal{X}}_{\boldsymbol{m}}(\boldsymbol{x}|\boldsymbol{x'})$.
The second law of thermodynamics under feedback control is given by the inequality of the partial entropy changes and the mutual information change $\Delta I$, $$\begin{aligned}
&\sigma^{\mathcal{X}}_{\rm sys}+ \sigma^{\mathcal{X}}_{\rm bath} \geq \Delta I, \\
&\sigma^{\mathcal{X}}_{\rm sys}= H(\boldsymbol{X'})-H(\boldsymbol{X}),\\
&\sigma^{\mathcal{X}}_{\rm bath}= \sum_{\boldsymbol{x}, \boldsymbol{x'}, \boldsymbol{m}} p_{\boldsymbol{X}, \boldsymbol{X'}, \boldsymbol{M}} (\boldsymbol{x}, \boldsymbol{x'}, \boldsymbol{m})\ln \frac{T^{\mathcal{X}}_{\boldsymbol{m}}(\boldsymbol{x'}|\boldsymbol{x})}{T^{\mathcal{X}}_{\boldsymbol{m}}(\boldsymbol{x}|\boldsymbol{x'})}, \\
&\Delta I = I(\boldsymbol{X'};\boldsymbol{M} ) -I(\boldsymbol{X};\boldsymbol{M}).\end{aligned}$$
We here introduce [*the feedback reversible manifold*]{} that backward dynamics from $\boldsymbol{X'}$ to $\boldsymbol{X}$ are driven by the transition probability $T^{\mathcal{X}}$, $$\begin{aligned}
\mathcal{M}_{\rm FR}= \{q_{\boldsymbol{X},\boldsymbol{X'}, \boldsymbol{M}} | q_{\boldsymbol{X},\boldsymbol{X'}, \boldsymbol{M}} (\boldsymbol{x},\boldsymbol{x'}, \boldsymbol{m}) =q_{\boldsymbol{X'} \boldsymbol{M}} (\boldsymbol{x'}, \boldsymbol{m}) T^{\mathcal{X}}_{\boldsymbol{m}}(\boldsymbol{x}|\boldsymbol{x'})\},\end{aligned}$$ where $q_{\boldsymbol{X'} \boldsymbol{M}} (\boldsymbol{x'}, \boldsymbol{m})= \sum_{\boldsymbol{x}} q_{\boldsymbol{X},\boldsymbol{X'}, \boldsymbol{M}} (\boldsymbol{x},\boldsymbol{x'}, \boldsymbol{m})$. The feedback reversible manifold is equivalent to the reversible manifold $\mathcal{M}_{\rm R} = \mathcal{M}_{\rm FR}$, if we consider the time evolution from $\boldsymbol{Z} = \{\boldsymbol{X}, \boldsymbol{M} \}$ to $\boldsymbol{Z'} = \{\boldsymbol{X'}, \boldsymbol{M}\}$. If the joint probability $p_{\boldsymbol{X}, \boldsymbol{X'}, \boldsymbol{M}}$ is on this manifold, dynamics of $\mathcal{X}$ are reversible in time under feedback control. In the case $q^{\mathcal{X}*}_{\boldsymbol{X}, \boldsymbol{X'}, \boldsymbol{M}}(\boldsymbol{x}, \boldsymbol{x'}, \boldsymbol{m})=T^{\mathcal{X}}_{\boldsymbol{m}}(\boldsymbol{x}|\boldsymbol{x'})p_{ \boldsymbol{X'}, \boldsymbol{M}}( \boldsymbol{x'}, \boldsymbol{m})$, the following Pythagorean theorem holds for any $q_{\boldsymbol{X}, \boldsymbol{X'}, \boldsymbol{M}} \in \mathcal{M}_{\rm FR}$, $$\begin{aligned}
D(p_{\boldsymbol{X}, \boldsymbol{X'}, \boldsymbol{M}}||q_{\boldsymbol{X}, \boldsymbol{X'}, \boldsymbol{M}}) = D(p_{\boldsymbol{X}, \boldsymbol{X'}, \boldsymbol{M}}||q^{\mathcal{X}*}_{\boldsymbol{X}, \boldsymbol{X'}, \boldsymbol{M}}) + D(q^{\mathcal{X}*}_{\boldsymbol{X}, \boldsymbol{X'}, \boldsymbol{M}}||q_{\boldsymbol{X}, \boldsymbol{X'}, \boldsymbol{M}}).
\label{Pythagorean}\end{aligned}$$ The second term $D(q^{\mathcal{X}*}_{\boldsymbol{X}, \boldsymbol{X'}, \boldsymbol{M}}||q_{\boldsymbol{X}, \boldsymbol{X'}, \boldsymbol{M}}) =D(p_{\boldsymbol{X'}, \boldsymbol{M}}||q_{\boldsymbol{X'}, \boldsymbol{M}})$ can be interpreted as the degree of freedom in the probability distribution of $\boldsymbol{M}$ and $\boldsymbol{X'}$.
The second law of thermodynamics under feedback control can be obtained from the projection onto the local reversible manifold, because the Kullback-Leibler divergence $ D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||q^{\mathcal{X}*}_{\boldsymbol{Z}, \boldsymbol{Z'}})$ is equal to the partial entropy production, $$\begin{aligned}
D(p_{\boldsymbol{X}, \boldsymbol{X'}, \boldsymbol{M}}||q^{\mathcal{X}*}_{\boldsymbol{X}, \boldsymbol{X'}, \boldsymbol{M}}) = \sigma^{\mathcal{X}}_{\rm partial} = \sigma^{\mathcal{X}}_{\rm sys}+ \sigma^{\mathcal{X}}_{\rm bath}- \Delta I.\end{aligned}$$ Then, the second law of thermodynamics under feedback control can also be related to the optimization problem $$\begin{aligned}
\sigma^{\mathcal{X}}_{\rm partial} = {\rm min}_{q_{\boldsymbol{X}, \boldsymbol{X'}, \boldsymbol{M}} \in \mathcal{M}_{\rm FR}} D(p_{\boldsymbol{X}, \boldsymbol{X'}, \boldsymbol{M}} ||q_{\boldsymbol{X}, \boldsymbol{X'}, \boldsymbol{M}} ).
\label{opt2}\end{aligned}$$ If and only if the path probability is on the feedback reversible manifold, the partial entropy production vanishes.
III. Detailed calculation of the integrated information
-------------------------------------------------------
We here show the relationship between the integrated information and non-additivity of the partial entropy productions Eq. (26) in the main text. The violation of the additivity of the partial entropy productions $\sigma_{\rm tot}^{\mathcal{Z}}- \sigma_{\rm partial}^{\mathcal{X}}-\sigma_{\rm partial}^{\mathcal{Y}} $ is calculated as $$\begin{aligned}
&\sigma_{\rm tot}^{\mathcal{Z}} - \sigma_{\rm partial}^{\mathcal{X}}-\sigma_{\rm partial}^{\mathcal{Y}} \nonumber\\
=&D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||q^*_{\boldsymbol{Z}, \boldsymbol{Z'}}) - D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||q_{\boldsymbol{Z}, \boldsymbol{Z'}}^{\mathcal{X}*}) - D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||q_{\boldsymbol{Z}, \boldsymbol{Z'}}^{\mathcal{Y}*}) \nonumber\\
=& \sum_{\boldsymbol{z}, \boldsymbol{z'}} p_{\boldsymbol{Z}, \boldsymbol{Z'}} (\boldsymbol{z}, \boldsymbol{z'}) \left[ \ln \frac{ p_{\boldsymbol{Z}, \boldsymbol{Z'}} (\boldsymbol{z}, \boldsymbol{z'})}{T (\boldsymbol{z}|\boldsymbol{z'})p_{\boldsymbol{Z'}}(\boldsymbol{z'})} + \ln \frac{T^{\mathcal{X}}_{\boldsymbol{y'};\boldsymbol{y}}(\boldsymbol{x}|\boldsymbol{x'}) p_{\boldsymbol{Z'}, \boldsymbol{Y}}(\boldsymbol{z'}, \boldsymbol{y}) }{p_{\boldsymbol{X'}| \boldsymbol{Z},\boldsymbol{Y}'}({\boldsymbol{x'}}| \boldsymbol{z},\boldsymbol{y}') p_{\boldsymbol{Z}, \boldsymbol{Y'}}(\boldsymbol{z}, \boldsymbol{y'}) } + \ln \frac{T^{\mathcal{Y}}_{\boldsymbol{x'};\boldsymbol{x}}(\boldsymbol{y}|\boldsymbol{y'}) p_{\boldsymbol{Z'}, \boldsymbol{X}}(\boldsymbol{z'}, \boldsymbol{x}) }{p_{\boldsymbol{Y'}| \boldsymbol{Z},\boldsymbol{X}'}({\boldsymbol{y'}}| \boldsymbol{z},\boldsymbol{x}') p_{\boldsymbol{Z}, \boldsymbol{X'}}(\boldsymbol{z}, \boldsymbol{x'}) } \right] \nonumber\\
=& \sum_{\boldsymbol{z}, \boldsymbol{z'}} p_{\boldsymbol{Z}, \boldsymbol{Z'}} (\boldsymbol{z}, \boldsymbol{z'}) \left[ \ln \frac{ T (\boldsymbol{z'}|\boldsymbol{z})p_{\boldsymbol{Z}}(\boldsymbol{z})}{T (\boldsymbol{z}|\boldsymbol{z'})p_{\boldsymbol{Z'}}(\boldsymbol{z'})} + \ln \frac{T^{\mathcal{X}}_{\boldsymbol{y'};\boldsymbol{y}}(\boldsymbol{x}|\boldsymbol{x'}) p_{ \boldsymbol{Y}|\boldsymbol{Z'}}(\boldsymbol{y}|\boldsymbol{z'}) p_{\boldsymbol{Z'}}(\boldsymbol{z'}) }{T^{\mathcal{X}}_{\boldsymbol{y};\boldsymbol{y}'}(\boldsymbol{x}'|\boldsymbol{x}) p_{ \boldsymbol{Y'}|\boldsymbol{Z}}(\boldsymbol{y'}|\boldsymbol{z}) p_{\boldsymbol{Z}}(\boldsymbol{z})} + \ln \frac{T^{\mathcal{Y}}_{\boldsymbol{x'};\boldsymbol{x}}(\boldsymbol{y}|\boldsymbol{y'}) p_{\boldsymbol{X}|\boldsymbol{Z'}}( \boldsymbol{x}|\boldsymbol{z'})p_{\boldsymbol{Z'}}(\boldsymbol{z'}) }{T^{\mathcal{Y}}_{\boldsymbol{x};\boldsymbol{x'}}(\boldsymbol{y}'|\boldsymbol{y}) p_{\boldsymbol{X'}|\boldsymbol{Z}}( \boldsymbol{x'}|\boldsymbol{z})p_{\boldsymbol{Z}}(\boldsymbol{z}) } \right] \nonumber\\
=& D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||p_{\boldsymbol{X}'|\boldsymbol{Z}}p_{\boldsymbol{Y}'|\boldsymbol{Z}}p_{\boldsymbol{Z}}) - D(p_{\boldsymbol{Z}, \boldsymbol{Z'}}||p_{\boldsymbol{X}|\boldsymbol{Z'}}p_{\boldsymbol{Y}|\boldsymbol{Z'}}p_{\boldsymbol{Z'}}) - \left[ \sum_{\boldsymbol{z}, \boldsymbol{z'}} p_{\boldsymbol{Z}, \boldsymbol{Z'}} (\boldsymbol{z}, \boldsymbol{z'}) \ln \frac{T (\boldsymbol{z}|\boldsymbol{z'})T^{\mathcal{X}}_{\boldsymbol{y};\boldsymbol{y}'}(\boldsymbol{x}'|\boldsymbol{x}) T^{\mathcal{Y}}_{\boldsymbol{x};\boldsymbol{x}'}(\boldsymbol{y'}|\boldsymbol{y}) }{T (\boldsymbol{z'}|\boldsymbol{z}) T^{\mathcal{X}}_{\boldsymbol{y'};\boldsymbol{y}}(\boldsymbol{x}|\boldsymbol{x'}) T^{\mathcal{Y}}_{\boldsymbol{x'};\boldsymbol{x}}(\boldsymbol{y}|\boldsymbol{y'}) } \right] \nonumber\\
=& \Phi_{\rm bath} +\Phi_{\rm SI} - \Phi_{\rm SI}^{\dagger}.\end{aligned}$$
IV. Detailed calculations of examples in the main text
------------------------------------------------------
We start with the joint distribution $$\begin{aligned}
p_{{\boldsymbol Z},{\boldsymbol Z'}}^{\hat{\boldsymbol{\theta}}}(\boldsymbol{s})=& \exp \left[ \sum_{i} s_i \hat{\theta}^i +\sum_{i<j} s_i s_j \hat{\theta}^{ij} + \sum_{i<j<k} s_i s_j s_k \hat{\theta}^{ijk} +\sum_{i<j<k<l} s_i s_j s_ks_l \hat{\theta}^{ijkl} - \phi_{{\boldsymbol Z},{\boldsymbol Z'}} (\boldsymbol{\hat{\theta}}) \right],\end{aligned}$$ where $\boldsymbol{s}=(s_1,s_2, s_3,s_4)=(x,y,x',y')$ is the spin notation with $s_i \in \{0,1\}$, and $\phi_{{\boldsymbol Z},{\boldsymbol Z'}} (\boldsymbol{\hat{\theta}})$ is the normalization constant. The joint distribution $p^{\hat{\boldsymbol{\theta}}}_{{\boldsymbol Z},{\boldsymbol X'}}({\boldsymbol z},{\boldsymbol x'})$ is calculated as $$\begin{aligned}
\ln p^{\hat{\boldsymbol{\theta}}}_{{\boldsymbol Z},{\boldsymbol X'}}({\boldsymbol z},{\boldsymbol x'}) =& \sum_{i \neq 4} s_i \hat{\theta}^i +\sum_{i<j| i\neq4, j \neq 4} s_i s_j \hat{\theta}^{ij} + s_1 s_2 s_3 \hat{\theta}^{123}+\psi_{{\boldsymbol Y'}} ({\boldsymbol z},{\boldsymbol x'}| \boldsymbol{\hat{\theta}}) - \phi_{{\boldsymbol Z},{\boldsymbol Z'}} (\boldsymbol{\hat{\theta}}), \nonumber \\
\psi_{{\boldsymbol Y'}} ({\boldsymbol z},{\boldsymbol x'}| \boldsymbol{\hat{\theta}}) :=& \ln \left[\exp \left(\hat{\theta}^4 + \sum_{i} s_i \hat{\theta}^{i4} + \sum_{i<j} s_i s_j \hat{\theta}^{ij4} + s_1 s_2 s_3 \hat{\theta}^{1234} \right) +1 \right].\end{aligned}$$ Then the conditional probability $p^{\hat{\boldsymbol{\theta}}}_{{\boldsymbol X'}|{\boldsymbol Z}}({\boldsymbol x'}|{\boldsymbol z})$ is calculated as $$\begin{aligned}
\ln p^{\hat{\boldsymbol{\theta}}}_{{\boldsymbol X'}|{\boldsymbol Z}}({\boldsymbol x'}|{\boldsymbol z})=& s_3 \hat{\theta}^3 +s_1 s_3 \hat{\theta}^{13} +s_2 s_3 \hat{\theta}^{23} + s_1 s_2 s_3 \hat{\theta}^{123}- \phi_{{\boldsymbol X'}|{\boldsymbol Z}} (s_1,s_2|\boldsymbol{\hat{\theta}}), \nonumber \\
\phi_{{\boldsymbol X'}|{\boldsymbol Z}} (s_1,s_2|\boldsymbol{\hat{\theta}}) :=& \ln \left[\exp \left(\hat{\theta}^3 + s_1 \hat{\theta}^{13}+ s_2 \hat{\theta}^{23} + s_1 s_2 \hat{\theta}^{123} \right) +1 \right].
\label{bi1}\end{aligned}$$ The conditional probability $p^{\hat{\boldsymbol{\theta}}}_{{\boldsymbol X'}|{\boldsymbol Z},{\boldsymbol Y'} }({\boldsymbol x'}|{\boldsymbol z},{\boldsymbol y'})$ is calculated as $$\begin{aligned}
\ln p^{\hat{\boldsymbol{\theta}}}_{{\boldsymbol X'}|{\boldsymbol Z},{\boldsymbol Y'} }({\boldsymbol x'}|{\boldsymbol z},{\boldsymbol y'}) =& s_3 \hat{\theta}^3 +s_1 s_3 \hat{\theta}^{13} +s_2 s_3 \hat{\theta}^{23} +s_3 s_4 \hat{\theta}^{34} + s_1 s_2 s_3 \hat{\theta}^{123} \nonumber \\
&+s_1 s_3 s_4 \hat{\theta}^{134} +
s_2 s_3 s_4 \hat{\theta}^{234} +s_1 s_2 s_3 s_4 \hat{\theta}^{1234}- \phi_{{\boldsymbol X'}|{\boldsymbol Z},{\boldsymbol Y'}} (s_1,s_2,s_4|\boldsymbol{\hat{\theta}}), \nonumber \\
\phi_{{\boldsymbol X'}|{\boldsymbol Z},{\boldsymbol Y'}} (s_1,s_2,s_4|\boldsymbol{\hat{\theta}}) :=& \ln \left[\exp \left(\hat{\theta}^3 + s_1 \hat{\theta}^{13} + s_2 \hat{\theta}^{23}+s_4 \hat{\theta}^{34}+ s_1 s_2 \hat{\theta}^{123}+s_1 s_4 \hat{\theta}^{134} +s_2 s_4 \hat{\theta}^{234} +s_1 s_2 s_4 \hat{\theta}^{1234} \right) +1 \right].
\label{bi2}\end{aligned}$$ The bipartite condition $\mathcal{C}_{\rm BI}$ is given by $p^{\hat{\boldsymbol{\theta}}}_{{\boldsymbol X'}|{\boldsymbol Z},{\boldsymbol Y'} } =p^{\hat{\boldsymbol{\theta}}}_{{\boldsymbol X'}|{\boldsymbol Z}}$. From Eqs. (\[bi1\]) and (\[bi2\]), we obtain the condition of $\mathcal{C}_{\rm BI}$ $$\begin{aligned}
\mathcal{C}_{\rm BI}: \hat{\theta}^{34}=\hat{\theta}^{134}=\hat{\theta}^{234}= \hat{\theta}^{1234}=0.\end{aligned}$$ In the same way, we obtain the condition of $\mathcal{C}_{\rm BI}^*$ $$\begin{aligned}
\mathcal{C}_{\rm BI}^*: \hat{\theta}^{12}=\hat{\theta}^{123}=\hat{\theta}^{124}= \hat{\theta}^{1234}=0.\end{aligned}$$ To clarify the relationship between $\mathcal{C}_{\rm BI}$ and $\mathcal{C}_{\rm BI}^*$, we can consider the permutation $(\alpha(1),\alpha(2),\alpha(3),\alpha(4)) = (3,4,1,2)$. The condition of $\mathcal{C}_{\rm BI}^*$ is given by the condition of $\mathcal{C}_{\rm BI}$ with the permutation $\alpha$, $$\begin{aligned}
\mathcal{C}_{\rm BI}^*: \hat{\theta}^{\alpha(3)\alpha(4)}=\hat{\theta}^{\alpha(3)\alpha(4)\alpha(1)}=\hat{\theta}^{\alpha(3)\alpha(4)\alpha(2)}= \hat{\theta}^{\alpha(3)\alpha(4)\alpha(1)\alpha(2)}=0.\end{aligned}$$
Next, we discuss the reversible manifold $\mathcal{M}_{\rm R}$. The transition probability $T({\boldsymbol z'}|{\boldsymbol z})=p^{\hat{\boldsymbol{\theta}}}_{{\boldsymbol Z'}|{\boldsymbol Z}}({\boldsymbol z'}|{\boldsymbol z})$ is calculated as $$\begin{aligned}
\ln T ({\boldsymbol z'}|{\boldsymbol z})=& s_3 \hat{\theta}^3 + s_4 \hat{\theta}^4 +\sum_{i<4} s_i s_4 \hat{\theta}^{i4}+\sum_{i<3} s_i s_3 \hat{\theta}^{i3} \nonumber \\
&+ \sum_{i<j<k} s_i s_j s_k \hat{\theta}^{ijk}+\sum_{i<j<k<l} s_i s_j s_k s_l \hat{\theta}^{ijkl} - \phi_{{\boldsymbol Z'}|{\boldsymbol Z}} (s_1, s_2| \boldsymbol{\hat{\theta}} ), \nonumber\end{aligned}$$ $$\begin{aligned}
&\phi_{{\boldsymbol Z'}|{\boldsymbol Z}} (s_1, s_2|\boldsymbol{\hat{\theta}} ) &\nonumber \\
& := \ln \left[\exp (\hat{\theta}^3 + \hat{\theta}^4 +s_1 \hat{\theta}^{14}+s_2 \hat{\theta}^{24}+\hat{\theta}^{34} +s_1 \hat{\theta}^{13} +s_2 \hat{\theta}^{23} + s_1 s_2 \hat{\theta}^{123}+ s_1 s_2 \hat{\theta}^{124} + s_1 \hat{\theta}^{134} + s_2 \hat{\theta}^{234} +s_1 s_2 \hat{\theta}^{1234} ) \right. \nonumber \\
&\left. +\exp (\hat{\theta}^3 +s_1 \hat{\theta}^{13} +s_2 \hat{\theta}^{23} + s_1 s_2 \hat{\theta}^{123}) +\exp (\hat{\theta}^4 +s_1 \hat{\theta}^{14}+s_2 \hat{\theta}^{24} + s_1 s_2 \hat{\theta}^{124} )+1 \right].
\label{sup3}\end{aligned}$$ The conditional probability $p^{{\boldsymbol{\hat{\theta}}}}_{{\boldsymbol Z}|{\boldsymbol Z'}}({\boldsymbol z}|{\boldsymbol z'})$ is given by $$\begin{aligned}
\ln p^{{\boldsymbol{\hat{\theta}}}}_{{\boldsymbol Z}|{\boldsymbol Z'}}({\boldsymbol z}|{\boldsymbol z'}) =& s_1 \hat{\theta}^1 + s_2 \hat{\theta}^2 +\sum_{1<i} s_1 s_i \hat{\theta}^{1i}+\sum_{2<i} s_2 s_i \hat{\theta}^{2i} \nonumber \\
&+ \sum_{i<j<k} s_i s_j s_k \hat{\theta}^{ijk}+\sum_{i<j<k<l} s_i s_j s_ks_l \hat{\theta}^{ijkl} - \phi_{{\boldsymbol Z}|{\boldsymbol Z'}} (s_3, s_4| {\boldsymbol{ \hat{\theta}}}), \nonumber\end{aligned}$$ $$\begin{aligned}
& \phi_{{\boldsymbol Z}|{\boldsymbol Z'}} (s_3, s_4| {\boldsymbol{ \hat{\theta}}}) &\nonumber \\
& := \ln \left[\exp (\hat{\theta}^1 + \hat{\theta}^2 +s_3 \hat{\theta}^{23}+s_4 \hat{\theta}^{24}+\hat{\theta}^{12} +s_3 \hat{\theta}^{13} +s_4 \hat{\theta}^{14} + s_3 s_4\hat{\theta}^{134}+ s_3 s_4 \hat{\theta}^{234} + s_3 \hat{\theta}^{123} + s_4 \hat{\theta}^{124} +s_3 s_4 \hat{\theta}^{1234} ) \right. \nonumber \\
&\left. +\exp (\hat{\theta}^1 +s_3 \hat{\theta}^{13} +s_4 \hat{\theta}^{14} + s_3s_4 \hat{\theta}^{134}) +\exp (\hat{\theta}^2 +s_3 \hat{\theta}^{23}+s_4 \hat{\theta}^{24} + s_3 s_4 \hat{\theta}^{234} + s_3 \hat{\theta}^{123} )+1 \right].
\label{sup4}\end{aligned}$$ The reversible manifold is defined as $\mathcal{M}_{\rm R}= \{ p^{{\boldsymbol{{\theta}}}}_{{\boldsymbol Z}, {\boldsymbol Z'}} | p^{{\boldsymbol{{\theta}}}}_{{\boldsymbol Z},{\boldsymbol Z'}}({\boldsymbol z},{\boldsymbol z'})= p^{{\boldsymbol{{\theta}}}}_{{\boldsymbol Z'}}({\boldsymbol z'})T({\boldsymbol z}|{\boldsymbol z'}) \}$ with $ p^{{\boldsymbol{{\theta}}}}_{{\boldsymbol Z'}}({\boldsymbol z'})= \sum_{{\boldsymbol z}} p^{{\boldsymbol{{\theta}}}}_{{\boldsymbol Z},{\boldsymbol Z'}}({\boldsymbol z},{\boldsymbol z'})$. The equations (\[sup3\]) and (\[sup4\]) yield $$\begin{aligned}
\mathcal{M}_{\rm R}= \left\{p^{{\boldsymbol{{\theta}}}}_{{\boldsymbol Z}, {\boldsymbol Z'}} \left| \theta^1=\hat{\theta}^3, \:\theta^2=\hat{\theta}^4, \:\theta^{23}=\hat{\theta}^{14}, \: \theta^{24}= \hat{\theta}^{24}, \: \theta^{12}= \hat{\theta}^{34},
\: \theta^{13}= \hat{\theta}^{13}, \: \theta^{14}= \hat{\theta}^{23},\right. \right. \nonumber \\ \: \left. \theta^{134}= \hat{\theta}^{123}, \: \theta^{234}= \hat{\theta}^{124}, \: \theta^{123}= \hat{\theta}^{134}, \: \theta^{124}= \hat{\theta}^{234}, \: \theta^{1234}= \hat{\theta}^{1234} \right\}.\end{aligned}$$ Under the both bipartite conditions $\mathcal{C}_{\rm BI}$ and $\mathcal{C}_{\rm BI}^*$, we obtain $$\begin{aligned}
\mathcal{M}_{\rm R}= \left\{p^{{\boldsymbol{{\theta}}}}_{{\boldsymbol Z}, {\boldsymbol Z'}} \left| \theta^1=\hat{\theta}^3, \:\theta^2=\hat{\theta}^4, \:\theta^{23}=\hat{\theta}^{14}, \: \theta^{24}= \hat{\theta}^{24},
\: \theta^{13}= \hat{\theta}^{13}, \: \theta^{14}= \hat{\theta}^{23} \right\} \right. .\end{aligned}$$
Next, we discuss the local reversible manifold $\mathcal{M}^{\mathcal{X}}_{\rm LR}$. Then the transition probability $T^{\mathcal{X}}_{\boldsymbol{y}}(\boldsymbol{x'}|\boldsymbol{x})= p^{{\boldsymbol{{\theta}}}}_{{\boldsymbol X'}|{\boldsymbol Z}}({\boldsymbol x'}|{\boldsymbol z})$ is given by Eq. (\[bi1\]). The conditional probability $p^{{\boldsymbol{{\theta}}}}_{{\boldsymbol X}|{\boldsymbol Z'}}({\boldsymbol x}|{\boldsymbol z'})$ is calculated as $$\begin{aligned}
\ln p^{{\boldsymbol{{\theta}}}}_{{\boldsymbol X}|{\boldsymbol Z'}}({\boldsymbol x}|{\boldsymbol z'}) =& s_1 \hat{\theta}^1 +s_1 s_3 \hat{\theta}^{13} +s_1 s_4 \hat{\theta}^{14} + s_1 s_3 s_4 \hat{\theta}^{134}- \phi_{{\boldsymbol X}|{\boldsymbol Z'}} (s_3,s_4|\boldsymbol{\hat{\theta}}), \nonumber \\
\phi_{{\boldsymbol X}|{\boldsymbol Z'}} (s_3,s_4|\boldsymbol{\hat{\theta}}) :=& \ln \left[\exp \left(\hat{\theta}^1 + s_3 \hat{\theta}^{13}+ s_4 \hat{\theta}^{14} + s_3 s_4 \hat{\theta}^{134} \right) +1 \right].
\label{sup6}\end{aligned}$$ The local reversible manifold is defined as $\mathcal{M}^{\mathcal{X}}_{\rm LR}: q_{{\boldsymbol X}|{\boldsymbol Z'}}({\boldsymbol x}|{\boldsymbol z'})= p^{{\boldsymbol{{\theta}}}}_{{\boldsymbol X}|{\boldsymbol Z'}}({\boldsymbol x}|{\boldsymbol z'}) = T^{\mathcal{X}}_{\boldsymbol{y'}}({\boldsymbol x}|{\boldsymbol x'})$. The equations (\[bi1\]) and (\[sup6\]) yield $$\begin{aligned}
\mathcal{M}_{\rm LR}^{\mathcal X}= \left\{p^{{\boldsymbol{{\theta}}}}_{{\boldsymbol Z}, {\boldsymbol Z'}} \left| \theta^1=\hat{\theta}^3, \:\theta^{13}=\hat{\theta}^{13}, \:\theta^{14}=\hat{\theta}^{23} , \: \theta^{134}= \hat{\theta}^{123} \right\} \right. .\end{aligned}$$ In the same way, we obtain the condition of $\mathcal{M}_{\rm LR}^{\mathcal Y}$ $$\begin{aligned}
\mathcal{M}_{\rm LR}^{\mathcal Y}= \left\{p^{{\boldsymbol{{\theta}}}}_{{\boldsymbol Z}, {\boldsymbol Z'}} \left| \theta^2=\hat{\theta}^4, \:\theta^{24}=\hat{\theta}^{24}, \:\theta^{23}=\hat{\theta}^{14} , \: \theta^{234}= \hat{\theta}^{124} \right\} \right. .\end{aligned}$$ To clarify the relationship between $\mathcal{M}_{\rm LR}^{\mathcal X}$ and $\mathcal{M}_{\rm LR}^{\mathcal Y}$, we can consider the permutation $(\alpha'(1),\alpha'(2),\alpha'(3),\alpha'(4)) = (2,1,4,3)$. The condition of $\mathcal{M}_{\rm LR}^{\mathcal Y}$ is given by the condition of $\mathcal{M}_{\rm LR}^{\mathcal Y}$ with the permutation $\alpha'$, $$\begin{aligned}
\mathcal{M}_{\rm LR}^{\mathcal Y}= \left\{p^{{\boldsymbol{{\theta}}}}_{{\boldsymbol Z}, {\boldsymbol Z'}} \left| \theta^{\alpha'(1)}=\hat{\theta}^{\alpha'(3)}, \:\theta^{{\alpha'(1)}{\alpha'(3)}}=\hat{\theta}^{{\alpha'(1)}{\alpha'(3)}}, \:\theta^{{\alpha'(1)}{\alpha'(4)}}=\hat{\theta}^{{\alpha'(2)}{\alpha'(3)}} , \: \theta^{{\alpha'(1)}{\alpha(4)}{\alpha'(3)}}= \hat{\theta}^{{\alpha'(2)}{\alpha'(1)}{\alpha'(3)}} \right\} \right..\end{aligned}$$ Under the both bipartite conditions $\mathcal{C}_{\rm BI}$ and $\mathcal{C}_{\rm BI}^*$, we obtain $$\begin{aligned}
\mathcal{M}_{\rm LR}^{\mathcal X}= \left\{p^{{\boldsymbol{{\theta}}}}_{{\boldsymbol Z}, {\boldsymbol Z'}} \left| \theta^1=\hat{\theta}^3, \:\theta^{13}=\hat{\theta}^{13}, \:\theta^{14}=\hat{\theta}^{23} \right\} \right.,\\
\mathcal{M}_{\rm LR}^{\mathcal Y}= \left\{p^{{\boldsymbol{{\theta}}}}_{{\boldsymbol Z}, {\boldsymbol Z'}} \left| \theta^2=\hat{\theta}^4, \:\theta^{24}=\hat{\theta}^{24}, \:\theta^{23}=\hat{\theta}^{14} \right\} \right..\end{aligned}$$
V. Example: the single spin model
---------------------------------
We here consider the entropy production $\sigma_{\rm tot}^{\mathcal{Z}}$ for a simple model of the single spin. The spin state at time $t$ is $\boldsymbol{z} =x \in \{0,1 \}$ and the spin state at time $t+dt$ is $\boldsymbol{z}' =x' \in \{0,1 \}$.
The stochastic evolution of the single spin is given by the transition probability $T(x'|x)$. If we start with the master equation $$\begin{aligned}
\frac{d}{dt} p (x';t) = \sum_{x} \left[ W(x \to x';t) p(x;t) - W(x' \to x;t) p(x';t) \right],\end{aligned}$$ the transition probability is given by $$\begin{aligned}
T(x'|x) =
\begin{cases}
(1- W(0 \to 1;t) dt) & (x=0, x'=0), \\
W(0 \to 1;t) dt & (x=0, x'=1),\\
W(1 \to 0;t) dt & (x=1, x'=0), \\
(1- W(1 \to 0;t) dt) & (x=1, x'=1).
\end{cases}\end{aligned}$$ The path probability $p_{X,X'}(x,x')$ is given by $$\begin{aligned}
p_{X,X'}(x,x')= T(x'|x)p (x;t) =
\begin{cases}
(1- W(0 \to 1;t) dt) p (0;t)& (x=0, x'=0), \\
W(0 \to 1;t) dtp (0;t) & (x=0, x'=1),\\
W(1 \to 0;t) dt (1-p (0;t))& (x=1, x'=0),\\
(1- W(1 \to 0;t) dt) (1-p (0;t))& (x=1, x'=1).
\end{cases}
\label{supspin}\end{aligned}$$ Here we introduce the exponential family $$\begin{aligned}
p_{X,X'}(x,x')&= \exp (\hat{\theta}^1 x + {\hat{\theta}}^{2} x' + {\hat{\theta}}^{12} xx' - \phi_{X,X'}({\hat{\theta}}^1, {\hat{\theta}}^2, {\hat{\theta}}^{12}) ), \nonumber\\
\phi_{X,X'}({\hat{\theta}}^1, {\hat{\theta}}^2, {\hat{\theta}}^{12} )&= \ln \left[1+ \exp ({\hat{\theta}}^1)+\exp ({\hat{\theta}}^2)+\exp ({\hat{\theta}}^1+{\hat{\theta}}^2 + {\hat{\theta}}^{12}) \right].\end{aligned}$$ The path probability can be written as $$\begin{aligned}
p_{X,X'}(x,x')=
\begin{cases}
\exp (- \phi_{X,X'}({\hat{\theta}}^1, {\hat{\theta}}^2, {\hat{\theta}}^{12}) ) & (x=0, x'=0), \\
\exp ({\hat{\theta}}^{2} - \phi_{X,X'}({\hat{\theta}}^1, {\hat{\theta}}^2, {\hat{\theta}}^{12}) ) & (x=0, x'=1),\\
\exp ({\hat{\theta}}^{1} - \phi_{X,X'}({\hat{\theta}}^1, {\hat{\theta}}^2, {\hat{\theta}}^{12}) ) & (x=1, x'=0), \\
\exp ({\hat{\theta}}^{1} + {\hat{\theta}}^2 + {\hat{\theta}}^{12} - \phi_{X,X'}({\hat{\theta}}^1, {\hat{\theta}}^2, {\hat{\theta}}^{12}) ) & (x=1, x'=1).
\end{cases}
\label{supspin2}\end{aligned}$$ From Eqs. (\[supspin\]) and (\[supspin2\]), we obtain the relationship between $({\hat{\theta}}^1, {\hat{\theta}}^2, {\hat{\theta}}^{12})$ and $(W(0 \to 1;t), W(1 \to 0;t), p (0;t))$ as $$\begin{aligned}
\phi_{X,X'}({\hat{\theta}}^1, {\hat{\theta}}^2, {\hat{\theta}}^{12})
&= \ln \frac{1}{p_{X,X'}(0,0)} \nonumber \\
&= -\ln [(1- W(0 \to 1;t) dt) p (0;t)], \\
{\hat{\theta}}^{1} &= \phi_{X,X'}({\hat{\theta}}^1, {\hat{\theta}}^2, {\hat{\theta}}^{12}) + \ln[W(1 \to 0;t) dt (1-p (0;t))]\nonumber\\
&= \ln \frac{p_{X,X'}(1,0)}{p_{X,X'}(0,0)}\nonumber\\
&=\ln \frac{W(1 \to 0;t) dt (1-p (0;t))}{(1- W(0 \to 1;t) dt) p (0;t)}, \\
{\hat{\theta}}^{2} &= \phi_{X,X'}({\hat{\theta}}^1, {\hat{\theta}}^2, {\hat{\theta}}^{12}) + \ln[W(0 \to 1;t) dtp (0;t)] \nonumber\\
&= \ln \frac{p_{X,X'}(0,1)}{p_{X,X'}(0,0)} \nonumber\\
&= \ln \frac{W(0 \to 1;t) dt}{1- W(0 \to 1;t) dt}, \\
{\hat{\theta}}^{12} &= \phi_{X,X'}({\hat{\theta}}^1, {\hat{\theta}}^2, {\hat{\theta}}^{12})- {\hat{\theta}}^{1} -{\hat{\theta}}^{2} + \ln[(1- W(1 \to 0;t) dt) (1-p (0;t))] \nonumber\\
&=\ln \frac{p_{X,X'}(0,0)p_{X,X'}(1,1)}{p_{X,X'}(0,1)p_{X,X'}(1,0)}\nonumber\\
&= \ln \frac{[1- W(0 \to 1;t) dt][1- W(1 \to 0;t) dt] }{[W(0 \to 1;t) dt][W(1 \to 0;t) dt]}.\end{aligned}$$ Here, we introduce the reversible manifold $\mathcal{M}_{\rm R}$. The transition probability is given by $$\begin{aligned}
T(x'|x) &= \exp ({\hat{\theta}}^2 x' + {\hat{\theta}}^{12} xx' - \phi_{X'|X} (x| {\hat{\theta}}^2, {\hat{\theta}}^{12})), \nonumber\\
\phi_{X'|X}(x| {\hat{\theta}}^2, {\hat{\theta}}^{12} )&= \ln \left[1+\exp ({\hat{\theta}}^2 + {\hat{\theta}}^{12}x) \right].\end{aligned}$$ We here consider the reversible manifold defined as $$\begin{aligned}
\mathcal{M}_{\rm R} = \{ q_{X,X'} | q_{X,X'}(x, x') = q_{X'}(x')T(x|x') \}.\end{aligned}$$ If we use the expression of the exponential family for $q_{X,X'}$, the reversible manifold is given by $$\begin{aligned}
\mathcal{M}_{\rm R} = \{ \exp ({\theta}^1 x + {\theta}^{2} x' + {\theta}^{12} xx' - \phi_{X,X'}({\theta}^1, {\theta}^2, {\theta}^{12}) ) | {\theta}^1= {\hat{\theta}}^2, {\theta}^{12}= {\hat{\theta}}^{12}\}.\end{aligned}$$ In our main result, the entropy production is given by the following optimization problem $$\begin{aligned}
\sigma_{\rm tot}^{\mathcal{Z}}={\rm min}_{q_{X,X'} \in \mathcal{M}_{\rm R}}D(p_{X,X'}||q_{X,X'} ).\end{aligned}$$ We here obtain the following Pythagorean theorem for $q_{X,X'} \in \mathcal{M}_{\rm R} $, $$\begin{aligned}
D(p_{X,X'}||q_{X,X'} )&=D(p_{X,X'}||q^*_{X,X'} )+ D(q^*_{X,X'}||q_{X,X'} ), \nonumber\\
q^*_{X,X'}(x, x') &= \exp (\hat{\theta}^2 x + {\theta}^{2*} x' + \hat{\theta}^{12} xx' - \phi_{X,X'}(\hat{\theta}^2, {\theta}^{2*}, \hat{\theta}^{12}) ), \end{aligned}$$ with the constraint $$\begin{aligned}
\sum_{x} q^*_{X,X'}(x, x') &= \sum_{x} p_{X, X'}(x, x').
\label{supconst}\end{aligned}$$ From the nonnegativity of the Kullback-Leibler divergence $D(q^*_{X,X'}||q_{X,X'} ) \geq 0$, the solution of the optimization problem is given by $$\begin{aligned}
D(p_{X,X'}||q_{X,X'} )& \geq D(p_{X,X'}||q^*_{X,X'} ), \\
q^*_{X,X'} &= {\rm argmin}_{q_{X,X'} \in \mathcal{M}_{\rm R}}D(p_{X,X'}||q_{X,X'} ).\end{aligned}$$
By using the expression by $({\theta}_1,{\theta}_2,{\theta}_{12})$, this optimization problem can be written as $$\begin{aligned}
\sigma_{\rm tot}^{\mathcal{Z}}={\rm min}_{{\theta}^2} \mathbb{E}[x({\hat{\theta}}^1-{\hat{\theta}}^2) +x' ({\hat{\theta}}^2 - {\theta}^2) - \phi_{X,X'}({\hat{\theta}}^1, {\hat{\theta}}^2, {\hat{\theta}}^{12}) + \phi_{X,X'}({\hat{\theta}}^2, {\theta}^2, {\hat{\theta}}^{12}) ],
\label{supopt}\end{aligned}$$ where $\mathbb{E}$ is the expected value $\mathbb{E}[f] = \sum_{x,x'}p_{X,X'}(x,x')f(x,x')$. The constraint Eq. (\[supconst\]) is calculated as $$\begin{aligned}
\exp \left[({\hat{\theta}}^2 - {\theta}^{2*} )x' - \phi_{X,X'}({\hat{\theta}}^1, {\hat{\theta}}^2, {\hat{\theta}}^{12}) +\phi_{X,X'}({\hat{\theta}}^2, {\theta}^{2*}, {{\hat{\theta}}}^{12}) \right] &= \exp \left[ \phi_{X|X'}(x'|{\hat{\theta}}^2, {\hat{\theta}}^{12} )-\phi_{X|X'}(x'|{\hat{\theta}}^1, {\hat{\theta}}^{12} ) \right].\end{aligned}$$ where $\phi_{X|X'}(x'|{\hat{\theta}}^1, {\hat{\theta}}^{12} )$ is defined as $$\begin{aligned}
\phi_{X|X'}(x'|{\hat{\theta}}^1, {\hat{\theta}}^{12} )&= \ln \left[1+\exp ({\hat{\theta}}^1 + {\hat{\theta}}^{12}x') \right]. \end{aligned}$$
Under the constraint, the optimization problem Eq. (\[supopt\]) is calculated as $$\begin{aligned}
\sigma_{\rm tot}^{\mathcal{Z}}&= \mathbb{E} \left[x({\hat{\theta}}^1-{\hat{\theta}}^2) +x' ({\hat{\theta}}^2 - {\theta}^{2*}) - \phi_{X,X'}({\hat{\theta}}^1, {\hat{\theta}}^2, {\hat{\theta}}^{12}) + \phi_{X,X'}({\hat{\theta}}^2, {\theta}^{2*}, {\hat{\theta}}^{12}) \right] \nonumber\\
&= \mathbb{E} \left[x({\hat{\theta}}^1-{\hat{\theta}}^2) + \phi_{X|X'}(x'|{\hat{\theta}}^2, {\hat{\theta}}^{12} )-\phi_{X|X'}(x'|{\hat{\theta}}^1, {\hat{\theta}}^{12} )\right].
\label{supen}\end{aligned}$$ We can check the equivalence between Eq. (\[supen\]) and the original definition of the entropy production as follows, $$\begin{aligned}
\sigma_{\rm tot}^{\mathcal{Z}}&= \sum_{x,x'}T(x'|x)p_{X}(x) \ln \frac{T(x'|x)p_{X}(x)}{T(x|x')p_{X'}(x')} \nonumber \\
&= \mathbb{E} \left[\ln \frac{T(x'|x)p_{X}(x)}{T(x|x')p_{X'}(x')} \right] \nonumber \\
&= \mathbb{E} \left[\ln \frac{p_{X|X'}(x|x')}{\exp ({\hat{\theta}}^2 x + {\hat{\theta}}^{12} xx' - \phi_{X'|X} (x'| {\hat{\theta}}^2, {\hat{\theta}}^{12})) } \right] \nonumber \\
&= \mathbb{E} \left[\ln \frac{\exp ({\hat{\theta}}^1 x + {\hat{\theta}}^{12} xx' - \phi_{X|X'} (x'| {\hat{\theta}}^2, {\hat{\theta}}^{12}))}{\exp ({\hat{\theta}}^2 x + {\hat{\theta}}^{12} xx' - \phi_{X'|X} (x'| {\hat{\theta}}^2, {\hat{\theta}}^{12})) } \right]\nonumber \\
&= \mathbb{E} \left[x({\hat{\theta}}^1-{\hat{\theta}}^2) + \phi_{X|X'}(x'|{\hat{\theta}}^2, {\hat{\theta}}^{12} )-\phi_{X|X'}(x'|{\hat{\theta}}^1, {\hat{\theta}}^{12} )\right],\end{aligned}$$ where we used $ \phi_{X'|X} (x'| {\hat{\theta}}^2, {\hat{\theta}}^{12})= \phi_{X|X'} (x'| {\hat{\theta}}^2, {\hat{\theta}}^{12})$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider the Cauchy problem associated with the Zakharov-Kuznetsov equation, posed on ${\mathbb T}^2$. We prove the local well-posedness for given data in $H^s({\mathbb T}^2)$ whenever $s> 5/3$. More importantly, we prove that this equation is of quasi-linear type for initial data in any Sobolev space on the torus, in sharp contrast with its semi-linear character in the ${\mathbb R}^2$ and ${\mathbb R}\times {\mathbb T}$ settings.'
address:
- |
IMPA\
Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brazil.
- |
IMECC-UNICAMP\
13083-859, Campinas, São Paulo, Brazil
- ' Université de Cergy-Pontoise, Cergy-Pontoise, F-95000, UMR 8088 du CNRS '
- ' Université de Cergy-Pontoise, Cergy-Pontoise, F-95000, UMR 8088 du CNRS '
author:
- 'F. Linares'
- 'M. Panthee'
- 'T. Robert'
- 'N. Tzvetkov'
title: 'On the Periodic Zakharov-Kuznetsov equation'
---
[^1]
Introduction
============
In this work we consider the initial value problem (IVP) associated with the following bi-dimensional dispersive model $$\label{zk-1}
\begin{cases}
{\partial}_tw+{\partial}_x^3w+{\partial}_x{\partial}_y^2w +w{\partial}_xw=0, \hskip15pt \mathbf{x}=(x,y)\in {\mathbb T}^2, \;\;t\in{\mathbb R},\\
w(0,x,y)=w_0(x,y),
\end{cases}$$ where $w:{\mathbb R}\times{\mathbb T}^2\to {\mathbb R}$, ${\mathbb T}^2:={\mathbb R}^2/(2\pi{\mathbb Z})^2$.\
The model presented in is a bi-dimensional generalization of the Korteweg-de Vries (KdV) equation, and was introduced in [@ZK] to describe the propagation of nonlinear ion-acoustic waves in magnetized plasma (for its rigorous derivation we refer to [@LLS]). This model is widely known as the Zakharov-Kuznetsov (ZK) equation and is extensively studied in the literature, see for example [@Fa; @G-H; @LPS10; @LP09; @LS; @M-P; @RV] and the references therein.\
If is posed on ${\mathbb R}^2$, it is known that it is locally well-posed in $H^s({\mathbb R}^2)$, $s>1/2$ (see [@M-P; @G-H]). Furthermore, if is posed on ${\mathbb R}\times{\mathbb T}$ then it is well-posed in $H^s({\mathbb R}\times{\mathbb T})$, $s\geq 1$ (see [@M-P]). In all these results the dispersion in the $x$ direction plays a crucial role, the solutions are constructed by the Picard iteration and the flow map is smooth on $H^s$.\
Our interest here is in studying the local well-posedness issue to the IVP for given data in the periodic Sobolev space $H^s({\mathbb T}^2)$, $s\in {\mathbb R}$. We note that for $s>2$ one can ignore the dispersive effects and solve as a quasi-linear hyperbolic equation leading to the local well-posedness of in $H^s({\mathbb T}^2)$, $s>2$. Our first goal is to prove a well-posedness result in spaces of lower regularity.
\[local-zk\] Let $s>5/3$. Then for every $w_0\in H^s({\mathbb T}^2)$ there exist a time $T=T(\|w_0\|_{H^s})$ and a unique solution $w\in C([0,T]: H^s({\mathbb T}^2))$ to the IVP such that $w, \partial_x w, \partial_y w \in L_T^1L_{xy}^{\infty}$. Moreover, the map that takes the initial data to the solution $w_0\mapsto w\in C([0,T]: H^s({\mathbb T}^2))$ is continuous.
This local well-posedness result for the IVP associated with the ZK equation posed in a purely periodic domain is the first one exploiting in a non trivial way the dispersive effect. The method of proof of Theorem \[local-zk\] is nowadays standard (see e.g. [@BGT; @IK; @CK; @KK; @KTz] ). It uses in a crucial way short time Strichartz estimates for the linear part of the equation, see Lemma \[lem-zk1\]. Our short time Strichartz estimate is derived by purely local in space considerations in the spirit of [@BGT]. It may be that global in space considerations based on number theory arguments may improve the result of Lemma \[lem-zk1\]. To the best of our knowledge, the works [@KPV91; @LV] are the first papers where short time Strichartz estimates on a compact spatial domain were considered. We also expect that further improvements can be obtained by using the ideas introduced in [@IKT].\
Our main purpose in this work is to show that the flow map constructed in Theorem \[local-zk\] is not locally uniformly continuous, which highlights the quasilinear behaviour of the ZK equation (\[zk-1\]) when considered with periodic boundary conditions. For previous related contributions, we refer to [@KochTzvetkov2003BO; @KochTzvetkov2008; @Robert2018].
\[lemma1.1\] Let $s>5/3$. There exist two positive constants $c$ and $C$ and two sequences $(u_m)$ and $(v_m)$ of solutions to (\[zk-1\]) in $C([0,1] : H^s({\mathbb T}^2))$ such that $$\label{unif_bound}
\sup_{t\in[0,1]}\|u_m(t)\|_{H^s}+\|v_m(t)\|_{H^s} \leq C,$$ and satisfying initially $$\label{init_bound}
\lim_{m\rightarrow +\infty}\|u_m(0)-v_m(0)\|_{H^s}=0,$$ but such that for every $t\in[0,1]$, $$\label{failure_bound}
\liminf_{m\rightarrow +\infty}\|u_m(t)-v_m(t)\|_{H^s}\geq ct.$$
As mentioned above the result of Theorem \[lemma1.1\] shows that the ZK equation behaves quite differently in the case of periodic boundary conditions in the $x$ variable since in the case when ZK is posed on ${\mathbb R}^2$ or ${\mathbb R}\times{\mathbb T}$ a result as Theorem \[lemma1.1\] cannot hold true. As in the previous related results in the proof of Theorem \[lemma1.1\] we exploit resonant interactions. We use the resonances coming from the zero $x$ frequency. Let us recall that in the context of the KdV or the KP equations we can eliminate this interaction by the use of a simple gauge transform (see *e.g.* [@JB_kp; @Robert2017]). We are not aware of such a gauge transform in the context of the ZK equation and we suspect that the situation is more intricate compared to KdV or KP. It is however not excluded that one may reduce the ZK equation on the torus to a semi-linear problem by the use of a suitably chosen gauge transform, at least for some class of initial data.\
Let us finally mention that since we work in the periodic setting, in the proof of Theorem \[lemma1.1\] we cannot use the localization in space arguments as in [@KochTzvetkov2003BO; @KochTzvetkov2008]. This makes that in order to construct the approximation of the true solution, we exploit a curvature property of the resonance function associated with the ZK equation (see also (\[eq-curv-reson\]) and Remark \[FIN\] below).
Preliminary estimates {#sec-3}
=====================
In this section we derive some preliminary estimates that are useful to prove the local well-posedness result of this work. We begin with the preparation to obtain the Strichartz estimate in localized form.
For $N\in 2^{{\mathbb N}}\cup \{0\}$, we define a projection operator $P_N$ as a Fourier multiplier operator by $$\label{eq-d2}
\begin{cases}
\widehat{P_Nf}({\mathbf m}) := \chi_{[N;2N[}(|{\mathbf m}|)\widehat{f}({\mathbf m}),~N\geq 1\\
\widehat{P_0f}({\mathbf m}) := \chi_{[0;1[}(|{\mathbf m}|)\widehat{f}({\mathbf m}),~N=0,
\end{cases}$$ where ${\mathbf m}=(m_1,m_2)\in{\mathbb Z}^2$, and $\chi_I$ is the indicator function of the interval $I$. In terms of the projection operator, we can write an equivalence for the Sobolev norms on the torus in the following way $$\label{eq-n2}
\|f\|_{H^s({\mathbb T}^2)}^2 \sim \sum_{N\geq 0} (1\vee N)^{2s}\|P_Nf\|_{L^2({\mathbb T}^2)}^2,$$ where we use the notation $a\vee b = \max(a,b)$.
Consider now the linear problem $$\label{zk-6}
\begin{cases}
{\partial}_tw+{\partial}_{x}^3w +{\partial}_x{\partial}_{y}^2w =0, \quad (x,y)\in {\mathbb T}^2,\; t\in {\mathbb R},\\
w(0,x, y) = w_0(x,y),
\end{cases}$$ and let $W(t)$ defined by $$\label{w-t}
W(t)w_0({\mathbf x}) =\frac{1}{(2\pi)^2} \sum_{{\mathbf m}\in{\mathbb Z}^2}e^{i({\mathbf m}\cdot{\mathbf x}+(m_1^3+m_1m_2^2)t)}\widehat{w_0}({\mathbf m}),$$ be the unitary group that describes the solution to .
In what follows we prove the localized version of the Strichartz estimate satisfied by the unitary group $W(t)$.
\[lem-zk1\] Let $W(t)$ be as defined in and $P_N$ be the operator defined in . Then for any $w_0\in L^2({\mathbb T}^2)$ and time interval $I\subset {\mathbb R}$ with $|I|\sim (1\vee N)^{-2}$, $$\label{eq-zk2a}
\|W(t)P_Nw_0\|_{L^{2}_{I} L_{xy}^{\infty}} \lesssim (1\vee N)^{-1/3} \, \|w_0\|_{L^2_{xy}},$$ and $$\label{eq-l3zk}
\|W(t) w_0\|_{L^{2}_{ [0, 1]} L_{xy}^{\infty}} \lesssim \, \|w_0\|_{H^{\frac23+}}.$$
We start by proving . Since it is straightforward for $N=0$, we assume $N\geq 1$ in the following. For ${\mathbf x}=(x,y)$, using the definition of the group $W(t)$ and the projection operator $P_N$, we have that $$\label{eq-p1}
W(t)P_Nw_0 = \frac{1}{(2\pi)^2 }\sum_{{\mathbf m}\in {\mathbb Z}^2}e^{i({\mathbf m}\cdot{\mathbf x}+(m_1^3+m_1m_2^2)t)}\chi_{[N;2N[}(|{\mathbf m}|)\widehat{w_0}({\mathbf m}).$$ In order to decouple the frequencies in the time oscillation, we will use the symmetrization argument of [@G-H]. First, to have good localizations in frequency after symmetrizing, we observe that if $|{\mathbf m}|< 2N$ then $|m_1\pm\frac{m_2}{\sqrt{3}}|< 4N$. Since $P_N$ is a bounded operator with norm one, it suffices to prove that (\[eq-zk2a\]) holds with $P_N$ replaced with $$\widetilde{P_N}w_0 = \sum_{{\mathbf m}\in{\mathbb Z}^2}\widetilde{\phi_{4N}}(m_1+\frac{m_2}{\sqrt{3}})\widetilde{\phi_{4N}}(m_1-\frac{m_2}{\sqrt{3}})\widehat{w_0}(m_1,m_2)e^{i{\mathbf m}\cdot{\mathbf x}},$$ where $\widetilde{\phi_{4N}}$ is a smooth cut-off defined as follows : let $\phi\in C_0^{\infty} (-2,2)$ be such that $\phi(r)=1$ for $|r|\leq 1$. Then write $$\widetilde{\phi_N}(x) = \phi\left(\frac{x}{N}\right).$$ Indeed, with this definition and the remark above, we have $P_N = \widetilde{P_N}P_N$.
We can now define the operator $$\mathbf{T} = \chi_I(t)W(t)\widetilde{P_N}$$ where $\chi_I$ stands for the indicator function of the interval $I$.
To prove that $\mathbf{T}$ is bounded from $L^2({\mathbb T}^2)$ to $L^{2}(I : L^{\infty}({\mathbb T}^2))$ with norm less than $N^{-\frac13}$, the classical $TT^*$ argument reduces the problem to show that $$g\in L^{2}(I:L^1({\mathbb T}^2))\mapsto \mathbf{T}\mathbf{T}^*(g)(t)=\int_{{\mathbb R}}\chi_I(t)\chi_I(t')W(t-t')\widetilde{P_N}^2g(t')dt'\in L^{2}(I:L^{\infty}({\mathbb T}^2))$$ is bounded with norm less than $N^{-\frac23}$. This last operator can be written as an integral operator, whose kernel is given by $$\label{eq-p3.2}
K_N(t, t', {\mathbf x},{\mathbf x}'):= \chi_{I}(t)\chi_{I}(t')\sum_{{\mathbf m}\in{\mathbb Z}^2}\widetilde{\phi_{4N}}(m_1+\frac{m_2}{\sqrt{3}})^2\widetilde{\phi_{4N}}(m_1-\frac{m_2}{\sqrt{3}})^2 e^{i[{\mathbf m}\cdot( {\mathbf x}-{\mathbf x}')+(m_1^3+m_1m_2^2)(t-t')]}.$$ Note that the time localizations imply that $|t-t'|\lesssim N^{-2}$. Therefore the whole matter reduces to proving the following pointwise estimate on the kernel $$\label{eq-p4}
\Big|\sum_{{\mathbf m}\in{\mathbb Z}^2}\widetilde{\phi_{2N}}(m_1+\frac{m_2}{\sqrt{3}})^2\widetilde{\phi_{2N}}(m_1-\frac{m_2}{\sqrt{3}})^2e^{i({\mathbf m}\cdot {\mathbf x}+(m_1^3+ m_1m_2^2)t)}\Big|\lesssim |t|^{-2/3},$$ for any ${\mathbf x}\in {\mathbb T}^2$ and $N^{-3}\lesssim |t|\lesssim N^{-2}$.
Indeed, with (\[eq-p4\]) at hand, the bound on $\mathbf{T}\mathbf{T}^*$ follows from Young’s inequality and the fact that $$\||t|^{-2/3}\chi_{[N^{-3};N^{-2}]}(t)\|_{L^{1}}\lesssim N^{-\frac23},$$ whereas the contribution of $|t-t'|\lesssim N^{-3}$ is estimated roughly with the trivial bound $|K_N|\lesssim N^2$.
To obtain (\[eq-p4\]), we use Poisson summation formula [@SW] $$\label{eq-p5}
\sum_{{\mathbf m}\in{\mathbb Z}^2}F({\mathbf m}) = \sum_{{\mathbf n}\in{\mathbb Z}^2}\widehat{F}(2\pi{\mathbf n}), \qquad \forall\; F\in \mathcal{S}({\mathbb R}^2).$$
Using , the sum in the LHS of becomes $$\sum_{{\mathbf n}\in {\mathbb Z}^2}\int_{{\mathbb R}^2}e^{-i2\pi{\mathbf n}\cdot\xi} \widetilde{\phi_{4N}}(\xi_1+\frac{\xi_2}{\sqrt{3}})^2\widetilde{\phi_{4N}}(\xi_1-\frac{\xi_2}{\sqrt{3}})^2 e^{i({\mathbf x}\cdot\xi+(\xi_1^3+\xi_1\xi_2^2t)}d\xi.$$
Here we symmetrize the linear evolution : by using the linear transformation in [@G-H Section 2.1] we can write the integrals within the sum above as $$c\int_{{\mathbb R}^2} \widetilde{\phi_{4N}}(\eta_1)^2\widetilde{\phi_{4N}}(\eta_2)^2 e^{i(\eta_1\frac{(x-2\pi n_1)+\sqrt{3}(y-2\pi n_2)}{2}+\eta_2\frac{(x-2\pi n_1)-\sqrt{3}(y-2\pi n_2)}{2}+\frac12(\eta_1^3+\eta_2^3)t}d\eta.$$
Thus (\[eq-p4\]) can be expressed as $$\label{eq-p6}
\Big|\sum_{{\mathbf n}\in{\mathbb Z}^2} F_N\left(\frac{x+\sqrt{3}y-2\pi (n_1 +\sqrt{3}n_2)}{2}\right)F_N\left(\frac{x-\sqrt{3}y-2\pi(n_1-\sqrt{3}n_2)}{2}\right)\Big|\lesssim |t|^{-2/3},$$ where $$F_N(X)=\int_{{\mathbb R}} \widetilde{\phi_{4N}}(\xi)^2e^{i(\xi X+\frac12 \xi^3t)}d\xi.$$ First, note that from the time and frequency localizations, we have $|\xi^2 t|\lesssim 1$, thus for $|X|$ large enough the phase in the above integral has no stationary point. As a result, we can bound $|F_N(X)|$ for $|X|> C$ for some (large) constant $C>0$ by successive integrations by parts. Indeed, by writing $F_N$ as an abstract oscillatory integral ${\displaystyle \int_{{\mathbb R}}\psi(\xi)e^{i\Phi(\xi)}d\xi}$, using the definition of $\widetilde{\phi_{4N}}$ and the localizations in $\xi$ and $t$ we get that for any $k\in{\mathbb N}$ we have $\|\psi^{(k)}\|_{L^1}\lesssim N^{1-k}$ and $\|\Phi'\|_{L^{\infty}}\sim |X|$, $\|\Phi''\|_{L^{\infty}}\lesssim N^{-1}$ and $\|\Phi^{(3)}\|_{L^{\infty}}\lesssim N^{-2}$. Thus for $|X|>C$, $$\int_{{\mathbb R}} \psi(\xi)e^{i\Phi(\xi)}d\xi =- \int_{{\mathbb R}}\left[(i\Phi')^{-1}\left\{(i\Phi')^{-1}\left((i\Phi')^{-1}\psi\right)'\right\}'\right]'e^{i\Phi(\xi)}d\xi=O(N^{-2}|X|^{-3}).$$ Now let us come back to (\[eq-p6\]). Let $M>0$ be some large constant. Using that $x$ and $y$ lie in a compact set, if $|n_1+\sqrt{3}n_2|\vee |n_1-\sqrt{3}n_2|>M$ then we can use the previous bound for the corresponding term in the sum of (\[eq-p6\]) along with the rough bound $\|F_N\|_{L^{\infty}}\lesssim N$ for the second term, to get $$\begin{gathered}
\Big|\sum_{|n_1+\sqrt{3}n_2|\vee |n_1-\sqrt{3}n_2|>M} F_N\left(\frac{x+\sqrt{3}y-2\pi (n_1 +\sqrt{3}n_2)}{2}\right)F_N\left(\frac{x-\sqrt{3}y-2\pi(n_1-\sqrt{3}n_2)}{2}\right)\Big|\\
\lesssim \sum_{|n_1+\sqrt{3}n_2|\vee |n_1-\sqrt{3}n_2|>M} N^{-1}(|n_1+\sqrt{3}n_2|\vee |n_1-\sqrt{3}n_2|)^{-3}\\
\lesssim N^{-1}\sum_{|n_1+\sqrt{3}n_2|\vee |n_1-\sqrt{3}n_2|>M} [(1\vee|n_1|)(1\vee|n_2|)]^{-3/2}.\end{gathered}$$ In view of the last bound, to prove (\[eq-p6\]) it remains to treat the contribution of the sum for $|n_1+\sqrt{3}n_2|<M$ and $|n_1-\sqrt{3}n_2|<M$. In this case we have in particular $|{\mathbf n}|\leq 2M$ so that the sum is finite (uniformly in $N$), thus it is enough to get the bound $$\label{eq-p6bis}
\|F_N\|_{L^{\infty}}\lesssim |t|^{-1/3}.$$ We first write $F_N$ as $$F_N(X)=\int_{{\mathbb R}}e^{i(X\xi+\frac{\xi^3}{2}t)}d\xi-\int_{{\mathbb R}}\left[1-\widetilde{\phi_{4N}}^2(\xi)\right]e^{i(X\xi+\frac{\xi^3}{2}t)}d\xi=I+II.$$ The first term is $$I = \frac{c}{t^{\frac13}}\int_{{\mathbb R}} e^{i(\sqrt[3]{2}Xt^{-1/3}\eta+\eta^3)} d\eta.$$ From [@KPV91] we have that the last integral is bounded (see also Lemma 7.2 in [@LP-15] or Lemma 3.6 in [@KPV-93]), so it remains to estimate the second term. We observe that the phase $\Phi(\xi)=X\xi+\frac{\xi^3}{2}t$ satisfies $|\Phi''(\xi)|= 3|t\xi|\gtrsim N|t|$ on the support of $(1-\widetilde{\phi_{4N}}^2)$. Thus we can conclude from van der Corput lemma (see *e.g.* [@Stein Chapter 8]) that $$|II|\lesssim (N|t|)^{-\frac12}\left\{\|1-\widetilde{\phi_{4N}}^2\|_{L^{\infty}}+\|(1-\widetilde{\phi_{4N}}^2)'\|_{L^1}\right\}\lesssim |t|^{-\frac13},$$ where in the last step we have used the lower bound on $|t|$. This proves (\[eq-p6bis\]).
The proof of the estimate follows from [*the short time Strichartz estimate*]{} using a Littlewood-Paley procedure. In fact, splitting the interval $[0, 1]$ on $(1\vee N)^2$ intervals of size $(1\vee N)^{-2}$ we obtain, using [*the short time Strichartz estimate*]{} (and that $P_N^2=P_N$) $$\label{eq-p10}
\|W(t)P_Nw_0\|_{L^{2}([0, 1]; L_{xy}^{\infty})}^{2}
\leq c (1\vee N)^2(1\vee N)^{-\frac23}\|P_Nw_0\|_{L_{xy}^2}^{2}.$$ Summing the last estimate over $N\geq 0$ and using the equivalence of norms, we conclude that $$\label{eq-p12}
\|W(t)w_0\|_{L^{2}([0, 1]; L_{xy}^{\infty})}
\leq c \|w_0\|_{H^{\frac23+}}.$$ This completes the proof of Lemma \[lem-zk1\].
Next, we move to prove a version of the Kato-Ponce Commutator estimate in $\mathbb T^2$, which is an extension of the one proved for the ${\mathbb T}$-case in [@IK].
First, let us define $$H^{\infty}(\mathbb T^2)=\bigcap_{s\in{\mathbb R}}H^{s}({\mathbb T}^2).$$ For $s\in{\mathbb R}$ we also define the Fourier multiplier $$\widehat{J^sf}({\mathbf m}):=\widehat{J^s_{{\mathbb T}^2}f}({\mathbf m}) = (1+|{\mathbf m}|^2)^{\frac{s}2}\widehat{f}({\mathbf m}).$$
\[lem2\] Let $s\geq 1$ and $ f, g \in H^{\infty}(\mathbb T^2)$. Then $$\label{eq2.5}
\begin{split}
\|J^s(fg)-fJ^sg&\|_{L^2(\mathbb T^2)}\le c\,\Big\{\|J^sf\|_{L^2(\mathbb T^2)}\|g\|_{L^{\infty}(\mathbb T^2)}\\
&+
(\|f\|_{L^{\infty} (\mathbb T^2)}+\|\nabla f\|_{L^{\infty} (\mathbb T^2)})\,\|J^{s-1}g\|_{L^2(\mathbb T^2)}\Big\}.
\end{split}$$
Lemma \[lem2\] is a bidimensional generalization of [@IK Lemma 9.A.1 (b)], it follows from the Kato-Ponce commutator estimate on ${\mathbb R}^2$ [@KP] through the same argument as in [@IK].
[*A PRIORI*]{} estimates: proof of the Theorem \[local-zk\] {#sec-4}
===========================================================
In this section we use the estimates established in the previous section to establish some more estimates and prove the main result regarding local well-posedness. We begin by proving an [*a priori*]{} estimate that plays a fundamental role in our argument.
\[lem2-zk\] Let $w_0\in H^{\infty}({\mathbb T}^2)$ and $w$ be the corresponding smooth solution to the IVP . Then for any $T\in [0, 1]$ and $s\geq 1$, we have $$\label{ap-1zk}
\|w\|_{L_T^{\infty}H^s({\mathbb T}^2)}\leq c_s\exp\big(c_s(\|w\|_{L_T^1L_{xy}^{\infty}}+\|\nabla w\|_{L_T^1L_{xy}^{\infty}})\big)\|w_0\|_{H^s({\mathbb T}^2)}.$$
We apply the operator $J^s$ to , multiply the resulting equation by $J^sw$ and then integrate by parts, to obtain $$\label{est-2zk}
\frac12\frac{d}{dt}\int_{{\mathbb T}^2}(J^sw)^2\,dxdy +\int_{{\mathbb T}^2}J^s(w\partial_xw)J^sw\, dxdy=0.$$
Using the commutator notation $[A, B]f = A(Bf) -B(Af)$, integration by parts and the Cauchy-Schwarz inequality, we obtain from that $$\label{eq2.8}
\begin{split}
\frac12\frac{d}{dt}\int_{{\mathbb T}^2}(J^sw)^2dxdy &= -\int_{{\mathbb T}^2}\{J^s(\partial_xw)-wJ^s\partial_xww +wJ^s\partial_xw\}J^sw\,dxdy\\
&= -\int_{{\mathbb T}^2}([J^s, w]\partial_xw)J^su\, dxdy+\frac12\int_{{\mathbb T}^2}\partial_x w(J^sw)^2 \,dxdy\\
&\leq \|[J^s, w]\partial_xw\|_{L^2({\mathbb T}^2)} \|J^sw\|_{L^2({\mathbb T}^2)} +\frac12\|\partial_xw\|_{L^{\infty}({\mathbb T}^2)}\|J^sw\|_{L^2({\mathbb T}^2)}^2.
\end{split}$$
In the light of estimate in Lemma \[lem2\], we have $$\label{eq2.9}
\begin{split}
\|[J^s, w]\partial_xw\|_{L^2({\mathbb T}^2)}\leq c&\big\{\|J^sw\|_{L^2({\mathbb T}^2)}\|\partial_x w\|_{L^{\infty}({\mathbb T}^2)}\\
&+(\|w\|_{L^{\infty}({\mathbb T}^2)}+\|\nabla w\|_{L^{\infty}({\mathbb T}^2)})\|J^{s-1}\partial_xw\|_{L^2({\mathbb T}^2)}\big\}.
\end{split}$$
Inserting in , we obtain, after simplification $$\label{eq2.10}
\frac{d}{dt}\|w(t)\|_{H^s({\mathbb T}^2)}^2 \leq c\big(\|w\|_{L^{\infty}({\mathbb T}^2)} + \|\nabla w\|_{L^{\infty}({\mathbb T}^2)}\big)\|w(t)\|_{H^s({\mathbb T}^2)}^2.$$
Using Gronwall’s inequality, yields $$\|w\|_{L_T^{\infty}H^s({\mathbb T}^2)}^2\leq \|w_0\|_{H^s({\mathbb T}^2)}^2\exp\Big(c\int_0^T\big(\|w\|_{L^{\infty}({\mathbb T}^2)} + \|\nabla w\|_{L^{\infty}({\mathbb T}^2)}\big)dt\Big),$$ which gives the required estimate .
Following the approach in [@CK] we obtain the following estimate that will be useful in our argument.
\[lem3-zk\] Let $F\in L^1([0,T]:L^2({\mathbb T}^2))$. Then any solution of the equation $$\label{eqn-12}
\partial_t w + \partial_x^3w+\partial_x\partial_y^2w+\partial_x F(w)=0, \hskip15pt (x,y)\in{\mathbb T}^2, \;\;t\in{\mathbb R},$$ satisfies $$\label{ineq-zk}
\|w\|_{L^1_TL^{\infty}_{xy}}\lesssim T^{\frac12}\,\Big(\|w\|_{L^{\infty}_T H^{\frac23+}}+\|F\|_{L^1_T L^2_{xy}}\Big).$$
Let $P_N$ be the projection operator defined in (\[eq-d2\]), and fix such a number $N\in 2^{{\mathbb N}}\cup\{0\}$. We divide the interval $[0;T]$ in subintervals $[a_k, a_{k+1})$ of size $T(1\vee N)^{-2}$ for $k=1, \dots, (1\vee N)^2$. Then
$$\label{ineq1-zk}
\begin{split}
\|P_N w\|_{L^1_T L^{\infty}_{xy}} &\le \underset{k=1}{\overset{(1\vee N)^2}{\sum}} \|\chi_{[a_k, a_{k+1})}(t) P_N w\|_{L^1_T L^{\infty}_{xy}}\\
&\lesssim \underset{k=1}{\overset{(1\vee N)^2}{\sum}} \big(\int_{a_k}^{a_{k+1}}\,dt\big)^{1/2} \|\chi_{[a_k, a_{k+1})}(t) P_N w\|_{L^{2}_T L^{\infty}_{xy}}\\
&\lesssim (T(1\vee N)^{-2})^{1/2} \underset{k=1}{\overset{(1\vee N)^2}{\sum}} \|\chi_{[a_k, a_{k+1})}(t) P_N w\|_{L^{2}_T L^{\infty}_{xy}}.
\end{split}$$
On the other hand, using Duhamel’s formula, for $t\in [a_k, a_{k+1})$, we obtain from $$\label{ineq2-zk}
w(t)= W(t-a_k)\, w(a_k) -\int_{a_k}^t W(t-s) [\partial_x F(w)]\, ds.$$
Using Lemma \[lem-zk1\] it follows from that $$\label{ineq3-zk}
\begin{split}
\|\chi_{[a_k, a_{k+1})}(t) P_N w\|_{L^{2}_T L^{\infty}_{xy}} \lesssim (1\vee N)^{-\frac13} \|P_N w(a_k)\|_{L^2_{xy}}
+ (1\vee N)^{-\frac13} N \|\chi_{[a_k, a_{k+1})}(t) P_N F\|_{L^1_T L^2_{xy}}.
\end{split}$$
Combining and , we have $$\label{ineq4-zk-s}
\begin{split}
\|P_N w\|_{L^1_T L^{\infty}_{xy}} &\lesssim (T(1\vee N)^{-2})^{1/2} \underset{k=1}{\overset{(1\vee N)^2}{\sum}} \Big((1\vee N)^{-\frac13} \|P_N w(a_k)\|_{L^2_{xy}}\Big ) \\
&\hskip15pt +\,(T(1\vee N)^{-2})^{1/2} \underset{k=1}{\overset{(1\vee N)^2}{\sum}} (1\vee N)^{-\frac13} N \|\chi_{[a_k, a_{k+1})}(t) P_N F\|_{L^1_T L^2_{xy}}\\
&\lesssim (T(1\vee N)^{-2})^{1/2} (1\vee N)^{-\frac13} \underset{k=1}{\overset{(1\vee N)^2}{\sum}} \|P_N w(a_k)\|_{L^2_{xy}}\\
&+(T(1\vee N)^{-2})^{1/2}(1\vee N)^{-\frac13} N \|P_N F\|_{L^1_T L^2_{xy}}.
\end{split}$$
Simplifying further, one obtains from that $$\label{ineq4-zk}
\begin{split}
\|P_N w\|_{L^1_T L^{\infty}_{xy}} &\lesssim T^{\frac12} \Big ( (1\vee N)^{\frac23} (1\vee N)^{-2} \underset{k=1}{\overset{(1\vee N)^2}{\sum}} \|P_N w(a_k)\|_{L^2_{xy}}
+(1\vee N)^{-\frac13} \|P_N F\|_{L^1_T L^2_{xy}}\Big)\\
&\lesssim T^{\frac12} (1\vee N)^{-\epsilon} \Big ( (1\vee N)^{-2} \underset{k=1}{\overset{(1\vee N)^2}{\sum}}
\| (1\vee N)^{\frac23+\epsilon} P_N w(a_k)\|_{L^2_{xy}}
+\|P_N F\|_{L^1_T L^2_{xy}}\Big)
\end{split}$$ for $0<\epsilon < \frac13$.
Thus inequality follows from summing over $N$.
Now, we record a key estimate to prove the local well-posedness result for the IVP .
\[lem4-zk\] Let $w$ be a solution of with $w_0\in H^{\infty}({\mathbb T}^2)$ defined in $[0,T]$. Then for any $s>5/3$, there exist $T=T(\|w_0\|_{H^s})$ and a constant $c_T(\|w_0\|_{H^s}, s)$ such that $$\label{apriori-zk}
g(T):=\int_0^T \big(\|w(t)\|_{L^{\infty}_{xy}}+\|\nabla w\|_{L^{\infty}({\mathbb T}^2)}\big)\,dt\le c_T.$$
Observe that $w$, $\partial_xw$ and $\partial_y w$ satisfy Lemma \[lem3-zk\] with $F$ given, respectively, by $\frac12w^2$, $\frac12\partial_x(w^2)$ and $\frac12\partial_y(w^2)$. Hence, for $s'> 2/3$, using for $w$, $\partial_x w$ and $\partial_y w$ with respective $F$, we obtain $$\label{ineq5-zk}
\begin{split}
g(T) &\le c\, T^{\frac12}\,\Big( \|J^{s'} w\|_{L^{\infty}_TL^2_{xy}}+ \|J^{s'} \partial_x w\|_{L^{\infty}_TL^2_{xy}}+ \|J^{s'} \partial_y w\|_{L^{\infty}_TL^2_{xy}}\\
&\hskip10pt + \int_0^T \| w^2\|_{L^2_{xy}}\,dt+ \int_0^T\|\partial_x (w^2)\|_{L^2_{xy}}\,dt+ \int_0^T\|\partial_y (w^2)\|_{L^2_{xy}}\,dt\Big).
\end{split}$$
It follows that, for $s>5/3$, $$\label{ineq6-zk}
\|J^{s'} w\|_{L^{\infty}_TL^2_{xy}}+ \|J^{s'} \partial_x w\|_{L^{\infty}_TL^2_{xy}}+ \|J^{s'} \partial_y w\|_{L^{\infty}_TL^2_{xy}} \le c\, \|J^s w\|_{L^{\infty}_TL^2_{xy}}.$$
On the other hand, $$\label{ineq7-zk}
\begin{split}
\int_0^T &( \| w^2\|_{L^2_{xy}}+\|\partial_x (w^2)\|_{L^2_{xy}}+\|\partial_y (w^2)\|_{L^2_{xy}})\,dt\\
&\le c\, \;\|w\|_{L^{\infty}_TL^2_{xy}}\int_0^T (\|w(t)\|_{L^{\infty}_{xy}}+\|\partial_x w(t)\|_{L^{\infty}_{xy}}+\|\partial_y w(t)\|_{L^{\infty}_{xy}}\big)\,dt\\
&\le c\, \;\|w\|_{L^{\infty}_T{H^s}}\int_0^T (\|w(t)\|_{L^{\infty}_{xy}}+\|\nabla w\|_{L^{\infty}_{xy}}\big)\,dt.
\end{split}$$
Combining , , with Lemma \[lem2-zk\] we obtain the inequality $$\label{apriori2-zk}
g(T) \le cT^{\frac12}\,\|w_0\|_{H^s}\,\exp{(c\, g(T))}(1+g(T)).$$
From we can deduce the required result analogously as in [@CK], so we omit the details.
The main tools in the proof are the results from Lemmas \[lem2-zk\] and \[lem4-zk\].
Let $s>5/3$ and consider an initial datum $w_0\in H^s({\mathbb T}^2)$. As already mentioned, for $s>2$, we can prove the local well-posedness in $H^s$ of (1.1) by solving it as a quasi-linear hyperbolic PDE. So, we may consider $w_0\in H^s$, $5/3<s\leq 2$. Density of $H^{\infty}$ in $H^s$ allows one to find $w_0^{\epsilon}\in H^{\infty}$ such that $\|w_0^{\epsilon}-w_0\|_{H^s}\to 0$. Moreover, one has $\|w_0^{\epsilon}\|_{H^s}\leq c \|w_0\|_{H^s}$.
For $0<\epsilon<1$, let $w^{\epsilon}$ be the solution to the IVP corresponding to the initial data $w_0^{\epsilon}\in H^{\infty}$ on $[0, \tau]$, $\tau>0$. One can use Lemma \[lem4-zk\] to extend $w^{\epsilon}$ on a time interval $[0,T]$, $T=T(\|w_0\|_{H^s} )>0$ and to show that there exists a constant $c_T$ such that $$\label{pf-3}
\int_0^T \big(\|w^{\epsilon}(t)\|_{L^{\infty}_{xy}}+\|\partial_x w^{\epsilon}(t)\|_{L^{\infty}_{xy}}+\|\partial_y w^{\epsilon}(t)\|_{L^{\infty}_{xy}}\big)\,dt\le c_T.$$ Also, from Lemma \[lem2-zk\], one has $$\label{pf-4}
\sup_{0\leq t \leq T}\|w^{\epsilon}\|_{H^s({\mathbb T}^2)}\leq c_T.$$
Now, using Gronwall’s inequality and the estimate , one can show that $$\label{pf-5}
\sup_{0\leq t \leq T}\|w^{\epsilon}-w^{{\epsilon}'}\|_{L^2({\mathbb T}^2)}\to 0\qquad {\rm{as}}\quad \epsilon,\,\epsilon '\to 0.$$
In view of and , one can get, for $s'<s$, $w\in C([0, T];H^{s'})\cap L^{\infty}([0, T]; H^s)$ such that $w^{\epsilon}\to w$ in $C([0, T]; H^{s'})$. Indeed, implies that as $\epsilon\to 0$, $w^{\epsilon}\to w$ in $C([0, T], L^2)$. In the light of estimate we note that, $w^{\epsilon}\in L^{\infty}([0, T], H^s)$. Hence, by weak\* compactness, $w\in L^{\infty}([0, T], H^s)$.
Once again, one can use Gronwall’s inequality, to prove that $w$ is the unique solution to the IVP . An usual Bona–Smith [@BS] argument can be used to prove the continuity of the solution $w(t)$ and the continuity of the flow-map in $H^s$. For a detailed exposition of this argument, we refer to a recent work [@KP16].
Failure of the local uniform continuity of the flow map {#sec-5}
========================================================
In this section we prove the failure of the local uniform continuity of the flow-map for the IVP associated to the ZK equation. Let $\varphi(m, n) = m^3+mn^2$ be the time frequency of the solution to the linear problem associated to (see also ) with spatial frequency ${\mathbf m}=(m,n)$.
Following the idea in [@KochTzvetkov2003BO; @KochTzvetkov2008; @Robert2018], we will construct a family of approximate solutions whose frequencies lie in the region where the resonance relation $$\label{resonance}
R(m,m_1,n,n_1):=\varphi(m, n)-\varphi(m-m_1, n-n_1)-\varphi(m_1, n_1),$$ vanishes. Recall that the resonance relation can explicitly be written as $$\label{reso}
\begin{split}
R(m,m_1,n,n_1)&:= 3mm_1(m-m_1)+\big[mn^2-m_1n_1^2 -(m-m_1)(n-n_1)^2\big]\\
& = 3mm_1(m-m_1) +2nn_1(m-m_1) +m_1n^2-mn_1^2.
\end{split}$$ In particular, note that for any choice of $(m,n)\in {\mathbb Z}^2$, we have $R(m,0,n,2n)=0$. Consequently, we will exploit this resonant interaction to construct the solutions in Theorem \[lemma1.1\].
Let us fix $s>5/3$. For $\theta \in [-1,1]$ and $m\in{\mathbb N}^*$, let us define the family of functions on $[0,1]\times{\mathbb T}^2$ by $$\begin{gathered}
\label{utheta}
u_{\theta,m}(t,x,y):= \theta m^{-1}\cos(2y) + \cos\left(\frac{\theta}{2}t\right)m^{-s}\cos\left(mx-y+\varphi(m,-1)t\right)\\+\sin\left(\frac{\theta}{2}t\right)m^{-s}\sin\left(mx+y+\varphi(m,1)t\right)+r_{\theta,m}(t,x,y).\end{gathered}$$ Note that all three modes above are solutions to the linear part of (\[zk-1\]) modulated by a time oscillating factor, and the third one corresponds to the main part of the nonlinear interaction of the first two (see below). The last term is given by $$\begin{gathered}
\label{remainder}
r_{\theta,m} =m^{-s}R(m,0,-1,2)^{-1}\cos\left(\frac{\theta}{2}t\right)\cos\left(mx-3y+\varphi(m,-1)t\right)\\ + m^{-s}R(m,0,1,-2)^{-1}\sin\left(\frac{\theta}{2}t\right)\sin\left(mx+3y+\varphi(m,1)t\right).\end{gathered}$$ It allows to cancel the remaining interactions (due to the requirement of working with non localized real-valued solutions) in order for these approximate solutions to satisfy the equation up to a sufficiently small error. Indeed, we have the estimate $$\label{approx-sol}
\|({\partial}_t+{\partial}_x\Delta)u_{\theta,m}+u_{\theta,m}{\partial}_xu_{\theta,m}\|_{H^s}\lesssim m^{(-1-s)\vee (1-2s)},$$ which holds uniformly in $(\theta,t)\in[-1,1]\times[0,1]$ (and $a\vee b$ stands for the maximum between $a$ and $b$).
To prove (\[approx-sol\]), let us write $u_i$, $i=1,2,3$ for the first three terms in the definition of $u_{\theta,m}$ (\[utheta\]), then the term inside the norm on the left-hand side of (\[approx-sol\]) reads $$({\partial}_t+{\partial}_x\Delta)(u_2+u_3+r_{\theta,m}) + u_1{\partial}_x(u_2+u_3) +\widetilde{r},$$ with $\widetilde{r}=u_1{\partial}_x r_{\theta,m} + (u_2+u_3+r_{\theta,m}){\partial}_x u_{\theta,m}$.
A key ingredient here is that $$\label{eq-curv-reson}
R(m,0,-1,2)=R(m,0,1,-2)=-8m,$$ so that $\widetilde{r}=O_{L^2}(m^{-1-s}+m^{1-2s})$, where we use the notation $O_{L^2}(m^{\alpha})$ for functions having an $L^2({\mathbb T}^2)$-norm bounded by a constant times $m^{\alpha}$, uniformly in $(\theta,t)\in[-1,1]\times[0,1]$.
Next, straightforward computations give $$\begin{gathered}
\label{comput1}
({\partial}_t+{\partial}_x\Delta)(u_2+u_3) = - \frac{\theta}{2}\sin\left(\frac{\theta}{2}t\right)m^{-s}\cos\left(mx-y+\varphi(m,-1)t\right)\\ + \frac{\theta}{2}\cos\left(\frac{\theta}{2}t\right)m^{-s}\sin\left(mx+y+\varphi(m,1)t\right),\end{gathered}$$ and $$\begin{gathered}
\label{comput2}
u_1{\partial}_x(u_2+u_3)= -\frac{\theta}{2}\cos(\frac{\theta}{2}t)m^{-s}\left[\sin(mx+y+\varphi(m,-1)t)+\sin(mx-3y+\varphi(m,-1)t)\right]\\
+\frac{\theta}{2}\sin(\frac{\theta}{2}t)m^{-s}\left[\cos(mx-y+\varphi(m,1)t)+\cos(mx+3y+\varphi(m,1)t)\right].\end{gathered}$$ Since $\varphi(m,-1)=\varphi(m,1)$ (as the symbol is even in $n$), we see that the first and third terms in the nonlinear interaction (\[comput2\]) are counterbalanced by the linear evolution (\[comput1\]). To cancel the two remaining terms, we finally compute $$\begin{gathered}
({\partial}_t+{\partial}_x\Delta)r_{\theta,m} = \frac{\theta}{2}\cos(\frac{\theta}{2}t)m^{-s}\sin(mx-3y+\varphi(m,-1)t)\\
-\frac{\theta}{2}\sin(\frac{\theta}{2}t)m^{-s}\cos(mx+3y+\varphi(m,1)t)+O_{L^2}(m^{-1-s}),\end{gathered}$$ by using the definition of $\varphi$ and $R$. This proves (\[approx-sol\]).
With this estimate at hand, we can control the difference between $u_{\theta,m}$ and the genuine solution arising from $u_{\theta,m}(0)$ (provided by Theorem \[local-zk\]) via a standard energy estimate. Let us write $u$ for this solution. First, we have that $u$ exists on the whole time interval $[0,1]$. Indeed, in Theorem \[local-zk\], it is not hard to keep track of the size of $T=T(\|u(0)\|_{H^s})$ : in view of (\[apriori2-zk\]), we can take ${\displaystyle T\sim (1+\|u(0)\|_{H^s})^{-\frac12}}$, thus $T\sim 1$ for $u(0)=u_{\theta,m}(0)$. Next, let us define $v = u - u_{\theta,m}$, then $v$ solves $$({\partial}_t +{\partial}_x\Delta)v +v{\partial}_x v +{\partial}_x(u_{\theta,m}v) + G = 0,$$ where $G$ is the term in (\[approx-sol\]). Proceeding then as in Lemma \[lem2-zk\] at the $L^2$ level, we get $$\frac{d}{dt}\|v(t)\|_{L^2}^2\lesssim \left(\|u_{\theta,m}(t)\|_{L^{\infty}}+\|{\partial}_xu_{\theta,m}(t)\|_{L^{\infty}}\right)\|v(t)\|_{L^2}^2 + \|v(t)\|_{L^2}\|G(t)\|_{L^2},$$ which, along with the definition of $u_{\theta,m}$ and (\[approx-sol\]) and that $v(0)=0$, leads to the bound $$\sup_{t\in[0,1]}\|v(t)\|_{L^2}\lesssim m^{(1-2s)\vee (-1-s)}.$$ Interpolating with the trivial bound $$\|v(t)\|_{H^{s+1}}\lesssim \|u(t)\|_{H^{s+1}}+\|u_{\theta,m}(t)\|_{H^{s+1}}\lesssim m,$$ which is a consequence of Lemmata \[lem2-zk\] and \[lem4-zk\] and the definition of $u_{\theta,m}$, we get at last $$\label{bound-th2}
\|v(t)\|_{H^s}\lesssim m^{-\varepsilon},$$ for some $\varepsilon>0$.
Finally, the proof of Theorem \[lemma1.1\] is completed as follows : take $u_m$ (respectively $v_m$) to be the solution of the IVP (\[zk-1\]) with initial condition $u_m(0,x,y) = u_{1,m}(0,x,y)$ (respectively $v_m(0,x,y) = u_{-1,m}(0,x,y)$) given by Theorem \[local-zk\] (which are well-defined on the time interval $[0,1]$ as before), then (\[unif\_bound\]) is again a consequence of Lemmata \[lem2-zk\] and \[lem4-zk\], whereas (\[init\_bound\]) is straightforward from the definition of $u_{\theta,m}(0)$. To prove (\[failure\_bound\]), we use (\[bound-th2\]) to get the lower bound $$\|u_m(t)-v_m(t)\|_{H^s}\geq \|u_{1,m}(t)-u_{-1,m}(t)\|_{H^s}+O(m^{-\varepsilon}).$$ From the definition of the approximate solution and with the use of (\[eq-curv-reson\]), we finally get $$\|u_{1,m}(t)-u_{-1,m}(t)\|_{H^s}=2\left|\sin\left(\frac{t}{2}\right)\right|\|\sin(mx+y+\varphi(m,1)t)\|_{L^2}+O(m^{-1})\gtrsim |t|+O(m^{-1}).$$ This completes the proof of Theorem \[lemma1.1\].
\[FIN\]
1. It is not difficult to see that $u_{\theta,m}$ is still a sufficiently good approximate solution even for lower values of $s$ : one can check that the last bound holds with $\varepsilon>0$ for any $s>1/2$. However, since there exists no flow map defined on $H^s({\mathbb T}^2)$ when $1/2<s\leq 5/3$, we restricted our construction to those $s>5/3$ treated by Theorem \[local-zk\].
2. The construction above is independent of the choice of the periods in $x$ and $y$, thus one can repeat the argument for any torus ${\mathbb T}^2_{\lambda} = ({\mathbb R}/2\pi\lambda_1{\mathbb Z})\times({\mathbb R}/2\pi\lambda_2{\mathbb Z})$ and recover the quasi-linear behaviour likewise. In particular, there is another interesting resonant interaction : $R(m,m_1,\sqrt{3}m,-\sqrt{3}m_1)=0$. To perform the same construction as above, this requires to work on a torus ${\mathbb T}^2_{\lambda}$ whose periods $\lambda=(\lambda_1,\lambda_2)$ satisfy the condition ${\displaystyle \sqrt{3}\frac{\lambda_2}{\lambda_1}\in\mathbb{Q}}$.
3. The frequency $m_1=0$ in the aforementioned resonant interaction corresponds to the $x$-mean value ${\displaystyle \int_{{\mathbb T}}u(t,x,y)\,dx}$, which is preserved by the flow : $$\int_{{\mathbb T}} u(t,x,y)\,dx = \int_{{\mathbb T}}u_0(x,y)\,dx,~\forall (t,y)\in{\mathbb R}\times{\mathbb T}.$$ We do not know the existence of a gauge transformation allowing to get rid of the contribution of this frequency. This is in contrast with the periodic KP type equations (see *e.g.* [@JB_kp; @Robert2017]), where ${\displaystyle \int_{{\mathbb T}}u_0(x,y)\,dx}$ must be independent of $y$ to define the anti-derivative, and where the Galilean transform reduces the problem to initial data with this constant being equal to zero.
4. At last, let us notice that the above construction uses in a crucial way the curvatureproperty (\[eq-curv-reson\]), meaning that for a given resonant interaction $R(m,m_1,n,n_1)=0$ we need $R(m, m+ m_1, n, n+ n_1)$ to be sufficiently large.
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[^1]: MP was partially supported by FAPESP (2016/25864-6) and CNPq (305483/2014-5) Brasil.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Random walk on the irreducible representations of the symmetric and general linear groups is studied. A separation distance cutoff is proved and the exact separation distance asymptotics are determined. A key tool is a method for writing the multiplicities in the Kronecker tensor powers of a fixed representation as a sum of non-negative terms. Connections are made with the Lagrange-Sylvester interpolation approach to Markov chains.'
address: |
University of Southern California\
Los Angeles, CA 90089-2532
author:
- Jason Fulman
title: Separation cutoffs for random walk on irreducible representations
---
[^1]
[^2]
Introduction
============
The study of convergence rates of random walk on a finite group is a rich subject; three excellent surveys are [@Al], [@Sal] and [@D1]. In recent papers ([@F1],[@F3],[@F6]), the author posed and studied a dual question, namely what can be said about the convergence rate of random walk on $Irr(G)$, the set of irreducible representations of a finite group $G$.
To define the random walk on $Irr(G)$, let $\eta$ be a (not necessarily irreducible) representation of $G$ whose character is real valued. From an irreducible representation $\lambda$, one transitions to the irreducible representation $\rho$ with probability $$K(\lambda,\rho):= \frac{d_{\rho} m_{\rho}(\lambda \otimes \eta)}{ d_{\lambda} d_{\eta}}.$$ Here $d_{\lambda}$ denotes the dimension of $\lambda$ and $m_{\rho}(\lambda \otimes \eta)$ denotes the multiplicity of $\rho$ in the tensor product (also called the Kronecker product) of $\lambda$ and $\eta$. Whereas random walk on $G$ has the uniform measure as a stationary distribution, random walk on $Irr(G)$ has the Plancherel measure as a stationary distribution. The Plancherel measure assigns a representation $\lambda$ probability $\frac{d_{\lambda}^2}{|G|}$.
There are many motivations for the study of these random walks: six motivations (with literature references and discussion) appear in the introduction of the recent paper [@F6]. There is no need to repeat the discussion here, but let us just mention that similar processes have been studied for compact Lie groups and Lie algebras ([@ER],[@BBO]), and that the decomposition of iterated tensor products of finite groups has been studied by combinatorialists ([@F3],[@GC],[@GK]). Moreover, these random walks arise in quantum computing ([@MR],[@F6]) and have been used to derive the first error bounds in limit theorems for the distribution of random character ratios ([@F1], [@F4]). As is clear from [@F6] and this paper, these random walks are also a tractable testing ground for results in Markov chain theory.
To illustrate a result in this paper, we discuss the case of random walk on $Irr(S_n)$. To do this, recall that two commonly used distances between probability distributions $P,Q$ on a finite set $X$ are total variation distance $$||P-Q|| := \frac{1}{2} \sum_{x \in X} |P(x)-Q(x)|$$ and separation distance $$s(P,Q) := \max_{x \in X} \left[ 1 -
\frac{P(x)}{Q(x)} \right].$$ The following recent result gave a sharp total variation distance convergence rate estimate.
\[TVbound\] ([@F6]) Let $G$ be the symmetric group $S_n$ and let $\pi$ be the Plancherel measure of $G$. Let $\eta$ be the $n$-dimensional defining representation of $S_n$. Let $K^r$ denote the distribution of random walk on $Irr(G)$ after $r$ steps, started from the trivial representation.
1. If $r=\frac{1}{2}n\log(n)+cn$ with $c \geq 1$ then $$||K^r-\pi|| \leq \frac{e^{-2c}}{2}.$$
2. If $r=\frac{1}{2}n\log(n)-cn$ with $0 \leq c \leq \frac{1}{6}
\log(n)$, then there is a universal constant $a$ (independent of $c,n$) so that $$||K^r-\pi|| \geq 1 -ae^{-4c}.$$
This paper gives precise separation distance asymptotics. Letting $s(r)$ denote the separation distance after $r$ steps, it will be shown that for the walk in Theorem \[TVbound\] and $c$ fixed in $\mathbb{R}$, $$s(n \log(n)+cn) = 1-e^{-e^{-c}}(1+e^{-c}) + O \left(
\frac{\log(n)}{n} \right).$$ This expression goes to $0$ as $c
\rightarrow \infty$ and to $1$ as $c \rightarrow -\infty$ and a cutoff (defined precisely in Section \[prelim\]) occurs since $cn=o(n
\log(n))$. Note that whereas the total variation cutoff occurs at time $\frac{1}{2}n \log(n)$, the separation distance cutoff occurs at time $n \log(n)$. The proof of the separation distance asymptotics (and also the corresponding result for $GL(n,q)$) consists of two steps:
1. One must determine at which element $\lambda$ of $Irr(G)$ the separation distance is attained. This is equivalent to finding the $\lambda$ which minimizes $\frac{m_{\lambda}(\eta^r)}{d_{\lambda}}$. This is tricky since the usual formula for multiplicities in tensor products involves character values and so both positive and negative terms. Our solution is to give a subtle rewriting of this expression as a sum of non-negative terms.
2. Once one knows at which representation the separation distance is attained, one needs a formula for the separation distance. For the cases in this paper we indicate how to do this using combinatorial arguments and the diagonalization (i.e. eigenvalues and eigenvectors) of the random walk on $Irr(G)$.
A rather remarkable fact is that for the cases studied in this paper, one can use Lagrange-Sylvester interpolation to carry out Step 2 knowing only the eigenvalues (and not the eigenvectors) of random walk on $Irr(G)$. A similar trick had been usefully applied in the one-dimensional setting of birth-death chains ([@Br], [@DSa]), but the state spaces $Irr(S_n)$ and $Irr(GL(n,q))$ are high-dimensional so it is interesting that the trick can be extended to this context. A sequel will treat combinatorial examples where similar ideas can be applied.
The organization of this paper is as follows. Section \[prelim\] gives background from Markov chain theory and recalls the diagonalization of the Markov chain on $Irr(G)$. Section \[symirrep\] derives separation distance asymptotics for random walk on $Irr(S_n)$ when $\eta$ is the defining representation (whose character on a permutation is the number of fixed points). Section \[glirrep\] obtains separation distance asymptotics for random walk on $Irr(GL(n,q))$ when $\eta$ is the representation whose character is $q^{d(g)}$, where $d(g)$ is the dimension of the fixed space of $g$. Section \[lagrange\] discusses Lagrange-Sylvester interpolation, giving eigenvector-free proofs of some results of Sections \[symirrep\] and \[glirrep\].
Preliminaries {#prelim}
=============
This section collects some background on finite Markov chains, using random walk on $Irr(G)$ as a running example. Let $X$ be a finite set and $K$ a matrix indexed by $X \times X$ whose rows sum to 1. Let $\pi$ be a distribution such that $K$ is reversible with respect to $\pi$; this means that $\pi(x) K(x,y) = \pi(y) K(y,x)$ for all $x,y$ and implies that $\pi$ is a stationary distribution for the Markov chain corresponding to $K$.
As an example, the Markov chain on $Irr(G)$ defined in the introduction is reversible with respect to the Plancherel measure $\pi$. To see this let $\chi$ denote the character of a representation, and recall the formula $$m_{\rho}(\lambda \otimes
\eta) = \frac{1}{|G|} \sum_{g \in G} \chi^{\lambda}(g) \chi^{\eta}(g)
\overline{\chi^{\rho}(g)}.$$ The equation $\pi(\lambda)K(\lambda,\rho)=\pi(\rho) K(\rho,\lambda)$ follows because $\eta$ was assumed to be real valued; in fact this was the reason for imposing this condition on $\eta$.
Define $\langle f,g \rangle = \sum_{x \in X} f(x) g(x) \pi(x)$ for real valued functions $f,g$ on $X$, and let $L^2(\pi)$ denote the space of such functions. Then when $K$ is considered as an operator on $L^2(\pi)$ by $$Kf(x) := \sum_y K(x,y) f(y),$$ it is self adjoint. Hence $K$ has an orthonormal basis of eigenvectors $f_i(x)$ with $Kf_i(x) = \beta_i
f_i(x)$, where both $f_i$ and $\beta_i$ are real. It is easily shown that the eigenvalues satisfy $-1 \leq \beta_{|X|-1} \leq \cdots \leq
\beta_1 \leq \beta_0=1$. One calls $K$ ergodic if $|\beta_{|X|-1}|,|\beta_1|<1$.
As an example, Lemma \[diaggroup\] determines an orthonormal basis of eigenvectors for the Markov chains on $Irr(G)$.
\[diaggroup\] ([@F2], Proposition 2.3) Let $K$ be the Markov chain on $Irr(G)$ defined using a representation $\eta$ whose character is real valued. The eigenvalues of $K$ are indexed by conjugacy classes $C$ of $G$:
1. The eigenvalue parameterized by $C$ is $\frac{\chi^{\eta}(C)}{d_{\eta}}$.
2. An orthonormal basis of eigenfunctions $f_C$ in $L^2(\pi)$ is defined by $f_C(\rho) = \frac{|C|^{1/2} \chi^{\rho}(C)}{d_{\rho}}$.
For instance when $G=S_n$ and $\eta$ is the n-dimensional defining representation, the eigenvalues are $\frac{i}{n}$ where $0 \leq i \leq
n-2$ or $i=n$, with multiplicity equal to the number of conjugacy classes of permutations with $i$ fixed points. Similarly, suppose that $G=GL(n,q)$ and that $\eta$ is the representation whose character is the number of fixed vectors of $g$ in its natural action on the n-dimensional vector space $V$. Then the eigenvalues are $q^{-i}$ for $i=0,\cdots,n$, with multiplicity equal to the number of conjugacy classes of elements of $GL(n,q)$ with an $n-i$ dimensional fixed space.
A common way to quantify convergence rates of Markov chains is using total variation distance. Given probabilities $P,Q$ on $X$, one defines the total variation distance between them as $||P-Q||=\frac{1}{2} \sum_{x \in X} |P(x)-Q(x)|$. It is not hard to see that $$||P-Q|| = \max_{A \subseteq X} |P(A)-Q(A)| .$$ Let $K_x^r$ be the probability measure given by taking $r$ steps from the starting state $x$. Researchers in Markov chains are interested in the behavior of $||K_x^r - \pi||$.
Lemma \[genbound\] relates total variation distance to the spectrum of $K$. Part 1 is the usual method for computing the power of a diagonalizable matrix. Part 2 is proved in [@DH] and upper bounds $||K_x^r - \pi||$ in terms of eigenvalues and eigenvectors and is effective in many examples; it was crucial in the proof of Theorem \[TVbound\] stated in the introduction.
\[genbound\]
1. $K^r(x,y) = \sum_{i=0}^{|X|-1} \beta_i^r f_i(x) f_i(y) \pi(y)$ for any $x,y \in X$.
2. $$4 ||K_x^r - \pi||^2 \leq \sum_y \frac{|K^r(x,y) - \pi(y)|^2}{\pi(y)} = \sum_{i=1}^{|X|-1} \beta_i^{2r} |f_i(x)|^2 .$$ Note that the final sum does not include $i=0$.
Another frequently used method to quantify convergence rates of Markov chains is to use separation distance, introduced in [@AD1],[@AD2]. The separation distance between probabilities $P,Q$ on $X$ is defined as $$s(P,Q) = \max_{x \in X} \left[ 1 -
\frac{P(x)}{Q(x)} \right].$$ Since $||P-Q|| = \sum_{x: Q(x) \geq P(x)}
[Q(x)-P(x)]$, it is straightforward that $||P-Q|| \leq
s(P,Q)$. Specializing to random walk on $Irr(G)$ started at the trivial representation $\hat{1}$, one has that $$s(K_{\hat{1}}^r,\pi)
= \max_{\lambda} \left[ 1 - \frac{|G|
K^r(\hat{1},\lambda)}{d_{\lambda}^2} \right].$$ Lemma 3.2 of [@F6] gives that $K^r(\hat{1},\lambda) = \frac{d_{\lambda}}{(d_{\eta})^r}
m_{\lambda}(\eta^r)$. Thus the separation distance is attained at the $\lambda$ which minimizes $\frac{m_{\lambda}(\eta^r)}{d_{\lambda}}$.
Finally, let us give a precise definition of the cutoff phenomenon. A nice survey of the subject is [@D2]; we use the definition from [@Sal]. Consider a family of finite sets $X_n$, each equipped with a stationary distribution $\pi_n$, and with another probability measure $p_n$ that induces a random walk on $X_n$. One says that there is a total variation cutoff for the family $(X_n,\pi_n)$ if there exists a sequence $(t_n)$ of positive reals such that
1. $\lim_{n \rightarrow \infty} t_n = \infty$;
2. For any $\epsilon \in (0,1)$ and $r_n = [(1+\epsilon)t_n]$, $\lim_{n \rightarrow \infty} ||p_n^{r_n}-\pi_n||=0$;
3. For any $\epsilon \in (0,1)$ and $r_n = [(1-\epsilon)t_n]$, $\lim_{n \rightarrow \infty} ||p_n^{r_n}-\pi_n||=1$.
For the definition of a separation cutoff, one replaces $||p_n^{r_n}-\pi_n||$ by $s(p_n^{r_n},\pi_n)$.
The symmetric group {#symirrep}
===================
This section studies the random walk $K$ on $Irr(S_n)$ defined from the representation $\eta$ whose character is the number of fixed points. Although not needed for the results in this section, it should be mentioned that when $Irr(S_n)$ is viewed as the partitions of $n$, the random walk $K$ has a description in terms of removing and then reattaching a corner box at each step (see [@F6] for a proof).
The primary purpose of this section is to determine the asymptotic behavior of the separation distance $$s(r) = \max_{\lambda} \left[ 1 -
\frac{K^r(\hat{1},\lambda)} {\pi(\lambda)} \right].$$
The first step in studying $s(r)$ is determine for which $\lambda$ the maximum is attained. Part 1 of Lemma \[genbound\] and Lemma \[diaggroup\] imply that $$\frac{K^r(\hat{1},\lambda)}
{\pi(\lambda)} = \sum_{i=0}^n \left( \frac{i}{n} \right)^r \sum_{g \in
S_n: fp(g) = i} \frac{\chi^{\lambda}(g)}{d_{\lambda}},$$ where $fp(g)$ is the number of fixed points of $g$. However since characters can take both positive and negative values, it is not clear which $\lambda$ minimizes this expression.
Theorem \[symsumpos\] will circumvent this difficulty by giving an expression for $\frac{K^r(\hat{1},\lambda)} {\pi(\lambda)}$ as a sum of non-negative terms. This result was first derived in our earlier paper [@F5] using the Robinson-Schensted-Knuth correspondence. The proof presented here is different and uses instead inclusion-exclusion and the branching rules for the irreducible representations of $S_n$. As will be seen in Section \[glirrep\], it generalizes perfectly to the group $GL(n,q)$.
As a first step, the following lemma is useful. In its statement, and throughout this section, we assume familiarity with the concept of standard tableaux as in Chapters 2 and 3 of [@Sag].
\[sumfix\] Let $d_{\lambda/\mu}$ denote the number of standard tableaux of shape $\lambda/\mu$. Then $$\sum_{g
\in S_n: fp(g)=i} \chi^{\lambda}(g) = \frac{n!}{i!}
\sum_{j=0}^{n-i} \frac{(-1)^j}{j!} d_{\lambda/(n-i-j)} .$$
Let $Fix(g)$ denote the set of fixed points of a permutation $g$. Then $$\begin{aligned}
& & \sum_{g \in S_n: fp(g)=i} \chi^{\lambda}(g)\\
& = & {n \choose i} \sum_{g: Fix(g)=\{n-i+1,\cdots,n\}}
\chi^{\lambda}(g)\\ & = & {n \choose i} \sum_{j=0}^{n-i} (-1)^j
\sum_{A \subseteq \{1,\cdots,n-i\} \atop |A|=j} \sum_{g: Fix(g)
\supseteq A \cup \{n-i+1,\cdots,n\}} \chi^{\lambda}(g)\\ & = & {n
\choose i} \sum_{j=0}^{n-i} (-1)^j {n-i \choose j} \sum_{g: Fix(g)
\supseteq \{n-i-j+1,\cdots,n\}} \chi^{\lambda}(g)\\ & = & {n \choose
i} \sum_{j=0}^{n-i} (-1)^j {n-i \choose j} (n-i-j)! \langle
Res^{S_n}_{S_{n-i-j}}[\chi^{\lambda}],\hat{1} \rangle \\ & = &
\frac{n!}{i!} \sum_{j=0}^{n-i} \frac{(-1)^j}{j!}
d_{\lambda/(n-i-j)}. \end{aligned}$$ The first and third equalities are since character values are invariant under conjugacy. The second equality is the inclusion-exclusion principle (Chapter 10 of [@VW]). In the fourth equality, $Res^{S_n}_{S_{n-i-j}}[\chi^{\lambda}]$ denotes the restriction of $\chi^{\lambda}$ from $S_n$ to $S_{n-i-j}$. The final equality follows from the branching rules for irreducible representations of symmetric groups [@Sag]. Note also that when $i=0$, the set $\{n-i+1,\cdots,n\}$ should be interpreted as the empty set.
In what follows $P(a,r,n)$ will denote the probability that when $r$ balls are dropped at random into $n$ boxes, there are exactly $a$ occupied boxes. Lemma \[combin1\] gives an explicit expression for $P(a,r,n)$. This expression is an exercise on page 103 of [@Fe], but since the proof is simple and motivates an analogous result in Section \[glirrep\], we include it.
\[combin1\] ([@Fe]) $$P(a,r,n) = {n \choose a} \sum_{b=n-a}^n (-1)^{b-(n-a)} {a \choose n-b} \left(1 - \frac{b}{n} \right)^r.$$
Clearly $P(a,r,n)$ is ${n \choose a}$ multiplied by the probability that the occupied boxes are the first $a$ boxes. By the principle of inclusion and exclusion, this is $${n \choose a}
\sum_{s=0}^a (-1)^{a-s} {a \choose s} P_{\leq}(s)$$ where $P_{\leq}(s)$ is the probability that the set of occupied boxes is contained in $\{1,\cdots,s\}$. Noting that $P_{\leq}(s)= \left(
\frac{s}{n} \right)^r$, the result follows from the change of variables $b=n-s$.
Theorem \[symsumpos\] gives the needed expression for $\frac{K^r(\hat{1},\lambda)} {\pi(\lambda)}$ as a sum of non-negative quantities.
\[symsumpos\] Let $d_{\lambda/\mu}$ denote the number of standard tableaux of shape $\lambda/\mu$. Then $$\frac{K^r(\hat{1},\lambda)} {\pi(\lambda)} =
\sum_{a=0}^n P(a,r,n)(n-a)! \frac{d_{\lambda/(n-a)}}{d_{\lambda}}.$$
As noted earlier, part 1 of Lemma \[genbound\] and Lemma \[diaggroup\] imply that $$\frac{K^r(\hat{1},\lambda)}
{\pi(\lambda)} = \sum_{i=0}^n \left( \frac{i}{n} \right)^r \sum_{g \in
S_n: fp(g) = i} \frac{\chi^{\lambda}(g)}{d_{\lambda}}.$$ By Lemma \[sumfix\] this is $$\sum_{i=0}^n \left( \frac{i}{n}
\right)^r \frac{n!}{i!} \sum_{j=0}^{n-i} \frac{(-1)^j}{j!}
\frac{d_{\lambda/(n-i-j)}}{d_{\lambda}}.$$ Letting $a=i+j$, this becomes $$\begin{aligned}
& & \sum_{i=0}^n \sum_{a=i}^n (-1)^{a-i}
\frac{n!}{i!} \frac{1}{(a-i)!} \left( \frac{i}{n} \right)^r
\frac{d_{\lambda/(n-a)}}{d_{\lambda}}\\ & = & \sum_{a=0}^n
\sum_{i=0}^a (-1)^{a-i} \frac{n!}{i!} \frac{1}{(a-i)!} \left(
\frac{i}{n} \right)^r
\frac{d_{\lambda/(n-a)}}{d_{\lambda}}. \end{aligned}$$ Letting $b=n-i$, this becomes $$\sum_{a=0}^n {n \choose a} \sum_{b=n-a}^n (-1)^{b-(n-a)} {a \choose n-b} \left(1 - \frac{b}{n} \right)^r (n-a)!
\frac{d_{\lambda/(n-a)}}{d_{\lambda}}.$$ The result follows from Lemma \[combin1\].
\[whichmax\] The quantity $\frac{K^r(\hat{1},\lambda)} {\pi(\lambda)}$ is minimized for $\lambda=(1^n)$, corresponding to the sign representation.
By Theorem \[symsumpos\] one wants to find $\lambda$ minimizing $$\frac{K^r(\hat{1},\lambda)} {\pi(\lambda)} =
\sum_{a=0}^n P(a,r,n)(n-a)! \frac{d_{\lambda/(n-a)}}{d_{\lambda}}.$$ Note that the $a=n-1$ and $a=n$ terms in this expression are independent of $\lambda$. Moreover, all other terms are non-negative, and vanish when $\lambda$ is the sign representation (which corresponds to the partition all of whose parts have size 1). The result follows.
Next we use Theorem \[symsumpos\] and Corollary \[whichmax\] to derive both a formula and a precise asymptotic expression for the separation distance of the Markov chain $K$.
\[asym\] Let $s(r)$ be the separation distance between $K^r$ started at the trivial representation and the Plancherel measure $\pi$.
1. $$s(r) = \sum_{i=0}^{n-2} (-1)^{n-i} {n \choose i} (n-i-1)
\left( \frac{i}{n} \right)^r .$$
2. For $c$ fixed in $\mathbb{R}$ and $n \rightarrow \infty$, $$s(n
\log(n)+cn) = 1-e^{-e^{-c}}(1+e^{-c}) + O \left( \frac{\log(n)}{n}
\right).$$
(First proof) Theorem \[symsumpos\] and Corollary \[whichmax\] imply that $$\begin{aligned}
s(r) & = & 1 - \sum_{a=0}^n P(a,r,n) (n-a)! \frac{ d_{(1^n)/(n-a)}}{d_{(1^n)}}\\
& = & 1 - P(n,r,n) - P(n-1,r,n). \end{aligned}$$ By Lemma \[combin1\] this is equal to $$\begin{aligned}
& & 1 - \sum_{b=0}^n
(-1)^b {n \choose b} \left(1-\frac{b}{n} \right)^r - n \sum_{b=1}^n
(-1)^{b-1} {n-1 \choose n-b} \left(1-\frac{b}{n} \right)^r\\ & = &
\sum_{b=1}^n (b-1) {n \choose b} (-1)^b \left( 1 - \frac{b}{n}
\right)^r. \end{aligned}$$ Letting $i=n-b$ proves the first assertion.
For the second assertion, we use asymptotics of the coupon collector’s problem: it follows from Section 6 of [@CDM] that when $n
\log(n)+cn$ balls are dropped into $n$ boxes, the number of unoccupied boxes converges to a Poisson distribution with mean $e^{-c}$, and that the error term in total variation distance is $O(\frac{\log(n)}{n})$. The chance that a Poisson random variable with mean $e^{-c}$ takes value not equal to 0 or 1 is $1-e^{-e^{-c}}(1+e^{-c})$, which completes the proof.
There is a second proof of Theorem \[asym\], which uses a connection with the top to random shuffle of the symmetric group. We prefer the first proof as the ideas are more elementary (one doesn’t need the RSK correspondence) and generalize to $GL(n,q)$ (see Section \[glirrep\]).
(Second proof) By Corollary \[whichmax\], $s(r) = 1 - n!
K^r(\hat{1},(1^n))$. Theorem 3.1 of [@F3] gives that for any shape $\lambda$, one has that $K^r(\hat{1},\lambda)$ is equal to the chance of obtaining a permutation with Robinson-Schensted-Knuth (RSK) shape $\lambda$ after $r$ top to random shuffles started from the identity. The only permutation with RSK shape $(1^n)$ is the “longest” permutation $\pi$, defined by $\pi(i)=n-i+1$ for all $i$. It follows from Corollary 2.1 of [@DFP] that the separation distance for the top to random shuffle is attained at this $\pi$. Thus the chain $K$ and the top to random shuffle have the same separation distance $s(r)$, so the result follows from page 142 of [@DFP].
From the second proof the reader might think that the theory of the chain $K$ can be entirely understood in terms of the top to random shuffle. This is untrue. For example if one measures convergence to the stationary distribution using total variation distance, the top to random shuffle takes $n \log(n)+cn$ steps to be close to random [@AD1], but the chain $K$ requires only $\frac{1}{2} n \log(n)+cn$ steps [@F6].
As a final result, we use Corollary \[whichmax\] and the cycle index of the symmetric group to give a third proof of part 1 of Theorem \[asym\].
By Corollary \[whichmax\], $s(r) = 1 - \frac{K^r(\hat{1},(1^n))}{\pi(1^n)}$. Part 1 of Lemma \[genbound\] and Lemma \[diaggroup\] imply that $$\frac{K^r(\hat{1},\lambda)}
{\pi(\lambda)} = \sum_{i=0}^n \left( \frac{i}{n} \right)^r \sum_{g \in
S_n: fp(g) = i} \frac{\chi^{\lambda}(g)}{d_{\lambda}}.$$ Specializing to $\lambda=(1^n)$ implies that $$s(r) = - \sum_{i=0}^{n-1} \left(
\frac{i}{n} \right)^r \sum_{g \in S_n:fp(g)=i} sign(g).$$ Here $sign(g)=(-1)^{n-c(g)}$, where $c(g)$ is the number of cycles of $g$.
A classic result in combinatorics (see [@W]) is the “cycle index” of the symmetric group, which states that $$1 + \sum_{n \geq
1} \frac{u^n}{n!} \sum_{g \in S_n} \prod_{j \geq 1} x_j^{n_j(g)} =
\exp \left( \sum_{m \geq 1} \frac{x_m u^m}{m} \right).$$ Here $n_j(g)$ is the number of cycles of length $j$ of $g$. Making the substitutions $x_1=-x$, $x_i=-1$ for $i \geq 2$ and replacing $u$ by $-u$, the cycle index implies that $$\begin{aligned}
1 + \sum_{n \geq 1}
\frac{u^n}{n!} \sum_{g \in S_n} sign(g) \cdot x^{fp(g)} & = & \exp \left( xu
- \sum_{m \geq 2} \frac{(-u)^m}{m} \right)\\ & = & \exp(xu-u)
\exp(\log(1+u))\\ & = & \frac{e^{xu}(1+u)}{e^u}. \end{aligned}$$ Taking the coefficient of $\frac{u^nx^i}{n!}$ on both sides shows that if $0 \leq i \leq n-1$, then $$\begin{aligned}
\sum_{g \in S_n:
fp(g)=i} sign(g) & = & \frac{n!}{i!} \left[ \frac{(-1)^{n-i}}{(n-i)!}
+ \frac{(-1)^{n-i-1}}{(n-i-1)!} \right]\\ & = & (-1)^{n-i+1} {n
\choose i} (n-i-1). \end{aligned}$$ The result now follows from the previous paragraph.
The general linear group {#glirrep}
========================
This section studies random walk on $Irr(GL(n,q))$ in the case that $\eta$ is the representation of $GL(n,q)$ whose character is $q^{d(g)}$, where $d(g)$ is the dimension of the fixed space of $g$. As in Section \[symirrep\], we aim to determine the asymptotic behavior of the separation distance $$s(r) = \max_{\Lambda} \left[ 1 -
\frac{K^r(\hat{1},\Lambda)}{\pi(\Lambda)} \right].$$
The first step is to find the irreducible representation $\Lambda$ for which the maximum is attained. Part 1 of Lemma \[genbound\] and Lemma \[diaggroup\] imply that $$\frac{K^r(\hat{1},\Lambda)}{\pi(\Lambda)} = \sum_{i=0}^n q^{-r(n-i)}
\sum_{g \in GL(n,q):d(g)=i} \frac{\chi^{\Lambda}(g)}{d_{\Lambda}}.$$ Since characters can take both positive and negative values, it is not at all clear which $\Lambda$ minimizes this expression. As in the symmetric group case, the key is to find a way to write $\frac{K^r(\hat{1},\Lambda)}
{\pi(\Lambda)}$ as a sum of non-negative terms.
To begin we recall some facts about the representation theory of $GL(n,q)$. A full treatment of the subject with proofs appears in [@M], [@Z]. As usual a partition $\lambda=(\lambda_1,\cdots,\lambda_m)$ is identified with its geometric image $\{(i,j): 1 \leq i \leq m, 1 \leq j \leq \lambda_i \}$ and $|\lambda|=\lambda_1+\cdots+\lambda_m$ is the total number of boxes. Let ${\mathbb{Y}}$ denote the set of all partitions, including the empty partition of size 0.
Given an integer $1 \leq k < n$ and two characters $\chi_1,\chi_2$ of the groups $GL(k,q)$ and $GL(n-k,q)$, their parabolic induction $\chi_1 \circ \chi_2$ is the character of $GL(n,q)$ induced from the parabolic subgroup of elements of the form $$P = \left \{ \left( \begin{array} {cc} g_1 & * \\
0 & g_2 \end{array} \right) : g_1 \in GL(k,q), g_2 \in GL(n-k,q) \right \}$$ by the function $\chi_1(g_1) \chi_2(g_2)$.
A character is called cuspidal if it is not a component of any parabolic induction. Let ${\it C}_m$ denote the set of cuspidal characters of $GL(m,q)$ and let ${\it C} = \bigcup_{m \geq 1} {\it C}_m$; it is known that $|C_m|=\frac{1}{m} \sum_{d|m} \mu(d) (q^{m/d}-1)$ where $\mu$ is the Moebius function. The unit character of $GL(1,q)$ plays an important role and will be denoted $e$; it is one of the $q-1$ elements of ${\it C}_1$. Given a family ${\Lambda}: {\it C} \mapsto
{\mathbb{Y}}$ with finitely many non-empty partitions ${\Lambda}(c)$, its degree $||\Lambda||$ is defined as $\sum_{m \geq 1} \sum_{c \in {\it
C}_m} m \cdot |{\it \Lambda}(c)|$. A fundamental result is that the irreducible representations of $GL(n,q)$ are in bijection with the families of partitions of degree $n$, so we also let $\Lambda$ denote the corresponding representation.
Let $\vec{e_1},\cdots,\vec{e_n}$ be the standard basis of the vector space $V$ on which $GL(n,q)$ acts (so the kth component of $\vec{e_j}$ is $\delta_{j,k}$). Define $H(k,q)$ as the subgroup of $GL(n,q)$ consisting of $g$ which fix all of $\vec{e_1},\cdots,\vec{e_k}$. Equivalently, the elements of $H(k,q)$ are block matrices of the form $$\left(
\begin{array}{c c} I_k & X \\ 0_{n-k} & Y \end{array} \right)$$ where $I_k$ is a $k$ by $k$ identity matrix, $X$ is any $k$ by $n-k$ matrix with entries in $\mathbb{F}_q$, and $Y$ is any element of $GL(n-k,q)$. Thus $|H(k,q)|= q^{k(n-k)} |GL(n-k,q)|$. For $\Lambda$ an element of $Irr(GL(n,q))$, it will be helpful to let $$c_k(\Lambda) = \sum_{g \in H(k,q)} \chi^{\Lambda}(g).$$ Then $c_k(\Lambda)$ is non-negative, since it is the product of $|H(k,q)|$ and the multiplicity of the trivial representation of $H(k,q)$ in the restricted representation $Res^{GL(n,q)}_{H(k,q)}[\Lambda]$.
It will also be convenient to let ${{\left [{n \atop k} \right]}}$ denote the q-binomial coefficient $\frac{(q^n-1) \cdots (q-1)}{(q^k-1) \cdots (q-1)
(q^{n-k}-1) \cdots (q-1)}$, which is equal to the number of $k$ dimensional subspaces of an $n$ dimensional vector space over a finite field $\mathbb{F}_q$.
\[sumfixq\] Let $\Lambda$ be an irreducible representation of $GL(n,q)$. Then $$\sum_{g \in GL(n,q) \atop d(g)=i} \chi^{\Lambda}(g) = {{\left [{n \atop i} \right]}} \sum_{j=0}^{n-i} {{\left [{n-i \atop j} \right]}} (-1)^j q^{{j \choose 2}} c_{i+j}(\Lambda).$$
Let $Fix(g)$ denote the fixed space of $g$. Also if $A$ is a set of vectors, $\langle A \rangle$ will denote their span. Then $$\begin{aligned}
\sum_{g \in GL(n,q) \atop
d(g)=i} \chi^{\Lambda}(g) & = & {{\left [{n \atop i} \right]}} \sum_{g \in GL(n,q) \atop
Fix(g) = \langle \vec{e_1},\cdots,\vec{e_i} \rangle} \chi^{\Lambda}(g)\\ & = &
{{\left [{n \atop i} \right]}} \sum_{j=0}^{n-i} \sum_{W \supseteq
\langle \vec{e_1},\cdots,\vec{e_i} \rangle \atop dim(W)=i+j} (-1)^j q^{{j \choose
2}} \sum_{g \in GL(n,q) \atop Fix(g) \supseteq W} \chi^{\Lambda}(g)\\
& = & {{\left [{n \atop i} \right]}} \sum_{j=0}^{n-i} {{\left [{n-i \atop j} \right]}} (-1)^j q^{{j \choose 2}}
\sum_{g \in GL(n,q) \atop Fix(g) \supseteq
\langle \vec{e_1},\cdots,\vec{e_{i+j}} \rangle} \chi^{\Lambda}(g)\\ & = & {{\left [{n \atop i} \right]}}
\sum_{j=0}^{n-i} {{\left [{n-i \atop j} \right]}} (-1)^j q^{{j \choose 2}}
c_{i+j}(\Lambda). \end{aligned}$$
The first and third equalities used the fact that if $W_1,W_2$ are subspaces of $V$ of equal dimension, then $\{ g:Fix(g)=W_1 \}$ and $\{g:Fix(g)=W_2 \}$ are conjugate in $GL(n,q)$. The second equality used Moebius inversion on the lattice of subspaces of a vector space (Chapter 25 of the text [@VW]). The third equality also used the fact that the number of $i+j$ dimensional subspaces of $V$ containing $\langle \vec{e_1},\cdots,\vec{e_i} \rangle$ is ${{\left [{n-i \atop j} \right]}}$.
In what follows we let $P_q(a,r,n)$ denote the probability that the span of $\vec{v_1},\cdots,\vec{v_r}$ is $a$ dimensional, where the $r$ vectors are chosen uniformly at random from an n-dimensional vector space over $\mathbb{F}_q$. Lemma \[countsub\] gives a formula for $P_q(a,r,n)$.
\[countsub\] $$P_q(a,r,n) = {{\left [{n \atop a} \right]}} \sum_{b=n-a}^{n} (-1)^{b-(n-a)} q^{{b-(n-a) \choose 2}} {{\left [{a \atop n-b} \right]}} q^{-rb}.$$
Clearly $P_q(a,r,n)$ is ${{\left [{n \atop a} \right]}}$ multiplied by the chance that the span of $\vec{v_1},\cdots,\vec{v_r}$ is exactly the $a$ dimensional subspace consisting of vectors whose last $n-a$ coordinates are 0. One applies Moebius inversion on the lattice of subspaces of an n-dimensional vector space over $\mathbb{F}_q$ (Chapter 25 of [@VW]) to conclude that $$P_q(a,r,n) = {{\left [{n \atop a} \right]}}
\sum_{s=0}^a \sum_{W \subseteq \langle \vec{e_1},\cdots,\vec{e_a} \rangle \atop
dim(W)=s} (-1)^{a-s} q^{{a-s \choose 2}} P_{\leq(W)}.$$ Here $\vec{e_1},\cdots,\vec{e_n}$ is the standard basis of $V$ and $P_{\leq}(W)$ is the probability that the span of $\vec{v_1},\cdots,\vec{v_r}$ is contained in $W$. Clearly $P_{\leq}(W) = q^{-r(n-dim(W))}$. Thus $$P_q(a,r,n) = {{\left [{n \atop a} \right]}}
\sum_{s=0}^a {{\left [{a \atop s} \right]}} (-1)^{a-s} q^{{a-s \choose 2}} q^{-r(n-s)},$$ and the result follows by the change of variables $b=n-s$.
Theorem \[sympsumposq\] is a key result of this section; it expresses $\frac{K^r(\hat{1},\Lambda)}{\pi(\Lambda)}$ as a sum of non-negative terms.
\[sympsumposq\] $$\frac{K^r(\hat{1},\Lambda)}{\pi(\Lambda)} =
\sum_{a=0}^n P_q(a,r,n) \frac{c_a(\Lambda)}{d_{\Lambda}}.$$
Part 1 of Lemma \[genbound\] and Lemma \[diaggroup\] imply that $$\frac{K^r(\hat{1},\Lambda)}{\pi(\Lambda)} = \sum_{i=0}^n
q^{-r(n-i)} \sum_{g \in GL(n,q):d(g)=i}
\frac{\chi^{\Lambda}(g)}{d_{\Lambda}}.$$ By Lemma \[sumfixq\], this is $$\sum_{i=0}^n
q^{-r(n-i)} {{\left [{n \atop i} \right]}} \sum_{j=0}^{n-i} {{\left [{n-i \atop j} \right]}} (-1)^j q^{{j
\choose 2}} \frac{c_{i+j}(\Lambda)}{d_{\Lambda}}.$$ Letting $a=i+j$, this becomes $$\begin{aligned}
& & \sum_{i=0}^n q^{-r(n-i)} {{\left [{n \atop i} \right]}} \sum_{a=i}^n
{{\left [{n-i \atop a-i} \right]}}(-1)^{a-i} q^{{a-i \choose 2}}
\frac{c_{a}(\Lambda)}{d_{\Lambda}} \\ & = & \sum_{a=0}^n
\frac{c_{a}(\Lambda)}{d_{\Lambda}} \sum_{i=0}^a q^{-r(n-i)} {{\left [{n \atop i} \right]}}
{{\left [{n-i \atop a-i} \right]}}(-1)^{a-i} q^{{a-i \choose 2}}. \end{aligned}$$ Setting $b=n-i$ this becomes $$\begin{aligned}
& & \sum_{a=0}^n
\frac{c_{a}(\Lambda)}{d_{\Lambda}} \sum_{b=n-a}^n q^{-rb} {{\left [{n \atop b} \right]}}
{{\left [{b \atop a-(n-b)} \right]}}(-1)^{a-(n-b)} q^{{a-(n-b) \choose 2}}\\ & = &
\sum_{a=0}^n \frac{c_a(\Lambda)}{d_{\Lambda}} {{\left [{n \atop a} \right]}}
\sum_{b=n-a}^{n} (-1)^{b-(n-a)} q^{{b-(n-a) \choose 2}} {{\left [{a \atop n-b} \right]}}
q^{-rb}. \end{aligned}$$ The result now follows from Lemma \[countsub\].
\[whichmaxq\] Suppose that $GL(n,q) \neq GL(1,2)$.
1. The quantity $\frac{K^r(\hat{1},\Lambda)}{\pi(\Lambda)}$ is minimized for any $\Lambda$ which satisfies $\Lambda(e)=\emptyset$, and such $\Lambda$ exist.
2. The separation distance $s(r)$ of $K$ started at the trivial representation is $$1 - P_q(n,r,n) = \sum_{b=1}^n (-1)^{b+1} q^{{b \choose 2}} {{\left [{n \atop b} \right]}} q^{-rb}.$$
The formula for $|C_m|$ stated earlier in this section implies the existence of $\Lambda$ with $\Lambda(e)=\emptyset$. Proposition 5.4 of [@F6] states that for any $\Lambda$, if $K^r(\hat{1},\Lambda)>0$, then the largest part of $\Lambda(e)$ is at least $n-r$. It follows that if $\Lambda(e)=\emptyset$ then $K^{n-1}(\hat{1},\Lambda)=0$. By Theorem \[sympsumposq\] and the fact that $P_q(a,n-1,n)>0$ for $0 \leq a \leq n-1$, it follows that if $\Lambda(e)=\emptyset$ then $c_a(\Lambda)=0$ for $0 \leq a \leq
n-1$. Thus if $\Lambda(e)=\emptyset$, only the $a=n$ term in Theorem \[sympsumposq\] can be non-vanishing, but this term is independent of $\Lambda$, since $\frac{c_n(\Lambda)}{d_{\Lambda}}=1$ for all $\Lambda$. This implies the first part of the lemma since all terms in Theorem \[sympsumposq\] are nonnegative. It also implies that $s(r)=1-P_q(n,r,n)$, and the equality in the second assertion follows from Lemma \[countsub\].
Theorem \[asympq\] bounds the separation distance $s(r)$ and determines its exact asymptotic behavior.
\[asympq\] Suppose that $GL(n,q) \neq GL(1,2)$.
1. If $r<n$, then $s(r)=1$. If $c \geq 0$, then $$\frac{1}{q^{c+1}}-\frac{4}{q^{2c+3}} \leq s(n+c) \leq \frac{2}{q^{c+1}}.$$
2. Let $c \geq 0$ be fixed. Then $$\lim_{n \rightarrow \infty} s(n+c) = 1 -
\prod_{m=1}^{\infty} ( 1 - q^{-(c+m)}).$$
The first two sentences of the proof of Corollary \[whichmaxq\] implies that if $r<n$, then $s(r)=1$. To upper bound $s(n+c)$, one checks that since $q \geq 2$, the sum in the second part of Corollary \[whichmaxq\] is alternating with terms of decreasing magnitude. Thus the sum is upper bounded by its first term, ${{\left [{n \atop 1} \right]}} q^{-(n+c)}$, which is easily seen to be less than $2q^{-(c+1)}$, since $q \geq 2$. For the lower bound, note that the first term in the second part of Corollary \[whichmaxq\] is at least $q^{-(c+1)}$ and that the second term $-\frac{{{\left [{n \atop 2} \right]}}}{q^{2n+2c-1}}$ is at least $-4q^{-(2c+3)}$ since $q \geq 2$. This proves the first part of the theorem.
To prove the second part, rewrite the expression for $s(n+c)$ in part 2 of Corollary \[whichmaxq\] as $$- \sum_{b=1}^n \frac{(-1)^b
(1-1/q^n)(1/q-1/q^n) \cdots (1/q^{b-1}-1/q^n)}{q^{(c+1)b} (1-1/q)
\cdots (1-1/q^b)}.$$ It is straightforward to see that as $n
\rightarrow \infty$ this converges to $$- \sum_{b=1}^{\infty}
\frac{(-1)^b}{q^{(c+1)b} q^{{b \choose 2}} (1-1/q) \cdots
(1-1/q^b)}.$$ An identity of Euler (Corollary 2.2 in [@An]) states that $$1 + \sum_{b=1}^{\infty} \frac{t^b}{q^{{b \choose 2}} (1-1/q)
\cdots (1-1/q^b)} = \prod_{m=0}^{\infty} (1+tq^{-m})$$ for $|t|<1,|q|>1$. The result follows by applying this identity with $t=-1/q^{(c+1)}$.
Lagrange-Sylvester Interpolation {#lagrange}
================================
For certain one dimensional random walks, namely stochastically monotone birth-death chains, Lagrange-Sylvester interpolation allows one to study separation distance knowing only the eigenvalues (and not the eigenvectors) of the Markov chain; see [@Br], [@DSa] and the remarks following Proposition \[noeigenvec\]. The purpose of this section is to give examples of higher dimensional state spaces (namely $Irr(S_n)$ and $Irr(GL(n,q))$) where the methodology is useful.
To begin we review the Lagrange-Sylvester interpolation approach to diagonalizable matrices, and hence to reversible Markov chains. A textbook discussion in the matrix setting appears in [@Ga], and the paper [@Br] uses the language of Markov chains.
As usual, $K$ is a Markov chain on a finite set $X$ and is reversible with respect to a distribution $\pi$. If $\pi(x)>0$ for all $x$, then letting $A$ be a diagonal matrix whose $(x,x)$ entry is $\pi(x)$, it follows that $A^{1/2} K A^{-1/2}$ is symmetric. Hence $K$ is conjugate to a diagonal matrix $D$, whose entries $d_1,\cdots,d_n$ are the eigenvalues of $K$. Thus if $f,g$ are polynomials with $f(d_i)=g(d_i)$ for $i=1,\cdots,n$ then $f(K)=g(K)$.
Let $\lambda_1,\cdots,\lambda_m$ be the distinct eigenvalues of $K$ (so $m \leq |X|$). Define $g_r(s) = s^r$ and $$f_r(s) = \sum_{i=1}^m
\lambda_i^r \left[ \prod_{j \neq i}
\frac{s-\lambda_j}{\lambda_i-\lambda_j} \right] .$$ Since $f_r(\lambda_i)=g_r(\lambda_i)$ for $i=1,\cdots,m$, it follows that $f_r(D)=g_r(D)$. Thus $f_r(K)=g_r(K)$ which gives that $$K^r =
\sum_{i=1}^m \lambda_i^r \left[ \prod_{j \neq i}
\frac{K-\lambda_j I}{\lambda_i-\lambda_j} \right] .$$ As noted in [@Br], expanding this expresses $K^r$ in terms of $I,K,\cdots,K^{m-1}$ as follows: $$K^r = \sum_{a=1}^m K^{a-1}
(-1)^{m-a} \sum_{i=1}^m \lambda_i^r \prod_{j \neq i} (\lambda_i
- \lambda_j)^{-1} \sum_{\alpha \in c(m,i,m-a)} ( \prod_{s \in
\alpha} \lambda_s ).$$ Here $c(m,i,m-a)$ consists of the ${m-1
\choose a-1}$ subsets of size $m-a$ from $\{j: 1 \leq j \leq m, j \neq
i \}$.
For the next proposition it is useful to define the distance $dist(x,y)$ between $x,y \in X$ as the smallest $r$ such that $K^r(x,y)>0$. For the special case of birth-death chains on the set $\{0,1,\cdots,d\}$, Proposition \[noeigenvec\] appears in [@DF] and [@Br].
\[noeigenvec\] Let $K$ be a reversible ergodic Markov on a finite set $X$. Let $1,\lambda_1,\cdots,\lambda_d$ be the distinct eigenvalues of $K$ (so $d+1 \leq |X|$). Suppose that $x,y$ are elements of $X$ with $dist(x,y)=d$. Then for all $r \geq 0$, $$1 - \frac{K^r(x,y)}{\pi(y)} = \sum_{i=1}^d \lambda_i^r
\left[ \prod_{j \neq i}
\frac{1-\lambda_j}{\lambda_i-\lambda_j} \right].$$
Since $dist(x,y)=d$, the Lagrange-Sylvester expansion of $K^r$ in terms of $I,K,\cdots,K^d$ gives that $$K^r(x,y) = K^d(x,y) \left( \prod_{j} (1 - \lambda_j)^{-1} - \sum_{i=1}^d \lambda_i^r (1-\lambda_i)^{-1} \prod_{j \neq i} (\lambda_i - \lambda_j)^{-1} \right).$$ By Lemma \[genbound\], ergodicity of $K$ implies that $\pi(y) =
lim_{r \rightarrow \infty} K^r(x,y)$. Thus $$\pi(y) = K^{d}(x,y) \prod_{j}
(1-\lambda_j)^{-1},$$ which implies that $$\frac{K^r(x,y)}{\pi(y)} = 1 -
\sum_{i=1}^d \lambda_i^r \left[ \prod_{j \neq i}
\frac{1-\lambda_j}{\lambda_i-\lambda_j} \right].$$
[**Remarks:**]{}
1. Let $K$ be a reversible Markov chain on a finite set $X$. As in Proposition \[noeigenvec\], let $1,\lambda_1,\cdots,\lambda_d$ be the distinct eigenvalues of $K$ (so $d+1 \leq |X|$). From the expansion of $K^r$ in terms of $I,K,\cdots,K^d$ it follows that $dist(x,y) \leq d$ for any states of the chain. Thus the hypothesis of Proposition \[noeigenvec\] is an extremal case.
2. Let $K$ be a birth-death chain on the set $\{0,\cdots,d\}$, with transition probabilities $$\begin{aligned}
a_x & = & K(x,x-1) , \
x=1,\cdots, d \\ b_x & = & K(x,x), \ x=0, \cdots, d \\ c_x & = &
K(x,x+1), \ x=0, \cdots, d-1. \end{aligned}$$ Suppose that $a_x>0$ for $0<x \leq d$ and that $c_x>0$ for $0 \leq x<d$. Such chains are reversible with respect to the stationary distribution $$\pi(x) = Z
\prod_{i=1}^x \frac{c_{i-1}}{a_i},$$ where $Z$ is a normalizing constant. Supposing further that $c_x+a_{x+1} \leq 1$ for $0 \leq x<
d$ (such chains are called monotone chains), then [@DF] showed that the separation distance for the chain started at 0 is equal to $$s(r) = 1 - \frac{K^r(0,d)}{\pi(d)}.$$ Applying Proposition \[noeigenvec\] with $x=0,y=d$ one recovers the lovely result of Diaconis and Fill [@DF] expressing the separation distance entirely in terms of the eigenvalues of $K$: $$s(r) = \sum_{i=1}^d
\lambda_i^r \left[ \prod_{j \neq i} \frac{1-\lambda_j}
{\lambda_i-\lambda_j} \right].$$ This fact was used in [@DSa] to give a necessary and sufficient spectral condition for the existence of a separation cutoff for monotone birth death chains.
Next we use Proposition \[noeigenvec\] to give eigenvector-free proofs of the formulas for separation distance for the random walks on $Irr(S_n)$ and $Irr(GL(n,q))$ analyzed in Sections \[symirrep\] and \[glirrep\]. It should be emphasized that as in the proofs of Sections \[symirrep\] and \[glirrep\], one still needs to know at what representation the separation distance is attained.
(Fourth proof of part 1 of Theorem \[asym\]) By Corollary \[whichmax\], the separation distance is equal to $$s(r)
= 1 - \frac{K^r(\hat{1},(1^n))}{\pi((1^n))},$$ where $(1^n)$ is the partition corresponding to the sign representation. As was mentioned at the beginning of Section \[symirrep\] and proved in [@F6], the Markov chain $K$ has a description as a random walk on partitions in which one removes and adds a box at each step. From that description it is clear that the trivial representation (corresponding to the partition $(n)$) and the sign representation $(1^n)$ are distance $n-1$ apart. By part 1 of Lemma \[diaggroup\], the chain $K$ has $n$ distinct eigenvalues, namely $\frac{i}{n}$ where $0 \leq i \leq n-2$ or $i=n$. Thus Proposition \[noeigenvec\] implies that $$\begin{aligned}
s(r) & = & \sum_{i=0}^{n-2} \left(
\frac{i}{n} \right)^r \prod_{j \neq i \atop 0 \leq j \leq n-2}
\frac{\left( 1-\frac{j}{n} \right)}{\left( \frac{i}{n} - \frac{j}{n}
\right)} \\
& = & \sum_{i=0}^{n-2} \left( \frac{i}{n} \right)^r \prod_{j \neq i \atop 0 \leq j \leq n-2} \frac{n-j}{i-j} \\
& = & \sum_{i=0}^{n-2} \left( \frac{i}{n} \right)^r \frac{n!}{n-i} \prod_{j \neq i \atop 0 \leq j \leq n-2} \frac{1}{i-j} \\
& = & \sum_{i=0}^{n-2} (-1)^{n-i} {n \choose i} (n-i-1) \left( \frac{i}{n} \right)^r. \end{aligned}$$
A similar argument works for the general linear case.
(Second proof of part 2 of Corollary \[whichmaxq\]) By part 1 of Corollary \[whichmaxq\], the separation distance is equal to $$s(r) = 1 - \frac{K^r(\hat{1},\Lambda)}{\pi(\Lambda)},$$ where $\Lambda$ is any representation satisfying $\Lambda(e)=\emptyset$. By part 1 of Lemma \[diaggroup\], the chain $K$ has $n+1$ distinct eigenvalues, namely $q^{-i}$ for $0 \leq i \leq n$. By the first remark after Proposition \[noeigenvec\], this implies that $dist(\hat{1},\Lambda) \leq n$. Proposition 5.4 of [@F6] states that for any $\Lambda$, if $K^r(\hat{1},\Lambda)>0$ then the largest part of $\Lambda(e)$ is at least $n-r$; it follows that $dist(\hat{1},\Lambda) \geq n$. Thus $dist(\hat{1},\Lambda)=n$, and so Proposition \[noeigenvec\] can be applied with $x=\hat{1},y=\Lambda$. Using the notation $(1/q)_k = (1-1/q) \cdots
(1-1/q^k)$, one obtains that $$\begin{aligned}
s(r) & = &
\sum_{i=1}^n q^{-ir} \prod_{j \neq i \atop 1 \leq j \leq n} \frac{1-q^{-j}}{q^{-i}-q^{-j}} \\
& = & \sum_{i=1}^n q^{-ir} \frac{(1/q)_n}{(1-q^{-i})} \prod_{j=1}^{i-1}
\frac{1}{q^{-i}-q^{-j}} \prod_{j=i+1}^n \frac{1}{q^{-i}-q^{-j}}\\ & = &
\sum_{i=1}^n q^{-ir} \frac{(1/q)_n}{(1-q^{-i})} \frac{(-1)^{i-1} q^{{i
\choose 2}}}{(1/q)_{i-1}} \frac{q^{i(n-i)}}{(1/q)_{n-i}}\\ & = &
\sum_{i=1}^n (-1)^{i+1} q^{{i \choose 2}} {{\left [{n \atop i} \right]}}
q^{-ir}. \end{aligned}$$
[AAA]{}
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[^1]: The author was partially supported by NSF grant DMS-050391.
[^2]: Version of March 8, 2007
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The $\Delta a$ photometric system provides an efficient observational method to identify and distinguish magnetic and several other types of chemically peculiar (CP) stars of spectral types B to F from other classes of stars in the same range of effective temperatures. We have developed a synthetic photometric system which can be used to explore the capability of model atmospheres with individual element abundances to predict photometric $\Delta a$ magnitudes which measure the extent of the flux depression around 5200 [Å]{} found in different types of CP stars. In this first paper, we confirm the observed dependency of the $a$ index as a function of various colour indices sensitive to the effective temperature of stars as well as its average scatter expected from surface gravity variations within the main sequence band. The behaviour of the so-called “normality line” of $\Delta a$ systems used in photometric observations of CP stars is well reproduced. The metallicity dependence of the normality line of the $\Delta a$ system was computed for several grids of model atmospheres where the abundances of elements heavier than He had been scaled $\pm$0.5dex with respect to the solar value. We estimate a lowering of $\Delta a$ magnitudes for CP stars within the Magellanic Clouds by $\sim -3$ mmag relative to those in the solar neighbourhood assuming an average metallicity of $[{\rm Fe}/{\rm H}]= -0.5$ dex. Using these results on the metallicity bias of the $\Delta a$ system we find the observational systems in use suitable to identify CP stars in other galaxies or distant regions of our own galaxy and capable to provide data samples on a statistically meaningful basis. In turn, the synthetic system is suitable to test the performance of model atmospheres for CP stars. This work will be presented in follow-up papers of this series.'
date: 'Submitted 2002 October 23.'
title: 'The 5200 [Å]{} flux depression of chemically peculiar stars: I. Synthetic $\Delta a$ photometry - the normality line.'
---
\[firstpage\]
stars: atmospheres — stars: chemically peculiar
Introduction {#Sect1}
============
The chemically peculiar (CP) stars of the upper main sequence have been targets for astrophysical studies since the discovery of these objects by the American astronomer Antonia Maury (1897). Most of this early research was devoted to the detection of peculiar features in their spectra and photometric behaviour. The main characteristics of the classical CP stars are: peculiar and often variable line strengths, quadrature of line variability with radial velocity changes, photometric variability with the same periodicity and coincidence of extrema. Slow rotation was inferred from the sharpness of spectral lines. Overabundances of several orders of magnitude compared to the Sun were derived for Si, Cr, Sr, Eu, and for other heavy elements.
Babcock (1947) discovered a global dipolar magnetic field in the star 78 Virginis followed by a catalogue of similar stars (Babcock 1958) in which also the variability of the field strength in many CP stars — including even a reversal of magnetic polarity — was discovered. Stibbs (1950) introduced the Oblique Rotator concept of slowly rotating stars with non-coincidence of the magnetic and rotational axes. This model reproduces variability and reversals of the magnetic field strength. Due to the chemical abundance concentrations at the magnetic poles also spectral and the related photometric variabilities are easily understood, as well as radial velocity variations of the appearing and receding patches on the stellar surface.
Preston (1974) divided the CP stars into the following groups:
- CP1: Am/Fm stars without a strong global magnetic field; weak lines of Ca[ii]{} and Sc[ii]{}, otherwise strong overabundances;
- CP2: “classical” CP stars with strong magnetic fields (they are also known as the magnetic CP or mCP stars);
- CP3: HgMn stars, basically non-magnetic;
- CP4: He-weak stars, some of these objects show a detectable magnetic field.
Kodaira (1969) was the first who noticed significant flux depressions at 4100Å, 5200Å, and 6300Å in the spectrum of HD 221568. Jamar (1977, 1978) investigated similar features in the ultraviolet region at 1400Å, 1750Å, and 2750Å. All features were found to be only visible in magnetic CP stars. Maitzen (1976) introduced the narrow band, three filter $\Delta a$ system in order to investigate the flux depression at 5200Å. It samples the depth of this flux depression by comparing the flux at the center (5220 [Å]{}, $g_{\rm 2}$), with the adjacent regions (5000 [Å]{}, $g_{\rm 1}$ and 5500 [Å]{}, $y$) using a band-width of 130 [Å]{} (for $g_{\rm 1}$ and $g_{\rm 2}$) and the band-width of 230 [Å]{} for the Strömgren y filter. The respective index was introduced as: $$a = g_{\rm 2} - (g_{\rm 1} + y)/2$$ Since this quantity is slightly dependent on temperature (increasing towards lower temperatures), the intrinsic peculiarity index had to be defined as $$\Delta a = a - a_{\rm 0}[(b - y); (B - V); (g_{\rm 1} - y)]$$ i.e. the difference between the individual $a$-value and the $a$-value of non-peculiar stars of the same colour. The locus of the $a_{\rm 0}$-values for non-peculiar objects has been called normality line. It was shown (e.g. Vogt et al. 1998) that virtually all peculiar stars with magnetic fields (CP2 stars) have positive $\Delta a$ values up to $+75$mmag whereas Be/Ae and $\lambda$ Bootis stars exhibit significantly negative ones (Maitzen & Pavlovski 1989a,b,c). Extreme cases of the CP1 and CP3 group may exhibit marginally peculiar positive $\Delta a$ values (Maitzen & Vogt 1983).
Several attempts have been made to explain the origin of this feature. Adelman & Wolken (1976) and Adelman, Shore & Wolken (1976) investigated bound-free discontinuities, Jamar, Macau-Hercot & Praderie (1978) proposed autoionisation transitions of Si[ii]{}, whereas enhanced line absorption was discussed by Maitzen (1976) and Maitzen & Muthsam (1980). The latter presented a comparison of synthesised flux distributions and observed spectrophotometry (the first attempt in this direction was by Leckrone, Fowler & Adelman 1974). From their synthetic spectra they recovered a narrow and deep feature at about 5175Å and a broad component centred at about 5275Å. They were not able to reproduce the flux depression for effective temperatures higher than 8000K. More recently, Adelman et al. (1995) have used model atmospheres of Kurucz (1993a,b) with enhanced metallicity (i.e. a solar element abundance distribution where all elements heavier than He had been scaled by +0.5 or +1.0 dex) and concluded that at least part of the 5200 [Å]{} feature in magnetic CP stars may be due to differential line blanketing. In a follow-up work, Adelman & Rayle (2000) have extended this study to a larger group of stars and found that solar composition models may successfully predict the flux distribution of normal and many CP3 (HgMn) stars, while they fail to do so for a number of CP2 stars, i.e. the group of stars showing the largest flux depression in the 5200 [Å]{} region and thus the largest magnitudes in $\Delta a$.
One of the main conclusions drawn from these previous works has been the necessity to build specific model stellar atmospheres for CP stars using state-of-the-art opacity data. The increase in available computer power and advances in computational algorithms during the last two decades have now brought this problem into the realm of workstations and personal computers. Moreover, stellar atmosphere modelling can take advantage of data bases for atomic line transitions devoted to stellar atmosphere applications such as Kurucz (1992) and the VALD project (Kupka et al. 1999; Ryabchikova et al. 1999). Among the current projects for the computation of model atmospheres there are several which can calculate models with individual abundances on workstations or personal computers. Two of them are based on an opacity sampling approach. ATLAS12 by Kurucz (1996), for which a first application was presented by Castelli & Kurucz (1994), is particularly suitable for B to K type stars at or close to the main sequence. The marcs project in Uppsala (Bengt Gustafsson and his group) is aimed at the cool part of the Hertzsprung-Russell diagram and can produce model atmospheres for stars with spectral types later than A0 (Gustafsson et al. 2003). However, for the computation of small grids of model atmospheres with individual abundances, which are required when varying or $\log(g)$ during the initial analysis of a single star or several sufficiently similar stars, the opacity distribution function (ODF) approach remains preferable due to its higher computational efficiency. Piskunov & Kupka (2001) have presented a new software toolkit based on this approach using a modified version of the ATLAS9 code of Kurucz (1993b). It is suitable for spectral types from early B type to early F type stars at or close to the main sequence.
Our study has been initiated to synthesise and reproduce the characteristics of the 5200Å flux depression which is measured via the $\Delta a$ photometric system with the currently available stellar atmosphere models. Our general aim is to explain the various aspects and characteristics of the $\Delta a$ photometric system (and thus the 5200Å flux depression), i.e. the dependency on the metallicity, surface gravity and effective temperature.
The new synthetic photometric $\Delta a$ system is introduced in Sect. \[Sect2\]. We discuss the chosen filter system and its properties with respect to systems that have been used for observations. We compare the $\Delta a$ calibration relations obtained from grids of standard model atmospheres with observed relations as a function of several photometric indicators of . We also discuss luminosity and metallicity effects and compute the expected zero point shift of the normality line $a_0$ of $\Delta a$ photometry when applying the latter to the Magellanic Clouds. Following the standard used in the literature we give values for $\Delta a$ in units of mmag throughout this paper, while other photometric quantities are given in mag with normalisations as described in Sect. \[Sect2\]. Whenever the unit of mmag is used, it is explicitly mentioned so in the text to avoid confusion. In the concluding Sect. \[Sect3\] we summarise the success of the synthetic $\Delta a$ system in reproducing the normality line and the usefulness of $\Delta a$ photometry to study CP stars in environments different from the solar neighbourhood. We also provide an outlook for the next papers of this series in which the software and methods of Piskunov & Kupka (2001) will be used. In two follow-up papers we will discuss the capability and limitations of homogeneous model atmospheres with individual abundances for reproducing observed $\Delta a$ indices of different types of CP stars.
Synthetic photometry {#Sect2}
====================
The synthetic $\Delta a$ filter system
--------------------------------------
Relyea & Kurucz (1978) have discussed in detail how to calculate synthetic colours from model atmosphere fluxes computed with the ATLAS code. The main idea is to convolve the emergent surface fluxes predicted from model atmospheres with several functions representing filter transmission, relative absorption of all other devices in the optical path (including telescope mirrors), and detector sensitivity.
[ ]{}
[ =.48 ]{} [ =.48 ]{}
Figure \[Fig.filters\] shows the response functions of our synthetic photometric system. The steep decay of the response function of the 1P21 RCA photomultiplier from 58% at 5000 [Å]{} to 29% at 5500 [Å]{} and a mere 10% at 6000 [Å]{} explains the smaller effective sensitivity of the system in the $y$-band relative to the $g_1$-band in comparison with the transmission functions of the filters themselves. We note here that the transmission functions for the $g_1$ and $g_2$ filters were taken from calibration measurements of one of the filter sets used in earlier observations (System “2” in Maitzen & Vogt 1983), while the $y$ filter transmission (difference less than 1% to that of Maitzen & Vogt 1983) and the mirror reflection efficiency and detector response functions were taken from the [uvby]{} code of Kurucz (1993b). Maitzen & Vogt (1983) used the response function of an EMI 6256 photomultiplier for their calibrations which has essentially the same shape but a different absolute sensitivity as the 1P21 one in the relevant wavelength region. This offset in the absolute sensitivity does not affect our synthetic magnitudes since we only use differences between individual filters.
We note that all $\Delta a$ measurements used for comparisons in this paper were observed within the classical photomultiplier system. The new CCD $\Delta a$ system (Maitzen, Paunzen & Rode 1997; Bayer et al. 2000; Maitzen et al. 2001; Paunzen & Maitzen 2001, 2002; Paunzen et al. 2002) in turn has been modified in order to compensate for the different response functions of a photomultiplier and a CCD. A classical photomultiplier is almost insensitive at approximately 6000[Å]{} whereas a CCD is very sensitive in the red region. This implies an almost linear increase of the response function from $g_1$ to $y$. In the new CCD system, the FWHM of the $g_1$ and $y$ filters are 222[Å]{} and 120[Å]{}, respectively. This guarantees that the total flux of each filter after the convolution with the response function is comparable with that one of the “old” system.
Calibration relations {#Sect_calib}
---------------------
The overall success of the ATLAS9 model atmospheres of Kurucz (1993b) to reproduce photometric colours and spectrophotometric fluxes of standard stars of spectral types B and A (Castelli & Kurucz 1994; Smalley & Dworetsky 1995; Castelli, Gratton & Kurucz 1997; Castelli 1998) and also for some of the CP stars (Adelman & Rayle 2000) promotes them as a logical choice when testing a synthetic photometric system. We have thus computed several grids of ATLAS9 model atmospheres with the Stellar Model Grid Tool (SMGT; see Heiter et al. 2002 for a description) using the line opacities from Kurucz (1993a) and the ATLAS9 code of Kurucz (1993b), unaltered except for the convection treatment (Smalley & Kupka 1997; Heiter et al. 2002). In fact, both Smalley & Kupka (1997) and Heiter et al. (2002) recommend the use of a convection model in ATLAS9 which predicts inefficient convection for mid to late A stars ( $\leqslant 8500$ K). We thus used the model of Canuto & Mazzitelli (1991) which allows a better reproduction of Strömgren colours of A stars than the original models of Kurucz (1993b), as shown in Smalley & Kupka (1997). For models with $> 8500$ K, where convection has only negligible influence on temperature gradients and colours, our models are virtually identical to those from the original grids published by Kurucz (1993b). Anyway, from a comparison we did with model atmospheres based on different convection models we conclude that for any of the convection treatments available for ATLAS9 (cf. Castelli et al. 1997; Heiter et al. 2002) the $a$ values change only by 0 to $+3$ mmag for the coolest models in our grids and remain completely unaltered for models with $> 8500$ K. Thus, no important bias is introduced into our calibration tests by selecting a particular convection model. The ATLAS9 grids of Kurucz (1993b) ones have a spacing of $\Delta \log(g)\,=\,0.5$ dex which is slightly too coarse for our purpose. Hence, we also included intermediate models in our computations by using a spacing of $\Delta \log(g)\,=\,0.25$ dex. We studied a $\log(g)$–range from 2.5 dex to 4.5 dex and a –range from 7000 K to 15000 K (with a spacing of 250 K as in Kurucz 1993b). Model atmospheres assuming one of the following three metallicities were investigated: $-0.5$, 0, and $+0.5$ dex, where \[M/H\]=0dex represents solar abundance and elements heavier than He are scaled by $\pm 0.5$ dex in the other cases. A constant value of 2 km s$^{-1}$ as in the standard grid of Kurucz (1993b) was used for the microturbulence.
To transform the theoretical colours into the frame of observed colours Kurucz (1993b) corrected the zero-point of his synthetic $uvby$ system so as to match $c_1$, $m_1$, and $(b-y)$ of Vega. A similar procedure is necessary to compare calculated $a$ values from our synthetic $\Delta a$ system directly to observations. We have added $0.6$ to the synthetic $a$ value computed from convolving the effective transmission functions with the fluxes from ATLAS9 model atmospheres. The numerical values we obtain are then close to Maitzen & Vogt (1983, see their Table 1 and Equation 1). We obtain $a \sim 0.594$ for a main sequence star with $(b-y) = 0.000$ and solar metallicity from our synthetic photometric system. For the $\Delta a$ system itself the specific value of $a$ and thus any zero-point correction related to it are irrelevant, because the quantity of interest is the difference of a measured $a$ value to the normality line $a_0$. Hence, it is much more important to show that our synthetic $\Delta a$ system recovers the observed dependencies of the $\Delta a$ index from different indicators of effective temperature, surface gravity, and metallicity.
Figure \[Fig.colour\_calib\] illustrates the temperature (and gravity) dependence of the $a$ index as a function of various experimental indicators of . For $(B-V)$ and $(g_1-y)$ the entire temperature range is displayed while a cut-off was introduced for the other two cases through requiring that $(b-y) \geqslant -0.02$ (i.e. $T_{\rm eff}
\lesssim 11000$ K) and $[$u-b$] \leqslant 1.38$ (i.e. $T_{\rm eff} \gtrsim
9250$ K). Models with $\log(g)=4$ dex have been connected with straight lines to provide a proxy for the normality line $a_0$ of the synthetic photometric system. Note that the output colours have been rounded to 1 mmag accuracy as in Kurucz (1993b). The actual run of the colours is continuous and smaller magnitude differences can hardly be assigned a real physical meaning within the current state of modelling.
For our comparison of the synthetic $\Delta a$ system to observational ones we have used the results for Galactic field stars (Maitzen & Vogt 1983; Vogt et al. 1998). More than 1140 bright normal, peculiar and related stars have been observed within four different $\Delta a$ systems. These four systems are mainly distinguished by slightly different filter transmission curves (Fig. 3 in Maitzen & Vogt 1983). The differences for the $\Delta a$ values were found to be in the range of 2 to 3 mmag. They find the following relation for the normality linegg: $$a_0 = G_0 + G_1(b-y) + G_2 (b-y)^2$$ with $G_{0} = 0.594$, and where $0.086 < G_{1} < 0.105$ as well as $-0.050 < G_{2} < -0.150$ hold for $-0.120 < (b-y) < +0.200$. The 1$\sigma$ level around the normality line was found to be between 2.9 and 5.1 mmag. These values are a superimposition of the internal measurement errors and the (observed) natural bandwidth. On the other hand, the colours from model atmospheres ranging the main sequence band from $\log(g)$ of \[3.5,4.5\] dex for all models with a of \[7000,15000\] K and solar metallicity, and for which $(b-y)\geq-0.02$, yield a $G_{0} = 0.591$ (with less than 0.5 mmag error), while $G_{1} = 0.0985 \pm 0.0056$ and $G_{2} = -0.044 \pm 0.030$, when fitting a least square parabola through the model colours (see Fig. \[Fig.colour\_calib\]). Hence, the $(b-y)$ dependence of the experimental $\Delta a$ systems investigated in Maitzen & Vogt (1983) is reproduced very well.
For the \[$u-b$\] relation, Maitzen (1985) lists $G_{1}$=0.024 for 22 bright unreddened stars with a 1$\sigma$ level of 4.5 mmag. For this correlation, the results in Fig. \[Fig.colour\_calib\] imply a more flat dependence of $G_{1} = 0.0098 \pm 0.0018$ (and a $G_{2}$ of $-0.0022 \pm 0.00097$). However, the rather small slope is very sensitive to the precise definition of the sample: including giants with $\log(g)\geqslant 2.5$ dex would raise $G_{1}$ to 0.0161, whereas reducing the range of \[$u-b$\] from an upper limit of 1.38 to 1.20 while keeping only the main sequence band models with $\log(g) \geqslant
3.5$ dex, as in the first case, would increase it to $G_{1} = 0.0211 \pm
0.0028$. Thus, within the overall uncertainties expected for such kind of a weak dependence on \[$u-b$\] the latter is reproduced sufficiently well.
Figure \[Fig.colour\_calib\] also shows the correlation of $a$ with the temperature indicators $(B-V)$ and $(g_1-y)$. Their dependency can easily be studied as before, but is unlikely to reveal more information beyond the uncertainties introduced by the colour transformation required to compare the different filter systems used in observations and the synthetic systems of Kurucz (1993b).
Luminosity effects {#Sect_lumin}
------------------
Because each of the colour relations presented in Fig. \[Fig.colour\_calib\] is also affected by surface gravity within the range of effective temperatures populated by the CP stars, we have looked at the direct dependence of $a$ on as well (see Fig. \[Fig.lumin\_calib\]). Clearly, models with lower surface gravity have higher $a$ values. The effect of surface gravity on the $a$ index is largest for the late B stars with effective temperatures around $\sim$10500 K. The width of the band of standard stars as induced by surface gravity for a given metallicity is between 2 and 5 mmag. This confirms the results of the previous subsection and is in agreement with the observational data quoted therein.
We note here that the step size in $\log(g)$ for model sequences shown in both Fig. \[Fig.lumin\_calib\] and \[Fig.metal\_calib\] is 0.25. However, as the output of the photometric indices has been truncated to 1 mmag, it turns out that the $\Delta a$ dependence on $\log(g)$ is too weak to show up more prominently. Hence, many models overlap in the Figures due the assumed output accuracy. The sensitivity of $\Delta a$ to $\log(g)$ slightly depends on and metallicity, as the number of apparent points in Figs. \[Fig.lumin\_calib\] and \[Fig.metal\_calib\] reveals as well.
Metallicity effects
-------------------
[ ]{}
[ ]{}
Metallicity has an effect on the $a$ index which is actually more important than that of luminosity (surface gravity). Fig. \[Fig.metal\_calib\] compares the main sequence band for solar metallicity with models having over- and underabundances of $\pm 0.5$ dex for all elements heavier than He. We conclude that an underabundance of $-0.5$ dex as in the Magellanic Clouds (Dirsch et al. 2000) yields a shift of the normality line of $-3$ mmag. Notice that Maitzen, Paunzen & Pintado (2001) found the first extragalactic CP stars in the Magellanic Cloud using the $\Delta a$ system. The size of this shift is quite constant over the entire range relevant for CP stars and also within the entire luminosity range expected for the main sequence band. On the other hand, an overabundance of $+0.5$ dex yields a larger shift of between $+3$ and $+6$ mmag with a maximum for the late B stars. This behaviour gives already some hint on the nature of the flux depression at 5200 [Å]{} in agreement with Adelman et al. (1995) and Adelman & Rayle (2000) who found that line opacities of ATLAS9 models with metal overabundances of $+0.5$ and $+1.0$ dex predict some extra line blanketing in this region. In turn, due to the good agreement of ATLAS9 model fluxes in this wavelength region with observations for mildly peculiar stars, which have underabundances or overabundances of up to about 0.5 dex (cf. Castelli & Kurucz 1994; Adelman & Rayle 2000), and due to the satisfactory agreement of our synthetic $\Delta a$ system with systems used in observations (see previous subsections), we can draw an important conclusion: application of $\Delta a$ photometry to the Magellanic Clouds will lead only to a small bias, with a size of about $-3$ mmag relative to the observations made for the solar neighbourhood, and the same will hold for more remote targets that have a similar metallicity range.
Conclusions and outlook {#Sect3}
=======================
In this first paper of our series we have established a synthetic photometric $\Delta a$ system and confirm the observed dependency of the $a$ index as a function of various colour indices sensitive to the effective temperature and surface gravity variations within the Strömgren $uvby\beta$ and Johnson $UBV$ photometric systems. Several calibration relations are presented to confirm that the new synthetic $\Delta a$ system provides a normality line and features an average scatter along the main sequence for normal type stars which is very close to the observed relations, if fluxes from ATLAS9 model atmospheres are used as input data for the synthetic photometric system. The metallicity dependence of the normality line of the $\Delta a$ system was computed for several grids of model atmospheres for which the abundances of elements heavier than He had been scaled by $\pm$0.5dex in order to test for the effects of over- and underabundances. We estimate a lowering of $\Delta a$ by $\sim -3$ mmag assuming an average metallicity of $[{\rm Fe}/{\rm H}]=-0.5$ dex compared to the Sun. This is a typical value as found for Magellanic Clouds for which the first CP stars have already been detected using the $\Delta a$ system. Thus, $\Delta a$ photometry is a viable tool to identify CP stars in samples with metallicities slightly different from the solar ones and it is well suited to draw statistically meaningful conclusions about their distribution. It is hence a recommendable method to find CP stars in other galaxies, too.
We intend to publish two follow-up papers. In these papers we will present model atmospheres computed with individual abundances for a representative sample of CP as well as $\lambda$ Bootis stars. For these objects we will either confirm or redetermine the input parameters (effective temperature, surface gravity, overall metallicity and microturbulence) found in the literature through comparisons with photometric, spectrophotometric, and high resolution spectra ($R$$\approx$20000) spectroscopic data. The final models obtained from this procedure will be used to compute synthetic $\Delta a$ values which will be compared with individual photoelectric observations. The observed behaviour of $\Delta a$ will be shown to be very well reproduced for several types of CP stars. Furthermore, a detailed statistical analysis of the relative abundances for each star will be given. This will illustrate which species contribute in the different filters.
The first follow-up paper (paper [ii]{}) will discuss the models for Am and cool CP2 stars with effective temperatures below about 10000 K. The second one will deal with a discussion of hotter CP stars with spectral types earlier than A0 (paper [iii]{}). The atmospheres of these objects are different from those of A stars. Among others, convection cannot play an essential role any more and the effects of stratification become even more important than for cooler objects.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank our referee, Dr. F. Castelli for helpful comments and improvements. This research was performed within the projects [*P13936-TEC*]{} and [*P14984*]{} of the Austrian Fonds zur Förderung der wissenschaftlichen Forschung (FwF). We acknowledge use of the SIMBAD and GCPD astronomical data base. For support during the final stage of writing this paper FK acknowledges support through the UK PPARC under grant PPA/G/O/1998/00576.
[99]{} Adelman S. J., Rayle K. E., 2000, A&A, 355, 308 Adelman S. J., Wolken P. R., 1976, ApJ, 207, 159 Adelman S. J., Shore S. N., Wolken P. R., 1976, in Weiss W.W., Jenkner H., Wood H.J., eds, Proc. IAU Coll. No. 32, Physics of Ap-Stars, University Vienna, p. 189 Adelman S. J., Pyper D. M., Lopez-Garcia Z., Caliskan H., 1995, A&A, 296, 467 Babcock H. W., 1947, ApJ, 105, 105 Babcock H. W., 1958, ApJS, 3, 141 Bayer C., Maitzen H. M., Paunzen E., Rode-Paunzen M., Sperl M., 2000, A&AS, 147, 99 Canuto V. M., Mazzitelli I., 1991, ApJ, 370, 295 Castelli F., 1998, Memorie della Societ[á]{} Astronomica Italiana, Vol. 69, p.165 Castelli F., Kurucz R. L., 1994, A&A, 281, 817 Castelli F., Gratton R., Kurucz R. L., 1997, A&A, 318, 841, erratum: 1997, A&A, 324, 432 Dirsch B., Richtler T., Gieren W. P., Hilker M., 2000, A&A, 360, 133 Gustafsson B., Edvardsson B., Eriksson K., Gr[å]{}e-J[ø]{}rgensen U., Mizuno-Wiedner M., Plez B., 2003, A Grid of Model Atmospheres for Cool Stars, in Hubeny I., Mihalas D., Werner K., eds, Workshop on Stellar Atmosphere Modeling, to be published in ASP Conference Series in 2003 Heiter U., Kupka F., van ’t Veer-Menneret C., et al., 2002, A&A, 392, 619 Jamar C., 1977, A&A, 56, 413 Jamar C., 1978, A&A, 70, 379 Jamar C., Macau-Hercot D., Praderie F., 1978, A&A, 63, 155 Kodaira K., 1969, ApJ, 157, L59 Kupka F., Piskunov N. E., Ryabchikova T. A., Stempels H. C., Weiss W. W., 1999, A&AS, 138, 119 Kurucz R. L., 1992, Rev. Mexicana Astron. Astrofis., 23, 45 Kurucz R. L., 1993a, Kurucz CD-ROMs 2-12 (Cambridge:SAO) Kurucz R. L., 1993b, Kurucz CD-ROM 13 (Cambridge:SAO) Kurucz R. L., 1996, in Adelman S.J., Kupka F., Weiss W.W., eds, Model Atmospheres and Spectrum Synthesis, ASP Conf. Ser. 108, San Francisco, p. 160 Leckrone D. S., Fowler J. W., Adelman S. J., 1974, A&A, 32, 237 Maitzen H. M., 1976, A&A, 51, 223 Maitzen H. M., 1985, A&AS, 62, 129 Maitzen H. M., Muthsam H. M., 1980, A&A, 83, 334 Maitzen H. M., Pavlovski K., 1989a, A&A, 219, 253 Maitzen H. M., Pavlovski K., 1989b, A&AS, 77, 351 Maitzen H. M., Pavlovski K., 1989c, A&AS, 81, 335 Maitzen H. M., Vogt N., 1983, A&A, 123, 48 Maitzen H. M., Paunzen E., Rode M., 1997, A&A, 327, 636 Maitzen H. M., Paunzen E., Pintado O. I., 2001, A&A, 371, L5 Maury A., 1897, Ann. Astron. Obs. Harvard Vol. 28, Part 1 Paunzen E., Maitzen, H. M., 2001, A&A, 373, 153 Paunzen E., Maitzen, H. M., 2002, A&A, 385, 867 Paunzen E., Pintado O. I., Maitzen, H. M., 2002, A&A, 395, 823 Piskunov N. E., Kupka F., 2001, ApJ, 547, 1040 Preston G. W., 1974, ARA&A, 12, 257 Relyea L. J., Kurucz R. L. 1978, ApJS 37, 45 Ryabchikova T. A., Piskunov N. E., Stempels H. C., Kupka F., Weiss W. W., 1999, Phys. Scripta, T83, 162 Smalley B., Dworetsky M. M., 1995, A&A, 293, 446 Smalley B., Kupka F., 1997, A&A, 328, 349 Stibbs D. W. N., 1950, MNRAS, 110, 395 Vogt N., Kerschbaum F., Maitzen H. M., Faúndez-Abans M., 1998, A&AS, 130, 455
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'An incremental observer generation for modular systems is presented in this paper. It is applied to verification and enforcement of current-state opacity and current-state anonymity, both of which are security/privacy notions that have attracted attention recently. The complexity due to synchronization of subsystems, but also the exponential observer generation complexity, are tackled by local observer generation and an incremental abstraction. Observable events are hidden and abstracted step by step when they become local after synchronization with other subsystems. For systems with shared unobservable events, complete observers can not be generated before some local models are synchronized. At the same time, observable events should be abstracted when they become local, to avoid state space explosion. Therefore, a new combined incremental abstraction and observer generation is proposed. This requires some precaution (detailed in the paper) to be able to accomplish local abstractions before shared unobservable events are removed by observer generation. Furthermore, it is shown how current state opacity and anonymity can be enforced by a supervisor. This is achieved by a natural extension of the verification problem to a supervisory control problem based on forbidden states and incremental abstraction. Finally, a modular and scalable building security problem with arbitrary number of floors and elevators is presented, for which the efficiency of the incremental abstraction is demonstrated.'
author:
- 'Mona Noori-Hosseini, Bengt Lennartson and Christoforos N. Hadjicostis'
bibliography:
- 'Automation-mona.bib'
date: 'Received: date / Accepted: date'
title: 'Incremental Observer Reduction Applied to Opacity Verification and Synthesis [^1] [^2]'
---
Introduction {#intro}
============
With the rapid growth of large communication networks and online services, and their diverse applications, ranging from modern technologies in defense and e-banking to health care and autonomous vehicles, security and privacy concerns on their information flow are raised. This means that unauthorized people should not acquire the information flow in these services, for instance in terms of being able to track and identify real time location information about the users. There are various notions on security and privacy for different applications based on their vulnerability to intruders. Next we provide some background information and related literature on topics that are relevant to the developments in this paper.
#### Opacity verification
One category of security notions, [@Focardi] concerns the information flow from the system to the outside observer, which is called opacity [@Saboori_2007; @lesage16]. Opacity is a general and formal security property that has been widely investigated for discrete event systems (DESs) for finite automata [@Saboori_2007; @bryan08; @Saboori_2008; @Saboori:2014], but also for [P]{}etri nets [@Bryans_2005; @Tong_2016; @tong:giua:17]. A system is opaque if, for any secret behavior, there exists at least one non-secret behavior that looks indistinguishable to the intruder [@Saboori_2007; @laf18]. The security notion is investigated for automata [@lesage16] using either state-based predicates [@Saboori_2007; @hadji11; @lesage16; @Tong:giua:2017], or language-based predicates [@Badouel:2007; @Saboori_2008; @Cassez_2009; @Lin_2011; @Tong_2016].
Depending on the modeling formalism of the system and the secret, there are different opacity notions, such as current-state opacity, initial-state opacity, and k-step opacity [@lesage16]. [@wu2013] show that there exists a polynomial-time transformation between different notions of opacity for finite automata and regular languages. Current-state opacity [@Saboori:2014; @Tong:giua:2017] requires that the sequence of observable events seen by the intruder never allows the external observer to unambiguously determine that the current state of the system falls within a given set of secret states. A number of examples and applications are presented in [@hadji11]. A privacy notion that is adapted from current-state opacity is proposed in [@bryan08; @Lin_2011]. It is called anonymity, and in [@Wu_2014] it is used for location privacy and is called current-state anonymity. The servers that access the user’s location information are then regarded as intruders.
An intruder with partial observation can be modeled as an observer of the system, meaning that it has full knowledge about the system structure, while it is only able to see the observable events of the system. Observers achieved by subset construction [@cl:int:2008] are deterministic finite automata that estimate the set of possible current states for verifying properties of interest. There are several works that exploit observer generation for opacity verification [@Saboori_2011; @Saboori_2013; @wu2013; @wu:abs:ver:2018].
#### Opacity enforcement
Ensuring opacity on a system is usually performed by exploiting supervisory control [@rw:con:1989] as in [@Takai_2008] and [@Takai_2009]. Given a system that is not current-state opaque with respect to a secret, it is required to design a maximally permissive supervisor that restricts the behavior of the system to turn it into a current-state opaque system. The design of supervisors to enforce opacity is also sometimes called opacity enforcement. In [@Badouel:2007], the language-based opacity and a set of intruders having different observations are considered. The work by [@dubreil08; @dubreil10] is also focused on language-based opacity enforcement for one intruder.
Enforcing opacity using supervisory control techniques is also investigated by [@Saboori_2008]. They propose methods for designing optimal supervisors to enforce two different opacity properties, with the assumption that the supervisor can observe all controllable events [@saboori:2012]. In the work by [@Yin_2016; @Tong:giua:2018], to enforce current-state opacity, the assumption that all controllable events should be observable is relaxed. In [@Wu_2014_automatica; @Ji_2018] a novel enforcement mechanism is proposed, based on the use of insertion functions that change the output behavior of the system, by inserting additional observable events.
#### Modularity and abstraction
To verify or synthesize a supervisor to enforce current-state opacity/anonymity in a modular system, it is required to generate the system’s observer. Given the exponential complexity of observer generation, as well as the complexity of interacting subsystems, especially for large complex modular systems, state space explosion often occurs while performing verification or synthesis. For this reason, reduction methods play an important role in making the procedure feasible. [@HadjAlouane_2017] use a binary decision diagram technique [@Bryant_1992] to abstract graphs of moderate size, as a method for the verification of three different opacity variants. Moreover, they prove that opacity properties are preserved by composition, which guarantees that local verification of these properties can also be performed.
In [@Zhang_2017], a bisimulation-based method to verify the infinite-step opacity of nondeterministic finite transition systems is proposed. Since this abstraction is based on strong bisimulation it has a minor reduction capability compared to abstractions where local events are hidden, such as weak bisimulation [@m:com:1989] and branching bisimulation [@glabbeek96]. Recently the authors have proposed an abstraction method for current-state opacity verification of modular systems [@mona_18] based on a similar abstraction, called visible bisimulation equivalence [@bl_18]. Both state labels and transition labels (events) are then integrated in the same abstraction method. This abstraction has the benefit that temporal logic properties are preserved in the abstraction, and the opacity verification in [@mona_18] is formulated as a temporal logic safety problem.
#### Incremental observer abstraction
In the abstraction, local events (only included in one subsystem) are hidden and then abstracted such that temporal logic properties related to specific state labels are still preserved. When subsystems are synchronized more local events are obtained, which also means that more events can be hidden and abstracted. This hiding/abstraction method is repeated until all subsystems have been synchronized. The result is an incremental abstraction technique where state space explosion is avoided when a reasonable number of events are local or at least only shared with a restricted number of subsystems. Most real systems have this event structure, and still some events can be shared by all subsystems. This incremental abstraction technique for modular systems can be traced back to [@graf:96], but its application to local events was more recently proposed in [@flordal09], where it was called compositional verification. In [@mona_18] this incremental abstraction is adapted to opacity verification, and it shows great computational time improvement compared to standard methods.
#### Nonblocking transformation
In this paper, both current state opacity verification and current state anonymity verification are formulated based on state labels in transition systems. Non-safe states in corresponding local observers are then naturally considered as forbidden states. By introducing simple detector automata, the problem is easily transformed to a nonblocking problem. For this modular system, the efficient conflict equivalence abstraction in [@Malik04] and [@flordal09] is used, since it preserves the nonblocking properties of the original modular observer. The reason for evaluating this abstraction is that it is known to be more efficient than visible bisimulation. This abstraction has independently been proposed for opacity verification by [@sahar_opacity:2019] and [@mona_opacity:2019]. In both reports, the abstraction gives an enormous reduction in computation time, compared to opacity verification without abstraction. In our work, the procedure is evaluated on a scaleable building security problem, including an arbitrary number of floors and elevators.
#### Observer abstraction including shared unobservable events
Two main extensions are also included in this paper, first the nontrivial introduction of shared unobservable events. It means that complete local observers can not be computed before some local models are synchronized. The reason is that shared unobservable events can not be reduced in the observer generation before they have become local after synchronization. At the same time, observable events should be abstracted when they become local, to avoid state space explosion. The proposed solution is to extend the incremental abstraction with an incremental observer generation, such that a switch between abstraction and observer generation can be performed when subsystems are synchronized. This requires some precaution to be able to accomplish local abstractions before shared unobservable events are removed by the observer generation. Some minor restrictions are included to be able to prove that the combined incremental observer generation and abstraction works correctly. This procedure includes additional temporary state labels, which motivates the more general and flexible visible bisimulation abstraction.
#### Incremental supervisor abstraction for opacity enforcement
To enforce opacity and anonymity it is also shown how an observer based maximally permissive supervisor can be generated by incremental abstraction. This supervisor generation follows naturally as an extension of the original forbidden state formulation of opacity and anonymity verification. The incremental abstraction is based on a supervision/synthesis equivalence proposed by [@fmf:sup:2007; @sahar14; @sahar17] as a natural extension of conflict equivalence [@Malik04].
#### Main contributions
To summarize, the main contributions of this paper are: 1) a transition system based formulation of modular observers applied to current state opacity and current state anonymity verification, 2) a simple transformation of the modular observer verification problem to a nonblocking problem based on simple detector automata, a generic technique that can be applied to many verification and synthesis problems, for instance abstraction based diagnosability verification [@mona_diag:2019], 3) a combined incremental observer generation and abstraction for modular systems including shared unobservable events, 4) an incremental abstraction based synthesis of observer based maximally permissive supervisors for current state opacity and anonymity, 5) a modular formulation of a scaleable building security problem including an arbitrary number of floors and elevators, and finally 6) a demonstration on how efficient the proposed incremental abstraction of observers for current state opacity and current state anonymity verification and synthesis works for large modular systems.
The remainder of the paper is organized as follows. After some preliminaries introduced in Section 2, the problem statement is presented in Section 3. Efficient generation of modular observers is shown in Section 4, followed by some specific results on current-state opacity/anonymity for modular systems in Section 5. In Section 6, it is shown how a combined incremental observer generation and abstraction can be achieved for systems including shared unobservable events. Section 7 presents a scaleable floor/elevator building for which the efficiency of the proposed incremental abstraction is demonstrated. In Section 8, an incremental abstraction based supervisor generation for current state opacity and anonymity is developed, followed by some concluding remarks in Section 9.
Preliminaries {#sprel}
=============
A [*transition system*]{} $G$ is defined by a 6-tuple $G=\langle X,\Sigma,T, I,AP,\lambda \rangle$ where $X$ is a set of states, $\Sigma$ is a finite set of events, $T\subseteq X\times \Sigma\times X$ is a transition relation, where $t=(x,a,x')\in T$ includes the source state $x$, the event label $a$, and the target state $x'$ of the transition $t$. A transition $(x,a,x')$ is also denoted . $I\subseteq X$ is a set of possible initial states, $AP$ is a set of atomic propositions, and is a state labeling function.
A subset $\mc L \subseteq \Sigma^*$ is called a *language*. Moreover, for the event set $\Omega \subseteq \Sigma$, the [*natural projection*]{} $P:\Sigma^*\ra \Omega^*$ is inductively defined as $P(\veps)=\veps$, $P(a) = a$ if $a\in\Omega$, $P(a) = \veps$ if $a\in\Sigma\setm\Omega$, and $P(sa)=P(s)P(a)$ for $s\in\Sigma^*$ and $a\in \Sigma$. In the composition of subsystems, see , events that are not included in any synchronization with other subsystems are called [*local events*]{}. Such events are central in the abstraction of observers.
#### Modeling $\veps$ transitions.
The transition system $G$ is now extended to include transitions labeled by the empty string $\veps$. In this paper, the $\veps$ label will explicitly be used for [*local unobservable events*]{}. If nothing special is pointed out, it means that such local unobservable events are replaced by $\veps$ and therefore not included in the alphabet $\Sigma$, while the total alphabet is extended to $\Sigma\cup \{\veps\}$. A sequence of $\veps$ transitions $x=x_0\trans{\veps} x_1\trans{\veps} \cdots \trans{\veps} x_n=x'$, $n\geq 0$, is denoted $x\transd{\veps}x'$. A corresponding sequence, including possible $\veps$ transitions before, after and in between events in a string $s\in \Sigma^*$, is denoted $x\transd{s}x'$. The of a state $x$ is defined as $R_\veps(x)=\{x'\st x\transd{\veps} x'\}$, and for a set of states $Y\subseteq X$ we write $R_\veps(Y)=\bigcup_{x\in Y} R_\veps(x)$.
A nondeterministic transition system generally includes a set of initial states, transitions, and/or alternative transitions with the same event label. A transition function for an event $a\in\Sigma$ in a nondeterministic transition system is defined as $\delta(Y,a)=R_\veps(\{x'\st (\exists x \in R_\veps(Y)) \;x\trans{a}x'\in T\})$. An extended transition function is then inductively defined, for $s\in \Sigma^*$ and $a\in \Sigma$, as $\delta(I,sa)=\delta(\delta(I,s),a)$ with the base case $\delta(I,\veps)=R_\veps(I)$. Furthermore, the language for a nondeterministic transition system is defined as $\mc L(G)=\{s \in \Sigma^* |(\exists x\in I) \, \delta(x,s)\neq \varnothing \}$.
#### Local transitions and hidden $\tau$ events
To obtain efficient abstractions, a special $\tau$ event label is used for transitions with local observable events. The lack of communication with other subsystems means that the $\tau$ event is hidden from the rest of the environment. The closure of $\tau$-transitions in a finite path $x=x_0 \trans{\tau}x_1\trans{\tau} \cdots\trans{\tau}x_n=x'$, $n\geq 0$ is denoted .
Note the difference between $\veps$ and $\tau$ events. Unobservable local events are replaced by $\veps$ before an observer is generated, which removes any $\veps$ transitions. Observable local events are then replaced by $\tau$ to model that they are hidden before performing any abstraction. In process algebra, the replacement of any specific event by the event $\tau$ is called [*hiding*]{}, cf. [@m:com:1989]. A transition system $G$ where the events in $\Sigma^h$ are hidden and replaced by $\tau$ is denoted $G^{\Sigma^h}$.
#### Partition $\Pi$ and block $\Pi(x)$
To obtain abstracted transition systems, states $x,y\in X$ that can be considered to be equivalent in some sense, denoted , are merged into equivalence classes , also called blocks. These blocks, which are non-overlapping subsets of $X$, divide the state space into the [*quotient set*]{} , also called a partition $\Pi$ of $X$. The block/equivalence class including state $x$ is denoted . A partition $\Pi_1$ that is [*finer*]{} than a partition $\Pi_2$, denoted $\Pi_1\preceq\Pi_2$, means that $\Pi_1(x)\subseteq \Pi_2(x)$ for all $x\in X$. The partition $\Pi_2$ is then said to be [*coarser*]{} than $\Pi_1$.
#### Invisible, visible and stuttering transitions
For a given state partition $\Pi$, a transition is invisible if , while a transition is visible if or . A path is called a [*stuttering transition*]{}, denoted , if , and or $\Pi(x_n)\neq$ $\Pi(x')$. This means that the first $n$ transitions are invisible, while the last one is visible. A [*block stuttering transition*]{} corresponding to is denoted .
#### Visible bisimulation
Different types of bisimulations, used for abstraction, are either defined for labeled transition systems, only including event labels (often called actions) on the transitions, or for Kripke structures, only including state labels [@baier08]. In this work, shared events are required for synchronization of subsystems, while state labels are used to model security properties. Recently, [@bl_18] introduced an abstraction for transition systems including both event and state labels, called visible bisimulation. It is directly defined as an equivalence relation based on block stuttering transitions, and more specifically on the set of event-target-blocks $\gmb(x) = \{\transtau{a}\Pi(x') \st x\transtau{a}x'\}$ that defines all possible stuttering transitions from an arbitrary state $x$.
\[dbs\] [Given a transition system $G=\langle X,$ $\Sigma,T,I,AP, \lambda \rangle$ and the state label partition $\Pi_\lambda(x)=\{y\in X\st \lambda(x)=\lambda(y) \}$, a partition $\Pi$, for all $x\in X$ determined by the greatest fixpoint of the fixpoint equation $$\Pi(x) = \{y\in X \st \Pi\preceq\Pi_\lambda\AND \gmb(x) = \gmb(y) \},$$ is a [*visible bisimulation (VB) equivalence*]{}, and states $x,y\in \Pi(x)$ are visibly bisimilar, denoted $x\sim y$. ]{}
#### Quotient transition system
Blocks are the states in abstracted transition systems, and the notion partition $\Pi$ is used in the computation of this model, while the resulting reduced model takes the equivalence perspective. It is therefore called [*quotient transition system*]{}, and for a given partition $\Pi$ it is defined as $G\qsb=\langle X\qsb,\Sigma,T\qssb,$ $I\qssb,AP,\lambda\qssb \rangle$, where is the set of block states (equivalence classes), is the set of block transitions, here specifically defined for VB, is the set of initial block states, and is the block state label function, where it is assumed that , $\forall y \in \ec{x}$.
Visibly bisimilar states $x\sim y$ in $G$ are also visibly bisimilar to the block state $\ec{x}$ in $G\qsb$, $\ec{x}\sim x$ for all $x\in X$. Furthermore, $G$ and $G\qsb$ are VB equivalent, denoted $G\sim G\qsb$. Combining hiding of a set of events $\Sigma^h$ for a system $G$, followed by the generation of the quotient transition system, results in the [*abstracted transition system*]{} $G^{\Sigma^h}\!\smm \qsb \define G^{\mathcal A^{\Sigma^h}}$. This also means that $G^{\Sigma^h}\!\smm\sim G^{\mathcal A^{\Sigma^h}}$.
#### Synchronous composition
The definition of the synchronous composition in [@h:com:1985] is adapted to $\tau$ events, where such events in different subsystems are not synchronized, although they share the same event label. They are simply considered as local events, which is natural since the hiding mechanism where an event is replaced by the invisible $\tau$ event is only applied to local events. This results in the following definition of the synchronous composition, including $\tau$ event labels.
\[dsynch\]
Any transitions with $\veps$ labels, representing local unobservable events, are handled in the same way as $\tau$ event labels, representing observable local events, since both stand for local events. On the other hand, before subsystems are synchronized, local observers will in this work be generated. This means that any $\veps$ transitions will be removed before synchronization.
#### Nonblocking and controllable supervisor
In order to determine whether a system satisfies a given specification or not, the system has to be [*verified*]{}, and if it fails, the system is restricted by *synthesizing* a *supervisor*. This means that states from which it is not possible to reach a desired marked state, called *blocking states*, are removed. Furthermore, any uncontrollable events that can be executed by the plant are not allowed to be disabled by the supervisor [@rw:con:1989; @Wonham_2017]. Thus, a supervisor is synthesized to avoid blocking states and disabling uncontrollable events. Such a nonblocking and controllable supervisor is also maximally permissive, meaning that it restricts the system as little as possible.
Problem statement {#prob_f}
=================
The focus of this paper is to generate reduced observers that still preserve relevant properties, to be able to verify different security notions. It is also shown how supervisors can be generated, avoiding states that do not satisfy desired properties. This section presents the main problem statements of the paper, the incremental generation of reduced observers, and some security notions that will be analyzed by such reduced observers. First observers only involving local unobservable events are considered, where all such local events are immediately replaced by $\veps$. The more complex case, where some unobservable events are shared between different subsystems, means on the other hand that the shared unobservable events can not be replaced by $\veps$ before they have become local due to synchronization.
Incremental abstraction for modular systems
-------------------------------------------
A transition system, including a number of subsystems $G_i$, $i\in\mathbb{N}^+_n$ that are interacting by synchronous composition, is defined as $$\label{modular}
G\sa= \,\saa\parallel_{i\in\mathbb{N}^+_n}\smm G_i=G_1\synch G_2\synch\cdots\synch G_n.$$ A straightforward approach to analyze such a modular system is to compute the explicit monolithic transition system $G$. However, there are limitations on memory and computation time in the generation and analysis of such monolithic systems. An alternative approach is to avoid building the explicit monolithic system, by analyzing each individual subsystem first. In this case, local events of each subsystem are hidden and abstracted based on the desired property to be preserved. Moreover, after every synchronization of subsystems more local events may appear and thus, additional abstraction is possible. This step by step combined hiding, abstraction and synchronization is here called *incremental abstraction*. In [@flordal09], this approach is proposed for verification, and is called *compositional verification*.
Incremental observer generation including abstraction {#incobs}
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The focus of this paper is on verification of security properties, while a simple extension towards synthesis is shown in the end of the paper. The security properties are analyzed by constructing an observer, where only observable events are involved. The generated observer is deterministic and computed by subset construction [@hmu:int:2001].
Since the observer generation as well as the synchronization of the subsystems have exponential complexity, the incremental abstraction mentioned above is of interest. This approach can be applied if the observer generation is divided into local observers that are synchronized. When all unobservable events are local, no shared unobservable events are involved, it is shown in Section \[obs:gen\] that an observer of the monolithic system $G$, denoted $\mathcal O(G)$, also can be computed by the synchronous composition of the local observers of its subsystems. Thus, $$\label{obs_eq}
\mathcal O(G) = \,\saa\parallel_{i\in\mathbb{N}^+_n}\smm \mathcal O(G_i).$$ The security properties considered in this work result in observer states that are either safe or non-safe. Introducing the state label $N$ for the non-safe states, visible bisimulation can be used in an incremental abstraction, still preserving the separation between the two types of states.
For two synchronized subsystems, $G_1\synch G_2$, the sets of local events in $G_1$ and $G_2$ are $\Sigma^h_1$ and $\Sigma^h_2$, respectively, and the events in $\Sigma^{h}_{12}$ are the shared events between the two subsystems that become local after the synchronization, see also . Thus, the set $\Sigma^{h}= \Sigma^{h}_1 \saa\dot\cup \Sigma^{h}_2 \saa\dot\cup \Sigma^{h}_{12}$ includes all events that can be hidden after the synchronization. Using the notations $G^{\Sigma^h}$ for hiding the events in $\Sigma^h$, $G^{\mathcal A^{\Sigma^h}}$ for abstraction including hiding, and the equivalence $G^{\Sigma^h}\!\smm\sim G^{\mathcal A^{\Sigma^h}}$, it is also shown in Section \[obs:gen\] that an abstraction of $\mathcal O(G_1 \synch G_2)^{\Sigma^h} $, including the local observer generation in , can be incrementally generated as
$$\label{abs_eq_old}
\mathcal O(G_1 \synch G_2)^{\Sigma^h} \!
\sim \big (\mathcal O(G_1)^{\mathcal A^{\Sigma^h_1}} \smm \synch \mathcal O(G_2)^{\mathcal A^{\Sigma^h_2}} \big )^{\mathcal A^{\Sigma^h_{12}}}.$$
Repeating this incremental abstraction procedure when more subsystems are included still implies that only observers of individual subsystems $ \mathcal O(G_i)$ are required. Furthermore, the repeated abstraction means that often systems with a moderate state space are synchronized, especially when a number of local events are obtained after each synchronization.
Since only includes one type of state label ($N$), it can also be expressed in terms of marked and non-marked states. Therefore, the problem can also be identified as a non-blocking problem, and more efficient abstractions (coarser state partitioning) than visible bisimulation can be used. This is further described in Section \[obs:gen\].
When no explicit set of hidden events is included in the abstraction operator $\mc A$, the default set of events to be hidden is assumed to be all local observable events. Assuming that this set is $\Sigma^h$ for transition system $G$, it means that $\mc O(G)^{\mathcal A^{\Sigma^h}}$ is often simplified to $\mc O(G)^{\mathcal A}$, where we also note that the observer is generated before the abstraction is performed.
Incremental observer generation with shared unobservable events {#subsec:abs}
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For systems also including shared unobservable events, such events can not be replaced by $\veps$ due to the synchronization with other subsystems. This means that a complete observer can not be computed by composing local observers as in before subsystems have been synchronized such that no shared unobservable events remain. On the other hand it is shown in Section \[mixed\] that observers can also be computed incrementally, such that shared unobservable events have to be retained, while transitions with local unobservable events can be removed in a partial observer generation.
To clarify this partial observer generation, the more detailed observer operator $\mathcal O_{\Sigma^\veps}(G)$ is introduced, where the subscript $\Sigma^{\veps}$ includes the set of local unobservable events that are replaced by $\veps$ before the observer generation. Similar to the sets of hidden events in , the sets of local unobservable events in $G_1$ and $ G_2$ are $\Sigma^{\veps}_1$ and $\Sigma^{\veps}_2$, respectively, and the events in the set $\Sigma^{\veps}_{12}$ are the shared unobservable events in $G_1$ and $G_2$ that become local after the synchronization $G_1\synch G_2$, see also . Thus, the set $\Sigma^{\veps}= \Sigma^{\veps}_1 \saa\dot\cup \Sigma^{\veps}_2 \saa\dot\cup \Sigma^{\veps}_{12}$ includes all unobservable events that can be replaced by $\veps$ when the observer is generated after the synchronization. In Section \[mixed\] it is shown that an observer alternatively can be generated incrementally as
$$\label{obs_eq_s}
\mathcal O_{\Sigma^{\veps}}(G_1\synch G_2) = \mathcal O_{\Sigma^{\veps}_{12}}\big ( \mathcal O_{\Sigma^\veps_1}(G_1)\synch \mathcal O_{\Sigma^\veps_2}(G_2)\big ),$$
where the shared unobservable events in $\Sigma^{\veps}_{12}$ are preserved until they become local. Also observe the special case with no shared unobservable event ($\Sigma^{\veps}_{12}=\varnothing$), where (\[obs\_eq\_s\]) simplifies to (\[obs\_eq\]). Furthermore, the observer generation, combined with the incremental abstraction, results in the equivalence
$$\label{abs_eq}
\mathcal O_{\Sigma^{\veps}}(G_1 \synch G_2)^{\Sigma^h} \!\sim \mathcal O_{\Sigma^{\veps}_{12}}\big (\mathcal O_{\Sigma^{\veps}_1}(G_1)^{\mathcal A^{\Sigma^h_1}} \smm\synch \mathcal O_{\Sigma^{\veps}_2}(G_2)^{\mathcal A^{\Sigma^h_2}} \big )^{\mathcal A^{\Sigma^h_{12}}}.$$
Note that the observer generation is always performed before corresponding abstraction. Observable and unobservable events are here incrementally replaced by $\tau$ and $\veps$, respectively, when they become local. The mix between step-wise abstraction and partial observer generation means that some events are replaced by $\veps$ first after one or more abstractions. To be able to construct correct partial observers, this implies that some restrictions must be included in the incremental abstractions. This is solved in Section \[mixed\] by introducing additional temporary state labels (other than labels for non-safe states).
When no explicit set of unobservable events is included in the observer operator $\mc O$, the default set is assumed to be all local unobservable events. Assuming that this set is $\Sigma^\veps$ for transition system $G$, it implies that ${\mc O}_{\Sigma^\veps}(G)$ is often simplified to $\mc O(G)$, where the observer is generated after the events in $\Sigma^\veps$ have been replaced by $\veps$.
\[ex\_sig\] This example illustrates the incremental replacement of local events by $\veps$ or $\tau$ in . The events $a$, $b$, $c$ and $d$ are observable, while the events $u$ and $v$ are unobservable. shows that the events $a$, $d$ and $v$ are shared. To generate the local observers $\mathcal O(G_i)$, $i=1,\dots,3$, local unobservable events are replaced by $\veps$, and $\Sigma^{\veps}_1=\{u\}$, $\Sigma^{\veps}_2=\varnothing$, and $\Sigma^{\veps}_3=\varnothing$. Although event $v$ is unobservable, it is shared between $G_1$ and $G_3$ and is not replaced by $\veps$ at this level. However, it becomes local after the synchronization $G_1 \synch G_3$, which means $\Sigma^{\veps}_{13}=\{v\}$. Moreover, $\Sigma^{\veps}_{12}=\varnothing$ and $\Sigma^{\veps}_{23}=\varnothing$. In the hiding process of local observable events before abstraction, the sets of hidden events are $\Sigma^{h}_1=\{b\}$, $\Sigma^{h}_2=\{c\}$, $\Sigma^{h}_3=\varnothing$, $\Sigma^{h}_{12}=\{a\}$, $\Sigma^{h}_{13}=\varnothing$, and $\Sigma^{h}_{23}=\{d\}$.
Opacity and privacy {#ssecprop}
-------------------
The two security and privacy properties that are studied in this work are current-state opacity (CSO) and current-state anonymity (CSA). It is assumed that an intruder knows the model of the system and has access to the observable events. Thus, an intruder can generate an observer of the system, and security and privacy violation can be formulated as the existence of non-safe states in this observer.
In CSO verification, the states of the observer that exclusively include secret states are called *non-safe* states. By definition, a system is *current-state opaque*, if there is no non-safe state in the observer. On the other hand, a system is *current-state anonymous*, if there is no singleton state in the observer. The singleton states are considered as non-safe states in CSA verification. In Section \[op\_an\], both opacity and anonymity notions for modular systems are described.
Moreover, for the synthesis of current-state opaque/anonymous systems, that is limited to systems including only local unobservable events, uncontrollable events are introduced such that an efficient supervision equivalence abstraction can be used to find the supervisor.
Efficient generation of observers {#obs:gen}
=================================
Since the computation of an observer has exponential complexity [@cl:int:2008], it is shown in this section how the incremental abstraction in can be used to significantly lower the computational complexity. All unobservable events are in this section assumed to be local and can therefore immediately be replaced by $\veps$. Based on this assumption, it is shown how local observers can be directly generated before the incremental abstraction is applied.
Incremental observer abstraction for modular systems {#obs:gen:sub}
----------------------------------------------------
For a nondeterministic transition system $G$, where unobservable (local) events have been replaced by $\veps$, a deterministic transition system with the same language as $\mc L(G)$, called an *observer* $\mathcal O (G)$, is generated by subset construction [@hmu:int:2001], where $\mathcal O (G) = \langle \what X,\Sigma,\what T,\what I,AP,\what \lambda \rangle$, and $\what X=\{Y\in 2^X\st$ $ (\exists s\in \mc L(G)) \,Y=\delta(I,s)\}$, $\what T= \{ Y\trans{a} Y' \st $ , and . The relation between $\what \lambda (Y)$ and $\lambda(x)$ is application dependent, see Section \[op\_an\], but the default assumption is that $\what \lambda (Y)=\bigcup_{x\in Y}\lambda(x)$. An obvious alternative is $\what \lambda (Y)=\bigcap_{x\in Y}\lambda(x)$, an interpretation that is applied in CSO.
Introduce the transition function $\what \delta(Y,a)\define\delta(Y,a)$ and the extended transition function, inductively defined as $\what \delta(\what I,sa)=\what\delta(\what\delta(\what I,s),a)$ with the base case $\what \delta(\what I,\veps)$ $=\what I$. It is then easily shown that $\what \delta(\what I,s)=\delta(I,s)$, see [@hmu:int:2001]. This means that $\mc L(\mathcal O(G))=\mc L(G)$.
For a modular system with partial observation and no shared unobservable events, the monolithic observer can be computed by first generating local observers for each subsystem before they are synchronized. This is possible, since the same monolithic observer is obtained when synchronization is made before and after observer generation. This was shown for automata by [@fabre:2012] and [@Pola_2017]. A minor extension to transition systems is presented in the following lemma. The first automata related part of the proof is included due to its simplicity compared to earlier formulations.
\[tobs\] [Let $G_i=\langle X_i,\Sigma_i,T_i,I_i,AP_i, \lambda_i \rangle$, $i=1,2$, be two nondeterministic transition systems with no shared unobservable events, where the alphabet $\Sigma_i$ only includes observable events. Then, the observer for the synchronized system]{} $${\mathcal O}{(G_1\synch G_2)} = \mathcal O(G_1) \synch \mathcal O(G_2).$$
[Proof:]{}
Consider the language of the synchronized system $\mc L(G_1\synch G_2)$ and the projection $P_i:(\Sigma_1\cup \Sigma_2)^* \ra \Sigma^*_i$ for $i=1,2$. After a string $s\in\mc L(G_1\synch G_2)$ has been executed, the set of reachable states can be expressed as $Y_1\times Y_2$, where $
Y_i=\{x\st (\exists x_0\in I_i)\,x_0\stackrel{P_i(s)\yspa{0.8ex}}{\Longrightarrow} x\}, \;i=1,2.
$ Assume that there are transitions $x_i\transd{a}x'_i$ in $G_i$ for $i=1,2$, where $a\in \Sigma_1\cap \Sigma_2$, $x_i\in Y_i$, and $x'_i\in Y'_i$. Then there is a corresponding transition $(x_1,x_2)\transd{a}(x'_1,x'_2)$ in $G_1\synch G_2$. Thus, subset construction of $G_1\synch G_2$ generates the transition $Y_1\times Y_2\trans{a}Y'_1\times Y'_2$. Since $Y_i$ and $Y'_i$ are also states in $\mathcal O (G_i)$, the corresponding transition in $\mathcal O (G_1) \synch \mathcal O (G_2)$ is $(Y_1, Y_2)\trans{a}(Y'_1,Y'_2)$.
With similar arguments for $a\in \Sigma_1\setminus \Sigma_2$ and $a\in \Sigma_2\setminus \Sigma_1$, we find that for a given string $s\in\mc L(\mathcal O {(G_1 \synch G_2))}= \mc L(\mathcal O (G_1) \synch \mathcal O (G_2))$, the reachable states included in the block states of $\mathcal O {(G_1 \synch G_2)}$ and $\mathcal O (G_1) \synch \mathcal O (G_2)$ are the same. Indeed, the bijective function $
f:2^{X_1\times X_2}\rightarrow 2^{X_1}\times 2^{X_2}, \txt{0.8}{where} f(Y_1\times Y_2)=(Y_1,Y_2)
$ for $Y_i\in 2^{X_i}, \;i=1,2$, shows that the states in the two transition systems are isomorphic. The states and transitions are therefore structurally equal.
In , the union of the state labels is taken in the synchronization. Together with the default assumption on union of state labels in observer block states, the state label of the synchronized block state $(Y_1,Y_2)=f(Y_1\times Y_2)$ becomes $\bigcup_{x_1\in Y_1}\lambda(x_1)\,\cup\, \bigcup_{x_2\in Y_2}\lambda(x_2)$. The alternative interpretation for CSO, where union is replaced by intersection in the observer generation, gives $\bigcap_{x_1\in Y_1}\lambda(x_1)\,\cup\, \bigcap_{x_2\in Y_2}\lambda(x_2)=\bigcap_{x_1\in Y_1}$ $\bigcap_{x_2\in Y_2}\big(\lambda(x_1)\,\cup\, \lambda(x_2)\big )$. The second formulation corresponds to synchronization before observer generation. The interpretation for CSA is shown in .
#### Online estimation
This lemma also has implications on online estimation of a modular system. Clearly, online estimation can be implemented by running local observers combined with online synchronization. Alternatively, one can simply maintain local sets of consistent estimates, which get synchronised when necessary (the latter approach avoids building and storing the local observers ahead of time, by essentially exploring only the observer states that are visited due to the particular sequence of observations that is seen). For either approach, the lemma results in a dramatic simplification on the complexity of online estimation.
In the following proposition, abstraction is added to the result of Lemma \[tobs\]. The proposition is valid for any abstraction that is congruent with respect to (wrt) synchronization and hiding. The basic idea behind this incremental abstraction can be traced back to [@Malik04] and [@flordal09].
\[prop:obs\] [Let $G_1$ and $G_2$ be two nondeterministic transition systems with no shared unobservable events but hidden observable events in the set $\Sigma^h\define\Sigma^{h}_1 \saa\dot\cup \Sigma^{h}_2 \saa\dot\cup \Sigma^{h}_{12}$, where $\Sigma^h_i$ includes local events in $G_i$, $i=1,2$, and $\Sigma^{h}_{12}$ includes shared events in $G_1$ and $G_2$. For an arbitrary abstraction equivalence $G^{\Sigma^h}\!\smm\sim G^{\mathcal A^{\Sigma^h}}$ that is congruent wrt synchronization and hiding, the abstraction of the following observer can be incrementally generated as]{} $$\mathcal O(G_1 \synch G_2)^{\Sigma^h} \!
\sim \big (\mathcal O(G_1)^{\mathcal A^{\Sigma^h_1}} \smm \synch \mathcal O(G_2)^{\mathcal A^{\Sigma^h_2}} \big )^{\mathcal A^{\Sigma^h_{12}}}.$$
[Proof:]{} [Combining Lemma \[tobs\] with hiding of the local observable events in $G_1$ and $G_2$, we find that $\mathcal O(G_1 \synch G_2)^{\Sigma_1^h\saa\dot\cup \Sigma^{h}_2} =
\big(\mathcal O(G_1) \synch \mathcal O(G_2)\big)^{\Sigma_1^h\saa\dot\cup \Sigma^{h}_2}=
\mathcal O(G_1)^{\Sigma^h_1} \synch \mathcal O(G_2)^{\Sigma^h_2}.$ For an arbitrary equivalence $G\sim H$, congruence wrt synchronization means that $G\synch R \sim H\synch R$. Thus, $
\mathcal O(G_1 \synch G_2)^{\Sigma_1^h\saa\dot\cup \Sigma^{h}_2} \sim
\mathcal O(G_1)^{\mathcal A^{\Sigma^h_1}} \smm \synch \mathcal O(G_2)^{\Sigma^h_2} \sim
\mathcal O(G_1)^{\mathcal A^{\Sigma^h_1}} \smm \synch \mathcal O(G_2)^{\mathcal A^{\Sigma^h_2}}.
$ Now, also hiding the shared events in $\Sigma^h_{12}$, combined with congruence wrt hiding ($G\sim H$ implies $G^{\Sigma^h} \sim H^{\Sigma^h}$) and one more abstraction, we finally obtain $
\mathcal O(G_1 \synch G_2)^{\Sigma_1^h\saa\dot\cup \Sigma^{h}_2\saa\dot\cup \Sigma^{h}_{12}} \sim
\big (\mathcal O(G_1)^{\mathcal A^{\Sigma^h_1}} \smm \synch \mathcal O(G_2)^{\mathcal A^{\Sigma^h_2}} \big )^{\mathcal A^{\Sigma^h_{12}}}.
$ ]{}
Incremental observer algorithm {#handling}
------------------------------
Based on , an incremental observer generation including abstraction is presented in Algorithm 1 for modular systems without any shared unobservable events.
#### Heuristics
In the selection of the sets $\Omega_1$ and $\Omega_2$ and corresponding transition systems $G_{\Omega_1}$ and $G_{\Omega_2}$, to be abstracted in Algorithm 1, a natural approach is to first select a group of transition systems with few transitions. Among them, the two systems with the highest proportion of local events are chosen to be abstracted. In this way, a significant reduction of states and transitions is achieved by the abstractions, and the intermediate system after the synchronization $G_\Omega:=G_{\Omega_1}^{\mathcal A} \synch G_{\Omega_2}^{\mathcal A}$ also becomes smaller.
Algorithm 1, including these heuristics, is a minor adaption of a method suggested by [@flordal09] for incremental verification. They call it compositional verification, and the focus is on nonblocking and controllability properties, while the formulation here is adapted to incremental observer generation and specific observer properties based on transition systems. The main reason why this algorithm is presented here is that the nontrivial extension in , on observer abstraction for modular systems with shared unobservable events, can be computed in the same way. The difference is mainly that an additional observer operation is added on line 8.
Transformation from forbidden state to nonblocking verification {#s:extobs}
---------------------------------------------------------------
The security related verification and synthesis problems considered in this paper are all related to identification of specific non-safe observer state properties, see Section \[ssecprop\]. In CSO, observer states that exclusively include secret states from the original system are non-safe, and in CSA, singleton observer states are considered as non-safe states. Non-safe states in an observer may formally be considered as [*forbidden states*]{}, and the verification as a forbidden state problem. This verification problem can be solved by introducing the state label $N$ for the non-safe states, and then use visible bisimulation as abstraction in Algorithm 1.
#### Extended local observers
Since the problem only includes two types of states, safe and non-safe, an alternative to generic state labels and visible bisimulation is to transform the forbidden state problem to a [*nonblocking problem*]{}. All forbidden (non-safe) states in each individual observer $\mathcal O(G_i)$ are then augmented with a self-loop. For CSO these self-loops are labeled by $w_i$, $i = 1,\dots,n$, and the resulting local observers are called $\mathcal O_{w_i}(G_i)$. For each such observer, a two-state [*detector automaton*]{} $G_i^d$, shown in , is then introduced. It includes a marked state with a self-loop on the set of observable events $\Sigma_i$ in $G_i$ and a transition via the event $w_i$ to a non-marked state. The [*extended local observer*]{} $$\mathcal O_e(G_i) = \mathcal O_{w_i}(G_i)\synch G_i^d$$ then obtains non-marked blocking states added to every occurrence of a $w_i$ self-loop in $\mathcal O_{w_i}(G_i)$. Thus, every forbidden state in $\mathcal O(G_i)$ results in a direct transition to a blocking state in the extended local observer, while all original states in $\mathcal O_{w_i}(G_i)$ become marked in $\mathcal O_e(G_i)$. The reason is that no state in $\mathcal O_{w_i}(G_i)$ is explicitly marked, meaning that every state is implicitly considered to be marked in the synchronization. If any blocking states remain in the total extended observer O\_e(G)=O\_e(G\_1)O\_e(G\_2)O\_e(G\_n), this observer is blocking, and the observer $\mathcal O(G)$ includes one or more non-safe states from a CSO point of view.
In the case of CSA, the transformation is simplified by choosing the same self-loop label $w$ for all observers $\mathcal O(G_i)$, $i = 1,\dots,n$, and the same $w$ label in every detector automaton $G_i^d$. This means that a blocking state may be reached first when all local observers have reached a non-safe state, a fact that is further motivated in .
#### Abstraction preserving nonblocking
Conflict equivalence, introduced by [@Malik04], preserves the nonblocking property of a transition system. This means that a system is nonblocking if and only if its conflict equivalence abstraction is also nonblocking. This abstraction, denoted $\mc A_c$, generally generates more efficient reductions compared to the visible bisimulation abstraction, here denoted $\mc A_v$. The reason is that only the nonblocking property is preserved by $\mc A_c$, while visible bisimulation, including divergence sensitivity, preserves temporal logics similar to [@bl_18].
By introducing the extended observer $\mc O_e$ as observer operator in Algorithm 1, an incremental observer based on the abstraction $\mc A_c$ is efficiently computed. This is possible, since conflict equivalence is congruent wrt hiding and synchronization [@Malik04]. The DES software tool Supremica [@olj:aff:sup:2006] includes an incremental conflict equivalence implementation based on [@flordal09]. As an alternative, Algorithm 1 can also be implemented based on the visible bisimulation abstraction $\mc A_v$ and the original local observers including the non-safe state label $N$. Note that this abstraction is also congruent wrt hiding and synchronization [@bl:des:2019].
The following example illustrates the transformation of a CSO verification problem to a nonblocking problem. Furthermore, the efficiency of the conflict equivalence and the visible bisimulation abstractions is demonstrated.
\[ex\_obs\_reduc\] Consider the subsystem $G_i$, $i\in\mathbb{N}^+_n$ in , where $v_i$ is a local unobservable event and therefore replaced by $\veps$ before observer generation. The events $a_i$ and $c_i$ are local observable and the events $b_i$ and $b_{i+1}$ are observable but shared between neighbor subsystems, except the local events $b_1$ and $b_{n+1}$. The local observer $\mathcal O(G_i)$ is also shown in .
The transition system $G_i$ is assumed to have one secret state, state $2$. Thus, the observer state $2$ is non-safe from a CSO point of view. This non-safe state with state label $N$ in $\mathcal O(G_i)$ is a forbidden state to which a $w_i$ self-loop is added in $\mathcal O_{w_i}(G_i)$ in . Including the detector automaton $G_i^d$ as depicted in , gives the extended local observer $\mathcal O_e(G_i)=\mathcal O_{w_i}(G_i) \synch G_i^d$ where the $w_i$ self-loop is replaced by a $w_i$ transition to a blocking state, also shown in .
In the complexity of the incremental extended observer $\mathcal O_e(G)^{\mathcal A_c}$, including abstraction based on conflict equivalence, is compared with the extended observer without abstraction $\mathcal O_e(G) \sa= \,\saa\parallel_{i\in\mathbb{N}^+_n}\smm \mathcal O_e(G_i)$ for different number of subsystems $n$. The result shows the strength of including the incremental abstraction, where the number of states $|X|$ and transitions $|T|$ including abstraction is constant independent of $n$, due to the specific structure of the problem.
Somewhat surprisingly, the incremental visible bisimulation abstraction $\mc A_v$ gives an even larger reduction down to only $2$ states and $2$ transitions, independent of $n$. This is shown in [@mona_18]. The reason why the conflict equivalence abstraction $\mc A_c$ does not achieve such an extreme reduction in this example is that the extended local observer $\mathcal O_e(G_i)$ includes an additional blocking state as a marker for the non-safe state. Thus, it is clear that for systems with special structures as the one in this example, visible bisimulation can be even more efficient than conflict equivalence.
Opacity and anonymity for modular systems {#op_an}
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So far we have shown how a specific type of states in an observer called non-safe states can be identified in an efficient way for modular systems. In this section a more detailed definition of these non-safe states is given. This is done for the security and privacy problems current state opacity and anonymity, focusing on modular structures.
A centralized architecture is considered, including one single intruder of the system. It is assumed that the intruder has full knowledge of the system structure. However, it can only observe a subset of the system events, included in the set of observable events. Based on its observations, the intruder is assumed to be able to construct an observer of the system, where only observable events are included as transition labels.
Current state opacity and anonymity
-----------------------------------
In current-state opacity (CSO) [@Saboori:2014; @lesage16], the goal is to evaluate if it is possible to estimate any secret states in a system based on its observable events. For a transition system $G$, let $X_S\subseteq X$ be the set of secret states. This system is then said to be [*opaque*]{} if for every string of observable events $s\in \mc L(G)$, each corresponding state set $Y=\delta(I,s)$ that includes secret states also includes at least one non-secret state from the set $X\smm\setminus \smm X_S$. Note that the natural projection on observable events that often is used in opacity definitions, see for instance [@lesage16], is not involved in this section, since the strings $s\in\mc L(G)$ only include observable events. The unobservable events are replaced by $\veps$.
\[d:cso\]
Since the block states $Y=\delta(I,s)$ in this CSO definition are states in the corresponding observer $\mc O(G)$, the following proposition follows immediately.
\[p:cso\]
According to this proposition, a transition system $G$ is current-state non-opaque if and only if at least one state in the observer $\mathcal O(G)$ includes only secret states from $G$, and is therefore [*non-safe*]{}. The label $N$ is a state label for all non-safe states in $\mc O(G)$.
#### Current state anonymity
With the increasing popularity of location-based services for mobile devices, privacy concerns about the unwanted revelation of user’s current location are raised. For this reason the notion of CSO is adapted, and a new related notion called *current state anonymity* (CSA) is introduced [@Wu_2014]. CSA captures the observer’s inability to know for sure the current locations of moving patterns.
\[d:csa\]
In the same way as for CSO, the block states $Y=\delta(I,s)$ in this CSA definition are states in the corresponding observer $\mc O(G)$, which directly implies the following proposition.
\[p:csa\]
According to this proposition, a transition system $G$ is current-state non-anonymous, if and only if at least one block state $Y$ in the observer $\mathcal O(G)$ is a singleton and is therefore [*non-safe*]{}. Obviously, anonymity is evaluated by verifying that no observer block state is a singleton state. This is natural, since more than one system state in each observer block state implies an uncertainty in determining the exact location of a moving pattern. Finally, in the same way as for CSO, the label $N$ is a state label for all non-safe (non-anonymous) block states in $G$ and corresponding states in $\mc O(G)$.
The following example shows the observers for CSO and CSA, including their different $N$ (non-safe) state label interpretations.
\[ex\_abs\_oa\] Consider the transition system $G$ in where the secret state set $X_S=\{0,1,2\}$. The event $u$ is unobservable and is therefore replaced by $\veps$ before the observer generation, where the source and target states of the $\veps$ transition are merged. Although the observers are structurally equal, depending on the verification problem, the interpretation differs concerning the non-safe states and therefore the state labeling. The block state $\{1,2\}$ in the observer $\mathcal O_{\tiny\mbox{CSO}}(G)$ has label $N$, because both states $1$ and $2$ are secret states. On the other hand, the corresponding state in $\mathcal O_{\tiny\mbox{CSA}}(G)$ does not have state label $N$, as it is not a singleton state.
Since $G$ does not include any subsystems, all events can be considered as local, since no synchronization between local subsystems is performed. The observable events $a$ and $b$ are therefore hidden by relabeling them with $\tau$. In the visible bisimulation (VB) abstraction $\mathcal O_{\tiny\mbox{CSO}}(G)^{\mathcal A^{\{a,b\}}}$, the $N$-labeled states with a $\tau$ transition in between are merged, while no reduction is achieved for $\mathcal O_{\tiny\mbox{CSA}}(G)^{\mathcal A^{\{a,b\}}}$. We notice that only states with the same state label (label $N$ or no label) are merged in the VB abstraction.
To summarize this subsection, a block state $Y$ in the observer $\mathcal O(G)$ is *non-safe* and is augmented with state label $N$, in CSO verification when $Y\subseteq X_S$, and in CSA verification when $|Y|=1$. These results are now generalized to modular systems.
Current state opacity and anonymity for modular systems
-------------------------------------------------------
For a modular transition system $G\sa= \,\saa\parallel_{i\in\mathbb{N}^+_n}\smm G_i$, CSO requires a modified definition of secret states. No shared unobservable events also means that both CSO and CSA can be expressed in terms of safety of the local block states $Y_i$ in $G_i$.
Before CSO is defined for modular systems, consider an $n$-dimensional cross product $Y=Y_1\times Y_2 \times \cdots \times Y_n$, where the $i$-th set $Y_i$ is replaced by the set $Z_i$. This modified cross product is denoted R(Y,Z\_i)=Y\_1Y\_[i-1]{} Z\_i Y\_[i+1]{} Y\_n
\[d:csoo\] [Consider a modular transition system $G\sa= \,\saa\parallel_{i\in\mathbb{N}^+_n}\smm G_i$, where $X_{S_i}$ is the set of secret states for subsystem $G_i$. The set of secret states for $G$ with state space $X=X_1\times\cdots\times X_n$ is then defined as $$X_S=\bigcup_{i=1}^n R(X,X_{S_i})$$ For a string of observable events $s\in \mc L(G)$, the block state $Y=\delta(I,s)$ of the modular system $G$ is [*safe*]{} if $Y\nsubseteq X_S$, and $G$ is [*current state opaque*]{} if for all strings $s\in \mc L(G)$, all corresponding block states $Y=\delta(I,s)$ are safe. ]{}
The only difference between this CSO definition and is the more complex structure of the set of secret states $X_S$, which is illustrated in the following example.
\[e:cso\_mod\] Consider the subsystems $G_1$ and $G_2$ in , where the events $a$ and $b$ are observable, and $u$ and $v$ are unobservable. Since $v$ is shared, local observers can not be generated. Thus, $G_1$ and $G_2$ are first synchronized before the unobservable events are replaced by $\veps$, and the observer $\mc O(G_1\synch G_2)$ is generated where the three block states are $Y=\{(0,0)\}$, $Y'=\{(1,1),(1,2)\}$, and $Y''=\{(2,3),(3,4)\}$. The local event $u$ in $G_2$ gives the two states in $Y'$, and the shared event $v$ gives the two states in $Y''$.
The set of secret states $X_S=X_{S_1}\times X_{2} \cup X_{1}\times X_{S_2}=\{1,2\}\times\{0,1,2,3,4\}\cup \{0,1,2,3,4\}\times \{4\}$ implies that $Y'\subseteq X_S$ and $Y''\subseteq X_S$, while $Y\nsubseteq X_S$. Thus, $Y'$ and $Y''$ are non-safe states, which means that the $G_1\synch G_2$ is current state non-opaque. Due to the union in the definition of secret states in , it is enough that $G_1$ is in the local secret state $1$ to make $Y'$ a non-safe state in the composed system. The state $Y''$ is non-safe due to a more complex behavior, where first $G_1$ is in the local secret state $2$ and $G_2$ is in the non-secret state $3$. After the shared unobservable event $v$ has been executed, the opposite occurs where $G_2$ is instead in a local secret state (state $4$), and $G_1$ is in a non-secret state (state $3$).
Both in state $Y'$ and $Y''$ an intruder is able to detect that one of the subsystems is in a secret state. In the non-safe state $Y'$, the intruder knows that $G_1$ is in a secret state. A safe state implies that both a secret and a non-secret state can be occupied. Thus, there are one or more states in a safe block state where no subsystem is in a secret state. In $Y''$ one of the subsystems is in a secret state, but the shared unobservable event $v$ does not make it possible to determine if it is $G_1$ or $G_2$, only that one of them is in its secret state.
In the next example, it is also shown how the CSO definition in can be simplified for modular systems where all unobservable events are local.
\[e:cso\_modd\] Consider the subsystems $G_1$ and $G_2$ in , where the shared unobservable event $v$ in $G_2$ is replaced by the local unobservable event $w$. No shared unobservable events give the local observers in . Synchronization of these observers results in the three states $(Y_1,Y_2)\define (\{0\},\{0\})$, $(Y'_1,Y'_2)\define (\{1\},\{1,2\})$, and $(Y''_1,Y''_2)\define (\{2,3\},\{3,4\})$ in $\mc O(G_1)\synch \mc O(G_2)$.
Taking the union of the state labels in the synchronization according to , the second state $(Y'_1,Y'_2)$ becomes non-safe since the second state $Y'_1=\{1\}$ in $G_1$ is non-safe, while the rest of the states are safe. This result coincides with , which in the same way as in shows that replacing the shared unobservable event $v$ with the local event $w$ in $G_2$ means that only the second state $Y'$ in $\mc O(G_1\synch G_2)$ is non-safe.
This example illustrates that the existence of non-safe block states for modular systems with only local unobservable events can be decided based on the existence of local non-safe block states. The following proposition confirms this statement.
\[p:cso\_mod\]
[Proof:]{}
($\IMP$) Assume by contradiction that $(\forall i\in \mathbb{N}^+_n) \,Y_i\nsubseteq X_{S_i}$, which means that $(\forall i\in \mathbb{N}^+_n) \,Y_i\cap X_i\setm X_{S_i} \neq \varnothing$. By the notation , this assumption yields $Y\cap X\setminus X_S$ $=Y\cap X\setminus \bigcup_{i=1}^n R(X,X_{S_i}) =Y \cap\, \bigcap_{i=1}^n R(X,X_i\setminus X_{S_i}) =\bigcap_{i=1}^n R(Y,Y_i\cap X_i\setminus X_{S_i}) \neq \varnothing$, and we conclude that $Y\nsubseteq X_S$, which is a contradiction.
($\Leftarrow$) Since $(\forall j\in \mathbb{N}^+_n) \,Y_j\subseteq X_j$, it follows that $Y=R(Y,Y_i)\subseteq R(X,Y_i)$. Furthermore, assuming that $(\exists i\in \mathbb{N}^+_n) \,Y_i\subseteq X_{S_i}$, and selecting $i$ such that $Y_i\subseteq X_{S_i}$, it implies that $Y\subseteq R(X,Y_i) \subseteq R(X,X_{S_i}) \subseteq \bigcup_{i=1}^n R(X,X_{S_i}) = X_S$.
A block state $Y=Y_1\times \cdots \times Y_n$ in $G=\parallel_{i\in\mathbb{N}^+_n}\smm G_i$ is also a state in the observer $\mc O(\parallel_{i\in\mathbb{N}^+_n}\smm G_i)$, and according to the block state $Y_i$ is then also a state in the corresponding local observer $\mc O(G_i)$. Thus, a modular system $G$ without shared unobservable events is CSO if there is no non-safe state in any local observer. The system is still opaque if the states in the global observer that include non-safe components are not reachable.
The interpretation of CSO from an intruder perspective was discussed in . For a system to be in a non-safe state $Y$, it is obviously enough, according to this example and , that one of the subsystems is in a secret state for every state $x\in Y$. Consider, for instance, that each subsystem models one moving pattern, say a person. It is then not necessary that all persons are in local secret states to get a global non-safe state. It can be enough with one person, depending on the availability of observable events. In another scenario, where only one person is involved, the person being in a secret state in one subsystem means that the local states of the other subsystems then represent the absence of the single person. This modeling scenario is applied in the multiple floor/elevator building in .
Generally, the modular system model in is very flexible. The definition of secret states is often natural, as we have tried to motivate above, but alternative definitions are possible, for instance by replacing the union operator with the intersection operator, such that $X_S=\bigcap_{i=1}^n R(X,X_{S_i})=X_{S_1}\times \cdots \times X_{S_n}$. A similar type of intersection is applied when current state anonymity is analyzed for modular systems.
#### Current state anonymity for modular systems
In the following definition, all subsystems must be in a singleton state for the whole system to break the location privacy for moving patterns.
\[d:csaa\] [Consider a modular system $G\sa= \,\saa\parallel_{i\in\mathbb{N}^+_n}\smm G_i$. For a string of observable events $s\in \mc L(G)$, the global block state $Y=\delta(I,s)$ is [*non-safe*]{} if $Y=x\in X$, $Y$ is a tuple of singleton states. Furthermore, $G$ is [*current state anonymous*]{} if for all strings $s\in \mc L(G)$, all corresponding global block states $Y=\delta(I,s)$ are safe, no global state is a tuple of singleton states. ]{}
According to this definition, a global block state is only non-safe when all local states are singleton states. The motivation for this interpretation is that the synchronized subsystems together are assumed to model a map. Location privacy is then violated if it is possible to get a specific location on such a map. This corresponds to a global singleton state, involving singleton states for all subsystems.
For a string of observable events $s\in \mc L(G)$, the global block state $Y=\delta(I,s)$ in $G\sa= \,\saa\parallel_{i\in\mathbb{N}^+_n}\smm G_i$ is also a state in the observer $\mc O(G)$. Furthermore, for a modular system without shared unobservable events, implies that $\mc O(G)=\parallel_{i\in\mathbb{N}^+_n}\smm \mc O(G_i)$. The state $Y$ can then also be expressed as $Y=Y_1\times \cdots \times Y_n$, where $Y_i$ is a state in the corresponding local observer $\mc O(G_i)$.
One clear difference between CSO and CSA is, however, the handling of state labels in the synchronization of the local observers $\parallel_{i\in\mathbb{N}^+_n}\smm \mc O(G_i)$. In the case of CSA, it is then necessary to take the [*intersection*]{} of the actual non-safe state labels $N$ from the individual subsystems, to generate a correct global state label according to . Thus, in current state anonymity for modular systems, the union of non-safe state labels $N$ in the synchronous composition of observers is replaced by the intersection of these state labels from the individual subsystems. The following example illustrates the differences between CSA and CSO.
First, consider the observer $\mc O(G)$ in . The structure of this observer is the same as for CSA, but the non-safe states are different. Concerning CSA, the first state $Y$ is non-safe, since it is a singleton state, while the states $Y'$ and $Y''$ are safe.
In no shared unobservable event is involved, and therefore the observer can be generated as $\mc O(G_1)\synch \mc O(G_2)$. In the local observers the non-safe states are assigned to the singleton states. After synchronization, taking the intersection of the $N$ state labels, only the first state in $\mc O(G_1)\synch \mc O(G_2)$ is non-safe. This is confirmed by noting that it is also the only singleton state in $\mc O(G_1)\synch \mc O(G_2)$.
Transformation of current state opacity and anonymity to nonblocking problems {#subsec:anon_nonblock}
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In it was shown how CSO and CSA verification can be transformed to nonblocking problems, when all unobservable events are local. More specifically, a system $G$ is CSO if the observer $\mathcal O_e(G)$ in is nonblocking. For CSA, the same self-loop label $w$ in all observers $\mathcal O_e(G_i)$, $i = 1,\dots,n$ means that a blocking state is only reached when all local observers have reached a non-safe singleton state. This models the intersection of all non-safe state labels $N$ in the synchronization of the individual subsystems, that is required to reach a global non-safe state in the case of CSA.
Other types of opacity {#subsec:LBO}
----------------------
Opacity can also be defined based on languages, see [@dubreil08], [@Badouel:2007], and [@Lin_2011]. For a system $G$ with a set of initial states $I$ and a language $\mc L(G,I)$, two sublanguages are introduced, a secret language $L_S\subseteq \mc L(G,I)$ and a non-secret language $L_{NS} \subseteq \mc L(G,I)$, where $L_{S}\cap L_{NS}=\varnothing$. Unobservable events are here not replaced by $\veps$. Instead, a projection $P$ from all events to the observable events is introduced. The system $G$ is then *language-based opaque* if $L_S\subseteq P^{-1}[P(L_{NS})]$.
To verify language-based opacity (LBO), this formulation can be transformed to CSO as in [@wu2013], and then verified based on the techniques proposed in this paper. This includes a modular formulation of the transformation from LBO to CSO. Furthermore, two notions of initial state opacity (ISO) and initial/final state opacity (IFO), as presented in [@wu2013], can also be transformed to LBO and then to a CSO problem.
Observer abstraction for systems with shared unobservable events {#mixed}
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For modular systems with partial observation and shared unobservable events, the observer generation including incremental abstraction presented in and Algorithm 1 must be reformulated. The problem is that the complete observer can not be computed by only synchronizing the local observers as in . The reason is that shared unobservable events can not be replaced by $\veps$ before they have become local after synchronization.
Incremental observer generation
-------------------------------
To highlight this complication, the more expressive observer operator $\mc O_{\Sigma^\veps}(G)$ is used, where we remind that the subscript $\Sigma^{\veps}$ includes the set of local unobservable events that are replaced by $\veps$ before the observer generation. First, the following lemma shows that an observer can also be generated incrementally. An observer $\mc O_{\Sigma^\veps_1}(G)$ is then computed, assuming that the events in $\Sigma^\veps_1$ are local and unobservable. When additional local events in $\Sigma^{\veps}_{2}$ are considered, an update of this observer as $\mc O_{\Sigma^\veps_2}(\mc O_{\Sigma^\veps_1}(G))$ is shown to give the same result as generating the observer $\mc O_{\Sigma^\veps_1 \dot\cup \Sigma^\veps_2}(G)$ in one step.
\[lemma:obs\] [Consider a nondeterministic transition system $G$ with a state set $X$, an initial state set $I$, a set of observable events $\Sigma^o$, and a set of unobservable events $\Sigma^{uo}$. Let $\Sigma^\veps_1\subseteq \Sigma^{uo}$, $\Sigma^\veps_2\subseteq \Sigma^{uo}$, and $\Sigma^\veps= \Sigma^\veps_1\,\dot\cup \,\Sigma^\veps_2$ be sets of unobservable events that are replaced by $\veps$ before corresponding observer generation. Then ]{} $$\mc O_{\Sigma^{\veps}}(G) = \mc O_{\Sigma^{\veps}_2}(\mc O_{\Sigma^{\veps}_1}(G)).$$
[Proof:]{}
For the total event set $\Sigma=\Sigma^o\cup\Sigma^{uo}$, consider the language $\mc L(G) \subseteq \Sigma^*$ and the projections $P_1:\Sigma^* \ra (\Sigma \setminus\Sigma^\veps_1)^*$, $P_2:(\Sigma \setminus \!\Sigma^\veps_1)^* \ra (\Sigma \setminus\!(\Sigma^\veps_1\dot\cup \Sigma^\veps_2))^*$, and $P:\Sigma^* \ra (\Sigma \setminus\!(\Sigma^\veps_1\dot\cup \Sigma^\veps_2))^*$. After a string $s\in\mc L(G)$ has been executed, the block state in the observer $\mc O_{\Sigma^\veps_1} (G)$ can be expressed as $Y_{1} = \{x\in X\st (\exists x_0\in I) $$x_0\stackrel{P_{1}(s)}{\Longrightarrow} x\}$. The corresponding string $t=P_1(s)$, executed by the observer $\mc O_{\Sigma^{\veps}_2}(\mc O_{\Sigma^{\veps}_1}(G))$, results in the block state $Y_{2} = \{Y_{1}\in 2^X\st \what I_{1}\stackrel{P_2(t)\yspa{0.1ex}}{\Longrightarrow} Y_{1} \}$, where $\what I_1$ is the initial state of $\mc O_{\Sigma^\veps_1} (G)$. Moreover, the projection $P(s)$ generates the block state $Y = \{x\in X\st (\exists x_0\in I)\,x_0\stackrel{P(s)}{\Longrightarrow} x\}$ in the observer $\mc O_{\Sigma^\veps} (G)$. The bijective function $f:2^{2^X}\ra 2^X$, where $f(Y_{2})=\bigcup_{Y_{1}\in Y_{2}} Y_{1}$, together with the fact that $P_2(t)=P_2(P_1(s))=P(s)$, finally means that $Y=f(Y_2)$ for any string $s\in\mc L(G)$. Hence, the states in the observers $\mc O_{\Sigma^{\veps}_2}(\mc O_{\Sigma^{\veps}_1}(G))$ and $\mc O_{\Sigma^{\veps}}(G)$ are isomorphic, and the observers are therefore structurally equal.
The equality also includes the state labels. First consider the default assumption on union of state labels in observer block states, and the block state relation $Y=f(Y_2)=\bigcup_{Y_1\in Y_2} Y_1$. Then the state label of a block state $Y_2$ in $\mc O_{\Sigma^{\veps}_2}(\mc O_{\Sigma^{\veps}_1}(G))$ is $
\bigcup_{Y_1\in Y_2} \bigcup_{x\in Y_1} \lambda(x)=\bigcup_{x\in\left(\bigcup_{Y_1\in Y_2} Y_1\right)} \lambda(x)= \bigcup_{x\in Y} \lambda(x),
$ which is the state label of the corresponding block state $Y=f(Y_2)$ in $\mc O_{\Sigma^{\veps}}(G)$. Thus, the state labels for the two observers coincide. The alternative interpretation for CSO, where union is replaced by intersection in the observer generation, gives the same result, since then $
\bigcap_{Y_1\in Y_2} \bigcap_{x\in Y_1} \lambda(x)=\bigcap_{x\in\left(\bigcup_{Y_1\in Y_2} Y_1\right)} \lambda(x)= \bigcap_{x\in Y} \lambda(x).
$ For the CSA interpretation we refer to .
Combing this lemma with for $G=G_1\synch G_2$, assuming that $\Sigma^{\veps}= \Sigma^{\veps}_1\, \dot\cup \, \Sigma^{\veps}_2 \, \dot\cup \, \Sigma^{\veps}_{12}$, where $\Sigma^{\veps}_{12}$ includes the shared unobservable events in $G_1$ and $G_2$ that become local after the synchronization $G_1\synch G_2$, we find that O\_[\^]{}(G\_1 G\_2) = O\_[\^\_[12]{}]{}( O\_[\^\_1 \^\_2]{}(G\_1G\_2))= O\_[\^\_[12]{}]{}(O\_[\^\_1]{}(G\_1)O\_[\^\_2]{}(G\_2)) Based on this result it is obvious that, in the case of unobservable shared events, the equality $\mc O_{\Sigma^\veps}(G_1 \synch G_2) = \mc O_{\Sigma^\veps_1}(G_1)\synch \mc O_{\Sigma^\veps_2}(G_2)$ does not always apply. This fact was also recently highlighted by an example in [@Masopust_2018].
Combined incremental observer generation and abstraction
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The challenge is now to combine the incremental observer generation in with the incremental abstraction in Algorithm 1, here based on visible bisimulation since conflict equivalence is not applicable. If some unobservable events are shared and therefore cannot immediately be replaced by $\veps$, while some observable events are local and can be replaced by $\tau$ and then abstracted, the question is if it is possible to perform abstraction before observer generation. The next example illustrates that this is not always possible. Before this example, two important remarks are given. \[(i)\]
Initial local observers are always assumed to be generated before any hiding and abstraction. This means that every transition system $G$ in this section is by default an observer, although not explicitly expressed to simplify the notation. Thus, $G$ is assumed to be deterministic (except in the final ), and any non-safe states are labeled by $N$.
Hiding and abstraction are always performed on deterministic systems. Hence, alternative choices, including $\tau$ events after hiding, are interpreted as deterministic choices in observer generation. Restrictions will also be included such that repeated observer generation and abstraction (presented later in this section) still means that $G$ can be regarded as a deterministic transition system, although it may include alternative choices involving $\tau$ events.
\[ex:abs\_obs\] Consider the deterministic transition system $G$ in , where the events $a$ and $b$ are observable, while $u$ is unobservable, and the first three states have label $N$. In the observer $\mc O_{\{u\}}(G)^{\mc A^{\{a,b\}}}$, the unobservable event is first replaced by $\veps$ and the CSO observer is generated followed by visible bisimulation abstraction, while the observable events are first hidden and abstracted in $\mc O_{\{u\}}(G^{\mc A^{\{a,b\}}})$, followed by the CSO observer generation.
State labels are preserved by visible bisimulation abstraction. This implies that abstracted block states with label $N$ only include states that before the abstraction were also labeled by $N$. Thus, observer generation before abstraction results in the correct solution, and the model $\mc O_{\{u\}}(G)^{\mc A^{\{a,b\}}}$ in shows that the states $0$ and $1$ are non-safe.
Making the abstraction before the observer generation generates in this example a different and therefore incorrect result. The reason is that the states are then merged in wrong blocks in $G^{\mc A^{\{a,b\}}}$, where state $2$ is incorrectly joined with the states $1$ and $0$. In the observer generation, where the non-safe block state $\{0,1,2\}$ is merged with the safe state $3$, the result is that the two states $0$ and $1$ incorrectly become safe.
#### Avoiding abstractions that influence observer generation
Shared unobservable events mean that we need to repeat the observer generation when subsystems have been synchronized, since additional local unobservable events result in more $\veps$ transitions. At the same time, observable events must be abstracted when they become local, to avoid state space explosion. Thus, it is necessary to switch between abstraction and observer generation when subsystems are synchronized. According to this is not possible without introducing some abstraction restrictions.
To avoid that any abstraction of a transition system influences later observer generations, additional state labels are added, assuming that some unobservable events have not yet been replaced by $\veps$. Unique state labels are then added to $\Sigma^{uo}$ source and target states.
\[d:uoST\] For a deterministic transition system $G$, with a state set $X$ and a set $\Sigma^{uo}$ of unobservable events, the states in the set $$X_{st}^{\Sigma^{uo}} = \{ x,x'\in X \st (\exists u\in {\Sigma_{uo}})\,x\trans{u}x'\}$$ are called $\Sigma^{uo}$ [*source and target states*]{} (STSs).
Adding unique state labels for all $\Sigma^{uo}$ STSs means that visible bisimulation abstraction can be performed before observer generation. This is now illustrated for the transition system in .
\[ex:uoST\] The same transition system $G$ as in Example \[ex:abs\_obs\] is considered, where unique state labels $\lambda^u_s$ and $\lambda^u_t$ are added to the source and target states of the transition $2\trans{u}3$. These $\Sigma^{uo}$ STS labels prevent the visible bisimulation abstraction $G^{\mc A^{\{a,b\}}}$ to merge state $2$ with the states $0$ and $1$, since they have now different state labels. The succeeding CSO observer generation $\mc O_{\{u\}}(G)^{\mc A^{\{a,b\}}}$ merges the source and target states of the transition $2\trans{u}3$, and the obsolete $\Sigma^{uo}$ STS labels are removed. The resulting observer $\mc O_{\{u\}}(G)^{\mc A^{\{a,b\}}}$ in now coincides with the correct observer $\mc O_{\{u\}}(G)^{\mc A^{\{a,b\}}}$ in .
#### Future nondeterministic choices
Generally, an observer transforms a nondeterministic system to a language equivalent deterministic system. Based on Algorithm 1, the first operation generates an observer for every subsystem, which implies that any multiple initial states are replaced by one initial state in every initial observer, before any abstraction is made. Furthermore, nondeterministic choices are removed in the initial observer generation. However, additional nondeterministic choices may appear after additional $\veps$ transitions have been introduced. The following example illustrates this phenomenon and resulting complications.
\[x:nondet\] Consider the deterministic transition system $G$ in , where the events $a$, $b$ and $c$ are observable, while $u$ is unobservable. Additional $\Sigma^{uo}$ STS labels $\lambda^u_s$ and $\lambda^u_t$ are therefore also introduced in $G$. Making an abstraction before the observer generation implies that the event $a$ can not be replaced by $\tau$, since a future nondeterministic choice occurs when $u$ is replaced by $\veps$. Thus, consider the abstraction $G^{\mc A^{\{b,c\}}}$ in where $b$ and $c$ have been replaced by $\tau$, and the block state $\{3,4\}$ has been generated. Generating the CSO observer $\mc O_{\{u\}}(G^{\mc A^{\{b,c\}}})$, followed by one more abstraction $\mc O_{\{u\}}(G^{\mc A^{\{b,c\}}})^{\mc A^{\{a\}}}$, verifies that this observer coincides with the observer $\mc O_{\{u\}}(G)^{\mc A^{\{a,b,c\}}}$ that is always correct, since no abstraction is made before the observer generation.
This example indicates that abstraction can also be included before future nondeterministic choices have been removed by observer generation. Unfortunately, additional complications sometimes show up, in this example when event $c$ is replaced by event $b$ in $G$. Starting with visible bisimulation abstraction means that $G^{\mc A^{\{b\}}}$ and $\mc O_{\{u\}}(G^{\mc A^{\{b\}}})^{\mc A^{\{a\}}}$ coincide with $G^{\mc A^{\{b,c\}}}$ and $\mc O_{\{u\}}(G^{\mc A^{\{b,c\}}})^{\mc A^{\{a\}}}$, respectively, in . Thus, the observer where abstraction is involved before observer generation incorrectly includes a non-safe state, see , while the correct observer $\mc O_{\{u\}}(G)^{\mc A^{\{a,b\}}}$ in has no non-safe state.
This example illustrates that even if an event label is preserved, which in the future will result in a nondeterministic choice, in this example event $a$, incorrect results may occur. Because correct results are often achieved, special rules can be established, but it is hard to define exactly when such rules give correct results.
Our conclusion is therefore that abstractions of subsystems should not be performed before future nondeterministic choices have been removed by synchronization of subsystems and observer generation. Thus, it is necessary to exactly define a future nondeterministic choice.
\[d:FNC\] For a deterministic transition system $G$ with a state set $X$, a set $\Sigma^o$ of observable events, and a set $\Sigma^{uo}$ of unobservable events, consider the set of transition relation sets T\_[nc]{}{(x,a,Y’)X\^o\^[uo]{} 2\^X\
&& (x’,x”Y’)(s\_u,s\_v(\^[uo]{}{a})\^\*) xx’xx”x’x”}. The tuple $(x,a,x')$, where $x\in Y'$ and $(x,a,Y')\in T_{nc}$, is called a [*future nondeterministic choice*]{} (FNC) transition.
First, we observe that the complexity of finding FNC transitions is in worst case $O(|X||T|)$. However, the strings $s_u$ and $s_v$ are normally short, which means that worst case complexity is irrelevant. More importantly, the evaluation of FNC transitions is made on the original submodules $G_i$ in , which are expected to be relatively small.
The exclusion of FNC transitions is only a minor restriction. First, note that unobservable identical events, not yet replaced by $\veps$, but generating FNC transitions, can without loss of generality be renamed to avoid such transitions that later nevertheless will be replaced by $\veps$ transitions. Observe that nondeterministic choice is the main reason for the well known worst case exponential state space complexity in observer generation. Thus, it is indeed recommended to accomplish this renaming, which of course also influences other submodels, including the same shared unobservable events.
For observable events, the expectation is that any nondeterministic choice is modeled explicitly, without adding unobservable events before a nondeterministic choice (as event $u$ in ). Explicit nondeterminism is then removed in the initial observer generation. Note that in this special case, where $s_u$ and $s_v$ are empty strings for $(x,a,Y')\in T_{nc}$, this relation can also be expressed by the function $Y'=\delta(x,a)$.
To summarize, the assumption from now on is that all nondeterminism, except possible future $\veps$ transitions, is taken care of in the initial observer generation.
#### Incremental observer generation and abstraction
Based on the definitions of $\Sigma^{uo}$ STSs and FNC transitions, we are now ready to prove . This lemma states that a deterministic system including observer generation is equivalent to a system where abstraction is performed before the observer generation. The abstraction is assumed to be based on visible bisimulation.
We remind again that hiding and abstraction are always assumed to be performed on deterministic systems. Hence, alternative choices including $\tau$ events are interpreted as deterministic choices in observer generation. Also note that CSO and CSA observer generation only considers the non-safe $N$ labels, not the unique $\Sigma_{uo}$ STS labels, which are only included to avoid abstractions that influence the resulting state labels in the observer generation, see . When states in $\veps$ transitions are merged in the observer generation, related unique STS labels are also removed, since they are then obsolete.
Due to the length of the following proof, an example is given after the proof where the relation between specific state sets and partitions is illustrated.
\[lemma:red\] [Consider a deterministic transition system $G= \langle X,\Sigma^o\cup\Sigma^{uo},T,I,AP,\lambda\rangle$ where $\Sigma^o$ is a set of observable events and $\Sigma^{uo}$ is a set of unobservable events. Let $\Sigma^h$ and $\Sigma^\veps$ be sets of local events that are replaced by $\tau$ and $\veps$, respectively. Moreover, assume that unique state labels for all $\Sigma^{uo}$ STSs are included in $G$, while no FNC transition exists in $G$. Then $$\mc O_{\Sigma^\veps}(G)^{\Sigma^h} \sim \mc O_{\Sigma^\veps}(G^{\mc A^{\Sigma^h}}),$$ where the abstraction is based on visible bisimulation.]{}
[Proof:]{}
The state set $X$ is naturally divided into $X^o$, the set of states with observable transitions to and/or from states in $X^o$, and the set of $\Sigma^{uo}$ STSs $X^{\Sigma_{uo}}$, $X= X^o{\mathbin{\dot{\cup}}} X^{\Sigma_{uo}}$. Note that border states in between observable and unobservable transitions belong to $X^{\Sigma_{uo}}$. The main point of this proof is to show the separation between the abstraction, mainly acting on $X^o$, and the observer generation that only affects $X^{\Sigma_{uo}}$.
Before the observer generation, the unobservable events in $\Sigma^\veps$ are replaced by $\veps$, and the transition relations in $T$ can then be divided into three parts: $T^{\Sigma^o}$ including observable transitions, $T^{\Sigma^\veps}$ only including $\veps$ transitions, and $T^{\Sigma^{uo}\setminus \Sigma^\veps}$ including remaining transitions with unobservable events, $T=T^{\Sigma^o}{\mathbin{\dot{\cup}}} T^{\Sigma^\veps}{\mathbin{\dot{\cup}}} $ $T^{\Sigma^{uo}\setminus\Sigma^\veps}$.
The observer generation $\mc O_{\Sigma^\veps}(\cdot)$ only affects the states in $X^{\Sigma_{uo}}$. The reason is that the transition function $G$ is deterministic before the introduction of $\veps$ transitions, and it does not include any FNC transitions. It means that for any choice of $\Sigma^\veps$, the transition relations in $T^{\Sigma^o}{\mathbin{\dot{\cup}}} T^{\Sigma^{uo}\setminus \Sigma^\veps}$ are all deterministic. More specifically, introducing the set X\^[\_[uo]{}]{}\_[O\_[\^]{}]{}{Y2\^X (xX) (a \^o\^[uo]{}\^)(x’X\^[\_[uo]{}]{})\
&&xx’ Y=R\_(x’) } the observer state set can then be expressed as $X_{\mc O_{\Sigma^\veps}}=X^o{\mathbin{\dot{\cup}}} X^{\Sigma_{uo}}_{\mc O_{\Sigma^\veps}}$. Since the states in $X^{\Sigma_{uo}}_{\mc O_{\Sigma^\veps}}$ are block states, the states in $X^o$ are now also considered as singleton block states. The fact that the observer generation only affects the states in $X^{\Sigma_{uo}}$ is also valid for the observer state label updates for CSO and CSA. It is however critical that both the source and target states of $\veps$ transitions are included in $X^{\Sigma_{uo}}$, see .
The structure of the observer state set $X_{\mc O_{\Sigma^\veps}}=X^o{\mathbin{\dot{\cup}}} X^{\Sigma_{uo}}_{\mc O_{\Sigma^\veps}}$ also implies that hiding of events for observable transitions in $T^{\Sigma^o}$ can be performed either before or after the observer generation. The only difference between $T^{\Sigma^o}$ and the observer based version $T_{\mc O_{\Sigma^\veps}}^{\Sigma^o}$ is that any target states $x'\in X^{\Sigma_{uo}}$ for transitions in $T^{\Sigma^o}$ are replaced by block target states $Y\in X^{\Sigma_{uo}}_{\mc O_{\Sigma^\veps}}$ for transitions in $T_{\mc O_{\Sigma^\veps}}^{\Sigma^o}$. This does not influence the hiding mechanism that only changes the observable events in $\Sigma^h$ to $\tau$. Thus, $\mc O_{\Sigma^\veps}(G)^{\Sigma^h}=\mc O_{\Sigma^\veps}(G^{\Sigma^h})$.
Now, consider the abstraction part, where the visible bisimulation partition $\Pi$ for $G^{\Sigma^h}$ is divided in the same way as the state set $X=X^o{\mathbin{\dot{\cup}}} X^{\Sigma_{uo}}$ such that $\Pi\define \Pi^o {\mathbin{\dot{\cup}}} \Pi^{\Sigma^{uo}}$. Since all states in $X^{\Sigma^{uo}}$ have unique state labels, corresponding block states in $\Pi^{\Sigma^{uo}}$ are singletons. This means that the abstraction only acts on the partition $\Pi^o$, resulting in possible non-singleton block states. Also note that the block states in $\Pi$ are the states of $G^{\mc A^{\Sigma^h}}$.
The states $X_{\mc O_{\Sigma^\veps}}=X^o{\mathbin{\dot{\cup}}} X^{\Sigma_{uo}}_{\mc O_{\Sigma^\veps}}$ of the observer $\mc O_{\Sigma^\veps}(G^{\Sigma^h})$ only change the states in the unobservable part from $X^{\Sigma_{uo}}$ to $X^{\Sigma_{uo}}_{\mc O_{\Sigma^\veps}}$. Assume first that the unique STS labels still remain when $\veps$ transitions are merged in the observer generation. Every block state in $X^{\Sigma_{uo}}_{\mc O_{\Sigma^\veps}}$ then has a unique state label. This implies that the visible bisimulation partition after the observer generation can be expressed as $\Pi_{\mc O_{\Sigma^\veps}} = \Pi^o{\mathbin{\dot{\cup}}} X^{\Sigma_{uo}}_{\mc O_{\Sigma^\veps}}$, where once again the abstraction only acts on the partition $\Pi^o$. Thus, the abstraction only influences $X^o$ and generates the same partition $\Pi^o$ both with and without observer generation. The abstraction is therefore completely separated from observer generation.
Finally, the removal of the unique but obsolete $\Sigma^\veps$ STS labels after the observer generation generally results in a coarser partition $\Pi_{\mc O} \succeq \Pi_{\mc O_{\Sigma^\veps}}$. In the search for the coarsest partition $\Pi_{\mc O}$ the question is then if the same visible bisimulation partition is achieved starting from the state space $X_{\mc O_{\Sigma^\veps}}=X^o{\mathbin{\dot{\cup}}} X^{\Sigma_{uo}}_{\mc O_{\Sigma^\veps}}$ of the observer $\mc O_{\Sigma^\veps}(G)^{\Sigma^h}=\mc O_{\Sigma^\veps}(G^{\Sigma^h})$ or the state space $ \Pi_{\mc O_{\Sigma^\veps}} = \Pi^o{\mathbin{\dot{\cup}}} X^{\Sigma_{uo}}_{\mc O_{\Sigma^\veps}}$ of the observer $\mc O_{\Sigma^\veps}(G^{\mc A^{\Sigma^h}})$? Since all individual states of a block in a visible bisimulation partition have the same set of event-target-blocks according to , a coarser partition just means that more states have the same set of event-target-blocks. Thus, it is no restriction to start with a state space where the states in each block state have the same set of event-target-blocks as in $\Pi_{\mc O_{\Sigma^\veps}}$. A coarser partition $\Pi_{\mc O}$ only implies that some block states in $\Pi_{\mc O_{\Sigma^\veps}}$ will be merged, more exactly those that generate the same set of event-target-blocks when the obsolete $\Sigma^\veps$ STS labels are removed, see further details in [@bl:des:2019] and .
To summarize, the visible bisimulation partition for $\mc O_{\Sigma^\veps}(G^{\mc A^{\Sigma^h}})$ is the same as for $\mc O_{\Sigma^\veps}(G)^{\Sigma^h}$ also for the coarser partition $\Pi_{\mc O}$, where the unique state labels for the $\Sigma^\veps$ STSs are removed, and therefore $\mc O_{\Sigma^\veps}(G)^{\Sigma^h} \sim \mc O_{\Sigma^\veps}(G^{\mc A^{\Sigma^h}})$.
\[ex:uoSTT\] Consider the deterministic transition system $G$ in (assumed to be an observer), where $\Sigma^o=\Sigma^h=\{a,b\}$, $\Sigma^{uo}=\{u,v,w\}$, and $\Sigma^\veps=\{u,v\}$. Based on the notations in , the involved state sets are $X^o=\{0,1,4,8,9\}$, $X^{\Sigma^{uo}}=\{2,3,5,6,7\}$, and $X^{\Sigma^{uo}}_{\mc O_{\Sigma^\veps}}=\{ \{2,3\},\{5,6\},\{7\} \}$. The involved visible bisimulation partitions are $\Pi^o=\{\{0,1\},\{4\},$ $\{8,9\}\},\Pi^{\Sigma^{uo}}=\{\{2\},\{3\},\{5\},\{6\},\{7\}\}$, and $\Pi^{\Sigma_{uo}}_{\mc O_{\Sigma^\veps}}=X^{\Sigma^{uo}}_{\mc O_{\Sigma^\veps}}$. The block states in $\Pi_{\mc O_{\Sigma^\veps}} = \Pi^o{\mathbin{\dot{\cup}}} \Pi^{\Sigma_{uo}}_{\mc O_{\Sigma^\veps}}=\{\{0,1\},\{2,3\},\{4\},$ $\{5,6\},\{7\},\{8,9\}\}$ are the states of $\mc O_{\Sigma_\veps}(G^{\mc A^{\Sigma^h}})$, see . In the observer, the obsolete $\Sigma^\veps$ STS labels have been removed. Taking this into account, the visible bisimulation partitions for $\mc O_{\Sigma^\veps}(G)^{\Sigma^h}$ and $\mc O_{\Sigma^\veps}(G^{\mc A^{\Sigma^h}})$ are equal, both being $\Pi_{\mc O}=\{\{0,1\},\{2,3,4\},\{5,6\},$ $\{7\},\{8,9\}\}$. This confirms , which says that these two observers are visible bisimulation equivalent.
Based on Lemmas \[tobs\]-\[lemma:red\] and , the following theorem shows how an abstracted observer for a modular system including shared unobservable events can be generated incrementally by combining observer generation and abstraction.
\[theorem\_imp\] [ Let $G_1$ and $G_2$ be two nondeterministic transition systems with hidden observable events in the set $\Sigma^h\define\Sigma^{h}_1 \saa\dot\cup \Sigma^{h}_2 \saa\dot\cup \Sigma^{h}_{12}$, where $\Sigma^h_i$ includes local events in $G_i$, $i=1,2$, and $\Sigma^{h}_{12}$ includes shared events in $G_1$ and $G_2$. Furthermore, let $\Sigma^{uo}$ be the set of unobservable events for $G_1\synch G_2$, where $\Sigma^\veps\define\Sigma^{\veps}_1 \saa\dot\cup \Sigma^{\veps}_2 \saa\dot\cup \Sigma^{\veps}_{12}\subseteq \Sigma^{uo}$ and the events in $\Sigma^{\veps}_i$ are local for $G_i$, $i=1,2$, while the events in $\Sigma^{\veps}_{12}$ are shared events in $G_1$ and $G_2$. Also, assume that unique state labels for $\Sigma^{uo}\setm (\Sigma^{\veps}_1{\mathbin{\dot{\cup}}} \Sigma^{\veps}_2)$ STSs are included in $G_i$, while no FNC transition exists in $G_i$ for $i=1,2$. Then, a visible bisimulation abstraction of the observer $\mc O_{\Sigma^{\veps}}(G_1 \synch G_2)^{\Sigma^h}$ can be incrementally generated as]{} $$\mc O_{\Sigma^{\veps}}(G_1 \synch G_2)^{\Sigma^h} \sim \mc O_{\Sigma^{\veps}_{12}}( \mc O_{\Sigma^{\veps}_1}(G_1)^{\mc A^{\Sigma^h_1}} \synch \mc O_{\Sigma^{\veps}_2}(G_2)^{\mc A^{\Sigma^h_2}})^{\mc A^{\Sigma^h_{12}}}.$$
[Proof:]{} [Based on , , and $G=G_1\synch G_2$ in we find that $$\mc O_{\Sigma^{\veps}}(G_1 \synch G_2)^{\Sigma^h_1 {\mathbin{\dot{\cup}}} \Sigma^h_2} \sim
\mc O_{\Sigma^{\veps}_{12}}\big( (\mc O_{\Sigma^{\veps}_1}(G_1) \synch \mc O_{\Sigma^{\veps}_2}(G_2))^{\mc A^{\Sigma^h_1 {\mathbin{\dot{\cup}}} \Sigma^h_2}} \big).$$ Together with and hiding of the events in $\Sigma^h_{12}$ on the left side, as well as an additional abstraction $\mc A^{\Sigma^h_{12}}$ on the right side, this gives $$\mc O_{\Sigma^{\veps}}(G_1 \synch G_2)^{\Sigma^h_1 {\mathbin{\dot{\cup}}} \Sigma^h_2 {\mathbin{\dot{\cup}}} \Sigma^h_{12}} \sim
\mc O_{\Sigma^{\veps}_{12}}\big( \mc O_{\Sigma^{\veps}_1}(G_1)^{\mc A^{\Sigma^h_1}} \synch \mc O_{\Sigma^{\veps}_2}(G_2)^{\mc A^{\Sigma^h_2}}\big)^{\mc A^{\Sigma^h_{12}}},$$ which proves the theorem. ]{}
The first local observer generations $\mc O_{\Sigma^{\veps}_i}(G_i)$, $i=1,2$, remove multiple initial states, non-deterministic choices and local unobservable events. The achieved observers guarantee that the following local hiding and abstraction is performed on deterministic systems. The critical point investigated in , including some minor restrictions, is that local abstractions can be accomplished before shared unobservable events are removed by the observer generation $\mc O_{\Sigma^{\veps}_{12}}$. This is followed by an additional abstraction ${\mc A^{\Sigma^h_{12}}}$ of the shared observable events that are only involved in $G_1\synch G_2$.
This procedure is repeated in the same way as in Algorithm 1, which means that the state space explosion can be reduced significantly by the incremental abstraction, also for modular systems with shared unobservable events.
#### Algorithm
An algorithm for incremental observer generation and abstraction in the presence of shared unobservable events, presented in for two subsystems, is generalized in the same way as in Algorithm 1. The only differences are that on line 2 unique state labels for source and target states of shared unobservable transitions are introduced after the observer generation, and line 8 is replaced with $$G_\Omega:=\mc O ((G_{\Omega_1})^{\mc A} \synch (G_{\Omega_2})^{\mc A}).$$ In all abstractions, new local observable events are hidden, and in all observer generations, new local unobservable events are replaced by $\veps$. After the observer generation on line 8, all obsolete state labels are also removed, as mentioned before .
Finally, note that no observer generation is necessary when there are no shared unobservable events in $G_{\Omega_1}$ and $G_{\Omega_2}$. A complement in the heuristics on selection of the sets $\Omega_1$ and $\Omega_2$ is therefore to also focus on subsystems that have shared unobservable events as early as possible. In this way, extra observer generations can be significantly reduced.
Opacity verification of a multiple floor/elevator building {#sec:ex}
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In order to demonstrate the practical use of the modular and incremental verification procedure, a CSO problem is formulated based on an $n$-story building with $m$ elevators on each floor. The model is inspired by an analytical and monolithic building example in [@dubreil10].
First, for better understanding of the problem, explicit results for the special case $n=2$ and $m=2$ are presented in the following example.
Transition systems for a two-story building with floor models $F^1$ and $F^2$ and elevator models $E^1$ and $E^2$ are shown in . Each floor consists of two corridors and two elevators. Elevator entrances are located in states $2$ and $4$ in $F^i$ and $E^j$ for $i,j=1,2$. There are card readers in the corridors and elevators, which are represented by the events $c^i_j$ and $e^i_j$, respectively. The subscript $j$ indicates the corridor and elevator and the superscript $i$ corresponding floor. The events $u_j$ and $d_j$ indicate the upward and downward movement of the $j$-th elevator, respectively. The shared elevator event $e^i_j$ coordinates floor $i$ with elevator $j$, and the alternative choices with equal events, $c^i_1$ in state $2$ and $c^i_2$ in state $4$, plus arbitrary initial states ($I^{F^i}=X^{F^i}$ and $I^{E^j}=X^{E^j}$) generate the non-determinism of the system.
The observer of the $i$-th floor $\mc O(F^i)$ is also shown in , where opacity depends on the choice of secret states. Assume that there are only secret states in the second floor model $F^2$, which means that there can only be non-safe states in the observer $\mc O(F^2)$. The opacity of the total system $F_1\synch F_2\synch E_1\synch E_2\synch$ is therefore determined by the state labels of $\mc O(F^2)$. If the secret state is $X^{F^2}_S=\{1\}$ or $X^{F^2}_S=\{3\}$, the system is opaque, since all block states in $\mc O(F^2)$ then include non-secret states, meaning that all states are safe. On the other hand, $X^{F^2}_S=\{1,3\}$ results in a non-opaque system, since the state $\{1,3\}$ in $\mc O(F^2)$ then becomes non-safe.
#### Building with $n$ floors and $m$ elevators
Consider an $n$-story building with $m$ elevators on each floor. The transition system models of floors and elevators are depicted in . There are $n$ floor models $F^i$, $i\in\mathbb{N}^+_n$ and $m$ elevator models $E^j$, $j\in\mathbb{N}^+_m$. Each floor consists of corridors, rooms and elevators. Elevator entrances are located in states $2,4,\dots,2m-2,2m$ in $F^i$ and states $2,4,\dots,2n-2,2n$ in $E^j$. Corridors are connected through doors that open using card readers. The card readers are installed at the entrances of the elevators. Passing through the entrances of corridors and elevators are shown by events $c$ and $e$, respectively. Note that $c^i_j$ ($e^i_j$) indicates the $j$-th corridor (elevator) on the $i$-th floor. The events $u_j$ and $d_j$ represent the upward and downward movement of the $j$-th elevator, respectively.
The shared elevator event $e^i_j$ coordinates floor $i$ with elevator $j$, and in state $2j$ on floor $i$ an alternative choice with equal event, $c^i_j$, as well as arbitrary initial states generate the system non-determinism. The staff moving patterns can be tracked by observing the records of their ID cards that are read by the card readers. All floors have similar structure, but have different secret states which are places in the building that have a storage for secret documents.
#### Two scenarios
One member of the staff wants to place a secret document in one of the secret locations in the building. There is an intruder that knows the structure of the system and has access to the records of card readers. The question is then if the intruder can have the knowledge that the secret document is in that specific location. In this case, an opaque system means that even a very careless staff with no specific strategy can place the secret document in any of the secret places, without being concerned of the secret being revealed. Most opacity examples have only one strategy for reaching to an opaque solution, while in this example, no specific strategy is needed as long as the system is opaque.
As an alternative scenario, consider the case when some card readers do not work due to power failure. Since corresponding doors in that case cannot be opened, related transitions are then removed.
#### Results
The CSO verification results for the building in , with different number of floors and elevators, are presented in for non-opaque systems. Results for the alternative scenario, where some doors can not be opened due to power failure, are shown in . Restrictions are then introduced such that all systems become opaque.
In the first column of both tables, the pair $(n,m)$ shows the number of floors and elevators. The second column in includes the set of secret states on each floor. The set of secret states in are the same as in , but omitted due to shortage of space. The column in both tables that is indicated with a $\star$ sign, shows the floors with local non-safe states, and in also the corridors where card readers do not work.
The number of states $|\what X|$ and transitions $|\what T|$, as well as the elapsed time for the verification $t_e$, are then presented for observers with abstraction $\mathcal O(G)^{\mathcal A}$ and without abstraction $\mathcal O(G)$. The verification is based on the transformation to a nonblocking problem, including conflict equivalence abstraction, as presented in .
The results in Tables \[tab:comp:trans\] and \[tab:comp:opaq\] clearly demonstrate the strength of including the incremental abstraction. The number of states in for three floors and four elevators is more than 1000 times larger when abstraction is not included, and no solution is obtained without abstraction for the larger systems. We also observe that for the non-opaque case in the computation does not continue when one of the non-safe states has been found, which makes it much faster than the verification of opaque systems, where the whole reachable state space (although abstracted) needs to be evaluated.
Opacity and anonymity enforcement
=================================
For a transition system $G$, with non-safe block states defined for CSO in and CSA in , it will now be shown how opacity/anonymity can be enforced by a supervisor $S$. Only safe block states will then be reachable in the controlled (closed loop) system. Two assumptions that simplify the computation of the supervisor are introduced:
(i) All unobservable events are assumed to be uncontrollable. Since only observable events are then controllable and therefore can be disabled by the supervisor, it can be generated by the observer $\mc O(G)$ that is relevant for the actual security problem. The assumption is the same as in [@saboori:2012], but a simpler CSO formulation is presented here.
(ii) All unobservable events are assumed to be local. This means that the observer $\mc O(G)$ of a modular system can be obtained by synchronizing local observers , and efficient abstractions can be used in the supervisor synthesis.
The input to the supervisor $S$ from the transition system $G$ only includes observable events, and $S$ restricts the behavior of the controlled system by disabling some controllable events [@rw:con:1989; @kg:mod:1995]. Since all controllable events here are also assumed to be observable, the information to and from the supervisor only involves observable events. Thus, the nondeterministic transition system $G$ is perfectly represented by its observer $\mc O(G)$ in the synthesis and implementation of the supervisor $S$. This means that the closed loop system, where $S$ is also represented as an automaton, can be modeled as $\mc O(G)\synch S$.
Observer based supervisor generation
------------------------------------
The non-safe block states in $G$ that must be avoided by the supervisor are determined by the observer $\mc O(G)$, according to Props. \[p:cso\] and \[p:csa\]. These states are now called [*forbidden states*]{} and included in the set $\what X_f$. Since no other state labels are considered, the observer is simplified from an arbitrary transition system to the automaton $\mathcal O (G) = \langle \what X,\Sigma,\what T,\what I \rangle$. Furthermore, a restricted observer automaton is introduced $$\mc O(G)_{\setminus \what X_f}=\langle\what X\setm \what X_f , \Sigma, \what T_{\setminus \what X_f} ,\what I\setm \what X_f \rangle$$ where $\what T_{\setminus \what X_f} = \{(Y,a,Y') \smm\in \smm\what T \st Y, Y' \notin \what X_f\}$. Hence, $\mc O(G)_{\setminus \what X_f}$ is the automaton where all forbidden states are excluded.
If all events in $\mc O(G)$ are controllable, the automaton $S=\mc O(G)_{\setminus \what X_f}$ is the maximally permissive supervisor for the closed loop system $\mc O(G) \synch S$. When also some observable transitions are uncontrollable, these events can not be disabled by the supervisor. Not only the forbidden states must then be excluded in $S$, but also states from which there are uncontrollable transitions to forbidden states. The following proposition shows how the supervisor is then obtained by generating an extended set of forbidden states $\what X^e_f$. It is a minor modification and special case of the supervisor synthesis method presented in [@mf:csdes:2008].
\[pro:cce\]
[Proof (sketch):]{} [Since the supervisor $S=\mc O(G)_{\setminus \what X^e_f}$ is a subautomaton of $\mc O(G)$ and both are deterministic, it follows that the supervisor is a model of the closed loop system, $\mc O(G)\synch S=S$. In the supervisor, only those states are removed from which it is possible to reach a forbidden state by only executing uncontrollable transitions. Thus, the supervisor $S$ is maximally permissive. Furthermore, all transitions entering $\what X^e_{f}$ are controllable, and since these transitions are removed from $S$, only controllable events are disabled. Hence, the supervisor is controllable. ]{}
\[ex:synth\] Consider the transition system $G$ and its observer $\mc O(G)$ in . The secret states are $X_S=\{2,5\}$ and the uncontrollable event $\Sigma_u=\{d\}$ (denoted by exclamation mark). Both CSO and CSA result in the forbidden (non-safe) state set $\what X_f=\{\{2\},\{5\}\}$. Since the event $d$ is uncontrollable, the set of extended forbidden states in the observer $\what X^e_f=\{\{2\},\{3,4\},\{5\}\}$. Thus, the only remaining state in the supervisor $S=\mc O(G)_{\setminus \what X^e_f}$ is $\what X \setm \what X^e_f=\{\{0,1,2\}\}$, and the disabled events in this block state are $b$ and $c$, while the event $a$ is enabled.
Incremental supervisor generation by nonblocking preserving abstraction
-----------------------------------------------------------------------
In it was shown how a forbidden state verification problem can be transformed to a nonblocking problem. More specifically, an observer $\mc O(G)$ has forbidden states if and only if the extended observer $\mc O_e(G)$ in has blocking states. To formulate this as a supervisor synthesis problem, the added $w$ self-loops in the local observers at the forbidden states are considered to be uncontrollable. This means that the transitions to the blocking states in the extended observer are uncontrollable. The source states at these uncontrollable transitions (the originally forbidden states), as well as the blocking states, will then be excluded in the nonblocking synthesis.
Since the extended observer $\mc O_e(G)$ is modular, consisting of synchronized local extended observers, the same type of abstraction as in can be applied also before the nonblocking and controllable supervisor is computed. Instead of conflict equivalence that preserves nonblocking properties for verification purpose, a somewhat finer equivalence is required to be able to make synthesis on the abstracted observer. In [@flordal09; @sahar14; @sahar17], a supervisor synthesis equivalence is proposed by which an incremental abstraction is performed similar to Algorithm 1 in . This compositional synthesis, which is also implemented in the DES software tool Supremica [@olj:aff:sup:2006], is applied to the modular extended observer $\mc O_e(G)$ in , including the uncontrollable events mentioned above.
\[ex:building:synth\] Consider again the $2$-story building with two elevators and the floor observers depicted in . The set of secret states for the floors are $X^{F^1}_S=\{3\}$ and $X^{F^2}_S=\{1,2'\}$. As can be seen in , the floor observers have no state exclusively including $3$ , while there is a state in the second floor observer that exclusively includes state $2'$. This state is a CSO non-safe state, which makes the second floor non-opaque. To make the whole system opaque, a supervisor $S$ is generated that restricts the second floor observer from entering the non-safe state. The resulting supervisor is shown in , where the synchronous composition $\mc O(F^2)\synch S$ does not include the non-safe state $2'$.
shows $|\what X|$, $|\what T|$ and the elapsed time $(t^S_e)$ for generating both compositional and monolithic supervisors, for the non-opaque building examples explained in Section \[sec:ex\]. The model has the same structure with similar set of secret states $X_S$ on each floor. The resulting computation times for the larger examples are about 3-6 times faster using the incremental synthesis procedure.
Conclusions
===========
To tackle the exponential observer generation complexity for current-state opacity/ anonymity verification and enforcement, but also the complexity that arises when modular subsystems are synchronized, an incremental local observer abstraction is proposed. The two notions of current-state opacity and current-state anonymity are formulated based on state labels in transition systems that are naturally extended to modular systems. Non-safe states in corresponding local observers are then considered as forbidden states. By introducing simple detector automata, this problem is easily transformed to a nonblocking problem for which efficient existing abstraction methods are applied. This transformation to a nonblocking problem by detector automata is a generic technique with a great potential. A recent alternative example is incremental abstraction for verification of diagnosability [@mona_diag:2019].
The main theoretical development in this paper is the combined incremental abstraction and observer generation for systems with shared unobservable events. Due to the need for additional temporary state labels, the more general but less efficient visible bisimulation is then used as abstraction. An interesting alternative is to introduce arbitrary state labels in the more efficient abstractions that also have been used in this paper, but then only applied to transition systems without shared unobservable events. The efficiency of the proposed methods is demonstrated through a modular multiple floor/elevator building example. Both verification and supervisor synthesis to enforce a secure system are investigated. The results show a great improvement when abstraction is included especially for verification, while there is hope for some further improvements in the observer based supervisor synthesis.
[^1]: This work was partly carried out within the project SyTec – Systematic Testing of Cyber-Physical Systems, a Swedish Science Foundation grant for strong research environment.\
The support is gratefully acknowledged.
[^2]: Some of the results in this paper were presented in preliminary form at the 14th International Workshop on Discrete Event Systems (WODES), Sorrento Coast, Italy, May 2018 [@mona_18].
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'J. Babington'
- 'B. A. van Tiggelen'
date: 'Received: date / Revised version: date'
title: 'Magneto-Electric response functions for simple atomic systems'
---
[leer.eps]{} gsave 72 31 moveto 72 342 lineto 601 342 lineto 601 31 lineto 72 31 lineto showpage grestore
Introduction {#intro}
============
Magneto-electric effects are by now a well established phenomena both theoretically and experimentally. The tendency has been to focus on relatively large molecular systems where DFT calculations [@Rizzo:2003; @Rizzo:2009] apply and experimental values have been measured [@Rikken:2002]. In addition to being associated with important optical phenomena [@Rizzo:2009; @Graham:1983], their existence has played an important role in the Casimir physics [@Feigel:2004zz; @Rikken:2006; @PhysRevE.76.066605], in particular if it is possible in certain circumstances to find a contribution from the quantum vacuum to a bodies momentum. For these reasons we consider it an interesting question to ask what are the simplest models that describe possible and display magneto-electric effects.
In this article we calculate the magneto-electric response function for the two simplest bound state systems - the harmonic oscillator and the hydrogen atom. The principal difference between these two atomic systems is that the harmonic oscillator is strongly bound whereas the hydrogen atom with its Coulomb potential is weakly bound. This manifests itself in the accessibility of different energy eigenstates in the perturbation theory
This article is organised as follows. In Section \[sec:gf\] we define the atomic system and formulate the response for an arbitrary binding potential. As a tractable example that has a closed form, the harmonic oscillator is chosen and its magneto-electric response function calculated explicitly. In Section \[sec:hydrogen\] the response function for hydrogen is presented. It is then compared to the appropriate DFT result and its relation to experimental values. Finally, in Section \[sec:summary\] we summarise our results and provide some comment on their validity and applicability.
General formulation {#sec:gf}
===================
We will now derive the ME response function for a two-body charge neutral composite system in static external fields $(\mathbf{E}^0,\mathbf{B}^0)$ using the Hamiltonian formalism (the boldface used here is to indicate they are external fields). One can also use the path integral approach and the coupled classical Lorentz force equations to obtain information about the response function. The path integral gives a correct response function at high frequencies but it is not reliable at low frequencies as well as suffering from the wrong analytic structure in the complex frequency plane [@Wen:2004ym]. A consideration of the coupled Lorentz force equations produces similar difficulties and so we use the Hamiltonian method exclusively in this paper.
The system we consider is illustrated in Figure \[fig:atomicsystem\]. Two equal but opposite electrical charges with coordinates $(q^i_1,q^i_2)$ and masses $(m_1,m_2)$ interacting with a classical (c-number) gauge field $(\phi , A^i)$ that are the combined contribution of the static external fields and the fluctuating source that is used to probe the system. A binding potential $V(q_1 -q_2)$ holds the charges together giving a bound state that is overall charge neutral. The Hamiltonian that describes this system is given by $$\begin{aligned}
H&=&\frac{1}{2m_1}(p_1-eA(q_1))^2 +e \phi(q_1) \nonumber \\
&+&\frac{1}{2m_2}(p_2+eA(q_2))^2 -e \phi(q_2)
+ V(q_1 - q_2).\label{eq:hamiltonian1}\end{aligned}$$ To calculate the response function of this body due to a high frequency electromagnetic field, it is necessary to pass to a new set of coordinates that consists of both the centre of mass coordinate $X^i$ and the separation vector $x^i$. These are shown in Figure \[fig:atomicsystem\]. The new variables are then defined by $X := (m_1 q_1 + m_2 q_2)/M $, $x:= (q_1 - q_2)$, $M:= m_1+m_2$, $m:= m_1m_2/M$, and $m_{\Delta}=m_2-m_1$. Correspondingly, we change from the two particles momenta $(p_1,p_2)$ to $(P,p)$ where $P$ is the conjugate momenta of the centre of mass coordinate $X$ and $p$ is likewise the conjugate momenta to the separation vector $x$. It is also necessary to implement this change of coordinates on the gauge field; it can then be expanded about the centre of mass coordinate as $$\begin{aligned}
A(X+ (m_2/M)x )&=& A(X)+\left(\frac{m_2}{M}\right) x^i\nabla_i A(X) \nonumber \\
&+&\left(\frac{1}{2}\right)\left(\frac{m_2}{M}\right)^2x^ix^j\nabla_i\nabla_j A(X), \\
A(X- (m_1/M)x )&=& A(X)-\left( \frac{m_1}{M}\right)x^i\nabla_i A(X) \nonumber \\
&+&\left(\frac{1}{2}\right)\left(\frac{m_1}{M}\right)^2x^ix^j\nabla_i \nabla_j A(X) ,\end{aligned}$$ and similarly for the scalar potential $\phi$. We will work to first order in the spatial derivatives of the gauge potential and thereby neglect the last terms in the above expansion. This corresponds to electric and magnetic fields that can vary in time, but that are spatially constant (so the approach is restricted to wavevectors that are less than the inverse of the size of the atomic system).
Using the Lagrangian as an intermediate step in performing the change of coordinates, we recognize here that $E(t,X)=-\nabla \phi (t,X) - \partial_{t} A(t,X)$, whilst the derivative of the vector potential once projected with the Levi-Civita tensor will give the magnetic field. The Hamiltonian in the new coordinates after making this expansion reads $$\begin{aligned}
H &=& \frac{1}{2M}P^2+\frac{1}{2m}p^2+V (x) -ex^i\cdot E_i (t,X) \nonumber \\
&&-e\left(\frac{m_{\Delta}}{M^2}\right) P^i (x\cdot \nabla) A_i(t,X) \nonumber \\
&&-e\left(\frac{m_{\Delta}}{Mm}\right) p^i (x\cdot \nabla )A_i(t,X) \nonumber \\
&&+\frac{e^2}{2m}\left(\frac{m_{\Delta}}{M}\right)^2 x^ix^j\nabla_i A_k(t,X)\nabla_j A_k(t,X) \nonumber \\
&&+\mathcal{O}(\nabla^2 A). \label{eq:hamiltonian2} \end{aligned}$$ The ME activity results from a source magnetic field inducing electrical polarisation. We can now define the magneto-electric response of the bound state system by promoting all of the canonical degrees of freedom to operators. It is defined by $$\langle e\delta \hat{x}_i (t) \rangle := \int dt^{\prime}\chi_{ij} ^{EB}(t-t^{\prime})\delta B_j(t^{\prime},X),$$ where the lhs is the standard expectation value of the fluctuating electric dipole moment induced on the rhs by an externally applied fluctuating magnetic field (i.e. a test source) that is in general time and space dependent. In terms of correlation functions, it is given by the retarded two-point function $$\chi_{ij} ^{EB}(t-t^{\prime}) =-i\theta(t-t^{\prime})\langle \Omega \vert [e\delta \hat{x}_i (t),\delta \hat{O}_j(t^{\prime}) ] \vert \Omega \rangle ,$$ where the operator $\hat{O}_j(t^{\prime})$ couples to the fluctuating magnetic field $\delta B_j(t^{\prime},X)$, and is to be found from the microscopic theory given by the Hamiltonian Equation (\[eq:hamiltonian2\]). The ground state $\vert \Omega \rangle$ that we will use will be specified later in this section. Taking the Fourier transform of this one finds in frequency space $$\begin{aligned}
\chi_{ij} ^{EB}(\omega) &=&\langle \Omega \vert e\hat{x}_i \frac{1}{\hat{H}-E_0-\hbar \omega}\hat{O}_j \vert \Omega \rangle \nonumber \\
&&+\langle \Omega \vert e\hat{x}_i \frac{1}{\hat{H}-E_0+\hbar \omega}\hat{O}_j \vert \Omega \rangle^{\ast},\end{aligned}$$ where $E_0$ is the ground state energy of the system. To obtain the detailed form of the response function, we need to specify the form of the operator $\hat{O}_j$. One can see from Equation (\[eq:hamiltonian2\]) that the operator has two contributions. One is linear and the other is quadratic in the gauge potential. To calculate the operator we make the specific gauge choice for the gauge potential $$A=\frac{1}{2}(\mathbf{B}^0 +\delta B(t) )\wedge X .$$ With this choice the Hamiltonian is then a function of only gauge invariant quantities. As a final step to fully specify the Hamiltonian we include the static electric field by substituting $E_i(t,X)=\mathbf{E}_i^0$. The final form of the Hamiltonian that will be used for subsequent calculations is given by $$\begin{aligned}
H = \frac{1}{2M}\hat{P}^2 +\frac{1}{2m}\hat{p}^2+V (\hat{x}) - e\hat{x}^i\cdot \mathbf{E}^0_i \nonumber \\
-e\left(\frac{m_{\Delta}}{2M^2}\right) (\hat{x}\wedge \hat{P} )\cdot (\mathbf{B}^0 +\delta B(t) ) \nonumber \\
-e\left(\frac{m_{\Delta}}{2Mm}\right) \hat{L} \cdot (\mathbf{B}^0 +\delta B(t)) \nonumber \\
+\frac{e^2}{4m}\left(\frac{m_{\Delta}}{M}\right)^2 \hat{x}^i\hat{x}^j\mathbf{B}_m^0 \delta B_n(t)[\delta_{ij}\delta_{mn}-\delta_{im}\delta_{jn}], \label{eq:hamiltonian3} \end{aligned}$$ where $L=x\wedge p$ is the orbital angular momentum about the centre of mass origin. One can now just read off the operator that couples to the fluctuating magnetic field $$\begin{aligned}
\hat{O}_i &=& -\left(\frac{em_{\Delta}}{2mM}\right)\hat{L}_i \nonumber \\
&&+\frac{e^2}{4m}\left(\frac{m_{\Delta}}{M}\right)^2 \hat{x}^a\hat{x}^b [\delta_{ab}\delta_{ij}-\delta_{ai}\delta_{bj}]\textbf{B}^0_j.\label{eq:Moperator}\end{aligned}$$ There is also a contribution from the term linear in the centre of mass momenta. However, this is zero when evaluated between the ground states (zero momentum plane- wave eigenfunctions). This amounts to a choice of reference frame (the centre of mass frame) and can be used to define the components of the static external electromagnetic fields as well. Indeed, we have not specified so far the nature of the ground state $\vert \Omega \rangle$. For computational convenience we work with a perturbed ground state due to the presence of static external electric field $$\begin{aligned}
\vert \Omega \rangle &=& \vert 0 \rangle - e\mathbf{E}_i^0\sum^{\infty}_{n = 1}\frac{1}{E_n-E_0}\vert n \rangle \langle n\vert \hat{x}^i \vert 0\rangle .
\label{eq:ground state}\end{aligned}$$ This choice corresponds to the physical situation where we put the two particle system in the static external electric field *first*, let the system settle down and then apply the static external magnetic field. From Equation (\[eq:Moperator\]) we see there are two separate contributions to the response function. The first is due to a coupling with the angular momentum operator which we write as $\chi_{ij} ^{EB}(\omega, \hat{L} )$; the second is due to a quadrupole moment coupling which we write as $\chi_{ij} ^{EB}(\omega, \hat{x}^2 )$. Then the full response function is just given by their sum $\chi_{ij} ^{EB}(\omega )=\chi_{ij} ^{EB}(\omega, \hat{L} )+\chi_{ij} ^{EB}(\omega, \hat{x}^2 )$. First consider evaluating the contribution due to the angular momentum operator $$\begin{aligned}
\chi_{ij} ^{EB}(\omega, \hat{L} ) &=&-\frac{e^2\Delta m}{mM}\langle \Omega \vert \hat{x}_i \frac{1}{\hat{H}-E_0-\hbar \omega} \hat{L}_j \vert \Omega \rangle \nonumber \\
&&-\frac{e^2\Delta m}{mM}\langle \Omega \vert \hat{x}_i \frac{1}{\hat{H}-E_0+\hbar \omega} \hat{L}_j \vert \Omega \rangle^{\ast} .\end{aligned}$$ The next step is to expand the denominators in terms of the static magnetic field. From Equation (\[eq:hamiltonian3\]) the perturbation of the Hamiltonian is given by $\delta H =-(em_{\Delta}/mM)\hat{L}_i\textbf{B}_i^0$, therefore we find (keeping only the terms linear in the static magnetic field) $$\begin{aligned}
\chi_{ij} ^{EB}(\omega , \hat{L}) =-\left(\frac{e^2m_{\Delta} }{mM}\right)\left(\frac{em_{\Delta}}{mM}\right) \nonumber \\
\times\langle \Omega \vert \hat{x}_i \frac{1}{\hat{H}_0-E_0-\hbar \omega}
(\hat{L}_k\textbf{B}_k^0)\frac{1}{\hat{H}_0-E_0-\hbar \omega} \hat{L}_j \vert \Omega \rangle \nonumber \\
-\left(\frac{e^2m_{\Delta} }{mM}\right)\left(\frac{em_{\Delta}}{mM}\right) \nonumber \\
\times \langle \Omega \vert \hat{x}_i \frac{1}{\hat{H}_0-E_0+\hbar \omega}
(\hat{L}_k\textbf{B}_k^0)\frac{1}{\hat{H}_0-E_0+\hbar \omega} \hat{L}_j \vert \Omega \rangle^{\ast} .\end{aligned}$$ Inserting two complete sets of states gives $$\begin{aligned}
\chi_{ij} ^{EB}(\omega ,\hat{L}) =-\left(\frac{e^3m_{\Delta}^2 }{4m^2M^2}\right)\textbf{B}_k^0\sum_{m,n} \nonumber \\
(\langle \Omega \vert \hat{x}_i \vert m \rangle
\langle m \vert \hat{L}_k \vert n \rangle
\langle n \vert \hat{L}_j \vert \Omega \rangle \nonumber \\
\times \frac{1}{E_m-E_0-\hbar \omega} \frac{1}{E_n-E_0-\hbar \omega} \nonumber \\
+(\langle \Omega \vert \hat{x}_i \vert m \rangle
\langle m \vert \hat{L}_k \vert n \rangle \langle n \vert \hat{L}_j \vert \Omega \rangle )^{\ast} \nonumber \\
\times \frac{1}{E_m-E_0+\hbar \omega} \frac{1}{E_n-E_0+\hbar \omega} ). \label{eq:MEresponseinter}\end{aligned}$$
For the quadrupole moment contribution we have $$\begin{aligned}
\chi_{ij} ^{EB}(\omega, \hat{x}^2) =\left(\frac{e^3m_{\Delta}^2 }{4mM^2}\right)
\langle \Omega \vert \hat{x}_i \left(\frac{1}{\hat{H}-E_0-\hbar \omega}\right) \nonumber \\
\times (\hat{x}^2\mathbf{B}_j^0-\mathbf{B}^0\cdot \hat{x}\hat{x}_j) \vert \Omega \rangle \nonumber \\
+\left(\frac{e^3m_{\Delta}^2 }{4mM^2} \right)
\langle \Omega \vert \hat{x}_i \left(\frac{1}{\hat{H}-E_0+\hbar \omega}\right) \nonumber \\
\times (\hat{x}^2\mathbf{B}_j^0-\mathbf{B}^0\cdot \hat{x}\hat{x}_j) \vert \Omega \rangle^{\ast}.
\label{eq:quad}
\end{aligned}$$ Inserting a single complete set of states gives $$\begin{aligned}
\chi_{ij} ^{EB}(\omega, \hat{x}^2) =\left(\frac{e^3m_{\Delta}^2 }{4mM^2}\right) \sum_n
\left(\frac{1}{E_n-E_0-\hbar \omega}\right) \nonumber \\
\times \langle \Omega \vert \hat{x}_i \vert n \rangle \langle n \vert (\hat{x}^2\mathbf{B}_j^0-\mathbf{B}^0\cdot \hat{x}\hat{x}_j) \vert \Omega \rangle \nonumber \\
+\left(\frac{e^3m_{\Delta}^2 }{4mM^2} \right) \sum_n \left(\frac{1}{E_n-E_0+\hbar \omega}\right)
\nonumber \\
\times(\langle \Omega \vert \hat{x}_i \vert n \rangle \langle n \vert (\hat{x}^2\mathbf{B}_j^0-\mathbf{B}^0\cdot \hat{x}\hat{x}_j) \vert \Omega \rangle)^{\ast}.
\label{eq:quad2}\end{aligned}$$ When the perturbed ground state given by Equation (\[eq:ground state\]) is substituted into Equations (\[eq:MEresponseinter\]) and (\[eq:quad2\]) we arrive at a final expression for the magneto-electric response function. To linear order in the static external fields Equation (\[eq:MEresponseinter\]) becomes
$$\begin{aligned}
\chi_{ij} ^{EB}(\omega ,\hat{L}) =\left(\frac{e^4m_{\Delta}^2 }{4m^2M^2}\right)\textbf{B}_k^0 \mathbf{E}_l^0\sum_{m,n} \sum_{s\neq 0} \nonumber \\
(\langle 0 \vert \hat{x}_i \vert m \rangle
\langle m \vert \hat{L}_k \vert n \rangle
\langle n \vert \hat{L}_j \vert s \rangle \langle s \vert \hat{x}^l \vert 0 \rangle \nonumber \\
\times\frac{1}{E_s-E_0} \frac{1}{E_m-E_0-\hbar \omega} \frac{1}{E_n-E_0-\hbar \omega} ) \nonumber \\
+(\langle 0 \vert \hat{x}_i \vert m \rangle
\langle m \vert \hat{L}_k \vert n \rangle \langle n \vert \hat{L}_j \vert s \rangle \langle s \vert \hat{x}^l \vert 0 \rangle )^{\ast} \nonumber \\
\times\frac{1}{E_s-E_0} \frac{1}{E_m-E_0+\hbar \omega} \frac{1}{E_n-E_0+\hbar \omega} ). \label{eq:MEresponseL}\end{aligned}$$
The corresponding form that Equation (\[eq:quad2\]) takes is $$\begin{aligned}
\chi_{ij} ^{EB}(\omega, \hat{x}^2) =-\left(\frac{e^4m_{\Delta}^2 }{4mM^2}\right)\mathbf{B}_k^0\mathbf{E}_l^0
\nonumber \\
\times \sum_{n}\sum_{s \neq 0} \left( \frac{1}{E_s-E_0} \frac{1}{E_n-E_0-\hbar \omega} \right) \nonumber \\
\times ((\langle 0 \vert \hat{x}_i \vert n \rangle \langle n \vert (\hat{x}^2\delta_{kj} -\hat{x}_k\hat{x}_j) \vert s \rangle \langle s \vert \hat{x}^l \vert 0 \rangle \nonumber \\
+\langle s \vert \hat{x}_i \vert n \rangle \langle n \vert (\hat{x}^2\delta_{kj} -\hat{x}_k\hat{x}_j) \vert 0 \rangle \langle 0 \vert \hat{x}^l \vert s \rangle )\nonumber \\
(\langle 0 \vert \hat{x}_i \vert n \rangle \langle n \vert (\hat{x}^2\delta_{kj} -\hat{x}_k\hat{x}_j) \vert s \rangle \langle s \vert \hat{x}^l \vert 0 \rangle \nonumber \\
+\langle s \vert \hat{x}_i \vert n \rangle \langle n \vert (\hat{x}^2\delta_{kj} -\hat{x}_k\hat{x}_j) \vert 0 \rangle \langle 0 \vert \hat{x}^l \vert s \rangle)^{\ast}).
\label{eq:MEresponsequadrupole}\end{aligned}$$
To go further, it is necessary to specify a binding potential so that the energy eigenvalues and eigenfunctions can be deduced.
The harmonic oscillator binding potential
-----------------------------------------
As the simplest example one might consider, the harmonic oscillator with $V(\hat{x})=m\omega^2_0\hat{x}^2/2$. This model is both possible to solve analytically and relevant phenomenologically in displaying the properties associated with real matter. A notable feature here is that since the potential is strongly confining, the perturbed ground state due to the external static electric field has only the first excited state surviving in the summation. Using the operator algebra of the oscillator and the defining relations $$\begin{aligned}
\hat{x}_i &=& \sqrt{\frac{\hbar}{2m\omega_0}}(\hat{a}^{\dagger}_i+\hat{a}_i) \\
\hat{L}_i &=& -i\hbar/2 \epsilon_{ijk}(\hat{a}_j^{\dagger}\hat{a}_k-\hat{a}_j\hat{a}^{\dagger}_k) \\
&=&-i\hbar \epsilon_{ijk}\hat{a}_j^{\dagger}\hat{a}_k,\end{aligned}$$ the matrix elements can be evaluated explicitly in Equation (\[eq:MEresponseL\]) $$\begin{aligned}
\chi_{ij} ^{EB}(\omega , \hat{L}) &=&\left(\frac{e^3m_{\Delta}^2 }{m^2M^2}\right)\left(\frac{\hbar^2e}{2m\omega_0^2}\right)\textbf{B}_k^0
\mathbf{E}_l^0\epsilon_{kib}\epsilon_{jbl} \nonumber \\
&&\times (\frac{1}{E_1-E_0-\hbar \omega} \frac{1}{E_1-E_0-\hbar \omega} \nonumber \\
&&+\frac{1}{E_1-E_0+\hbar \omega} \frac{1}{E_1-E_0+\hbar \omega} ). \nonumber \\
&=&\left(\frac{e^4m_{\Delta}^2 }{\omega^2_0m^3M^2}\right)(
\mathbf{E}_i^0\textbf{B}_j^0 -(\textbf{E}^0 \cdot \mathbf{B}^0)\delta_{ij}) \nonumber \\
&&\times \frac{\omega^2+\omega_0^2}{(\omega_0^2- \omega^2)^2}. \label{eq:MEresponseLHO}
\end{aligned}$$ In an analogous use of the operator algebra the quadrupole contribution Equation (\[eq:MEresponsequadrupole\]) can be similarly evaluated $$\begin{aligned}
\chi_{ij} ^{EB}(\omega, \hat{x}^2) &=& -\left(\frac{e^4m_{\Delta}^2 }{4\omega^2_0m^3M^2}\right)
\left(\frac{1}{\omega_0^2- \omega^2} \right) \nonumber \\
&&\times (4\mathbf{E}_i^0\textbf{B}_j^0- \mathbf{E}_j^0\textbf{B}_i^0-(\mathbf{E}^0 \cdot\textbf{B}^0)\delta_{ij}).
\label{eq:MEresponsequadHO}
\end{aligned}$$ Note here that both contributions have an anti-symmetric term upon writing the tensor structure in the external fields as a sum of symmetric and antisymmetric pieces, which will have a relevance to our later discussion. Both the angular momentum and quadrupolar contribution are of the same order as can be seen from their multiplicative coefficients. The full response function then takes the final form $$\begin{aligned}
\chi_{ij} ^{EB}(\omega)=\chi_{ij} ^{EB}(\omega, \hat{L})+ \chi_{ij} ^{EB}(\omega, \hat{x}^2) \nonumber \\
= -\frac{e^4m_{\Delta}^2 }{\omega^2_0m^3M^2}
[ - \frac{\omega^2+\omega_0^2}{(\omega_0^2- \omega^2)^2}(\mathbf{E}_i^0\textbf{B}_j^0 -(\textbf{E}^0 \cdot \mathbf{B}^0)\delta_{ij}) \nonumber \\
+ \frac{1}{\omega_0^2- \omega^2} (\mathbf{E}_i^0\textbf{B}_j^0- \frac{1}{4} \mathbf{E}_j^0\textbf{B}_i^0-\frac{1}{4}(\mathbf{E}^0 \cdot\textbf{B}^0)\delta_{ij}) ]. \nonumber \\
\label{eq:MEresponseFinal}\end{aligned}$$
The non-relativistic hydrogen atom {#sec:hydrogen}
==================================
We now consider the second simplest system, namely the hydrogen atom. This is a weakly bound system with a Coulomb potential given by $V(r)=-e^2/(4\pi \epsilon_0 r)$ in spherical polar coordinates. Since the proton is very much more massive than the electron, we can make the replacement $m_{\Delta}/M \rightarrow 1$, leaving just the reduced mass in all expressions (which is just the electron mass). Equation (\[eq:MEresponseinter\]) can be evaluated now using the hydrogenic eigenstates $\vert n, L, m \rangle $ and the energy spectrum $E_n=E_1/n^2$. One finds $$\begin{aligned}
\chi_{ij} ^{EB}(\omega , \hat{L} ) =-\left(\frac{e^3}{m^2}\right)\textbf{B}_k^0\prod^{2}_{a=1}\sum^{\infty}_{n_a=1}\sum^{n_a-1}_{L_a=0}
\sum^{L_a}_{m_a = -L_a} \nonumber \\
\left( \frac{1}{E_{n_1}-E_1-\hbar \omega} \frac{1}{E_{n_2}-E_1-\hbar \omega} \right) \nonumber \\
\times (\langle \Omega \vert \hat{x}_i \vert n_1,L_1,m_1 \rangle \langle n_1,L_1,m_1 \vert \hat{L}_k \vert n_2,L_2,m_2 \rangle
\nonumber \\
\times \langle n_2,L_2,m_2 \vert \hat{L}_j \vert \Omega \rangle \nonumber \\
+\left(\frac{1}{E_{n_1}-E_1+\hbar \omega} \frac{1}{E_{n_2}-E_1+\hbar \omega} \right) \nonumber \\
(\langle \Omega \vert \hat{x}_i \vert n_1,L_1,m_1 \rangle \langle n_1,L_1,m_1 \vert \hat{L}_k \vert n_2,L_2,m_2 \rangle \nonumber \\
\times \langle n_2,L_2,m_2 \vert \hat{L}_j \vert \Omega \rangle)^{\ast}. \label{eq:MEresponseLH}\end{aligned}$$ For the quadrupole contribution Equation (\[eq:quad2\]) becomes $$\begin{aligned}
\chi_{ij} ^{EB}(\omega , \hat{x}^2 ) =+\left(\frac{e^3}{4m^2}\right)\textbf{B}_k^0\sum_{n_1,L_1,m_1}( \langle \Omega \vert \hat{x}_i \vert n_1,L_1,m_1 \rangle\nonumber \\
\times \langle n_1,L_1,m_1 \vert \hat{x}^2\delta_{kj}-\hat{x}_k\hat{x}_j \vert \Omega \rangle \frac{1}{E_{n_1}-E_1-\hbar \omega}
\nonumber \\
+(\langle \Omega \vert \hat{x}_i \vert n_1,L_1,m_1 \rangle\nonumber \\
\times \langle n_1,L_1,m_1 \vert \hat{x}^2\delta_{kj}-\hat{x}_k\hat{x}_j \vert \Omega \rangle )^{\ast}\frac{1}{E_{n_1}-E_1+\hbar \omega} . \nonumber \\ \label{eq:MEresponseHQ}\end{aligned}$$ Because the electron is weakly bound the perturbed ground state due to the external static electric field requires a full summation over the principal quantum number $$\begin{aligned}
\vert \Omega \rangle &=& \vert 1,0,0 \rangle - e\mathbf{E}_l^0\sum^{\infty}_{n = 2}\sum^{1}_{m=-1}\frac{1}{E_n-E_1}\vert n, 1, m \rangle \nonumber \\
&&\times \langle n, 1,m \vert \hat{x}_l \vert 1,0,0\rangle ,\label{eq:groundstateh}\end{aligned}$$ in contrast to the strongly bound harmonic oscillator potential. In Equation (\[eq:MEresponseLH\]) the only non-zero elements are when $L_1=L_2=1$ (i.e. just standard selection rules or addition of angular momentum) and thus reduces to $$\begin{aligned}
\chi_{ij} ^{EB}(\omega , \hat{L}) =\left(\frac{e^4}{m^2}\right)\textbf{B}_k^0\mathbf{E}_l^0 \sum^1_{m_1=-1}\sum^1_{m_2=-1}\sum^{\infty}_{n=2}\sum^{1}_{m=-1} \nonumber \\
\langle 1,0,0 \vert \hat{x}_i \vert n,1,m_1 \rangle \langle n, 1,m \vert \hat{x}_l \vert 1,0,0\rangle \langle 1,m_1 \vert \hat{L}_k \vert 1,m_2 \rangle
\nonumber \\
\times \langle 1,m_2 \vert \hat{L}_j \vert 1,m \rangle \frac{1}{(E_{n}-E_1-\hbar \omega)^2} \frac{1}{E_n-E_1} \nonumber \\
+( \langle 1,0,0 \vert \hat{x}_i \vert n,1,m_1 \rangle\langle n, 1,m \vert \hat{x}_l \vert 1,0,0\rangle
\langle 1,m_1 \vert \hat{L}_k \vert 1,m_2 \rangle \nonumber \\
\times \langle 1,m_2 \vert \hat{L}_j \vert 1,m \rangle)^{\ast}\frac{1}{(E_{n}-E_1+\hbar \omega )^2}\frac{1}{E_n-E_1} . \nonumber \\ \label{eq:MEresponseHL2}\end{aligned}$$ Turning now to the quadrupole case, Equation (\[eq:MEresponseHQ\]) is slightly more complicated as the expectation values of the quadrupole operator requires performing some addition of angular momenta. Indeed, the quadrupole operator can be written as a linear superposition of spherical harmonics $Y_{2,m}$ and $Y_{0,0}$. This implies non-trivial overlaps of matrix elements such that the summation over the $L$ eigenvalue will not completely reduce to a single value as in the previous case but rather one finds $$\begin{aligned}
\chi_{ij} ^{EB}(\omega , \hat{x}^2 ) &=&-\left(\frac{e^4}{4m^2}\right)\textbf{B}_k^0\textbf{E}_l^0\sum^{\infty}_{n_1=2}\sum^{\infty}_{n = 2}\sum^{3}_{L=1}\sum^{2}_{L_1=1}\sum^{L}_{m=-L} \nonumber \\
&&\times \sum^{L_1}_{m_1=-L_1}\left(\frac{1}{E_n-E_1} \frac{1}{E_{n_1}-E_1-\hbar \omega} \right) \nonumber \\
&&\times( \langle 1,0,0 \vert \hat{x}_i \vert n_1,L_1,m_1 \rangle \langle n,L, m \vert \hat{x}_l \vert 1, 0,0 \rangle \nonumber \\
&& \times \langle n_1,L_1,m_1 \vert \hat{x}^2\delta_{kj}-\hat{x}_k\hat{x}_i \vert n, L, m \rangle \nonumber \\
&&\langle 1,0, 0 \vert \hat{x}_l \vert n,L,m \rangle \langle n,L, m \vert \hat{x}_i \vert n_1,L_1,m_1 \rangle \nonumber \\
&&\times \langle n_1,L_1,m_1 \vert \hat{x}^2\delta_{kj}-\hat{x}_k\hat{x}_j \vert 1, 0, 0 \rangle ) \nonumber \\
&&+\left(\frac{1}{E_n-E_1} \frac{1}{E_{n_1}-E_1+\hbar \omega} \right) \nonumber \\
&&\times( (\langle 1,0,0 \vert \hat{x}_i \vert n_1,L_1,m_1 \rangle \langle n,L, m \vert \hat{x}_l \vert 1, 0,0 \rangle \nonumber \\
&& \times \langle n_1,L_1,m_1 \vert \hat{x}^2\delta_{kj}-\hat{x}_k\hat{x}_i \vert n, L, m \rangle )^{\ast}\nonumber \\
&&+(\langle 1,0, 0 \vert \hat{x}_l \vert n,L,m \rangle \langle n,L, m \vert \hat{x}_i \vert n_1,L_1,m_1 \rangle \nonumber \\
&&\times \langle n_1,L_1,m_1 \vert \hat{x}^2\delta_{kj}-\hat{x}_k\hat{x}_j \vert 1, 0, 0 \rangle)^{\ast} ). \label{eq:MEresponseHQ2}\end{aligned}$$ Equations (\[eq:MEresponseHL2\]) and (\[eq:MEresponseHQ2\]) are the main results of this paper. We see that like for the harmonic oscillator, there is a $1/\omega^2$ behaviour at large frequencies. To avoid the singularity when $\hbar \omega = E_n-E_1$, we supplement a phenomenological line width $\Gamma$ so that any given excited energy level can decay to a lower eigenstate by the replacement $E_n\rightarrow E_n +i \Gamma$. We have evaluated Equations (\[eq:MEresponseHL2\]) and (\[eq:MEresponseHQ2\]) analytically using as an expansion in the principal quantum number. The real part of the susceptibility is plotted in Figures \[fig:plot1\] and \[fig:plot2\], as a function of frequency both for the low and high frequency limits, and close to the resonance respectively. To do this we have used numerical data of typical field strengths of $\vert \textbf{B}_k^0\vert=10T$, $\vert \textbf{E}_k^0\vert=10^5Vm^{-1}$, a resonant frequency of $\omega_0=10^{16}Hz$ which corresponds to the energy difference between the first two levels in hydrogen, and a spontaneous decay rate of $\Gamma \sim 10^8 s^{-1}$. In Figure \[fig:plot3\] the imaginary part of the susceptibility is plotted close to the resonance. Of course, since the decay rate is very much smaller than the resonant frequencies there is a huge enhancement in the susceptibility close to resonance.
It is worth pointing out here an issue of the validity of our calculation with respect to the photoelectric effect [@Loudon2000]. Given that we are driving an atomic system with an electromagnetic wave at some frequency we may wonder if it is valid at high frequencies. In this regime the associated wave vector is also high and the photon can probe the shorter length scales and transfer more momentum to the electron. However, if one considers the differential cross section for photo-electric effect in hydrogen one knows at high frequency it behaves as $1/\omega^9$. Therefore at high frequency the atom will remain intact and the previous analysis of the magneto-electric response should remain valid.
Size of the effect and comparisons
----------------------------------
It is instructive to give a numerical estimate of the size of the response function as compared to the standard electric susceptibility. To do this, consider the harmonic oscillator Equations (\[eq:MEresponseLHO\]) and (\[eq:MEresponsequadHO\]) at zero frequency. Then up to numerical factors (i.e. simple dimensional analysis) we have $$\begin{aligned}
\vert \chi_{ij} ^{EB}(\omega)/(\epsilon_0 c) \vert & \sim &\left(\frac{e^2 }{\epsilon_0 m\omega^2_0}\right)\left(\frac{e^2}{cm^2\omega_0^2}\vert \textbf{B}_k^0\vert \vert
\mathbf{E}_l^0\vert
\right) \nonumber \\
& \sim &\left(\frac{e^2 }{\epsilon_0 m\omega^2_0}\right)\beta
\label{eq:MEresponse6}
\end{aligned}$$ The last factor, $\beta$, can be seen too be a dimensionless number, whilst the first is the standard static electric susceptibility. Putting in the number leads to a factor of $\beta\sim 10^{-12}$. This will also be approximately the same for the hydrogen atom if the optical transition frequencies are chosen to coincide.
We can compare this scale to experimental [@Rikken:2002] and DFT values [@Rizzo:2003] found for the change in refractive $\Delta n$ as compared to the absence of static external electromagnetic fields. For certain large complex molecules values of $\Delta n =(N/V)(\chi/(\epsilon_0 c)) \sim 10^{-11}$ are found experimentally [@Rikken:2002], where $N/V$ is the number density of the sample. For the helium atom a DFT calculation [@Rizzo:2003] gives a refractive index difference of $\Delta n_{helium} \sim 10^{-17}$ for a sample with number density $N/V \sim 10^{25}m^{-3}$ evaluated at a wavelength of $\lambda = 632.8 nm$. The hydrogen estimate (for the same number density) from our calculations taking into account the number density scaling is $\Delta n_{hydrogen}=(N/V)(\chi/(\epsilon_0 c))\sim (N/V)(e^2/\epsilon_0 m\omega^2_0)\beta\sim 10^{-18}$. It is also worth mentioning that $\vert \chi_{ij} ^{EB}(\omega) \vert $, as for the harmonic oscillator, will scale as size of the system since it is proportional to the standard polarisability. For the helium atom, the static polarisability is $\alpha_{helium}(0) = 0.22\times 10^{-40}$Coulomb meter$^2$/Volt, which when divided by $\epsilon_0$ gives a volume of $16.6 a_0^3$. For the hydrogen atom the corresponding volume is $4a_0^3$ from which we find a scaling factor of approximately four between the hydrogen and helium. The precise form however would have to be fitted empirically with the help of DFT calculations in order to match on to experimental values such as found in [@Rikken:2002].
Summary {#sec:summary}
=======
We have presented an exact quantum mechanical perturbation theory calculation of the magneto-electric response function for atomic systems with the simplest binding potentials, namely the harmonic oscillator and the Coulomb potential. We have deduced analytic forms for the magneto-electric response tensor as a function of frequency that can be calculated exactly. A common feature is that at high frequency they have $1/\omega^2$ behaviour whilst at low frequency they tend to a constant value. It would be interesting to try and apply the same method exactly to the helium atom.
There are interesting implications of this calculation. Concerning the Feigel effect [@Feigel:2004zz], where a net momentum density for a medium (Equation (21) in [@Feigel:2004zz]) with an magneto-electric response function is developed, the result found there is fourth power divergence in frequency, which is then simply cut-off. In fact it is only the anti-symmetric part of the magneto-electric response tensor that contributes to the momentum. This neglected however the dependence on frequency and was treated as a constant. What we see now is that this divergence will be softened to a quadratic divergence though it will not be simply washed away altogether. This shows that the assumption of a cut-off at high frequencies for ME is not justified and that the divergence has to be resolved by other means, such as done recently in [@Kawka2010].
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Geert Rikken for useful discussions. This work was supported by the ANR contract PHOTONIMPULS ANR-09-BLAN-0088-01.
A. Rizzo and S. Coriani, J. Chem. Phys. **119**, 11064 (2003).
A. Rizzo, D. Shcherbin and K. Ruud, Can. J. Chem. **87**:1352-1361 (2009).
T. Roth and G. L. J. A. Rikken, Phys. Rev. Lett. **88**, 063001 (2002).
E. B. Graham and R. E. Raab, Proc. R. Soc. Lond. Ser. A. **390**. 73 (1983).
A. Feigel, Phys. Rev. Lett. [**92**]{} (2004) 020404. B. A. van Tiggelen, G. L. J. A. Rikken and V. Krstic, Phys. Rev. Lett. [**96**]{} (2006) 130402.
O. J. Birkeland and I. Brevik, Phys. Rev. E [**76**]{}, 6, (2007) 066605.
X. G. Wen, “Quantum field theory of many-body systems", OUP (2004).
R. Loudon, “The Quantum Theory of Light", OUP (2000).
S. Kawka and B. A. van Tiggelen, EPL. [**89**]{}, 11002 (2010).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Let $G$ be a real connected Lie group with a left invariant metric $d$, $\mathfrak{g}$ its Lie algebra. In this paper we present a set of interesting upper and lower bounds for $|d\exp_{x}(y)|,\ x,y \in \mathfrak{g}$. If $\textrm{ad}_x$ is diagonalizable, these bounds only depend on eigenvalues of $\textrm{ad}_x$, but in general they are functions of the singular values $\textrm{ad}_x$.'
address: |
Department of Mathematics\
Albion College\
Albion, MI 49224
author:
- Reza Bidar
bibliography:
- 'aomsample.bib'
title: Estimates for the norm of the derivative of Lie exponetial map for connected Lie groups
---
Introduction
============
Let $G$ be a connected real Lie group with a left invariant Riemannian metric $d$, and $\mathfrak{g}$ be its Lie Algebra as an inner product space, and $\exp: \mathfrak{g} \rightarrow G$ the Lie exponential map. For $g \in G$ let $l_g$ denote the left multiplication by $g$. One important question that arise about the exponential map would be asking if there are conditions under which the exponential map is quasi-isometry. This is trivially true if the universal covering of $G$ is $\mathds{R}^n$. The other conditions that might be worthy of investigation are when $G$ is compact, semi-simple, solvable or nilpotent. In this paper we present general lower and upper bounds for the norm of the differential of the exponential map which provides valuable information regarding the behavior of the exponential map. These bounds show that in general, the exponential map for these types of Lie groups is not a quasi-isometry.\
Given a non-zero vector $x \in \mathfrak{g}$, it is well known that the differential of the exponential map at $x$ is given by $$d\exp_{x}=dl_{\exp(x)} \frac{1-e^{-\textrm{ad}_x}}{\textrm{ad}_x}$$ Rossman [@Lie; @Groups-An; @Intro p.15]. Since the metric $d$ is left invariant, it follows that for any vector $y \in \mathfrak{g}$ $$\left|d\exp_{x}(y)\right|=\left| \frac{1-e^{-\textrm{ad}_x}}{\textrm{ad}_x}(y)\right|\, .$$ Thus the problem of finding upper and lower bounds for $\left|d\exp_{x}(y)\right|$ would be equivalent to finding estimates for the norm of the image of $$\frac{1-e^{-\textrm{ad}_x}}{\textrm{ad}_x} \ ,$$ which can be regarded as a compact operator on $\mathfrak{g}$ as a finite dimensional Hilbert space. When $\textrm{ad}_x$ is diagonalizable it is possible to bound the differential of exponential map by the biggest and the smallest eigenvalues of $(1-e^{-\textrm{ad}_x})/\textrm{ad}_x$ as stated in the following theorem.
\[thm-main\] Let $x \in \mathfrak{g}$ be non-zero and such that $\emph{ad}_x$ is diagonalizable. Let $\hat{x}=x/|x|$, $\lambda_1,\cdots,\lambda_{p} \in \mathds{C}$ be non-zero eigenvalues of $\emph{ad}_{\hat{x}}$, and
$$\tilde{\lambda}_{\min}(|x|)=\min \left\{1,\left| \frac{1-e^{-\lambda_{1}|x|}}{\lambda_{1}|x|}\right|,\cdots, \left|\frac{1-e^{-\lambda_{p}|x|}}{\lambda_{p}|x|}\right|\right\},$$
$$\tilde{\lambda}_{\max}(|x|)=\max \left\{1,\left| \frac{1-e^{-\lambda_{1}|x|}}{\lambda_{1}|x|}\right|,\cdots, \left|\frac{1-e^{-\lambda_{p}|x|}}{\lambda_{p}|x|}\right|\right\}\,.$$
Then there exist positive constants $C,D$, only depending on $\hat{x}$, such that for any unit vector $y \in \mathfrak{g}$ $$C\tilde{\lambda}_{\min}(|x|) \leq \left| d\exp_{x} (y) \right| \leq D \tilde{\lambda}_{\max}(|x|)\, .$$
For a fix unit vector $\hat{x}$, if $d\exp_{\hat{x}}$ is invertible, it’s easy to find a constant $C_0$ such that for all $t \geq 1$: $$\left|1-e^{-\lambda_{j}t}\right| \geq C_0,\ 1 \leq j \leq p \, .$$ This leads us to the following corollary:
Assume $\hat{x}$ is a unit vector, $\emph{ad}_{\hat{x}}$ diagonalizable, $d\exp_{\emph{ad}_{\hat{x}}}$ invertible. Then there exist a positive constant $C$ such that for all $t \geq 1$ and any unit vector $y \in \mathfrak{g}$ $$\left| d\exp_{t\hat{x}} (y) \right| \geq \frac{C}{t}\, .$$
If $\textrm{ad}_x$ is not diagonalizable, the bounds in above theorem will not work in gerenal. However, by the minimax theorem for singular values (see Proposition \[mini-max\]), maximum and minimum of $|d\exp_x(y)|$ taken over all unit vectors $y$ are indeed the smallest and the largest singular values of $(1-e^{-\textrm{ad}_x})/\textrm{ad}_x$. This would lead us to the bounds stated in the following theorem.
\[thm-MAIN\] Let $x \in \mathfrak{g}$ be non-zero, $\hat{x}=x/|x|$, and $\lambda_1,\cdots,\lambda_{p}$ be non-zero eigenvalues of $\emph{ad}_{\hat{x}}$, $\tilde{\lambda}_{\min}(|x|), \tilde{\lambda}_{\max}(|x|)$ be defined as in Theorem 1, $\mathfrak{g}_{\mathds{C}}=\mathfrak{g}\otimes \mathds{C}$ be the complexification of $\mathfrak{g}$.\
Let $\displaystyle \delta_0=\max \left\{ \left|\emph{ad}_{x_1}(y_1)\right|:\ x_1,y_1 \in\mathfrak{g}_{\mathds{C}},\ |x_1|,|y_1|=1 \right\}$.\
We have the following bounds:
1. For all unit vectors $y$, $$\left( \frac{e^{\delta_0|x|}-1}{\delta_0|x|} \right)^{1-n}\prod_{j=1}^p\left| \frac{1-e^{-\lambda_{j}|x|}}{\lambda_{j}|x|}\right| \leq \left| d\exp_{x} (y) \right| \leq \frac{e^{\delta_0|x|}-1}{\delta_0|x|}$$
2. For every $x$, there are unit vectors $y_0,y_1 \in \mathfrak{g}$ such that $$\left| d\exp_{x} (y_0) \right| \leq \tilde{\lambda}_{\min}(|x|),\ \tilde{\lambda}_{\max}(|x|) \leq \left| d\exp_{x} (y_1) \right|$$
When $\textrm{ad}_x$ is nilpotent, $(1-e^{-\textrm{ad}_x})/\textrm{ad}_x$ would a polynomial in terms of $|x|$ which leads us to the bounds stated in the following proposition.
\[exp-nilpotent\] Let $\mathfrak{g}$ be $p$-step nilpotent. Given a non-zero $x \in \mathfrak{g}$, $|d\exp_x|=O(x^{p-1})$. There exist a polynomial $Q$ with $\deg(Q)\leq (n-1)(p-1)$, such that for all non-zero $x \in \mathfrak{g}$, $$\min_{|y|=1} |d\exp_x(y)| \geq \frac{1}{Q(|x|)}\, .$$
Preliminaries
=============
To derive estimates on the norm of the differential of the exponential map, we will need some facts from functional calculus. Let $\mathcal{H}$ be a complex Hilbert space. Singular values of a linear bounded operator $T$ defined on Hilbert space $\mathcal{H}$ are the square roots of non-negative eigenvalues of the self-adjoint operator $T^{*}T$. Given a linear operator $T$, we enumerate the singular values $\{s_j(T) \},\ j=1,2,\cdots$ in a non-increasing order, and the eigenvalues $\{\lambda_j(T) \},\ j=1,2,\cdots$ so that the moduli are non-increasing. For a normal operator $T$, singular values are the absolute of eigenvalues.
Assume that $\mathcal{H}$ is finite dimensional, $\dim \mathcal{H}=n$. We will use the following facts:
- As a direct consequence of the *Spectral Mapping Theorem* (Rudin [@Functional-Rudin Theorem 10.33]) for every complex function $f$ which is holomorphic on a domain including all eigenvalues of $T$: $$\label{SpectralMT}
\lambda_j(f(T))=f(\lambda_j(T)),\ 1 \leq j \leq n\, .$$
- If $T$ is diagonal with respect to an orthonormal basis, then $T$ is normal and thus $s_j(T)=|\lambda_j(T)|, \ 1 \leq j \leq n$.
Singular values of a linear operator defined on a finite dimensional Hilbert space are related to the norm of the operator on some subspaces of $\mathcal{H}$. This is known and minimax principle for singular values, Bhatia [@Matrix-Analysis p.75]:
\[mini-max\] Given any operator on a finite dimensional Hilbert space $\mathcal{H}$, $\dim \mathcal{H}=n$, $$\begin{split}
s_j(T)& = \max_{\mathcal{M}:\dim \mathcal{M}=j} \min_{x \in \mathcal{M}, |x|=1} |T(x)| \\
& =\min_{\mathcal{N}:\dim \mathcal{N}=n-j+1} \max_{x \in \mathcal{N}, |x|=1} |T(x)|
\end{split}$$ for $1 \leq j \leq n$.
In particular, the minimax principle implies that: $$\max_{|x|=1} |T(x)|=s_1(T),\quad \min_{|x|=1} |T(x)|=s_n(T)\, .$$
We also need the following proposition, often known as Weyl’s inequality, Birman [@Spectral-Self-Adjoint p.258]:
Let $\{\lambda_k(T) \}$ be the sequence of eigen-values of a compact operator $T$ on a Hilbert space $\mathcal{H}$, enumerated so that the moduli are non-increasing, and $\{s_k(T) \}$ be its singular values in a non-increasing order. Then $$\label{Weyl}
\prod_{1}^r |\lambda_k(T)| \leq \prod_{1}^r s_k(T), \quad r=1,2,\cdots$$
Letting $r=1$, we conclude $$\label{eigen<sing}
|\lambda_1(T)| \leq |s_1(T)|\, .$$ In addittion, if $\mathcal{H}$ is finite dimensional, $\dim \mathcal{H}=n$, the identity $$\label{singular-eigen-det}
\prod_{1}^n |\lambda_k(T)| = \prod_{1}^n s_k(T) = |\det(T)|$$ together with the inequality \[Weyl\] for $r=n-1$, implies that $$\label{eigen>sing}
s_n(T)\leq |\lambda_n(T)|\, .$$
Main Results
============
To apply the results of the previous section for bounding the derivative of the exponential map we need to work with a complex Lie algebra. For this reason we consider the complexification $\mathfrak{g}_{\mathds{C}}=\mathfrak{g}\otimes \mathds{C}$. The inner product of $\mathfrak{g}$ induced by the metric $d$, may be extended to an inner product in $\mathfrak{g}\otimes \mathds{C}$. Moreover $(1-e^{-\textrm{ad}_x})/\textrm{ad}_x$ can be regarded as a linear operator on $\mathfrak{g}_{\mathds{C}}$. Since the problem of estimating the norm the exponential map reduces to estimating the norm of a linear map over the Lie algebra, we can do estimates in $\mathfrak{g}_{\mathds{C}}$ and then derive the bounds for the real case. We will use the following notations:
Throughout this section $\mathfrak{g}_{\mathds{C}}=\mathfrak{g} \otimes \mathds{C}$ is the complexification of $\mathfrak{g}$ and we consider $\textrm{End}_{\mathds{C}}(\mathfrak{g}_{\mathds{C}})$ as a Banach space with the operator norm.
1. For any $a \in \mathds{C}$, $x \in \mathfrak{g}$, $T \in \textrm{End}_{\mathds{C}}(\mathfrak{g}_{\mathds{C}})$, $|a|$, $|x|$, and $\| T \|$ represent the absolute value of $a$, norm of vector $x$, and the operator norm of the linear map $T$, respectively. $\| T\| _{\mathfrak{g}}$ represents the norm of $T|_{\mathfrak{g}}$.
2. For non-zero $x$, $\hat{x}=x/|x|$, $\lambda_1,\cdots,\lambda_n,\ |\lambda_1|\geq \cdots \geq |\lambda_n|=0$ are the eigenvalues, and $s_1\geq \cdots \geq s_n$ are the singular values of $\textrm{ad}_{\hat{x}}$. $\tilde{\lambda}_1(x),\cdots, \tilde{\lambda}_n(x)$ are the corresponding eigenvalues of $(1-e^{-\textrm{ad}_x})/\textrm{ad}_x$. $\tilde{s}_1(x),\cdots, \tilde{s}_n(x)$ are singular values of $(1-e^{-\textrm{ad}_x})/\textrm{ad}_x$ in a non-increasing order. In addition, we admit the convention $(1-e^0)/0=1$.
3. Let $\mathcal{B}$ be a given basis in $\mathfrak{g}$. For an endomorphism $T \in \textrm{End}_{\mathds{C}}(\mathfrak{g}_{\mathds{C}})$, $[T]_{\mathcal{B}}$ is the matrix representation of $T$ in basis $\mathcal{B}$. For a $n \times n$ square complex matrix $A$, $A_{\mathcal{B}} \in \textrm{End}_{\mathds{C}}(\mathfrak{g}_{\mathds{C}})$ is the unique endomorphism such that $[A_{\mathcal{B}}]_{\mathcal{B}}=A$.
In order to prove Theorem 1 we need the following lemma:
\[lem1\] Let $V$ be a finite dimensional complex inner product space, and $P,Q,T \in \textrm{End}_{\mathds{C}}(V)$, $P$ and $Q$ invertible. Then $$\min_{|y|=1}|PTQ(y)| \geq \min_{|y|=1}|P(y)| \cdot \min_{|y|=1}|T(y)| \cdot \min_{|y|=1} |Q(y)|.$$
Without loss of generality we can assume $T$ is invertible (other wise the right side of inequality is zero). We have: $$\min_{|y|=1}|TQ(y)|= \min_{|y|=1}\left|T\left(\frac{Q(y)}{|Q(y)|}\right)|Q(y)|\right| \geq \min_{|y|=1}|T(y)| \cdot \min_{|y|=1} |Q(y)|\ ,$$ repeating the argument one more time gives the desired inequality.
We may proceed to prove Theorem \[thm-main\]:
Assume $\textrm{ad}_x$ is diagonalizable, let $\mathcal{B}$ be an eigenbasis for $\textrm{ad}_{\hat{x}}$, and $\mathcal{F}$ an arbitray orthonormal basis. Let $P$ be the change of basis matrix from $\mathcal{F}$ to $\mathcal{B}$. Then $[\textrm{ad}_x]_{\mathcal{F}}=P^{-1}[\textrm{ad}_x]_{\mathcal{B}}P=P^{-1}\textrm{diag}(\lambda_{1}|x|,\cdots,\lambda_n|x|)P$ and thus $$\left[\frac{1-e^{-\textrm{ad}_x}}{\textrm{ad}_x}\right]_{\mathcal{F}}=
P^{-1} \textrm{diag}\left(\frac{1-e^{-\lambda_{1}|x|}}{\lambda_{1}|x|},\cdots,\frac{1-e^{-\lambda_{n}|x|}}{\lambda_{n}|x|}\right) P$$ So we have: $$\begin{split}
& \max_{|y|=1,\, y\in \mathfrak{g}}\left| d\exp_{x} (y) \right| \leq
\left\|\frac{1-e^{-\textrm{ad}_x}}{\textrm{ad}_x}\right\| \\
& \leq \|P_{\mathcal{F}}^{-1}\| \left\| \textrm{diag}\left(\frac{1-e^{-\lambda_{1}|x|}}{\lambda_{1}|x|},\cdots,\frac{1-e^{-\lambda_{n}|x|}}{\lambda_{n}|x|}\right)_{\mathcal{F}}\right\| \|P_{\mathcal{F}}\|
\\
& =\|P_{\mathcal{F}}^{-1}\|\|P_{\mathcal{F}}\| \tilde{\lambda}_{\max}(|x|) \, ,
\end{split}$$ In addition, using Lemma \[lem1\] it follows that: $$\begin{split}
& \min_{|y|=1,\, y\in \mathfrak{g}}\left| d\exp_{x} (y) \right| \geq \min_{|y|=1,\, y\in \mathfrak{g}_{\mathds{C}}}
\left|\frac{1-e^{-\textrm{ad}_x}}{\textrm{ad}_x}(y)\right| \\
& \geq \min_{|y|=1} |P_{\mathcal{F}}^{-1}(y)| \cdot \min_{|y|=1} \left|\textrm{diag}\left(\frac{1-e^{-\lambda_{1}|x|}}{\lambda_{1}|x|},\cdots,\frac{1-e^{-\lambda_{n}|x|}}{\lambda_{n}|x|}\right)_{\mathcal{F}}(y)\right| \cdot
\\
& \, \quad \min_{|y|=1} |P_{\mathcal{F}}(y)|= \min_{|y|=1} |P_{\mathcal{F}}^{-1}(y)| \tilde{\lambda}_{\min}(|x|) \min_{|y|=1} |P_{\mathcal{F}}(y)|\, .
\end{split}$$
Let $\displaystyle \delta_0=\max \left\{ \left|\emph{ad}_{x_1}(y_1)\right|:\ x_1,y_1 \in\mathfrak{g}_{\mathds{C}},\ |x_1|,|y_1|=1 \right\}$ then $\|\textrm{ad}_x\| \leq \delta_0 |x|$. Noting $$\frac{1-e^{-\textrm{ad}_x}}{\textrm{ad}_x}=\sum_{k=0}^{\infty} \frac{(-1)^k}{(k+1)!}\textrm{ad}^k_x$$ and the absolute convergence of the above power series, we find that: $$\begin{split}
\left\| \frac{1-e^{-\textrm{ad}_x}}{\textrm{ad}_x} \right\|& =
\left\| \sum_{k=0}^{\infty} \frac{(-1)^k}{(k+1)!} \textrm{ad}_x^k
\right\| \leq \sum_{k=0}^{\infty} \frac{1}{(k+1)!} \|\textrm{ad}_x^k\|
\\
& \leq \sum_{k=0}^{\infty} \frac{1}{(k+1)!} \|\textrm{ad}_x\|^k = \frac{e^{\|\textrm{ad}_x\|}-1}{\|\textrm{ad}_x\|} \leq \frac{e^{\delta_0|x|}-1}{\delta_0|x|}
\end{split}$$ proving the upper bound.
Applying the equation \[singular-eigen-det\] we colnculde: $$\begin{split}
\left| d\exp_{x} (y) \right| & \geq \tilde{s}_n(x) \geq \frac{\prod_1^n {|\tilde{\lambda}_k(x)|}}{(\tilde{s}_1(x))^{n-1}} = \frac{\prod_1^p {|\tilde{\lambda}_j(x)|}}{(\tilde{s}_1(x))^{n-1}} \\
& \geq \left( \frac{e^{\delta_0|x|}-1}{\delta_0|x|} \right)^{1-n} \prod_1^p {|\tilde{\lambda}_j(x)|}\, .
\end{split}$$ By the identity \[SpectralMT\]: $$\tilde{\lambda}_j(x)= \frac{1-e^{-\lambda_{j}|x|}}{\lambda_{j}|x|},\ 1\leq j \leq p \, ,$$ completing the lower bound proof.
The second statement in the theorem follows from the inequalities \[eigen<sing\] and \[eigen>sing\] applied to $T=(1-e^{-\textrm{ad}_x})/\textrm{ad}_x$.
Let $x \in \mathfrak{g}$ be non-zero, and Let $\displaystyle \delta_0$ be defined as in the statement of Theorem \[thm-MAIN\]. We have: $$\begin{split}
\|d\exp_x \| & = \left\| \frac{1-e^{-\textrm{ad}_x}}{\textrm{ad}_x}\right\|_{\mathfrak{g}} \leq \left\| \frac{1-e^{-\textrm{ad}_x}}{\textrm{ad}_x}\right\| = \left\| \sum_{k=0}^{p-1} \frac{(-1)^k}{(k+1)!} |x|^k \textrm{ad}_{\hat{x}}^k \right\| \\
& \leq \sum_{k=0}^{p-1} \frac{1}{(k+1)!} |x|^k \|\textrm{ad}_{\hat{x}}\|^k \leq \sum_{k=0}^{p-1} \frac{1}{(k+1)!} |x|^k \delta_0^k =O(|x|^{p-1})\, .
\end{split}$$ Let $$Q_1(|x|)=\sum_{k=0}^{p-1} \frac{1}{(k+1)!} |x|^k \delta_0^k, \quad Q=Q_1^{n-1}$$ then $\tilde{s}_1(x) \leq Q_1(|x|)$. Noting $\prod_1^n {|\tilde{\lambda}_k(x)|}=1$ we have: $$\min_{|y|=1} \left| d\exp_{x} (y) \right| \geq \tilde{s}_n(x) \geq \frac{\prod_1^n {|\tilde{\lambda}_k(x)|}}{(\tilde{s}_1(x))^{n-1}} \geq \frac{1}{Q(|x|)}\, .$$
[9]{}
R. Bhatia, *Matrix Analysis*, Springer-Verlag (1997), 357pp.
M. S. Birman, M. Z. Solomjak *Spectral Theory of Self-adjoint operators in Hilbert Space*, D. Reidel Publishing Company (1987), 300pp.
W. Rudin, *Functional Analysis*, McGraw-Hill, Inc. (1991), 424pp.
W. Rossmann, *Lie Groups: An Introduction Through Linear Groups*, Oxford University Press (2002), 273pp.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Atom- and site-resolved experiments with ultracold atoms in optical lattices provide a powerful platform for the simulation of strongly correlated materials. In this letter, we present a toolbox for the preparation, control and site-resolved detection of a tunnel-coupled *bilayer* degenerate quantum gas. Using a collisional blockade, we engineer occupation-dependent inter-plane transport which enables us to circumvent light-assisted pair loss during imaging and count $n=0$ to $n=3$ atoms per site. We obtain the first number- and site-resolved images of the Mott insulator “wedding cake" structure and observe the emergence of antiferromagnetic ordering across a magnetic quantum phase transition. We are further able to employ the bilayer system for spin-resolved readout of a mixture of two hyperfine states. This work opens the door to direct detection of entanglement and Kosterlitz-Thouless-type phase dynamics, as well as studies of coupled planar quantum materials.'
author:
- 'Philipp M. Preiss'
- Ruichao Ma
- 'M. Eric Tai'
- Jonathan Simon
- Markus Greiner
bibliography:
- 'bilayer\_bib.bib'
title: 'Quantum gas microscopy with spin, atom-number and multi-layer readout'
---
### Introduction
Reduced and mixed dimensionality in solid state systems is at the heart of exceptional material properties. Prominent examples include bilayer graphene [@Ohta2006; @Novoselov2006], exciton condensation in bilayer systems [@Eisenstein2004] and unconventional superconductors where superconductivity may originate from couplings in multilayer systems [@Nakosai2012].
Experiments with ultracold atoms offer a clean and dissipation-free platform for the quantum simulation of such condensed matter systems [@Bloch2012]. Recently developed microscopy techniques with single-atom and single-site resolution provide direct access to local observables [@Bakr2010; @Sherson2010], but so far have been constrained to two-dimensional systems. Here, we present a scheme for high-fidelity fluorescence imaging of a *bilayer* system with single-site resolution. We realize full control over the resonantly coupled bilayer system and observe coherent dynamics between the two planes, confirming the suitability of our setup for investigations of strongly interacting bilayer materials.
![Preparation of a bilayer system. A degenerate gas of $^{87}$Rb is loaded into two adjacent axial planes, near the focus of a high-resolution imaging system. In combination with gravity, the axial lattice (spacing $1.5~\mu$m) and a $6\times$ superlattice (spacing $9.2~\mu$m) result in a double-well geometry with tunable tunnel-coupling $J$ and offset $\Delta$. Both lattices along $z$ are generated by reflecting beams off the flat surface of the hemispheric last lens of the imaging system. A square lattice projected through the objective onto the $xy$-plane realizes a bilayer Bose-Hubbard system. The optical molasses used for fluorescence imaging are also reflected at the surface, resulting in a standing wave pattern along $z$. \[schematics\]](preiss_fig1.pdf){width="48.00000%"}
The bilayer system can be used to extend the ability of site-resolved optical lattice experiments to detect many-body ordering: Typically, the atomic hyperfine spin cannot be resolved during readout in quantum gas microscopes, leading to complications in the study of spin systems [@Fukuhara2013; @Fukuhara2013a]. More severely, only the parity of a lattice site occupation is accessible in fluorescence imaging due to light-assisted collisions and pairwise atom loss (“parity projection") [@Schlosser2001; @Nelson2007; @Bakr2010; @Sherson2010]. Several schemes for atom counting have been developed, providing global or averaged number statistics via spin-changing collisions [@Folling2006] or an interaction blockade in double-wells [@Cheinet2008]. We use this interaction blockade to engineer occupation-dependent transport between the two planes of the bilayer system. We circumvent parity projection and resolve lattice occupation numbers $n=0$ to $n=3$ in one plane. Our technique allows for the first site-resolved, atom-number sensitive images of the Mott insulator “wedding cake" structure and of many-body ordering across a magnetic quantum phase transition. Alternatively, we apply a magnetic field gradient to obtain spin-dependent transport between the planes and demonstrate spin-resolved readout in a mixture of two hyperfine spin states.
The extension of quantum gas microscopy beyond the projection onto the atom number parity will give access to new observables. For one-and two-dimensional systems, our techniques can yield complete number statistics and provide spectra of high-order density-density correlations and distribution functions. Such observables can be used to directly characterize quantum phases, for example the Tonks-Girardeau gas through its local and non-local pair correlations [@Sykes2008] or the dynamics of quantum phase transitions through counting statistics of excitations [@Smacchia2014].
### Preparation of a resonant bilayer system
The experimental setup, capable of single-atom resolved *in situ* imaging of an ultracold cloud of $^{87}$Rb, has been described in previous work [@Bakr2009]. Our experiments begin with a three-dimensional Bose-Einstein condensate, which we compress in the direction of gravity (the $z$-direction) in two-dimensional layers at the focus of the imaging system. The confinement in the $z$-direction is provided by the “axial lattice," with a spacing of $ d=1.5~\mu$m and a corresponding recoil energy of $E_{r}=2\pi \times \frac{h}{8 m_{Rb} d^2} \approx 2\pi \times 250$ Hz.
To bring two adjacent planes of the axial lattice into resonance, we employ a $6\times$ superlattice with a spacing of $9.2~\mu$m. Both lattices are generated by reflecting beams off the flat surface of the last, hemispherical lens of the imaging system at large angles from normal ($75^\circ$ and $87.6^\circ$). The relative phase between the two lattices at the position of the atoms can be tuned by changing the angle of incidence of the superlattice. In combination with the constant gradient of $g= 2\pi \times 3.2~\frac{\text{kHz}}{\text{axial plane}}$ from gravity, we realize a resonant double-well system in the $z$-direction (Fig. \[schematics\]). The residual offset $\Delta$ between the two axial planes of interest can be tuned by varying the depth of the superlattice, while the inter-plane tunnel-coupling $J$ is controlled by the depth of the axial lattice. Other axial planes are sufficiently offset in energy that they remain entirely unpopulated.
By controlling the initial position of the condensate, we can deterministically load one or two adjacent axial planes with arbitrary ratios of atom numbers. We use our imaging system to project a two-dimensional square lattice with spacing $a=680$ nm onto the $xy$-plane, realizing a bilayer Bose-Hubbard system.
### Imaging two planes with single-site resolution
To image atoms in the bilayer system with single-site resolution, we use a sequential readout with two exposures. Separate high-resolution images of the two planes are obtained by shifting the focus of the imaging optics and tuning the fluorescence rate of atoms in different planes in real time.
![Site-resolved imaging of a bilayer quantum gas using two exposures. Image \#1 (bottom row) comprises atoms in both planes, while image \#2 (top row) contains atoms only in plane I, directly revealing the corresponding atom positions. We obtain the atom distribution in plane II from image \#1 after calculating and subtracting the background contributed by atoms in plane I. The procedure is illustrated here for decoupled Mott insulators in both planes, with up to $n=2$ atoms per site. Framed panels denote the fitted atom distribution. \[imaging\]](preiss_fig2.pdf){width="45.00000%"}
At the beginning of the imaging process, the atoms are localized and pinned in a deep optical lattice $60~\text{GHz}$ blue-detuned w.r.t the $^{87}$Rb D1 line at $795~\text{nm}$. They are simultaneously cooled by an optical molasses on the D2 line ($780~\text{nm}$), and the scattered photons are collected to form images of the atom distribution. The molasses beams are reflected at an angle of $82^\circ$ from normal off the flat surface of the hemisphere, forming a standing wave of period $2.8~\mu$m along the $z$-direction. This modulation of the molasses intensity corresponds to a modulation of fluorescence rate for atoms in different planes. By changing the angle of incidence of the molasses with a galvanometer, we are able to tune the ratio of fluorescence rates in planes I and II in real time. For the first exposure, we image both planes at a fluorescence ratio $1 \colon 2$ for $500~\text{ms}$: Atoms in plane II with higher fluorescence rate are at the focus of the imaging system, while atoms in plane I contribute a weak out-of-focus background. Next, we remove atoms contained in plane II from the system. To this end, we increase the fluorescence ratio to $1 \colon 3$ and apply a higher molasses power for $300~\text{ms}$. Due to the high scattering rate, atoms in plane II are now heated rather than cooled and ejected from the lattice, while atoms in plane I remained pinned. Simultaneously, we remove a 26 mm thick glass plate from the imaging path to shift the focus of the imaging system to plane I. At this point, plane II has been cleared of atoms and we take the second exposure to image atoms in plane I for $500~\text{ms}$ at an intermediate molasses power.
The atom distributions in both planes are obtained after post-processing: We first determine the positions of atoms in the second image, containing only atoms in plane I. Subsequently we reconstruct the background contributed by these atoms to the first image by convolving the extracted atom distribution with the measured point spread function of atoms one plane away from the focus. This simulated background is scaled to match the fluorescence rate of atoms in plane I during the first exposure and subtracted from the first image. A fit to the processed image then returns the atom positions in plane II. Figure \[imaging\] illustrates the image analysis for decoupled Mott insulators in the two axial planes.
The fidelity of the readout process is limited by our ability to hold atoms in plane I while imaging and ejecting atoms in plane II. For optimized parameters, the lifetime of atoms in plane I during the first exposure and ejection process is $27(2)$ s, resulting in a combined atom loss of $3.0\%$ in plane I prior to imaging. The efficiency of the ejection of atoms from plane II is $99(1)\%$, leading to an occasional unwanted background from atoms in plane II in the second image. In combination, these effects lead to an imaging fidelity of $95\%$ in plane I and $99\%$ in plane II. The slight reduction in imaging fidelity for plane I primarily affects measurements in the Mott insulator phase, where quantum fluctuations of the local atom number are suppressed. Here, imaging errors effectively introduce an additional entropy of $0.2\,k_B$ per site, complicating access to physics at very low entropies, such as spin dynamics in the superexchange regime [@Fukuhara2013]. The current limitations on the imaging fidelity might be overcome by increasing the photon collection efficiency and choosing shorter exposure times, or by using additional molasses beams on the D1 transition to decouple the cooling mechanism from the imaging process [@McGovern2011].
### Coherent dynamics between resonant axial planes
We characterize the bilayer system by studying the double-well dynamics in the axial lattice direction. Our experiments begin with a single-layered Mott insulator in plane I in a deep two-dimensional optical lattice, initially decoupled from plane II. Tunneling in the plane of the Mott insulator is negligible on time scales of our experiment and we concentrate on dynamics in the $z$-direction.
We first set the depth of the superlattice to bring planes I and II near resonance. The axial lattice depth is then reduced from initially $250~E_{r}$ to $8~E_{r}$ in 2 ms, enabling inter-plane tunneling at a rate $J\approx 2 \pi \times 37$ Hz. Figure \[spectrum\] a) shows offsets $\Delta$ at which particles may resonantly tunnel from plane I to plane II. The first resonance, at $\Delta \approx - 2 \pi \times 1.7$ kHz corresponds to tunneling from the ground band in plane I to the first excited band in plane II. The second resonance, at $\Delta=0$ corresponds to atoms tunneling within the ground band from plane I to plane II. For both processes, the on-site interaction shift of $U\approx 2 \pi \times 300$ Hz between singly and doubly occupied sites is well-resolved. Second-order tunneling is expected to occur on a much slower timescale of $\frac{2J^2}{U}\approx 2 \pi \times 9$ Hz and cannot be detected due to our parity projecting readout [@Folling2007]. The horizontal scale is calibrated by measuring the offset $\Delta$ at various superlattice depths via photon-assisted tunneling [@Ma2011].
We observe coherent Rabi oscillations between the two axial planes at the respective resonances for singly and doubly occupied sites, as shown in Fig. \[spectrum\] b). The extracted resonant oscillation frequencies of the populations are 118(7) Hz and 280(30) Hz respectively.The dynamics in singly occupied double-wells are in agreement with a numerical solution of the Schrödinger equation for single-particle wavefunctions and tunneling rates at an axial lattice depth of $6.5~E_{r}$, while the rate of oscillations on doubly occupied sites is enhanced by more than the factor $\sqrt{2}$ expected from bosonic enhancement. We attribute this to the deformation of the potential when the tilt is applied using the superlattice, which can result in reductions in both the effective double-well barrier height [@Meinert2013] and the effective double-well spacing. The observed damping of the Rabi oscillations is caused by inhomogeneities in the double-well potential across the two-dimensional plane.
![Inter-plane tunneling dynamics. **a)** Spectrum for inter-plane tunneling. Sites with single $(n=1,~\text{orange})$ and double $(n=2,~\text{blue})$ occupancy are initially prepared in plane I. After reducing the axial lattice depth to $7.9~E_{r}$ ($J\approx 2 \pi \times 37$ Hz) in 2 ms and allowing atoms to tunnel for 8 ms, we measure $p_{\text{odd}}$, the probability to detect a single atom in plane I, as a function of offset $\Delta$. Resonances correspond to tunneling into the ground band or first excited band of plane II, as indicated by sketches. The interaction shift between $n=1$ and $n=2$ is clearly resolved ($U\approx 2 \pi \times 300$ Hz). Solid lines are Lorentzian fits to the data. **b)** Rabi oscillations for $n=1$ and $n=2$ at their respective resonant offsets ($\Delta = 0$ and $\Delta = 2 \pi \times 300$ Hz). Here, the axial lattice depth is ramped to $6.5~E_{r}$ in 0.5 ms, giving $J\approx 2 \pi \times 55$ Hz. Solid lines are damped sinusoidal fits from which the resonant Rabi frequencies are extracted. All errorbars in this letter reflect 1$\sigma$ statistical errors in the region-averaged mean $p_{\text{odd}}$. \[spectrum\]](preiss_fig3.pdf){width="47.00000%"}
### Beyond parity imaging
![Number-resolved observation of many-body ordering. After preparing a many-body state in plane I, occupation-sensitive transport of atoms to plane II allows the detection of occupancies $n=0$ to $n=2$. **a)** Processed single-shot image of the “wedding cake" structure of a two-shell Mott insulator. **b)** One-dimensional quantum phase transition from a paramagnetic phase to an antiferromagnetic phase. An increasing tilt is applied horizontally along three decoupled chains of length eight, tuning the system from unity filling (top) via the formation of doublon-hole pairs (middle) to a density-wave ordered state (bottom).\[wedding cake\]](preiss_fig4.pdf){width="45.00000%"}
Our technique of resonant population transfer to a second axial plane can be used to circumvent the limitations imposed by parity projection in optical lattice microscope experiments. We start by preparing a single-layered Mott insulator with singly and doubly occupied sites in plane I. With the axial tunnel coupling enabled ($J\approx 2 \pi \times 48$ Hz), we sweep the offset from $\Delta=2.1~U$ to $\Delta=0$ in 75 ms, across the ground band tunneling resonance for doubly occupied sites at $\Delta \approx U$. On doubly occupied sites, a single atom transitions at an offset corresponding to the on-site interaction $U$. The transfer of a second atom is suppressed by a collisional interaction blockade, leaving one atom in plane I and one atom in plane II [@Cheinet2008]. Atoms on singly occupied sites distribute over planes I and II with roughly equal probabilities. At the end of the sweep we image both planes and obtain the distribution of holes, single atoms and doublons in the initial Mott insulator by adding the atom distributions from both planes. The reconstructed “wedding cake" structure of a Mott insulator is shown in Fig. \[wedding cake\] a), combining single-site resolution [@Bakr2010; @Sherson2010] and atom-number sensitive detection [@Gemelke2009; @Campbell2006]. Taking into account the defect probability of the initial Mott insulator and the bilayer imaging fidelity, we achieve $96(1)\%$ efficiency of separating the doublons into two planes.
![Resolving up to three atoms per site in a binary readout scheme. **a)** Occupations $n=1$ to $n=3$ in plane I are mapped to different distributions in plane I and II by inter-plane transfer and subsequent parity projection. **b)** Averaged $p_{\text{odd}}$ in plane I (blue) and plane II (green) after preparing a three-shell Mott insulator in plane I and mapping the occupation onto the two planes. Mott-insulating regions from $n=3$ at the trap center to $n=1$ near the trap perimeter are resolved. Curves are fits with (concatenated) error functions. \[binary\_readout\]](preiss_fig5.pdf){width="45.00000%"}
We employ our imaging technique to detect many-body ordering across a magnetic quantum phase transition described in previous work [@Simon2011; @Meinert2013]. Our experiments start with a $n=1$ Mott insulator in plane I, decoupled into one-dimensional chains along the $x$-direction. Using a magnetic field gradient along the chains , we drive a quantum phase transition from a paramagnetic state (unity filling) to an antiferromagnetic (density-wave ordered) state. We image the atom distribution at various points along the transition, carrying out the beyond-parity readout scheme as above. The formation of doublon-hole pairs and antiferromagnetic ordering is visible in single-shot reconstructions of the atom distribution in Fig. \[wedding cake\] b). In contrast to the previous global detection of antiferromagnetic order [@Simon2011], the ability to resolve individual doublon-hole pairs enables direct measurement of the N[é]{}el order parameter, and detailed studies of phenomena such as frustration and the dynamics of defect or domain formation in the underlying model.
A further generalization of our readout scheme allows the unambiguous detection of atom numbers $n=0$ to $n=3$. Using each side of the double-well as a “bit" that is either bright (odd occupancy) or dark (even occupancy) after parity projection, four different number states can be encoded. Figure \[binary\_readout\] a) illustrates the mapping for this “binary readout". After preparing a three-shell Mott insulator in plane I, we ramp the offset $\Delta$ through the resonances for $n=2$ and $n=3$ atoms per site, stopping before reaching the $n=1$ resonance at $\Delta=0$. Figure \[binary\_readout\] b) shows *p*$_{\text{odd}}$ after the ramp vs. radial distance for both planes. All plateaus of constant atom number from $n=3$ at the center of the cloud to $n=1$ on the outside edge can be identified.
The fidelity of the atom-number sensitive readout is currently limited by the small energy scales for dynamics in the $z$-direction. The relatively small interaction (${U\approx 2 \pi \times 300}$ Hz) and the large spacing of the axial lattice ($1.5~\mu$m) lead to slow dynamics and sensitivity to lattice inhomogeneity. By using a Feshbach resonance and a smaller axial lattice spacing, the robustness of the mapping process onto axial planes could be further improved.
### Spin-resolved readout
The presence of a spin degree of freedom greatly extends the capability of experiments with ultracold atoms to simulate condensed matter Hamiltonians. Our bilayer system may be used to observe spin ordering in such systems, providing spin-sensitive readout in a two-dimensional mixture of two spin states. The scheme is illustrated in Fig. \[spin\_dep\_readout\]: A mixture of two appropriately chosen hyperfine states initially resides in plane I. To map out the distribution of both spin states in plane I, we enable transport between the two planes, after motion within the planes has been frozen out by a deep lattice. A magnetic field gradient applied in the $z$-direction causes atoms in one hyperfine state to transfer to plane II, while atoms in the second hyperfine state experience a force in the opposite direction and remain in plane I. The hyperfine spin degree of freedom is thus mapped to the two planes of the axial lattice, and both spin states can be imaged simultaneously.
We demonstrate spin-resolved readout for the two hyperfine states $|F,m_F\rangle=|1,-1\rangle$ (labeled $|\mathord{\uparrow}\rangle$) and $|2,-2\rangle$ ($|\mathord{\downarrow}\rangle$), for which $g_{F} m_{F}=+\frac{1}{2}$ and $g_{F} m_{F}=-1$, respectively. Initially, we prepare a $n=1$ Mott insulator of atoms in $|\mathord{\uparrow}\rangle$ in plane I, and transfer atoms to the $|\mathord{\downarrow}\rangle$ state using a resonant microwave pulse in a bias field of $1.5$ G. After adiabatically transferring the atoms from the axial lattice into a single well of the superlattice, we ramp up a magnetic field gradient of $30$ G/cm in the $z$-direction in $70$ ms. When the axial lattice is then ramped back on, the magnetic field gradient causes atoms in state $|\mathord{\downarrow}\rangle$ to transfer to plane II, while atoms in state $|\mathord{\uparrow}\rangle$ remain in plane I. Figure \[spin\_dep\_readout\] shows the population of both planes after mapping versus microwave pulse duration. The sinusoidal variation in anti-phase demonstrates the mapping of spin to plane degree of freedom. We estimate a fidelity of $97(1)\%$ for the correct sorting of hyperfine spin into different axial lattice planes from the offset and amplitude of the fit.
Unlike previous experiments, in which only one of two spin states could be imaged *in situ*, our technique gives access to the full spin distribution in an interacting many-body system. This scheme will enable further studies of two-component systems, such as impurity dynamics [@Fukuhara2013] and collective excitations [@Kleine2008].
![Spin-resolved readout. **a)** Procedure for mapping the hyperfine spin onto different axial planes: After reducing the tunnel barrier in a double-well with an arbitrary spin in plane I to zero, a B-field gradient separates the $|\mathord{\uparrow}\rangle$ and $|\mathord{\downarrow}\rangle$ components into different axial planes. **b)** A resonant microwave pulse is applied to a $n=1$ Mott insulator in state $|\mathord{\uparrow}\rangle$ in plane I. The sinusoidal variation in hyperfine spin is mapped onto occupation in plane I (blue) and plane II (green). Curves are fits with sine functions.\[spin\_dep\_readout\]](preiss_fig6.pdf){width="45.00000%"}
### Summary
We have demonstrated single-atom and single-site resolved imaging of a bilayer quantum degenerate gas. We engineer occupation-sensitive transport between resonant layers with an optical superlattice and the use of a collisional interaction blockade. This transport and readout scheme circumvents the problem of parity projection in quantum gas microscopes, allowing the unambiguous identification of atom numbers up to $n=3$. We have obtained the first number- and site-resolved images of the Mott-insulator “wedding-cake" structure and directly imaged many-body ordering across a quantum phase transition. Making use of a magnetic field gradient in the $z$-direction, we have demonstrated spin-dependent transport and spin- and site-resolved readout of a two-species mixture. This approach will facilitate the observation of antiferromagnetic ordering in the Fermi-Hubbard model and spin-dependent phenomena such as spin-charge separation [@Kleine2008].
Direct extensions to our imaging scheme will enable site-resolved detection in more than two planes. In particular, the use of additional molasses beams on the D1 transition for cooling [@McGovern2011] while collecting fluorescence photons only from molasses on the D2 transition will lead to even more spatially selective readout.
Our techniques for the preparation and readout of resonant bilayer systems open numerous possibilities for further studies of low-dimensional phenomena: Interfering two planar superfluids should enable *in situ* observation of phase evolution in two dimensions and the dynamics of the Kosterlitz-Thouless phase transition [@Mathey2011], while many-body entanglement can be measured in a system of two copies of a planar system [@Daley2012]. Finally, our technique could be used to reduce the entropy in two-dimensional Mott insulators by “filling" empty sites (defects) with atoms by merging with a reservoir plane [@Tichy2012].
We thank Andrew Daley for helpful discussions. This work is supported by grants from NSF through the Center for Ultracold Atoms, the Army Research Office with funding from the DARPA OLE program and a MURI program, an Air Force Office of Scientific Research MURI program, and the Gordon and Betty Moore Foundation’s EPiQS Initiative. M.E.T is supported by the U.S. Department of Defense through the NDSEG program.
#### Note added:
Spin-resolved detection of individual atoms in a single one-dimensional Hubbard chain was recently reported in Ref. [@Fukuhara2015]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Inference algorithms based on evolving interactions between replicated solutions are introduced and analyzed on a prototypical NP-hard problem - the capacity of the binary Ising perceptron. The efficiency of the algorithm is examined numerically against that of the parallel tempering algorithm, showing improved performance in terms of the results obtained, computing requirements and simplicity of implementation.'
author:
- 'Roberto C. Alamino, Juan P. Neirotti and David Saad'
bibliography:
- 'traj\_v02.bib'
title: 'Replication-based Inference Algorithms for Hard Computational Problems'
---
Introduction
============
One of the main contributions of statistical physics to application domains such as information theory and theoretical computer science has been the introduction of established methods that facilitate the analysis of typical properties of very large systems in the presence of disorder. For instance, in information theory applications, these large systems correspond to (mostly binary) transmissions where the disorder is manifested through transmission noise or the manner by which the message is generated or encoded. Established approaches in the statistical physics community such as the replica and cavity methods [@mezard09] have proved to be useful tools in describing typical properties of error-correcting codes [@Saad01; @Alamino07; @Alamino07b], the analysis of optimization problems such as the traveling salesman [@Mezard86], K-satisfiability [@Braunstein05] and graph coloring [@vanMourik02; @mulet02] to name but a few.
Another important contribution, which complements the ones mentioned above, was in the development of algorithmic tools to find microscopic solutions in specific problem instances. One of the most celebrated inference methods, the message-passing (MP) or belief propagation algorithm, had been developed independently in the information theory [@Gallager63], machine learning [@Pearl88] and statistical physics [@Mezard86] communities until the links between them have been identified [@Kabashima98; @Opper01] and established [@yedidia05]. Subsequently, a number of successful inference methods have been devised using insights gained from statistical physics [@mezard02b; @mezard09].
In MP algorithms, the system to be solved is mapped onto a bipartite factor graph, where on the one hand factor nodes correspond to observed (given) information or interaction between variables; while on the other hand are variable nodes, to be estimated on the basis of approximate marginal pseudo-posteriors. The latter are obtained by a set of consistent marginal conditional probabilities (messages) passed between variable and factor nodes [@mezard09]. Unfortunately there are many caveats to the MP procedure, especially in the presence of closed loops in the factor graph, which may give rise to inconsistent messages and non-convergence. It can be shown that MP converges to the correct solution when the factor graph is a tree but there is no such guarantee for more general graphs, although MP does result in good solution in many other cases too.
There are two main general difficulties in using MP algorithms to problems represented by densely connected graphs. The first is that the computational cost grows exponentially with the degree, making the computation impractical; while the second arises from the existence of many short loops that result in recurrent messages and lack of convergence. These problems have been solved in specific cases, especially in the case of real observations and continuous noise models by aggregating messages [@Kabashima03]. One of the shortcomings identified in [@Kabashima03] was non-convergence when prior knowledge on the noise process is inaccurate or unknown; which typically results in multiple solutions and conflicting messages.
While MP would be successfully applied if a weighted average over *all* possible states could be carried out, it is clear that such average is infeasible. Inspired by the state-space representation obtained using the replica method [@Mezard86; @nishimori01; @mezard09] whereby state vectors are organized in an ultrametric structure, two of us suggested an MP algorithm based on averaging messages over a structured solutions space [@Neirotti05]. The approach is based on using an infinite number of copies (or *real replica*, not to be confused with those employed in the replica method) of the variables exposed to the same observations (factor nodes). The replicated variable systems facilitate a broader exploration of solution space as long as these replica are judiciously distributed according to the solution-space structure implied by the statistical mechanics analysis. The variable vectors inferred by these algorithms are then combined by taking either weighted or white average to obtain the marginal pseudo-posterior of the various variables.
The approach has been successful in addressing the Code Division Multiple Access (CDMA) problem as well as the Linear Ising Perceptron capacity problem [@Neirotti07], even in cases where prior information is absent. It is worthwhile noting that a seed of this replication philosophy can be found in several previous algorithms such as: (a) *query by committee* [@Seung92], where the potential solutions (system replica) are used for choosing the best most informative next example and later combines the solutions using a majority voting. (b) An analytical approach [@deDominicis80] aimed at obtaining solutions for the Sherrington-Kirkpatrick model via averages over the Thouless-Anderson-Palmer equations. (c) A study of the p-spin model meta-stable states by considering averages over a small number of real replica [@Kurchan93]. (d) The *Parallel Tempering* (PT) algorithm, also known as replica exchange MCMC sampling, which relies on many replica searching the space at different temperatures [@Swendsen86; @Marinari1992]; the latter, due to its good performance and relation to the approach we advocate, will be explained in more detail later on and will be used for comparison with the method develop here. It is interesting to note that approaches based on averaging multiple interacting solutions have also been successfully tried in neighboring disciplines, for example for decoding in the context of error-correcting codes [@Shrinivas2011].
While this approach has been successful in addressing inference problems in the case of real observations and continuous noise models, it is less clear how it could be extended to accommodate more general cases. In this work we will present an alternative method for carrying out averages over the replicated solutions, which can be applied to more general cases. Generally, like most MP-based algorithms, the approach is based on solutions being calculated iteratively using a pair of coupled self-consistent equations. We will study the properties of the new algorithm, its advantages and limitations, on an exemplar problem of the Binary Ising Perceptron (BIP) [@Krauth89; @Engel01] that has been used as a benchmark also in other works on advanced inference methods [@Braunstein06].
One obvious obstacle in most MP algorithms is that the iterative dynamics can be trapped in suboptimal minima; in addition, the algorithm itself can either create spurious suboptimal minima in the already complex solution space or change the height of the energy barriers between the existing ones. We will show that our replica-based MP algorithm fails under naive averaging of the replica for the BIP capacity problem, explain analytically why it happens and show that in the limit of a large number of replica, averages flow to the clipped Hebb algorithm [@Engel01]. We will then propose an alternative approach and show how replication can indeed improve performance if carried out appropriately.
In section \[section:BIP\] we will explain the exemplar problem to be used in this study; we will then review the non-replicated MP solution to the BIP capacity problem under the approximation for densely connected systems in section \[section:MP\] and provide an analytical solution to the naively replicated MP algorithm, showing here its equivalence with the clipped Hebb rule. Section \[section:OMP\] will point to the main reason for the failure of the naive replica-averaging approach and argue that an online version of the MP algorithm, which is derived and presented, can solve it. By replicating the new online MP (OnMP) algorithm and using the extra degrees of freedom that it provides, we show how it outperforms the non-replicated MP algorithm, termed offline MP (OffMP) algorithm. Section \[section:CP\] compares the replicated OnMP (rOnMP) with a benchmark parallel algorithm, namely the PT algorithm. Finally, conclusions and future directions are discussed in section \[section:Conclusions\].
Exemplar Problem - the Binary Ising Perceptron {#section:BIP}
==============================================
To extend the replica-based inference method [@Neirotti07] we would like to use an exemplar problem that is particularly difficult, not only in the worst-case scenario but also typically, where both observations and noise model are not real-valued. In addition, we would like to examine a case where exact results have been obtained by the replica theory; this provides helpful insight in devising the corresponding algorithm by suggesting a possible structure for the solution space as well as an analytical tool to assess the efficacy of the algorithm.
One prototypical NP-complete problem [@Pitt88] that was shown to be computationally hard even in the typical case, which was solved exactly using the replica method, is the capacity of the Binary Ising Perceptron (BIP) [@Krauth89]. This is due to the complex structure of its solution space studied in [@Obuchi09], showing a non-trivial topology even in the replica symmetric (RS) phase.
The BIP [@Engel01] represents a process whereby $K$-dimensional binary input vectors ${{\boldsymbol s}}_\mu\in{{\left\{\pm1\right\}}}^K$ are received, where the input vector index $\mu=1,...,N$, represents each of the $N$ example vectors. The corresponding outputs for each one of them is determined by the binary classification $$y_\mu = {{\mbox{sgn }}}{{\left(\frac1{\sqrt{K}}\sum_{k=1}^K s_{\mu k} b_k\right)}},$$ where ${{\boldsymbol b}}=(b_1,...,b_K)\in{{\left\{\pm1\right\}}}^K$ is called the unknown binary variables (also referred to as the perceptron’s variable vector); the prefactor $\sqrt{K}$ is for scaling purposes, so that the argument of the sign function remains order $O(1)$ as $K\rightarrow\infty$.
The capacity problem for a BIP is a storage problem, although it can alternatively be seen as a compression task [@Hosaka2002]. In the simplest version of the problem, a dataset $D={{\left\{({{\boldsymbol s}}_\mu,y_\mu)\right\}}}_{\mu=1}^N$ consisting of $N$ pairs of inputs and outputs (also called *examples*) is randomly generated and a perceptron with an appropriate variable vector ${{\boldsymbol b}}$ should be found, such that when presented with an input pattern ${{\boldsymbol s}}_\mu$, it reproduces the corresponding output $y_\mu$. That is the equivalent of compressing the information contained in the set of classifications ${{\left\{y_\mu\right\}}}$, comprising $N$ bits, into a vector ${{\boldsymbol b}}$ with only $K$ bits. One is usually interested in the typical case, which is calculated by averaging over all possible datasets $D$ drawn at random from a certain probability distribution.
Typical performances are algorithm dependent and are measured by counting the fraction of correctly stored patterns as a function of the number of examples in the dataset. One very convenient measure used in statistical physics calculations is the average value of the energy function $$E (\hat{{{\boldsymbol b}}})= 1-\prod_{\mu=1}^N \Theta{{\left(y_\mu \frac1{\sqrt{K}} \sum_{k=1}^K s_{\mu k} \hat{b}_k\right)}},
\label{equation:Energy}$$ with $\Theta(\cdot)$ being the Heaviside step function and $\hat{{{\boldsymbol b}}}$ the inferred variable vector. This measure gives 0 if all examples are correctly learned and 1 otherwise, i.e., it is an indicator that all the patterns were perfectly memorized. The maximum value of $\alpha=N/K$ for which this cost function is 1 (when averaged over all possible datasets) is the *achieved capacity* of the algorithm and a measure of its overall performance.
Although the achieved capacity varies between algorithms, there is an absolute upper bound, the critical capacity $\alpha_c$, above which no algorithm can memorize the whole set of examples in the typical case (although it might be possible for specific instances); this reflects the information content limit of the perceptron itself.
The critical capacity was calculated by Krauth and Mézard using the one-replica symmetry breaking (1RSB) ansatz [@Krauth89] with the result of $\alpha_c\approx0.83$. Taking into consideration that the problem is computationally hard, the challenge then becomes to find an algorithm which infers appropriate ${{\boldsymbol b}}$ values in typical specific instances of $D$ as close as possible to $\alpha_c$, where the corresponding computational complexity scales polynomially with the system size.
Naive Message Passing {#section:MP}
=====================
The inference problem we aim to address is finding the most appropriate value of the variable vector ${{\boldsymbol b}}$ capable of reproducing the classifications given the examples dataset $D$. Firstly, one needs to determine a quality measure that quantifies the appropriateness of a solution. The most commonly used error measure in similar estimation problems is the expected error per variable, or bit-error-rate in the information theory literature, the minimization of which leads to a solution based on the Marginal Posterior Maximiser (MPM) estimator given by $$\hat{b}_k = \mbox{argmax}_{b_k\in{{\left\{\pm1\right\}}}} \, \sum_{b_{l\neq k}} {{\mathcal{P}{{\left({{\boldsymbol b}}|D\right)}}}} = {{\mbox{sgn }}}{{\left<b_k\right>_{{{\mathcal{P}{{\left({{\boldsymbol b}}|D\right)}}}}}}},$$ which means that one estimates ${{\boldsymbol b}}$ bitwise, such that each component $\hat{b}_k$ corresponds to the variable value that maximizes the marginal distribution per variable given the dataset $D$. The MP equations allow one to carry out an approximate Bayesian inference procedure to find this estimator.
It is important to remember that there might not exist a variable vector capable of reproducing the whole dataset. In this case, the dataset is *unrealizable* by the BIP, although one can still identify the most probable candidate. In the BIP capacity problem, unrealizable datasets exist since they are generated randomly, not by a teacher perceptron as is the case in some generalization problems. Each variable in the set $D$ is drawn from an independent distribution and therefore one can write the posterior distribution of the variable vector as $${{\mathcal{P}{{\left({{\boldsymbol b}}|D\right)}}}} = {{\mathcal{P}{{\left({{\boldsymbol b}}|{{\left\{y_\mu\right\}}},{{\left\{{{\boldsymbol s}}_\mu\right\}}}\right)}}}} \propto {{\mathcal{P}{{\left({{\left\{y_\mu\right\}}}|{{\boldsymbol b}},{{\left\{{{\boldsymbol s}}_\mu\right\}}}\right)}}}}{{\mathcal{P}{{\left({{\boldsymbol b}}\right)}}}},$$ where ${{\mathcal{P}{{\left({{\left\{y_\mu\right\}}}|{{\boldsymbol b}},{{\left\{{{\boldsymbol s}}_\mu\right\}}}\right)}}}}$ factorizes as the examples are sampled identically and independently $${{\mathcal{P}{{\left({{\left\{y_\mu\right\}}}|{{\boldsymbol b}},{{\left\{{{\boldsymbol s}}_\mu\right\}}}\right)}}}}=\prod_{\mu=1}^N {{\mathcal{P}{{\left(y_\mu|{{\boldsymbol b}},{{\boldsymbol s}}_\mu\right)}}}}.$$ From the Bayesian point of view, ${{\mathcal{P}{{\left({{\boldsymbol b}}\right)}}}}$ is interpreted as the (factorized) prior distribution of possible variable vectors. As there is no noise involved in the capacity problem, the likelihood factor is simply given by $${{\mathcal{P}{{\left(y_\mu|{{\boldsymbol b}},{{\boldsymbol s}}_\mu\right)}}}} = \frac12+\frac{y_\mu}2 {{\mbox{sgn }}}\xi_\mu,$$ defining $$\xi_\mu = \frac1{\sqrt{K}}\sum_{k=1}^K s_{\mu k} b_k.$$ As for each instance the dataset is fixed, we will omit in the following expressions the explicit reference to the input vectors ${{\boldsymbol s}}_\mu$ in the posterior distribution for brevity.
The resulting MP equations are self-consistent coupled equations of marginal conditional probabilities which are iterated until convergence (or up to a cutoff number of iterations). These equations are obtained by applying Bayes theorem to each one of the so-called $Q$-messages and $R$-messages $$\begin{aligned}
Q^{t+1}_{\mu k} {{\left(b_k\right)}} &= \mathcal{P}^{t+1} {{\left(b_k|{{\left\{y_{\nu\neq\mu}\right\}}}\right)}}
\propto {{\mathcal{P}{{\left(b_k\right)}}}} \prod_{\nu\neq\mu} \mathcal{P}^{t+1} {{\left(y_\nu|b_k,{{\left\{y_{\sigma\neq\nu}\right\}}}\right)}}\\
R^{t+1}_{\mu k} {{\left(b_k\right)}} &= \mathcal{P}^{t+1} {{\left(y_\mu|b_k,{{\left\{y_{\nu\neq\mu}\right\}}}\right)}}
=\sum_{{{\left\{b_{l\neq k}\right\}}}} {{\mathcal{P}{{\left(y_\mu|{{\boldsymbol b}}\right)}}}}\prod_{l\neq k} \mathcal{P}^t
{{\left(b_l|{{\left\{y_{\nu\neq\mu}\right\}}}\right)}},\end{aligned}$$ where ${{\mathcal{P}{{\left(b_k\right)}}}}$ is the prior distribution over the $k$-th entry of the variable vector and $t$ stands for the current iteration step.
As $b_k\in{{\left\{\pm1\right\}}}$, one can write $$Q^t {{\left(b_k\right)}} = \frac{1+m^t_{\mu k}b_k}2 \qquad \text{and} \qquad
R^t {{\left(b_k\right)}} \propto \frac{1+\hat{m}^{t-1}_{\mu k}b_k}2.$$
The variables $m_{\mu k}$ may be interpreted as magnetization related to the cavity field in analogy to spin lattices in magnetic fields. The interpretation of the $\hat{m}_{\mu k}$ variables is less intuitive. Substituting the $R$-messages into the $Q$-messages and summing over the two possible values of $b_k$ we finally reproduce the MP equations in their well-known form $$\begin{aligned}
\label{equation:orsp_1}
\hat{m}^t_{\mu k} &= \frac{\sum_{b_k} b_k \mathcal{P}^{t+1}{{\left(y_\mu|b_k,{{\left\{y_{\nu\neq\mu}\right\}}}\right)}}}
{\sum_{b_k}\mathcal{P}^{t+1}{{\left(y_\mu|b_k,{{\left\{y_{\nu\neq\mu}\right\}}}\right)}}},\\
\label{equation:orsp_2}
m^t_{\mu k} &= \tanh{{\left[\sum_{\nu\neq\mu} {{\mbox{atanh }}}\hat{m}^t_{\nu k}\right]}}
\approx \tanh{{\left(\sum_{\nu\neq\mu} \hat{m}^t_{\nu k}\right)}},\end{aligned}$$ the approximation in the last equation is possible since $\hat{m}_{\mu k} \sim O(1/\sqrt{K})$ as we will see later.
Once convergence is attained, the value for the variable vector can be estimated by $$\begin{aligned}
\hat{b}_k &= {{\mbox{sgn }}}m_k,\\
\label{equation:magk}
m_k &= \tanh{{\left(\sum_\nu \hat{m}^t_{\nu k}\right)}},\end{aligned}$$ or $$\hat{b}_k = {{\mbox{sgn }}}{{\left(\sum_\nu \hat{m}^t_{\nu k}\right)}}.$$
As mentioned in section \[section:BIP\], the factor graph representing the BIP is densely connected, but an expansion for large $K$ suggested by Kabashima [@Kabashima03] helps to simplify the equations away from criticality. However, for the BIP this expansion requires extra care due to the discontinuity of the sign function. To address this problem, we developed a different approach to carry out this expansion which can be generalized to accommodate other types of perceptrons with minimal modifications; it can be applied to either continuous or discontinuous activation functions, with or without noise. Equation (\[equation:orsp\_2\]) for $m_{\mu k}$ is not modified, but $\hat{m}_{\mu k}$ is expanded in powers of $1/\sqrt{K}$, giving rise to a different expression (detailed derivation is provided in appendix \[appendix:MPE\]): $$\label{equation:MPFE}
\hat{m}_{\mu k} =\frac{2 s_{\mu k} y_\mu }{\sqrt{K}}\frac{\mathcal{N}_{\mu k}}{1+{{\mbox{erf}}}{{\left(y_\mu u_{\mu k}/\sqrt{2\sigma^2_{\mu k}}\right)}}},$$ where $$\begin{aligned}
\mathcal{N}_{\mu k} &= \frac1{\sqrt{2\pi\sigma^2_{\mu k}}} \exp{{\left(-\frac{u_{\mu k}^2}{2\sigma^2_{\mu k}}\right)}},\\
\sigma^2_{\mu k} &= \frac1K \sum_{l\neq k} (1-m_{\mu l}^2),\\
u_{\mu k} &= \frac1{\sqrt{K}} \sum_{l\neq k} m_{\mu l} s_{\mu l}.\end{aligned}$$
However, this version of the algorithm is unable to memorize large numbers of examples. Simulation results show that, even for small system sizes ($K\sim10$) it cannot memorize more than a single pattern on average. This is a consequence of the fact that the dynamical map defined by the MP equations becomes trapped in the many suboptimal minima of the energy landscape.
In principle, one should be able to correct this by replicating the system and distributing the $n$ replica randomly in solution space, let each one carry out the inference task independently and compare their final fixed points. An idea along these lines, with a small number of real replica searching the space in parallel, was tested with some success in [@Kurchan93]; where the replica helped change the landscape to facilitate jumps over barriers between metastable states. However, the corresponding algorithm was not very efficient computationally. Also the replicated version of MP we tested failed and the observed performance coincided with that of the non-replicated version.
To understand the reasons for the failure of the naively replicated algorithm, we solved the replicated version of the algorithm analytically. We consider the case where a simple white average of the $n$ replica is used for inferring the variable vector value $$\hat{b}_k = {{\mbox{sgn }}}{{\left(\frac1n \sum_{a=1}^n b_k^a\right)}}~.$$
We can then evaluate the MP equations using a saddle point method when $n,N,K\rightarrow\infty$. The detailed calculation is given in appendix \[appendix:Hebb\] giving rise to a surprisingly simple final result $$\label{equation:Hebb}
\hat{b}_k = {{\mbox{sgn }}}{{\left(\sum_{\mu=1}^N y_\mu s_{\mu k}\right)}}.$$
Simplicity is not the only surprising aspect of this result. Those familiar with past research in machine learning will readily recognize this equation as the clipped version of the Hebb learning rule [@Koehler90]. Unfortunately, this is not good news as the maximum attainable capacity by this algorithm has been already calculated analytically to be $\frac{N_H}{K} \equiv\alpha_H = \approx0.11$ [@Sompolinsky86; @vanHemmen87]. Worse yet, the achieved capacity of the clipped-Hebb rule quickly deteriorates as $K$ increases, converging to zero asymptotically.
The flow of the replicated algorithm towards the clipped Hebb rule points out some other weaknesses of the MP algorithm. It is not difficult to appreciate that MP results in a clipped rule as the final estimate of the variable vector is obtained by clipping the fixed point of the magnetization; this implies that it suffers from all pathologies present in clipped rules such as suboptimal solutions.
Another characteristic that is highlighted by this result is the fact that, like the Hebb rule, the MP approach is an offline (batch) learning algorithm in the sense that it does not depend on the order of presentation of the examples. This is true both for the non-replicated and replicated algorithms. This suggests that one could introduce an extra source of stochasticity by devising an online version of the MP, which could allow for the algorithm to overcome the energy barriers that trap it in local minima. Different orders of examples correspond to different paths in solution space which, combined, could potentially explore it much more efficiently. The examples order is an extra degree of freedom that cannot be exploited in offline algorithms. In the following section we show that by pursuing this idea, we find a replicated version of MP which does not only perform better than the offline one (OffMP), but also offers many additional advantages.
Online Message Passing {#section:OMP}
======================
The results of the previous section indicate that replication of the OffMP algorithm does not offer any significant improvement in performance in the BIP capacity problem. The online version of the MP algorithm introduced here allows one to exploit the order of presentation of examples as a mechanism to avoid algorithmic trapping in local minima. This algorithm will then be used in its replicated version with a polynomial number of replica $n$ with respect to the number of examples $N$.
In order to develop an online version of the MP algorithm we rely on a large $K$ expansion. When $K\rightarrow\infty$, one can derive the equations for the magnetization (mean values) of the inferred variable vector, equation (\[equation:magk\]), as $$\label{eq:m_online}
\begin{split}
m_{k} &= \tanh {{\left(\sum_{\nu} \hat{m}_{\nu k}\right)}}\\
&= \tanh {{\left(\sum_{\nu\neq\mu} \hat{m}_{\nu k}+\hat{m}_{\mu k}\right)}}\\
&\approx \tanh {{\left(\sum_{\nu\neq\mu} \hat{m}_{\nu k}\right)}} + \hat{m}_{\mu k}{{\left[1-\tanh^2 {{\left(\sum_{\nu\neq\mu}
\hat{m}_{\nu k}\right)}} \right]}}\\
&= m_{\mu k} + {{\left[1-{{\left(m_{\mu k}\right)}}^2\right]}}\hat{m}_{\mu k}.
\end{split}$$
Equation (\[eq:m\_online\]) singles out the $\mu$-th example similarly to the OffMP derivation. However, in the online interpretation it is considered a *new example*, being presented sequentially after all previous $\mu-1$ examples have been learned. Then, $m_k$ can be interpreted as the updated magnetization, while $m_{\mu k}$ is the magnetization linked to the cavity field induced by the previous examples, before example $\mu$ is included. In the bipartite interpretation of the model this is akin to the introduction of new a factor node, exploiting conditional probabilities calculated with respect to the previous $\mu-1$ examples. To make this interpretation more explicit, we add a time label to the obtained equation by changing $m_k$ to $m_k(t)$, $m_{\mu k}$ to $m_k(t-1)$ and considering the $\mu$-th example as the example being presented at time $t$. The online MP algorithm can finally be written as $$m_k(t) = m_k(t-1) + \frac{s_{tk}y_t}{\sqrt{K}} F_k(t),$$ with the so-called *modulation function* given by $$F_k(t) = 2 {{\left[1-m^2_k(t-1)\right]}} \frac{\mathcal{N}_{t k}}{1+{{\mbox{erf}}}{{\left(y_t u_{t k}/\sqrt{2\sigma^2_{t k}}\right)}}},$$ where $$\begin{aligned}
\sigma^2_{t k} &= \frac1K \sum_{l\neq k} {{\left[1-m_l^2(t-1)\right]}},\\
u_{t k} &= \frac1{\sqrt{K}} \sum_{l\neq k} s_{t l} m_l(t-1).\end{aligned}$$
The performance of the OnMP algorithm without replication is shown in fig. \[figure:OnxOff\] for $K=21$ averaged over 200 different sets of examples. The vertical axis shows $\rho=1-{{\left<E\right>_{}}}$, the average value of the function that indicates perfect learning. However, because $E\in{{\left\{0,1\right\}}}$ we have to estimate the variance by repeating the average several times and calculating an average over averages. For the graph of fig. \[figure:OnxOff\], this was done 200 times for each particular dataset and the corresponding error bars are smaller than the size of the symbols. We see that, contrary to the OffMP, the online version is now able to memorize perfectly a larger number of examples on average, with a larger achieved capacity. The slow decay to zero is to be interpreted as a finite size effect, which is however difficult to since increasing the system size $K$ leads to a deterioration of performance instead of a sharper transition.
![Non-replicated online version of the MP algorithm. While the offline MP cannot learn perfectly more than one single example, we see that the OnMP can, already without replication, memorize perfectly a larger number of examples. The vertical axis $\rho$ is the average value of the indicator function that gives 1 if all patterns are memorized and zero otherwise. The horizontal axis is the capacity $\alpha=N/K$.[]{data-label="figure:OnxOff"}](fig1.eps){width="12cm"}
Let us now replicate this algorithm. For $N$ examples, there are $N!$ possible orders of presentation, but we will choose only a number $n$ of these sequences, with $n$ being of polynomial order in $N$. We will see that this is enough to improve considerably the performance of the algorithm. We compare two versions of the replicated algorithm with white and weighted average over replica. Both versions work by exposing the $n$ replica independently to different orders of examples. To minimize residual effects, we allow a relearning procedure with $L\sim 10$ relearning cycles while keeping the same order of presentation. As the MP algorithm relies on clipping, it shows a poorer performance when the number of examples is small, especially for the even cases where parity effects are amplified. This effect however disappears as $N$ grows larger.
The difference between the white and weighted algorithms lies in how the final estimate for the variable vector is calculated. Respectively, we have $$\begin{aligned}
\hat{b}_k^{white} &= {{\mbox{sgn }}}{{\left(\frac1n \sum_{a=1}^n b_k^a\right)}}\\
\hat{b}_k^{weighted} &= {{\mbox{sgn }}}{{\left(\sum_{a=1}^n w^a b_k^a\right)}},\end{aligned}$$ where $$w^a \propto e^{-\beta E({{\boldsymbol b}}^a)},$$ and the energy of each replica is calculated as in equation (\[equation:Energy\]). The parameter $\beta$ works as an inverse temperature and is given a high value in order to select lower energy states. Clearly, when $\beta=0$, the white and weighted averages are the same.
We compared the performance of the two versions of the rOnMP against the non-replicated one. Both perform much better than the non-replicated algorithm. The difference between weighted and white averages in related problems had already been studied in relation to the TAP equations via the replica approach yielding similar results [@deDominicis80]; this indicates that similar problems appear in the corresponding dynamical maps. Contrary to our expectations, though, we have not found any difference in performance between the weighted and white averaged algorithms. This seems to indicate that even selection of the best performers as done by the weighted average is not enough to prevent the algorithm of being trapped in suboptimal solutions, which can only be avoided by increasing the number of replica.
It is interesting to note that a variational approach carried out by Kinouchi and Caticha [@Kinouchi96] was successful in finding the optimal online learning rule for a perceptron, in the sense that it will saturate the Bayes’ generalization bound calculated by Opper and Haussler [@Opper91].
Although the perceptron generalization problem is different from the capacity problem, as in the former the dataset is clearly realizable having been generated by a corresponding perceptron, which might not be the case for the latter; up to the critical capacity one can assume that the set of random examples, in the typical case, is indeed realizable. In fact, this is usually one of the underlying assumptions when attempting to solve the capacity problem. This means that we can use the same algorithms to carry out both tasks.
The precise form for the parallel variational optimal (VO) algorithm for the BIP was derived in [@Neirotti10] and is given by $${{\boldsymbol b}}(t+1) = {{\boldsymbol b}}(t)+ \frac{{{\boldsymbol s}}_{t}y_t}{\sqrt{N}} F(t),$$ where the modulation function is $$F(t) = 2 \sqrt{\frac{Q(t)}{R(t)^2}} {{\left[1-R(t)^2\right]}} \frac{\mathcal{N}_t}{1+{{\mbox{erf}}}{{\left(R(t)\phi(t)/\sqrt{2(1-R(t)^2)}\right)}}},$$ with $$R(t) = \frac{{{\boldsymbol b}}_0\cdot{{\boldsymbol b}}(t)}{|{{\boldsymbol b}}_0||{{\boldsymbol b}}(t)|}, \qquad
Q(t) = \frac{{{\boldsymbol b}}(t)^2}{N}, \qquad
\phi(t) = h(t)y_t, \qquad
h(t) = \frac{{{\boldsymbol b}}\cdot{{\boldsymbol s}}_t}{|{{\boldsymbol b}}|}~;$$ where ${{\boldsymbol b}}_0$ is a teacher perceptron, which in the capacity case would correspond to the correct inferred variable vector, the true value of which we do not know. In employing the VO algorithm, an assumption that the overlaps are self-averaging has been used. Therefore, a sensible way to obtain a value that could be used as a good estimate of ${{\boldsymbol b}}_0$ is to run the algorithm many times in parallel and average all values of ${{\boldsymbol b}}(t)$ at each iteration. Like in our algorithm, this average can be either white or weighted.
A notable characteristic of the above set of equations is their similarity with our equations for the OnMP if one substitutes $$m_k\rightarrow b_k, \qquad m_k^2\rightarrow R^2, \qquad R\phi\rightarrow yu, \qquad 1-R^2\rightarrow\sigma^2,$$ respectively. In fact, the asymptotic behavior of the VO guarantees that even the square-root amplitude appearing in front of the modulation function tends to the same value as in the OnMP, making the two sets isomorphic under this substitution. This striking relation between both algorithms is a strong indication that our algorithm must also be capable of achieving the optimal capacity and saturates Bayes’ generalization bound [@Opper91].
Performance {#section:CP}
===========
In this section we compare the performance of the rOnMP with that of the PT algorithm. The reason for choosing PT is that it is a well established parallel algorithm with good performance in searching for solutions in the BIP capacity problem. Other derivatives of BP-based algorithms have been used to solve the BIP capacity problem, for instance Survey Propagation [@Braunstein05; @Braunstein06]; the latter also aims to address the fragmentation of solution space but employs a different approach. The results reported [@Braunstein05; @Braunstein06] show that solutions can be found very close to the theoretical limits even for large systems but additional practical techniques and considerations should be used to successfully obtain solutions. As our aim in this work is to show how replication can improve significantly the performance of MP algorithms, we use the PT algorithm as the preferred benchmark method due to its simpler implementation.
Parallel Tempering (PT) or replica exchange Monte Carlo algorithm [@Marinari1992; @Swendsen86] was introduced as a tool for carrying out simulations of spin glasses. Like the BIP capacity problem, spin glasses have a complicated energy landscape with many peaks and valleys of varying heights and PT has been successfully applied to that and many other similar problems where the extremely rugged energy landscape causes other methods to underperform [@Neirotti00; @Neirotti00b].
In many cases searching for the low energy states is done by gradient descent methods. In statistical physics, simulated annealing is a principled and useful alternative to gradient descent by allowing for a stochastic search while slowly decreasing the temperature; it is particularly effective in the cases where the landscape has one or very few valleys. However, to guarantee convergence to an optimal state the temperature should be lowered very slowly and most applications use a much faster cooling rate. In the case of spin glasses, this causes the algorithm to be easily trapped in local minima.
The idea behind the PT algorithm is to introduce a number of replica of the system that search the solution space in parallel at different temperatures using a simple Metropolis-Hastings procedure. The higher the temperature, the easier it is for the replica to jump over energy barriers, but convergence becomes increasingly compromised. However, jumping over barriers allows for the exploration of a large part of solution space and the PT algorithm cleverly exploits this by comparing, at chosen time intervals, the energy of the present random walker at two different successive temperatures. If the higher-temperature random walker reaches a state of smaller energy than the one at a lower temperature, they are exchanged, otherwise there is an exponentially small probability for this exchange to take place; this probability is given by the ratio of the Boltzmann weights as in the usual Metropolis-Hastings algorithm.
As time proceeds, the lowest energy walker corresponds to the lowest temperature replica. After convergence or when a certain number of iterations have been done, results for all temperatures can be obtained. PT has a very high performance for the BIP capacity problem and can achieve very high storage capacities. The disadvantage comes from the fact that PT uses much more information than is needed to solve the BIP capacity problem and is therefore computationally expensive.
Figure \[figure:PT\] shows the performance of the rOnMP compared to PT for a system size $K=21$. We see that already with $n=10000$ replica rOnMP has a better performance than PT, which was run up to the point when there was no extra improvement. We observed that by increasing the number of replica we can reach better performances although the improvement in the performance becomes more modest for higher values; studies with a large number of replica $n\sim 10^5$ seem to indicate that the critical capacity can indeed be achieved for $n$ sufficiently large. However, the computing time increases as well and more lengthy and detailed analysis are necessary to get precise results.Further experiments also seem to indicate that in the many replica case the algorithm’s performance does not deteriorate with increasing $K$; but clearly, the corresponding computing time increases as well.
![Results from the Parallel Tempering algorithm (circles) versus the replicated online MP (triangles) with 10000 replica. The system size is $K=21$. The graph, presented with error bars over 10000 trials, shows the superiority of the MP already for this number of replica.[]{data-label="figure:PT"}](fig2.eps){width="12cm"}
In addition to the better performance, rOnMP has several other advantages over PT. Firstly, the running time for achieving a similar performance is lower. Secondly, and more importantly, PT depends on a complicated fine-tuning of the number of replica at different temperatures and how these are spaced. Different ranges of temperatures and spacings between them give different results and these require optimization trials. On the other hand, the application of rOnMP is much more straightforward and depends only on the number of replica.
Conclusions {#section:Conclusions}
===========
The main objective for this work was to show that parallelizing message passing algorithms, via *replication* of the variable system, can lead to a dramatic improvement of their performance. Replication is based on insights and concepts from statistical physics, especially in the subfield of disordered systems. The binary Ising perceptron (BIP) capacity problem was chosen as a difficult benchmark problem due to its complex solution space and its discrete output and noise model; both make the inference problem particularly difficult.
Firstly, we showed analytically that the offline version of the MP algorithm for the BIP capacity problem results in the clipped Hebb rule estimator in the thermodynamic limit and when the number of replica is large. This shows a fundamental limitation of the MP procedure and motivated us to search for an online version of it; after establishing the single system version it has been extended to accommodate a replicated version. Both non-replicated and the replicated versions were shown to have superior performance to that of the OffMP.
There are two important aspects of replicated algorithms we would like to point out, namely the way the search is carried out in solution space and how to combine the search results to obtain a unified estimate. We devised a method to make replicated variable systems follow different paths in the solution space by using different orders of example presentations, which is only possible in online algorithms. We also tried two different ways to combine the results, white and weighted averaging, the latter using the Boltzmann factors of each replica. We found no difference between both approaches, indicating that weighting the averages is not sufficient to avoid local minima.
Finally, we compared the results of the weighted rOnMP algorithm with those of the Parallel Tempering algorithm, showing that our replicated version of MP performs much better than PT.
Showing that replication in online MP improves its efficiency paves the way to using similar approaches to address other hard computational problems. We are currently exploring the applicability of techniques developed here to address other problems in physics and in information theory. There are still many issues that should be studied concerning these algorithms. One of them, which is currently underway, is finding an efficient way to choose the order of examples, which can be seen as a query learning procedure. However, query learning for the particular problem studied here corresponds to sampling from a fragmented solution space that corresponds to a the replica symmetry breaking solution space and demands the introduction of a carefully constructed interaction between the replicated solutions, which we currently investigate.
Acknowledgments {#acknowledgments .unnumbered}
===============
Support by the Leverhulme trust (F/00 250/M) is acknowledged.
Message Passing Expansion for the Binary Ising Perceptron {#appendix:MPE}
=========================================================
Consider the first MP equation (\[equation:orsp\_1\]), repeated below for convenience $$\label{equation:orsp_1_app}
\hat{m}^t_{\mu k} = \frac{\sum_{b_k} b_k \mathcal{P}^{t+1}{{\left(y_\mu|b_k,{{\left\{y_{\nu\neq\mu}\right\}}}\right)}}}
{\sum_{b_k}\mathcal{P}^{t+1}{{\left(y_\mu|b_k,{{\left\{y_{\nu\neq\mu}\right\}}}\right)}}}.$$
We denote the numerator of this expression simply by $A$, ignoring for brevity the dependence on the indices. By introducing a variable $\xi$ to represent the field $\xi_\mu$ using a Dirac delta, we can write $$\begin{split}
A &= \frac{y_\mu}{2^K} \int\frac{d\xi d\hat{\xi}}{2\pi} \, e^{i\xi\hat{\xi}} ({{\mbox{sgn }}}\xi)
{{\left[\prod_{l\neq k} \sum_b (1+m_{\mu l} b) \exp{{\left(-i\hat{\xi}\frac{s_{\mu l} b}{\sqrt{K}}\right)}}\right]}}\\
& \quad\times \sum_b b \, \exp{{\left(-i\hat{\xi}\frac{s_{\mu k} b}{\sqrt{K}}\right)}}.
\end{split}$$
Summing over $b\in{{\left\{\pm 1\right\}}}$ one obtains $$\begin{split}
\sum_b (1+m_{\mu k} b) \exp{{\left(-i\hat{\xi}\frac{s_{\mu k} b}{\sqrt{K}}\right)}}
&= 2{{\left[\cos{{\left(\frac{\hat{\xi}}{\sqrt{K}}\right)}}-im_{\mu k} s_{\mu k} \sin{{\left(\frac{\hat{\xi}}{\sqrt{K}}\right)}}\right]}}\\
&\approx 2{{\left[1-im_{\mu k} s_{\mu k}\frac{\hat{\xi}}{\sqrt{K}}-\frac{\hat{\xi}^2}{2K}\right]}},
\end{split}$$ where, in the last line, we expand the trigonometric functions to their first non-trivial orders in $1/\sqrt{K}$, already taking into consideration the large $K$ scenario. Doing the same expansion to the second summation one obtains $$\begin{split}
\sum_b b \, \exp{{\left(-i\hat{\xi}\frac{s_{\mu k} b_k}{\sqrt{K}}\right)}} &= -2i s_{\mu k} \sin{{\left(\frac{\hat{\xi}}{\sqrt{K}}\right)}}\\
&\approx -2i s_{\mu k}\frac{\hat{\xi}}{\sqrt{K}},
\end{split}$$
These approximations allow one to rewrite the expression for $A$ as $$\begin{split}
A &= \frac{-iy_\mu s_{\mu k}}{\sqrt{K}} \int\frac{d\xi d\hat{\xi}}{2\pi} \, e^{i\xi\hat{\xi}} ({{\mbox{sgn }}}\xi) \hat{\xi}
\exp{{\left[\sum_l \ln {{\left(1-\frac{\hat{\xi}^2}{2K}-im_{\mu l} s_{\mu l}\frac{\hat{\xi}}{\sqrt{K}}\right)}}\right]}}\\
&\approx \frac{-iy_\mu s_{\mu k}}{\sqrt{K}} \int\frac{d\xi}{2\pi} \, ({{\mbox{sgn }}}\xi) \int d\hat{\xi}\,\hat{\xi}
\exp{{\left[-\frac{\hat{\xi}^2\sigma^2_{\mu k}}{2}+i\hat{\xi}{{\left(\xi-u_{\mu k}\right)}}\right]}},
\end{split}$$ where $$\sigma^2_{\mu k} = \frac1K \sum_{l\neq k} (1-m_{\mu l}^2), \qquad
u_{\mu k} = \frac1{\sqrt{K}} \sum_{l\neq k} m_{\mu l} s_{\mu l}.$$
The resulting integral is trivial and, by following the analogous steps for the denominator, we finally reach the result given by expression (\[equation:MPFE\]).
Analytical Derivation of the Replicated Naive MP Algorithm {#appendix:Hebb}
==========================================================
Upon replication of the variable system such that the final estimate of the variable vector is inferred by a white average of the $n$ replica $$\hat{b}_k = {{\mbox{sgn }}}{{\left(\frac1n \sum_{a=1}^n b_k^a\right)}},$$ one can take the limit $n\rightarrow\infty$ to calculate a closed expression for it. The MP equations (\[equation:orsp\_1\]) and (\[equation:orsp\_2\]) remain the same, but the likelihood term has to include the contribution of the replica as $${{\mathcal{P}{{\left(y_\mu|{{\boldsymbol b}}\right)}}}} = \sum_{{{\left\{{{\boldsymbol b}}^a\right\}}}} {{\mathcal{P}{{\left(y_\mu|{{\boldsymbol b}},{{\left\{{{\boldsymbol b}}^a\right\}}}\right)}}}}{{\mathcal{P}{{\left({{\left\{{{\boldsymbol b}}^a\right\}}}|{{\boldsymbol b}}\right)}}}},$$ $$\begin{aligned}
{{\mathcal{P}{{\left(y_\mu|{{\boldsymbol b}}\right)}}}} &= \frac1{2^{n+1}} {{\left[1+y_\mu\,{{\mbox{sgn }}}{{\left(\frac1{\sqrt{K}} \sum_{k=1}^K s_{\mu k} b_k\right)}}\right]}}
\prod_a{{\left[1+y_\mu\,{{\mbox{sgn }}}{{\left(\frac1{\sqrt{K}} \sum_{k=1}^K s_{\mu k} b^a_k\right)}}\right]}},\\
{{\mathcal{P}{{\left({{\left\{{{\boldsymbol b}}^a\right\}}}|{{\boldsymbol b}}\right)}}}} &\propto \prod_{k} \frac12 {{\left[1+b_k\,{{\mbox{sgn }}}{{\left(\frac1n \sum_{a=1}^n b_k^a\right)}}\right]}}.\end{aligned}$$
In the last equation we ignore the normalization. For the calculation to be carried out rigorously, the normalization should be taken into account in what follows. However, careful calculations show that it does not change the saddle point result. The above expressions can be substituted in the first of the MP equations (\[equation:orsp\_1\]). Let us concentrate on the numerator of Eq. (\[equation:orsp\_1\]), which can be written as $$\begin{split}
A &\propto \int {{\left[\frac{d\xi d\hat{\xi}}{2\pi} e^{i\xi\hat{\xi}}\right]}}
{{\left[\prod_a \frac{d\xi^a d\hat{\xi}^a}{2\pi} e^{i\xi^a\hat{\xi}^a}\right]}} {{\left(1+y_\mu\,{{\mbox{sgn }}}\xi\right)}}
\prod_a {{\left(1+y_\mu\,{{\mbox{sgn }}}\xi^a\right)}}\\
& \quad\times\sum_{{{\boldsymbol b}}} b_k {{\left[\prod_{l\neq k}\frac12 {{\left(1+b_l m_{\mu l}\right)}}\right]}}
\exp{{\left[-\frac{i\hat{\xi}}{\sqrt{K}}\sum_{j=1}^K s_{\mu j} b_j\right]}}\\
& \quad\times\sum_{{{\left\{{{\boldsymbol b}}^a\right\}}}} \prod_j \frac12 {{\left[1+b_j \,{{\mbox{sgn }}}{{\left(\frac1n \sum_{a=1}^n b_j^a\right)}}\right]}}
\exp{{\left[-\frac{i}{\sqrt{K}}\sum_a \hat{\xi}^a \sum_{j=1}^K s_{\mu j} b_j^a\right]}}.
\end{split}$$
To decouple the replicated systems, we introduce the $K$ variables $$\lambda_k = \frac1n \sum_a b_k^a,$$ via Dirac deltas. By defining the notation $$D{{\left[\xi,\hat{\xi}\right]}} \equiv {{\left[\frac{d\xi d\hat{\xi}}{2\pi} e^{i\xi\hat{\xi}}\right]}}
{{\left[\prod_a \frac{d\xi^a d\hat{\xi}^a}{2\pi} e^{i\xi^a\hat{\xi}^a}\right]}}, \qquad
D{{\left[\lambda,\hat{\lambda}\right]}} \equiv {{\left[\prod_k \frac{d\lambda_k d\hat{\lambda}_k}{2\pi/n} e^{in\lambda_k\hat{\lambda}_k}\right]}},$$ and summing over ${{\boldsymbol b}}$’s we obtain $$\begin{split}
A &\propto \int D{{\left[\lambda,\hat{\lambda}\right]}} D{{\left[\xi,\hat{\xi}\right]}} {{\left(1+y_\mu\,{{\mbox{sgn }}}\xi\right)}}
\prod_a {{\left(1+y_\mu\,{{\mbox{sgn }}}\xi^a\right)}}\\
& \quad\times{{\left[\prod_{a,j} \cos{{\left(\hat{\lambda}_j+\frac{\hat{\xi}^a s_{\mu j}}{\sqrt{K}}\right)}}\right]}}
{{\left[-i\sin{{\left(\frac{\hat{\xi} s_{\mu k}}{\sqrt{K}}\right)}}+{{\mbox{sgn }}}\lambda_k\cos{{\left(\frac{\hat{\xi} s_{\mu k}}{\sqrt{K}}\right)}}\right]}}\\
& \quad\times\prod_{l\neq k} {{\left(1+m_{\mu l}\,{{\mbox{sgn }}}\lambda_l\right)}}{{\left[ \cos{{\left(\frac{\hat{\xi}s_{\mu l}}{\sqrt{K}}\right)}}
-i\,{{\mbox{sgn }}}\lambda_l\sin{{\left(\frac{\hat{\xi}s_{\mu l}}{\sqrt{K}}\right)}}\right]}}.
\end{split}$$
One can now expand the arguments of the $\cos$ and $\sin$ functions in powers of $1/\sqrt{K}$ to obtain $$\begin{split}
A &\propto \int D{{\left[\lambda,\hat{\lambda}\right]}} D{{\left[\xi,\hat{\xi}\right]}} {{\left(1+y_\mu\,{{\mbox{sgn }}}\xi\right)}}
\prod_a {{\left(1+y_\mu\,{{\mbox{sgn }}}\xi^a\right)}}\\
& \quad\times \exp{{\left[\sum_{a,j} \ln {{\left(\cos\hat{\lambda}_j
-\frac{\hat{\xi}^a s_{\mu j}}{\sqrt{K}}\sin\hat{\lambda}_j -\frac{(\hat{\xi}^a)^2}{2K} \cos\hat{\lambda}_j\right)}}\right]}}
{{\left(-i\frac{\hat{\xi}s_{\mu k}}{\sqrt{K}}+{{\mbox{sgn }}}\lambda_k\right)}}\\
& \quad\times\exp{{\left[\sum_{l\neq k}\ln{{\left(1+m_{\mu l}\,{{\mbox{sgn }}}\lambda_l\right)}}
+\sum_{l\neq k} \ln{{\left(1-\frac{\hat{\xi}^2}{2K}-\frac{i\hat{\xi}}{\sqrt{K}}\,{{\mbox{sgn }}}\lambda_l\right)}}\right]}}.
\end{split}$$
The integrals over the $\xi$ variables are easy to calculate, leading to the following expression at leading order in $1/\sqrt{K}$ $$\label{eq:saddlepointintegral}
A \propto \int {{\left[\prod_j \frac{d\lambda_j d\hat{\lambda}_j}{2\pi/n}\right]}} {{\mbox{sgn }}}\lambda_k \, e^{n\Phi},$$ where $$\Phi = i\sum_j \lambda_j\hat{\lambda}_j +\frac1n \sum_{l\neq k} \ln{{\left(1+m_{\mu l} {{\mbox{sgn }}}\lambda_l\right)}} +\sum_j \ln \cos\hat{\lambda}_j
+\frac1n\sum_{c=0}^n \ln I_c,$$ with $$\begin{aligned}
I_a &= 1+y_\mu {{\mbox{erf}}}{{\left(\frac{u_{\mu}}{\sqrt{2\sigma_\mu^2}}\right)}}, \qquad a=1,...,n\\
u_\mu &= -\frac{i}{\sqrt{K}}\sum_j s_{\mu j} \tan \hat{\lambda}_j,\\
\sigma^2_\mu &= \frac1K \sum_j {{\left(1+\tan^2 \hat{\lambda}_j\right)}},\\
I_0 &= 1+y_\mu{{\mbox{erf}}}{{\left(\frac{u^0_{\mu k}}{\sqrt{2}}\right)}},\\
u^0_{\mu k} &= \frac1{\sqrt{K}} \sum_{l\neq k} s_{\mu l}\,{{\mbox{sgn }}}\lambda_l.\end{aligned}$$
Following the same calculations for the denominator, one can see that for large $n$ the variables $\hat{m}_{\mu k}$ are given by ${{\mbox{sgn }}}\lambda^*_k$, where $\lambda^*_k$ is defined by the saddle point of the integral (\[eq:saddlepointintegral\]) which is a solution of the simultaneous equations $${{\frac{\partial \Phi}{\partial \lambda_j}}}={{\frac{\partial \Phi}{\partial \hat{\lambda}_j}}}=0.$$
Differentiating $\Phi$ we finally find the result $$\hat{m}_{\mu k}=y_\mu s_{\mu k},$$ resulting in the estimate (\[equation:Hebb\]) of the variable vectors that also corresponds to the clipped Hebb rule.
|
{
"pile_set_name": "ArXiv"
}
|
IFIC/06-03\
hep-th/0602043
[**Compact multigluonic scattering amplitudes with heavy scalars and fermions**]{}
[**Paola Ferrario $^{(a)}$[^1]**]{}, [**Germán Rodrigo $^{(a)}$[^2]**]{} and [**Pere Talavera $^{(b)}$[^3]**]{}
${}^{(a)}$ Instituto de Física Corpuscular, CSIC-Universitat de València,\
Apartado de Correos 22085, E-46071 Valencia, Spain.\
${}^{(b)}$ Departament de Física y Enginyeria Nuclear, Universitat Politècnica de Catalunya, Jordi Girona 1-3, E-08034 Barcelona, Spain.\
[**Abstract**]{}
> 10000 Combining the Berends-Giele and on-shell recursion relations we obtain an extremely compact expression for the scattering amplitude of a complex scalar-antiscalar pair and an arbitrary number of positive helicity gluons. This is one of the basic building blocks for constructing other helicity configurations from recursion relations. We also show explicity that the all positive helicity gluons amplitude for heavy fermions is proportional to the scalar one, confirming in this way the recently advocated SUSY-like Ward identities relating both amplitudes.
IFIC/06-03\
February 6, 2006
Motivation
==========
To achieve a successful physics program at LHC there must be a good control over all the possible expected backgrounds. These, among other processes, require the evaluation of multipartonic scattering amplitudes at higher orders in the perturbative expansion. Without this information the identification of any signal of new physics is only partial. Despite their relevance, Yang-Mills scattering amplitudes are very poorly known, mainly because the number of Feynman diagrams increases exponentially with the number of the external fields involved in the processes. This is one of the main reason for elaborating other techniques to obtain the amplitudes. Between them one of the most successful is the use of recursion relations within the helicity amplitude formalism. The helicity amplitude formalism [@Jacob:1959at; @Bjorken:1966kh; @Mangano:1990by] has been proven to be an elegant and efficient tool to calculate multipartonic scattering amplitudes. Recursion relations extensively used in the literature at tree [@Berends:1987me; @Dixon:1996wi] and one-loop level [@Bern:1994zx; @Bern:1994cg] to calculate multipartonic scattering amplitudes.
Based on old insights, [@Nair:1988bq], in Ref. [@Witten:2003nn] Witten presents the idea of a weak-weak duality between supersymmetric ${\cal N} =4$ Yang-Mills and topological B string theories in twistor space. Inspired by, but independent of these findings, a new method for the evaluation of scattering amplitudes in gauge theories has been proposed [@Cachazo:2004kj], the so called CSW. It is based on the recursive use of off-shell Maximal Helicity Violating amplitudes (MHV) [@Parke:1986gb] as basic vertices for new amplitudes. Recent works have accomplished interesting progress since the original formulation, and the method has been refined by introducing more efficient recursion relations [@Britto:2004ap; @Britto:2005fq], the so called BCFW, and extending this approach to the one-loop level [@Bern:2005cq; @Bern:2005hs].
Extending the BCFW formalism to massive particles, on-shell recursion relations at tree-level have been introduced in Ref. [@Badger:2005zh] for massive scalars, and in Ref. [@Badger:2005jv] for vector boson and fermions. Scattering amplitudes with heavy scalars and up to four gluons of positive helicity were first derived in Ref. [@Bern:1996ja]. In Ref. [@Badger:2005zh] all the helicity configurations with up to four gluons have been computed by using the on-shell recursion relations. These results have been extended to amplitudes with an arbitrary number of gluons of identical helicity or one gluon of opposite helicity in Ref. [@Forde:2005ue]. The approach of Forde and Kosower [@Forde:2005ue] is based on a basic ansatz for the all positive helicity amplitude which is shown to fullfil the BGKS [@Badger:2005zh] recursion relations, and which is used to construct the rest of the helicity configurations. Using off-shell recursion relations [@Berends:1987me], multigluonic scattering amplitudes with heavy fermions and an arbitrary number of gluons of positive helicity have been calculated in Ref. [@Rodrigo:2005eu].
We are concern in this note with two kind of multigluonic scattering amplitudes, and more in concrete with their relation: the first involves heavy fermions and are interesting by its own, due the expected rich phenomenology driven by the heavy quarks at LHC. The second of the amplitudes, with complex colored massive scalars, are of use in the unitarity method for computing massless loop amplitudes in nonsupersymmetric gauge theories [@Bern:2005cq]. In a recent paper [@Schwinn:2006ca] it has been demonstrated that a SUSY-like model Ward identities relate both amplitudes with heavy scalars and fermions. The apparent quite different structure of the results presented in Refs. [@Forde:2005ue] and [@Rodrigo:2005eu] makes however quite difficult to test explicitly that relationship, apart for amplitudes with a few gluons due to their simplicity.
Is our aim to show explicitly inside QCD, that for a given helicity configuration the multigluonic massive heavy quark and the massive heavy scalar amplitudes are related by a simple overall kinematical factor. For this we construct in Sec. (2) the off-shell massive scalar amplitude. In Sec. (3) we review the equivalent fermionic amplitude and present the relation with the scalar case. Finally Sec. (4) contains our summary. Some notations and definitions issues are gathered in an Appendix.
Scalar amplitudes
=================
The colour ordered off-shell current of an on-shell complex scalar of four-momentum $p_1$ and $(n-2)$-gluons of four-momenta $p_2$ to $p_{n-1}$ and positive helicity is given in terms of the off-shell scalar current with less gluons, and the off-shell gluonic current $J^\mu$ of the rest of the gluons: S(1\_s;2\^+,…,n-1\^+) = - \_[k=1]{}\^[n-2]{} S(1\_s;2\^+,…,k\^+) p\_[1,k]{} J(k+1\^+,…,n-1\^+) , \[scalarrecursion\] where $p_{1,k}=p_1+p_2+\ldots +p_k$ and $S(1_s) = 1$. We also define $y_{1,k}=p_{1,k}^2-m^2$. For all gluons of positive helicity the gluonic current has the form [@Berends:1987me]: J\^(i\^+,…,j\^+) = , where = , with $\xx{i,i} = 1$ . The null vector $\xi$ is the reference gauge vector which is assumed to be the same for all the gluons. Then from , we get S(1\_s;2\^+,…,n-1\^+) = - \_[k=1]{}\^[n-2]{} S(1\_s;2\^+,…,k\^+) . \[recursion2\] To obtain the recursion relation in we apply the Berends-Giele rules [@Berends:1987me] and consider the $\phi g \phi^\dagger$ vertex V(p\_1,k\^,p\_2) = (p\_2-p\_1)\^ , where $p_1$, $k$ and $p_2$ are the four-momenta of the scalar, the gluon and the antiscalar respectively, and the $\sqrt{2}$ comes from the normalization conventions used in colour ordered Feynman rules. Four-point vertices do not contribute to the current with all the gluons of the same helicity, since J(i\^+,…,j\^+)J(k\^+,…,l\^+) = 0 .
Let’s anticipate our result for the scalar current with an arbitrary number of gluons: S(1\_s;2\^+,…,n-1\^+) &=& S(1\_s;2\^+, …, n-2\^+)\
&+& A\_n(1\_s;2\^+,…,n-1\^+;n\_s) , \[offshell\] where && A\_n(1\_s;2\^+,…,n-1\^+;n\_s) = i { \[2|[p/]{}\_1 [p/]{}\_[23]{}|n-1\]\
&& + \_[j=1]{}\^[n-5]{} \[2|[p/]{}\_1 [p/]{}\_[23]{}|w\_1\] } , \[onshell\] with $w_1<w_2< \dots < w_j$ and $w_k \in [4,\ldots,n-2]$, is the corresponding on-shell amplitude which is obtained from the off-shell current by removing the propagator of the off-shell antiscalar, and imposing momentum conservation: $y_{1,n-1}=0$. The well known one-, two-, and three-gluon on-shell scattering amplitudes are A\_3(1\_s;2\^+;3\_s) = i , A\_4(1\_s;2\^+,3\^+;4\_s) = i , A\_5(1\_s;2\^+,3\^+,4\^+;5\_s) = i .
To obtain these results we have performed the following transformation in the first term of : = ( m\^2 \[2 | [p/]{}\_[3,n-1]{}|+ | [p/]{}\_1 | 2\] ) , together with = |3\] - z |2\] , & |= |3 . The only term that contributes to the recursion relation is the one where the scalar and the first gluon are factorized in the left side: A\_n(1\_s\^+;2\^+, …, n-1\^+;n\_[s]{}) = A\_3(1\_s\^+;\^+; -\_[12s]{}) A\_[n-1]{}(\_[12s]{};\^+, …, n-1\^+;n\_[s]{}) . \[bgks\] Thus, we have && A\_n(1\_s;2\^+,…,n-1\^+;n\_s) = i { |n-1\]\
&& + \_[j=1]{}\^[n-6]{} |w\_1\] } , \[scalarbcfw\] with the gauge choice $\xi=\hat{3}$ in the left amplitude, and $w_k \in [5,\ldots,n-2]$. For the channel under consideration $z=- y_{12}/[2|{p\hspace{-.42em}/\hspace{-.07em}}_1|3\ra$. Using this value for the shifted four-momenta the following relation holds after some algebra \[2|[p/]{}\_1 [/]{}\_3 [/]{}\_[12]{} [/]{}\_[34]{} = \[2| [p/]{}\_1 [p/]{}\_[23]{} (y\_[1,4]{} - [p/]{}\_4 [p/]{}\_[1,3]{}) . \[resimple\] Then, with the help of it becomes almost trivial to demonstrate that fullfils the on-shell recursion relation in . The first term in the rhs of generates all the terms that do not contain the $1/y_{1,4}$ propagator, the second term instead initiates the spinorial chains for which $w_1=4$. This fact also explains why the number of terms contributing to the amplitude doubles each time that we add one extra gluon. On the other hand, it is worth to notice that we can bring the lhs of the rhs of into the form \[2|[p/]{}\_1 [p/]{}\_[23]{} = \[2| (y\_[1,3]{}- [p/]{}\_3 [p/]{}\_[12]{}) . This suggest that we can either extend the sum in down to $w_k=3$, or even better, we can regroup all the terms in the sum into a single one by going upwards. Our final result for the amplitude with all gluons of positive helicity becomes in this way extremely compact: && A\_n(1\_s;2\^+,…,n-1\^+;n\_s) = i m\^2 . \[recompact\]
Fermionic amplitudes
====================
The all positive helicity gluon amplitudes with a heavy fermion-antifermion pair have been calculated in Ref. [@Rodrigo:2005eu]. We have worked out further these expressions with the help of and in order to obtain a more compact formulae to compare with the heavy scalar amplitude. With our spinor choice the on-shell helicity conserving amplitude vanishes, and for the helicity flip amplitude we find in a straightforward way the following relationship to the scalar amplitude A\_n(1\_q\^+;2\^+, …, n-1\^+;n\_[|[q]{}]{}\^+) &=& A\_n(1\_s;2\^+, …, n-1\^+;n\_s) , with $\beta_+$ as given in the Appendix. This represents an explicit and independent confirmation of the SUSY-like Ward identities found recently in Ref. [@Schwinn:2006ca], that relate several multigluonic amplitudes of heavy scalars and fermions. Since we have obtained a very compact expression for the scalar amplitude, the same simple result holds for the case of heavy fermions.
Summary
=======
Combining off-shell and on-shell recursion relations we have obtained an extremely compact expression for the scattering amplitude of a colored scalar-antiscalar pair and an arbitrary number of gluons of positive helicity at tree-level. We think that Eq. (\[recompact\]) is the most reduced expression one can obtain for such process. This result is the main input to obtain other helicity configurations from recursion relations. Due to its simplicity, we expect also that these other amplitudes can be calculated more efficiently and will be written in a more compact way than previously published. SUSY-like Ward identities might also help to extend these simple results to amplitudes with heavy fermions, or viceversa. In particular, we have tested explicity the validity of these identities relating scalar and fermionic amplitudes with an arbitrary number of positive helicity gluons. Eventhough these kind of relations are so far valid just at tree level.
Acknowledgements {#acknowledgements .unnumbered}
================
This work has been partially supported by Ministerio de Educación y Ciencia (MEC) under grants FPA2004-00996 and FPA 2004-04582-C02-01, Acciones Integradas DAAD-MEC (contract HA03-164), Generalitat Valenciana (GV05-015), by the European Commission RTN Program (MRTN-CT-2004-005104, MRTN-CT-2004-503369), and by the Generalitat de Catalunya (CIRIT GC 2001SGR-00065).
Spinors and heavy four-momenta
==============================
We follow the conventions of Ref. [@Rodrigo:2005eu], and denote by $p_1^\mu$ and $p_n^\mu$, with $p_1^2=p_n^2=m^2$, the four-momenta of the heavy particles. In terms of two light-like vectors ($\bar{p}_1^2=\bar{p}_n^2=0$) these four-momenta can be written as p\_1\^&=& \_+ |[p]{}\_1\^+ \_- |[p]{}\_n\^ ,\
p\_n\^&=& \_- |[p]{}\_1\^+ \_+ |[p]{}\_2\^ , where $\beta_\pm=(1\pm \beta)/2$ with $\beta=\sqrt{1-4m^2/s_{1n}}$ the velocity of the heavy particles, and $s_{1n} = (p_1+p_n)^2$. Among other advantages, this transformation preserves momentum conservation such that $p_1+p_n=\bar{p}_1+\bar{p}_n$. Furthermore, in the massless limit we have: $p_1\to \bar{p}_1$ and $p_2\to \bar{p}_2$.
If the heavy particles are fermions, we use the following choice of spinors |[u]{}\_(p\_1,m) = n\^| ([p/]{}\_1+m) , v\_(p\_n,m) = ([p/]{}\_n-m) |1\^ , where $| i^\pm \rangle = | \bar{p}_i^\pm \rangle$ are the Weyl spinors of the light-like vectors.
[90]{}
M. Jacob and G. C. Wick, Annals Phys. [**7**]{} (1959) 404 \[Annals Phys. [**281**]{} (2000) 774\]. J. D. Bjorken and M. C. Chen, Phys. Rev. [**154**]{} (1966) 1335. M. L. Mangano and S. J. Parke, Phys. Rept. [**200**]{} (1991) 301. F. A. Berends and W. T. Giele, Nucl. Phys. B [**306**]{} (1988) 759. L. J. Dixon, arXiv:hep-ph/9601359. Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Nucl. Phys. B [**425**]{} (1994) 217 \[arXiv:hep-ph/9403226\]. Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Nucl. Phys. B [**435**]{} (1995) 59 \[arXiv:hep-ph/9409265\].
V. P. Nair, Phys. Lett. B [**214**]{} (1988) 215. E. Witten, Commun. Math. Phys. [**252**]{} (2004) 189 \[arXiv:hep-th/0312171\]. F. Cachazo, P. Svrcek and E. Witten, JHEP [**0409**]{} (2004) 006 \[arXiv:hep-th/0403047\]. S. J. Parke and T. R. Taylor, Phys. Rev. Lett. [**56**]{} (1986) 2459. R. Britto, F. Cachazo and B. Feng, Nucl. Phys. B [**715**]{} (2005) 499 \[arXiv:hep-th/0412308\]. R. Britto, F. Cachazo, B. Feng and E. Witten, Phys. Rev. Lett. [**94**]{} (2005) 181602 \[arXiv:hep-th/0501052\]. Z. Bern, L. J. Dixon and D. A. Kosower, arXiv:hep-ph/0507005. Z. Bern, L. J. Dixon and D. A. Kosower, Phys. Rev. D [**71**]{} (2005) 105013 \[arXiv:hep-th/0501240\]. S. D. Badger, E. W. N. Glover, V. V. Khoze and P. Svrcek, JHEP [**0507**]{} (2005) 025 \[arXiv:hep-th/0504159\]. S. D. Badger, E. W. N. Glover and V. V. Khoze, JHEP [**0601**]{} (2006) 066 \[arXiv:hep-th/0507161\]. Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Phys. Lett. B [**394**]{} (1997) 105 \[arXiv:hep-th/9611127\]. D. Forde and D. A. Kosower, arXiv:hep-th/0507292. G. Rodrigo, JHEP [**0509**]{} (2005) 079 \[arXiv:hep-ph/0508138\]. C. Schwinn and S. Weinzierl, arXiv:hep-th/0602012.
[^1]: E-mail: [email protected]
[^2]: E-mail: [email protected]
[^3]: E-mail: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'M. Ribó'
- 'E. Ros'
- 'J. M. Paredes'
- 'M. Massi'
- 'J. Martí'
date: 'Received 2 August 2002 / Accepted 27 August 2002'
title: |
EVN and MERLIN observations of microquasar candidates\
at low galactic latitudes
---
Introduction \[sec:intro\]
==========================
Microquasars are stellar-mass black holes or neutron stars that mimic, on smaller scales, many of the phenomena seen in AGN and quasars. Microquasars have been found in X-ray binary systems, where a compact object accretes matter from a companion star. Radio Emitting X-ray Binaries (REXBs) with relativistic radio jets, like , , or , are good examples of microquasars (see Mirabel & Rodríguez [@mirabel99] for a detailed review), while other well known sources like could turn out to be new relativistic jet sources (Massi et al. [@massi02]). With the recent addition of (Paredes et al. [@paredes00]), (Stirling et al. [@stirling01]) and (Hannikainen et al. [@hannikainen01]) to the microquasar group, the current known number of this kind of sources is 14, among $\sim50$ REXBs within the $\sim280$ known X-ray binaries (Liu et al. [@liu00]; Liu et al. [@liu01]). Recent studies of microquasars can be found in Castro-Tirado et al. ([@castro01]).
As pointed out in Sect. 1 of Paredes et al. ([@paredes02], hereafter Paper I), the number of known microquasars remains still small, especially when trying to study them from a statistical point of view. An interesting aspect of microquasars is their possibility of being related to unidentified high-energy $\gamma$-ray sources, as suggested by Paredes et al. ([@paredes00]). Moreover, some parameters like the jet velocity, seem to be related to the mass of the compact object (i.e., with its potential well). Nevertheless, the lack of meaningful statistical studies because of the small population of microquasars with known jet velocities, prevents any definitive statements of this kind being made. Therefore, it is worth searching for new microquasars in order to increase the known population.
[@lr@[$^\mathrm{h}$]{}r@[$^\mathrm{m}$]{}r@[$\rlap{.}^\mathrm{s}$]{}l@[ ]{}r@r@r@[$\rlap{.}\arcsec$]{}l|lr@[$^\mathrm{h}$]{}r@[$^\mathrm{m}$]{}r@[$\rlap{.}^\mathrm{s}$]{}l@[ ]{}r@r@r@[$\rlap{.}\arcsec$]{}l@]{} &\
1RXS name & & & & &\
& 00&14&42&12822 & +58&02&01&2460 & J0007+5706$^\mathrm{a}$ & 00&07&48&47110 & +57&06&10&4540\
& & $\pm$0&00013 & & $\pm$0&0022 & & &\
& 01&31&07&23210 & +61&20&33&3752 & J0147+5840$^\mathrm{a}$ & 01&47&46&54380 & +58&40&44&9750\
& & $\pm$0&00014 & & $\pm$0&0016 & & &\
& 06&21&47&75264 & +17&47&35&0818 & J0630+1738$^\mathrm{b}$ & 06&30&07&25870 & +17&38&12&9300\
& & $\pm$0&00006 & & $\pm$0&0017 & & &\
& 07&22&59&68188 & $-$07&31&34&8009 & $^\mathrm{c}$ & 07&24&17&2912 & $-$07&15&20&339\
& & $\pm$0&00011 & & $\pm$0&0022 & & &\
$^\mathrm{d}$ & 07&24&17&2912 & $-$07&15&20&339 & J0730$-$116 & 07&30&19&1125 & $-$11&41&12&601\
& & $\pm$0&0007 & & $\pm$0&010 & & &\
[ ]{}
Position from Patnaik et al. ([@patnaik92]).
Position from Browne et al. ([@browne98]).
This position was obtained from VLA observations (listed in the line below).
The entries in this line correspond to VLA observations (see Paper I).
In this context, the sources to search for are REXBs, which are in fact microquasar candidates. To this end, a cross-identification between the X-ray ROSAT all sky Bright Source Catalog (RBSC) (Voges et al. [@voges99]) and the NRAO VLA Sky Survey (NVSS) (Condon et al. [@condon98]) was made for sources with $|b|<5\degr$ under very restrictive selection criteria, and the obtained results have been presented in Paper I. A sample containing 13 sources was obtained, and 6 of them were observed at radio wavelengths in order to obtain radio spectra and their variability, as well as accurate radio positions, which allowed the authors to discover the corresponding optical counterparts for all of them. At the end of this study, two of the sources, namely and were classified as promising microquasar candidates. The remaining four sources were classified as weaker candidates for a number of reasons. shows a highly inverted spectrum at high radio frequencies and an optical counterpart slightly extended. is an object extended in the optical. presents a one-sided radio jet at arcsecond scales, supporting the possibility of being extragalactic. And finally, is a known quasar.
In this paper we present Very Long Baseline Interferometry (VLBI) observations of these six sources, aimed at revealing possible jet-like features at milliarcsecond scales. We describe the observations and the data reduction in Sect. \[sec:obs\], present the results and a discussion in Sect. \[sec:results\], and we summarize our findings in Sect. \[sec:summary\].
Observations and data reduction \[sec:obs\]
===========================================
We observed the six sources studied in Paper I simultaneously with the Multi-Element Radio-Linked Interferometer Network (MERLIN) and the European VLBI Network (EVN) on February 29th/March 1st 2000 (23:30–23:05 UT) at 5 GHz. Since some of the target radio sources were faint, we scheduled the observations introducing phase-reference calibrators with cycle times of around 7 min (compatible with the expected coherence times). Apart from the calibrators presented in Table \[table:merlinpos\], was used as calibrator for , and and for . We also observed the fringe-finder and the MERLIN flux density calibrator . Single dish flux density measurements were carried out with the MPIfR 100 m antenna in Effelsberg, Germany.
MERLIN \[subsec:obsmerlin\]
---------------------------
MERLIN is a connected radio interferometer across England, with baselines reaching up to 217 km length. This array observed with 2-bit sampling at dual polarisation with two blocks of 16 channels, each channel of 1 MHz bandwidth. We analysed the left hand circular polarisation data excluding one channel at both edges of the band, yielding a final bandwidth of 14 MHz. The correlator integration time was of 4 s. The MERLIN data reduction was carried out at Jodrell Bank Observatory, using standard procedures within the Astronomical Image Processing System ([aips]{}, developed and maintained by the US National Radio Astronomy Observatory). We did not detect . All other sources were detected, and accurate positions for the target radio sources were obtained via phase-referencing. These positions, presented in Table \[table:merlinpos\], were used later as [*a priori*]{} information for the VLBI correlation. The position given in Table \[table:merlinpos\] for the quasar was obtained from the VLA observations presented in Paper I. The position of is deduced from the phase-reference offset relative to provided by MERLIN.
To image the sources, we averaged the data in frequency and exported them to be processed into the difference mapping software [difmap]{} (Shepherd et al. [@shepherd94]), where we time-averaged the data in 32 s bins after careful editing.
EVN \[subsec:obsevn\]
---------------------
The EVN observations were performed with the following array (name, code, location, diameter): Effelsberg, EB, Germany, 100 m; Jodrell Bank, JB, U.K., 25 m; Cambridge, CM, U.K., 32 m; Westerbork, WB, The Netherlands, 14$\times$25 m; Medicina, MC, Italy, 32 m; Noto, NT, Italy, 32 m; Shanghai, SH, China, 25 m; Toruń, TR, Poland, 32 m; and Onsala85, ON, Sweden, 25 m. Data were recorded in MkIV mode with 2-bit sampling at 256 Mbps with left hand circular polarization. A bandwidth of 64 MHz was used, divided into 8 intermediate frequency (IF) bands.
The data were processed at the EVN MkIV correlator at the Joint Institute for VLBI in Europe (JIVE), in Dwingeloo, The Netherlands. The correlator integration time was of 4 s. A first post-processing analysis was also carried out at JIVE. The data were processed using [aips]{}. A first [*a priori*]{} visibility amplitude calibration was performed using antenna gains and system temperatures measured at each antenna. The fringe fitting ([fring]{}) of the residual delays and fringe rates was performed for all the radio sources. No fringes were found for and . Fringes for many baselines were missing for , and .
To improve the fringe detection on all baselines for and we used the delay, rate, and phase solutions from their corresponding phase-reference calibrators (, 118 separation, and , 159 separation) and interpolated them to the target sources using the [aips]{} task [clcal]{}. We fringe-fitted the target sources again using narrower search windows and obtained solutions for all baselines. A similar attempt on (with respect to and ) was unfruitful. Effelsberg was used as reference antenna throughout the [aips]{} data reduction process.
We then averaged the data in frequency and exported them to be imaged and self-calibrated in [difmap]{}. The [*a priori*]{} visibility amplitude calibration was not sufficient to reliably image the weakest radio sources. We improved that by first imaging in [difmap]{} the calibrator sources , , and , with appropriate amplitude self-calibration. We deduced correction factors for each antenna, these being consistent for the three radio sources within 2%. We corrected the amplitude calibration back in [aips]{} and exported the data again into [difmap]{}, where the final imaging was performed after editing and averaging of the visibilities in 32 s blocks.
Combining EVN and MERLIN \[subsec:evn+merlin\]
----------------------------------------------
The EVN and MERLIN arrays have one common baseline, between JB and CM, which allows to combine both data sets and map them together. We processed the data within [aips]{} in order to combine both arrays. The B1950.0 $(u,v)$ coordinates of the MERLIN data had to be corrected to the ones of the EVN for the same reference system (J2000.0) with [uvfix]{}. Then, the MERLIN data were self-calibrated with the EVN images (see below), and the phase solutions were limited to the longest MERLIN baselines. The MERLIN data were imaged and the peak-of-brightness of both data sets were checked to be similar. As a next step, the EVN data were averaged in frequency to correspond to the MERLIN data. The [aips]{} headers of both data sets were modified conveniently to match together, and the weighting of both data sets was also modified to be equal with [wtmod]{}. Finally, both data sets were concatenated (using [dbcon]{}) and exported to [difmap]{} to be imaged with different data weighting in $(u,v)$ distance (tapering) after time averaging in 32 s bins.
Flux density measurements at the 100 m antenna in Effelsberg \[subsec:effelsberg\]
----------------------------------------------------------------------------------
We interleaved cross-scans (in azimuth and elevation) with the 100 m Effelsberg antenna to measure the radio source flux densities (A. Kraus, private communication). We fitted a Gaussian function to the flux-density response for every cross-scan, and we averaged the different Gaussians. We linked the flux density scale by observing primary calibrators such as , , or (see e.g. Kraus [@kraus97]; Peng et al. [@peng00]). We list the single dish flux density measurements in the second column of Table \[table:param\]. The flux density values for the main VLBI calibrators were of $7.5\pm0.1$ Jy for 3C 286, and $5.9\pm0.2$ Jy for DA 193.
Results and discussion \[sec:results\]
======================================
We present all the imaging results in Figs. \[fig:pmr1\]–\[fig:pmr10\], and the image parameters in Table \[table:param\]. The total flux density values for the different images diverge from each other and from the single dish measurements, due to the amplitude self-calibration process in all cases. Therefore, those values should be considered with care. The minimum contours in the images are those listed as $S_\mathrm{min}$ in Table \[table:param\], while consecutive higher contours scale with $3^{1/2}$. Here follows a detailed discussion on each source.
[@lr@[$\,\pm\,$]{}lcr@[$\,\times\,$]{}lcccc@]{} & & &\
1RXS name & & & & P.A. & $S_\mathrm{tot}$ & $S_\mathrm{peak}$ & $S_\mathrm{min}$\
& & & \[mas\] & \[mas\] & & \[mJy\] & \[mJy beam\] & \[mJy beam\]\
& 6.5&0.5$^\mathrm{b}$ & M & 57 & 51 & 37 & 6.2 & 5.8 & 0.1\
& & E+M (10) & 7.7 & 7.2 & $-$4 & 10.1 & 9.9 & 0.18\
& & E & 1.77 & 0.86 & $-$20 & 11.5 & 7.0 & 0.15\
& 20.1&0.6 & M & 71 & 39 & $-$63 & 17.5 & 17.9 & 0.5\
& & E+M (15) & 7.8 & 6.2 & $-$75 & 19.2 & 18.7 & 0.8\
& & E & 1.04 & 0.99 & $-$1 & 17.6 & 11.6 & 0.4\
& & — & & & & &\
& 7.1&0.7$^\mathrm{b}$ & M & 88 & 39 & 24 & 5.8 & 5.6 & 0.2\
& & E+M (8) & 13.1 & 11.1 & $-$24 & 6.0 & 6.8 & 0.3\
& & E & 8.8 & 4.1 & 47 & 7.0 & 6.4 & 0.3\
& 67.9&1.1 & M & 115 & 66 & 4 & 66.0 & 62.9 & 0.7\
& & E+M (15) & 9.5 & 7.3 & $-$61 & 52.1 & 41.9 & 0.9\
& & E & 5.74 & 1.12 & 11 & 46.9 & 36.2 & 0.9\
& 282.2&4.1 & M & 120 & 64 & 8 & 301.0 & 285.7 & 0.9\
& & E+M (10) & 11.7 & 9.8 & $-$61 & 287.0 & 263.4 & 2.0\
& & E & 5.73 & 1.02 & 11 & 248.0 & 184.8 & 0.7\
[ ]{}
M: MERLIN. E+M: EVN+MERLIN, FWHM of the tapering function (weighting of visibilities) in parenthesis. E: EVN.
Values with low SNR in the Gaussian fits.
and its two-sided jet \[subsec:pmr1\]
--------------------------------------
As can be seen in our images, shown in Fig. \[fig:pmr1\], this source appears point-like at MERLIN resolution, partially resolved in the tapered EVN+MERLIN image and clearly resolved at EVN scales. In this last case, it shows a two-sided jet-like structure roughly in the north-south direction, with brighter components towards the south. The trends are visible in the closure phases, giving us confidence that the structure observed is not a consequence of sidelobes or imaging artifacts. In the tapered EVN+MERLIN images, the structure extends up to 20–30 mas outside of the core, and more clearly towards the north. This discrepancy could be due to calibration problems.
Model fitting of the EVN visibilities with circular Gaussians provides a parametrization of the inner structure. Five components reproduce the visibilities. The central one has 7.8 mJy, with a FWHM of 0.4 mas. Towards the north, one component (N2) of 0.6 mJy at 3.6 mas (P.A. $-10\degr$, FWHM 0.7 mas) and another one (N1) of 0.5 mJy at 8.6 mas (P.A. $2\degr$, extended over 3 mas) are needed. Brighter components are present southwards, one (S2) of 1.1 mJy at 5.0 mas (P.A. $177\degr$, FWHM of 0.8 mas) and the other one (S1) of 0.4 mJy at 13.7 mas (P.A. $177\degr$, FWHM below 0.3 mas).
If we assume that components S1 and N1 correspond to a pair of plasma clouds ejected at the same epoch near the compact object and perpendicularly to the accretion disk, we can estimate some parameters of the jets by using the following equation: $$\beta\cos\theta={{\mu_{\rm a}-\mu_{\rm r}}\over{\mu_{\rm a}+\mu_{\rm r}}}={{d_{\rm a}-d_{\rm r}}\over{d_{\rm a}+d_{\rm r}}}~~,
\label{eqdist}$$ $\beta$ being the velocity of the clouds in units of the speed of light, $\theta$ the angle between the direction of motion of the ejecta and the line of sight and $\mu_{\rm a}$ and $\mu_{\rm r}$ the proper motions of the approaching and receding components, respectively (Mirabel & Rodríguez [@mirabel99]). Although we do not know the epoch of ejection of the clouds, we can cancel the time variable by using the relative distances to the core $d_{\rm a}$ and $d_{\rm r}$, as expressed in Eq. \[eqdist\]. Since both variables, $\beta$ and $\cos\theta$, take values between 0 and 1, it is clear that knowing $\beta\cos\theta$ allows us to compute a lower limit for the velocity ($\beta_{\rm min}$) and an upper limit for the angle ($\theta_{\rm
max}$). The same applies for the S2 and N2 components. In Table \[table:components\] we list the positions of the components obtained from model fitting, together with the derived values from $\beta_{\rm min}$ and $\theta_{\rm max}$ for each one of the pairs. The slightly different results obtained using pair 1 or 2, could be due to the fact that the position for the S1 component obtained with model fitting happens to be at the lower part of this elongated component, hence increasing $\beta\cos\theta$, or to intrinsic different velocities for each one of the pairs. Hereafter we will use $\beta>0.20\pm0.02$ and $\theta<78\pm1\degr$.
A similar approach to obtain the jet parameters of the source can be performed thanks to the brightness asymmetry of the components using the following equation (Mirabel & Rodríguez [@mirabel99]): $$\beta\cos\theta={\big({S_{\rm a}/{S_{\rm r}}}\big)^{1/(k-\alpha)}-1 \over \big({S_{\rm a}/{S_{\rm r}}}\big)^{1/(k-\alpha)}+1}~~,
\label{eqflux}$$ where $S_{\rm a}$ and $S_{\rm r}$ are the flux densities of the approaching and receding components, respectively, $k$ equals 2 for a continuous jet and 3 for discrete condensations, and $\alpha$ is the spectral index of the emission ($S_{\nu}\propto \nu^{+\alpha}$). However, the equation above is only valid when the components are at the same distance from the core. If this is not the case (i.e., $d_{\rm r}<d_{\rm a}$), the ratio $S_{\rm a}/S_{\rm r}$ will be lower than the one that should be used in Eq. \[eqflux\] (because the flux density decreases with increasing distance from the core). In consequence, Eq. \[eqflux\] only allows us to obtain a lower limit for $\beta\cos\theta$.
[@ccr@[ ]{}|@[ ]{}cc@]{} Comp. & Distance & P.A. & $\beta_{\rm min}$ & $\theta_{\rm max}$\
& \[mas\] & \[\] & & \[\]\
N1 & $8.6\pm0.3$ & 2 & $0.23\pm0.02$ & $77\pm1$\
N2 & $3.6\pm0.2$ & $-$10 & $0.16\pm0.02$ & $81\pm1$\
S2 & $5.0\pm0.1$ & 177 & $0.16\pm0.02$ & $81\pm1$\
S1 & $13.7\pm0.3$ & 177 & $0.23\pm0.02$ & $77\pm1$\
In order to use this approach we will consider $k=3$, because the components seem to be discrete condensations, and $\alpha=-0.20\pm0.05$, according to the overall spectral index reported in Paper I, since we do not have spectral index information of the components. The use of the flux densities obtained after model fitting gives $\beta\cos\theta>0.09\pm0.04$ for the S2–N2 pair and $\beta\cos\theta>-0.07\pm0.08$ for the S1–N1 pair. This last value is certainly surprising, although it can be explained by the fact that the S1 flux density obtained after model fitting does not account for the total flux of this plasma cloud. In fact, better estimates of the flux densities can be obtained summing together the flux densities of the [clean]{} components obtained within each one of the four plasma clouds. This yields to $\beta\cos\theta>0.13\pm0.05$ for the S2–N2 pair and $\beta\cos\theta>0.17\pm0.02$ for the S1–N1 pair, in good agreement with the values computed using the distances from the components to the core, shown in Table \[table:components\].
If we compare the VLA position reported in Paper I with the MERLIN position in Table \[table:merlinpos\] we can see that they are different. In fact, taking into account the errors, the MERLIN$-$VLA position offsets can be expressed as: $\Delta\alpha\cos\delta=16\pm10$ mas and $\Delta\delta=27\pm10$ mas. Assuming that the difference in position is due to intrinsic proper motions and considering the time span between both observations, 224 days, we obtain: $\mu_{\alpha\cos\delta}=26\pm16$ mas yr$^{-1}$ and $\mu_{\delta}=44\pm16$ mas yr$^{-1}$. Although a proper motion of the source would clearly indicate its microquasar nature, because no proper motion would be detected in an extragalactic source, we must be cautious with this result. First of all, the phase-reference sources where different in the VLA and MERLIN observations, as well as the observing frequencies from which positions were estimated. On the other hand, none of these results exceeds the $3\sigma$ value, preventing to state that a proper motion has been detected.
Summarizing, our results indicate that this source exhibits relativistic radio jets with $\beta>0.20$ and, therefore, together with the results reported in Paper I, we consider as a very promising microquasar candidate.
and its one-sided jet \[subsec:pmr3\]
--------------------------------------
We show in Fig. \[fig:pmr3\] the images obtained after our observations. Although the source appears compact at MERLIN and EVN+MERLIN scales, there is a weak one-sided radio jet towards the northwest in the EVN image. Model fitting with circular Gaussian components can reproduce the observed visibilities as follows: a central 15.4 mJy component with 0.64 mas FWHM, and a northwest 2.1 mJy component with a FWHM of 0.74 mas, located at 1.8 mas in P.A. $-73\degr$. As can be seen, this last component is located at a distance of $\sim2$ times the beam size from the core.
Using the fact that we do not detect a counter-jet, we can use Eq. \[eqflux\] replacing $S_{\rm r}$ with the $3\sigma$ level value. This, of course, will only provide a lower limit to $\beta\cos\theta$, expressed as follows: $$\beta\cos\theta> {\big({S_{\rm a}/3\sigma}\big)^{1/(k-\alpha)}-1 \over \big({S_{\rm a}/3\sigma}\big)^{1/(k-\alpha)}+1}~~.
\label{eqsigma}$$ Using $S_{\rm a}=2.1$ mJy, $3\sigma=0.30$ mJy (the $1\sigma$ value has been taken as the root mean square noise in the image), $\alpha=-0.05\pm0.05$ (see Paper I), and $k=3$ to be consistent with the lowest limit, we obtain $\beta\cos\theta>0.31\pm0.05$ ($\beta>0.31\pm0.05$ and $\theta<72\pm3$). Hence, a lower limit of $\beta\geq0.3$ is obtained, pointing towards relativistic radio jets as the origin of the elongated radio emission present in the EVN image. Although the one-sided jet morphology at mas scales is found mostly in extragalactic sources, it is also present in some galactic REXBs, like (Mioduszewski et al. [@mioduszewski01]) or (Massi et al. [@massi01]). Hence, we cannot rule out a possible galactic nature on the basis of the detected morphology.
As done for the previous source, we can compare the VLA position with the MERLIN one, and find that they differ in $\Delta\alpha\cos\delta=39\pm10$ mas and $\Delta\delta=-0.8\pm10$ mas. If the offsets are real, this would imply $\mu_{\alpha\cos\delta}=64\pm16$ mas yr$^{-1}$ and $\mu_{\delta}=-1\pm16$ mas yr$^{-1}$. Hence, it seems possible that we have detected a proper motion in right ascension at a $4\sigma$ level. However, we must be cautious since, as in the previous case, the phase-reference sources and observing frequencies were different in each observation.
Overall, these results are indicative of relativistic radio jets, and hence, together with the results reported in Paper I, allow us to classify this source as a promising microquasar candidate.
, a non-detected source \[subsec:pmr4\]
---------------------------------------
This radio source was marginally seen in the cross-scans carried out in Effelsberg. In fact, this is compatible with the low flux density of $2.3\pm0.4$ mJy at 1.4 GHz listed in the NVSS. The source was not detected with MERLIN or with the EVN, mainly due to problems with the phase-reference sources, as pointed out in Sect. \[subsec:obsevn\]. In fact, as discussed in Paper I, VLA A configuration observations showed a flux density of 0.4 mJy at 5 GHz and an inverted spectrum with a spectral index up to $\alpha=+1.6$, suggesting thermal radio-emission resolved at higher angular resolutions. However, we cannot exclude the possibility of having a highly variable source.
, a compact source \[subsec:pmr7\]
----------------------------------
As can be seen in our images, shown in Fig. \[fig:pmr7\], the radio source is compact on all scales. In fact, model fitting of the EVN visibilities converges to a point-like radio source (the FWHM of a circular Gaussian tends to zero). We must note that when this source was observed it was below the horizon in SH. Therefore, the obtained beam size for the EVN image is larger than the ones obtained for the other sources, as can be seen in Table \[table:param\], hence providing lower angular resolution than in the other cases. A comparison between the VLA and MERLIN positions reveals that they are perfectly compatible within the errors, suggesting an extragalactic nature or a small proper motion if it turns out to be a galactic microquasar. Although the compactness of the source is not indicative of a galactic or an extragalactic origin, the extended optical counterpart, reported in Paper I, suggests an extragalactic origin for this source. In fact, the detected radio variability from 8 to 11 mJy within a week reported in Paper I, is compatible with the compactness in an extragalactic object (see IDV phenomenon, Wagner & Witzel [@wagner95]).
and its bent one-sided jet \[subsec:pmr9\]
-------------------------------------------
The images obtained at different resolutions are plotted in Fig. \[fig:pmr9\]. The MERLIN image presents a compact structure with some elongation eastwards, while the EVN and combined EVN+MERLIN images show a clear one-sided jet towards the east, with a slight bent towards the south at larger core separations. The closure phases clearly show that the source departs from symmetry, with preferred emission to the east, both in the MERLIN and the EVN data sets. Two distinct components are present in the EVN image, at 9 and 17 mas from the compact core (P.A. of 89 and 113$\degr$, respectively). Those components can be model fitted with elliptical Gaussians, yielding flux densities of 3.7 and 2.8 mJy, respectively, for a core of 40.3 mJy.
Using again the fact that we do not detect a counter-jet, we can use Eq. \[eqsigma\] with $S_{\rm a}=3.7$ mJy (the closest component to the core in the EVN image), $3\sigma=0.54$ mJy (as previously done, the $1\sigma$ value has been taken as the root mean square noise in the image), $\alpha=-0.25\pm0.2$ (see Paper I), and $k=3$ to be consistent with the discrete nature of the components, to obtain $\beta\cos\theta>0.29\pm0.05$, and hence $\beta>0.29\pm0.05$ and $\theta<73\pm3\degr$. Therefore, these results point towards relativistic radio jets as the origin of the elongated radio emission present in the images.
As pointed out for , the one-sided jet morphology does not rule out a microquasar nature for this object. However, the bending of the jet at such small angular scales resembles the ones seen in blazars. In fact, as reported in Paper I, a one-sided arcsecond scale jet is also present in VLA A configuration observations at 1.4 GHz, an unusual feature in the already known microquasars.
A comparison between the VLA and MERLIN positions reveals that they agree within the errors, suggesting an extragalactic nature or a small proper motion if it turns out to be a galactic microquasar.
Overall, these results are indicative of relativistic radio jets, although this source shows characteristics more similar to blazars than to microquasars.
, a quasar with a bent one-sided jet \[subsec:pmr10\]
-----------------------------------------------------
This radio source has recently been (March 2002) classified as a quasar in the SIMBAD database, and it is not any more a microquasar candidate. It is catalogued in the Parkes-MIT-NRAO (Griffith et al. [@griffith94]) and the Texas Survey (Douglas et al. [@douglas96]). It is listed as in the NED database, and is the source in Perlman et al. ([@perlman98]), who reported a faint and quite broad H$\alpha$ emission line (rest-frame $W_{\lambda}=30.3$ Å, FWHM=4000 km s$^{-1}$), and classified it as a Flat Spectrum Radio Quasar (FSRQ) with $z=0.270$. Nevertheless, we have reported here our observational results for this source, since it was a candidate when we performed the observations. It presents (Fig. \[fig:pmr10\]) a one-sided pc-scale jet oriented towards the northeast, changing from a P.A. of $\sim50\degr$ at 5 mas from the core (EVN image) to $20\degr$ up to 200 mas, at MERLIN scales.
Model fitting of the EVN visibilities with circular Gaussians reveals a compact core (0.7 mas FWHM) with 245.4 mJy, and two distinct components, one with 11.9 mJy at 6.2 mas (P.A. $46\degr$, 1.9 mas FWHM, present in the right panel image of Fig. \[fig:pmr10\]) and another one with the size of the beam, a flux density of 1.7 mJy at a distance of 29.6 mas in P.A. $27\degr$ (visible in the middle panel image of Fig. \[fig:pmr10\]).
[@ll@cl@[ ]{}cl@[ ]{}c@[ ]{}cl@]{} 1RXS name & & & & Notes\
& Structure & $\alpha$ & Structure & $I$ mag. & Structure & $\beta$ & $\theta[\degr]$ &\
& compact & $-0.2$ & point-like & 19.9 & two-sided jet & $>0.20$ & $<78$ & Promising candidate\
& compact & $-0.1$ & point-like & 17.9 & one-sided jet & $>0.31$ & $<72$ & Promising candidate\
& compact & $+1.6^*$ & extended$^*$ & 17.5 & not detected$^*$ & — & — & Thermal source ?\
& compact & $+0.1$ & extended$^*$ & 17.6 & compact & — & — & Galaxy ?\
& one-sided jet$^*$ & $-0.2$ & point-like & 16.8 & bent one-sided jet$^*$ & $>0.29$ & $<73$ & Blazar ?\
& compact & $+0.1$ & point-like & 17.2 & bent one-sided jet$^*$ & $>0.51$ & $<60$ & Quasar (FSRQ)\
Using again the fact that we do not detect a counter-jet, we can use Eq. \[eqsigma\] with $S_{\rm a}=11.9$ mJy (the closest component to the core), $3\sigma=0.45$ mJy, $\alpha=0.07\pm0.02$ (see Paper I), and $k=3$ to be consistent with the discrete nature of the components, to obtain $\beta\cos\theta>0.51\pm0.04$. Hence, an upper limit of $\theta<60\pm3\degr$ and a lower limit of $\beta>0.51\pm0.04$ is obtained, pointing towards relativistic radio jets as the origin of the elongated radio emission present in the images of this already identified quasar.
Summary \[sec:summary\]
=======================
We have presented EVN and MERLIN observations of the six sources studied by Paredes et al. ([@paredes02]) in their search for microquasar candidates at low galactic latitudes. The first one, namely , displays a two-sided radio jet, which after analysis implies $\beta>0.20\pm0.02$ and $\theta<78\pm1\degr$. , displays a one-sided radio jet, requiring $\beta>0.31\pm0.05$ and $\theta<72\pm3\degr$. The third one, namely , was not detected due to its low flux density and/or to phase-referencing problems. appeared compact at all scales. The fifth one, namely , displays a bent one-sided radio jet, implying $\beta>0.29\pm0.05$ and $\theta<73\pm3\degr$. Finally, shows also a bent one-sided jet, requiring $\beta>0.51\pm0.04$ and $\theta<60\pm3\degr$.
After a detailed analysis of our data, we show in Table \[table:summary\] a summary of the results obtained after the VLA and optical observations (Paper I), and the EVN+MERLIN observations reported here. As can be seen, the first two sources, and , are promising microquasar candidates. is probably of thermal nature due to the highly inverted spectrum at high radio frequencies, while is probably an extragalactic object due to the extended nature of the optical counterpart. shows properties common to blazars, while is an already identified quasar. We note that is bright enough at radio wavelengths to attempt an absorption experiment, that could allow to determine if this source is galactic or not. In any case, optical spectroscopic observations of the first five sources are in progress, to clearly unveil their galactic or extragalactic nature.
We acknowledge R. Porcas and W. Alef for their useful comments and suggestions after reading through a draft version of this paper. We acknowledge useful comments from L. F. Rodríguez, the referee of this paper. We are very grateful to S. T. Garrington, M. A. Garrett, D. C. Gabuzda, and C. Reynolds for their valuable help in the data reduction process. This paper is based on observations with the 100-m telescope of the MPIfR (Max-Planck-Institut für Radioastronomie) at Effelsberg. We thank the staff of the JIVE correlator and of the observing telescopes, especially A. Kraus for the single dish flux density measurements at the 100 m antenna in Effelsberg. The European VLBI Network is a joint facility of European, Chinese and other radio astronomy institutes funded by their national research councils. MERLIN is operated as a National Facility by the University of Manchester at Jodrell Bank Observatory on behalf of the UK Particle Physics & Astronomy Research Council. The EVN observations were carried out thanks to the TMR Access to Large-scale Facilities programme under contract No. ERBFMGECT950012. Part of the data reduction was done at JIVE with the support of the European Community - Access to Research Infrastructure action of the Improving Human Potential Programme under contract No. HPRI-CT-1999-00045. M. R., J. M. P. and J. M. acknowledge partial support by DGI of the Ministerio de Ciencia y Tecnología (Spain) under grant AYA2001-3092, as well as partial support by the European Regional Development Fund (ERDF/FEDER). During this work, M. R. has been supported by two fellowships from CIRIT (Generalitat de Catalunya, ref. 1998 BEAI 200293 and 1999 FI 00199). J. M. has been aided in this work by an Henri Chrétien International Research Grant administered by the American Astronomical Society, and has been partially supported by the Junta de Andalucía. This research has made use of the NASA’s Astrophysics Data System Abstract Service, of the SIMBAD database, operated at CDS, Strasbourg, France, and of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
Browne, I. W. A., Wilkinson, P. N., Patnaik, A. R., & Wrobel, J. M. 1998, MNRAS, 293, 257
Castro-Tirado, A. J., Greiner, J., & Paredes, J. M. 2001, Microquasars, Proc. of the Third Microquasar Workshop on ‘Galactic Relativistic Jet Sources’, (Kluwer Academic Publishers, Dordrecht, The Netherlands), Ap&SS, 276
Condon, J. J., Cotton, W. D., Greisen, E. W., et al. 1998, AJ, 115, 1693
Douglas, J. N., Bash, F. N, Bozyan, F. A, Torrence, G. W., & Wolfe, C. 1996, AJ, 111, 1945
Griffith, M. R., Wright, A. E., Burke, B. F., & Ekers, R. D. 1994, ApJS, 90, 179
Hannikainen, D., Campbell-Wilson, D., Hunstead, R., et al. 2001, in Proc. of the Third Microquasar Workshop ‘Galactic Relativistic Jet Sources’, ed. A. J. Castro-Tirado, J. Greiner, & J. M. Paredes (Kluwer Academic Publishers), Ap&SS, 276, 45
Kraus, A. 1997, PhD Thesis, Friedrich-Wilhelms-Universität Bonn, Germany
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Massi, M., Ribó, M., Paredes, J. M., et al. 2002, Sub-arcsecond radio jets in the high mass X-ray binary LS I +61 303, in Proceedings of the 6th European VLBI Network Symposium, ed. E. Ros, R. W. Porcas, A. P. Lobanov, & J. A. Zensus, (Max-Planck-Institut für Radioastronomie, Bonn, Germany), 279
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|
{
"pile_set_name": "ArXiv"
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|
---
abstract: 'The HARP experiment was designed to study hadron production in proton–nucleus collisions in the energy range of 1.5 [$\mbox{GeV}/c$]{}–15 [$\mbox{GeV}/c$]{}. The experiment was made of two spectrometers, a forward dipole spectrometer and a large-angle solenoid spectrometer. In the large-angle spectrometer the main tracking and particle identification is performed by a cylindrical Time Projection Chamber (TPC) which suffered a number of shortcomings later addressed in the analysis. In this paper we discuss the effects of time-dependent (*dynamic*) distortions of the position measurements in the TPC which are due to a build-up of ion charges in the chamber during the accelerator spill. These phenomena have been studied both theoretically and experimentally, and a correction procedure has been developed. First, the dynamics of the positive ion cloud and of the full electrostatics of the field-cage system have been modelled with a phenomenological approach and a general correction procedure has been developed and applied to all data settings. Then, the correction procedure has been benchmarked experimentally by means of recoil protons in elastic scattering reactions, where the track coordinates are precisely predictable from simple kinematical considerations. After application of the corrections for dynamic distortions the corrected data have a performance equal to data where the dynamic distortions are absent. We describe the theoretical model, the comparison with the measurements, the distortion correction method and the results obtained with experimental data.'
address:
- 'Institute for Nuclear Research, Moscow, Russia'
- ' Université de Genève, Switzerland'
- ' Current address University of Glasgow, UK'
- 'INFN, Bari, Italy'
- 'INFN, Trieste, Italy'
- 'Laboratori Nazionali di Legnaro dell’ INFN, Legnaro, Italy'
- 'CERN, Geneva, Switzerland'
- 'P. N. Lebedev Institute of Physics (FIAN), Russian Academy of Sciences, Moscow, Russia'
- 'On leave of absence from Ecoanalitica, Moscow State University, Moscow, Russia'
- 'Faculty of Physics, St. Kliment Ohridski University, Sofia, Bulgaria'
- 'ITEP, Moscow, Russian Federation'
author:
- 'A. Bagulya'
- 'A. Blondel'
- 'S. Borghi'
- 'G. Catanesi'
- 'P. Chimenti'
- 'U. Gastaldi'
- 'S. Giani'
- 'V. Grichine'
- 'V. Ivanchenko'
- 'D. Kolev'
- 'J. Panman'
- 'E. Radicioni'
- 'R. Tsenov'
- 'I. Tsukerman'
title: 'Dynamic Distortions in the HARP TPC: observations, measurements, modelling and corrections'
---
Introduction {#sec:intro-harp}
============
The HARP experiment [@ref:harp:spsc99; @ref:harp:detector] was designed to study hadron production in proton–nucleus collisions in the energy range of 1.5 [$\mbox{GeV}/c$]{}–15 [$\mbox{GeV}/c$]{}. The main aim of the experiment is to provide pion production data for the calculation of neutrino fluxes in conventional neutrino beams at accelerators, to provide data for extended air shower simulations and for prediction of the atmospheric neutrino flux, as well as to provide input to the quantitative design of a future neutrino factory.
The experiment was made of two spectrometers:
- A forward dipole spectrometer with planar drift chambers for the particle tracking and a time-of-flight (TOF) scintillator wall, a Cherenkov detector and an electromagnetic calorimeter for particle identification (PID).
- A large-angle solenoid spectrometer where the main tracking and PID is performed by a cylindrical Time Projection Chamber (TPC) occupying most of the radial space of the solenoid magnet. The TPC provides track, momentum and vertex measurements for all outgoing charged particles in the angular range from 20$^\circ$ to 135$^\circ$ with respect to the beam axis. In addition, it provides particle identification by recording the particle’s energy loss in the gas ([$\mbox{d}E/\mbox{d}x$]{}). The PID capabilities of the TPC detector are complemented by a set of multi-gap RPCs (Resistive Plate Chambers) serving as TOF detectors and surrounding the TPC.
Data analysed with the large-angle spectrometer have been published in Refs. [@ref:tantalum; @ref:harp:art1; @ref:harp:art2; @ref:harp:art3]
The HARP TPC {#sec:intro-tpc}
============
The schematic layout of the HARP TPC is shown in Fig. \[fig:tpc\]. The TPC is positioned inside the solenoid magnet, providing a magnetic volume with a diameter of 0.9 [$\mbox{m}$]{}, a length of 2.25 [$\mbox{m}$]{}and a field of 0.7 [$\mbox{T}$]{}in the main sensitive volume. The magnet was previously used for the R&D of the ALEPH experiment’s TPC [@ref:tpc90] and later modified for HARP. The downstream end of the return yoke was left open to minimize materials encountered by secondary particles emerging from the TPC in the direction of the forward spectrometer. At the upstream end there is a small cylindrical hole in the end-cap yoke for the passage of the incident beam and to leave space to insert the inner trigger cylinder (ITC) and target holder inside the inner field cage (IFC). The drift volume is 1541 [$\mbox{mm}$]{}long with a nominal electric field gradient of 111 V/[$\mbox{cm}$]{}. Given the drift velocity of the chosen gas mixture under these operating conditions, the maximum drift time is approximately 30 [$\mu \mbox{s}$]{}. The induced charge from the gas amplification at the anode wires is measured using a pad plane, subdivided in six sectors; the anode wires are strung onto the six spokes defining the sectors. The pads are organized in 20 concentric rows, each pad being connected to an individual pre-amplifier. The pad dimensions are 6.5 [$\mbox{mm}$]{} $\times$ 15 [$\mbox{mm}$]{}and the number of pad ranges from 11 per row per sector at the inner radius to 55 at the outer radius. The pad-charges are sampled into charge time-series by one Flash-ADC (FADC) per pad, with a sampling interval of 100 [$\mbox{ns}$]{}. The total Data Acquisition (DAQ) readout time is 500 [$\mu \mbox{s}$]{}to 1000 [$\mu \mbox{s}$]{}per event depending on the event size.
During the analysis, after unpacking the FADC values, time-series are organized in $R\phi$ clusters. The clusters are assigned to tracks by a tree-based algorithm for pattern recognition which used a general three-dimensional binary search method for fast look-up of clusters [@ref:uiterwijk].
Once clusters are assigned to a track, a helix fit is performed. The fitting procedure is based on the algorithm developed by the ALEPH Collaboration [@ref:aleph] with slight modifications [@ref:morone], [*e.g.*]{} the possibility to fit tracks which spiral for more than 2$\pi$ [@ref:silvia:thesis]. The fit consists of two consecutive steps: a circle-fit in the $x$–$y$ plane[^1], based on a least-square method [@ref:chernov], and a subsequent straight line fit in the $z$–$s_{xy}$ plane[^2]. The two fitting steps allow the five parameters which uniquely define the helix to be determined. The code uses the same naming and sign conventions as in the TASSO and ALEPH software [@ref:aleph] with a particle direction associated to the motion along the helix itself.
The analysis revealed that the TPC suffered from a number of operational problems which were discovered, one after the other, during and after the data taking:
1. large excursions of the gains of the pad pre-amplifiers;
2. a relatively large number of dead or noisy pads;
3. large pad gain variations with time;
4. static distortions caused by the inhomogeneity of the magnetic field, an accidental HV mismatch between the inner and outer field cage;
5. cross-talk between pads caused by capacitive coupling between signal lines in the multilayer printed boards;
6. dynamic distortions caused by build-up of ion-charge density in the drift volume during the 400 ms long beam spill.
The corrections for the first four effects are described in Refs. [@ref:harp:detector; @ref:tantalum; @ref:silvia:thesis]. A detailed discussion of the cross-talk effect can be found in Ref. [@ref:lara]. The dynamic distortion effect and its corrections are presented in this paper.
Evidence for TPC *Dynamic Distortions* and overall characteristics {#sec:evidence}
==================================================================
In a situation where an incoming beam hits a target at the centre of a rotationally symmetric TPC one expects the tracks of produced particles to a good approximation to emerge from the centre of the TPC. We then define [$d_0$]{}to be the impact point of the tracks in the $xy$ plane, i.e. the minimum distance between the track and the $z$-beam axis in the $xy$ plane. By convention, its sign indicates whether the helix encircles the $z$-beam axis (positive sign) or not (negative sign). The distribution of [$d_0$]{}is then expected to be symmetric around the origin. The presence of distortions in the TPC can modify this distribution. Thus, the distribution of the distance of closest approach of the helix to the nominal axis of the TPC is a measure of distortions in the TPC. The [$d_0$]{}distribution of TPC tracks was found to show a difference between the peaks for tracks with opposite curvature, wih a separation depending on the beam tuning and intensity [@ref:silvia:thesis]. This effect is shown in Fig. \[fig:distortion:evidence\]. The two panels of the figure show two different runs of the same setting (8 [$\mbox{GeV}/c$]{}on a 5% $\lambda_{\mathrm{I}}$ Be target) taken close to each other in time, one just before and the other after re-tuning of the beam.
The presence of the effect persisted even after correcting for static distortion due to voltage misalignment between the inner and outer field cages, the effect of which was shown to be much less important.
A better measure of the impact parameter, namely the track impact distance with respect to the trajectory of the incoming beam particle, [$d'_0$]{}[^3], was found to be a very sensitive probe to measure the distortion strength. The difference between [$d'_0$]{}and [$d_0$]{}may be large due to the relatively large width of the beam spot at the target ($\approx$5 [$\mbox{mm}$]{}) and due to the fact that the beam was not always centred.
The influence of the distortions can be monitored using its average value $\langle {\ensuremath{d'_0}\xspace}\rangle$ as shown in Ref. [@ref:silvia:thesis]. In Fig. \[fig:distortion:spilldependence\] this quantity is displayed separately for positively and negatively charged pion tracks as a function of the event number within the spill [$N_{\mathrm{evt}}$]{}. A similar benchmark was used in Ref. [@ref:star] for distortions observed in the STAR TPC. Due to the sign-convention, the dynamic distortions shift the [$d'_0$]{} value for particle tracks of positive and negative charge in opposite direction. The dependence of [$d'_0$]{}shows that a distortion effect builds up during the spill, testifying its *dynamic* character. The absence of distortions at the beginning of the spill, and the increasingly larger distortions with opposite sign for oppositely charged particles during the spill, explains the peak structure initially observed in the [$d_0$]{}distribution. The curvature of high momentum tracks changes sign and therefore their [$d'_0$]{}migrates to large values with opposite sign.
The evolution of the [$d'_0$]{}distortion during the spill suggests a slowly drifting cloud of positive ions to be the cause of the track distortions. In addition, the absence of a saturation plateau suggests that the ion build-up does not reach its maximum before the end of the spill. It is therefore expected that a cloud of positive ions generated around the wires of the anode grid of the TPC grows progressively with time during the spill. The ion cloud would have approximately a torical shape with inner and outer diameter limited by the inner and outer field cages of the TPC. The front of the ion cloud does not reach the TPC HV cathode before the end of the spill. At the end of the beam spill the production of positive ions stops. During the inter-spill time the full cloud drifts to the HV cathode and vanishes completely before the start of the following spill. The minimum time between spills is ensured to be 2 s by the operation of the PS.
It was observed that the overall effect of the dynamic distortion in $R\phi$ was opposite in sign to the one induced by the static distortions due to the aforementioned HV mismatch. Via a straightforward $E \times B$ calculation it is possible to conclude that the radial component of the electric field due to dynamic distortions has to be directed predominantly outwards.
The hypothesis that dynamic distortions are caused by the build-up of positive ions in the drift volume during the 400 [$\mbox{ms}$]{}long beam spill makes it easier to understand why changes in the beam parameters (intensity, steering, focus) cause an increase or decrease in the dynamic distortions: the large amount of material around the target is likely to produce many low-energy secondary particles very close to the inner field cage whenever it is hit by a sizable beam halo. After the end of the beam spill, the initial conditions are re-established by the fact that the inter-spill time is large enough to drain all ions to the HV cathode.
Although charge build-up is a common phenomenon in TPCs, in the case of HARP the large amount of material before and around the target – very close to the beam axis – and the way the beam has often been tuned (too intense and/or not enough collimated) in some of the setting makes its occurrence rather difficult to deal with. In addition, the strongly inhomogeneous distortion of the electric field (larger at smaller radius) complicates the dynamic modelling of the actual field lines and of the ionization charge trajectories.
Benchmarking the distortion effects using elastic scattering {#sec:elastics-effect}
============================================================
Elastic scattering interactions of protons and pions on hydrogen provide events where the kinematics are fully determined by the scattering angle of the forward scattered beam particle. These kinematic properties were exploited to provide a known *beam* of protons pointing into the TPC sensitive volume. Data sets taken with liquid hydrogen targets at beam momenta from 3 [$\mbox{GeV}/c$]{}to 8 [$\mbox{GeV}/c$]{}were used for this analysis.
A good fraction of forward elastically-scattered protons or pions enter into the acceptance of the forward spectrometer, where the full kinematics of the event can be constrained. In particular, the direction and momentum of the recoil proton can be precisely predicted. Selecting events with one and only one track in the forward direction and requiring that the measured momentum and angle are consistent with an elastic reaction already provides an enriched sample of elastic events. By requiring that only one barrel RPC hit is recorded at the position predicted for an elastic event (the precision of the prediction from the forward spectrometer is within the RPC pad size) and within a time window consistent with a proton time-of-flight, we obtained a $\simeq$99% pure sample of recoil protons in the TPC volume and with known momentum vector.
At beam momenta in the range 3 [$\mbox{GeV}/c$]{}–8 [$\mbox{GeV}/c$]{}the protons which are tagged by accepted forward beam particles point into the TPC with angles of $\approx 70^{\circ}$ with respect to the beam direction. The beam counters provide a direct measurement of the incoming particle direction and of the scattering vertex coordinates in the target transverse plane. Once a clean sample of elastic-scattering events is isolated, by using detectors that are all independent from the TPC, the absolute efficiency of the track finding and fitting procedure can be measured (this allows the Monte Carlo calculations of the TPC detection efficiency to be benchmarked), and both the direction of emission and the momentum at the scattering vertex of the proton which traverses the TPC at large angle are determined for each event. After correction for energy loss and multiple scattering the complete track trajectory is determined. It can be used both for comparison with the track points of the proton track reconstructed from the TPC data not corrected for distortions, to measure directly the effect of dynamic distortions and for benchmarking the proton track reconstructed from TPC data after corrections of dynamic distortions [@ref:INFN-LNL]. This fact has been used for a direct measurement of the distortions, as it will be described in Sec. \[sec:elastics-shift\].
By comparison with the momentum vector predicted with the elastic scattering kinematics, it was verified with the data that the value of the polar angle [$\theta$]{}is not modified by the dynamic distortions. However, the momentum measurement is affected. By disregarding the impact point of the incoming beam particle during the fit, the curvature of the track in the TPC gas volume can be measured directly. One observes that the momentum and the value of [$d'_0$]{}are biased as a function of [$N_{\mathrm{evt}}$]{}as shown in Fig. \[fig:el-p-bias-spill\]. The [$N_{\mathrm{evt}}$]{}dependence for [$d'_0$]{}does not show a significant difference between the lower and higher momentum part of the spectrum.
The analysis of the elastic scattering events sets very stringent constraints on the maximum effect of distortions of all kinds on the measurements of kinematic quantities with the TPC. This method has provided solid estimates of the systematic errors associated with distortions *as a function of event-in-spill*. In particular, it has been used to estimate the overall systematic error on the momentum determination [@ref:tantalum], [@ref:harp:MomCalib].
Ion Cloud Distortion Dynamics {#sec:ion-cloud}
=============================
Given the beam intensity and the data acquisition rate with the 5% interaction length targets, it follows that HARP operated under conditions of high dead time (higher than 90%). Hence, within one setting, the parameter [$N_{\mathrm{evt}}$]{}is a good measure of the time the event was taken after the start of the spill. (Depending on the precise data-taking conditions, the DAQ recorded about one event per ms.)
The beam instrumentation allows a precise evaluation of the direction, intensity and particle type of the impinging particles to be made. It is therefore possible to show correlations of beam properties and dynamic distortions. The beam is poorly focused during several settings[^4]. It is observed that a badly tuned beam, with large number of halo particles which hit the target support material (representing several nuclear interaction lengths $\lambda_{\mathrm{I}}$) induces a large number of particles during the data-acquisition dead time. The contrary also holds: dynamic distortions disappear or are strongly reduced for settings where the beam is well focused, and can change from run to run during a setting when the beam was re-tuned between these runs. Moreover, it is observed that the dependencies of the distortions on the azimuthal angle $\phi$ (observed for some settings) are correlated with an off-axis tuning of the beam.
Beam particles hitting the beam entrance hole and the target support material produce secondaries which enter the TPC and produce ionization charges. Notice that a large number of low-energy (and therefore highly ionizing) electrons are expected to enter the active gas volume. This kind of phenomenon is likely to produce many more electron-ion pairs than the typical triggered event in the target. The produced ionization electrons drift towards the amplification region and then their number is multiplied near the pad plane with an amplification factor of the order of $10^5$, producing an equivalent number of argon ions. Any inefficiency of the gating grid at the level of $10^{-3}$ or even $10^{-4}$ allows an overwhelming number of ions to reach the drift region and to start travelling in the TPC gas volume towards the cathode, forming at the same time a positive charge cloud. This charge cloud distorts the (otherwise uniform) drift field.
From measurements of positive ion mobility in argon based gas mixtures [@ref:alice], the velocity of the ions is computed to be about 2 mm/ms (four orders of magnitude lower than the velocity of drift electrons in the TPC).
To help visualize the build-up and motion of the ion cloud, one should keep in mind that the pad plane is at $z \approx -500$ [$\mbox{mm}$]{}, the thin targets are at $z \approx 0$ [$\mbox{mm}$]{}, the nose (Stesalite end disc) of the inner field cage is at about $z \approx
250$ [$\mbox{mm}$]{}, and the total drift region is $\approx$ 1500 [$\mbox{mm}$]{}long (hence ending at $z \approx 1000$ [$\mbox{mm}$]{}). Those numbers imply that, in a spill of about 400 [$\mbox{ms}$]{}, the ion cloud just reaches the $z$ position of the target.
The constant increase of distortions during the spill is easily explained: the drift electrons generated by tracks of triggered events have to cross an increasing number of ions produced by the beam, as the thickness of the ion cloud to be traversed increases from 0 to about 600 [$\mbox{mm}$]{}from the start to the end of a 400 ms long beam spill. Before the period of linear increase in strength of the distortions, one expects a short period of zero distortions: the ion cloud produced in the amplification region first has to reach the drift volume. This period is estimated to be about 25 [$\mbox{ms}$]{}. Between the period of zero effect and linear growth one expects a smooth transition given by the ion diffusion and the difference in length of the drift path in the regions around the anode and gating grid wires.
These expectations can be tested by comparing the distortions affecting tracks generated at different values of $z$ in the TPC. Tracks within a limited angular range approximately perpendicular to the beam direction are used for this analysis since their trajectory lies within a small range in $z$. (Tracks between $\pm 30$ degrees with respect to the normal are accepted.) If the dynamics of the ion cloud is correctly described with the considerations given above, tracks generated in the Stesalite end disc of the IFC (large positive $z$) and in the target should be affected by the same distortions at any time of the spill, because their drift electrons have to cross the same ion cloud. Fig. \[fig:distortion:all\] demonstrates that the distortions observed for these two groups of tracks are indeed identical.
On the other hand, tracks generated in a long target (e.g. a two interaction length long aluminum target of $\approx$80 [$\mbox{cm}$]{}) at negative $z$ values in the TPC (half-way between the pad plane and $z=0$), should show a saturation of the distortions before the end of spill. Figure \[fig:distortion:saturation\] indeed shows that the distortion of tracks produced at $z \approx -250$ [$\mbox{mm}$]{}is no longer increasing after about 130 [$\mbox{ms}$]{}, consistent with the predictions given in Section \[sec:ion-cloud\]. The ion wavefront is expected to reach $z \approx -250$ mm after about 125 ms, and the thickness of ion cloud to be traversed by the drift electrons remains constant. (Given the rather uniform beam intensity during the spill, per unit time the same number of ions are produced at the pad plane as the number which cross the ideal $z = -250$ mm plane.) Fig. \[fig:distortion:saturation\] also shows that tracks produced at increasingly larger $z$ exhibit the distortion saturation at increasingly later times.
Ions are no longer generated in the amplification region after the end of the beam spill. Thus it is expected that the ion cloud remains of constant thickness (about 600 [$\mbox{mm}$]{}) between spills, and that it drifts into the direction of positive $z$, gradually freeing the active volume from distortions starting first with the negative $z$ region of the TPC. Cosmic-ray tracks recorded for calibration taken during the time between spills allow this behaviour to be studied. To be able to study the distortion effects, tracks are selected which approximately cross the IFC region. The distortions in the measurements of the trajectory on either side of the region of the IFC are of opposite sign (if expressed in Cartesian coordinates). As a measure of the distortions the variable $\Delta \phi^0$ is defined as the difference of the measured $\phi$ of the top half of the track compared to the $\phi$ measured for the complete cosmic-ray track. Figure \[fig:distortion:interspill\] shows the $\Delta \phi^0$ for two time periods with different delays from the end of the preceding spill. Indeed the two back-to-back segments of cosmic-ray tracks taken at negative $z$ become progressively less affected by distortions, and the distortion-free region expands with time (while the ions drift towards more positive $z$).
Subtle effects can be shown and explained by analyzing the cosmic-rays taken between spills. From Fig. \[fig:distortion:interspill\_largeZ\] it can be observed that tracks at very large positive $z\ge 550$ [$\mbox{mm}$]{}see a different distortion strength at the beginning of the inter-spill (first cosmic) with respect to the distortion seen after about additional 80 [$\mbox{ms}$]{}–100 [$\mbox{ms}$]{}after the end of the preceding spill. This occurs despite the fact that in both cases their drift electrons cross the same ion cloud thickness. The front of the cloud has meanwhile just moved by less than 200 [$\mbox{mm}$]{}, and did not reach yet $z=500$ [$\mbox{mm}$]{}. This holds true even if taking into account that the trigger system was programmed to provide a wait-state of about 140 [$\mbox{ms}$]{}between the end-of-spill and the first inter-spill cosmic. The reasons for this behaviour can be fully understood: during the 80 [$\mbox{ms}$]{}–100 [$\mbox{ms}$]{}difference between the first and last inter-spill cosmic-rays, a fraction of the ion cloud has passed the position of the inner field-cage and disc, entering a region where the electrostatic configuration of the field-cage is completely different. In the region where the electric field is only formed by the cylinder of the outer field-cage, the distortion field produced by the ions in the disc with $R$ smaller than the drifting electron clusters is only attractive towards the origin: the repulsion term of the inner field cage is missing, and the outward component of the ions at radii larger than the position of each drift electron is null. Therefore, there is a range in $z$ in which the the drift electrons feel an inward force, thus partially compensating the usual distortion at small R in the $z$ range, where both the ions and the inner field cage are present. This is why the cosmic-ray tracks crossing the TPC at large $z$ values are more distorted at times directly after the spill compared to later times. There is a $z$ value (taking in account the trigger shift), corresponding to the end/cap of inner field cage, where the distortions have a maximum, see Fig. \[fig:distortion:interspill\_largeZ\].
Experimental determination of the $R\phi$ distortion using elastic scattering {#sec:elastics-shift}
=============================================================================
When trying to measure the effect of distortion in an unbiased way it is important to avoid the use of reference quantities which can themselves be affected by the distortions. For example, a biased result is obtained if one measures first the track curvature using the distorted trajectory and if one then at a later stage uses this curvature in combination with fixed references such as vertex position and hits of RPCs to predict the undistorted position of charge clusters inside the gas volume. This approach was suggested in [@ref:tpc:dydak] and we show in the following that such a procedure can be avoided.
On the contrary, as already discussed in Sec. \[sec:elastics-effect\], elastic scattering off H$_2$ can be used to predict the complete undistorted trajectory without making use of quantities which are affected in any way by the distortions. By measuring the scattering angle ($\theta$) of the forward going particle with respect the direction of the beam particle (whose momentum is precisely selected by the beam setting) the four-momentum of the proton recoiling at large angle is derived from the elastic scattering kinematics. This provides a reference quantity suitable to actually [*measure*]{} the distortion. The knowledge of the four-momentum of the large-angle proton is the key to extend the method to directly determine the $R\phi$ displacement of the clusters. This approach avoids completely the introduction of dependencies on parameters affected by the distortions.
The full trajectory of the large-angle proton in the active region of the TPC is calculated by using the geometry of the detector as described in detail in the simulation program. The simulation program takes into account all the details of the materials traversed by the scattered proton. This creates for every individual pad row an *unbiased* reference sample as function of [$N_{\mathrm{evt}}$]{}free from [*a-priori*]{} assumptions.
The procedure was applied to the five reference hydrogen data sets available: 3 [$\mbox{GeV}/c$]{}, 5 [$\mbox{GeV}/c$]{}and 8 [$\mbox{GeV}/c$]{}with a 60 [$\mbox{mm}$]{}long target and 3 [$\mbox{GeV}/c$]{}and 8 [$\mbox{GeV}/c$]{}with a 180 [$\mbox{mm}$]{}long target. The average difference (along $r\phi$) of the position of the predicted trajectory and the measured $r\phi$ coordinate are shown in Fig. \[fig:rphi-measured\] as a function of [$N_{\mathrm{evt}}$]{} for data taken with the 180 [$\mbox{mm}$]{}hydrogen target in the 3 [$\mbox{GeV}/c$]{}beam. For each pad plane row a straight line fit of the distortion measurements during the whole spill is made. The slope of the best straight line fit is used as monitor of the growth of the distortion versus time, is called distortion strength, and is given in units of growth of the distortion per recorded event. Figure \[fig:distortion:slopes\] shows the results obtained for the 3 [$\mbox{GeV}/c$]{}beam impinging on the 180 [$\mbox{mm}$]{}target and the 5 [$\mbox{GeV}/c$]{}data taken with the 60 [$\mbox{mm}$]{}H$_2$ target. The distortion strength increases during the whole spill, consistent with the behaviour of [$d'_0$]{}shown in fig. \[fig:distortion:all\]. Most interestingly, Fig. \[fig:distortion:slopes\] also shows that the direction of the distortion changes sign from the inner TPC rows to the outer ones, and that there is a cross-over point of vanishing distortion. The change of sign can be explained qualitatively with electrostatic arguments taking into account the fact that the HV power supply keeps the inner and outer field cages at a constant voltage. These arguments will be worked out in detail below. One can further observe that the absolute value of the outward field component at row number one is larger than the absolute value of the inward field component at row number twenty.
Phenomenological Model
======================
Simple discrete model
---------------------
A phenomenological model can be constructed based on the fact that the field which is responsible for the force acting on each drift electron is equivalent to a vector sum of two field systems:
- a field system where ions, in a given angular section at $R$ values internal to the drift electron position contribute to attract the drift electrons inward;
- a field system where ions, in a given angular section at $R$ values external to the drift electron position, contribute to attract the drift electrons outward.
In this description, the fixed voltage of the inner field cage plays a crucial role: it breaks the circular symmetry by shielding the charges in its shadow. This model makes it possible to understand all the peculiar features of the TPC dynamic distortions:
A simple, approximate algorithm can be written, using a discrete representation at the row-level. Despite of its simplicity such a model can predict the basic features with surprising accuracy. Various models of the distribution of the density of ions as a function of $R$ and of the R dependence of the electric field can be readily modelled. These models predict the change of sign of the distortion strength between the first and last pad rows, as well as the position of the pad row with vanishing distortion. The system can be modelled with only a very small number of fixed geometrical parameters of the TPC and a free parameter describing the overall distortion strength. A comparison with the distortion measured with the data taken with the 180 [$\mbox{mm}$]{}hydrogen target exposed to the 3 [$\mbox{GeV}/c$]{}beam is shown in Fig. \[fig10\].
General Analytical Solution
---------------------------
The radial electric field distorting the trajectory of the drift electron is due to two components. Given any radius $r_e$ where a drift electron is supposed to travel parallel to $z$ in the TPC, a distortion term is directly due to the contribution of the positive ion cloud integrated from the first row up to $r_e$. The cylindrical shells of the ion cloud, external to $r_e$, do not contribute because of the Gauss theorem. A second distortion term is due to the induced excess of negative charge onto the conducting surface of the inner field cage. Such a field is equivalent to the one of a charged wire along the $z$ axis.
The two distortion terms have opposite sign. The repulsive (outward force) term prevails at small $R$. The attractive (inward force, same sign as the static distortion force) term prevails at large $R$. There must be a radial position with vanishing distortion effect. A simple discretization of the field due to the ions alone, at the radius $R=r_e$, can be written as:
$$E_r^1(R)=\frac{2\sum_{i}Q(R_i)}{ZR},\;\;R_i<R \ ,
\label{eq1}$$
because the field due to each discrete cylindrical shell is equivalent to the one generated by a wire length $Z$ charged with the charge $Q(R_i)$ contained in the volume of each shell. R is the radius at which the field has to be computed, which is at the same distance from all effective wires along $z$ [*generated*]{} by each cylindrical shell, according to the Gauss theorem with the assumption of uniform charge density along $z$ and over the cylindrical angle.
The ions are generated by the amplification of all the ionization electrons produced by all the charged particles that traverse the TPC during the spill. The ion cloud in the drift volume is populated by ions which cross the grid in front of the anode wires either because the grid was open or not perfectly closed. Interactions of beam and halo particles with the target and its surrounding materials produce secondary particles most of each reach the outer field cage of the TPC. Therefore, a $1/r$ dependence of the ion charge distribution in space is a very good initial estimate. This consideration gives:
$$E_r^1(R)=\frac{K_1}{R}\int_{R_1}^{R}{\frac{dr}{r + c}} \ ,
\label{eq2}$$
where $K_1$ is a charge normalization factor depending on the beam and target setting, $c$ is the parameter fixing the charge at any given radius (also depending on beam and target setting), $R_1$ is the internal radius of the ion cloud (close to the first pad row).
On the other hand, the field due to the induced negative charge $-Q_2$ on the inner field cage is given by:
$$E_r^2(R)=-\frac{K_2Q_2}{R} \ .
\label{eq3}$$
This relation is correct as long the inner field cage is surrounded by the ion cloud, as it is indeed the case, because the ion velocity is such that the ion wavefront reaches the target position $z = 0$ by the end of the spill. Thus the total radial distortion field can be expressed as:
$$E_r(R)=E_r^1+E_r^2=\frac{1}{R}\left(K_1\ln \left(\frac{R +
c}{R_1+c}\right)-K_2Q_2\right) \ .
\label{eq4}$$
The general expression can finally be written as:
$$E_r(R)=\frac{K}{R}\left(\ln \left(R + c \right)-y \right) \ ,
\label{eq5}$$
where $K$ and $y$ absorb the various constant factors. This equation contains the setting dependencies of the charge distribution and beam intensity in the form of the free parameters $K$, $y$ and $c$ and they can be used to parametrize and correct the dynamic distortions of any setting. The number of parameters can be reduced by the condition:
$$\int_{R_1}^{R_2}E_r(R)dr=0 \ ,
\label{eq6}$$
where $R_1$ is the conducting surface of the inner field cage and $R_2$ is the conducting surface of the outer field cage. This condition uses a natural assumption that the power supply manages to keep the voltage at the nominal value at each resistor partition (we assume here that the static distortions have been corrected).
The absolute scale to be used in Eq. \[eq5\] can be fixed with the help of experimental data. In the hydrogen settings, the direct measurement of the position of the clusters can be used, while in the data taken with the other targets the time-dependence of the distribution of $\langle {\ensuremath{d'_0}\xspace}\rangle$ is a good estimator.
Assuming a uniform distribution of the ion charge in the volume, Eq. \[eq6\] can be resolved analytically if one requires $c \ll R$ (justified by the charge generation mechanism around the beam), and if one neglects the difference between inner field cage radius and the inner radius of the ion cloud (which is in any case internal to the first pad ring radius) Then this gives a solution for the value of $y$:
$$y=\ln \left(R_0\right) \ \mbox{ where } R_0 = \sqrt{R_1R_2}\,
\label{eq7}$$
[*i.e.*]{} $R_0$ is the geometric average of the radial positions of the conducting surfaces and is the radius for which the resulting field is zero. Thus this could be predicted from first principles.
A detailed comparison with the measurements is shown in Fig. \[fig10\], using Eq. \[eq5\] and \[eq7\] and the geometric dimensions of the TPC surfaces and pad size. This shows a remarkable agreement between the distortion data for the 3 [$\mbox{GeV}/c$]{}hydrogen data and the prediction based on the uniform charge distribution. If one uses a non-uniform space charge distribution the numerical solution of Eq. \[eq6\] gives similar results.
Both with the numerical integration code and with the analytical calculation, the electrostatic problem has been solved also for different density distributions (and different $R$-dependencies of the electric field taking into account more or less strong edge effects). This may be useful in case of particular beam settings. For example, below is given the formula expressing the radial electric field for the case of a $1/R$ superficial density distribution on each cylindrical shell (i.e. constant charge in the volume of each cylindrical shell):
$$E_r(R)=E_0 \left(1 -\frac{R_2 - R_1}{R\ln(R/R_1)} \right) \ .
\label{eq8}$$
We have shown that the method works for other density distributions; however the density distribution used in the paper is the one resulting from the raw measurements of the pad occupancies.
Correction method {#sec:correction}
=================
With the models described above and the direct measurements a *distortion strength* as a function of row number is determined. The *strength* is measured as a residual, therefore it can be used as the basis for an $R\phi$ correction applied to clusters measured on tracks. However, before the correction can be used for each target and beam setting where the elastic scattering cannot be measured, one has to correlate the characterization with the behaviour of one or more global track parameters. The analysis of the elastic scattering data shows that the largest effect of the distortions is seen in the pad rows nearest to the centre. Therefore, one of the best candidates in this respect is [$d'_0$]{}which is easy to measure for each track and therefore gives statistically significant reasults for each data set.
The behaviour of [$d'_0$]{}as a function of the event-in-spill shows a first part with, essentially, no distortion, then a quadratic rise, followed by a linear behaviour until an upper limit is reached. Empirically, this behaviour can be understood from the previously described ion cloud dynamics. At the beginning of the spill the TPC is distortion-free; soon after the onset of the distortion the effect stabilizes into a linear increase until levelling-off at the point in time where all the distance travelled by the ionization charges is filled by the ion cloud. The saturation is in practice not reached for tracks emanating from the target during the spill. The intermediate region approxamated by the quadratic rise is understood as the onset of the effect when the front of the ion cloud enters in the drift region. Due to the different paths the ions travel in the amplification region, the front of the cloud is not sharp. The behaviour is seen in Fig. \[fig:be:dzeroprime\] (left panel). It is observed that the dependence of [$d'_0$]{}as a function of [$N_{\mathrm{evt}}$]{}shows the three regimes of the distortion as described above. Three calibration parameters are extracted with an iterative procedure: the value of [$N_{\mathrm{evt}}$]{}up to which there is no distortion, the value of [$N_{\mathrm{evt}}$]{}where the rise changes from quadratic to linear, and an overall scale factor. A time-dependent upper limit to the growth is also defined to take into account the fact that the distortion saturates at a different value of [$N_{\mathrm{evt}}$]{}depending on the $z$ position of the original ionization charge. This is not a free parameter.
To take into account the different characteristics of the initial charge distribution the data taken with the 3 [$\mbox{GeV}/c$]{}beam use the corrections determined using the 3 [$\mbox{GeV}/c$]{}hydrogen data, the 5 [$\mbox{GeV}/c$]{}beam corrections use the 5 [$\mbox{GeV}/c$]{}hydrogen data, while the 8 [$\mbox{GeV}/c$]{}hydrogen data are used for the higher momenta.
For a given setting all data are first reconstructed without any correction for dynamic distortions and with a default (setting-independent) correction for static distortions. The characteristics of the dependence of [$d'_0$]{}on [$N_{\mathrm{evt}}$]{}is then used to determine the initial values for the four parameters (three for the dynamic distortions and one for the static correction.) The row-by-row dependence is characterized by a set of 20 numbers (one of three sets as explained above). Then this set is multiplied by a single strength factor, depending on the value of [$N_{\mathrm{evt}}$]{}. As only additional complication, the strength factor has a $z$ and $R$ dependent ceiling to take into account the saturation.
The iterative procedure is terminated if the [$d'_0$]{}curves of positive and negative pions are equal within $\pm$2 [$\mbox{mm}$]{}over the whole spill. Typically, only one extra iteration is needed to obtain the required precision. This indicates that the characteristics of the [$d'_0$]{}distributions describe the overall distortion strength reliably. The result of the procedure is shown in Fig. \[fig:be:dzeroprime\]. The small difference between the positive and negative pions around ${\ensuremath{N_{\mathrm{evt}}}\xspace}=50$ has no effect on the measurement of the momentum, but shows that the simple parabolic model describing the period of gradual onset of the distortions is not completely accurate. The approximations used in the method are valid for values of $\langle {\ensuremath{d'_0}\xspace}\rangle$ not exceeding 20 mm. The shape of the $\langle {\ensuremath{d'_0}\xspace}\rangle$ distribution as a function of [$N_{\mathrm{evt}}$]{}shows clearly up to which value of event-in-spill the fitted parameters can be used. This maximum value is setting dependent, and is larger for beam settings which were better focused, for beams tuned at lower intensity and for targets of lower $z$. The target material dependence is introduced by the multiplicity of the interaction products, which is higher for higher $z$. In practice, this criterion does not represent a significant loss in final statistics of the data sets. On average more the 80% of the data can be reliably corrected. The data sets which have had to be truncated most turn out to be the ones which were not statistics limited in any way (e.g. the high $z$ data sets).
Performance of TPC after correction {#sec:corrected-performance}
===================================
In order to check the results of the corrections for the distortions effects a number of control distributions were evaluated for each analysed data-set.
One control plot is the overall [$p_{\mathrm{T}}$]{}distribution of all tracks as a function of [$N_{\mathrm{evt}}$]{}. Figure \[fig:invpt\] shows the distribution in $Q/{\ensuremath{p_{\mathrm{T}}}\xspace}$, where $Q$ is the measured charge of the particle, for six groups of tracks, each corresponding $50 n < {\ensuremath{N_{\mathrm{evt}}}\xspace}\le 50 (n+1)$ (for $n$ ranging from zero to five). The distributions have been normalized to an equal number of incident beam particles, with the first group as reference. In the left panel, no dynamic distortion correction have been applied and a clear difference of the distributions is visible. One should note that the momentum measurement as well as the efficiency is modified. The right panel shows the distributions after the corrections. The distributions are no longer distinguishable. To understand the asymmetry of positively and negatively charged tracks, one should keep in mind that no particle identification was performed. Thus both protons and pions contribute to the positives while the [$\pi^-$]{}’s are the only component of the negative particles.
A more direct test of the effect of the correction on the measurement of momentum is shown in Fig. \[fig:be:momentum\]. Four groups of tracks were selected, two classes of proton tracks and pions separated on the basis of their charge. A sample of relatively high momentum protons was selected using their range to set a lower limit. The protons were required to produce a hit in two RPC layers. A fixed window with relatively high values of [$\mbox{d}E/\mbox{d}x$]{}in the TPC ensured the particle identification as protons and limited the maximum momentum. Another window with higher values of [$\mbox{d}E/\mbox{d}x$]{}selects protons with a lower momentum. The pions are selected again by [$\mbox{d}E/\mbox{d}x$]{}, which is only possible for low momentum values (around 100 [$\mbox{MeV}$]{}). Positively charged and negatively charged pions are treated separately. The angle of the particles is restricted in a range with $\sin \theta
\approx 0.9$, ensuring a small range of [$p_{\mathrm{T}}$]{}. In the left panel (uncorrected data) one observes a variation of $\approx 5\%$ for the high [$p_{\mathrm{T}}$]{}samples. The corrected data stay stable well within 3%. The low [$p_{\mathrm{T}}$]{}pion data remain stable with or without correction. The width of the measured momentum distributions remains the same over the length of the spill, indicating that also the resolution is well corrected. It should also be noted that there is an effect on the efficiency. While the efficiency to find a collection of clusters as a track is not modified by the distortions, the requirement that the track is pointing to the target does introduce an efficiency loss for the uncorrected data. This loss is visible as an increase in the error bars on the measurements.
From the combination of the two sets of control plots one can conclude that the dynamic distortion corrections achieve a uniform efficiency and a constant measurement of momentum over the whole spill. Since the initial characterization of the TPC performance and calibration was determined using the first part of the spill which is not affected by dynamic distortions one expects that the calibration remains applicable. The systematic errors on these quantities remain approximately equal: although an additional correction would improve the situation, more events are now used with larger corrections applied to their tracks.
Results for 8.9 [$\mbox{GeV}/c$]{}Be data {#sec:cross-sections}
=========================================
Finally, a comparison of the end-product of the analysis, double-differential cross-sections, before and after the corrections can be made.
The measured double-differential cross-sections for the production of [$\pi^+$]{}and [$\pi^-$]{}in the laboratory system as a function of the momentum and the polar angle for each incident beam momentum were measured for many targets and beam momenta. These results are in agreement with what previously found using only the first part of the spill and using no dynamic distortions corrections. Of course, both analyses only use the data for which their calibrations are applicable. Thus a lower statistics sample is used for the uncorrected data. Making this comparison using the 8.9 [$\mbox{GeV}/c$]{}Be data has the advantage of using the data set with the highest statistics, thus achieving the best possible comparison. Figures \[fig:becomp1\] and \[fig:becomp2\] show the ratio of the cross sections without and with the correction factor for dynamic distortions in 8.9 [$\mbox{GeV}/c$]{}beryllium data. The error band in the ratio takes into account the usual estimate of momentum error and the error on efficiency, the other errors are almost fully correlated. The agreement is within $1 \sigma$ for most of the points, confirming the estimate of differential systematic error. The statistical error bar represents the statistics of the non-overlapping events.
Conclusions {#sec:conclusions}
===========
The HARP TPC suffers a rather large number of operational problems. The dynamic distortions observed for the particle trajectories were tackled after the other problems had been corrected. The overall characteristics of the effect of these distortions were described. Mainly the measurement of curvature and the extrapolation to the target were affected. It was shown that the origin of the distortions is fully understood both theoretically and experimentally. An experimental method to obtain a direct measurement of the distortions on the trajectory in space was developed. The [$d'_0$]{}variable has been identified to be a sensitive indicator of dynamic distortions both with H$_2$ targets and heavier targets. The effect of dynamic distortions on the particle trajectories in the TPC has been measured directly with H$_2$ targets by exploiting the forward spectrometer and the kinematics of elastic scattering. A simple model of the generation of dynamic distortions and a correction algorithm which depends on parameters that are controlled by the [$d'_0$]{}variable were developed. By monitoring the distortion strength with the [$d'_0$]{}observable the correction algorithm can be applied to all data sets taken with different targets. The TPC performance (momentum scale and absolute efficiency) were measured during the full spill by using data with hydrogen targets. The results of the corrections show that the performance of the TPC is restored for the vast majority of the data.
Acknowledgments
===============
We gratefully acknowledge the help and support of the PS beam staff and of the numerous technical collaborators who contributed to the detector design, construction, commissioning and operation. This work would not have been possible without the combined use of all the detector components of HARP. We would like to thank our colleagues from the HARP collaboration for their contribution to this work.
The experiment was made possible by grants from the Institut Interuniversitaire des Sciences Nucléaires, Ministerio de Educacion y Ciencia, Grant FPA2003-06921-c02-02 and Generalitat Valenciana, grant GV00-054-1, CERN (Geneva, Switzerland), the German Bundesministerium für Bildung und Forschung (Germany), the Istituto Nazionale di Fisica Nucleare (Italy), INR RAS (Moscow), the Particle Physics and Astronomy Research Council (UK) and the Swiss National Science Foundation, in the framework of the SCOPES programme. We gratefully acknowledge their support.
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V. Ammosov [*et al.*]{}, “The HARP Time Projection Chamber: Characteristics and Physics Performance”, Nucl. Instrum. Methods [**A588**]{} (2008) 294.
![ Schematic layout of the TPC. The beam enters from the left. Starting from the outside, first the return yoke of the magnet is seen, closed with an end-cap at the upstream end, and open at the downstream end. The inner field cage is visible as a short cylinder entering from the left. The ITC trigger counter and the target holder are inserted in the inner field cage. The RPCs (not drawn) are positioned between the outer field cage and the coil. The drift length is delimited on the right by the cathode plane and on the left by the anode wire planes. On the right the mechanical drawing of a sector of the TPC, the layout of the pads is indicated. []{data-label="fig:tpc"}](plots/fig_tpc_drawing_2.eps "fig:"){width="65.00000%"} ![ Schematic layout of the TPC. The beam enters from the left. Starting from the outside, first the return yoke of the magnet is seen, closed with an end-cap at the upstream end, and open at the downstream end. The inner field cage is visible as a short cylinder entering from the left. The ITC trigger counter and the target holder are inserted in the inner field cage. The RPCs (not drawn) are positioned between the outer field cage and the coil. The drift length is delimited on the right by the cathode plane and on the left by the anode wire planes. On the right the mechanical drawing of a sector of the TPC, the layout of the pads is indicated. []{data-label="fig:tpc"}](plots/fig_tpc_pads.eps "fig:"){width="30.00000%"}
![The $d_0$ distributions in a 8 [$\mbox{GeV}/c$]{}beam exposure on a 5% $\lambda_{\mathrm{I}}$ Be target. Left panel: run 9540. Right panel: run 9455. The histograms drawn with continuous lines are for positive particles and the the ones with dotted lines for negative particles. The two runs are taken close to each other in time, one just before and the other after re-tuning of the beam. []{data-label="fig:distortion:evidence"}](plots/fig2_be9450.eps "fig:"){width="45.00000%"} ![The $d_0$ distributions in a 8 [$\mbox{GeV}/c$]{}beam exposure on a 5% $\lambda_{\mathrm{I}}$ Be target. Left panel: run 9540. Right panel: run 9455. The histograms drawn with continuous lines are for positive particles and the the ones with dotted lines for negative particles. The two runs are taken close to each other in time, one just before and the other after re-tuning of the beam. []{data-label="fig:distortion:evidence"}](plots/fig3_be9455.eps "fig:"){width="45.00000%"}
![Example of monitoring of the dependence of the distortion on the time during the spill. [$d'_0$]{}as function of the event number in the spill for a 3 [$\mbox{GeV}/c$]{}beam exposure on a Ta target. Open circles: positively charged particles; closed circles: negatively charged particles.[]{data-label="fig:distortion:spilldependence"}](plots/fig4_ta3gev.eps){width="0.49\linewidth"}
![ Left panel: The shift in average momentum for elastic scattering data ( 3 [$\mbox{GeV}/c$]{}: open squares, 5 [$\mbox{GeV}/c$]{}: open circles) measured with elastic events as a function of the momentum predicted by the forward scattered track. It is as function of the event number in spill for different predicted momentum. The momentum estimator from the fit not constrained by the impact point of the incoming beam particle is used here. Right panel: The shift in average [$d'_0$]{}as a function of the event number in spill for elastic scattering data ( 3 [$\mbox{GeV}/c$]{}: filled and open boxes, 5 [$\mbox{GeV}/c$]{}: filled and open circles) measured with elastic events as a function of the momentum predicted by the forward scattered track. The open symbols show the data for momenta below 450 [$\mbox{MeV}/c$]{}and the filled symbols for momenta above 450 [$\mbox{MeV}/c$]{}. []{data-label="fig:el-p-bias-spill"}](plots/elas_evt_bias_p.eps "fig:"){width="0.45\linewidth"} ![ Left panel: The shift in average momentum for elastic scattering data ( 3 [$\mbox{GeV}/c$]{}: open squares, 5 [$\mbox{GeV}/c$]{}: open circles) measured with elastic events as a function of the momentum predicted by the forward scattered track. It is as function of the event number in spill for different predicted momentum. The momentum estimator from the fit not constrained by the impact point of the incoming beam particle is used here. Right panel: The shift in average [$d'_0$]{}as a function of the event number in spill for elastic scattering data ( 3 [$\mbox{GeV}/c$]{}: filled and open boxes, 5 [$\mbox{GeV}/c$]{}: filled and open circles) measured with elastic events as a function of the momentum predicted by the forward scattered track. The open symbols show the data for momenta below 450 [$\mbox{MeV}/c$]{}and the filled symbols for momenta above 450 [$\mbox{MeV}/c$]{}. []{data-label="fig:el-p-bias-spill"}](plots/elas_evt_lo_d0p.eps "fig:"){width="0.45\linewidth"}
![The average [$d'_0$]{}as a function of event number in spill for positively (open dots) and negatively (filled dots) charged particles for the hydrogen target for 3 [$\mbox{GeV}/c$]{}beam. Left panel: distortion of tracks originated in the target ($z \sim 0$ mm). Right panel: distortion of tracks generated in the Stesalite ($z \sim 268.5$ mm). Open circles: positively charged particles; closed circles: negatively charged particles. []{data-label="fig:distortion:all"}](plots/fig6_impactpnt_h_stesa.eps "fig:"){width="49.00000%"} ![The average [$d'_0$]{}as a function of event number in spill for positively (open dots) and negatively (filled dots) charged particles for the hydrogen target for 3 [$\mbox{GeV}/c$]{}beam. Left panel: distortion of tracks originated in the target ($z \sim 0$ mm). Right panel: distortion of tracks generated in the Stesalite ($z \sim 268.5$ mm). Open circles: positively charged particles; closed circles: negatively charged particles. []{data-label="fig:distortion:all"}](plots/fig7_impactpnt_h_stesa.eps "fig:"){width="49.00000%"}
![ Average [$d'_0$]{}measured as a function of the event number in spill using a long aluminium target. The data are divided into four regions of the $z$ of the interaction point. $z_0$: $-50 \ {\ensuremath{\mbox{cm}}\xspace}\le z < -30 \ {\ensuremath{\mbox{cm}}\xspace}$ (closed circles); $z_1$: $-30 \ {\ensuremath{\mbox{cm}}\xspace}\le z < -10 \ {\ensuremath{\mbox{cm}}\xspace}$ (open circles); $z_2$: $-10 \ {\ensuremath{\mbox{cm}}\xspace}\le z < +10 \ {\ensuremath{\mbox{cm}}\xspace}$ (closed squares); $z_3$: $+10 \ {\ensuremath{\mbox{cm}}\xspace}\le z < +30 \ {\ensuremath{\mbox{cm}}\xspace}$ (open squares). The data are shown for [$\pi^-$]{}tracks. The event number in spill (in first approximation corresponding to time) where the deviation of the average of the different series of points saturate clearly show the ion mobility. During the first $\approx$25 events (30 [$\mbox{ms}$]{}) no deviation is visible, consistent with the fact that the ions – created in the amplification region – have not yet reached the drift volume. During the following $\approx$50 events the derivative of the slope increases, showing the ion diffusion. []{data-label="fig:distortion:saturation"}](plots/k2k-target_d0p_pim.eps){width="45.00000%"}
![ The average $\Delta \phi_0$ for two time periods with different delays from the end of the preceding spill. Cosmic rays taken within the first 175 [$\mbox{ms}$]{}after the end of the spill are shown with closed circles, while the tracks taken between 250 [$\mbox{ms}$]{} and 300 [$\mbox{ms}$]{}after the end of the spill are shown with open circles. The distortions tend to zero at $z$ values which are already passed by the ion packet.[]{data-label="fig:distortion:interspill"}](plots/fig8_cosmic_deltaphi.eps){width="0.45\linewidth"}
![The dependence of the distortion for cosmic-ray tracks at large $z$ on the position of ion cloud during the inter-spill period. [*Dmin*]{} is the minimum distance of the trajectory of one arm of the cosmic-ray track with respect to the point closest to the origin evaluated for the complete cosmic-ray track. Cosmic rays taken within the first 175 [$\mbox{ms}$]{}after the end of the spill are shown with closed circles, while the tracks taken between 250 [$\mbox{ms}$]{} and 300 [$\mbox{ms}$]{}after the end of the spill are shown with open circles. []{data-label="fig:distortion:interspill_largeZ"}](plots/fig9_dmin_cosmic_interspill1.eps){width="0.45\linewidth"}
![The $R\phi$ distortion $\Delta (r \phi) = \langle(r \phi)_\mathrm{measured}
- (r \phi)_\mathrm{predicted}\rangle$ measured row–by–row as a function of event number in spill for the 3 [$\mbox{GeV}/c$]{}beam and 180 cm long H$_2$ target data. The four panels show data for four groups of five pad-rows each. The different symbols represent the individual pad-rows. []{data-label="fig:rphi-measured"}](plots/B_plots_bw_dphi_PV_H180_3_dfmm_S_0.eps "fig:"){width="0.45\linewidth"} ![The $R\phi$ distortion $\Delta (r \phi) = \langle(r \phi)_\mathrm{measured}
- (r \phi)_\mathrm{predicted}\rangle$ measured row–by–row as a function of event number in spill for the 3 [$\mbox{GeV}/c$]{}beam and 180 cm long H$_2$ target data. The four panels show data for four groups of five pad-rows each. The different symbols represent the individual pad-rows. []{data-label="fig:rphi-measured"}](plots/B_plots_bw_dphi_PV_H180_3_dfmm_S_1.eps "fig:"){width="0.45\linewidth"} ![The $R\phi$ distortion $\Delta (r \phi) = \langle(r \phi)_\mathrm{measured}
- (r \phi)_\mathrm{predicted}\rangle$ measured row–by–row as a function of event number in spill for the 3 [$\mbox{GeV}/c$]{}beam and 180 cm long H$_2$ target data. The four panels show data for four groups of five pad-rows each. The different symbols represent the individual pad-rows. []{data-label="fig:rphi-measured"}](plots/B_plots_bw_dphi_PV_H180_3_dfmm_S_2.eps "fig:"){width="0.45\linewidth"} ![The $R\phi$ distortion $\Delta (r \phi) = \langle(r \phi)_\mathrm{measured}
- (r \phi)_\mathrm{predicted}\rangle$ measured row–by–row as a function of event number in spill for the 3 [$\mbox{GeV}/c$]{}beam and 180 cm long H$_2$ target data. The four panels show data for four groups of five pad-rows each. The different symbols represent the individual pad-rows. []{data-label="fig:rphi-measured"}](plots/B_plots_bw_dphi_PV_H180_3_dfmm_S_3.eps "fig:"){width="0.45\linewidth"}
![The variation of the distortion strength as a function of the pad row number, fitted during the whole spill, left panel: 180 [$\mbox{mm}$]{}H$_2$ target, 3 [$\mbox{GeV}/c$]{}beam; right panel: 60 [$\mbox{mm}$]{}H$_2$ target, 5 [$\mbox{GeV}/c$]{}beam. []{data-label="fig:distortion:slopes"}](plots/B_plots_bw_slope_PV_H180_3_dfmm_S.eps "fig:"){width="0.45\linewidth"} ![The variation of the distortion strength as a function of the pad row number, fitted during the whole spill, left panel: 180 [$\mbox{mm}$]{}H$_2$ target, 3 [$\mbox{GeV}/c$]{}beam; right panel: 60 [$\mbox{mm}$]{}H$_2$ target, 5 [$\mbox{GeV}/c$]{}beam. []{data-label="fig:distortion:slopes"}](plots/B_plots_bw_slope_PV_H_5_dfmm_S.eps "fig:"){width="0.45\linewidth"}
![ The comparison of the measured distortion and the analytical models. The strength is shown with an arbitrary scale. Closed diamonds are the experimental data; long dashes show the analytical calculation for uniform space charge Eq. (\[eq7\]) and short dashes the numerical model. []{data-label="fig10"}](plots/plot_model_eq.eps){width="90.00000%"}
![ Average [$d'_0$]{}(dots for reconstructed positive tracks, squares for reconstructed negative tracks) as a function of event number in spill for 8.9 [$\mbox{GeV}/c$]{} Be data. (left panel uncorrected; right panel: dynamic distortion corrections applied.) After the “default” correction for the static distortions (equal for each setting) a small residual effect at the beginning of the spill is visible at ${\ensuremath{N_{\mathrm{evt}}}\xspace}=0$ (left panel). This is due to the fact that the inner and outer field cages are powered with individual HV supplies. A setting-by-setting correction compatible with the reproducibility of the power supplies is applied for the data of the right panel together with the dynamic distortion correction. The value of $\langle {\ensuremath{d'_0}\xspace}\rangle$ at $N_{evt}=0$ in the right panel has a small negative value as expected from the fact that the energy-loss is not described in the track-model used in the fit. The difference observed in the results for the two charges around $N_{evt}=50$ shows that even at the onset of the effect the model can correct the distortion within 1 mm. []{data-label="fig:be:dzeroprime"}](plots/Be5_nodyn_d0p.eps "fig:"){width="43.00000%"} ![ Average [$d'_0$]{}(dots for reconstructed positive tracks, squares for reconstructed negative tracks) as a function of event number in spill for 8.9 [$\mbox{GeV}/c$]{} Be data. (left panel uncorrected; right panel: dynamic distortion corrections applied.) After the “default” correction for the static distortions (equal for each setting) a small residual effect at the beginning of the spill is visible at ${\ensuremath{N_{\mathrm{evt}}}\xspace}=0$ (left panel). This is due to the fact that the inner and outer field cages are powered with individual HV supplies. A setting-by-setting correction compatible with the reproducibility of the power supplies is applied for the data of the right panel together with the dynamic distortion correction. The value of $\langle {\ensuremath{d'_0}\xspace}\rangle$ at $N_{evt}=0$ in the right panel has a small negative value as expected from the fact that the energy-loss is not described in the track-model used in the fit. The difference observed in the results for the two charges around $N_{evt}=50$ shows that even at the onset of the effect the model can correct the distortion within 1 mm. []{data-label="fig:be:dzeroprime"}](plots/Be5_dyn_d0p.eps "fig:"){width="43.00000%"}
![ Distribution in $1/{\ensuremath{p_{\mathrm{T}}}\xspace}$ for the 8.9 [$\mbox{GeV}/c$]{}Be data. The six curves show six regions in event number in spill (each in groups of 50 events in spill). Groups are labelled with the last event number accepted in the group, e.g. “50” stands for the group with event number from 1 to 50. The six groups are normalized to the same number of incoming beam particles, taking the first group as reference. Left panel: without dynamic distortion corrections; right panel: with dynamic distortion corrections. In the left panel only the first three groups of 50 events in spill are equivalent, while in the right panel all six groups are indistinguishable. []{data-label="fig:invpt"}](plots/Be5_pos9_nodyn.eps "fig:"){width="43.00000%"} ![ Distribution in $1/{\ensuremath{p_{\mathrm{T}}}\xspace}$ for the 8.9 [$\mbox{GeV}/c$]{}Be data. The six curves show six regions in event number in spill (each in groups of 50 events in spill). Groups are labelled with the last event number accepted in the group, e.g. “50” stands for the group with event number from 1 to 50. The six groups are normalized to the same number of incoming beam particles, taking the first group as reference. Left panel: without dynamic distortion corrections; right panel: with dynamic distortion corrections. In the left panel only the first three groups of 50 events in spill are equivalent, while in the right panel all six groups are indistinguishable. []{data-label="fig:invpt"}](plots/Be5_pos9_dyn.eps "fig:"){width="43.00000%"}
![ Momentum benchmarks. Left panel uncorrected; right panel: dynamic distortion corrections applied. The closed boxes show the average momentum observed for protons selected using their range (reaching the second RPC) and [$\mbox{d}E/\mbox{d}x$]{}; closed circles show protons selected within a high [$\mbox{d}E/\mbox{d}x$]{}region; open circles: [$\pi^-$]{} selected with [$\mbox{d}E/\mbox{d}x$]{}; open boxes: [$\pi^+$]{}selected with [$\mbox{d}E/\mbox{d}x$]{}. The angle of the particles is restricted in a range with $\sin \theta
\approx 0.9$. In the left panel (uncorrected data) one observes a variation of $\approx 5\%$ for the high [$p_{\mathrm{T}}$]{}samples. The corrected data stay stable well within 3%. The shaded bands show a $\pm1.5$% variation. The low [$p_{\mathrm{T}}$]{}data remain stable with or without correction. []{data-label="fig:be:momentum"}](plots/Be5_nodyn_p3.eps "fig:"){width="45.00000%"} ![ Momentum benchmarks. Left panel uncorrected; right panel: dynamic distortion corrections applied. The closed boxes show the average momentum observed for protons selected using their range (reaching the second RPC) and [$\mbox{d}E/\mbox{d}x$]{}; closed circles show protons selected within a high [$\mbox{d}E/\mbox{d}x$]{}region; open circles: [$\pi^-$]{} selected with [$\mbox{d}E/\mbox{d}x$]{}; open boxes: [$\pi^+$]{}selected with [$\mbox{d}E/\mbox{d}x$]{}. The angle of the particles is restricted in a range with $\sin \theta
\approx 0.9$. In the left panel (uncorrected data) one observes a variation of $\approx 5\%$ for the high [$p_{\mathrm{T}}$]{}samples. The corrected data stay stable well within 3%. The shaded bands show a $\pm1.5$% variation. The low [$p_{\mathrm{T}}$]{}data remain stable with or without correction. []{data-label="fig:be:momentum"}](plots/Be5_dyn_p3.eps "fig:"){width="45.00000%"}
![Ratio of the [$\pi^-$]{}production cross-sections measured without and with corrections for dynamic distortions in p–Be interactions at 8.9 [$\mbox{GeV}/c$]{}, as a function of momentum shown in different angular bins (shown in mrad in the panels). The error band in the ratio takes into account momentum error and the error on the efficiency, the other errors being correlated. The errors on the data points are statistical. []{data-label="fig:becomp1"}](plots/B_plots_bw_ratio_pos9_THETApiminus_rat.eps){width="70.00000%"}
![Ratio of the [$\pi^+$]{}production cross-sections measured without and with corrections for dynamic distortions in p–Be interactions at 8.9 [$\mbox{GeV}/c$]{}, as a function of momentum shown in different angular bins (shown in mrad in the panels). The error band in the ratio takes into account momentum error and the error on the efficiency, the other errors being correlated. The errors on the data points are statistical. []{data-label="fig:becomp2"}](plots/B_plots_bw_ratio_pos9_THETApiplus_rat.eps){width="70.00000%"}
[^1]: The Cartesian coordinates $x$ and $y$ are the coordinates perpendicular to the nominal magnetic field.
[^2]: The $s_{xy}$ coordinate is defined as the arc length along the circle in the $x$-$y$ plane between a point and the impact point.
[^3]: The [$d'_0$]{}sign indicates if the helix encircles the beam particle trajectory (positive sign) or not (negative sign)
[^4]: HARP uses the word [*setting*]{} to define a group of runs with the same beam momentum and polarity, target and trigger definition.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper, we review the theory of time space-harmonic polynomials developed by using a symbolic device known in the literature as the classical umbral calculus. The advantage of this symbolic tool is twofold. First a moment representation is allowed for a wide class of polynomial stochastic involving the Lévy processes in respect to which they are martingales. This representation includes some well-known examples such as Hermite polynomials in connection with Brownian motion. As a consequence, characterizations of many other families of polynomials having the time space-harmonic property can be recovered via the symbolic moment representation. New relations with Kailath-Segall polynomials are stated. Secondly the generalization to the multivariable framework is straightforward. Connections with cumulants and Bell polynomials are highlighted both in the univariate case and in the multivariate one. Open problems are addressed at the end of the paper.'
author:
- Elvira Di Nardo
title: 'On a representation of time space-harmonic polynomials via symbolic Lévy processes.'
---
[**]{}
[**keywords:**]{} Lévy process, time-space harmonic polynomial, Kailath-Segall polynomial, cumulant, umbral calculus.
Introduction
============
In mathematical finance, a Lévy process [@sato] is usually employed to model option pricing.
A Lévy process $X=\{X_t\}_{t \geq 0}$ is a stochastic process satisfying the following properties:
[*a)*]{}
: $X$ has independent and stationary increments;
[*b)*]{}
: $P[X(0)=0]=1$ on the probability space $(\Omega, {\cal F}, P);$
[*c)*]{}
: $X$ is stochastically continuous, i.e. for all $a > 0$ and for all $s \geq 0,$ $\lim_{t \rightarrow s} P(|X(t)-X(s)|>a)=0.$
The employment of Lévy processes in mathematical finance is essentially due to the property of manage continuous processes interspersed with jump discontinuities of random size and at random times, well fitting the main dynamics of a market. In order to include the risk neutrality, a martingale pricing could be applied to options. But Lévy processes do not necessarily share the martingale property unless they are centred. Instead of focusing the attention on the expectation, a different approach consists in resorting a family of stochastic processes, called [*polynomial processes*]{} and introduced very recently in [@cuchiero]. These processes are built by considering a suitable family of polynomials $\{P(x,t)\}_{t \geq 0}$ and by replacing the indeterminate $x$ with a stochastic process $X_t.$ Introduced in [@Seng00] and called [*time-space harmonic polynomials*]{} (TSH), the polynomials $\{P(x,t)\}_{t \geq 0}$ are such that $$E[P(X_t,t) \,\, |\; \mathfrak{F}_{s}] =P(X_s,s), \qquad \hbox{for} \,\, s \leq t
\label{TSH}$$ where $\mathfrak{F}_{s}=\sigma\left( X_\tau : \tau \leq s\right)$ is the natural filtration associated with $\{X_t\}_{t \geq 0}.$
As done in [@neveu] for the discretized version of a Lévy process, that is a random walk, TSH polynomials can be characterized as coefficients of the Taylor expansion $$\frac{\exp\{z X_t\}}{E[\exp\{z X_t\}]} = \sum_{k \geq 0} R_k(X_t,t)\frac{z^k}{k!}
\label{expmart}$$ in some neighborhood of the origin. The left-hand side of (\[expmart\]) is the so-called Wald’s exponential martingale [@Kuchler]. Wald’s exponential martingale is well defined only when the process admits moment generating function $E[\exp\{z X_t\}]$ in a suitable neighborhood of the origin. Different authors have tried to overcome this gap by using other tools. Sengupta [@Seng00] uses a discretization procedure to extend the results proved by Goswami and Sengupta in [@GS95]. Solé and Utzet [@Sole] use Ito’s formula showing that TSH polynomials with respect to Lévy processes are linked to the exponential complete Bell polynomials [@Comtet]. Wald’s exponential martingale (\[expmart\]) has been recently reconsidered also in [@Sengupta08], but without this giving rise to a closed expression for these polynomials.
The employment of the classical umbral calculus turns out to be crucial in dealing with (\[expmart\]). Indeed, the expectation of the polynomial processes $R_k(X_t,t)$ can be considered without taking into account any question involving the convergence of the right hand side of (\[expmart\]). Indeed the family $\{R_k(x,t)\}_{t \geq 0}$ is linked to the Bell polynomials which are one of the building blocks of the symbolic method. The main point here is that any TSH polynomial could be expressed as a linear combination of the family $\{R_k(x,t)\}$ and the symbolic representation of these coefficients is particularly suited to be implemented in any symbolic software. The symbolic approach highlights the role played by Lévy processes with regard to which the property (\[TSH\]) holds and makes clear the dependence of this representation on their cumulants.
The paper is organized as follows. Section 2 is provided for readers unaware of the classical umbral calculus. We have chosen to recall terminology, notation and the basic definitions strictly necessary to deal with the object of this paper. We skip any proof. The reader interested in is referred to [@Dinsen; @Dinardoeurop]. The theory of TSH polynomials is resumed in Section 3 together with the symbolic representation of Lévy processes closely related to their infinite divisible property. Umbral expressions of many classical families of polynomials as TSH polynomials with respect to suitable Lévy processes are outlined. The generalization to the multivariable framework is given in Section 4. This setting allows us to deal with multivariate Hermite, Euler and Bernoulli polynomials as well as with the class of multivariate Lévy-Sheffer systems introduced in [@DiNardoOliva3]. Open problems are addressed at the end of the paper.
The classical umbral calculus.
==============================
Let ${\mathbb R}[x]$ be the ring of polynomials with real coefficients[^1] in the indeterminate $x.$ The classical umbral calculus is a syntax consisting in an alphabet ${\mathcal{A}}=\{\alpha,\beta,\gamma, \ldots\}$ of elements, called *umbrae*, and a linear functional $E\,:\,{\mathbb R}[x][{\mathcal{A}}]{\longrightarrow}{\mathbb R}[x]$, called *evaluation*, such that $E[1]=1$ and $$E[x^n \, \alpha^i \, \beta^j \, \cdots \, \gamma^k ]=x^n \, E[\alpha^i] \, E[\beta^j] \, \cdots \, E[\gamma^k]
\quad \hbox{(uncorrelation property)}$$ where $\alpha, \beta, \ldots, \gamma$ are distinct umbrae and $n,i,j, \ldots,k$ are nonnegative integers.
A sequence $\{a_i\}_{i \geq 0} \in {\mathbb R}[x],$ with $a_0=1,$ is *umbrally represented*[^2] by an umbra $\alpha$ if $E[\alpha^i]=a_i,\;$ for all nonnegative integers $i.$ Then $a_i$ is called the $i$-th [*moment*]{} of $\alpha$. An umbra is *scalar* if its moments are elements of ${\mathbb R}$ while it is *polynomial* if its moments are polynomials of ${\mathbb R}[x].$ Special scalar umbrae are given in Table 1.
*Umbrae* *Moments*
------------------------- -------------------------------------------------------------------------------------------
Augmentation $\epsilon$ $E[\epsilon^i]=\delta_{0,i},$ with $\delta_{i,j}=1$ if $i=j,$ otherwise $\delta_{i,j}=0.$
Unity $u$ $E[u^i]=1$
Boolean unity $\bar{u}$ $E[\bar{u}^i] = i!$
Singleton $\chi$ $E[\chi]=1$ and $E[\chi^i]=0$, for all $i > 1$
Bell $\beta$ $E[\beta^i]=B_i,$ with $B_i$ the $i$-th Bell number
Bernoulli $\iota$ $E[\iota^i]={\mathfrak B}_i,$ with ${\mathfrak B}_i$ the $i$-th Bernoulli number
Euler $\varepsilon$ $E[\varepsilon^i]={\mathfrak E}_i,$ with ${\mathfrak E}_i$ the $i$-th Euler number
: Special scalar umbrae. The equalities on the right column refer to all nonnegative integer $i,$ unless otherwise specified.[]{data-label="table1"}
The core of this moment symbolic calculus consists in defining the [*dot-product*]{} of two umbrae, whose construction is shortly recalled in the following.
First let us underline that in the alphabet ${\mathcal{A}}$ two (or more) distinct umbrae may represent the same sequence of moments. More formally, two umbrae $\alpha$ and $\gamma$ are said to be [*similar*]{} when $E[\alpha^n]=E[\gamma^n]$ for all nonnegative integers $n,$ in symbols $\alpha \equiv \gamma.$ Therefore, given a sequence $\{a_n\},$ there are infinitely many distinct, and thus similar umbrae, representing the sequence.
Denote $\alpha'+\alpha''+\cdots+\alpha'''$ by the symbol $n {\mathbf{.}}\alpha,$ where $\{\alpha',\alpha'',\ldots,\alpha'''\}$ is a set of $n$ uncorrelated umbrae similar to $\alpha.$ The symbol $n {\mathbf{.}}\alpha$ is an example of *auxiliary umbra*. In a *saturated* umbral calculus, the auxiliary umbrae are managed as they were elements of ${\mathcal{A}}$ [@SIAM]. The umbra $n {\mathbf{.}}\alpha$ is called the [*dot-product*]{} of the integer $n$ and the umbra $\alpha$ with moments [@Dinardoeurop]: $$q_i(n)=E[(n {\mathbf{.}}\alpha)^i]=\sum_{k=1}^i (n)_k B_{i,k}(a_1, a_2, \ldots, a_{i-k+1}),
\label{(1)}$$ where $(n)_k$ is the lower factorial and $B_{i,k}$ are the exponential partial Bell polynomials [@Comtet].
In (\[(1)\]), the polynomial $q_i(n)$ is of degree $i$ in $n.$ If the integer $n$ is replaced by $t \in {\mathbb R},$ in (\[(1)\]) we have $q_i(t) = \sum_{k=1}^i (t)_k
B_{i,k}(a_1, a_2, \ldots, a_{i-k+1}).$ Denote by $t {\mathbf{.}}\alpha$ the auxiliary umbra such that $E[(t {\mathbf{.}}\alpha)^i] = q_i(t),$ for all nonnegative integers $i$. The umbra $t {\mathbf{.}}\alpha$ is the dot-product of $t$ and $\alpha.$ A kind of distributive property holds: $$(t + s) {\mathbf{.}}\alpha \equiv t {\mathbf{.}}\alpha + s {\mathbf{.}}\alpha^{\prime}, \quad s,t \in {\mathbb R}
\label{(distributive)}$$ where $\alpha^{\prime} \equiv \alpha.$ In particular if in (\[(1)\]) the integer $n$ is replaced by $-t,$ the auxiliary umbra $-t {\mathbf{.}}\alpha$ is such that $$-t {\mathbf{.}}\alpha + t {\mathbf{.}}\alpha^{\prime} \equiv {\epsilon},
\label{(inverse)}$$ where $\alpha^{\prime} \equiv \alpha.$ Due to equivalence (\[(inverse)\]), the umbra $-t {\mathbf{.}}\alpha$ is the [*inverse*]{}[^3] umbra of $t {\mathbf{.}}\alpha.$
Let us consider again the polynomial $q_i(t)$ and suppose to replace $t$ by an umbra $\gamma.$ The polynomial $q_i(\gamma) \in {\mathbb R}[x][{\mathcal{A}}]$ is an [*umbral polynomial*]{} with support [^4] $\hbox{\rm supp} \,
(q_i(\gamma))=\{\gamma\}.$ The [*dot-product of $\gamma$ and $\alpha$*]{} is the auxiliary umbra $\gamma {\mathbf{.}}\alpha$ such that $E[(\gamma {\mathbf{.}}\alpha)^i]=E[q_i(\gamma)]$ for all nonnegative integers $i.$ Two umbral polynomials $p$ and $q$ are said to be *umbrally equivalent* if $E[p]=E[q],$ in symbols $p \simeq q.$ Therefore equations (\[(1)\]), with $n$ replaced by an umbra $\gamma,$ can be written as the equivalences $$q_i(\gamma) \simeq (\gamma {\mathbf{.}}\alpha)^i \simeq \sum_{k=1}^i (\gamma)_k B_{i,k}(a_1, a_2, \ldots, a_{i-k+1}).
\label{(dotprodumbrae)}$$ Special dot-product umbrae are the $\alpha$-cumulant umbra $\chi {\mathbf{.}}\alpha$ and $\alpha$-partition umbra $\beta {\mathbf{.}}\alpha,$ that we will use later on. In particular any umbra is a partition umbra [@Dinardoeurop]. This property means that if $\{a_i\}$ is a sequence umbrally represented by an umbra $\alpha,$ then there exists a sequence $\{h_i\}$ umbrally represented by an umbra $\kappa_{{\scriptscriptstyle \alpha}},$ such that $\alpha \equiv \beta {\mathbf{.}}\kappa_{{\scriptscriptstyle \alpha}}.$ The umbra $\kappa_{{\scriptscriptstyle \alpha}}$ is similar to the $\alpha$-cumulant umbra, that is $\kappa_{{\scriptscriptstyle \alpha}} \equiv \chi {\mathbf{.}}\alpha,$ and its moments share the well-known properties of cumulants. [^5]
Dot-products can be nested. For example, moments of $(\alpha {\mathbf{.}}\varsigma) {\mathbf{.}}\gamma$ can be recursively computed by applying two times formula (\[(dotprodumbrae)\]). Parenthesis can be avoided since $(\alpha {\mathbf{.}}\varsigma) {\mathbf{.}}\gamma \equiv \alpha {\mathbf{.}}(\varsigma {\mathbf{.}}\gamma).$ In particular $\alpha {\mathbf{.}}\beta {\mathbf{.}}\gamma,$ with $\beta$ the Bell umbra, is the so-called [*composition umbra*]{}, with moments $$E[(\alpha {\mathbf{.}}\beta {\mathbf{.}}\gamma)^i]=\sum_{k=1}^i a_k B_{i,k}(g_1, g_2, \ldots, g_{i-k+1}),
\label{(2)}$$ where $\{a_i\}$ are moments of $\alpha$ and $\{g_i\}$ are moments of $\gamma.$ When the umbra $\alpha$ is replaced by $t \in {\mathbb R},$ then equation (\[(2)\]) gives the $i$-th moment of a compound Poisson random variable (r.v.) of parameter $t:$ $$E[(t {\mathbf{.}}\beta {\mathbf{.}}\gamma)^i]=\sum_{k=1}^i t^k B_{i,k}(g_1, g_2, \ldots, g_{i-k+1}).$$ There are more auxiliary umbrae that will employed in the following. For example, if $E[\alpha] \ne 0,$ the compositional inverse $\alpha^{\scriptscriptstyle <-1>}$ of an umbra $\alpha$ is such that $\alpha {\mathbf{.}}\beta {\mathbf{.}}\alpha^{\scriptscriptstyle <-1>} \equiv
\alpha^{\scriptscriptstyle <-1>} {\mathbf{.}}\beta {\mathbf{.}}\alpha \equiv \chi.$ The derivative of an umbra $\alpha$ is the umbra $\alpha_{\scriptsize D}$ whose moments are $E[\alpha_{\scriptsize D}^i] = i \, a_{i-1}$ for all nonnegative integers $i \geq 1.$ The disjoint sum $\alpha \dot{+} \gamma$ of $\alpha$ and $\gamma$ represents the sequence $\{a_i + g_i\}.$ Its main property involves the Bell umbra: $$\beta {\mathbf{.}}(\alpha \dot{+} \gamma) \equiv \beta {\mathbf{.}}\alpha + \beta {\mathbf{.}}\gamma.
\label{(disj)}$$
Symbolic Lévy processes.
------------------------
The family of auxiliary umbrae $\{t {\mathbf{.}}\alpha\}_{t \in I},$ with $I \subset {\mathbb R}^{+},$ is the umbral counterpart of a stochastic process $\{X_t\}_{t \in I}$ having all moments and such that $E[X_t^i]=E[(t {\mathbf{.}}\alpha)^i]$ for all nonnegative integers $i.$ This symbolic representation parallels the well-known infinite divisible property of a Lévy process, summarized by the following equality in distribution $$X_t \stackrel{d}{=} \underbrace{\Delta X_{t/n} + \cdots + \Delta X_{t/n}}_n
\label{(idp)}$$ with $\Delta X_{t/n}$ a r.v. corresponding to the increment of the process over an interval of amplitude $t/n.$ The $n$-fold convolution (\[(idp)\]) is usually expressed by the product of $n$ times a characteristic function $E[e^{{\mathfrak i} z X_t}] = E[e^{{\mathfrak i} z \Delta X_{t/n}}]^n$ with ${\mathfrak i}$ the imaginary unit. More generally one has $$E[e^{{\mathfrak i} z X_t}] = E[e^{{\mathfrak i} z X_1}]^t.
\label{(chfunc)}$$ Equation (\[(chfunc)\]) allows us to show that the auxiliary umbra $t {\mathbf{.}}\alpha$ is the symbolic version of $X_t.$ To this aim we recall that the formal power series $$f(\alpha, z) = 1 + \sum_{i \geq 1} a_i \frac{z^i}{i!}
\label{(fps)}$$ is the generating function of an umbra $\alpha,$ umbrally representing the sequence $\{a_i\}.$ Table 2 shows generating functions for some special auxiliary umbrae introduced in the previous section.
*Umbrae* *Generating functions*
------------------------------------------------------------- --------------------------------------------------------------------------------
Augmentation $\epsilon$ $f(\epsilon,z)=1$
Unity $u$ $f(u,z)=e^z$
Boolean unity $\bar{u}$ $f(\bar{u},z)= \frac{1}{1-z}$
Singleton $\chi$ $f(\chi,z)= 1 + z$
Bell $\beta$ $f(\beta,z)=\exp[e^z-1]$
Bernoulli $\iota$ $f(\iota,z) = z/(e^z-1)$
Euler $\eta$ $f(\eta,z) = 2 \, e^z / [e^z + 1]$
dot-product $n {\mathbf{.}}\alpha$ $f(n {\mathbf{.}}\alpha,z)= f(\alpha,z)^n$
dot-product $t {\mathbf{.}}\alpha$ $f(t {\mathbf{.}}\alpha,z)= f(\alpha,z)^t$
dot-product $\gamma {\mathbf{.}}\alpha$ $f(\gamma {\mathbf{.}}\alpha,z)= f(\gamma,\log f(\alpha,z))$
$\alpha$-cumulant $\chi {\mathbf{.}}\alpha$ $f(\chi {\mathbf{.}}\alpha, z) = 1 + \log[f(\alpha,z)]$
$\alpha$-partition $\beta {\mathbf{.}}\alpha$ $f(\beta {\mathbf{.}}\alpha,z) = \exp[f(\alpha,z) - 1]$
composition $\alpha {\mathbf{.}}\beta {\mathbf{.}}\gamma$ $f(\alpha {\mathbf{.}}\beta {\mathbf{.}}\gamma,z) = f[\alpha,f(\gamma,z) - 1]$
$\alpha$-partition $t {\mathbf{.}}\beta {\mathbf{.}}\gamma$ $f(t {\mathbf{.}}\beta {\mathbf{.}}\gamma,z) = \exp[t(f(\gamma,z) - 1)]$
derivative $\alpha_{\scriptscriptstyle D}$ $f(\alpha_{\scriptscriptstyle D},z) = 1 + z f(\alpha, z)$
: Generating functions for some special auxiliary umbrae.[]{data-label="table2"}
As for infinitely divisible stochastic processes (\[(idp)\]), one has $f(t {\mathbf{.}}\alpha, z)=f(\alpha,z)^t.$ It is well-known that the class of infinitely divisible distributions coincides with the class of limit distributions of compound Poisson distributions [@Feller]. By the symbolic method, any Lévy process is of compound Poisson type [@elviraannali]. This result is a direct consequence of the Lévy-Khintchine formula [@sato] involving the moment generating function of a Lévy process. Indeed, if $\phi(z,t)$ denotes the moment generating function of $X_t$ and $\phi(z)$ denotes the moment generating function of $X_1$ then $\phi(z,t)=\phi(z)^t$ from (\[(chfunc)\]). From the Lévy-Khintchine formula $\phi(z)=\exp[g(z)],$ with $$g(z)= z m + \frac{1}{2} s^2 z^2 + \int_{{\mathbb R}} \left(e^{zx} - 1 -
z \, x \, {\bf 1}_{ \{ |x| \leq 1\} } \right) {\rm d}(\nu(x)).
\label{(LK)}$$ The term $(m,s^2,\nu)$ is the Lévy triplet and $\nu$ is the Lévy measure. The function $\phi(z,t)$ shares the same exponential form of the moment generating function $f(t {\mathbf{.}}\beta {\mathbf{.}}\gamma,z)$ of a compound Poisson process, see Table 2. If $\nu$ admits all moments and if $c_0 = m+\int_{\{|x| \geq 1\}} x \, {\rm d}(\nu(x)),$ then the function $g(z)$ given in (\[(LK)\]) has the form $$g(z)= c_0 z + \frac{1}{2} s^2 z^2 + \int_{{\mathbb R}} \left(e^{zx} - 1 - z \, x \right) {\rm d}(\nu(x)).
\label{(LK1)}$$ Thanks to (\[(LK1)\]), the symbolic representation $t {\mathbf{.}}\beta {\mathbf{.}}\gamma$ of a Lévy process is such that the umbra $\gamma$ can be further decomposed in a suitable disjoint sum of umbrae. Indeed, assume
> [*i)*]{} $\varsigma$ an umbra with generating function $f(\varsigma,z)=1 + z^2/2,$\
> [*ii)*]{} $\, \eta$ an umbra with generating function $f(\eta,z) =
> \int_{{\mathbb R}} \left(e^{zx} - 1 - z \, x \right) {\rm d}(\nu(x)).$
Then a Lévy process is umbrally represented by the family $$\{t {\mathbf{.}}\beta {\mathbf{.}}(c_0 \chi \dot{+}
s \varsigma \dot{+} \eta)\}_{t \geq 0} \quad \hbox{or} \quad \{t {\mathbf{.}}\beta {\mathbf{.}}(c_0 \chi \dot{+}
s \varsigma ) + t {\mathbf{.}}\beta {\mathbf{.}}\eta\}_{t \geq 0},
\label{(symbLev)}$$ due to (\[(disj)\]). Symbolic representation (\[(symbLev)\]) is in agreement with Itô representation $X_t = W_t + M_t + c_0 t$ of a Lévy process with $W_t + c_0 t$ a Wiener process and $M_t$ a compensated sum of jumps of a Poisson process involving the Lévy measure. Indeed the Gaussian component is represented by the symbol $t {\mathbf{.}}\beta {\mathbf{.}}(c_0 \chi \dot{+} s \varsigma)$ as stated in [@dinardooliva2009], with $c_0$ corresponding to the mean and $s^2$ corresponding to the variance. The Poisson component is represented by $t {\mathbf{.}}\beta {\mathbf{.}}\eta,$ that is $t {\mathbf{.}}\beta {\mathbf{.}}\eta$ is the umbral counterpart of a random sum $S_N = Y_1 + \cdots + Y_N,$ with $\{Y_i\}$ independent and identically distributed r.v.’s corresponding to $\eta,$ associated to the Lévy measure, and $N$ a Poisson r.v. of parameter $t.$ The representation $\{t {\mathbf{.}}\beta {\mathbf{.}}(c_0 \chi \dot{+} s \varsigma \dot{+} \eta)\}_{t \geq 0}$ shows that the Lévy process is itself a compound Poisson process with $\{Y_i\}$ corresponding to the disjoint sum $(c_0 \chi \dot{+} s \varsigma \dot{+} \eta).$
More insights may be added on the role played by the umbra $c_0 \chi \dot{+} s \varsigma \dot{+} \eta$. Indeed the moment generating function of a Lévy process can be written as $\phi(z,t)= \exp[t \, g(z)]$ with $g(z)=\log \, \phi(z).$ So the function $g(z)$ in (\[(LK1)\]) is the cumulant generating function of $X_1$ and $\gamma \equiv c_0 \chi \dot{+} s \varsigma \dot{+} \eta$ is the symbolic representation of a r.v. whose moments are cumulants of $X_1.$ Therefore, in the symbolic representation $t {\mathbf{.}}\alpha$ of a Lévy process, introduced at the beginning of this section, the umbra $\alpha$ is the partition umbra of the cumulant umbra $\gamma \equiv c_0 \chi \dot{+} s \varsigma \dot{+} \eta$ that is $\alpha \equiv \beta {\mathbf{.}}\gamma.$
This remark suggests the way to construct the boolean and the free version of a Lévy process by using the boolean and the free cumulant umbra [@DinardoPetrulloSenato].
> [**Boolean Lévy process.**]{} Let $M(z)$ be the ordinary generating function of a r.v. $X,$ that is $M(z)= 1 + \sum_{i \geq 1} a_i z^i,$ where $a_i = E[X^i]$. The boolean cumulants of $X$ are the coefficients $b_i$ of the power series $B(z)= \sum_{i \geq 1} b_i z^i$ such that $M(z) = 1/[1-B(z)].$ Denote by $\bar{\alpha}$ the umbra such that $E[\bar{\alpha}^i] = i! a_i$ for all nonnegative integers $i.$ Then the umbra $\varphi_{\scriptscriptstyle\alpha}$ such that $\bar{\alpha} \equiv \bar{u} {\mathbf{.}}\beta {\mathbf{.}}\varphi_{\scriptscriptstyle\alpha}$ represents the sequence $\{b_i\}$ and is the $\alpha$-boolean cumulant umbra. Therefore the symbolic representation of a boolean Lévy process is $t {\mathbf{.}}\bar{u} {\mathbf{.}}\beta {\mathbf{.}}\varphi_{\scriptscriptstyle\alpha}.$
>
> [**Free Lévy process.**]{} The noncrossing (or free) cumulants of $X$ are the coefficients $r_i$ of the ordinary power series $R(z) = 1 +
> \sum_{i \geq 1} r_i z^i$ such that $M(z)=R[z M(z)].$ If $\bar{\alpha}$ is the umbra with generating function $M(z),$ then the $\bar{\alpha} \,$-free cumulant umbra $\mathfrak{K}_{\scriptscriptstyle \bar{\alpha}}$ represents the sequence $\{i! r_i\}.$ Assuming $\bar{\alpha}$ the umbral counterpart of the increment of a Lévy process over the interval $[0,1],$ then the symbolic representation of a free Lévy process is $t {\mathbf{.}}{\bar{\mathfrak{K}}}_{\scriptscriptstyle \alpha} {\mathbf{.}}\beta {\mathbf{.}}{(-1 {\mathbf{.}}{\bar{\mathfrak {K}}}_{\scriptscriptstyle
> \alpha})_{\scriptscriptstyle D}^{\scriptscriptstyle <-1>}}.$
Some more remarks on the parameters $c_0$ and $s$ may be added. The Lévy process in (\[(symbLev)\]) is a martingale if and only if $c_0 = 0,$ see Theorem 5.2.1 in [@Applebaum]. When this happens, $E[X_t]=0$ for all $t \geq 0$ and the Lévy process is said to be centered. Since the parameter $c_0$ allows the contribution of the singleton umbra $\chi$ in (\[(symbLev)\]), such an umbra plays a central role in the martingale property of a Lévy process. If $c_0=0,$ no contribution is given by $\chi$ which indeed does not admit a probabilistic counterpart.
If $s=0,$ the corresponding Lévy process is a [*subordinator*]{}, with almost sure non-decreasing paths. The subordinator processes are usually employed to scale the time of a Lévy process. This device is useful to widen or to close the jumps of the paths in market dynamics. Denote by $T_t$ the subordinator process of $X_t$ chosen independent of $X_t.$ The process $X_{T_t}$ is of Lévy type too. The symbolic representation of $T_t$ is $t {\mathbf{.}}\beta {\mathbf{.}}(c_0 \chi \dot{+} \eta^{\prime})$ so that $t {\mathbf{.}}\beta {\mathbf{.}}(c_0 \chi \dot{+} \eta^{\prime}) {\mathbf{.}}\beta {\mathbf{.}}(c_0 \chi \dot{+} s \varsigma + \eta)$ represents $X_{T_t}$ with $\eta$ and $\eta^{\prime}$ similar and uncorrelated umbrae. Despite its nested representation, the following result is immediately recovered: the process $X_{T_t}$ is a compound Poisson process $S_N$ with $Y_i$ a randomized compound Poisson r.v. of random parameter $\eta^{\prime},$ shifted of $c_0$ in its mean.
Time-space harmonic polynomials.
================================
Set ${\cal X}=\{\alpha\}.$ The conditional evaluation $E(\cdot \, \, \vline \,\, \alpha)$ with respect to $\alpha$ handles the umbra $\alpha$ as it was an indeterminate [@elviraannali]. In particular, $E(\cdot \, \, \vline \,\, \alpha): {\mathbb R}[x][{\cal A}] \rightarrow {\mathbb R}[{\cal X}]$ is such that $E(1 \,\, \vline \,\,\alpha)=1$ and $$E(x^m \alpha^n \gamma^i \xi^j\cdots \, \, \vline \,\, \alpha)=x^m \alpha^n
E[\gamma^i]E[\xi^j]\cdots$$ for uncorrelated umbrae $\alpha, \gamma, \xi, \ldots$ and for nonnegative integers $m,n,i,j,\ldots.$ As it happens in probability theory, the conditional evaluation is an element of ${\mathbb R}[x][{\mathcal{A}}]$ and, if we take the overall evaluation of $E(p\,\, \vline \,\, \alpha),$ this gives $E[p \,],$ with $p \in {\mathbb R}[x][{\mathcal{A}}],$ that is $E[E(p\,\,
\vline \,\, \alpha)]=E[p \,].$ Umbral polynomials $p,$ not having $\alpha$ in its support, are such that $E(p \,\, \vline \,\, \alpha)=E[p \,].$
Conditional evaluations with respect to auxiliary umbrae need to be handled carefully. For example, since $(n+1) {\mathbf{.}}\alpha \equiv n {\mathbf{.}}\alpha + \alpha^{\prime},$ the conditional evaluation with respect to the dot product $n {\mathbf{.}}\alpha$ is defined as $$E[(n+1) {\mathbf{.}}\alpha \,\, \vline \,\, n {\mathbf{.}}\alpha] = n {\mathbf{.}}\alpha + E[\alpha^{\prime}],$$ and more general, from (\[(distributive)\]) with $t$ and $s$ replaced by $n$ and $m,$ $$E( [(n+m) {\mathbf{.}}\alpha]^k \,\, | \,\, n {\mathbf{.}}\alpha) =
E([n {\mathbf{.}}\alpha + m {\mathbf{.}}\alpha^{\prime}]^k \,\, | \,\, n {\mathbf{.}}\alpha) = \sum_{j=0}^k \binom{k}{j} (n {\mathbf{.}}\alpha)^j E[(m {\mathbf{.}}\alpha^{\prime})^{k-j}],
\label{(condevalint)}$$ for all nonnegative integers $n$ and $m.$ Therefore, for $t \geq 0$ the conditional evaluation of $t {\mathbf{.}}\alpha$ with respect to the auxiliary umbra $s {\mathbf{.}}\alpha,$ with $0 \leq s \leq t,$ is defined according to (\[(condevalint)\]) such as $$E[(t {\mathbf{.}}\alpha)^k \,\, | \,\, s {\mathbf{.}}\alpha] = \sum_{j=0}^k \binom{k}{j} (s {\mathbf{.}}\alpha)^j
E([(t-s) {\mathbf{.}}\alpha^{\prime}]^{k-j}).$$ Equation (\[TSH\]) traces the way to extend the definition of polynomial processes to umbral polynomials.
Let $\{P(x,t)\} \in {\mathbb R}[x][{\cal A}]$ be a family of polynomials indexed by $t \geq 0.$ $P(x,t)$ is said to be a [TSH polynomial]{} with respect to the family of umbral polynomials $\{q(t)\}_{t \geq 0}$ if and only if $E \left[ P(q(t),t) \, \, \vline \, \, q(s) \right] = P(q(s),s)$ for all $0 \leq s \leq t.$
The main result of this section is the following theorem [@elviraannali].
\[UTSH2\] For all nonnegative integers $k,$ the family of polynomials[^6] $$Q_k(x,t)=E[(x - t {\mathbf{.}}\alpha)^k] \in {\mathbb R}[x]$$ is TSH with respect to $\{t {\mathbf{.}}\alpha\}_{t
\geq 0}.$
By expanding $Q_k(x,t)$ via the binomial theorem, one has $$Q_k(x,t)= \sum_{j=0}^k \binom{k}{j} x^j E[(-t {\mathbf{.}}\alpha)^{k-j}]$$ so that $$Q_k(t {\mathbf{.}}\alpha,t)= \sum_{j=0}^k \binom{k}{j} (t {\mathbf{.}}\alpha)^j E[(-t {\mathbf{.}}\alpha)^{k-j}].$$ Since $t {\mathbf{.}}\alpha$ is the symbolic version of a Lévy process, the property $$E[Q_k(t {\mathbf{.}}\alpha, t) \,\, | \,\, s {\mathbf{.}}\alpha] =
\sum_{j=0}^k \binom{k}{j} E[(t {\mathbf{.}}\alpha)^j|s {\mathbf{.}}\alpha] E[(-t {\mathbf{.}}\alpha)^{k-j}] =
Q_k(s {\mathbf{.}}\alpha, s)$$ parallels equation (\[TSH\]). In particular $\{Q_k(x,t)\}$ is a polynomial sequence umbrally represented by the polynomial umbra $x - t {\mathbf{.}}\alpha,$ which is indeed the TSH polynomial umbra with respect to $t {\mathbf{.}}\alpha.$ Polynomial umbrae of type $x + \alpha$ are Appell umbrae [@DNS]. Then $\{Q_k(x,t)\}$ is an Appell sequence and $$\frac{{\rm d}}{{\rm d} x} \, Q_k(x,t) = k \, Q_{k-1}(x,t), \qquad \hbox{for all integers
$k \geq 1.$}$$ The generating function of the TSH polynomial umbra $x -t {\mathbf{.}}\alpha$ is $$f(x - t {\mathbf{.}}\alpha, z)=\frac{\exp\{xz\}}{f(\alpha,z)^t}=\sum_{k \geq 0} Q_k(x,t) \frac{z^k}{k!}.
\label{(wald)}$$ By replacing $x$ with $t {\mathbf{.}}\alpha$ in (\[(wald)\]), Wald’s exponential martingale (\[expmart\]) is recovered. Equality of two formal power series is given in terms of equality of their corresponding coefficients, so that $E[R_k(X_t,t)] = E[Q_k(t {\mathbf{.}}\alpha,t)]$ by comparing (\[(wald)\]) with (\[expmart\]). Wald’s identity $\sum_{k \geq 0} E[R_k(X_t,t)] z^k/k!=1$ is encoded by the equivalence $t {\mathbf{.}}\alpha - t {\mathbf{.}}\alpha \equiv \epsilon$ obtained from $x -t {\mathbf{.}}\alpha$ when $x$ is replaced by $t {\mathbf{.}}\alpha.$
The next proposition gives the way to compute the coefficients of $Q_k(x,t)$ in any symbolic software.
If $\{a_n\}$ is the sequence umbrally represented by the umbra $\alpha$ and $\{Q_k(x,t)\}$ is the sequence of TSH polynomials with respect to $\{t {\mathbf{.}}\alpha\}_{t \geq 0},$ then $$Q_k(x,t) = \sum_{j,i=0}^{k} c^{(k)}_{i,j} \, t^i \, x^j,$$ with $$c^{(k)}_{i,j} = \binom{k}{j} \sum_{\lambda \vdash k-j} {\rm d}_{\lambda} (-1)^{2l(\lambda)+i} \, s[l(\lambda),i]
\, a_1^{r_1} a_2^{r_2} \cdots$$ where the sum is over all partitions[^7] $\lambda = (1^{r_1}, 2^{r_2},
\ldots) \vdash k-j, s[l(\lambda),i]$ denotes a Stirling number of first kind and $d_{\lambda} = i!/(r_1!
r_2! \cdots \, (1!)^{r_1}(2!)^{r_2} \cdots).$
More properties on the coefficients of $Q_k(x,t)$ are given in [@elviraannali].
Any TSH polynomial is a linear combination of $\{Q_k(x,t)\},$ which indeed are a bases of the space of TSH polynomials. The following theorem characterizes the coefficients of any TSH polynomial $P(x,t)$ in terms of the coefficients of $\{Q_k(x,t)\}.$
\[comb\] A polynomial $P(x,t)=\sum_{j=0}^k p_j(t) \, x^j$ of degree $k$ for all $t \geq 0$ is a TSH polynomial with respect to $\{t {\mathbf{.}}\alpha\}_{t \geq 0}$ if and only if $$p_j(t) = \sum_{i=j}^k \binom{i}{j} \, p_i(0) \, E[(-t{\mathbf{.}}\alpha)^{i-j}], \quad
\hbox{for $j=0,\ldots,k.$}$$
Cumulants.
----------
A different symbolic representation of TSH polynomials $\{Q_k(x,t)\}$ is $$Q_k(x,t)=E[(x - t {\mathbf{.}}\beta {\mathbf{.}}\kappa_{{\scriptscriptstyle \alpha}})^k],$$ with $\kappa_{{\scriptscriptstyle \alpha}}$ the $\alpha$-cumulant umbra. The umbra $-t {\mathbf{.}}\beta {\mathbf{.}}\kappa_{{\scriptscriptstyle \alpha}} \equiv t {\mathbf{.}}\beta {\mathbf{.}}(-1 {\mathbf{.}}\kappa_{{\scriptscriptstyle \alpha}}) \equiv t {\mathbf{.}}\beta {\mathbf{.}}\kappa_{(-1 {\mathbf{.}}{\scriptscriptstyle \alpha})}$ is the symbolic version of a Lévy process with sequence of cumulants of $X_1$ umbrally represented by $\kappa_{(-1 {\mathbf{.}}{\scriptscriptstyle \alpha})}.$ Therefore, also the polynomials $Q_k(x,t) = E[(x + t {\mathbf{.}}\beta {\mathbf{.}}\kappa_{{\scriptscriptstyle \alpha}})^k]$ are TSH with respect to Lévy processes umbrally represented by $\{t {\mathbf{.}}\beta {\mathbf{.}}\kappa_{(-1 {\mathbf{.}}{\scriptscriptstyle \alpha})}\}_{t \geq 0} \equiv \{t {\mathbf{.}}(-1 {\mathbf{.}}\alpha)\}_{t \geq 0}.$
Moments of polynomial umbrae $t {\mathbf{.}}\beta {\mathbf{.}}\gamma$ involve the exponential Bell polynomials. When $t$ is set equal to $1,$ then complete exponential Bell polynomials are recovered. More generally, the moments of $x - t {\mathbf{.}}\beta {\mathbf{.}}\kappa_{{\scriptscriptstyle \alpha}} \equiv x - t {\mathbf{.}}\alpha$ can be expressed by using exponential complete Bell polynomials too since $$x - t {\mathbf{.}}\alpha \equiv \beta {\mathbf{.}}[\chi {\mathbf{.}}(x - t {\mathbf{.}}\alpha)]
\equiv \beta {\mathbf{.}}\kappa_{{\scriptscriptstyle x - t {\mathbf{.}}\alpha}}
\equiv \beta {\mathbf{.}}(\kappa_{(x {\mathbf{.}}u)} \dot{+} \kappa_{(- t {\mathbf{.}}\alpha)})
\equiv \beta {\mathbf{.}}\kappa_{(x {\mathbf{.}}u)} + \beta {\mathbf{.}}\kappa_{(- t {\mathbf{.}}\alpha)}
\label{(TSHbell)}$$ where $\kappa_{{\scriptscriptstyle x - t {\mathbf{.}}\alpha}}$ is the cumulant umbra of $x - t {\mathbf{.}}\alpha,$ that could be replaced by $\kappa_{(x {\mathbf{.}}u)} \dot{+} \kappa_{(- t {\mathbf{.}}\alpha)}$ due to the additivity property of cumulants. The last equivalence in (\[(TSHbell)\]) follows from equivalence (\[(disj)\]). From equivalences (\[(TSHbell)\]), we have $$Q_k(x,t)=Y_k(x+h_1, h_2, \ldots, h_k),
\label{(complBell)}$$ with $Y_k$ exponential complete Bell polynomials and $\{h_i\}$ cumulants of $-t {\mathbf{.}}\alpha.$ Equation (\[(complBell)\]) has been proved in [@Sole] by using Teugel martingales.
For $Q_k(x,t),$ the Sheffer identity with respect to $t$ holds: $$Q_k(x,t+s)=\sum_{j=0}^k \binom{k}{j} P_j(s) Q_{k-j}(x,t),$$ where $P_j(s)=Q_{j}(0,s)$ for all nonnegative integers $j.$
Examples.
---------
The discretized version of a Lévy process is a random walk $S_n = X_1 + X_2+ \cdots
+ X_n,$ with $\{X_i\}$ independent and identically distributed r.v.’s. For the symbolic representation of a Lévy process we have dealt with, the symbolic counterpart of a random walk is the auxiliary umbra $n {\mathbf{.}}\alpha.$ Indeed the infinite divisible property (\[(idp)\]) is highlighted in the summation $\alpha^{\prime} + \alpha^{\prime \prime} + \cdots + \alpha^{\prime \prime \prime},$ encoded in the symbol $n {\mathbf{.}}\alpha,$ with $\alpha^{\prime}, \alpha^{\prime \prime}, \ldots, \alpha^{\prime \prime \prime}$ uncorrelated and similar umbrae. Nevertheless not all r.v.’s having the symbolic representation $n {\mathbf{.}}\alpha$ share the infinite divisible property. For example, the binomial r.v. has not the infinite divisible property [@sato], nevertheless its symbolic representation is of type $n {\mathbf{.}}\alpha$ where $\alpha \equiv \chi {\mathbf{.}}p {\mathbf{.}}\beta$ and $p \in (0,1).$ So the generality of the symbolic approach lies in the circumstance that if the parameter $n$ is replaced by $t,$ that is if the random walk is replaced by a Lévy process, more general classes of polynomials can be recovered for which many of the properties here introduced still hold. The following tables resume the TSH representation for different families of classical polynomials, see [@Roman]. In particular Table 3 gives the umbra corresponding to the r.v. $X_i$ of $S_n$ in the first column, its umbral counterpart in the second column and the associated TSH polynomial in the third column. In Table 4, the TSH polynomials given in Table 3 are traced back to special families of polynomials. In particular, with the polynomials ${\mathcal P}_k(x,t)$ we refer to $${\mathcal P}_k(x,t)=\sum_{j=1}^k Q_j(x,t) B_{k,j}(m_1, m_2, \ldots, m_{k-j+1})$$ for suitable $\{Q_j(x,t)\}$ and $\{m_i\}.$
$X_i$ Umbral counterpart Corresponding TSH polynomial
---------------------------- -------------------------------------------- -----------------------------------------------------------------------------------
Uniform $[0,1]$ $-1 {\mathbf{.}}\iota$ $E[(x + n {\mathbf{.}}\iota)^k]$
Bernoulli $p=\frac{1}{2}$ $\frac{1}{2}(-1 {\mathbf{.}}\epsilon + u)$ $E[(x + n {\mathbf{.}}\left[ \frac{1}{2}(-1 {\mathbf{.}}u + \epsilon)\right])^k]$
Bernoulli $p \in (0,1)$ $\chi {\mathbf{.}}p {\mathbf{.}}\beta$ $E[(x - n {\mathbf{.}}\chi {\mathbf{.}}p {\mathbf{.}}\beta)^k]$
Sum of $a \in {\mathbb N}$ $a {\mathbf{.}}(-1 {\mathbf{.}}\iota)$ $E[(x + (a\,n) {\mathbf{.}}\iota)^k]$
uniform r.v.’s on $[0,1]$
: TSH polynomials associated to special random walks[]{data-label="table3"}
---------------------------- ------------------------------------ ---------------------------------------------------------------------------
$X_i$ Special families Connection with
of polynomials TSH polynomials
Uniform $[0,1]$ Bernoulli $B_k(x,n)$ $B_k(x,t)=Q_k(x,t)$
Bernoulli $p=1/2$ Euler ${\mathcal E}_k(x,n)$ ${\mathcal E}_k(x,n)=Q_k(x,t)$
Bernoulli $p \in (0,1)$ Krawtchouk ${\mathcal K}_k(x,p,n)$ $(n)_k
{\mathcal K}_k(x,p,n)= {\mathcal P}_k(x,t)$
$m_i=E[((-1 {\mathbf{.}}\chi {\mathbf{.}}p {\mathbf{.}}\beta)^{<-1>})^i]$
Sum of $a \in {\mathbb N}$ pseudo-Narumi $N_k(x,an)$ $k! \, N_k(x,an) = {\mathcal P}_k(x,t)$
uniform r.v.’s on $[0,1]$ $m_i=E[(u^{<-1>})^i]$
---------------------------- ------------------------------------ ---------------------------------------------------------------------------
: Connection between special families of polynomials and TSH polynomials[]{data-label="table4"}
Next tables 5 and 6 give TSH polynomials for some special Lévy processes.
Lévy process Umbral representation TSH polynomial $Q_k(x,t)$
-------------------------------- --------------------------------------------------------- --------------------------------------------------------------------
Brownian motion
with variance $s^2$ $t {\mathbf{.}}\beta {\mathbf{.}}(s \varsigma)$ $E[(x - t {\mathbf{.}}\beta {\mathbf{.}}(s \varsigma))^k]$
Poisson process
with parameter $\lambda$ $(t \lambda) {\mathbf{.}}\beta$ $E[(x - (t \lambda) {\mathbf{.}}\beta)^k]$
Gamma process
with scale parameter $1$ $t {\mathbf{.}}\bar{u}$ $E[(x - t {\mathbf{.}}\bar{u})^k]$
and shape parameter $1$
Gamma process
with scale parameter $\lambda$ $(t \lambda) {\mathbf{.}}\bar{u}$ $E[(x - (t \lambda) {\mathbf{.}}\bar{u})^k]$
and shape parameter $1$
Pascal process
with parameter $d=p/q$ $t {\mathbf{.}}\bar{u} {\mathbf{.}}d {\mathbf{.}}\beta$ $E[(x - t {\mathbf{.}}\bar{u} {\mathbf{.}}d {\mathbf{.}}\beta)^k]$
and $p+q=1$
: TSH polynomials associated to special Lévy processes[]{data-label="table5"}
-------------------------------- ---------------------------------------------- ----------------------------------------------------------------
Lévy process Special family Connection with
of polynomials TSH polynomials
Brownian motion
with variance $s^2$ Hermite $H_k^{(s^2)}(x)$ $H_k^{(s^2)}(x)=Q_k(x,t)$
Poisson process $\tilde{C}_k(x, \lambda t)=$
with parameter $\lambda$ Poisson-Charlier $\tilde{C}_k(x, \lambda t)$ $=\sum_{j=1}^k s(k,j) Q_k(x,t)$
with $s(k,j)$ Stirling
numbers of first kind
Gamma process
with scale parameter $1$ Laguerre ${\mathcal L}_k^{t-k}(x)$ $k! (-1)^k {\mathcal L}_k^{t-k}(x)= Q_k(x,t)$
and shape parameter $1$
Gamma process
with scale parameter $\lambda$ actuarial $g_k(x,t)$ $g_k(x,t)= {\mathcal P}_k(x,t)$
and shape parameter $1$ $m_i=E[((\chi {\mathbf{.}}(- \chi))^{<-1>})^i]$
Pascal process $(-1)^k (t)_k M_k(x,t,p)=$
with parameter $d=p/q$ Meixner polynomials $= {\mathcal P}_k(x,t)$
and $p+q=1$ of first kind $M_k(x,t,p)$ $m_i=E[(\chi {\mathbf{.}}(-1 {\mathbf{.}}\chi + \chi / p))^i]$
-------------------------------- ---------------------------------------------- ----------------------------------------------------------------
: Special families of polynomials and TSH polynomials[]{data-label="table6"}
Orthogonality of TSH polynomials.
---------------------------------
A special class of TSH polynomials is the one including the Lévy-Sheffer polynomials, whose applications within orthogonal polynomials are given in [@Schoutens]. A sequence of polynomials $\{V_k(x,t)\}_{t \geq 0}$ [@ST] is a Lévy-Sheffer system if its generating function is such that $$\sum_{k \geq 0} V_k(x,t)\frac{z^k}{k!} = \left(g(z)\right)^t \exp\{x u(z)\},
\label{genfunlevshef}$$ where $g(z)$ and $u(z)$ are analytic functions in a neighborhood of $z = 0,$ $u(0) = 0,$ $g(0) = 1,$ $u'(0) \neq 0$ and $1/g(\tau(z))$ is an infinitely divisible moment generating function, with $\tau(z)$ such that $\tau(u(z)) = z.$ Assume $\alpha$ an umbra such that $f(\alpha,z)=g(z)$ and $\gamma$ an umbra such that $f(\gamma,z)=1+u(z).$ From (\[genfunlevshef\]), the Lévy-Sheffer polynomials are moments of $x {\mathbf{.}}\beta {\mathbf{.}}\gamma + t {\mathbf{.}}\alpha:$ $$V_k(x,t)=E[(x {\mathbf{.}}\beta {\mathbf{.}}\gamma + t {\mathbf{.}}\alpha)^k].
\label{(VKLS)}$$
\[aaa\] The TSH polynomials $Q_k(x,t)$ are special Lévy-Sheffer polynomials.
The proof of Theorem \[aaa\] is straightforward by choosing in (\[(VKLS)\]) as umbra $\alpha$ its inverse $-1 {\mathbf{.}}\alpha$ and as umbra $\gamma$ the singleton umbra $\chi.$ All the Lévy-Sheffer polynomials possess the TSH property. Indeed the following theorem has been proved in [@elviraannali].
The Lévy-Sheffer polynomials $\{V_k(x,t)\}_{t \geq 0}$ are TSH with respect to Lévy processes umbrally represented by $\{-t {\mathbf{.}}\alpha {\mathbf{.}}\beta {\mathbf{.}}\gamma^{{\scriptscriptstyle <-1>}}\}_{t \geq 0}.$
In particular, one has [@elviraannali] $$V_k(x,t) = \sum_{i=0}^k E[(x + t {\mathbf{.}}\beta {\mathbf{.}}\kappa_{({\alpha {\mathbf{.}}\beta {\mathbf{.}}\gamma^{{\scriptscriptstyle <-1>}}})})^i] B_{k,i}(g_1, \ldots, g_{k-i+1}),
\label{(LScomb)}$$ where $g_i = E[\gamma^i],$ for all nonnegative $i,$ and $\kappa_{({\alpha {\mathbf{.}}\beta {\mathbf{.}}\gamma^{{\scriptscriptstyle <-1>}}})}$ is the cumulant umbra of $\alpha {\mathbf{.}}\beta {\mathbf{.}}\gamma^{{\scriptscriptstyle <-1>}},$ with $\gamma^{{\scriptscriptstyle <-1>}}$ the compositional inverse of the umbra $\gamma.$ When the umbra $\alpha$ is replaced by its inverse and the umbra $\gamma$ by the singleton umbra, since $\chi^{{\scriptscriptstyle <-1>}} \equiv \chi,$ the only contribution in the summation (\[(LScomb)\]) is given by $i=k.$ So again equation (\[(LScomb)\]) reduces to $Q_k(x,t)=E[(x - t {\mathbf{.}}\alpha)^k]$ since $\kappa_{({\alpha {\mathbf{.}}\beta {\mathbf{.}}\gamma^{{\scriptscriptstyle <-1>}}})} \equiv \chi {\mathbf{.}}-1 {\mathbf{.}}\alpha.$
Within Lévy-Sheffer polynomials, the Lévy-Meixner polynomials are those orthogonal with respect to the Lévy processes $-t {\mathbf{.}}\alpha {\mathbf{.}}\beta {\mathbf{.}}\gamma^{{\scriptscriptstyle <-1>}},$ due to their TSH property. The orthogonal property is $$E \left[ V_n(-t {\mathbf{.}}\alpha {\mathbf{.}}\beta {\mathbf{.}}\gamma^{{\scriptscriptstyle <-1>}},t) V_m(-t {\mathbf{.}}\alpha {\mathbf{.}}\beta {\mathbf{.}}\gamma^{{\scriptscriptstyle <-1>}},t)\right] = c_m \delta_{n,m}.$$ According to [@Schoutens], all the polynomials in Table 6 are orthogonal. Their measure of orthogonality corresponds to the Lévy process $-t {\mathbf{.}}\alpha$ since $\chi^{{\scriptscriptstyle <-1>}} \equiv \chi$ and $-t {\mathbf{.}}\alpha {\mathbf{.}}\beta {\mathbf{.}}\gamma^{{\scriptscriptstyle <-1>}} \equiv -t {\mathbf{.}}\alpha.$
Kailath-Segall polynomials.
---------------------------
Equivalence (\[(TSHbell)\]) gives the connection between TSH polynomials and Kailath-Segall polynomials [@KS], which is a different class of polynomials strictly related to Lévy processes. Indeed, both have a representation in terms of partition umbra of a suitable polynomial umbra. Overlaps are removed by suitably choosing the indeterminates.
The $n$-th Kailath-Segall polynomial $P_n(x_1, \ldots, x_n)$ is a multivariable polynomial such that when the indeterminates are replaced by the sequence $X_t^{(1)}, \ldots,$ $ X_t^{(n)}$ of variations of a Lévy process $X_t$ $$X_t^{(1)} = X_t, \,\,\, X_t^{(2)} = [X,X]_t, \,\,\, X_t^{(n)} = \sum_{s \geq t}(\Delta X_s)^n
\;\; n \geq 3,$$ its iterated integrals are recovered $$P_t^{(0)} = 1, \,\,\, P_t^{(1)}=X_t, \,\,\, P_t^{(n)}=\int_0^t P_{s-}^{(n-1)} {\rm d}X_s, \;\; n \geq 2,$$ that is $P_t^{(n)} = P_n\left(X_t^{(1)}, \ldots, X_t^{(n)}\right).$ The following recursion formula is known as Kailath-Segall formula $$P_t^{(n)} = \frac{1}{n} \left( P_t^{(n-1)} X_t^{(1)} - P_t^{(n-2)} X_t^{(2)} + \cdots +
(-1)^{n+1} P_t^{(0)} X_t^{(n)} \right).
\label{(KS)}$$ When $X_t^{(1)}, \ldots, X_t^{(n)}$ are replaced by the power sums $S_1, \ldots, S_n$ in the indeterminates $x_1, \ldots, x_k,$ according to formula (1.2) in [@Taqqu] and Theorem 3.1 in [@Bernoulli], the corresponding polynomials $P_n(S_1, \ldots, S_n)$ are such that $$n! P_n(S_1, \ldots, S_n) = E[ (\beta {\mathbf{.}}[(\chi {\mathbf{.}}\chi) \sigma])^n],
\label{(KSPS)}$$ where $\sigma$ is the power sum umbra representing $\{S_j\}$ and the $\chi$-cumulant umbra $\chi {\mathbf{.}}\chi$ represents the sequence $\{(-1)^{i-1} (i-1)!\}.$
In order to recognize special TSH polynomials within the family $\{P_n\},$ two steps are necessary:
> [*i)*]{} Kailath-Segall polynomials need to be umbrally represented when the power sums $\{S_j\}$ are replaced by the indeterminates $\{x_i\};$\
> [*ii)*]{} the indeterminates $\{x_i\}$ need to be replaced by suitable terms involving $x$ and $t.$
For the first step, we will use equation (\[(KSPS)\]). Assume $p$ an umbra representing the sequence $\{E[(\chi {\mathbf{.}}x_i {\mathbf{.}}\beta)^i]\}.$ Observe that $E[(\chi {\mathbf{.}}x_i {\mathbf{.}}\beta)^i]=x_i$ for all nonnegative $i.$ Then from (\[(KSPS)\]) one has $n! P_n( x_1, \ldots, x_n) = E[(\beta {\mathbf{.}}[(\chi {\mathbf{.}}\chi) p \,])^n]$ so that the generating function of $P_n$ is $$f\left(\beta {\mathbf{.}}[(\chi {\mathbf{.}}\chi) p \,],z \right) = \exp \left( \sum_{n \geq 1} \frac{(-1)^{n+1}}{n} z^n x_n \right),$$ see also [@Yablonski]. The strength of this symbolic representation essentially relies on the properties of the partition umbra $\beta {\mathbf{.}}[(\chi {\mathbf{.}}\chi) p \,]$ reproducing those of Bell polynomials. For example, the following property of Kailath-Segall polynomials $$P_n(a x_1, a^2 x_2, \ldots, a^n x_n) = a^n P_n(x_1, x_2, \ldots, x_n), \quad a \in {\mathbb R}\label{(TSHBELL)}$$ is proved by observing that $\beta {\mathbf{.}}[a (\chi {\mathbf{.}}\chi) p \,] \equiv a (\beta {\mathbf{.}}[(\chi {\mathbf{.}}\chi) p \,]).$ For the next step, we need to characterize the indeterminates $x_1, x_2, \ldots$ such that $$\kappa_{(x {\mathbf{.}}u)} \, \dot{+} \, \kappa_{(- t {\mathbf{.}}\alpha)} \equiv (\chi {\mathbf{.}}\chi) p \Rightarrow E[(\kappa_{(x {\mathbf{.}}u)})^n] + E[(\kappa_{(- t {\mathbf{.}}\alpha)})^n] = (-1)^{n-1}(n-1)! \, x_n.
\label{(TSHKS)}$$ In the following we show some examples of how to perform this selection. These results extend the connections between TSH polynomials and Kailath-Segall polynomials analyzed in [@Sole].
[**Generalized Hermite polynomials:**]{}\
Since $E[(\kappa_{(x {\mathbf{.}}u)})^i]= x \, \delta_{i,1}$ and $E[(\kappa_{(- t {\mathbf{.}}\beta {\mathbf{.}}(s \varsigma))})^i] = s^2 \, t \, \delta_{i,2},$ then $$k! P_k(x, s^2 t, 0, \ldots, 0) = H_k^{(t)}(x)$$ where $\sum_{k \geq 0} H_k^{(t)}(x) z^k/k! = \exp\{xz - t z^2/ 2\}.$
[**Poisson-Charlier polynomials:**]{}\
Poisson-Charlier polynomials $\{\widetilde{C}_k(x,t)\},$ with generating function $$\sum_{k\geq 0} \widetilde{C}_k(x,t) \frac{z^k}{k!} = e^{-tz}(1+z)^x,$$ are umbrally represented by $$\widetilde{C}_k(x,\lambda t) = E[(x {\mathbf{.}}\chi - t {\mathbf{.}}\lambda {\mathbf{.}}u)^k],
\label{(PCTSH)}$$ see [@elviraannali]. Nevertheless (\[(PCTSH)\]) differs from the result of Theorem \[UTSH2\], the TSH property holds since $\{\widetilde{C}_k(x,\lambda t)\}$ are a linear combination of special $Q_k(x,t).$ Moreover representation (\[(PCTSH)\]) allows us the connection with Kailath-Segall polynomials, when the indeterminates $\{x_i\}$ are chosen such that $E[(\kappa_{(x {\mathbf{.}}\chi)})^i] + E[(\kappa_{(- t {\mathbf{.}}\lambda {\mathbf{.}}u)})^i] = (-1)^{i-1}(i-1)! \, x_i.$ Since $$E[(\kappa_{(x {\mathbf{.}}\chi)})^i] + E[(\kappa_{(- t {\mathbf{.}}\lambda {\mathbf{.}}u)})^i] = \left\{ \begin{array}{lc}
x - t \lambda, & i=1, \\
(-1)^{i-1} (i-1)! \, x^i, & i \geq 2,
\end{array} \right.$$ then $k! P_k(x - t \lambda,x,x, \ldots)=\widetilde{C}_k(x,\lambda t).$
[**Laguerre polynomials:**]{}\
Laguerre polynomials $\{\mathcal{L}_k^{t-k}(x)\}$ are TSH polynomials such that $$k! (-1)^k {\mathcal L}_k^{t-k}(x) = E[(x - t {\mathbf{.}}\bar{u})^k].$$ They can be traced back to Kailath-Segall polynomials if the indeterminates $\{x_i\}$ are characterized by $E[(\kappa_{(x {\mathbf{.}}u)})^i] + E[(\kappa_{(- t {\mathbf{.}}\bar{u})})^i] = (-1)^{i-1}(i-1)! \, x_i.$ Since $f(\kappa_{(- t {\mathbf{.}}\bar{u})},z) = 1 + t \, \log (1 -z)$ then $E[(\kappa_{(- t {\mathbf{.}}\bar{u})})^i] = - t (i-1)!.$ So $$P_k(x-t, t , - t, t, \ldots) = (-1)^k {\mathcal L}_k^{t-k}(x)$$ and from (\[(TSHBELL)\]) we have $P_k(t-x, t , t, \ldots) = {\mathcal L}_k^{t-k}(x).$
[**Actuarial polynomials:**]{}\
The actuarial polynomials $g_k(x,t)$ are a linear combination of suitable TSH polynomials $Q_k(x,t)$ (see Table 6) but they are moments of the polynomial umbra $\lambda t - x {\mathbf{.}}\beta,$ that is $g_k(x,t) = E[( \lambda t - x {\mathbf{.}}\beta)^k].$ In order to characterize the connection with Kailath-Segall polynomials, the indeterminates $\{x_i\}$ need to be characterized by $E[(\kappa_{(\lambda t {\mathbf{.}}u)})^i] + E[(\kappa_{(- x {\mathbf{.}}\bar{u})})^i] = (-1)^{i-1}(i-1)! \, x_i.$ As before $E[(\kappa_{(\lambda t {\mathbf{.}}u)})^i] = \lambda \, t \, \delta_{i,1},$ instead $E[(\kappa_{(- x {\mathbf{.}}\bar{u})})^i]=-x (i-1)!$ for all nonnegative integers $i$ as in the previous example. Therefore one has $k! (-1)^k P_k(x - \lambda t, x , x, \ldots) = g_k(x,t) .$
[**Meixner polynomials of first kind:**]{}\
Meixner polynomials of first kind $\{M_k(x,t,p)\}$ [@Schoutens] are a linear combination of suitable TSH polynomials $Q_k(x,t)$ (see Table 6) but they are moments of the following polynomial umbra $$(-1)^k (t)_k M_k(x,t,p) = E\left\{\left[x {\mathbf{.}}\left(-1 {\mathbf{.}}\chi + \frac{\chi}{p}\right)
- t {\mathbf{.}}\chi\right]^k\right\},$$ which allows us to find the connection with Kailath-Segall polynomials. Indeed for all nonnegative integers $i$ we have $$E\left[\left\{\kappa_{x {\mathbf{.}}\left(-1 {\mathbf{.}}\chi + \frac{\chi}{p}\right)}\right\}^i\right]= (-1)^{i-1} \, (i-1)! \, x \, \left( \frac{1}{p^i} - 1 \right)$$ and $$E\left[\left\{\kappa_{(\chi {\mathbf{.}}-t {\mathbf{.}}\chi)} \right\}^i\right] = (-1)^{i-1} (i-1)! \, t.$$ Then Kailath-Segall polynomials give the Meixner polynomials $(-1)^k (t)_k M_k(x,t,p)$ by choosing $$x_i = \left[ \left( \frac{1}{p^i} - 1 \right) x - t \right]$$ for $i=1,2,\ldots.$
Symbolic multivariate Lévy processes.
=====================================
In the multivariate case, the main device of the symbolic method here proposed relies on the employment of multi-indices of length $d$. Sequences like $\{g_{i_1, i_2, \ldots, i_d}\}$ are replaced with a product of powers $\mu_{1}^{i_1} \mu_{2}^{i_2} \cdots \mu_{d}^{i_d},$ where $(\mu_1, \mu_2, \ldots, \mu_d)$ are umbral monomials and $(i_1, i_2, \ldots, i_d)$ are nonnegative integers. Since the umbral monomials could not have disjoint support, then the evaluation $E$ does not necessarily factorizes on the product $\mu_{1}^{i_1} \mu_{2}^{i_2} \cdots \mu_{d}^{i_d},$ that is $$E[\mu_1^{i_1} \mu_2^{i_2} \cdots \mu_d^{i_d}]=E[{\boldsymbol{\mu}}^{{\boldsymbol{i}}}] = g_{{\boldsymbol{i}}}
\label{(multiindex)}$$ where ${\boldsymbol{i}}=(i_1, i_2, \ldots, i_d)$ and ${\boldsymbol{\mu}}=(\mu_1, \mu_2, \ldots, \mu_d).$ We assume $g_{\boldsymbol 0}=1$ with ${\boldsymbol 0}=(0,0,\ldots,0).$ Then $g_{{\boldsymbol{i}}}$ is called the multivariate moment of ${\boldsymbol{\mu}}.$ Table 7 shows some special $d$-tuples we will use later.
** *$d$-tuple* *Generating functions*
-------------------------------------------------- ------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------
Multivariate Unity ${\boldsymbol{u}}$ $(u, \ldots, u^{\prime})$ $f({\boldsymbol{u}},{\boldsymbol{z}}) = e^{z_1 + \cdots + z_d}.$
Multivariate Gaussian ${\boldsymbol{\varsigma}}$ $(\varsigma, \ldots, \varsigma^{\prime})$ $f({\boldsymbol{\varsigma}},{\boldsymbol{z}})=1 + \frac{1}{2} {\boldsymbol{z}}{\boldsymbol{z}}^T.$
Multivariate Bernoulli ${\boldsymbol{\iota}}$ $(\iota, \ldots, \iota)$ $f({\boldsymbol{\iota}},{\boldsymbol{z}})=\displaystyle{\frac{z_1 + \cdots + z_d}{e^{z_1 + \cdots + z_d} - 1}}.$
Multivariate Euler ${\boldsymbol{\eta}}$ $(\eta, \ldots, \eta)$ $f({\boldsymbol{\eta}},{\boldsymbol{z}})=\displaystyle{\frac{2 e^{(z_1 + \cdots + z_d)}}{e^{2(z_1 + \cdots + z_d)} + 1}}.$
: Generating functions of special $d$-tuples of umbral monomials[]{data-label="table7"}
The notions of similarity and uncorrelation are updated as follows. Two $d$-tuples ${\boldsymbol{\mu}}$ and ${\boldsymbol{\nu}}$ of umbral monomials are said to be similar if they represent the same sequence of multivariate moments. They are said to be uncorrelated if $E[{\boldsymbol{\mu}}^{{{\boldsymbol{i}}}_1} {\boldsymbol{\nu}}^{{{\boldsymbol{i}}}_2}] = E[{\boldsymbol{\mu}}^{{{\boldsymbol{i}}}_1}]
E[{\boldsymbol{\nu}}^{{{\boldsymbol{i}}}_2}].$
Multivariate Lévy processes are represented by $d$-tuples of umbral monomials.
\[def\_levy\_multi\] A stochastic process $\{{\boldsymbol{X}}_t\}_{t \geq 0}$ on ${\mathbb R}^d$ is a *multivariate Lévy process* if
[(i)]{}
: ${\boldsymbol{X}}_0 = \boldsymbol{0}$ a.s.
[(ii)]{}
: For all $n \geq 1$ and for all $\,0 \leq t_1 \leq t_2 \leq \ldots \leq t_n < \infty,$ the r.v.’s ${\boldsymbol{X}}_{t_2} - {\boldsymbol{X}}_{t_1}, {\boldsymbol{X}}_{t_3} - {\boldsymbol{X}}_{t_2}, \ldots$ are independent.
[(iii)]{}
: For all $s \leq t,$ ${\boldsymbol{X}}_{t + s} - {\boldsymbol{X}}_s \stackrel{d}{=} {\boldsymbol{X}}_t.$
[(iv)]{}
: For all $\varepsilon > 0,$ $\lim_{h \rightarrow 0} P(|{\boldsymbol{X}}_{t + h} - {\boldsymbol{X}}_t| > \varepsilon) = 0.$
[(v)]{}
: $t \mapsto {\boldsymbol{X}}_t (\omega)$ are right-continuous with left limits, for all $\omega \in \varOmega,$ with $\varOmega$ the underlying sample space.
As in the univariate case, the moment generating function of a multivariate Lévy process is $\varphi_{\scriptscriptstyle{{\boldsymbol{X}}_1}}({\boldsymbol{z}}) = E\left[e^{{\boldsymbol{z}}{\boldsymbol{X}}_1^{{\footnotesize T}}}\right],$ with ${\boldsymbol{z}}\in {\mathbb R}^d.$ Paralleling the univariate case, the generating function of a $d$-tuple ${\boldsymbol{\mu}}$ is $$f({\boldsymbol{\mu}}, {\boldsymbol{z}}) = 1 + \sum_{k \geq 1} \sum_{\substack{{\boldsymbol{i}}\in \mathbb{N}_0^d \\ |{\boldsymbol{i}}| = k}} g_{{\boldsymbol{i}}} \frac{{\boldsymbol{z}}^{{\boldsymbol{i}}}}{{\boldsymbol{i}}!}.$$ Choose the $d$-tuple ${\boldsymbol{\mu}}$ such that $f({\boldsymbol{\mu}},{\boldsymbol{z}})=\varphi_{{\boldsymbol{X}}_1}({\boldsymbol{z}}),$ that is $E[{\boldsymbol{\mu}}^{{\boldsymbol{i}}}]=E[{\boldsymbol{X}}_1^{{\boldsymbol{i}}}]$ for all ${\boldsymbol{i}}\in \mathbb{N}_0^d.$ The auxiliary umbra $n {\mathbf{.}}{\boldsymbol{\mu}}$ denotes the sum of $n$ uncorrelated $d$-tuples of umbral monomials similar to ${\boldsymbol{\mu}}.$ Its multivariate moment is [@dibruno] $$E[(n {\mathbf{.}}{\boldsymbol{\mu}})^{{\boldsymbol{i}}}] = \sum_{{\boldsymbol{\lambda}}\vdash {\boldsymbol{i}}} \frac{{\boldsymbol{i}}!}{\mathfrak{m}({\boldsymbol{\lambda}}) {\boldsymbol{\lambda}}!} \, (n)_{l({\boldsymbol{\lambda}})} \,
E[{\boldsymbol{\mu}}_{{\boldsymbol{\lambda}}}], \label{(auxmult)}$$ where $E[{\boldsymbol{\mu}}_{\scriptscriptstyle{{\boldsymbol{\lambda}}}}] = g_{{\boldsymbol{\lambda}}_1}^{r_1}
g_{{\boldsymbol{\lambda}}_2}^{r_2} \ldots$ and ${\boldsymbol{\lambda}}$ is a partition[^8] of the multi-index ${\boldsymbol{i}}$ of length $l({\boldsymbol{\lambda}}).$ By replacing the nonnegative integer $n$ with the real parameter $t$ in (\[(auxmult)\]) the resulting auxiliary umbra $t {\mathbf{.}}{\boldsymbol{\mu}}$ is the symbolic representation of the multivariate Lévy process $ {\boldsymbol{X}}_t.$
As in the univariate case, since ${\boldsymbol{\mu}}\equiv \beta {\mathbf{.}}\kappa_{{\boldsymbol{\mu}}}$ with $\kappa_{{\boldsymbol{\mu}}}$ the ${\boldsymbol{\mu}}$-cumulant umbra [@Bernoulli], a different representation for a multivariate Lévy process is $t {\mathbf{.}}\beta {\mathbf{.}}\kappa_{{\boldsymbol{\mu}}}.$ The cumulant $d$-tuple could be further specified by using the multivariate Lévy-Khintchine formula [@sato].
\[LK\_thm\_multi\] ${\boldsymbol{X}}= \{{\boldsymbol{X}}_t\}_{t \geq 0}$ is a Lévy process if and only if there exists ${\boldsymbol{m}}_1 \in {\mathbb R}^d,$ a symmetric, positive defined $d \times d$ matrix $\Sigma > 0$ and a measure $\nu$ on ${\mathbb R}^d$ with $$\nu(\{0\}) = 0 \mbox{ and } \int_{{\mathbb R}}(|{\boldsymbol{x}}|^2 \wedge 1) \nu(d {\boldsymbol{x}}) < \infty$$ such that $$\varphi_{\scriptscriptstyle{{\boldsymbol{X}}}}({\boldsymbol{z}}) = \exp\left\{t\left[\frac{1}{2} {\boldsymbol{z}}\Sigma {\boldsymbol{z}}^{T} + {\boldsymbol{m}}_1{\boldsymbol{z}}^{T} + \int_{{\mathbb R}^d}
(e^{{\boldsymbol{x}}{\boldsymbol{z}}^{T}} - 1 - {\boldsymbol{x}}{\boldsymbol{z}}^{T} {\boldsymbol 1}_{\{|{\boldsymbol{x}}|\leq 1\}}({\boldsymbol{x}}))\, \nu({\rm d}{\boldsymbol{x}})\right]\right\}.
\label{LK_multi}$$ The representation of $\varphi_{\scriptscriptstyle {\boldsymbol{X}}}({\boldsymbol{z}})$ in (\[LK\_multi\]) by $\boldsymbol{m_1},$ $\Sigma$ and $\nu$ is unique.
Set ${\boldsymbol{m}}_2 {\boldsymbol{z}}^{{\footnotesize T}} = \int_{{\mathbb R}^d}{\boldsymbol{z}}{\boldsymbol{x}}^{{\footnotesize T}} {\boldsymbol 1}_{\{|{\boldsymbol{x}}| > 1\}}({\boldsymbol{x}})\, \nu(d{\boldsymbol{x}})$ and ${\boldsymbol{m}}= {\boldsymbol{m}}_1
+ {\boldsymbol{m}}_2,$ then $$\varphi_{\scriptscriptstyle{{\boldsymbol{X}}}}({\boldsymbol{z}}) = \exp\bigg\{t\bigg[\frac{1}{2}{\boldsymbol{z}}\Sigma {\boldsymbol{z}}^{{\footnotesize T}} + {\boldsymbol{m}}{\boldsymbol{z}}^{{\footnotesize T}} + \int_{{\mathbb R}^d}(e^{{\boldsymbol{z}}{\boldsymbol{x}}^{{\footnotesize T}}} - 1 -{\boldsymbol{z}}{\boldsymbol{x}}^{{\footnotesize T}}) \, \nu(d{\boldsymbol{x}})\bigg]\bigg\},$$ that is, $$\varphi_{\scriptscriptstyle{{\boldsymbol{X}}}}({\boldsymbol{z}}) = \exp\bigg\{t\bigg[\frac{1}{2}{\boldsymbol{z}}\Sigma {\boldsymbol{z}}^{{\footnotesize T}} + {\boldsymbol{m}}{\boldsymbol{z}}^{{\footnotesize T}}\bigg]\bigg\} \exp\bigg\{t\bigg[\int_{{\mathbb R}^d}(e^{{\boldsymbol{z}}{\boldsymbol{x}}^{{\footnotesize T}}} - 1 -{\boldsymbol{z}}{\boldsymbol{x}}^{{\footnotesize T}}) \, \nu(d{\boldsymbol{x}})\bigg]\bigg\}.
\label{LK2_multi}$$
\[fund\] Every Lévy process $\{{\boldsymbol{X}}_t\}_{t \geq 0}$ on ${\mathbb R}^d$ is umbrally represented by the family of auxiliary umbrae $$\{t {\mathbf{.}}\beta {\mathbf{.}}(\chi {\mathbf{.}}{\boldsymbol{m}}\dot{+} {\boldsymbol{\varsigma}}C^{T} \dot{+} {\boldsymbol{\eta}})\}_{t \geq 0},\label{levymulti}$$ where $\beta$ is the Bell umbra, ${\boldsymbol{m}}\in {\mathbb R}^d,$ ${\boldsymbol{\varsigma}}$ is the multivariate umbral counterpart of a standard gaussian r.v., $C$ is the square root of the covariance matrix $\Sigma$ and ${\boldsymbol{\eta}}$ is the multivariate umbra associated to the Lévy measure.
Every auxiliary umbra $t {\mathbf{.}}\beta {\mathbf{.}}\kappa_{{\boldsymbol{\mu}}}$ is the symbolic version of a multivariate compound Poisson r.v. of parameter $t,$ that is a random sum $S_N = {\boldsymbol{Y}}_1 + \cdots + {\boldsymbol{Y}}_N$ of independent and identically distributed random vectors $\{ {\boldsymbol{Y}}_i\},$ whose index $N$ is a Poisson r.v. of parameter $t.$ Then the same holds for the Lévy process. The $d$-tuple $({\boldsymbol{\varsigma}}C^{T} \dot{+} \chi {\mathbf{.}}{\boldsymbol{m}}\dot{+} {\boldsymbol{\eta}})$ umbrally represents any of the random vectors $\{ {\boldsymbol{Y}}_i\}.$ Observe that $\chi {\mathbf{.}}{\boldsymbol{m}}$ has not a probabilistic counterpart. If ${\boldsymbol{m}}$ is not equal to the zero vector, this parallels the well-known difficulty to interpret the Lévy measure as a probability measure.
Multivariate TSH polynomials.
-----------------------------
The conditional evaluation with respect to an umbral $d$-tuple ${\boldsymbol{\mu}}$ has been introduced in [@DiNardoOliva3]. Assume ${\mathcal{X}} = \{\mu_1, \mu_2, \ldots, \mu_d\}.$ The conditional evaluation with respect to the umbral $d$-tuple ${\boldsymbol{\mu}}$ is the linear operator $$E(\;\cdot \; \vline \,\, {\boldsymbol{\mu}}): \, {\mathbb R}[x_1, \ldots, x_d][{\mathcal{A}}] \; \longrightarrow \; {\mathbb R}[\mathcal{X}]$$ such that $E(1 \,\, \vline \,\, {\boldsymbol{\mu}}) = 1$ and $$E(x_1^{l_1} \, x_2^{l_2} \ \cdots \, x_d^{l_d} \, {\boldsymbol{\mu}}^{{\boldsymbol{i}}} \, {\boldsymbol{\nu}}^{{\boldsymbol{j}}} \, {\boldsymbol{\eta}}^{{\boldsymbol{k}}} \cdots \, \, \vline \,\, {\boldsymbol{\mu}})
= x_1^{l_1} \, x_2^{l_2} \, \cdots \, x_d^{l_d} \, {\boldsymbol{\mu}}^{{\boldsymbol{i}}} \, E[{\boldsymbol{\nu}}^{{\boldsymbol{j}}}] \, E[{\boldsymbol{\eta}}^{{\boldsymbol{k}}}] \cdots
\label{(condeval1)}$$ for uncorrelated $d$-tuples ${\boldsymbol{\mu}}, {\boldsymbol{\nu}}, {\boldsymbol{\eta}}\ldots,$ multi-indices ${\boldsymbol{i}}, {\boldsymbol{j}}, {\boldsymbol{k}}\ldots \in \mathbb{N}_0^d$ and $\{l_i\}_{i=1}^d$ nonnegative integers. Since $f[(n + m) {\mathbf{.}}{\boldsymbol{\mu}}, {\boldsymbol{z}}] = f({\boldsymbol{\mu}}, {\boldsymbol{z}})^{n + m} = f(n {\mathbf{.}}{\boldsymbol{\mu}}, {\boldsymbol{z}}) \, f(m {\mathbf{.}}{\boldsymbol{\mu}}, {\boldsymbol{z}}),$ then $(n + m) {\mathbf{.}}{\boldsymbol{\mu}}\equiv n {\mathbf{.}}{\boldsymbol{\mu}}+ m {\mathbf{.}}{\boldsymbol{\mu}}^{\prime},$ with ${\boldsymbol{\mu}}$ and ${\boldsymbol{\mu}}^{\prime}$ uncorrelated $d$-tuples of umbral monomials. Then, for $E(\;\cdot \; \vline \,\, n {\mathbf{.}}{\boldsymbol{\mu}})$ we assume $E[\{(n + m) {\mathbf{.}}{\boldsymbol{\mu}}\}^{{\boldsymbol{i}}} \,\, \vline \,\, n {\mathbf{.}}{\boldsymbol{\mu}}] = E[\{n {\mathbf{.}}{\boldsymbol{\mu}}+ m {\mathbf{.}}{\boldsymbol{\mu}}^{\prime}\}^{{\boldsymbol{i}}} \,\,
\vline \,\, n {\mathbf{.}}{\boldsymbol{\mu}}]$ for all nonnegative integers $n, m$ and for all ${\boldsymbol{i}}\in \mathbb{N}_0^d.$ If $n \ne m,$ then $$E[\{(n + m) {\mathbf{.}}{\boldsymbol{\mu}}\}^{{\boldsymbol{i}}} \,\, \vline \,\, n {\mathbf{.}}{\boldsymbol{\mu}}] = E[\{n {\mathbf{.}}{\boldsymbol{\mu}}+ m {\mathbf{.}}{\boldsymbol{\mu}}\}^{{\boldsymbol{i}}} \,\,
\vline \,\, n {\mathbf{.}}{\boldsymbol{\mu}}],
\label{(aaa1)}$$ since $n {\mathbf{.}}{\boldsymbol{\mu}}$ and $m {\mathbf{.}}{\boldsymbol{\mu}}$ are uncorrelated auxiliary umbrae. We will use the same $d$-tuple ${\boldsymbol{\mu}}$ as in (\[(aaa1)\]) when no misunderstanding occurs. Thanks to equations (\[(condeval1)\]) and (\[(aaa1)\]), we have $$E\left[\{(n + m) {\mathbf{.}}{\boldsymbol{\mu}}\}^{{\boldsymbol{i}}} \,\, \vline \,\, n {\mathbf{.}}{\boldsymbol{\mu}}\right] = \sum_{{\boldsymbol{k}}\leq {\boldsymbol{i}}} \binom{{\boldsymbol{i}}}{{\boldsymbol{k}}} (n {\mathbf{.}}{\boldsymbol{\mu}})^{{\boldsymbol{k}}} E[(m {\mathbf{.}}{\boldsymbol{\mu}})^{{\boldsymbol{i}}- {\boldsymbol{k}}}],
\label{ce_dot_multi}$$ where ${\boldsymbol{k}}\leq {\boldsymbol{i}}\; \Leftrightarrow \; k_j \leq i_j \mbox{ for all } j = 1, \ldots, d$ and $\binom{{\boldsymbol{k}}}{{\boldsymbol{i}}} = \binom{k_1}{i_1} \cdots \binom{k_d}{i_d}.$ By analogy with (\[(aaa1)\]) and (\[ce\_dot\_multi\]), we have $t {\mathbf{.}}{\boldsymbol{\mu}}\equiv s
{\mathbf{.}}{\boldsymbol{\mu}}+ (t-s) {\mathbf{.}}{\boldsymbol{\mu}}$ and for $t \geq 0$ and $s \leq t$ $$E\left[(t {\mathbf{.}}{\boldsymbol{\mu}})^{{\boldsymbol{i}}} \,\, \vline \,\, s {\mathbf{.}}{\boldsymbol{\mu}}\right] = \sum_{{\boldsymbol{k}}\leq {\boldsymbol{i}}} \binom{{\boldsymbol{i}}}{{\boldsymbol{k}}} (s {\mathbf{.}}{\boldsymbol{\mu}})^{{\boldsymbol{k}}}
E[\{(t - s) {\mathbf{.}}{\boldsymbol{\mu}}\}^{{\boldsymbol{i}}- {\boldsymbol{k}}}].$$
\[UTSH2\_multi\] For all ${\boldsymbol{i}}\in \mathbb{N}_0^d,$ the family of polynomials $$Q_{{\boldsymbol{i}}}({\boldsymbol{x}},t) = E[({\boldsymbol{x}}- t {\mathbf{.}}{\boldsymbol{\mu}})^{{\boldsymbol{i}}}] \in {\mathbb R}[x_1, \ldots, x_d]
\label{(tshumbral_multi)}$$ is TSH with respect to $\{t {\mathbf{.}}{\boldsymbol{\mu}}\}_{t \geq 0}.$
The auxiliary umbra $-t {\mathbf{.}}{\boldsymbol{\mu}}$ denotes the inverse of $t {\mathbf{.}}{\boldsymbol{\mu}}$ that is $- t {\mathbf{.}}{\boldsymbol{\mu}}+ t {\mathbf{.}}{\boldsymbol{\mu}}\equiv {\boldsymbol{\epsilon}},$ where ${\boldsymbol{\epsilon}}$ is the $d$-tuple such that ${\boldsymbol{\epsilon}}=(\epsilon_1, \epsilon_2, \ldots, \epsilon_d)$ with $\{\epsilon_i\}$ uncorrelated augmentation umbrae. Coefficients of $Q_{{\boldsymbol{i}}}({\boldsymbol{x}},t)$ in (\[(tshumbral\_multi)\]) are such that $$Q_{{\boldsymbol{i}}}({\boldsymbol{x}},t) = \sum_{{\boldsymbol{k}}\leq {\boldsymbol{i}}} \binom{{\boldsymbol{i}}}{{\boldsymbol{k}}} {\boldsymbol{x}}^{{\boldsymbol{i}}- {\boldsymbol{k}}} E[(-t {\mathbf{.}}{\boldsymbol{\mu}})^{{\boldsymbol{k}}}]$$ so when ${\boldsymbol{x}}$ is replaced by $t {\mathbf{.}}{\boldsymbol{\mu}}$ their overall evaluation is zero. Properties on the coefficients of $Q_{{\boldsymbol{i}}}({\boldsymbol{x}},t)$ can be found in [@DiNardoOliva3]. Here we just recall a characterization of the coefficients of any multivariate TSH polynomial in terms of those of $Q_{{\boldsymbol{i}}}({\boldsymbol{x}},t).$
\[comb\_multi\] A polynomial $$P({\boldsymbol{x}},t) =\sum_{{\boldsymbol{k}}\leq {\boldsymbol{v}}} p_{\scriptscriptstyle{\boldsymbol{k}}}(t) \, {\boldsymbol{x}}^{{\boldsymbol{k}}}
\label{polynomial}$$ is a TSH polynomial with respect to $\{t {\mathbf{.}}{\boldsymbol{\mu}}\}_{t \geq 0}$ if and only if $$p_{\scriptscriptstyle{\boldsymbol{k}}}(t) = \sum_{{\boldsymbol{k}}\leq {\boldsymbol{i}}\leq {\boldsymbol{v}}} \binom{{\boldsymbol{i}}}{{\boldsymbol{k}}} \, p_{\scriptscriptstyle{\boldsymbol{k}}}(0) \,
E[(-t {\mathbf{.}}{\boldsymbol{\mu}})^{{\boldsymbol{i}}-{\boldsymbol{k}}}], \quad \hbox{ for } {\boldsymbol{k}}\leq {\boldsymbol{v}}.
\label{prop5_multi}$$
Table 8 and 9 give some examples of multivariate TSH polynomials and their connection with multivariate Lévy processes. The corresponding $d$-tuples are given in Table 7.
multivariate Lévy process Umbral representation TSH polynomial $Q_{{\boldsymbol{i}}}({\boldsymbol{x}},t)$
------------------------------------------------------------------------------- -------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------
Brownian motion
with covariance $\Sigma=C C^T$ $t {\mathbf{.}}\beta {\mathbf{.}}({\boldsymbol{\varsigma}}C^T)$ $E[({\boldsymbol{x}}- t {\mathbf{.}}\beta {\mathbf{.}}({\boldsymbol{\varsigma}}C^T))^{{\boldsymbol{i}}}]$
${\boldsymbol{X}}_t$ with ${\boldsymbol{X}}_1 \stackrel{d}{=} (U, \ldots, U)$
and $U$ uniform r.v. on $[0,1]$ $- t {\mathbf{.}}{\boldsymbol{\iota}}$ $E[({\boldsymbol{x}}+ t {\mathbf{.}}{\boldsymbol{\iota}})^{{\boldsymbol{i}}}]$
${\boldsymbol{X}}_t$ with ${\boldsymbol{X}}_1 \stackrel{d}{=} (Y, \ldots, Y)$
and $Y$ Bernoulli r.v. $\frac{1}{2} [t {\mathbf{.}}({\boldsymbol{u}}- 1 {\mathbf{.}}{\boldsymbol{\eta}})]$ $E\left\{ \left({\boldsymbol{x}}+ \frac{1}{2} [ t {\mathbf{.}}({\boldsymbol{\eta}}- {\boldsymbol{u}})] \right)^{{\boldsymbol{i}}}\right\}$
of parameter $1/2$
: TSH polynomials associated to special multivariate Lévy processes[]{data-label="table8"}
------------------------------------------------------------------------------- ----------------------------------------------------------------- -----------------------------------------------------------------------------------------------------
multivariate Lévy process Special family Connection with
of polynomials TSH polynomials
Brownian motion
with covariance $\Sigma=C C^T$ Hermite $H_{{\boldsymbol{i}}}^{(t^2)}({\boldsymbol{x}},\Sigma)$ $H_{{\boldsymbol{i}}}^{(t^2)}({\boldsymbol{x}},\Sigma)=Q_{{\boldsymbol{i}}}({\boldsymbol{x}},t)$
${\boldsymbol{X}}_t$ with ${\boldsymbol{X}}_1 \stackrel{d}{=} (U, \ldots, U)$
and $U$ uniform r.v. on $[0,1]$ Bernoulli $B_{{\boldsymbol{i}}}^{(t)}({\boldsymbol{x}})$ $B_{{\boldsymbol{i}}}^{(t)}({\boldsymbol{x}}) = Q_{{\boldsymbol{i}}}({\boldsymbol{x}},t)$
${\boldsymbol{X}}_t$ with ${\boldsymbol{X}}_1 \stackrel{d}{=} (Y, \ldots, Y)$
and $Y$ Bernoulli r.v. Euler $\mathcal{E}_{{\boldsymbol{i}}}^{(t)}({\boldsymbol{x}})$ $\mathcal{E}_{{\boldsymbol{i}}}^{(t)}({\boldsymbol{x}}) = Q_{{\boldsymbol{i}}}({\boldsymbol{x}},t)$
of parameter $1/2$
------------------------------------------------------------------------------- ----------------------------------------------------------------- -----------------------------------------------------------------------------------------------------
: Special families of polynomials and TSH polynomials[]{data-label="table9"}
Let us remark that Hermite polynomials $H_{{\boldsymbol{i}}}^{(t^2)}({\boldsymbol{x}},\Sigma)$ in Table 9 are a generalization of the polynomials $H_{{\boldsymbol{i}}}({\boldsymbol{x}})$ in [@Withers] whose moment representation is $H_{{\boldsymbol{i}}}({\boldsymbol{x}})=E[({\boldsymbol{x}}\Sigma^{-1} + {\mathfrak i} {\boldsymbol{Y}})^{{\boldsymbol{i}}}]$ with $E$ the expectation symbol, ${\boldsymbol{Y}}\simeq N({\bf 0}, \Sigma^{-1})$ and $\Sigma$ a covariance matrix of full rank $d$.
A generalization of Lévy-Sheffer system to the multivariate case has been introduced in [@DiNardoOliva3]. A sequence of multivariate polynomials $\{V_{{\boldsymbol{k}}}({\boldsymbol{x}},t)\}_{t \geq 0}$ is a multivariate Lévy-Sheffer system if $$1 + \sum_{k \geq 1} \sum_{\substack{{\boldsymbol{v}}\in \mathbb{N}_0^d \\ |{\boldsymbol{v}}| = k}} V_{{\boldsymbol{k}}}({\boldsymbol{x}},t)\frac{{\boldsymbol{z}}^{{\boldsymbol{k}}}}{{\boldsymbol{k}}!} = [g({\boldsymbol{z}})]^t
\exp\{ (x_1 + \cdots + x_d) [h({\boldsymbol{z}}) - 1] \},$$ where $g({\boldsymbol{z}})$ and $h({\boldsymbol{z}})$ are analytic in a neighborhood of ${\boldsymbol{z}}=\boldsymbol{0}$ and $$\left. \frac{\partial}{\partial z_i} h({\boldsymbol{z}}) \right|_{{\boldsymbol{z}}= \boldsymbol{0}} \neq 0 \quad \hbox{for
$i=1,2,\ldots,d$}.$$ If ${\boldsymbol{\mu}}$ and ${\boldsymbol{\nu}}$ are $d$-tuples of umbral monomials such that $f({\boldsymbol{\mu}},{\boldsymbol{z}})=g({\boldsymbol{z}})$ and $f({\boldsymbol{\nu}},{\boldsymbol{z}})=h({\boldsymbol{z}})$ respectively, then $$V_{{\boldsymbol{k}}}({\boldsymbol{x}},t) = E[(t {\mathbf{.}}{\boldsymbol{\mu}}+ (x_1 + \cdots + x_d) {\mathbf{.}}\beta {\mathbf{.}}{\boldsymbol{\nu}})^{{\boldsymbol{k}}}].
\label{(LS)}$$ The multivariate Lévy-Sheffer polynomials for the pair ${\boldsymbol{\mu}}$ and ${\boldsymbol{\nu}}$ are TSH polynomials with respect to a special symbolic multivariate Lévy process involving the multivariate compositional inverse of a $d$-tuple ${\boldsymbol{\nu}}.$ Assume ${\boldsymbol{\chi}}_{(i)}$ the $d$-tuple with all components equal to the augmentation umbra and only the $i$-th one equal to the singleton umbra, that is ${\boldsymbol{\chi}}_{(i)} = (\epsilon, \ldots, \chi, \ldots, \epsilon).$ The multivariate compositional inverse of ${\boldsymbol{\nu}}$ is the umbral $d$-tuple ${\boldsymbol{\nu}}^{{\scriptscriptstyle <-1>}}=({({\boldsymbol{\nu}}^{{\scriptscriptstyle <-1>}})}_1, \ldots, ({\boldsymbol{\nu}}^{{\scriptscriptstyle <-1>}})_d)$ such that $({\boldsymbol{\nu}}^{{\scriptscriptstyle <-1>}})_i {\mathbf{.}}\beta {\mathbf{.}}{\boldsymbol{\nu}}\equiv {\boldsymbol{\chi}}_{(i)}$ for $i = 1, \ldots, d.$
The multivariate Lévy-Sheffer polynomials for the pair ${\boldsymbol{\mu}}$ and ${\boldsymbol{\nu}}$ are TSH polynomials with respect to the symbolic multivariate Lévy process $$\{t {\mathbf{.}}(\mu_1 {\mathbf{.}}\beta {\mathbf{.}}{\boldsymbol{\nu}}^{{\scriptscriptstyle <-1>}}_1 + \cdots + \mu_d {\mathbf{.}}\beta {\mathbf{.}}{\boldsymbol{\nu}}^{{\scriptscriptstyle <-1>}}_d)\}_{t \geq 0}.$$
Conclusions and open problems.
==============================
In this paper, the review of a symbolic treatment of TSH polynomials, relied on the classical umbral calculus, is proposed. The main advantage of this symbolic presentation is the plainness of the overall setting which reduces to few fundamental statements, but also the availability of efficient routines [@dinardooliva2009] for the implementation of formulae as (\[(auxmult)\]), which is the key to manage the polynomials $Q_{{\boldsymbol{k}}}({\boldsymbol{x}},t).$
The main result of this presentation is that any univariate (respectively multivariate) TSH polynomial has the form $Q_k(x,t)$ (respectively $Q_{{\boldsymbol{k}}}({\boldsymbol{x}},t)$) or can be expressed as a linear combination of the polynomials $Q_k(x,t)$ with coefficients given by (\[prop5\_multi\]). Thanks to the umbral representation of multivariate Lévy-Sheffer systems, more families of umbral polynomials could be characterized, together with their orthogonality properties. This will be the object of future research and investigation.
In [@Barrieu], Barrieu and Shoutens have related the infinitesimal generator of a Markov process to a more general class of linear operators possessing the TSH property, both ascribable to special families of martingales. A stochastic Taylor formula is produced which results to be a generalization of a TSH polynomial due to the presence of a remainder term series. A symbolic representation of this new TSH function could open the way to a new classification of the corresponding operators by which to recover the martingale property on Lévy processes. Similarly, the extension to the more general class of Markov processes (a first attempt is given in [@Barrieu]) would move the employment of TSH functions beyond the field of applications strictly connected to the market portfolio. One step more consists in dealing with matrix-valued stochastic processes by replacing formal power series (\[(fps)\]) with hypergeometric functions, as done in [@lawi]. This would allow us a symbolic representation also for zonal polynomials whose computational handling is still an open problem.
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[^1]: The ring ${\mathbb R}[x]$ may be replaced by any ring in whatever number of indeterminates, as for example ${\mathbb R}[x, y, \ldots].$
[^2]: When no misunderstanding occurs, we use the notation $\{a_i\}$ instead of $\{a_i\}_{i \geq 0}$
[^3]: Since $-t {\mathbf{.}}\alpha$ and $t {\mathbf{.}}\alpha$ are two distinct symbols, they are considered uncorrelated, therefore $-t {\mathbf{.}}\alpha + t {\mathbf{.}}\alpha^{\prime} \equiv
-t {\mathbf{.}}\alpha + t {\mathbf{.}}\alpha \equiv {\epsilon}.$ When no confusion occurs, we will use this last similarity instead of (\[(inverse)\]).
[^4]: The support $\hbox{\rm supp} \, (p)$ of an umbral polynomial $p \in {\mathbb R}[x][{\mathcal{A}}]$ is the set of all umbrae occurring in it.
[^5]: For cumulants $\{C_i(Y)\}$ of a random variable $Y,$ the following properties hold for all nonnegative integers $i:$ (Homogeneity) $C_i(a Y) = a^i C_i (Y)$ for $a \in {\mathbb R},$ (Semi-invariance) $C_1(Y+a)=a + C_1(Y), C_i(Y+a) = C_i(Y)$ for $i \geq 2,$ (Additivity) $C_i(Y_1+Y_2) = C_i(Y_1) + C_i(Y_2),$ if $Y_1$ and $Y_2$ are independent random variables.
[^6]: When no confusion occurs, we will use the notation $x - t {\mathbf{.}}\alpha$ to denote the polynomial umbra $- t {\mathbf{.}}\alpha + x = x + (- t) {\mathbf{.}}\alpha.$
[^7]: Recall that a partition of an integer $i$ is a sequence $\lambda = (\lambda_1, \lambda_2, \ldots,
\lambda_m),$ where $\lambda_j$ are weakly decreasing positive integers such that $\sum_{j=1}^{m} \lambda_j = i.$ The integers $\lambda_j$ are named [*parts*]{} of $\lambda.$ The [*length*]{} of $\lambda$ is the number of its parts and will be indicated by $l(\lambda).$ A different notation is $\lambda = (1^{r_1},
2^{r_2}, \ldots),$ where $r_j$ is the number of parts of $\lambda$ equal to $j$ and $r_1 + r_2 + \cdots = l(\lambda).$ Note that $r_j$ is said to be the multiplicity of $j$. We use the classical notation $\lambda \vdash i$ to denote $\lambda$ is a partition of $i$.
[^8]: A partition ${\boldsymbol{\lambda}}$ of a multi-index ${\boldsymbol{i}},$ in symbols ${\boldsymbol{\lambda}}\vdash {\boldsymbol{i}},$ is a matrix ${\boldsymbol{\lambda}}= (\lambda_{ij})$ of nonnegative integers and with no zero columns in lexicographic order $\prec$ such that $\lambda_{r_1} + \lambda_{r_2} + \cdots + \lambda_{r_k} = i_r$ for $r = 1, 2, \ldots , d.$ The number of columns of ${\boldsymbol{\lambda}}$ is denoted by $l({\boldsymbol{\lambda}})$. The notation ${\boldsymbol{\lambda}}= ({\boldsymbol{\lambda}}_{1}^{r_1} , {\boldsymbol{\lambda}}_{2}^{r_2}, \ldots)$ represents the matrix ${\boldsymbol{\lambda}}$ with $r_1$ columns equal to ${\boldsymbol{\lambda}}_{1},$ $r_2$ columns equal to ${\boldsymbol{\lambda}}_{2}$ and so on, where ${\boldsymbol{\lambda}}_{1} \prec {\boldsymbol{\lambda}}_{2} \prec \ldots.$ We set $\mathfrak{m}({\boldsymbol{\lambda}}) = (r_1, r_2, \ldots),$ $\mathfrak{m}({\boldsymbol{\lambda}})! = r_1! r_2! \cdots$ and ${\boldsymbol{\lambda}}! = {\boldsymbol{\lambda}}_1! {\boldsymbol{\lambda}}_2! \cdots.$
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Much of the energy consumption in buildings is due to HVAC systems, which has motivated several recent studies on making these systems more energy-efficient. Occupancy and activity are two important aspects, which need to be correctly estimated for optimal HVAC control. However, state-of-the-art methods to estimate occupancy and classify activity require infrastructure and/or wearable sensors which suffers from lower acceptability due to higher cost. Encouragingly, with the advancement of the smartphones, these are becoming more achievable. Most of the existing occupancy estimation techniques have the underlying assumption that the phone is always carried by its user. However, phones are often left at desk while attending meeting or other events, which generates estimation error for the existing phone based occupancy algorithms. Similarly, in the recent days the emerging theory of Sparse Random Classifier (SRC) has been applied for activity classification on smartphone, however, there are rooms to improve the on-phone processing. We propose a novel sensor fusion method which offers almost 100% accuracy for occupancy estimation. We also propose an activity classification algorithm, which offers similar accuracy as of the state-of-the-art SRC algorithms while offering 50% reduction in processing.'
address:
- 'Autonomous Systems Program, CSIRO'
- 'Intelligent Efficiency, CSIRO'
author:
- Rajib Rana
- Brano Kusy
- Josh Wall
- Wen Hu
bibliography:
- 'occuActivity.bib'
title: 'Novel Methods for Activity Classification and Occupany Prediction Enabling Fine-grained HVAC Control.'
---
HVAC ,Sparse Random Classifier ,Sensor Fusion ,Smartphone ,Occupancy ,Physical Activity.
Introduction {#sec:intro}
============
HVAC is a dominant power consumer in commercial buildings. Much efforts have been invested in the past to efficiently control HVAC. Conventionally, most HVAC systems intake temperature and humidity to control cooling [@ashrae2004standard]. This limitation can often lead to inefficient energy usage. For example, a room might be cooled to a conservative $24^oC$ even when it is unoccupied. Similarly, conventional HVAC control strategy does not take into consideration human comfort factors, such as recent physical activity. This can cause discomfort. For example, upon returning to desk after climbing many stairs, worker would like the temperature to be cooler than the conservative set point of $24^oC$.
A number of studies have been conducted in the past that attempted to estimate occupancy using sensors spanning motion sensors, RFID and contact sensors. However, introducing additional sensors incur cost of installation and maintenance, especially for large commercial buildings. To alleviate this problem, a number of studies have attempted to use smartphone for occupancy estimation. For example, [@krioukov2012personal; @Balaji:2013:SOB:2517351.2517370; @christensen2014using] uses WiFi signal strength to localize the phone. The general control strategy of these methods are, if the phone location is estimated as the office, the HVAC is activated, otherwise the local unit is turned off. However, note that if the phone is left on desk while the person is away, all of these systems will still determine the office space as occupied. This will lead to significant energy wastage. We overcome this challenge in our paper.
A large body of literature can be found which use wearable sensors for activity detection. Given the current advancement in smartphone sensor technology, our objective is to piggyback the activity estimation on the smartphone to save additional expenses of wearable devices. In particular, we propose a *Sparse Random Classifier (SRC)* [@wright2009robust] based activity classifier, which pave the pathway to featureless classification that is suited to embedded smartphone platform. Notably, a number of attempts [@zhang2013human; @xu2012robust; @liu2009human; @xu2012co] have been taken in the past to use SRC for activity classification. However, the key limitations of these approaches is that higher accuracies are obtained when the feature dimension is significantly higher. Higher feature dimension warrants higher processing on embedded platforms, which could potentially interrupt the processing of other applications on the phone. In this paper we seek to address this challenge. We extend the theory of SRC and show that we can achieve similar accuracy as of the existing proposals, however, with half feature dimension.
Our contributions in this paper are as follows:
1. We propose a sensor fusion method that uses sensor feed from phone microphone and phone accelerometer to determine office occupancy when the phone is placed on the office desk. Privacy is a critical issue when using audio data. We preserve privacy by not storing or transiting raw audio data. Features are directly extracted from raw audio data and used for classification.
2. We also propose a extension of the Sparse Random Classifier for physical activity classification on smartphone. Our classifier achieves better accuracy with significantly smaller feature dimension.
3. Experimental results show that our proposed fusion algorithm achieves 100% accuracy in estimating office occupancy. Results also show that our activity classification algorithm offers greater than 95% accuracy with 50% smaller feature dimension compared to the existing methods.
This paper is organized as follows. In the next section (Section \[Sec:activity\]), after providing some background on SRC, we describe our proposed extension of SRC for activity classification. Then in Section \[sec:sensorFusion\], we first describe the Support Vector Regression model for occupancy inference using individual sensing modality. Then we describe our sensor fusion algorithm for occupancy estimation jointly using multi-modal sensing. We then present the experimental results in Section \[sec:results\] and finally conclude in Section \[sec:conclude\].
Sparse Random Classifer for Activity Classifcation {#Sec:activity}
==================================================
Sparse Random Classifier
------------------------
The Sparse Random Classifier(SRC) has been developed underpinning the theory of Compressive Sensing (CS) [@wright2009robust]. SRC has been heavily used for classification in the past [@sivapalan2011compressive; @chew2012sparse; @wei2013real; @xu2011dynamic; @Wei:2012:DSA:2185677.2185699]. The underlying assumption of SRC is that, given sufficient training samples of the $i$th activity class (such as walking or running and os on) $A_i = [v_{i,1} + v_{i,2}+ ... + v_{i,n_i}] \in \mathbb{R}^{m\times n_i}$, any test object $y\in \mathbb{R}^m$ from the same activity class will approximately lie in the linear span of the training samples associated with object $i$. Mathematical representation of this assumption using $\alpha_{i,j}$ as the coefficients can be given by . $$\begin{aligned}
y = A_i = [\alpha_{i,1}v_{i,1} + \alpha_{i,2}v_{i,2}+ ... + \alpha_{i,n_i}v_{i,n_i}] \label{eqn:basis}\end{aligned}$$ However, the membership $i$ of the test sample is unknown primarily, we define a new matrix $A$ for the entire training set was the concentration of the all $n$ training samples of $k$ object classes:
$$\begin{aligned}
A = [A_1,A_2,...,A_k] = [v_{1,1} + v_{1,2}+ ... + v_{k,n_k}] \end{aligned}$$
Then the linear representation of the training object $y$ can be rewritten in terms of training samples as
$$\begin{aligned}
y = A x_0 \in \mathbb{R}^m\end{aligned}$$
where $x_0 = [0,...,0,\alpha_{i,1} , ... , \alpha_{i,n_i},0,...,0] \in \mathbb{R}^n]$ is a coefficient vector whose entries are zero except those associated with $i$-class.
The solution $x_0$ can be obtained by solving the system of equation $y = Ax$ when $m>n$. However, in reality $m<n$ (we will explain this later in this section), therefore the system is underdetermined. Conventionally, this difficulty is resolved by solving the minimum $\ell_2$-norm solution: $$\begin{aligned}
\hat{x}_2 = \arg \min ||x||_2 \mbox{ s. t. } Ax = y, \label{eqn:l2norm}\end{aligned}$$ This optimization in can be solved easily by pseudo inverse of A, however, the solution $\hat{x}$ is not informative. This is because we expect the solution to be sparse where only the coefficients related to test object class are non-zero. However, the $\ell_2$ norm provides a dense solution with many non-zero entries spanning multiple classes. This motivates us to seek the sparsest solution to $y = Ax$ by solving the following optimization problem:
$$\begin{aligned}
\hat{x}_0 = \arg \min ||x||_0 \mbox{ s. t. } Ax = y,\end{aligned}$$
Here $||.||_0$ denotes the $\ell_0$-norm, which counts the number of nonzero entries in a vector. Solution to the above optimization problem provides the optimal solution, however, solving it for an underdetermined system is NP-hard.
Recent development of the theory of compressive sensing and sparse representation has shown that if the solution $x_0$ is sparse enough, $\ell_1$-minimization provides same solution as that of $\ell_0$-minimization: $$\begin{aligned}
\hat{x}_1 = \arg \min ||x||_1 \mbox{ s. t. } Ax = y,\end{aligned}$$ In the past a large body of studies have successfully used $\ell_1$-minimization to find the sparse solution [@rana2011adaptive; @ear_phone_ipsn; @east_ewsn; @shen2013nonuniform; @chou2012efficient; @adaptivecompressive; @shen2011non]. This problem can be solved in polynomial time by standard linear programming methods.
Novel Extension of Sparse Random Classifier for Activity Classification {#sec:dimensionalityReduction}
-----------------------------------------------------------------------
Recall from the previous section that $m<n$. Here, $n$ is the number of samples in an object instance. For large signals, for example for images, it is the number of pixels in the image vector after vectorization. For instance, if the face images are given at the typical resolution, 640$\times$480 pixels, the dimension $m$ is in the order of $10^5$. A higher dimension of the object offers extended processing complexity for regular computers, therefore, embedded platforms are out of question. In order to address this problem projections are taken to transform the images from image space to feature space:
$$\begin{aligned}
Ry = RAx
\label{eqn:dimensionReduction}\end{aligned}$$
Here $R \in \mathbb{R}^{d\times n}$ is the so called projection matrix with $d<<n$. In conventional SRC, Gaussian random numbers are used as the elements of $R$. However, we propose a novel construction of projection matrix. We first describe the projection matrix construction method in [@Carin:12]. The method assumes that the dictionary $A \in \mathbb{R}^{m \times n}$ and the number of projections $m$ are the inputs. The method first computes the singular value decomposition (SVD) of $A$: $$D = U \Lambda V^T
\label{sec:svdD}$$ where $^T$ denotes matrix transpose, $\Lambda \in \mathbb{R}^{n\times d}$ contains the singular values in its main diagonal, and $U \in \mathbb{R}^{n \times n}$ and $V \in \mathbb{R}^{d\times d}$ are orthonormal matrices. The method in [@Carin:12] is to *randomly* choose $m$ columns from the matrix $U$. Let $\tilde{U}_m$ be a $n \times m$ matrix formed by these $m$ randomly chosen columns from $U$. The method is to use $ \tilde{U}_m^T$ as the projection matrix. The rationale of the method is that the columns in $U$ are highly uncorrelated with the dictionary $D$, therefore the sensing matrix $\Psi D = \tilde{U}_m^T D$ will have low coherence.
In our proposed method, we choose the $m$ columns in $U$ corresponding to the *largest* $m$ singular values of $D$. We now explain why this is a better choice. To simplify notation, we assume that the SVD in has been permuted so that singular values appear in non-increasing order in the diagonal of $\Lambda$. With this notation, let $U_m$ denotes the sub-matrix containing the left-most $m$ columns of $U$; note that these $m$ columns correspond to the largest $m$ singular values of $D$. Our choice of projection matrix is therefore $U_m^T$.
To understand why $U_m$ is a better choice, note that the activity classification problem can be stated as estimating the unknown coefficient vector $s$ from the projection $y$ by solving $y = R A x_0$. We assume that the unknown coefficient vector $x_0$ comes from some probability distribution such that $\mathbb{E}[ x_0 x_0^T] = \mathbb{I}$ where $\mathbb{E}$ and $\mathbb{I}$ denote respectively the expectation operator and the identity matrix. It can be shown that the mean signal power $\mathbb{E}[y^T y]$ can be written as: $$\begin{aligned}
\mathbb{E}[y^T y] = trace(R U \Lambda^2 U^T R^T)\end{aligned}$$ If we impose the constraint that each row of the projection matrix $R$ has unit norm, then the $R$ that maximizes $\mathbb{E}[y^T y]$ is given by the first $m$ rows of $U^T$ (or $U_m^T$), i.e. the $m$ left singular vectors corresponding to the largest $m$ singular values. This shows that our choice of projection matrix maximises the signal power of $y$. A higher signal power typically translates to lower estimation error.
Occupancy Estimation Using Sensor Fusion {#sec:sensorFusion}
========================================
When the phone is carried, it can be trivially determined that it is accompanied by its owner. However, in office people often tend to put the phone on desk and work. Further, often they leave the desk while keeping in the phone on it. We consider determining occupancy in these two scenarios. We consider data from the *phone accelerometer* and the *phone microphone* to determine occupancy and use a number of assumptions to develop our occupancy estimation algorithm:
1. When people are typing it could create minor acceleration in the phone kept on the table.
2. When the person is not typing, the accelerometer may not be able to pickup the presence, however, sounds of general movement while seated can be picked up by the phone microphone.
3. When people are away from desk, the accelerometer can pickup acceleration from any devices, such as PC or laptop running on the desk.
Features
--------
### Accelerometer
We study three axes of the accelerometer separately. We use a segment size containing one second of data. We investigate mean, 25 and 75 percentile and *maximum* magnitude of each segment and do not observe any significant difference in classification for those choices. In the results section we use *maximum* magnitude of each segment as a feature.
### Sound
To preserve privacy we refrain using raw audio data for classification. We test the feasibility of signal energy and number of zero crossing as features and find number of zero crossing to be a better choice. Computation of signal energy involves determining a hamming window, which is a resource intensive operation and most importantly it does not provide classification accuracy as good as zero-crossing. We therefore use *number of zero crossings* as a feature. We investigate on two attributes of audio data to calculate number of zero crossings: 1. *sampling frequency* and 2. *segment size*. We test a range of sampling frequencies spanning 8kHz,16kHz,32kHz and 48kHz and find that 48kHz provides the best results. We test window sizes of 5s, 15s, 25s, 35s and 45s and observe that 5s window provides the best results (Results are shown in Section \[sec:results\]).
Occupancy modeling
------------------
Note that prior to applying fusion, we use the classical Support Vector Regression (SVR) for modeling occupancy individually from the audio and accelerometer features discussed in the previous section. The complete description of SVR is outside the scope of this paper. However, in this section we will provide intuition sufficient to understand the working principles of SVR.
### Support Vector Regression
Our approach to predicting gait velocity is based on learning the functional relationship between the transition times and gait velocity. To learn this relationship, we used a support vector regression model, which is widely used for prediction [@rana2013feasibility; @rana2011adaptive; @rana2013passive].
Consider a training set $\{(x_1,y_1),(x_2,y_2),...,(x_\ell,y_\ell)\}$, where $x_i$s are aggregated accelerometer or acoustic features and $y_i$s are the occupancy status. Support vector regression computes the function $f(x)$ that has the largest $\epsilon$ deviation from the actual observed $y_i$ for the complete training set.
Let us assume the relationship between the variables is linear of the form $y = \omega x + b$, where $\omega$ (weight vector) and $b$ are parameters to be estimated. Fig. \[fig:svrFig2\] shows a few possible linear relationships between the points $x$ and $y$. The solid line in Fig. \[fig:svrFig3\] shows the SVR line given by $f(x) = \omega x + b$. The cylindrical area between the dotted lines shows the region without regression error penalty. In the SVR literature this area is considered as the measure of complexity of the regression function used. Points lying outside the cylinder are penalized by an $\epsilon$-insensitive loss function [@Vapnik:1995:NSL:211359] given by $|\xi|_\epsilon$. $$\begin{aligned}
|\xi|_\epsilon := \begin{cases}
0 {\hspace{2 cm} \mbox if } |\xi| \leq \epsilon\\
|\xi| - \epsilon \mbox{ \hspace{1 cm} otherwise.}
\end{cases}
\label{eqn:lossFunction}\end{aligned}$$ Now lets us explain the implication of a few different values of $\omega$. In the extreme case when $\omega = 0$ (as in Fig. \[fig:svrFig2\]), the functional relationship between $x$ and $y$ is least complex or in other words there is no relationship between $x$ and $y$. Therefore the overall error is very high. Next Fig. \[fig:svrFig1\] represents the case where the training data fits the solid line quite well. The solid line represents the classical regression analysis, where the loss function is measured as the squared estimation error. Note that although the solid line fits the data well, the cylindrical area between the dotted line is small, which means that the model will not generalize as well in predicting new data. SVR seeks to find a balance between the flatness of the area amongst the dotted lines and the number of training mistakes (see Fig. \[fig:svrFig3\]).
![Feature Space Transformation.[]{data-label="fig:transformSVR"}](transformSVR){width="0.7\linewidth"}
Note that in many cases the relationship between the variables is non-linear as shown in the left diagram in Fig. \[fig:transformSVR\]. In those cases the SVR method needs to be extended, which is done by transforming $x_i$ into a feature space $\Phi(x_i)$. The feature space linearizes (right diagram in Fig. \[fig:transformSVR\]) the relationship between $x_i$ and $y_i$, therefore, the linear approach can be used to find the regression solution. A mapping function or so called Kernel function is used to transform into feature space. There are four different functions which are frequently used as kernels within support vector regression: linear, RBF (Radial Basis Function), polynomial, and sigmoid. When the feature set is small, the RBF kernel is preferable over others. We use only one feature of transition time, therefore we use the RBF Kernel. However, we empirically verify that the RBF kernel performs better than the linear kernel. There are two parameters, namely $\gamma$ and $C$ (refer to [@chang2011libsvm] for details) whose values need to be determined for best prediction. Here $C$ is the manually adjustable constant, and $\gamma$ is the kernel parameter which is formally defined as $K(x,y)= e^{-\gamma}||x-y||^2$. The overview of our SVR prediction framework is illustrated in Fig. \[fig:transformSVR\].
![SVR Prediction Framework.[]{data-label="fig:transformSVR"}](svrModel){width="0.7\linewidth"}
Sensor Fusion
-------------
Since both of the sensor data are simultaneous collected on the phone, we investigate the feasibility of sensor fusion to improve the accuracy of occupancy estimation. Our fusion algorithm is motivated by the Dynamic Weighted Majority (DWM) Voting [@kolter2003dynamic], which is a ensemble learning algorithm. In DWM a series of learning algorithms, namely experts, are used to improve the predictive performance. At the beginning all the experts have equal weights, however, weights are penalized due to wrong prediction. We present a slight modification of the algorithm: we use multiple modalities as experts and use support vector regression for perdition, for each modality. Unlike DWM, we do not remove experts. Our extended Dynamic Weighted Majority Algorithm is presented in Algorithm \[alg:dwm\].
At the start of the algorithm both modalities have equal weight of 1. We use a 5-Fold cross validation where in each fold the accelerometer and acoustic features are used separately to predict the occupancy using the SVR model. If the prediction is wrong, weight is reduced by half ($\beta = 0.5$) otherwise weight is kept the same. At the end of these 5-Fold cross validation, weights for both modalities are learned and later on used for testing. We learn the weights individually for each subject.
$\{x, y\}_1^n$: training data, feature vector and class label $\beta$ : factor for decreasing weights, $0 \leq \beta < 1$ $c \in N^*$: number of classes $\{e, w\}_1^m$: set of sensing modalities and their weights $\sigma \in \mathbb{R}^c$: sum of weighted predictions for each class $\Lambda, \lambda \in {1, . . . , c}$: global and local predictions
$\sigma_i \gets 0$ $\lambda \gets SVR(e_j,x_i)$ $w_j \gets w_j \beta$ $\sigma_\lambda \gets \sigma_\lambda + w_j$ $\Lambda = \arg\max_j \sigma_j$
Results {#sec:results}
=======
Activity Classification
-----------------------
### DataSets
We validate the performance of our proposed $\ell_1$ classifier, using publicly available dataset from University of California, Irvine (UCI) [@anguita2012human]. In order to create this datasets, experiments were carried out with a group of 30 volunteers within an age bracket of 19-48 years. Each person performed the six activities: *standing, walking, laying, walking, walking upstairs* and *walking downstairs*, wearing the a Samsung Galaxy S2 smartphone on the waist. The features selected for this database come from the accelerometer and gyroscope 3-axial raw signals tAcc-XYZ and tGyro-XYZ. These time domain signals (prefix ’t’ to denote time) were captured at a constant rate of 50 Hz. Then they were filtered using a median filter and a 3rd order low pass Butterworth filter with a corner frequency of 20 Hz to remove noise. Similarly, the acceleration signal was then separated into body and gravity acceleration signals (tBodyAcc-XYZ and tGravityAcc-XYZ) using another low pass Butterworth filter with a corner frequency of 0.3 Hz.
Subsequently, the body linear acceleration and angular velocity were derived in time to obtain Jerk signals (tBodyAccJerk-XYZ and tBodyGyroJerk-XYZ). Also the magnitude of these three-dimensional signals were calculated using the Euclidean norm (tBodyAccMag, tGravityAccMag, tBodyAccJerkMag, tBodyGyroMag, tBodyGyroJerkMag).
Finally a Fast Fourier Transform (FFT) was applied to some of these signals producing fBodyAcc-XYZ, fBodyAccJerk-XYZ, fBodyGyro-XYZ, fBodyAccJerkMag, fBodyGyroMag, fBodyGyroJerkMag. (Note the ’f’ to indicate frequency domain signals). In total there were *561* features made available for this dataset. The experiments have been video-recorded to facilitate the data labeling.
Activity Classification
------------------------
The key fundamental aspect of SRC is that it can perform classification with random features or in other words with raw sensor data. Therefore, it is possible to avoid the extensive feature extraction process. However, there is a trade-off between processing and accuracy while choosing between random and sophisticated features, which we present in Fig. \[fig:randomProjectionVersusSVM.eps\] and Fig. \[fig:rawDataClassification\]. We call our proposed extension of SRC, SRC-SVD. We benchmark the performance of SRC-SVD upon comparing with three other powerful alternatives: SRC (conventional implementation), Support Vector Machine (SVM) and k-nearest neighbor(kNN).
In Fig \[fig:randomProjectionVersusSVM.eps\], the 561 features are used for various classifiers. We apply dimension reduction (using ) to the signals. Due to our interest in smaller feature dimension, we present results for 95% to 80% of dimension reduction. We have a number of observations from this figure: 1. in general our proposed SRC-SVD performs the best compared to its alternatives. 2. SRC-SVD performs the best compared to the alternatives when dimension reduction is 95%. 3. Performance of SRC-SVD improves while dimension is increased from 5% to 15%, but plateaus beyond that point. The highest classification accuracy is 95%.
In Fig. \[fig:rawDataClassification\], we present the classification performance of SRC-SVD to classify using raw accelerometer data. We use all three axes of the accelerometer separately. Similar to Fig. \[fig:randomProjectionVersusSVM.eps\], we also compare the performance of SRC-SVD with SRC, SVM and kNN. We observe that 1. SRC-SVD performs the best compared to the other classification methods. 2. X-axis provides the best classification commonly for all classification methods. 3. Classification performance of SRC-SVD increases gradually when dimension increases from 5% to 35% and plateaus beyond that point. The highest classification accuracy achieved is 75%.
Occupancy Estimaiton
---------------------
### Experimental Settings
Experiments to determine occupancy were conducted with the assistance of three office colleagues. Each of them were given a Nexus 4S smartphone temporarily to collect data about their occupancy. They were allowed to use the phones for personal purpose (day to day usage). The subjects logged the information about their occupancy and location of the phone. We only used the log entries indicating phone on table and in/out of office to validate our occupancy algorithm. There was no information available about how the phone was placed on the table, such as how far from keyboard etc.
Occupancy Estimation Accuracy
-----------------------------
The results related to occupancy estimation accuracy are shown in Fig. \[fig:accuracyWithAccelerometer\] to Fig. \[fig:accuracyWithFusion\]. In Fig. \[fig:accuracyWithAccelerometer\] we present the accuracy when only accelerometer data is used. When the phone is on the desk, Z-axis is vertical to the desk surface. So, intuitively, most of the acceleration should be experienced along Z-axis. This is also reveled in this figure. For all our subjects, Z-axis provides better accuracy in both occupied and unoccupied cases.
Fig. \[fig:accuracyWithMic\] shows the occupancy estimation performance when using zero crossing as a feature of microphone data. The sound is recorded at 48KHz and we investigate the feasibility of various sampling window, such as 5s, 10s, 15s, and so on as shown in the figure. We observe that 1. microphone is particular suited to determine the occupied case. It performs badly for the unoccupied case and 2. the sampling window of 5s provides the best accuracy.
Given the differential performances of each modality, we investigate the feasibility of joint estimation using both modality. We define a fusion algorithm (Algorithm \[alg:dwm\]) which is motivated by the Dynamic Weighted Majority Voting. The sensor fusion provides significantly better performance compared to the individual modality in both occupied and unoccupied cases. It can be found from Fig. \[fig:accuracyWithFusion\] that for all three subjects and for both occupied and unoccupied cases, the accuracy is almost 100%. We use z-axis data from accelerometer and use a 5s sampling window for the microphone data (sampled at 48KHz) in the sensor fusion algorithm.
Conclusions {#sec:conclude}
===========
In this paper we have presented two novel methods for activity classification and occupancy estimation, which are two crucial components for intelligent HVAC control. Physical activities performed recently typically indicate the need for adjusting thermal preferences. Similarly, occupancy is very important in HVAC control. It can save power consumption substantially while maintaing thermal comfort if the HVAC duty cycling can be adapted correctly to the occupancy status.
The novelty of our activity classification algorithm lies in the extension of the Sparse Random Classifier. We introduce a novel construction of projection matrix that offers higher accuracy compared to the conventional implementation of Sparse Random Classifier (SRC). We report that the classification accuracy can be as high as 95% while using 50% smaller feature dimension compared to the existing implementation of SRC.
Our occupancy estimation is novel because, for the first time we propose the fusion of accelerometer and microphone data from smartphone to determine occupancy. We perform experiments with real subjects and experimental results reveal that we achieve almost 100% classification accuracy, which is substantially high. In these experiments accelerometer and microphone data were used when the subject reported that phone was on desk. One of the most important aspects of our experiments is that it was completely uncontrolled. There was no information available on how the phone was placed on the desk, i.e., how far it was placed from the keyboard or hand etc.
In our future studies, we want to validate the proposed sensor fusion algorithm with large number of subjects. In addition, we are also aiming to implement the activity classification algorithm on smartphone and conduct experiments with human subjects to validate its performance.
References
==========
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a general method of constructing maximally localized Wannier functions. It consists of three steps: (1) picking a localized trial wave function, (2) performing a full band projection, and (3) orthonormalizing with the Löwdin method. Our method is capable of producing maximally localized Wannier functions without further minimization, and it can be applied straightforwardly to random potentials without using supercells. The effectiveness of our method is demonstrated for both simple bands and composite bands.'
author:
- Junbo Zhu
- Zhu Chen
- Biao Wu
title: Construction of Maximally Localized Wannier Functions
---
[UTF8]{}[gbsn]{}
Wannier functions that are localized at lattice sites are an alternative representation of electronic states in crystalline solid to the Bloch wave representation [@Wannier37]. They offer an insightful picture of chemical bonds, play a pivotal role in the modern theory of polarization [@Resta94; @Vanderbilt93], and are the basis for an efficient linear-scaling algorithms in electric-structure calculations [@Goedecker99; @Galli96]. Wannier functions are also important in linking cold atom experiments in continuous light potentials with lattice Hamiltonians, such as the Bose-Hubbard model and Anderson random lattice [@Jaksch98; @White09; @Ceperley10].
Although there are explicit formula that transform Bloch waves to Wannier functions, the construction of Wannier functions is far from trivial. The primary reason is that there are infinite number of Wannier functions for a given band due to phase choices for Bloch waves whereas in practice the maximally localized Wannier functions (MLWFs) are sought and preferred [@Marzari97]. For one dimensional lattice, Kohn showed how to fix the Bloch wave phases to obtain these MLWFs [@Kohn59]. In 1971, Teichler found a general method to construct Wannier functions, which is insensitive to the phases of Bloch waves [@Teichler]. However, this method does not guarantee maximal localization and depends on initial trial function. A more sophisticated method involving numerical minimization of the spread of a Wannier function has been developed to compute MLWFs [@Marzari12]. There are also some new developments recently [@Nenciu2016; @Louie2016; @Levitt].
In this work we present a general method for computing MLWFs. Our method is somehow similar to Teichler’s method as it also involves projection and the Löwdin orthonormalization method [@Lowdin]. Nevertheless there is a significant difference so that our method can produce MLWFs and is insensitive to the initial trial wave functions. Compared to the method in Ref. [@Marzari12], our method does not need the minimization procedure. As our method uses the full band projection, it can be applied straightforwardly to random potential without using supercells. The effectiveness of our method is demonstrated for both simple bands and composite bands.
We consider a simple band that is isolated from other bands. Composite bands will be discussed later. Our method of constructing MLWFs consists of three steps:
1. Guess: choose a set of trial wave functions $\ket{g_n}$, which are localized at lattice sites.
2. Projection: $\ket{\xi_n}=P\ket{g_n}$, where $P=\sum_k \ket{\psi_k}\bra{\psi_k}$ with the summation over the whole first Brillouin zone. Note that $P$ is a full band projection and is different from the projection in Refs. [@Marzari97; @Teichler], which is at a given $k$ point. As a result of this crucial difference, the projected function $\ket{\xi_n}$ is localized at site $n$.
3. Orthonormalization: use the Löwdin orthonormalization method [@Lowdin] to transform $\ket{\xi_n}$ into a set of MLWFs $\ket{w_n}$. If one uses other methods such as Kohn’s method [@Kohn1973orth] to orthonormalize $\ket{\xi_n}$, the resulted Wannier function is unlikely maximally localized.
Here is why our method is effective and capable of producing MLWFs. We re-write the full band projection operator in terms of Wannier functions, $P=\sum_n\ket{w_n}\bra{w_n}$, where the summation is over all lattice sites. We thus have $$\ket{\xi_n}=\sum_m\ket{w_m}\braket{w_m | g_n}\approx
\sum_{m=\langle n\rangle}\ket{w_m}\braket{w_m | g_n}\,,$$ where $\langle n\rangle$ indicates that the summation is only over site $n$ and its nearest neighbors. It is clear that $\ket{\xi_n}$ is localized at lattice site $n$. However, these projected functions $\ket{\xi_n}$’s are not orthonormal. Any orthonormalization of $\ket{\xi_n}$’s will give us a set of Wannier functions. For example, one may use Kohn’s method [@Kohn1973orth]. We choose the Löwdin method [@Lowdin] as it can produce the MLWFs. According to Ref. [@Aiken], the Löwdin orthogonalization uniquely minimizes the functional measuring the least squares distance between the given orbitals and the orthogonalized orbitals. In our case, for the set of projected orbitals $\ket{\xi_n}$, the Löwdin method produces Wannier functions $\ket{w_n}$ that minimize $$\label{sum}
\sum_n\int dx |\braket{x|\xi_n}-\braket{x|w_n}|^2\,,$$ where the summation is over all lattice sites. For a periodic potential, this is equivalent to maximizing $\braket{w_n | \xi_n}=\braket{w_n | g_n}$. Therefore, if $\ket{g_n}$ is properly chosen, the resulted Wannier function is maximally localized. No further minimization such as the one in Ref. [@Marzari12] is needed.
![(color online) Wannier functions for one dimensional periodic potential $V(x)=\cos(2\pi x/a)$. (a) The Wannier functions obtained with our method with trial Gaussian functions of different widths $\alpha$. (b) The spread of the Wannier function obtained with our method as a function of the width of the Gaussian function $\alpha$. (c) The Wannier functions obtained with three different methods. The black line is our method; the blue line is obtained with the traditional method, where one fixes the phases of Bloch wave according to the prescription given in Ref. [@Kohn59]; the red one is obtained by orthonormalizing $\ket{\xi_n}$’s with Kohn’s method [@Kohn1973orth]. []{data-label="ssband"}](fig1){width="0.8\linewidth"}
To illustrate our method, we consider a single particle Hamiltonian with the periodic potential $V(x)=\cos(2\pi x/a)$, where $a$ is the lattice constant. We construct the Wannier function for its lowest band. As each potential well is symmetric with respect to its center (the lowest point of the well), we expect that the MLWF is also symmetric with its highest peak at the center. So, we choose $\braket{x|g_n}=e^{-(x-na)^2/2\alpha^2}/\sqrt{2\alpha^2\pi}$. It is clear that $\braket{w_n | g_n}$ is the largest for a narrowest Wannier function allowed within the band or MLWF. The resulted Wannier function should be insensitive to the width of $\braket{x|g_n}$. This is exactly what we see in Fig. \[ssband\](a). With three Gaussians of different widths $\alpha=a, 0.5a, 0.1a$ as trial functions, the resulted Wannier functions fall right on top of each other as seen in Fig. \[ssband\](a). We have computed the spread of the Wannier functions obtained with our method; they are almost identical for Gaussian trial functions of different widths as seen Fig. \[ssband\](b).
To further illustrate the effectiveness of our method, we have computed Wannier functions with two other methods. One is the traditional method, where one fixes the phases of the Bloch waves according to the prescription in Ref. [@Kohn59]. The other method is to orthonormalize $\ket{\xi_n}$’s with Kohn’s method [@Kohn1973orth]. They are compared to our results in Fig. \[ssband\](c), where we see that our result is in excellent agreement with the traditional method while the one obtained with Kohn’s method is much worse.
The Löwdin method can be implemented differently. However, as long as it is implemented correctly, the method transforms a given set of non-orthogonal vectors to a unique set of orthonormal vectors. Nevertheless, we show here explicitly how we implement it. We impose a periodic boundary condition with $N$ unit cells. As a result, the crystal wave vector $k$ takes $N$ discrete values ${k_1, k_2, \cdots, k_N}$. We let $\ket{\psi_j}=\ket{\psi_{k_j}}$ and $A_{nj}=\braket{\psi_j|g_n}$. The Löwdin orthonormalization is then implemented as =\_[mj]{} (AA\^)\^[-1/2]{}\_[nm]{}A\_[mj]{}. \[lowdin\] If the trial function $\ket{g_n}$ is translationally symmetric, $\braket{x|g_n}=\braket{x-r_n|g_0}$, we have $A_{nj}=e^{-ik_jr_n}A_{0j}$ and $(AA^\dagger)_{nm}=\sum_j e^{ik_j(r_m-r_n)}|\braket{\psi_j|g_0}|^2$.
Our method is applicable for composite bands. In composite bands, one or more Wannier functions have nodes. Therefore, to have largest $\braket{w_n | g_n}$ for MLWFs, we need to choose such $\ket{g_n}$’s that they have nodes at proper positions. The node positions can be determined by symmetries of the wells. In the worst case, we can determine these node positions by numerically computing the eigenstates of the local wells. Here we consider a two dimensional periodic potential $V(x,y)=V_0[\cos(2\pi x/a)+\cos(2\pi y/a)]$ and use its p-bands to illustrate the effectiveness of our method for composite bands. The two trial wave functions are chosen as $g_{1,2}=\sqrt{\frac{\pi}{2\alpha^4}} \eta_{1,2} e^{-(x^2+y^2)/2\alpha^2}$ with $\eta_1=x, \eta_2=y$, which are the two first excited states of a two dimensional harmonic oscillator. The results are plotted in Fig.\[fig:wanner2D\]. Shown in Fig.\[fig:wanner2D\](a) is a comparison between the trial function and the resulted Wannier function. Three Wannier functions obtained from different trial functions are shown in Fig.\[fig:wanner2D\](c) and (d). Our numerical calculations show that when the width of the trial function $\alpha$ is narrow enough, the resulted Wannier functions are almost identical to each other. When they are plotted in the figure, they fall right on top of each other. So, in Fig.\[fig:wanner2D\](c) and (d) only three Wannier functions are plotted. The spread of a Wannier function $\langle w | x^2+y^2 | w \rangle$ is plotted as the function of $\alpha$ in Fig.\[fig:wanner2D\](b), which shows that the Wannier function spread no longer changes when $\alpha$ is narrow enough.
![\[fig:wanner2D\] (color online) Wannier functions of the composite $p$-bands in a two dimensional lattice $V(x,y)=V_0[\cos(2\pi x/a)+\cos(2\pi y/a)]$ with $V_0=-30$. (a) The left is the trial wave function $g_2$ with $\alpha=0.5a$; the right is the Wannier function obtained with this trial function. The white dashed line marks the unit cell. (b) The spread $\langle w| x^2+y^2 | w\rangle$ of a Wannier function with respect to varying $\alpha$. The convergence to the black line is obvious. The black line is the width of the Wannier function $w(x,y)=w_1(x)w_2(y)$, where $w_1(x)$ as the $s$-band Wannier function of $\cos(2\pi x/a)$ and $w_2(y)$ as the $p$-band Wannier function of $\cos(2\pi y/a)$ are obtained with the traditional method. (c) Logarithmic plot of Wannier functions at $x=0$ along $y$-axis for different values of $\alpha$: $\alpha=0.3a$(red), $0.6a$ (green), $0.88a$ (blue). Note that the latter two almost overlap. (d) Logarithmic plot of Wannier functions at $y_0$ along the $x$-axis for different choices of $\alpha$: $\alpha=0.3a$(red), $0.6a$ (green), $0.88a$ (blue). $y_0$ is the highest peak position of the Wannier functions. ](wannier2D){width="0.85\linewidth"}
Our method is directly applicable to random potentials. The reason is that we use the full band projection. It can be constructed with all the energy eigenfunctions in a given band no matter the eigenfunctions are Bloch waves or not. It is well known that even in random potentials, there exist eigen-energy “band" that are isolated from other eigen-energies by gaps that are independent of the system size. For these bands, there exist Wannier functions [@Nenciu93]. Our method can be used to compute these Wannier functions by constructing the projection $P$ with the energy eigenstates of a given random energy band. According to Eq.(\[sum\]), the resulted Wannier functions maximize a sum, $\sum_n \braket{w_n|g_n}$. With a proper choice of $\ket{g_n}$’s, these Wannier functions can be regarded as maximally localized collectively. There already exist several methods to compute Wannier functions for random potentials. Our method has various advantages. The method in Ref. [@Kohn73] is only applicable to the case where the random potential is a perturbation to a periodic potential. Kivelson’s method [@Kivelson82] has difficulty for two or three dimensional systems. Our method is clearly more efficient than the one in Ref. [@Ceperley10]. We ourselves have recently proposed a method to compute Wannier functions for random potentials [@Zhu2016]. Our current method is certainly superior.
We now illustrate our method in disordered systems. We choose a disordered potential as a series of cosine-type wells of random depths, $V_n(x)=A_n[\cos(2\pi x/a)-1]$. Well depths $A_n=A[1+\eta\cdot\mathfrak{R}_n]$, where $A$ is a constant and $\mathfrak{R}_n$ denotes a sequence of random numbers between -0.5 and 0.5, and $\eta$ denotes the relative strength of disorder. For instance, we refer to $\eta=0.1$ as a 10% disorder. We use $A=5$ and $a=1$, and $\eta=0.3$ in the example shown in Fig. \[disordered\]. In our computation, we choose the Gaussian trial functions $\ket{g_n}$ for different wells despite the wells are different. As shown in Fig. \[disordered\], our method produces successfully exponentially localized Wannier functions. It is clear that a narrower trial Gaussian function leads to more localized Wannier functions. Numerical results indicate that Gaussian trial functions with $\sigma$ as small as 0.5$a$ is enough to construct MLWFs.
![(color online) Wannier functions constructed by our method for one dimensional 30% disordered potential, which is plotted as the black line at the top of (a). The initial trial Gaussian functions are the same for different wells. (a) The Wannier functions obtained with Gaussian functions of different widths $\alpha$. (b) The spread of the Wannier functions obtained with our method as a function of the width of the trial Gaussian function. []{data-label="disordered"}](fig3){width="0.9\linewidth"}
For bands of non-trivial topology, the existing methods [@Marzari97; @Teichler] face inherent singularity at special $k$ points. One has to find ingenious ways to circumvent the singularity to obtain exponentially localized Wannier functions [@Winkler2016]. Our method has the potential to circumvent the singularity without any special re-design as we use the full band projection instead of the projection at individual $k$ points.
Note that we have so far assumed that the system has time reversal symmetry. For these systems, its energy eigenfunctions can always be made real. It follows that the projection $P$ and all the matrices involved in the Löwdin method (Eq.\[lowdin\]) can also be made real. Therefore, the final Wannier functions $\ket{w_n}$ are also real as long as the trial functions are real. For systems where the time reversal symmetry is broken, $\ket{w_n}$ can be complex. In this case, the Löwdin method maximizes ${\rm Re}\{\braket{w_n|g_n}\}$ and we may not obtain MLWFs. We shall leave it for future discussion.
In sum, we have presented a simple and general method for constructing MLWFs. In our method, the full band projection is used on localized trial wave functions. If the trial functions are properly chosen to respect the local potential configuration and have good node positions, the ensuing Löwdin method orthonormalize them to MLWFs. No numerical minimization is needed. Our method can be directly applied to random potentials. The application of our method to a real material is left for future.
We thank Ji Feng and Xianqing Lin for helpful discussion. This work is supported by the National Basic Research Program of China (Grants No. 2013CB921903 and No. 2012CB921300) and the National Natural Science Foundation of China (Grants No. 11274024, No. 11334001, and No. 11429402).
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abstract: 'Classical scalar-response regression methods treat covariates as a vector and estimate a corresponding vector of regression coefficients. In medical applications, however, regressors are often in a form of multi-dimensional arrays. For example, one may be interested in using MRI imaging to identify which brain regions are associated with a health outcome. Vectorizing the two-dimensional image arrays is an unsatisfactory approach since it destroys the inherent spatial structure of the images and can be computationally challenging. We present an alternative approach—regularized matrix regression—where the matrix of regression coefficients is defined as a solution to the specific optimization problem. The method, called SParsity Inducing Nuclear Norm EstimatoR (SpINNEr), simultaneously imposes two penalty types on the regression coefficient matrix—the nuclear norm and the lasso norm—to encourage a low rank matrix solution that also has entry-wise sparsity. A specific implementation of the alternating direction method of multipliers (ADMM) is used to build a fast and efficient numerical solver. Our simulations show that SpINNEr outperforms others methods in estimation accuracy when the response-related entries (representing the brain’s functional connectivity) are arranged in well-connected communities. SpINNEr is applied to investigate associations between HIV-related outcomes and functional connectivity in the human brain.'
author:
- |
Damian Brzyski$^a$, Xixi Hu$^b$, Joaquin Goni$^c$, Beau Ances$^d$,\
Timothy W Randolph$^e$, Jaroslaw Harezlak$^f$\
$\mbox{}^{a}$[*Faculty of Pure and Applied Mathematics, Wroclaw University of Science and Technology, Wroclaw, Poland*]{}\
$\mbox{}^{b}$[*Department of Statistics, Indiana University, Bloomington, IN, USA*]{}\
$\mbox{}^{c}$[*Purdue University, West Lafayette, IN, USA*]{}\
$\mbox{}^{d}$[*Washington University School of Medicine, St. Louis, MO, USA*]{}\
$\mbox{}^{e}$[*Fred Hutchinson Cancer Research Center, Seattle, WA, USA*]{}\
$\mbox{}^{f}$[*Department of Epidemiology and Biostatistics, Indiana University, Bloomington, IN, USA*]{}\
bibliography:
- 'references.bib'
title: '**A Sparsity Inducing Nuclear-Norm Estimator (SpINNEr) for Matrix-Variate Regression in Brain Connectivity Analysis**'
---
\#1
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[**Nuclear and L1 Norms Marriage in Sparse and Low-Rank Regularized Matrix Estimation**]{}
[*Keywords:*]{} Nuclear plus L1 norm, Low-rank and sparse matrix, Spectral regularization, Penalized matrix regression, Clusters in brain network
Introduction
============
Regression problems where the response is a scalar and the predictors constitute a multidimensional array arise often in medical applications where a matrix or a high dimensional array of measurements is collected for each subject. For example, it is of clinical interest to understand associations between: (a) alcoholism and the electrical activity of different brain regions over time collected from electroencephalography (EEG) [@Li-dimension-2010]; (b) cognitive function and three-dimensional white-matter structure data collected from diffusion tensor imaging (DTI) [@Goldsmith-smooth-2014] for patients with multiple sclerosis (MS); and (c) cognitive impairment and brain’s metabolic activity data collected from three-dimensional positron emission tomography (PET) imaging [@Wang-regularized-2014]. Our work focuses on the problem of identifying brain network connections that are associated with neurocognitive measures for HIV-infected individuals. The outcome (response) is a continuous variable and the predictors are matrix representations of functional connectivity between the brain’s cortical regions.
Biophysical considerations motivate our interest in estimating a matrix of regression coefficients that has the following two properties: (i) it should be relatively sparse, since we aim to identify connections that most strongly predict the outcome; and more importantly, (ii) the response-related connections form clusters, since brain activity networks are known to consist of densely connected regions. These two properties translate to the coefficient matrix having relatively small clusters, or blocks of nonzero entries, which implies that it is low-rank. Hence, we aim to solve the matrix regression problem by estimating a coefficient matrix that is both sparse and low-rank. To further illustrate our approach, consider the three matrices in Figure \[fig\_LowRankSparseMatrices\]. The one in the left panel is sparse, but full-rank, the one on the right panel is low-rank, but not sparse, while the one in the middle panel is both low-rank and sparse, which is the structure we are interested in. To find such a solution, we propose a regularization method called *SParsity Inducing Nuclear Norm EstimatoR* (${\textrm{SpINNEr}}$).
[.32]{} ![Illustrative example of (a) a matrix that is sparse, but full-rank; (c) a matrix that is low-rank, but not sparse; and (b) a matrix that is both low-rank and sparse.[]{data-label="fig_LowRankSparseMatrices"}](./SparseMatrix.pdf "fig:"){width="1\linewidth"}
[.32]{} ![Illustrative example of (a) a matrix that is sparse, but full-rank; (c) a matrix that is low-rank, but not sparse; and (b) a matrix that is both low-rank and sparse.[]{data-label="fig_LowRankSparseMatrices"}](./LowRankAndSparse.pdf "fig:"){width="1\linewidth"}
[.32]{} ![Illustrative example of (a) a matrix that is sparse, but full-rank; (c) a matrix that is low-rank, but not sparse; and (b) a matrix that is both low-rank and sparse.[]{data-label="fig_LowRankSparseMatrices"}](./LowRankMatrix.pdf "fig:"){width="1\linewidth"}
Several regularization methods have been proposed for regression problems where the response is a scalar and the predictors constitute a multidimensional array or tensor. These methods fall mainly along two investigative directions. The first treats the multidimensional array of predictors as functional data. Among the early efforts, [@Reiss-functional-2010] extended their functional principal component regression method for one-dimensional signal predictors [@Reiss-functional-2007] to two-dimensional image predictors. Their method is based on B-splines with a penalty on the roughness of the coefficient function which encourages local structure but does not impose constraints on rank or sparsity. [@Wang-regularized-2014] developed a regularized wavelet-based approach that induces sparsity in the coefficient function. The second line of research treats images as tensors rather than as functional data. [@Zhou-tensor-2013] proposed a tensor regression framework that achieves dimension reduction through fixed-rank tensor decomposition [@Kolda-tensor-2009]. They obtain the estimates by a regularized maximum likelihood approach. For regression problems with matrix covariates, [@Zhou-regularized-2014] proposed spectral regularization where the penalty term is a function of the coefficient matrix singular values. Using $\ell_1$-norm of the singular values as penalty gives rise to a nuclear norm regression which induces a low-rank structure on the coefficient matrix. Our work builds upon [@Zhou-regularized-2014] by inducing sparsity of the coefficient matrix in terms of both its rank (low-rank) and the number of its nonzero entries (sparse). Under a Bayesian framework, [@Goldsmith-smooth-2014] used a prior distribution on latent binary indicators to induce sparsity and spatial contiguity of relevant image locations, and appealed to Gaussian Markov random field to induce smoothness in the coefficients.
The approaches summarized above are insufficient for finding a coefficient matrix that is both sparse and low-rank. More specifically, for regularization approaches based on functional regression, [@Reiss-functional-2010] do not impose conditions on sparsity rank, while [@Wang-regularized-2014] do not impose any constraint on rank. For approaches based on tensor decomposition, the method of [@Zhou-tensor-2013] can potentially induce sparsity via regularized maximum-likelihood, but the rank of the solution must be pre-specified and fixed prior to model fitting. In other words, the rank is not determined in a data-driven manner. [@Zhou-regularized-2014] lifted the fixed-rank constraint by using a nuclear norm—a convex relaxation of rank—as penalty, but the solution may not be sparse. Finally, [@Goldsmith-smooth-2014] impose sparsity and spatial smoothness, which implicitly reduces complexity (and possibly rank), but this approach assumes spatially adjacent regions are similarly associated with the response.
In contrast to all of these methods, ${\textrm{SpINNEr}}$ combines a nuclear-norm penalty with an $\ell_1$-norm penalty that simultaneously imposes low-rank and sparsity on the coefficient matrix. Specifically, the low-rank constraint induces accurate estimation of coefficients inside response-related blocks, while outside of these blocks the sparsity constraint encourages zeros. These blocks, however, are not presumed to consist of only spatially adjacent brain regions and so this sparse-and-low-rank approach is more flexible and physiologically meaningful.
While ${\textrm{SpINNEr}}$ seeks a singly regression coefficient matrix that is both sparse and low-rank, others have proposed estimating two structures: one low-rank and one sparse. For graphical models, in particular, [@Chandrasekaran-latent-2012] proposed a method for estimating a precision matrix that decomposes into the sum of a sparse matrix and a low-rank matrix when there are latent variables. Their estimation is based on regularized maximum likelihood where sparsity is induced by the $\ell_1$-norm and low-rank is induced by the nuclear norm. Building upon [@Chandrasekaran-latent-2012], [@Ciccone-robust-2019] imposed the additional constraint that the sample covariance matrix of the observed variables is close to the true covariance in terms of Kullback-Leibler divergence, and proposed a computational solution based on the alternating direction method of multipliers (ADMM) algorithm. While [@Chandrasekaran-latent-2012] and [@Ciccone-robust-2019] assume observations to be i.i.d., [@Foti-sparse-2016] extended the “sparse plus low-rank” framework to graphical models with time series data, whereas [@Basu-low-rank-2018] considered vector autoregressive models and directly imposed the decomposition on the transition matrix. In the matrix completion literature, the “sparse plus low-rank” decomposition has also been exploited for algorithmic concerns, enabling more efficient storage and computation [@Mazumder-spectral-2010; @Hastie-matrix-2015]. Our work is distinct from these proposals in that we obtain a single matrix that is simultaneously sparse and low-rank by imposing two penalties on the same matrix, rather than separately penalizing two components of a matrix.
The rest of the article is organized as follows. Section \[sec\_Problem\] further motivates and describes the objective of finding response-related clusters and translate it to a problem of finding a low-rank and sparse coefficient matrix. Section \[sec\_Method\] formulates the objective as an optimization problem, characterizes properties of its solution, and develops an algorithm for numerical implementation. Simulation experiments are summarized in Section \[sec\_SimulationExperiments\] and an application to brain imaging data is described in Section \[sec\_BrainDataResults\]. We conclude with discussion in Section \[sec\_Discussion\]. Technical derivations of the algorithm are presented in the Appendix.
Clusters recovery problem {#sec_Problem}
=========================
Statistical model
-----------------
Assume we observe a real-valued response, $y_i$, and a $p\times p$ matrix, $A_i$, for each subject, $i=1,\ldots,n$. We additionally assume a vector of $m$ covariates, $X_i$, such that the $n\times m$ matrix, $X$, with rows $X_i$, for each subject, has independent columns (hence $m\leq n$).
Motivated by brain imaging applications, $A_i$ is viewed as an adjacency matrix of connectivity information (structural or functional), each $X_i$ corresponds to a vector of demographic covariates and an intercept (i.e., the first entry of $X_i$ is 1). Additionally, we assume that there exists an (unknown) $p\times p$ matrix $B$ and (unknown) $m\times 1$ vector $\beta$ whose entries represent coefficients to be estimated in the regression equation $$\label{model}
y_i = \langle A_i, B \rangle + X_i \beta + \varepsilon_i, \quad \textrm{for } i=1,\ldots, n,$$ where $\langle A_i, B \rangle:= \big\langle {\operatorname{vec}}(A_i),\, {\operatorname{vec}}(B) \big\rangle = \operatorname{tr}\big(A_i{^\mathsf{T}}B\big)$ is the Frobenius inner product and $\varepsilon\sim\mathcal{N}(0,\sigma^2I_n)$. Unless $n$ is unusually large (greater than $p(p-1)/2$), this is an informal statement of the problem which does not have a unique solution for $B$ and $\beta$ without further constraints. The focus of this work is on the rigorous implementation of constraints that lead to a biologically meaningful regression model having a unique solution.
We will use the equivalent graph description of the problem to build the intuition behind the assumed model . In that interpretation, brain regions are viewed as $p$ nodes in a graph and the connectivity information of $i$th subject for these regions is represented by a $p\times p$ matrix $A_i$ with zeros on the diagonal. The off-diagonal entries, $A_i(j,l)$, of $A_i$ denote weights of connectivity between regions $j$ and $l$; positive as well as negative weights are acceptable. If $A_i(j,l)$ is positive, its value indicates how strongly regions $j$ and $l$ are connected, while the magnitude of negative entry indicates the level of dissimilarity. In , $B$ denotes the (unknown) $p\times p$ matrix of regression coefficients whose $(j,l)$ entry, $B_{j,l}$, represents the association between the response variable and the connectivity across regions $j$ and $l$. We assume $B$ is symmetric and note that its diagonal entries $B_{j,j}$ are not included in the model, since each connectivity matrix $A_i$ has zeros on the diagonal. The main goal, therefore, is to estimate the off-diagonal entries of $B$ in a manner that reveals brain subnetwork structure that is associated with the response. This structure is revealed by the clusters and hubs defined by the non-zero entries in $\hat{B}$, an estimate of $B$.
Response-related clusters
-------------------------
We are interested in identifying only significant brain-region connectivities and therefore we want to encourage the estimate, $\hat{B}$, to be sparse entry-wise. However, our most important goal is to protect the structure of response-related connectivities (i.e., the non-zero entries of $B$). Indeed, brain networks exhibit a so-called “rich club" organization, meaning that there are relatively small groups of densely connected nodes [@Richclub; @HubDet]. Recent studies have demonstrated that such hubs play an important role in information integration between different parts of the network [@Richclub]. This structure would be lost if the process of estimating $B$ only focused on sparsity. Thus, we want the estimation process to allow for a potentially cluster-structured form of $B$ so that it may more accurately reflect the association between brain connectivities and a phenotypic outcome. However, estimating $B$ using a two-step process would imply that we must detect individual response-impacting edges *prior to* forming clusters from the selected edges. Clearly some cluster-defining edges and/or hubs may be missed by this preliminary focus on sparsity.
As an illustration, consider a setting where there are many response-related connectivities between a few, say $k$, brain regions. Even if some of these effects are moderate, it is the entire cluster of regions that, as a whole, affects the response. However, if there are $(k-1)k/2$ connections in the cluster and only the strongest effects survive a sparsity-inducing lasso estimate [@LassoF] or other entry-wise thresholding techniques, then the “systems level" information is lost and inferring relevant information about the clusters may become impossible.
In our work, we introduce the notion of *response-related clusters* and focus on their selection rather than on accurate estimation of each individual effect which, in fact, would be impossible due to the limited sample size. Precisely, we define a set of nodes, $S$, to be a *response-related cluster* (RRC) if for any two distinct indices $j, l \in S$ there is a path of edges from $S$ connecting $j$ and $l$, namely the sequence of the elements $i_1, \ldots, i_k \in S$ such as $i_1 = j$, $i_k= l$ and $B_{i_h, i_{h+1}}\neq 0$ for $h=1,\ldots, k-1$. We define $S$ to be a *positive response-related cluster* if for every pair of its elements, $j$ and $l$, it holds that $B_{j,l}\geq 0$ (zeros are acceptable). Accordingly, $S$ is a *negative response-related cluster* if $B_{j,l}\leq 0$ for all its elements. Motivated by the rich-club pattern of brain connectivity, we will assume that relatively few clusters of brain nodes spanning the subnetworks are strongly associated with $y$. If the brain regions are arranged in a cluster-by-cluster ordering, this assumption is simply reflected in a block-diagonal pattern of the matrix of regression coefficients, $B$, with blocks corresponding to RRCs (Figure \[fig:signalForm\]).
[.34]{} ![The assumed form of $B$ after arranging nodes in the cluster-by-cluster ordering is presented in (a). Three RRC (two positive and one negative) of various connectivity patterns are present. Plots (b)-(d) present the equivalent graph representations of RRCs. Clusters are also easily recognizable in the $k$-rank best (with respect to the Frobenius norm) approximations of $B$, denoted by $r_k(B)$. Only one—the densest—cluster is showed by the 1-rank best approximation (e), while 3-rank and 6-rank approximations, (f) and (g), respectively, can reveal two and three clusters. Panel (g) shows that the low rank approximation of signal may reflect its structure well, although some edges may be lost (as the edge between nodes 1 and 7 in RRC$_1$) or falsely introduced (some edges in RRC$_2$ and RRC$_3$). []{data-label="fig:example"}](./exampleB.pdf "fig:"){width="1\linewidth"}
[.195]{} ![The assumed form of $B$ after arranging nodes in the cluster-by-cluster ordering is presented in (a). Three RRC (two positive and one negative) of various connectivity patterns are present. Plots (b)-(d) present the equivalent graph representations of RRCs. Clusters are also easily recognizable in the $k$-rank best (with respect to the Frobenius norm) approximations of $B$, denoted by $r_k(B)$. Only one—the densest—cluster is showed by the 1-rank best approximation (e), while 3-rank and 6-rank approximations, (f) and (g), respectively, can reveal two and three clusters. Panel (g) shows that the low rank approximation of signal may reflect its structure well, although some edges may be lost (as the edge between nodes 1 and 7 in RRC$_1$) or falsely introduced (some edges in RRC$_2$ and RRC$_3$). []{data-label="fig:example"}](./Hub1.pdf "fig:"){width="1\linewidth"}
[.195]{} ![The assumed form of $B$ after arranging nodes in the cluster-by-cluster ordering is presented in (a). Three RRC (two positive and one negative) of various connectivity patterns are present. Plots (b)-(d) present the equivalent graph representations of RRCs. Clusters are also easily recognizable in the $k$-rank best (with respect to the Frobenius norm) approximations of $B$, denoted by $r_k(B)$. Only one—the densest—cluster is showed by the 1-rank best approximation (e), while 3-rank and 6-rank approximations, (f) and (g), respectively, can reveal two and three clusters. Panel (g) shows that the low rank approximation of signal may reflect its structure well, although some edges may be lost (as the edge between nodes 1 and 7 in RRC$_1$) or falsely introduced (some edges in RRC$_2$ and RRC$_3$). []{data-label="fig:example"}](./Hub2.pdf "fig:"){width="1\linewidth"}
[.195]{} ![The assumed form of $B$ after arranging nodes in the cluster-by-cluster ordering is presented in (a). Three RRC (two positive and one negative) of various connectivity patterns are present. Plots (b)-(d) present the equivalent graph representations of RRCs. Clusters are also easily recognizable in the $k$-rank best (with respect to the Frobenius norm) approximations of $B$, denoted by $r_k(B)$. Only one—the densest—cluster is showed by the 1-rank best approximation (e), while 3-rank and 6-rank approximations, (f) and (g), respectively, can reveal two and three clusters. Panel (g) shows that the low rank approximation of signal may reflect its structure well, although some edges may be lost (as the edge between nodes 1 and 7 in RRC$_1$) or falsely introduced (some edges in RRC$_2$ and RRC$_3$). []{data-label="fig:example"}](./Hub3.pdf "fig:"){width="1\linewidth"}
[.32]{} ![The assumed form of $B$ after arranging nodes in the cluster-by-cluster ordering is presented in (a). Three RRC (two positive and one negative) of various connectivity patterns are present. Plots (b)-(d) present the equivalent graph representations of RRCs. Clusters are also easily recognizable in the $k$-rank best (with respect to the Frobenius norm) approximations of $B$, denoted by $r_k(B)$. Only one—the densest—cluster is showed by the 1-rank best approximation (e), while 3-rank and 6-rank approximations, (f) and (g), respectively, can reveal two and three clusters. Panel (g) shows that the low rank approximation of signal may reflect its structure well, although some edges may be lost (as the edge between nodes 1 and 7 in RRC$_1$) or falsely introduced (some edges in RRC$_2$ and RRC$_3$). []{data-label="fig:example"}](./exampleB1.pdf "fig:"){width="1\linewidth"}
[.32]{} ![The assumed form of $B$ after arranging nodes in the cluster-by-cluster ordering is presented in (a). Three RRC (two positive and one negative) of various connectivity patterns are present. Plots (b)-(d) present the equivalent graph representations of RRCs. Clusters are also easily recognizable in the $k$-rank best (with respect to the Frobenius norm) approximations of $B$, denoted by $r_k(B)$. Only one—the densest—cluster is showed by the 1-rank best approximation (e), while 3-rank and 6-rank approximations, (f) and (g), respectively, can reveal two and three clusters. Panel (g) shows that the low rank approximation of signal may reflect its structure well, although some edges may be lost (as the edge between nodes 1 and 7 in RRC$_1$) or falsely introduced (some edges in RRC$_2$ and RRC$_3$). []{data-label="fig:example"}](./exampleB3.pdf "fig:"){width="1\linewidth"}
[.32]{} ![The assumed form of $B$ after arranging nodes in the cluster-by-cluster ordering is presented in (a). Three RRC (two positive and one negative) of various connectivity patterns are present. Plots (b)-(d) present the equivalent graph representations of RRCs. Clusters are also easily recognizable in the $k$-rank best (with respect to the Frobenius norm) approximations of $B$, denoted by $r_k(B)$. Only one—the densest—cluster is showed by the 1-rank best approximation (e), while 3-rank and 6-rank approximations, (f) and (g), respectively, can reveal two and three clusters. Panel (g) shows that the low rank approximation of signal may reflect its structure well, although some edges may be lost (as the edge between nodes 1 and 7 in RRC$_1$) or falsely introduced (some edges in RRC$_2$ and RRC$_3$). []{data-label="fig:example"}](./exampleB6.pdf "fig:"){width="1\linewidth"}
A marriage of sparsity and low rank
-----------------------------------
As observed previously, most often we do not have a large enough set of samples to accurately estimate all entries of $B$. More precisely, suppose that we are considering the MLE of $B$ under the model , without any constraints imposed on the estimates. This leads to the problem of minimizing ${\sum_{i=1}^n\big(y_i -\langle A_i, B \rangle\big)^2}$ with respect to the $p$ by $p$ matrix $B$ (we exclude $X$ and $\beta$ for clarity). Such a problem does not have a unique solution unless we assume that all $p^2$ vectors $v_{j,k}: = [A_1(j,k),\ldots, A_n(j,k)]{^\mathsf{T}}$ are linearly independent, implying that $n\geq p^2$. If $p = 100$, which is rather a small number of regions compared to brain parcellations widely used in applications, this would necessitate observing data on at least $n=10,000$ subjects. Of course we can limit the degrees of freedom by assuming (very reasonably) that $B$ is symmetric but this still requires $n>p(p-1)/2=4,950$. We can reduce this number further by assuming there exist relatively few RRCs, so that $B$ is sparse (Figure \[fig:signalForm\]). Let $k$ and $s$ denote the number and the average size of RRCs, respectively. However, even with an oracle telling us precisely the locations of non-zeros and we restrict the estimation to these corresponding entries of $B$) there are still $O(s^2)$ observations required since we have, roughly, $ks^2/2$ entries to estimate (and this is the simplest scenario with all clusters having the same number of nodes).
Again consider a signal with a block pattern as in Figure \[fig:signalForm\]. For such matrices, the first a few eigenvectors (i.e., those corresponding to the largest eigenvalues) are of a special form. Assuming that they are unique (up to a change in sign), each of them may have its non-zeros located inside exactly one RRC. In our example (presented in Figure \[fig:example\]), the eigenvector $v_1$, corresponding to the largest eigenvalue $\lambda_1$, has all its non-zeros located inside RRC$_3$. Moreover, as a direct consequence of Perron-Frobenius theorem [@FrobTheorem] for irreducible matrices, all coefficients of $v_1$ located inside RRC$_3$ must be either
*strictly positive* or *strictly negative*. That is, the non-zero entries of $v_1$ coincide precisely with the indices of RRC$_3$ and the matrix $\lambda_1v_1v_1{^\mathsf{T}}$ (i.e., the best rank-one approximation of $B$) and corresponds to the entire corresponding block in Figure \[fig:1rank\]. In fact, the first several, say $\tilde{k} \ll p$, eigenvectors may effectively summarize the structure of signal via the best rank-$\tilde{k}$ approximation, $r_{\tilde{k}}(B): = \sum_{i=1}^{\tilde{k}}\lambda_iv_iv_i{^\mathsf{T}}$, of $B$ (Figure \[fig:6rank\]).
Returning to the calculations, if we restrict attention to the general structure of $B$ reflected by its first $\tilde{k}$ eigenvectors and assume that each of them has roughly $s$ non-zeros within one of the RRCs, then we obtain $O(s)$, namely $\tilde{k}s$, coefficients to estimate (according to an oracle). This refocus on “structured sparsity" not only reduces the computational requirements, but also adds to the physiological interpretation.
In summary, we propose a method called *SParsity Inducing Nuclear Norm EstimatoR* (${\textrm{SpINNEr}}$) that constructs a low-rank and sparse estimate of $B$ under the model . It provides a principled approach to estimating $B$ via: (i) exploiting its dominant eigenvectors to accurately estimate block-structured coefficients and, simultaneously, (ii) imposing sparsity outside of these blocks.
Methodology {#sec_Method}
===========
Penalized optimization
----------------------
With the goal of encouraging a regression coefficient (matrix) estimate to be both sparse and low-rank, ${\textrm{SpINNEr}}$ employs two types of matrix norms which cooperate together as penalties to regularize the estimate: an $\ell_1$ norm imposes entry-wise sparsity and a [*nuclear norm*]{} achieves a convex relaxation of rank minimization ([@Candes2009; @Rech2010]). The nuclear norm (also referred to as the trace norm) of $B$, denoted by $\|B\|_*$, is defined as a sum of the singular values of $B$.
For a pair of prespecified nonnegative tuning parameters $\lambda_N$ and $\lambda_L$, ${\textrm{SpINNEr}}$ is defined as a solution to the following optimization problem $$\label{SPINNER}
\big\{\hat{B}^{{\textnormal{\tiny {S}}}},\ \hat{\beta}^{{\textnormal{\tiny {S}}}} \big\}:={\underset{B, \beta}{\operatorname{argmin}}\;}\ \left\{\frac12\sum_{i=1}^n\Big(y_i -\langle A_i, B \rangle - X_i\beta \Big)^2+ \lambda_N \big\|B\big\|_* + \lambda_L\big\|{\operatorname{vec}}(W\circ B)\big\|_1\right\},$$ where $W$ is a $p\times p$ symmetric matrix of nonnegative weights. Here, $W\circ B$ denotes the Hadamard product (entrywise product) of $W$ and $B$, hence $\big\|{\operatorname{vec}}(W\circ B)\big\|_1 = \sum_{j,l = 1}^pW_{j,l}|B_{j,l}|$. By default, $W$ is the matrix with zeros on the diagonal and ones on the off-diagonal. Setting all $W_{j,j}$s as zeros protects the diagonal entries of $\hat{B}^{{\textnormal{\tiny {S}}}}$ from being shrunk to zero by the $\ell_1$ norm and leads to a more accurate recovery of a low-rank approximation of $B$ via the nuclear norm. Penalizing $\hat{B}^{{\textnormal{\tiny {S}}}}_{j,j}$s by $\ell_1$ norm in a situation when all $A_i$s have zeros on their diagonals would not be justified, since the diagonal $B_{j,j}$s (the nodes’ effects) are not included in model ; i.e., there is no information about them in $y$. We note that with the default $W$, $B\mapsto \big\|{\operatorname{vec}}(W\circ B)\big\|_1$ does not define norm, however it is a convex function (and a seminorm) so is a convex optimization problem.
${\textrm{SpINNEr}}$ is well-defined in the sense that a solution to exists for any tuning parameters $\lambda_N, \lambda_L$ (see the subsection \[subs:basic\]). If $\lambda_N = 0$, has infinitely many pairs $\big\{\hat{B}^{{\textnormal{\tiny {S}}}},\ \hat{\beta}^{{\textnormal{\tiny {S}}}} \big\}$ minimizing with a default selection of weights, since the diagonal entries of $\hat{B}^{{\textnormal{\tiny {S}}}}$ do not impact the objective function. The off-diagonal elements of $\hat{B}^{{\textnormal{\tiny {S}}}}$ (which in that case reduces to a lasso estimate) will, however, be unique with probability one if the predictor variables are assumed to be drawn from a continuous probability distribution [see, @tibshirani2012lasso]. In a situation of non-uniqueness, the name “${\textrm{SpINNEr}}$” will refer to the set of all solutions to .
Simplifying the optimization problem {#simpOpt}
------------------------------------
The problem in is defined as an optimization with respect to both $B$ and $\beta$, but this can be reformulated so that, in practice, we need only solve a minimization problem with respect to $B$. To see this, define the vector $w_B$ as $(w_B)_i: = y_i -\langle A_i, B \rangle$ and note that $$\hat{\beta}_{B} := {\underset{\beta}{\operatorname{argmin}}\;} \ \frac12\sum_{i=1}^n\Big( y_i-\langle A_i, B \rangle - X_i\beta \Big)^2 \\
= {\underset{\beta}{\operatorname{argmin}}\;} \big\| w_b - X\beta\big\|^2 = \big(X{^\mathsf{T}}X\big)^{-1}X{^\mathsf{T}}w_B.
$$ Since the penalty terms involving $B$ can be treated as additive constants, this solves with respect to $\beta$. Therefore, we can substitute $\hat{\beta}_{B}$ into and transform the problem into one involving only $B$. For this, denote the projection onto the orthogonal complement of the range of $X$ as $H: = \mathbf{I}_n - X\big(X{^\mathsf{T}}X\big)^{-1}X{^\mathsf{T}}$. Also, denote by $\mathcal{A}$ the $n$-row matrix of stacked vectors from $\{{\operatorname{vec}}(A_i)\}_{i=1}^n$. If we transform $y$ and $\mathcal{A}$ as $\widetilde{y}: = Hy$ and $\widetilde{\mathcal{A}}: = H\mathcal{A}$, then upon substitution of $\hat{\beta}_{B}$ into , we can rewrite the model-fit term as $$\begin{split}
&\sum_{i=1}^n\Big(y_i -\langle A_i, B \rangle - X_i\hat{\beta}_B \Big)^2 =\sum_{i=1}^n\Big(y_i -{\operatorname{vec}}(A_i){^\mathsf{T}}{\operatorname{vec}}(B) - X_i\hat{\beta}_B \Big)^2
=\big\|y - \mathcal{A}{\operatorname{vec}}(B) - X\hat{\beta}_B\big\|_2^2= \\
&\Big\|y - \mathcal{A}{\operatorname{vec}}(B) - X\big(X{^\mathsf{T}}X\big)^{-1}X{^\mathsf{T}}\big(y - \mathcal{A}{\operatorname{vec}}(B)\big)\Big\|_2^2
=\big\|Hy - H\mathcal{A}{\operatorname{vec}}(B)\big\|_2^2 =\big\|\widetilde{y} - \widetilde{\mathcal{A}}{\operatorname{vec}}(B)\big\|_2^2.
\end{split}$$ Hence, can be equivalently represented as $$\label{SPINNER3}
\left\{
\begin{array}{l}
\hat{B}^{{\textnormal{\tiny {S}}}}:={\underset{B}{\operatorname{argmin}}\;}\ \bigg\{ \big\|\widetilde{y} - \widetilde{\mathcal{A}}{\operatorname{vec}}(B)\big\|_2^2 + \lambda_N \big\|B\big\|_* + \lambda_L\big\|{\operatorname{vec}}(W\circ B)\big\|_1\bigg\}\\
\hat{\beta}^{{\textnormal{\tiny {S}}}}: = \big(X{^\mathsf{T}}X\big)^{-1}X{^\mathsf{T}}\big[y - \mathcal{A}{\operatorname{vec}}(\hat{B}^{{\textnormal{\tiny {S}}}})\big]
\end{array}
\right..$$
Basic properties {#subs:basic}
----------------
In view of the reformulation of ${\textrm{SpINNEr}}$ in we will, without loss of generality, exclude $X$ and $\beta$ from consideration and focus on the minimization problem with the objective function $$\label{objective}
F(B):\ =\ \underbrace{\frac12\sum_{i=1}^n\Big(y_i -\langle A_i, B \rangle \Big)^2}_{f(B)}\ +\ \underbrace{\lambda_N \big\|B\big\|_*}_{g(B)}\ +\ \underbrace{\lambda_L\big\|{\operatorname{vec}}(W\circ B)\big\|_1}_{h(B)}.$$ The following proposition clarifies that a ${\textrm{SpINNEr}}$ estimate is well-defined.
\[prop0\] For any pair of regularization parameters $\lambda_N\geq 0$ and $\lambda_L\geq 0$ there exists at least one solution to . The claim is still valid, if the function $g(B) + h(B)$ in is replaced by any nonnegative convex function $\tilde{g}$.
We use the concept of *directions of recession* [@Rockafellar1970]. In our situation, a matrix $C$ belongs to the set of directions of recession of $F$ if $F(B + \lambda C) \leq F(B)$ for any matrix $B$ and any scalar $\lambda \geq 0$. In particular, for $B=0$, the direction of recession of $F$ must satisfy $F(\lambda C) - F(0) \leq 0$ for any $\lambda \geq 0$. Therefore, $$\label{dirrec}
\frac12\sum_{i=1}^n\Big[\big(y_i -\lambda\langle A_i, C \rangle \big)^2 - y_i^2\Big]\ +\ \lambda\cdot\lambda_N \big\|C\big\|_*\ +\ \lambda\cdot\lambda_L\big\|{\operatorname{vec}}(W\circ C)\big\|_1 \leq 0.$$ Since the last two terms in are nonnegative, it also holds that $\sum_{i=1}^n\big[\big(y_i -\lambda\langle A_i, C \rangle \big)^2 - y_i^2\big] \leq 0$, hence $\lambda^2 \sum_{i=1}^n \langle A_i, C \rangle^2 - 2\lambda \sum_{i=1}^n y_i\langle A_i, C \rangle \leq 0$, for all $\lambda\geq 0 $. This can happen only when $\sum_{i=1}^n \langle A_i, C \rangle^2 = 0$, implying that $\langle A_i, C \rangle = 0$ for each $i$. Combining this with gives also $\lambda_N \big\|C\big\|_* = 0$ and $\lambda_L\big\|{\operatorname{vec}}(W\circ C)\big\|_1 = 0$. When $\lambda_N>0$ this implies $C=0$, but any selection of regularization parameters imply that $C$ must be a direction in which the objective function is constant. Therefore, applying Theorem 27.1(b) from [@Rockafellar1970], $F$ attains its minimum.
Since $B$ is assumed to be symmetric, it is natural to expect estimates of $B$ to have the same property, although, as yet, we have not enforced this condition on $\hat{B}^{{\textnormal{\tiny {S}}}}$. Fortunately, as shown next, we can always obtain a symmetric minimizer of $F$ in .
\[prop1\] Suppose that $W$ and all matrices $A_i$s are symmetric. Then, the set of solutions to the minimization problem with an objective function defined in contains a symmetric matrix. The claim is still valid if the function $g(B) + h(B)$ in is replaced by any nonnegative convex function $\tilde{g}$ such as $\tilde{g}(A{^\mathsf{T}}) = \tilde{g}(A)$ for any $p\times p$ matrix $A$.
From Proposition \[prop0\] we know that there exists a solution $B^* ={\underset{B}{\operatorname{argmin}}\;} F(B)$. Consider its symmetric part, $\widetilde{B}: = \frac12( B^* + {B^*}{^\mathsf{T}})$. By the symmetry of each $A_i$, for $f$ defined in $$f(\widetilde{B}) = \frac12\sum_{i=1}^n\Big(y_i -\frac12\langle A_i, B^* \rangle - \frac12\langle A_i, {B^*}{^\mathsf{T}} \rangle \Big)^2 =\frac12\sum_{i=1}^n\Big(y_i -\frac12\langle A_i, B^* \rangle - \frac12\langle A_i{^\mathsf{T}}, B^*\rangle \Big)^2 = f(B^*).$$ Now, $$\begin{split}
g(\widetilde{B}) + h(\widetilde{B}) &\ = \ \lambda_N \Big\| \frac12B^* + \frac12{B^*}{^\mathsf{T}}\Big\|_*\ +\ \lambda_L\Big\|\frac12{\operatorname{vec}}( W\circ B^*) + \frac12{\operatorname{vec}}(W\circ{B^*}{^\mathsf{T}})\Big\|_1\\
&\ \leq \ \frac{\lambda_N}2 \big\| B^* \big\|_*\, +\, \frac{\lambda_N}2 \big\|{B^*}{^\mathsf{T}}\big\|_*\ +\ \frac{\lambda_L}2\big\|{\operatorname{vec}}( W\circ B^*)\big\|_1\,+\,\frac{\lambda_L}2\big\|{\operatorname{vec}}(W\circ{B^*}{^\mathsf{T}})\big\|_1\\
&\ = \ \lambda_N \big\| B^* \big\|_* + \lambda_L\big\|{\operatorname{vec}}(W\circ B^*)\big\|_1\ =\ g(B^*) + h(B^*),
\end{split}$$ where the inequality follows from the fact that $g$ and $h$ are convex and the last equality holds since both these functions are invariant under transpose, provided that $W$ is symmetric. Consequently, we get $F(\widetilde{B})\leq F(B^*)$, hence $\widetilde{B}$ must be a solution.
In summary, this shows that the symmetric part of any solution to is also a solution. In particular, when the solution is unique, it is guaranteed to be a symmetric matrix.
\[permProp\] Let $\hat{B}$ be a solution to minimization problem with an objective, $F(B)$, defined in . We consider the modification of the data relying on the nodes reordering. Precisely, suppose that $\pi:\{1,\ldots, p\} \rightarrow \{1,\ldots, p\}$ is a given permutation with corresponding permutation matrix $P_{\pi}$, i.e. it holds $P_{\pi}v = [v_{\pi(1)},\ldots, v_{\pi(p)}]{^\mathsf{T}}$ for any column vector $v$. We replace the matrices $A_i$’s and $W$ in with matrices having rows and columns permuted by $\pi$, namely, $A^{\pi}_i: = P_{\pi}A_iP_{\pi}{^\mathsf{T}}$ and $W^{\pi}: = P_{\pi}WP_{\pi}{^\mathsf{T}}$. Then, $\hat{B}$ with rows and columns permuted by $\pi$, i.e. $\hat{B}^{\pi}: = P_{\pi}\hat{B}P_{\pi}{^\mathsf{T}}$, is a solution to the updated problem.
Suppose that $\hat{B}^{\pi}$ is not a solution, hence there exists matrix $C$ such as $$\begin{aligned}
\label{21012020}
\begin{split}
\frac12\sum\limits_{i=1}^n\big(y_i -\langle A^{\pi}_i,\, C \rangle \big)^2\ +\ \lambda_N & \big\|C\big\|_*\ +\ \lambda_L\big\|{\operatorname{vec}}(W^{\pi}\circ C)\big\|_1 < \\[-1ex]
&\frac12\sum\limits_{i=1}^n\big(y_i -\langle A^{\pi}_i,\, \hat{B}^{\pi} \rangle \big)^2\ +\ \lambda_N \big\|\hat{B}^{\pi}\big\|_*\ +\ \lambda_L\big\|{\operatorname{vec}}(W^{\pi}\circ \hat{B}^{\pi})\big\|_1.
\end{split}\end{aligned}$$ We have $\langle A^{\pi}_i, C \rangle = \langle P_{\pi}A_iP_{\pi}{^\mathsf{T}}, C \rangle = \operatorname{tr}\big(P_{\pi}A_iP_{\pi}{^\mathsf{T}}C\big) = \operatorname{tr}\big(A_iP_{\pi}{^\mathsf{T}}CP_{\pi}\big) = \langle A_i, P_{\pi}{^\mathsf{T}}CP_{\pi} \rangle = \langle A_i, \widetilde{C} \rangle$, for $\widetilde{C}:=P_{\pi}{^\mathsf{T}}CP_{\pi}$. Moreover, $\big\|{\operatorname{vec}}(W^{\pi}\circ C)\big\|_1 = \big\|{\operatorname{vec}}(P_{\pi}WP_{\pi}{^\mathsf{T}}\circ C)\big\|_1 = \big\|{\operatorname{vec}}(P_{\pi}WP_{\pi}{^\mathsf{T}}\circ P_{\pi}\widetilde{C}P_{\pi}{^\mathsf{T}})\big\|_1 = \big\|{\operatorname{vec}}\big(P_{\pi}(W\circ \widetilde{C})P_{\pi}{^\mathsf{T}}\big)\big\|_1 = \sum\limits_{j,l}\big|W_{\pi(j),\pi(l)}\widetilde{C}_{\pi(j),\pi(l)}\big|= \sum\limits_{j,l}\big|W_{j,l}\widetilde{C}_{j,l}\big|=\big\|{\operatorname{vec}}(W\circ \widetilde{C})\big\|_1$, where the third equation follows from the exchangeability of Hadamard product and permutation imposed on rows or columns of matrices (provided that the same permutation is used for two matrices). Since $C$ and $\widetilde{C}$ share the same singular values, it also holds $\|C\|_* = \|\widetilde{C}\|_*$. Consequently, the left-hand side of can be simply expressed as $F(\widetilde{C})$.
On the other hand we have $\langle A^{\pi}_i,\, \hat{B}^{\pi} \rangle = \operatorname{tr}\big( P_{\pi}A_iP_{\pi}{^\mathsf{T}} P_{\pi}\hat{B}P_{\pi}{^\mathsf{T}}\big)= \operatorname{tr}\big(A_i\hat{B}\big) = \langle A_i,\, \hat{B} \rangle$, since $P_{\pi}{^\mathsf{T}} P_{\pi} = \mathbf{I}$. As above, we can get rid of the permutation symbols inside the nuclear and $\ell_1$ norms, yielding $\|\hat{B}^{\pi}\|_* = \|\hat{B}\|_*$ and $\big\|{\operatorname{vec}}(W^{\pi}\circ \hat{B}^{\pi})\big\|_1 = \big\|{\operatorname{vec}}(W\circ \hat{B})\big\|_1$. Therefore, the right-hand side of becomes $F(\hat{B})$ and the inequality yields $F(\widetilde{C}) < F(\hat{B})$ which contradicts the optimality of $\hat{B}$ and proves the claim.
The above statement implies that ${\textrm{SpINNEr}}$ is invariant under the order of nodes in a sense that the rearrangement of the nodes simply corresponds to the rearrangement of rows and columns of an estimate. Consequently, there is no need for fitting the model again. More importantly, the *optimal order* of nodes, i.e. the permutation which reveals the assumed clumps structure (see, Section \[sec\_BrainDataResults\]), can be found at the end of the procedure based on the ${\textrm{SpINNEr}}$ estimate achieved for *any* arrangement of nodes, e.g. corresponding to the alphabetical order of node labels.
Numerical implementation {#sec:numSol}
------------------------
To build the numerical solver for the problem , we employed the Alternating Direction Method of Multipliers (ADMM) [@Gabay1976ADA]. The algorithm relies on introducing $p\times p$ matrices $C$ and $D$ as new variables and considering the constrained version of the problem (equivalent to ) with a separable objective function: $$\arraycolsep=1.4pt\def{1.2}{0.8}
{\underset{B, C, D}{\operatorname{argmin}}\;}\ \big\{f(B)\,+\,g(C)\,+\,h(D)\big\} \qquad \textrm{such that }\ \left\{\begin{array}{l}
D-B=0\\
D-C=0
\end{array}
\right..$$ The augmented Lagrangian with the scalars $\delta_1>0$, $\delta_2>0$ and dual variable ${Z}: = \left [\begin{BMAT}(c)[0.5pt,0pt,0.7cm]{c}{cc}{Z}_1 \\ {Z}_2 \end{BMAT} \right ]\in \mathbb{R}^{2p\times p}$ is $$L_{\delta}(B,C,D; {Z}) =f(B)+g(C)+h(D)+\langle{Z}_1,D-B\rangle+\langle{Z}_2,D-C\rangle+\frac{\delta_1}2\big\|D-B\big\|_F^2+\frac{\delta_2}2\big\|D-C\big\|_F^2.$$
ADMM builds the update of the current guess, i.e., the matrices $B^{[k+1]}$, $C^{[k+1]}$ and $D^{[k+1]}$, by minimizing $L_{\delta}(B,C,D; {Z})$ with respect to each of the primal optimization variables separately while treating all remaining variables as fixed. Dual variables are updated in the last step of this iterative procedure. Since $\langle{Z}_1,D-B\rangle+\frac{\delta_1}2\|D-B\|_F^2 =\frac{\delta_1}2\|D+\frac{{Z}_1}{\delta_1}-B\|_F^2 + const_1$ and $\langle{Z}_2,D-C\rangle+\frac{\delta_2}2\|D-C\|_F^2 =\frac{\delta_2}2\|D+\frac{{Z}_2}{\delta_2}-C\|_F^2 + const_2$, where $const_1$ does not depend on $B$ and $const_2$ does not depend on $C$, ADMM updates for the considered problem takes the final form $$\begin{aligned}
&B^{[k+1]}:=\ {\underset{B}{\operatorname{argmin}}\;}\bigg\{\,2f(B)\ +\ \delta^{[k]}_1\Big\|\,D^{[k]} + \frac{{Z}_1^{[k]}}{\delta^{[k]}_1}- B\,\Big\|_F^2\,\bigg\}\label{Update1}\\
&C^{[k+1]}:=\ {\underset{C}{\operatorname{argmin}}\;}\bigg\{\,2g(C)\ +\ \delta^{[k]}_2\Big\|\,D^{[k]} + \frac{{Z}_2^{[k]}}{\delta^{[k]}_2}- C\,\Big\|_F^2\,\bigg\}\label{Update2}\\
&D^{[k+1]}:=\ {\underset{D}{\operatorname{argmin}}\;}\bigg\{\,2h(D)\ +\ \delta^{[k]}_1\Big\|\,D + \frac{{Z}_1^{[k]}}{\delta^{[k]}_1} - B^{[k+1]}\,\Big\|_F^2\ +\ \delta^{[k]}_2\Big\|D + \frac{{Z}_2^{[k]}}{\delta^{[k]}_2} -C^{[k+1]}\Big\|_F^2\,\bigg\}\label{Update3}\\
&\left\{
\begin{array}{l}
{Z}_1^{[k+1]}: =\ {Z}_1^{[k]} + \delta^{[k]}_1\big(D^{[k+1]}-B^{[k+1]}\big)\\
{Z}_2^{[k+1]}: =\ {Z}_2^{[k]} + \delta^{[k]}_2\big(D^{[k+1]}-C^{[k+1]}\big)
\end{array}
\right..\end{aligned}$$
All of the subproblems , and have analytical solutions and can be computed very efficiently (see Section \[subs:subproblems\] in the Appendix). Here, the positive numbers $\delta^{[k]}_1$ and $\delta^{[k]}_2$ are treated as the step sizes. The convergence of ADMM is guaranteed under very general assumptions when these parameters are held constant. However, their selection should be performed with caution since they strongly impact the practical performance of ADMM [@Xu2017AdaptiveRA]. Our `MATLAB` implementation uses the procedure based on the concept of *residual balancing* [@Wohlberg2017ADMMPP; @Xu:2017] in order to automatically modify the step sizes in consecutive iterations and provide fast convergence. The stopping criteria are defined as the simultaneous fulfilment of the conditions $$\max\left\{\frac{\|C^{[k+1]}-B^{[k+1]}\|_F}{\|B^{[k+1]}\|_F}, \frac{\|D^{[k+1]}-B^{[k+1]}\|_F}{\|B^{[k+1]}\|_F}\right\}< \epsilon_P,\qquad \frac{\|D^{[k+1]}-D^{[k]}\|_F}{\|D^{[k]}\|_F}< \epsilon_D,$$ which we use as a measure stating that the primal and dual residuals are sufficiently small [@Wohlberg2017ADMMPP]. The default settings are $\epsilon_P: = 10^{-6}$ and $\epsilon_D: = 10^{-6}$.
Simulation Experiments {#sec_SimulationExperiments}
======================
We now investigate the performance of the proposed method, ${\textrm{SpINNEr}}$, and compare it with nuclear-norm regression [@Zhou-regularized-2014], lasso [@LassoF], elastic net [@elasticNet] and ridge [@ridge]. Without loss of generality we focus the simulations on the model where there are no additional covariates, i.e. ${y_i = \langle A_i, B \rangle + \varepsilon_i}$, for $i=1,\ldots, n$ and with $\varepsilon\sim\mathcal{N}(0,\sigma^2I_n)$.
Considered scenarios
--------------------
We consider three scenarios. In the first two scenarios, the observed matrices $\{A_i\}_{i=1}^n$ are synthetic and the “true" signal is defined by a pre-specified $B$: Scenario 1 considers the effects of signal strength (determined by $B$) and Scenario 2 studies power and the effects of sample size, $n$. In Scenario 3, the $A_i$’s are from real brain connectivity maps. Specifically:
1. \[S1\] For each $A_i$, its upper triangular entries are first sampled independently from $\mathcal{N}(0,1)$. Then these entries are standardized element-wise across $i$ to have mean 0 and standard deviation 1. The lower triangular entries are obtained by symmetry and the diagonal entries are set at 0. $B$ is block-diagonal $\{\mathbbm{1}_{8\times8}, -s\times\mathbbm{1}_{8\times8}, s\times\mathbbm{1}_{8\times8}, \mathbf{0}_{(p-24)\times(p-24)}\}$, where $p = 60$, $\mathbbm{1}_{8\times8}$ denotes an $8\times8$ matrix of ones, and $s = 2^k$ with $k\in \{-3,-2,\ldots,5\}$. The noise level is $\sigma=0.1$ and the number of observations is $n = 150$.
2. \[S2\] In this scenario, the $A_i$’s are obtained in the same manner as in \[S1\] except that we fix $s = 1$ and vary $n \in \{50, 100, \ldots, 300\}$.
3. \[S3\] The $A_i$’s are real functional connectivity matrices of 100 unrelated individuals from the Human Connectome Project (HCP) [@conn]. Data were preprocessed in FreeSurfer [@FreeS]. We removed the subcortical areas what resulted in a final number of $p = 148$ brain regions. As before, entries in $A_i$’s are standardized element-wise across $i$ before $y$ is generated. $B$ is block-diagonal $\{\mathbf{0}_{56\times56}, \mathbbm{1}_{6\times6}, \mathbf{0}_{6\times6}, -s\times\mathbbm{1}_{5\times5}, \mathbf{0}_{49\times49}, s\times\mathbbm{1}_{8\times8}, \mathbf{0}_{18\times18}\}$, $s = 2^k$ with $k\in \{-3,-2,\ldots,5\}$, and $\sigma = 0.1$.
For each setting in \[S1\] and \[S2\], the process of generating $A$ and $y$ is repeated $100$ times. For each setting in \[S3\], the process of generating $y$ is repeated $100$ times.
Simulation implementation {#subsec_SimulationImplementation}
-------------------------
For each simulation setting, we apply the following five regularization methods to estimate the matrix $B$: ${\textrm{SpINNEr}}$, elastic net, nuclear-norm regression, lasso, and ridge. ${\textrm{SpINNEr}}$ and elastic net both involve penalizing two types of norms, while the others use a single norm for the penalty term. The regularization parameters for these methods are chosen by five-fold cross-validation, where the fold membership of each observation is the same across methods.
For ${\textrm{SpINNEr}}$, we consider a $15\times 15$ two-dimensional grid of paired parameter values $(\lambda_L, \lambda_N)$. The smallest value for each coordinate is zero and the largest is the smallest value that produces $\hat{B}=0$ when the other coordinate is zero. The other 13 values for each coordinate are equally spaced on a logarithmic scale. The optimal $(\lambda_L^*, \lambda_N^*)$ is chosen as the pair that minimizes the average cross-validated squared prediction error. Elastic net regression requires the selection of two tuning parameters, $\alpha$ and $\lambda$. Their selection is implemented using the `MATLAB` package `glmnet` [@glmnet], in which we consider 15 equally-spaced (between 0 and 1) $\alpha$ values, where $\alpha = 0$ corresponds to ridge regression and $\alpha = 1$ corresponds to lasso regression. Then, for each $\alpha$, we let `glmnet` optimize over 15 automatically chosen $\lambda$ values, and pick the one that minimizes the average cross-validated squared prediction error. We treat nuclear-norm and lasso regression as special cases of ${\textrm{SpINNEr}}$ where one of the regularization parameters is set at 0, and the other one takes on the same 15 values as specified for the ${\textrm{SpINNEr}}$. The simplified versions of ADMM algorithm for these two special cases can be found in Appendix \[app:degenerate\]
. Ridge regression is implemented using `glmnet` where the optimization is done over 15 automatically chosen $\lambda$ values.
The default form of $W$ is used for ${\textrm{SpINNEr}}$ so that the diagonal elements of $B$ are not penalized. Since the connectivity matrices $A_i$ have zeros on their diagonals, the off-diagonal elements for the elastic net, lasso, and ridge regression are the same regardless of whether the diagonal is penalized and therefore we do not exclude diagonal elements from penalization (consequently, they are always estimated as zeros for these methods). As will be discussed in the following section, diagonal elements do not enter our evaluation criterion.
Simulation Results
------------------
We measure the performance of each method by the relative mean squared error between its estimator $\hat{B}$ and the true $B$, defined as $\text{MSEr} = \Vert \hat{B} - B\Vert_{F^{\star}}^2/\Vert B\Vert_{F^{\star}}^2$, where $\Vert\cdot\Vert_{F^{\star}}$ denotes the Frobenius norm of a matrix, excluding diagonal entries. In other words, $\Vert B\Vert_{F^{\star}}^2$ is the sum of squared off-diagonal entries of $B$.
### Scenario 1: Synthetic Connectivity Matrices with Varying Signal Strengths {#ssec:scenario1}
Figure \[fig\_Scenario1\] shows the relative mean squared errors of $\hat B$ for the five regularization methods as $\log_2(s) \in \{-3,-2,\ldots,5\}$ under \[S1\], where the nonzero entries in $B$ consist of blocks $\mathbbm{1}_{8\times8}$, $-s\times\mathbbm{1}_{8\times8}$, and $s\times\mathbbm{1}_{8\times8}$. We can see that ${\textrm{SpINNEr}}$ outperforms the other methods for all values of $s$ and produces a relative mean squared error much smaller than that of elastic net, lasso, and ridge. It is also observed that for ${\textrm{SpINNEr}}$, nuclear-norm regression, elastic net, and lasso, their relative mean squared errors are at the highest when $s = 1$. This can be explained as follows. When $s\ll 1$, the block $\mathbbm{1}_{8\times8}$ dominates the blocks $-s\times\mathbbm{1}_{8\times8}$ and $s\times\mathbbm{1}_{8\times8}$, making them more like noise terms so that effectively the number of response-relevant variables closer to 64, which is smaller than the number of observations $n = 150$. Similarly, when $s\gg 1$, the blocks $-s\times\mathbbm{1}_{8\times8}$ and $s\times\mathbbm{1}_{8\times8}$ dominate the block $\mathbbm{1}_{8\times8}$, effectively making the number of variables closer to 128, which is still smaller than 150. However, when $s = 1$, the total number of response-relevant variables is 192, which is larger than the number of observations, making the estimation of $B$ in this case more difficult.
[.325]{} ![Relative mean squared errors (MSEr) of estimators obtained from ${\textrm{SpINNEr}}$, elastic net (ElasNet), nuclear-norm regression (Nuclear), lasso, and ridge under different simulation scenarios. Each point represents the average MSEr over 100 replicates and error bars indicates $95\%$ confidence intervals. (a) MSEr against $\log_2(s)$ under simulation \[S1\]. (b) MSEr against sample size $n$ under simulation \[S2\]. (c) MSEr against $\log_2(s)$ under simulation \[S3\].[]{data-label="fig_SimulationResults"}](./scenario1.pdf "fig:"){width="1\linewidth"}
[.325]{} ![Relative mean squared errors (MSEr) of estimators obtained from ${\textrm{SpINNEr}}$, elastic net (ElasNet), nuclear-norm regression (Nuclear), lasso, and ridge under different simulation scenarios. Each point represents the average MSEr over 100 replicates and error bars indicates $95\%$ confidence intervals. (a) MSEr against $\log_2(s)$ under simulation \[S1\]. (b) MSEr against sample size $n$ under simulation \[S2\]. (c) MSEr against $\log_2(s)$ under simulation \[S3\].[]{data-label="fig_SimulationResults"}](./scenario2.pdf "fig:"){width="1\linewidth"}
[.325]{} ![Relative mean squared errors (MSEr) of estimators obtained from ${\textrm{SpINNEr}}$, elastic net (ElasNet), nuclear-norm regression (Nuclear), lasso, and ridge under different simulation scenarios. Each point represents the average MSEr over 100 replicates and error bars indicates $95\%$ confidence intervals. (a) MSEr against $\log_2(s)$ under simulation \[S1\]. (b) MSEr against sample size $n$ under simulation \[S2\]. (c) MSEr against $\log_2(s)$ under simulation \[S3\].[]{data-label="fig_SimulationResults"}](./scenario3.pdf "fig:"){width="1\linewidth"}
As $s$ increases to values greater than 1, ${\textrm{SpINNEr}}$ and nuclear-norm regression (the two methods that use a nuclear-norm penalty) exhibit substantial decrease in relative mean squared error. Elastic net and lasso, on the other hand, do not show a pronounced decrease. This demonstrates that when the true $B$ is both sparse and low-rank, and when the number of variables is comparable to the number of observations, encouraging sparsity alone is not sufficient for a regularized regression model to produce high estimation accuracy. Encouraging low-rank structure may be important. The behavior of ridge regression is different from the other four methods. As a shrinkage method that does not induce sparsity or low-rank structure, a ridge estimator’s MSEr varies little with $s$.
[.28]{} ![(b) True $B$ with $s = 8$ for simulation \[S1\]. Each of (a), (c), (d), (e), and (f) shows the estimated $\hat{B}$ from one simulation run for each of the five regularization methods. The same color bar scale is shared across all subfigures.[]{data-label="fig_SimulationBhats"}](./Bhat_E.pdf "fig:"){width="1\linewidth"}
[.28]{} ![(b) True $B$ with $s = 8$ for simulation \[S1\]. Each of (a), (c), (d), (e), and (f) shows the estimated $\hat{B}$ from one simulation run for each of the five regularization methods. The same color bar scale is shared across all subfigures.[]{data-label="fig_SimulationBhats"}](./B_true.pdf "fig:"){width="1\linewidth"}
[.28]{} ![(b) True $B$ with $s = 8$ for simulation \[S1\]. Each of (a), (c), (d), (e), and (f) shows the estimated $\hat{B}$ from one simulation run for each of the five regularization methods. The same color bar scale is shared across all subfigures.[]{data-label="fig_SimulationBhats"}](./Bhat_R.pdf "fig:"){width="1\linewidth"}
[.28]{} ![(b) True $B$ with $s = 8$ for simulation \[S1\]. Each of (a), (c), (d), (e), and (f) shows the estimated $\hat{B}$ from one simulation run for each of the five regularization methods. The same color bar scale is shared across all subfigures.[]{data-label="fig_SimulationBhats"}](./Bhat_L.pdf "fig:"){width="1\linewidth"}
[.28]{} ![(b) True $B$ with $s = 8$ for simulation \[S1\]. Each of (a), (c), (d), (e), and (f) shows the estimated $\hat{B}$ from one simulation run for each of the five regularization methods. The same color bar scale is shared across all subfigures.[]{data-label="fig_SimulationBhats"}](./Bhat_S.pdf "fig:"){width="1\linewidth"}
[.28]{} ![(b) True $B$ with $s = 8$ for simulation \[S1\]. Each of (a), (c), (d), (e), and (f) shows the estimated $\hat{B}$ from one simulation run for each of the five regularization methods. The same color bar scale is shared across all subfigures.[]{data-label="fig_SimulationBhats"}](./Bhat_N.pdf "fig:"){width="1\linewidth"}
In a more focused look at these regularization methods, Figure \[fig\_SimulationBhats\] displays the estimated $\hat{B}$ from a single simulation run—i.e., one representative set $\{y_i, A_i\}_{i=1}^n$—where the prescribed true $B$ consists of three signal-related blocks: $B_1 = \mathbbm{1}_{8\times8}$, $B_2 = -8\times\mathbbm{1}_{8\times8}$, and $B_3 = 8\times\mathbbm{1}_{8\times8}$, implying two positive and one negative RRC. Of these three blocks, $B_1$ is dominated by $B_2$ and $B_3$. We observe that while the elastic net estimate is sparse, and entries corresponding to these three blocks have the correct signs, the overall structure does not accurately recover the truth. For ridge regression, the estimate is neither sparse nor low-rank. For lasso, a relatively small tuning parameter was chosen by cross-validation and hence the estimate is not sparse, although the block structure of $\hat{B}_2$ and $\hat{B}_3$ is, to some extent, discernible. For nuclear-norm regression, while the blocks $\hat{B}_2$ and $\hat{B}_3$ are more pronounced, many entries outside these blocks are nonzero, especially along the rows and columns of $\hat{B}_2$ and $\hat{B}_3$. Conversely, ${\textrm{SpINNEr}}$ recovers $B_2$ and $B_3$ effectively, and although a few nonzero entries outside the three blocks are estimated to be non-zero, their magnitudes are small, hence producing an estimate having the smallest MSEr among the five methods. This example demonstrates how the simultaneous combination of low rank and sparsity penalization can recover this structure accurately while applying each penalty separately fails to do so.
### Scenario 2: Synthetic Connectivity Matrices with Varying Sample Sizes {#ssec:scenario2}
The results of simulation \[S1\] show that when the sample size is fixed at 150, the MSEr for ${\textrm{SpINNEr}}$ and nuclear-norm regression decreases substantially for $s>1$, while MSEr for all other models changes minimally, even for $s=128$.
Figure \[fig\_Scenario2\] display the MSEr for $\hat B$ from each of the five estimates for $n \in \{50, 100, \ldots, 300\}$ and $s=1$ (\[S2\]). The nonzero entries in $B$ consist of blocks $\mathbbm{1}_{8\times8}$, $-\mathbbm{1}_{8\times8}$, and $\mathbbm{1}_{8\times8}$, resulting in 3 RRC (each having 8 nodes) and 192 individual response-relevant variables. We can see that for all five methods, MSEr decreases with sample size, suggesting that each of these methods benefits from more information. However, the MSEr with ${\textrm{SpINNEr}}$ and nuclear-norm regression decreases much faster than for the other methods which do not involve a nuclear-norm penalty. More specifically, for elastic net and lasso, their when there are 300 observations are about the same as those from ${\textrm{SpINNEr}}$ and nuclear-norm regression when there are only 150 observations.
Further, as seen in \[S1\], it is the simultaneous combination of the nuclear norm and $\ell 1$ penalties that is most effective. In particular, although the decrease of MSEr for nuclear-norm regression appears to be at a rate that does not change much with $n$, the decrease for ${\textrm{SpINNEr}}$ from $n = 150$ to $n = 200$ is much more substantial than the decrease from $n = 100$ to $n = 150$, suggesting that once the sample size exceeds the number of variables (192 in this case) ${\textrm{SpINNEr}}$ may exhibit a leap in estimation accuracy. Moreover, as the sample size increases beyond 250, the relative mean squared error from ${\textrm{SpINNEr}}$ is nearly zero, while more than 300 observations for nuclear-norm regression to approach zero (when $n = 300$, MSEr is still approximately 0.2).
### Scenario 3: Real Connectivity Matrices with Varying Signal Strength {#ssec:scenario3}
Figure \[fig\_Scenario3\] shows MSEr from the five regularization methods under \[S3\], where the matrices $\{A_i\}_{i=1}^n$ represent functional connectivity among brain regions estimated from $n=100$ humans. To simulate signal, we once again considered 3 RRC by assigning nonzero entries in diagonal blocks of $B$, $\mathbbm{1}_{6\times6}$, $-s\times\mathbbm{1}_{5\times5}$, and $s\times\mathbbm{1}_{8\times8}$. In this scenario, ${\textrm{SpINNEr}}$ has lower MSEr than all other methods across all values of $s$. As in \[S1\], the MSEr for ${\textrm{SpINNEr}}$ and lasso are at their highest when $s = 1$, which gives $125$ response-relevant variables in a sample size of $n=100$.
When $s>1$, while the relative mean squared error for ${\textrm{SpINNEr}}$ decreases with $s$, the error curves for the other methods are relatively flat. This is similar to the results in Figure \[fig\_Scenario1\], except for the nuclear-norm regularization. A closer examination of the $\hat{B}$ from nuclear-norm regularization under \[S1\] (see Figure \[fig\_SimulationBhats\](f)) and \[S3\] (not shown) reveals that although the solutions are not sparse, the estimated blocks for RRCs are more pronounced under \[S1\] than under \[S3\], most likely because the response-relevant entries constitute a larger fraction of the true $B$ under \[S1\]. Finally, we note that the error bars in Figure \[fig\_Scenario3\] are narrower than in Figure \[fig\_Scenario1\] because under \[S1\], different synthetic connectivity matrices are generated across replicates, while under \[S3\], where we use real functional connectivity matrices, only the response values in $y$ differ across replicates.
In summary, all simulation scenarios examined here demonstrate that ${\textrm{SpINNEr}}$ significantly outperforms elastic net, nuclear-norm regression, lasso, and ridge in terms of MSEr.
Application in Brain Imaging {#sec_BrainDataResults}
============================
Here we report on the results of ${\textrm{SpINNEr}}$ as applied to a real brain imaging data set. The goal is to estimate the association of functional connectivity with neuropsychological (NP) language test scores in a cohort $n=116$ HIV-infected males. The clinical characteristics of this cohort are summarized in Table \[rda\_demographic\_data\_summary\].
[@ccccccccccc]{}\
\
**Characteristic** && && && && &&\
\
Age && 20 && 51 && 74 && 46.5 && 14.8\
Recent VL && 20 && 20 && 288000 && 9228 && 38921\
Nadir CD4 && 0 && 193 && 690 && 219.5 && 171\
Recent CD4 && 20 && 536 && 1354 && 559.1 && 286.5\
\
For each participant, their estimated resting state functional connectivity matrix, $A_i$, and age, $X_i$, are included in the regression model. Each functional connectivity matrix, $A_i$ was constructed according to the Destrieux atlas (aparc.a2009s) [@Destrieux], which defines $p=148$ cortical brain regions. The response variable, $y$, is defined as the mean of two word-fluency test scores: the *Controlled Oral Word Association Test-FAS* and the *Animal Naming Test*.
We hypothesize that brain connectivity is associated with $y$ via a subset of the $148\times 148$ brain region connectivity values. As in this is modeled as $$\label{modelRDA}
y_i\ =\ \langle A_i, B \rangle \, +\, [1\, X_i]\left [\begin{BMAT}(c)[0.5pt,0pt,0.7cm]{c}{cc}\beta_1 \\ \beta_2 \end{BMAT} \right ]\, +\, \varepsilon_i, \quad i=1,\ldots, n,\qquad \textrm{for}\quad \varepsilon_i\sim \mathcal{N}\big(0, \sigma^2\big).$$
The ${\textrm{SpINNEr}}$ estimate of $B$ comes from tuning parameters $\lambda_N = 12.1$ and $\lambda_L=2.4$. These were selected by the 5-fold cross-validation from $225$ grid points; i.e. all pairwise combinations of $15$ values of $\lambda_N$ and $15$ values $\lambda_L$ (see, subsection \[subsec\_SimulationImplementation\]). The connectivity matrices, the covariate of age and the response variable were all standardized across subjects before performing cross-validation. We attempted to fit both the lasso and nuclear-norm estimates by applying one-dimensional cross validation, but in each case the estimated $B$ matrices contained all zeros. Therefore, for displaying these latter estimates, we used the marginal tuning parameter values from the two-dimensional ${\textrm{SpINNEr}}$ cross validation.
Figure \[Results\] shows the three matrix regression estimates. The estimate from ${\textrm{SpINNEr}}$ is in Figure \[Results\](b) and is flanked by estimates from the lasso (with $\lambda_L=2.4$; Figure \[Results\](a)) and the nuclear-norm penalty ($\lambda_N = 12.1$; Figure \[Results\](c)). We also marked 7 main brain networks which were extracted and labeled in [@Yeo2011] (known as *Yeo seven-network parcellation*).
[.325]{} ![(b) presents ${\textrm{SpINNEr}}$ estimate with $\lambda_N = 12.1$ and $\lambda_L = 2.4$ selected via 5-fold cross-validation, (a) shows the lasso solution with $\lambda_L = 2.4$ and (c) corresponds to nuclear-norm solution with $\lambda_N = 12.1$. The black boxes show the Yeo’s parcellation into seven main brain networks. Nodes were permuted inside boxes in order to reveal the clusters.[]{data-label="Results"}](./RDA_HIV_RSN7_LASSO.pdf "fig:"){width="1\linewidth"}
[.325]{} ![(b) presents ${\textrm{SpINNEr}}$ estimate with $\lambda_N = 12.1$ and $\lambda_L = 2.4$ selected via 5-fold cross-validation, (a) shows the lasso solution with $\lambda_L = 2.4$ and (c) corresponds to nuclear-norm solution with $\lambda_N = 12.1$. The black boxes show the Yeo’s parcellation into seven main brain networks. Nodes were permuted inside boxes in order to reveal the clusters.[]{data-label="Results"}](./RDA_HIV_RSN7.pdf "fig:"){width="1\linewidth"}
[.325]{} ![(b) presents ${\textrm{SpINNEr}}$ estimate with $\lambda_N = 12.1$ and $\lambda_L = 2.4$ selected via 5-fold cross-validation, (a) shows the lasso solution with $\lambda_L = 2.4$ and (c) corresponds to nuclear-norm solution with $\lambda_N = 12.1$. The black boxes show the Yeo’s parcellation into seven main brain networks. Nodes were permuted inside boxes in order to reveal the clusters.[]{data-label="Results"}](./RDA_HIV_RSN7_NUCLEAR.pdf "fig:"){width="1\linewidth"}
The graph in Figure \[graphFigure\](b) reveals a very specific structure of the estimated associations. This structure is based on five brain regions which comprise the boundary between positive and negative groups of edges.
[.42]{} ![${\textrm{SpINNEr}}$ estimate restricted to 40 brain regions for which some response-related connectivities were found. (a) presents estimate after permuting nodes to achieve the cluster-by-cluster order. This was done based on the largest coefficients magnitudes of first a few left and right-singular vectors from singular value decomposition of $\hat{B}$, which indicate clusters indices. The corresponding graph representation is shown in (b).[]{data-label="graphFigure"}](./ImportantManualOrder.pdf "fig:"){width="1\linewidth"}
[.45]{} ![${\textrm{SpINNEr}}$ estimate restricted to 40 brain regions for which some response-related connectivities were found. (a) presents estimate after permuting nodes to achieve the cluster-by-cluster order. This was done based on the largest coefficients magnitudes of first a few left and right-singular vectors from singular value decomposition of $\hat{B}$, which indicate clusters indices. The corresponding graph representation is shown in (b).[]{data-label="graphFigure"}](./GraphPlot2.pdf "fig:"){width="1\linewidth"}
These five regions are spread across the brain from the frontal lobe (left and right suborbital sulcus, Ss\[L\] and Ss\[R\]) to the area located by the corpus callosum (left and right posterior-dorsal part of the cingulate gyrus, Gcp-d\[L\], Gcp-d\[R\]), and up to the medial part of the parietal lobe (left precuneus gyrus, Gp\[L\]). They span two response-related groups of brain regions. First with conductivities having negative associations with the response is represented by left orbital H-shaped sulci (Sos\[L\]), left and right gyrus rectus (Gr\[L\], Gr\[R\]) and left medial orbital sulcus, Som-o\[L\]. The second, showing the positive associations, contains left superior occipital sulcus and transverse occipital sulcus, Sst\[L\], right middle frontal sulcus (Sfm\[R\]), superior temporal sulcus, Sts\[R\], right vertical ramus of the anterior segment of the lateral sulcus Lfv\[R\] and right middle occipital gyrus, Gom\[R\]. Interestingly, there is also the third response-relevant group of brain regions clearly visible in Figure \[graphFigure\](b). It has a different structure than two aforementioned groups and forms star-shaped subgraph of negative effects, with the center in the occipital pole, Po\[L\]. Brain networks were visualized in Figure \[brainView\] by using BrainNet viewer (http://www.nitrc.org/projects/bnv/).
[.3]{} ![The brain network visualization of response-relevant connectivities found by ${\textrm{SpINNEr}}$. Different views on edges corresponding to positive entries of $\hat{B}^{{\textnormal{\tiny {S}}}}$ (PE) are presented in (a)–(c). Negative $\hat{B}^{{\textnormal{\tiny {S}}}}$ entries (NE) are shown in (d)–(f).[]{data-label="brainView"}](./SagittalP2-min.jpg "fig:"){width="1\linewidth"}
[.2]{} ![The brain network visualization of response-relevant connectivities found by ${\textrm{SpINNEr}}$. Different views on edges corresponding to positive entries of $\hat{B}^{{\textnormal{\tiny {S}}}}$ (PE) are presented in (a)–(c). Negative $\hat{B}^{{\textnormal{\tiny {S}}}}$ entries (NE) are shown in (d)–(f).[]{data-label="brainView"}](./AxialP-min.jpg "fig:"){width="1\linewidth"}
[.31]{} ![The brain network visualization of response-relevant connectivities found by ${\textrm{SpINNEr}}$. Different views on edges corresponding to positive entries of $\hat{B}^{{\textnormal{\tiny {S}}}}$ (PE) are presented in (a)–(c). Negative $\hat{B}^{{\textnormal{\tiny {S}}}}$ entries (NE) are shown in (d)–(f).[]{data-label="brainView"}](./CoronalP-min.jpg "fig:"){width="1\linewidth"}
[.3]{} ![The brain network visualization of response-relevant connectivities found by ${\textrm{SpINNEr}}$. Different views on edges corresponding to positive entries of $\hat{B}^{{\textnormal{\tiny {S}}}}$ (PE) are presented in (a)–(c). Negative $\hat{B}^{{\textnormal{\tiny {S}}}}$ entries (NE) are shown in (d)–(f).[]{data-label="brainView"}](./SagittalN2-min.jpg "fig:"){width="1\linewidth"}
[.2]{} ![The brain network visualization of response-relevant connectivities found by ${\textrm{SpINNEr}}$. Different views on edges corresponding to positive entries of $\hat{B}^{{\textnormal{\tiny {S}}}}$ (PE) are presented in (a)–(c). Negative $\hat{B}^{{\textnormal{\tiny {S}}}}$ entries (NE) are shown in (d)–(f).[]{data-label="brainView"}](./AxialN-min.jpg "fig:"){width="1\linewidth"}
[.31]{} ![The brain network visualization of response-relevant connectivities found by ${\textrm{SpINNEr}}$. Different views on edges corresponding to positive entries of $\hat{B}^{{\textnormal{\tiny {S}}}}$ (PE) are presented in (a)–(c). Negative $\hat{B}^{{\textnormal{\tiny {S}}}}$ entries (NE) are shown in (d)–(f).[]{data-label="brainView"}](./CoronalN-min.jpg "fig:"){width="1\linewidth"}
Discussion {#sec_Discussion}
==========
We have proposed a novel way of estimating the regression coefficients for the scalar-on-matrix regression problem and derived theoretical properties of the estimator, which takes the form of a matrix. One of the primary contributions of this work is that it provides an accurate estimation of this matrix via a combination of two penalty terms: a nuclear norm and a $\ell_1$ norm. This approach may be viewed as an extension of both low-rank and sparse regression estimation approximations resulting in matrix estimates dominated by blocks structure. Advantages of our approach include: the estimation of meaningful, connected-graph regression coefficient structure; a computationally efficient algorithm via ADMM; and the ability to choose optimal tuning parameters.
In our simulation studies, in Section \[sec\_SimulationExperiments\] the first scenario illustrates the advantages of ${\textrm{SpINNEr}}$ over several competing methods with respect to varying signal strengths. The second simulation scenario shows the performance of ${\textrm{SpINNEr}}$ across a range of sample sizes. The third scenario shows how ${\textrm{SpINNEr}}$ behaves when the data come from real structural connectivity matrices. In each case, ${\textrm{SpINNEr}}$ outperforms all other methods considered.
Finally, we applied ${\textrm{SpINNEr}}$ to an actual study of HIV-infected participants which aimed to understand the association of a language-domain outcome with functional connectivity. The estimated regression coefficients matrix revealed three response-related clusters of brain regions — the smaller one dominated by a star-shaped structure of positive effects and two larger clumps (one negative and one positive) which shared 5 common brain regions. From the perspective of the RRCs recovery — the notion which we introduce — this can be treated as finding the overlapping clusters. However, ${\textrm{SpINNEr}}$ can be also used to reveal much more complex structures than block diagonal matrices, which constitute a kind of the “model signals” for us. The class of sparse and low-rank signals includes also the matrices having symmetric, non-diagonal blocks, which may be important in some applications, like correlation matrices recovery.
Speed and stability are important considerations for the implementation of any complex estimation method. We have implemented the ${\textrm{SpINNEr}}$ estimation process using the ADMM algorithm by dividing the original optimization problem (for given tuning parameters $\lambda_N$ and $\lambda_L$) into three subproblems, deriving their analytical solutions and computing them iteratively until the convergence. For each such iteration, our implementation precisely selects the step sizes based on the idea of residual balancing which turns out to work very fast and stable in practice. The final solution is obtained after cross-validation applied for the optimal selection of tuning parameters.
We note that the weights matrix $W$ must be prespecified at the beginning of the ${\textrm{SpINNEr}}$ algorithm. The default setting (which we always used in this article) is a matrix of zeros on its diagonal and ones on off-diagonal entries. However, our implementation allows for an arbitrary choice of nonnegative weights. One may consider the selection based on the external information, if such is available, imposing weaker penalties for the entries being already reported as response-relevant in the particular application. The other possible strategy is an adaptive construction of $W$. It may rely on using the default $W$ first and update it, based on ${\textrm{SpINNEr}}$ estimate, so as the large magnitude of $\hat{B}^{{\textnormal{\tiny {S}}}}_{j,l}$ generates small value of $W_{j,l}$. This procedure emphasizes the findings and can potentially improve variable selection accuracy.
In the future we want to perform a valid inference on the estimated clusters. Additionally, as developed here, ${\textrm{SpINNEr}}$ addresses scalar-on-matrix regression models involving a continuous response. However, binary and count responses are often of interest. Indeed, an important problem that arises in studies of HIV-infected individuals is that of understanding the association of (binary) impairment status and neuro-connectivity. These more general settings will motivate future work in the estimation problem for scalar-on-matrix regression.
Declaration of interest {#declaration-of-interest .unnumbered}
=======================
The authors confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.
Acknowledgements {#acknowledgements .unnumbered}
================
DB, JH, TWR and JG were partially supported by the NIMH grant R01MH108467. D.B. was also funded by Wroclaw University of Science and Technology resources (8201003902, MPK: 9130730000). Data were provided \[in part\] by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University.
Appendix {#appendix .unnumbered}
========
Subproblems in ADMM algorithm {#subs:subproblems}
=============================
We use the letter $\mathcal{A}$ to denote the $n\times p^2$ matrix that collects the vectorized matrices, $A_1,\ldots, A_n$, in rows: $\mathcal{A}:= \big [{\operatorname{vec}}(A_1)|\ldots| {\operatorname{vec}}(A_n)\big ]{^\mathsf{T}}$. The submatrix of $\mathcal{A}$ built from columns that correspond to the upper-diagonal entries of matrices $A_i$s (without diagonal entries) is denoted as $\mathcal{A}_U$. Accordingly, the columns of $\mathcal{A}$ that correspond to the symmetric entries from the lower-diagonal part is denoted by $\mathcal{A}_L$. Our implementation is derived under the assumption that all the matrices $A_i$s are symmetric and have zeros on their diagonals. Therefore, $\mathcal{A}_U = \mathcal{A}_L$.
Analytical solution to
-----------------------
The considered update is $$\label{appendix1}
B^{[k+1]}:=\ {\underset{B}{\operatorname{argmin}}\;}\bigg\{\,\sum_{i=1}^n\Big(y_i -\langle A_i, B \rangle\Big)^2 \ +\ \delta^{[k]}_1\Big\|\, B - D^{[k]} - \frac{{Z}_1^{[k]}}{\delta^{[k]}_1}\,\Big\|_F^2\,\bigg\}.$$ We introduce a variable $\widetilde{B}:= B - D^{[k]} - \frac{{Z}_1^{[k]}}{\delta^{[k]}_1}$, which gives $B = \widetilde{B} + D^{[k]} + \frac{{Z}_1^{[k]}}{\delta^{[k]}_1}$. This yields the equivalent problem $$\label{eqRidge}
\widetilde{B}^*=\ {\underset{\widetilde{B}}{\operatorname{argmin}}\;}\bigg\{\,\sum_{i=1}^n\Big(\underbrace{y_i -\langle A_i, D^{[k]} + \frac{{Z}_1^{[k]}}{\delta^{[k]}_1}\rangle}_{\widetilde{y}_i} - \langle A_i, \widetilde{B} \rangle\Big)^2 \ +\ \delta^{[k]}_1\big\|\,\widetilde{B}\,\big\|_F^2\,\bigg\}$$ In terms of the stacked vectorized matrices in $\mathcal{A}$, takes the form of ridge regression with the objective $$\begin{split}
\big\|\widetilde{y} -& \mathcal{A}{\operatorname{vec}}(\widetilde{B})\big\|_2^2\ +\ \delta^{[k]}_1\big\|{\operatorname{vec}}(\widetilde{B})\big\|_2^2 =\\
&\big\|\widetilde{y} - \mathcal{A}_L{\operatorname{vec}}_L(\widetilde{B}) - \mathcal{A}_U{\operatorname{vec}}_U(\widetilde{B})\big\|_2^2\ +\ \delta^{[k]}_1\Big(\,\big\|{\operatorname{vec}}_L(\widetilde{B})\big\|_2^2\ +\ \big\|{\operatorname{vec}}_U(\widetilde{B})\big\|_2^2\ +\ \sum_{i=1}^p\widetilde{B}_{i,i}^{\,2}\,\Big),
\end{split}$$ where ${\operatorname{vec}}_U(\widetilde{B})$ and ${\operatorname{vec}}_L(\widetilde{B})$ are the vectors obtained from the upper and lower diagonal elements of $\widetilde{B}$, respectively.
\[prop2\] Suppose that all the matrices $A_i$’s have zeros on the diagonals and consider the minimization problem with $\delta^{[k]}_1>0$. Then, $\widetilde{B}^*$ has zeros on the diagonal.
Suppose that $\widetilde{B}^*_{k,k}\neq 0$ for some $k\in\{1,\ldots,p\}$ and construct matrix $\overline{B}$ by setting $\overline{B}_{k,k}:=0$ and $\overline{B}_{i,j}:=\widetilde{B}^*_{i,j}$ for $(i,j)\neq (k,k)$. Obviously, we have that $\langle A_i, \widetilde{B}^* \rangle = \langle A_i, \overline{B} \rangle$ for each $i$. Denoting the objective function in by $F$, we therefore get $$\begin{split}
F(\widetilde{B}^*) - F(&\overline{B})\ = \ \delta^{[k]}_1\big\|\,\widetilde{B}^*\,\big\|_F^2 - \delta^{[k]}_1\big\|\,\overline{B}\,\big\|_F^2 =\\
&\underbrace{\delta^{[k]}_1\sum_{(i,j)\neq(k,k)}(\widetilde{B}^*_{i,j})^2\, -\, \delta^{[k]}_1\sum_{(i,j)\neq(k,k)}\big(\overline{B}_{i,j}\big)^2}_{=\,0}\ +\ \delta^{[k]}_1\big(\widetilde{B}^*_{k,k}\big)^2\ >\ 0.
\end{split}$$ Consequently $F(\overline{B}) < F(\widetilde{B}^*)$, which contradicts the optimality of $\widetilde{B}^*$.
Proposition \[prop2\] together with Proposition \[prop1\] imply that $\widetilde{B}^*$ is a symmetric matrix with zeros on the diagonal, which allows us to confine the minimization problem by the conditions ${\operatorname{vec}}_L(\widetilde{B}) = {\operatorname{vec}}_U(\widetilde{B}): = c$ and $\widetilde{B}_{i,i}=0$. This yields $$\label{eqRidge2}
{\operatorname{vec}}_U(\widetilde{B}^*)=\ {\underset{c\in\mathbb{R}^{(p^2-p)/2}}{\operatorname{argmin}}\;}\bigg\{\,\big\|\widetilde{y} - 2\mathcal{A}_Uc\big\|_2^2\ +\ 2\delta^{[k]}_1\|c\|_2^2\bigg\}.$$ In summary, it suffices to solve the ridge regression problem to obtain $\widetilde{B}^*$ and then recover $B^{[k+1]}$ by setting $B^{[k+1]} = \widetilde{B}^* + D^{[k]} + \frac{{Z}_1^{[k]}}{\delta^{[k]}_1}$.
Now assume the (reduced) singular value decomposition (SVD) of $2\mathcal{A}_U$ is given; i.e., write $2\mathcal{A}_U = U\operatorname{diag}(d_1,\ldots, d_n)V{^\mathsf{T}}$, where $U\in \mathbb{R}^{n\times n}$ is an orthogonal matrix, $V\in \mathbb{R}^{(p^2-p)/2\times n}$ has orthogonal columns, and $d_1,\ldots, d_n$ are the $n$ singular values. The solution to can then be obtained as $$\label{ridgeSol}
{\operatorname{vec}}_U(\widetilde{B}^*)\ =\ 2\Big(4\mathcal{A}_U{^\mathsf{T}}\mathcal{A}_U + 2\delta^{[k]}_1\mathbf{I}\Big)^{-1}\mathcal{A}_U{^\mathsf{T}}\widetilde{y}\ =\ V\left(\left [\begin{BMAT}(c)[0.5pt,0pt,0.7cm]{c}{ccc}d_1/\big(d_1^2+2\delta^{[k]}_1\big) \\\vdots \\d_n/\big(d_n^2+2\delta^{[k]}_1\big) \end{BMAT} \right ]\circ \big[U{^\mathsf{T}}\widetilde{y}\big]\right),$$ where “$\circ$” denotes a *Hadamard product* (i.e., an entry-wise product of matrices). It is worth noting that the SVD of $\mathcal{A}_U$ need only be computed once, at the beginning of the numerical solver, since the left and right singular vectors, as well as the singular values, do not depend on the current iteration, nor do they depend on the regularization parameter in Therefore they can be used for the entire grid of regularization parameters in the ${\textrm{SpINNEr}}$ process. This significantly speeds up the computation.
Analytical solution to
-----------------------
We start with $$C^{[k+1]}:=\ {\underset{C}{\operatorname{argmin}}\;}\bigg\{\,\frac12\Big\|\,\underbrace{D^{[k]} + \frac{{Z}_2^{[k]}}{\delta^{[k]}_2}}_{M^{[k]}}\,-\, C\,\Big\|_F^2\ +\ \frac{\lambda_N}{\delta^{[k]}_2}\|C\|_*\,\bigg\}.$$ To construct a fast algorithm for finding the solution, we use the following well known result.
\[propOrt\] For any matrix $M$ with singular value decomposition $M = U\operatorname{diag}(s)V{^\mathsf{T}}$, the optimal solution to $${\underset{C}{\operatorname{argmin}}\;} \Big\{\,\frac12\|C - M\|_F^2\ +\ \lambda \|C\|_*\,\Big\}$$ shares the same singular vectors as $M$ and its singular values are $s_i^*= (s_i - \lambda)_+:=\max\{s_i - \lambda, 0\}$.
Now, let $M^{[k]} = U^{[k]}\operatorname{diag}(s^{[k]}){V^{[k]}}{^\mathsf{T}}$ be the SVD of $M^{[k]}$. Thanks to Proposition \[propOrt\], we can recover $C^{[k+1]}$ in two steps $$\left\{
\begin{array}{l}
S^*:= \operatorname{diag}\Big(\,\big[\ (s^{[k]}_1 - \frac{\lambda_N}{\delta^{[k]}_2})_+\ ,\ldots,\ (s^{[k]}_p - \frac{\lambda_N}{\delta^{[k]}_2})_+\,\big]{^\mathsf{T}}\,\Big)\\
C^{[k+1]} = U^{[k]}S^*{V^{[k]}}{^\mathsf{T}}
\end{array}
\right..$$
Analytical solution to
-----------------------
We use the following result.
Let $D$, $K$ and $L$ be matrices with matching dimensions. Then, $$\delta_1\big\|D -K\big\|_F^2\ +\ \delta_2\big\|D -L\big\|_F^2=(\delta_1+\delta_2)\bigg\|\,D -\frac{\delta_1K+\delta_2L}{\delta_1 + \delta_2}\,\bigg\|_F^2\ +\ \varphi(K, L, \delta_1, \delta_2),$$ where $\varphi(K, L, \delta_1, \delta_2)$ does not depend on $D$.
Simply observe that $$\begin{split}
&\nabla_D \bigg\{\delta_1\big\|D -K\big\|_F^2\ +\ \delta_2\big\|D -L\big\|_F^2 - (\delta_1+\delta_2)\Big\|D -\big(\delta_1K+\delta_2L\big)/\big(\delta_1+\delta_2\big)\Big\|_F^2\bigg\} =\\
&2\delta_1(D-K) + 2\delta_2(D-L) - 2(\delta_1+\delta_2)\Big[D - \big(\delta_1K+\delta_2L\big)/\big(\delta_1+\delta_2\big)\Big]\ =\\
&2\delta_1(D-K) + 2\delta_2(D-L) - 2(\delta_1+\delta_2)D + 2\big(\delta_1K+\delta_2L\big) =\ 0.
\end{split}$$ This proves the claim.
Denote $\Delta^{[k]}: = \delta_1^{[k]} + \delta_2^{[k]}$. The above proposition reduces problem to lasso regression under an orthogonal design matrix $$D^{[k+1]}=\ {\underset{D}{\operatorname{argmin}}\;}\bigg\{\,\frac12\bigg\|\,\underbrace{\big(\delta_1^{[k]}B^{[k+1]} + \delta_2^{[k]}C^{[k+1]}-{Z}_1^{[k]} - {Z}_2^{[k]}\big)/\Delta^{[k]}}_{Q^{[k+1]}}\,-\,D\,\bigg\|_F^2\ +\ \frac{\lambda_L}{\Delta^{[k]}}\Big\|{\operatorname{vec}}(W \circ D)\Big\|_1\,\bigg\}.$$ The closed-form solution in this situation is well known and can be formulated simply as $$D^{[k+1]}_{ij}= \operatorname{sgn}\Big(Q^{[k+1]}_{ij}\Big)\cdot\bigg(\big|Q^{[k+1]}_{ij}\big|\, -\, \frac{\lambda_LW_{ij}}{\Delta^{[k]}}\bigg)_+,\quad\textrm{for}\quad i,j\in\{1,\ldots,p\}.$$
Degenerate situations {#app:degenerate}
=====================
The case with $\lambda_L=0$
---------------------------
We consider the problem with $\lambda_L=0$. We introduce a new variable, i.e., a $p\times p$ matrix $C$, to create the (equivalent) constrained version of the problem with separable objective function: $${\underset{B, C}{\operatorname{argmin}}\;}\ \big\{f(B)\,+\,g(C)\big\} \qquad \textrm{s. t.}\ \ \ C-B=0.$$ The augmented Lagrangian with scalar $\delta>0$ and dual variable, ${Z}\in \mathbb{R}^{p\times p}$, for this problem is $$L_{\delta}(B,C; {Z}) =f(B)+g(C)+\langle{Z},\, C-B\rangle + \frac{\delta}2\big\|C-B\big\|_F^2,$$ and the ADMM updates for this case take the form $$\begin{aligned}
&B^{[k+1]}:=\ {\underset{B}{\operatorname{argmin}}\;}\bigg\{\,\sum_{i=1}^n\Big(y_i -\langle A_i, B \rangle\Big)^2\ +\ \delta^{[k]}_1\Big\|\,C^{[k]} + \frac{{Z}^{[k]}}{\delta^{[k]}_1}- B\,\Big\|_F^2\,\bigg\},\label{Update1nuc}\\
&C^{[k+1]}:=\ {\underset{C}{\operatorname{argmin}}\;}\bigg\{\,\frac12\Big\|\,B^{[k+1]}- \frac{{Z}^{[k]}}{\delta^{[k]}_1}-C\,\Big\|_F^2\ +\ \frac{\lambda_N}{\delta^{[k]}_1}\big\|C\big\|_*\,\bigg\},\label{Update2nuc}\\
&{Z}^{[k+1]}: =\ {Z}^{[k]} + \delta^{[k]}_1\big(C^{[k+1]}-B^{[k+1]}\big).\end{aligned}$$ For $\lambda_L=0$ the criterion reduces to the method described by Zhou and Li in [@Zhou-regularized-2014].
The case with $\lambda_N=0$
---------------------------
We consider the problem with $\lambda_N=0$. We again introduce a new variable, i.e., a $p\times p$ matrix $D$, to create the (equivalent) constrained version of the problem with separable objective function: $${\underset{B, D}{\operatorname{argmin}}\;}\ \big\{f(B)\,+\,h(D)\big\} \qquad \textrm{s. t.}\ \ \ D-B=0.$$ The augmented Lagrangian with scalar $\delta>0$ and dual variable, ${Z}\in \mathbb{R}^{p\times p}$, for this problem is $$L_{\delta}(B,C; {Z}) =f(B)+h(D)+\langle{Z}, D-B\rangle + \frac{\delta}2\big\|D-B\big\|_F^2,$$ and the ADMM updates for this case take the form $$\begin{aligned}
&B^{[k+1]}:=\ {\underset{B}{\operatorname{argmin}}\;}\bigg\{\,\sum_{i=1}^n\Big(y_i -\langle A_i, B \rangle\Big)^2\ +\ \delta^{[k]}_2\Big\|\,D^{[k]} + \frac{{Z}^{[k]}}{\delta^{[k]}_2}- B\,\Big\|_F^2\,\bigg\},\label{Update1nuc}\\
&D^{[k+1]}:=\ {\underset{D}{\operatorname{argmin}}\;}\bigg\{\,\frac12\Big\|\,B^{[k+1]}- \frac{{Z}^{[k]}}{\delta^{[k]}_2}-D\,\Big\|_F^2\ +\ \frac{\lambda_L}{\delta^{[k]}_2}\big\|{\operatorname{vec}}(W\circ D)\big\|_1\,\bigg\},\label{Update2nuc}\\
&{Z}^{[k+1]}: =\ {Z}^{[k]} + \delta^{[k]}_2\big(D^{[k+1]}-B^{[k+1]}\big).\end{aligned}$$ For $\lambda_N=0$ the criterion reduces to the lasso.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We show that if the propagating speed of gravitational waves (GWs) gradually diminishes during inflation, the power spectrum of primordial GWs will be strongly blue, while that of the primordial scalar perturbation may be unaffected. We also illustrate that such a scenario is actually a disformal dual to the superinflation, but it does not have the ghost instability. The blue tilt obtained is $0<n_T\lesssim 1$, which may significantly boost the stochastic GWs background at the frequency band of Advanced LIGO/Virgo, as well as the space-based detectors.'
author:
- 'Yong Cai$^{1}$[^1]'
- 'Yu-Tong Wang$^{1}$[^2]'
- 'Yun-Song Piao$^{1}$[^3]'
title: Propagating speed of primordial gravitational waves and inflation
---
Introduction
============
Recently, the LIGO Scientific Collaboration has observed a transient gravitational wave (GWs) signal with a significance in excess of 5.1$\sigma$ [@Abbott:2016blz], which is consistent with an event of the binary black hole coalescence. This discovery will be a scientific milestone for understanding our universe, if it is confirmed.
It is speculated that the stochastic GWs background contributed by the incoherent superposition of all merging binaries in the universe might be higher than expected previously [@TheLIGOScientific:2016wyq], which is potentially measurable around $25$Hz by the Advanced LIGO/Virgo detectors operating at their projected final sensitivity. However, some cosmological sources may also contribute a stochastic background of GWs at the corresponding frequency band, such as cosmic strings [@Damour:2000wa] and cosmological phase transitions [@Kamionkowski:1993fg][@Dev:2016feu].
It is well known that the standard slow-roll inflation predicts a nearly flat spectrum of scalar perturbation, as well as primordial GWs [@Starobinsky:1979ty][@Rubakov:1982]. Recently, the BICEP2/Keck data, combined with the Planck data and the WMAP data, have put the constraint $r<0.09$ (95% C.L.) [@Array:2015xqh] on the amplitude of primordial GWs on large scale, or at ultra-low frequency, which corresponds to $\Omega_{gw}\sim 10^{-15}$, but there is no strong limit for its tilt $n_T$. Actually, as long as its spectrum is blue enough, the stochastic GWs background from primordial inflation is also not negligible at the frequency band of Advanced LIGO/Virgo.
The slow-roll inflation model with $\epsilon = -{\dot H}/H^2\ll 1$ generally has $n_T=-2\epsilon < 0$. Thus $n_T>0$ requires either the superinflation [@Piao:2004tq][@Baldi:2005gk], also [@Liu:2013iha][@Cai:2015yza], which breaks the null energy condition (NEC), or an anisotropic stress source during inflation, e.g., the particle production [@Cook:2011sp][@Sorbo:2011ja][@Mukohyama:2014gba][@Namba:2015gja]. During the superinflation, the primordial GWs come from the amplification of vacuum tensor perturbations. However, since the almost scale-invariance of the scalar perturbation requires $|\epsilon|\sim 0.01 $, we generally have $|n_T|\sim {\cal
O}(0.01)$ for the superinflation. Obtaining a blue GWs spectrum $n_T>0.1$ without the ghost instability while reserving a scale-invariant scalar spectrum with slightly red tilt is still a challenge for the inflation scenario[^4], see e.g.[@Wang:2014kqa] for comments.
In Einstein gravity, the propagating speed $c_T$ of GWs is the same as the speed of light, thus can naturally be set as unity. Nevertheless, it might be modified when dealing with the extremely early universe, e.g., the low-energy effective string theory with higher-order corrections [@Met:1987][@Antoniadis:1993jc][@Kawai:1998ab][@Cartier:1999vk], see also [@Maeda:2004vm][@Nojiri:2015qyc]. Since the amplitude of the primordial GWs is determined by $c_T$ and the Hubble radius $\sim H^{-1}$, the running of $c_T$ will inevitably affect the power spectrum of primordial GWs (see also [@Giovannini:2015kfa] for the study from the point of view of the running of GWs’ refractive index $n$). It was found in [@Cai:2015ipa][@Cai:2015dta] that the oscillation of $c_T$ may leave some observable imprints in CMB B-mode polarization. The effect of the sound speed $c_S$ of scalar perturbation on the scalar spectrum has been investigated in e.g.[@Khoury:2008wj] [@Park:2012rh].
Here, we show that if the propagating speed $c_T$ of GWs gradually diminishes during inflation, the power spectrum of primordial GWs will be strongly blue, while the spectrum of scalar perturbation may be still that of slow-roll inflation. There is no the ghost instability. The blue tilt obtained is $0<n_T\lesssim 1$, which may significantly boost the stochastic GWs background within the window of Advanced LIGO, as well as those of the space-based detectors.
Inflation and $c_T$
===================
The model {#model}
---------
We follow the effective field theory of inflation [@Cheung:2007st], beginning with the Langrangian in unitary gauge S & = & [M\_p\^22]{}d\^4x , \[action\] where $M_p=1/\sqrt{8\pi G}$, $c_1(t)=2({\dot H}+3H^2)$, $c_2(t)=-2{\dot H}$, and a dot denotes the derivative with respect to cosmic time $t$. We will work in the inflation background with $0<\epsilon\ll 1$, which may be set by requiring $|{\dot H}|\ll H^2$ in (\[action1\]). The scalar perturbation at quadratic order is not affected by $\delta
K_{\mu\nu}\delta K^{\mu\nu}-\delta K^2$, see Appendix \[scalar\], and also [@Creminelli:2014wna], so its spectrum is determined by slow-roll parameters. However, the quadratic action of tensor perturbation is altered as[^5] S\^[(2)]{}\_=dd\^3x [M\_p\^2a\^2c\_T\^[-2]{}8]{}, \[paction\]where $\tau=\int dt/a$, and $\gamma_{ij}$ satisfies $\gamma_{ii}=0$ and $\partial_i \gamma_{ij}=0$.
The Fourier series of $\gamma_{ij}$ is \_[ij]{}(,)=e\^[-i]{} \_[=+,]{} \_(,) \^[()]{}\_[ij]{}(), in which $
\hat{\gamma}_{\lambda}(\tau,\mathbf{k})=
\gamma_{\lambda}(\tau,k)a_{\lambda}(\mathbf{k})
+\gamma_{\lambda}^*(\tau,-k)a_{\lambda}^{\dag}(-\mathbf{k})$, the polarization tensors $\epsilon_{ij}^{(\lambda)}(\mathbf{k})$ satisfy $k_{j}\epsilon_{ij}^{(\lambda)}(\mathbf{k})=0$, $\epsilon_{ii}^{(\lambda)}(\mathbf{k})=0$, and $\epsilon_{ij}^{(\lambda)}(\mathbf{k})
\epsilon_{ij}^{*(\lambda^{\prime}) }(\mathbf{k})=\delta_{\lambda
\lambda^{\prime} }$, $\epsilon_{ij}^{*(\lambda)
}(\mathbf{k})=\epsilon_{ij}^{(\lambda) }(-\mathbf{k})$, the annihilation and creation operators $a_{\lambda}(\mathbf{k})$ and $a^{\dag}_{\lambda}(\mathbf{k}^{\prime})$ satisfy $[
a_{\lambda}(\mathbf{k}),a_{\lambda^{\prime}}^{\dag}(\mathbf{k}^{\prime})
]=\delta_{\lambda\lambda^{\prime}}\delta^{(3)}(\mathbf{k}-\mathbf{k}^{\prime})$. The equation of motion for $u(\tau,k)$ is +(c\_T\^2k\^2- )u=0, \[eom1\] where (,k)= \_(,k) [z\_T]{}, z\_T= [aM\_p [ c\_T\^[-1]{} ]{}2]{}.\[zt\] Initially, the perturbations are deep inside the sound horizon, i.e., $c_T^2k^2 \gg \frac{d^2z_T/d\tau^2}{z_T}$, the initial state is the Bunch-Davies vacuum, thus $u\sim
\frac{1}{\sqrt{2c_T k} }e^{-i c_T k\tau}$. The power spectrum of primordial GWs is P\_T=\_[=+,]{} |\_ |\^2= |u |\^2, aH/(c\_Tk) 1.\[pt\]
The diminishment of $c_T$ may be regarded as c\_T= (-H\_[inf]{})\^p, \[cTn\]in which $p>0$, and $H_{inf}$ is the Hubble parameter during inflation, which is regarded as constant for simplicity. Additionally, Eq. (\[cTn\]) suggests ${{\dot c}_T\over
H_{inf}c_T}=-p$.
We set $dy=c_Td\tau$, thus Eq. (\[eom1\]) is rewritten as u\_[,yy]{}+(k\^2- )u=0, \[eom4\] where ${u}(y,k)= \gamma_{\lambda}(y,k) {z_T}$, $z_T= {aM_p {
c_T^{-1/2} }\over 2}$ and the subscript ‘$,y$’ denotes $d/dy$. Note here $u(y,k)$ and $z_T$ are different from those in Eq. (\[eom1\]), but $\gamma_{\lambda}$ is still the same. The solution of Eq.(\[eom4\]) is u\_k(y)=[2]{}H\^[(1)]{}\_(-ky),\[cq002\] where H\^[(1)]{}\_(-ky)-i([2-ky]{})\^[()]{}, and $\nu=1+{1\over 2(1+p)}$. Thus the spectrum (\[pt\]) is P\_T=[4k\^3\^2 M\_P\^2]{} [c\_T|u|\^2a\^2]{}= [2\^[-p1+p]{}]{}\^2 ([12(1+p)]{}) [2H\_[inf]{}\^2\^2 M\_P\^2 c\_T]{} (-ky)\^[p1+p]{}, \[nT\] where $y={c_T\tau\over {1+p} }={-}{c_T\over
{(1+p)}aH_{inf} }$. Therefore, n\_T=[p1+p]{} \[nTB\]is blue-tilt, which is $n_T\simeq p$ for $p\ll 1$ and $n_T\simeq 1$ for $p\gg 1$. Here, the running of $H_{inf}$ may contribute $-2\epsilon\sim -0.01$, which has been neglected.
Thus, we obtain a blue-tilt GWs spectrum with $0<n_T\leqslant 1$. Here, both the scalar perturbation and the background are unaffected by additional operator (\[action\]). The background is set by (\[action1\]), which is the slow-roll inflation with $0<\epsilon\ll 1$, so the scalar spectrum is flat with a slightly red tilt, which is consistent with the observations. It is noticed that based on the effective field theory of inflation, the introducing of other operators may also result in the blue-tilt GWs spectrum [@Cannone:2014uqa][@Baumann:2015xxa], however, in [@Cannone:2014uqa] $n_T>0.1$ requires that the graviton have a large mass $m_{graviton}\simeq H_{inf}$, while in [@Baumann:2015xxa] $|{{\dot c}_T\over H_{inf}c_T}|\ll 1$ was implicitly assumed.
It is well known that the blue-tilt GWs spectrum is the hallmark of the superinflation. Here, the scenario proposed is actually a disformal dual to the superinflation. We will discuss this issue in detail in Sec. \[app-disformal\].
The stochastic background of GWs {#twoB}
--------------------------------
We will focus on the stochastic background of GWs from such a scenario of inflation. The present observations are still not able to put stringent constraints on $c_T$ at present (see, e.g., [@Amendola:2014wma][@Raveri:2014eea], also [@Moore:2001bv] for the constraint on the phase velocity and [@Blas:2016qmn] for the group velocity). Future observations may put more stringent constraints on $c_T$ [@Nishizawa:2014zna][@Nishizawa:2016kba]. However, we will not get involved in this issue too much and we will assume that $c_T(t)$ will return to $c_T=1$ at certain time before the end of inflation. Conventionally, one define $$\label{density} \Omega_{\text{gw}}(k,
\tau_{0})=\frac{1}{\rho_{\text{c}}}\frac{d\rho_{\text{gw}}}{d\ln
k}=\frac{k^{2}}{12 a^2_0H^2_0}P_{T}T^2(k,\tau_0)\,,$$ where $\rho_{\text{c}}=3H^{2}_0/\big(8\pi G\big)$, $\tau_{0}=1.41\times10^{4}$ Mpc, $a_0=1$, $H_0=67.8$ km s$^{-1}$ Mpc$^{-1}$, the reduced Hubble parameter $h=H/\big(100\,
\text{km s}^{-1}\text{Mpc}^{-1}\big)$, and $\rho_{\text{gw}}$ is the energy density of relic GWs at present, so $\Omega_{gw}(k,
\tau_{0})$ reflects the fraction of $\rho_{\text{gw}}$ per logarithmic frequency interval. The transfer function is [@Turner:1993vb][@Boyle:2005se][@Zhang:2006mja] T(k,\_[0]{})=, \[Tk\] where $k_{\text{eq}}=0.073\,\Omega_{\text{m}} h^{2}$ Mpc$^{-1}$ is that of the perturbation mode that entered the horizon at the equality of matter and radiation. We have neglected the effects of the neutrino free-streaming on $T(k,\tau_0)$ [@Weinberg:2003ur], which is actually negligible. The underlying assumption on the thermal history of the post-inflation universe is able to affect $T(k,\tau_{0})$ significantly, see e.g.[@Kuroyanagi:2014nba], but we will only focus on the simplest case described by Eq.(\[Tk\]).
One generally parameterizes $P_T$ as P\_T = A\_T()\^[n\_T]{}, \[para1\] where $k_*=0.01$ Mpc$^{-1}$ is the pivot scale. However, if $n_T>0.4$, one will have $P_T>1$ at high-frequency region ($f>10^5$Hz). The GWs with $P_T\sim 1$ will induce the same-order scalar perturbation at nonlinear order, e.g.[@Wang:2014kqa], which will result in the overproduction of primordial black holes at the corresponding scale, which is inconsistent with their abundance. The upper bound put by the production of primordial black holes is $P_T<0.4$ [@Nakama:2015nea]. In addition, the indirect upper bound given by the combination of CMB with lensing, BAO and BBN observations is $\Omega_{gw}<3.8\times 10^{-6}$ [@Pagano:2015hma], which also puts a strong constraint on $n_T$, i.e., $n_T<0.36$ at $95\%$ C.L. for $r=0.11$ [@Lasky:2015lej], otherwise $\Omega_{gw}$ at higher frequency will exceed this bound.
However, in our scenario, $c_T(t)$ is assumed to return to unity at a certain time $t_c$ before the end of inflation, as has been mentioned. This means that the blue-tilt spectrum will acquire a cutoff around $k_c$, see Sec. \[app-cutoff\] for details, which may avoid the above constraints on $n_T$. We may parameterize the corresponding $P_T$ as P\_[T]{} = A\_T()\^[n\_T]{}, \[para2\] which is (\[para1\]) for $k\ll k_c$, and tends to a constant $A_T(\frac{k_c}{k_*})^{n_T}$ for $k\gg k_c$. Though we will use (\[para1\]) and (\[para2\]) since we are mainly interested in the boosted blue-tilted spectrum, we should point out that $P_T$ will decrease at $k>k_c$ or $k\gg k_c$ (which may be out of the range we are interested in), if we assume that $c_T$ will increase back to unity. In such case, $P_T$ may be parameterized as P\_T= A\_T ()\^[n\_T]{} [ 11+([k]{}/[k\_c]{})\^[n\_[Tc]{}]{}]{}, \[para3\] where $n_{Tc}> n_T$, so that when $k\gg k_c$, $P_T=A_T
({k_c}/{k_*})^{n_T}({k}/{k_c})^{n_T-n_{Tc}}$ has a red tilt. When $n_{Tc}=n_T$, (\[para3\]) is similar to (\[para2\]).
We plot the stochastic background of our GWs in Fig.\[fig01\]. It is obvious that a blue-tilt primordial GWs with $n_T\gtrsim
0.4$ is able to contribute a large stochastic GWs background within the windows of Advanced LIGO/Virgo, which may be greater than the contribution from the incoherent superposition of all binary black hole coalescence. $n_T\gtrsim 0.4$ requires $p\gtrsim 2/3$ in (\[cTn\]), which suggests that the diminishment of $c_T$ in units of Hubble time is not too fast. It is also interesting to notice that if such a GWs background could be detected by Advanced LIGO/Virgo in upcoming observing runs, it will also be able to be detected by the space-based interferometers at a lower frequency band, such as eLISA, and China’s Taiji program in space, see Fig.\[fig02\], as well as the PTA, e.g.[@Lasky:2015lej][@Liu:2015psa].
![The brown line is the stochastic GWs background from inflation with spectral index $n_T=0.45$ and tensor-to-scalar ratio $r=0.05$ at the CMB scale. O1, O2, and O5 curves, taken from [@TheLIGOScientific:2016wyq], are the current Advanced LIGO/Virgo sensitivity, the observing run (2016-2017) and (2020-2022) sensitivities at $1\sigma$ C.L., respectively. The blue curve is the GWs background generated by all binary black hole coalescence without excluding potentially resolvable binaries. []{data-label="fig01"}](ligo){width="55.00000%"}
![The green and the brown lines are the stochastic GWs backgrounds from inflation with $n_T=0.3$ in (\[para1\]) and $n_T=0.45$ in (\[para2\]), respectively. Both C1 and C4-lines are eLISA’s representative configurations given in [@Caprini:2015zlo]. The sensitivity curves of DECIGO and BBO are given in [@Kuroyanagi:2010mm]. The red dashed curve is Taiji’s sensitivity curve, see, e.g., [@Gao:2016tzv] for a preliminary report. Fig.\[fig01\] is actually the amplification of image at the frequency band 10-400 Hz in this figure.[]{data-label="fig02"}](omega){width="55.00000%"}
Disformal dual to superinflation {#app-disformal}
=================================
![This sketch illustrates the evolutions of the primordial perturbations during inflation in our scenario. The brown line is $\sim 1/aH$. The blue line is $\sim c_T/aH$, which is the sound horizon of GWs. We assume that $c_T$ decrease to some value less than unit and begin to increase later, so that it could return to unity and both horizons coincides before or near the end of inflation. []{data-label="fig03"}](aH.eps){width="55.00000%"}
The superinflation is the inflation with $\epsilon=-{\dot
H}/H^2<0$, i.e. ${\dot H}>0$, which breaks the NEC. The model we proposed in Sec. \[model\], i.e., inflation with a diminishing $c_T=(-H_{inf}\tau)^p$, is actually disformally dual to superinflation. This can be inferred from the evolution of the GWs sound horizon.
The perturbation mode outside the comoving sound horizon $1/(aH_{Per})$ of the perturbations[^6] (i.e., $k\ll a H_{Per}$) will freeze, while it will evolve inside $1/(aH_{Per})$. In inflation scenario, the spectrum of GWs generally has similar shape to that of the scalar perturbation, since both GWs and scalar perturbations have a comoving sound horizon $1/(aH_{Per})$ almost coincide with $1/(aH)$. Here, since the comoving sound horizon of GWs is $c_T/(aH)$, and its evolution is completely different from $1/(aH)$, the spectrum of GWs shows itself blue-tilt, see Fig.\[fig03\]. However, the tilt of $c_T/(aH)$-line in Fig.\[fig03\] is the same as that of the superinflation with $c_T=1$, see Fig.1 in [@Piao:2006jz]. This indicates the physical processes of horizon crossing of GWs modes are same in these two scenarios, thus will generate the same power spectra. In fact, these two scenarios can be connected by a disformal transformation. Bellow, we give the strict proof.
We make a disformal redefinition of the metric [@Creminelli:2014wna] g\_c\_T\^[-1]{}. \[gmunu1\] with c\_T\^[1/2]{}dt,()c\_T\^[-1/2]{}a(t), \[tildea\]which makes (\[paction\]) become S\^[(2)]{}=d d\^3x [M\_p\^2\^28]{}\[newaction\] with ${\tilde c_T}=1$.
Here, with $d\tilde{\tau}={d\tilde{t}/\tilde{a}}$, which implies =\^(-H\_[inf]{})\^p d=-(H\_[inf]{})\^p[(-)\^[p+1]{}p+1]{}, we have =c\_T\^[-1/2]{}a \~(-)\^[-[2+p2(1+p)]{}]{}. ==c\_T\^[-1/2]{}(H\_[inf]{}-)\~(-)\^[-[p2(1+p)]{}]{}. \[tildeH\]Thus the value of $\tilde H$ is gradually increasing. This suggests that after the disformal transformation the background is actually the superinflation with ${\tilde
\epsilon}=-p/(2+p)$, which satisfies $-1\lesssim
\tilde{\epsilon} <0$. The scenario with $\tilde{\epsilon}\ll -1$ is the slow expansion, which was implemented in [@Piao:2003ty].
The equation of motion for $u(\tilde{\tau},k)$ is +(k\^2- )u=0,\[eom2\] where ${u}(\tilde{\tau},k)=
\gamma_{\lambda}(\tilde{\tau},k) {\tilde{z}_T}$ and $\tilde{z}_T=
{\tilde{a}M_p/ 2}$. The initial state is still the Bunch-Davies vacuum $u\sim\frac{1}{\sqrt{2k} }e^{-ik\tilde{\tau}}$. The solution is u\_k()=[2]{} H\^[(1)]{}\_(-k), where H\^[(1)]{}\_(-k) -i([2-k]{})\^, and $\tilde{\nu}=1+{1\over2(1+p)}$. Thus the power spectrum is P\_T &= & \_[=+,]{} |\_ |\^2\
&=& [4k\^3\^2M\_p\^2\^2]{}(-k) [2\^[2+[11+p]{}]{}(-k)\^[2+[11+p]{}]{}]{}\
&=&[2\^2\^2 M\_p\^2]{} \^2([12(1+p)]{})(-k)\^[p1+p]{}\[Ptilde\]\
&=&[c\_Tk\^2\^3M\_p\^2]{}\^2([12(1+p)]{}) (kH\_[inf]{}\^p)\^[-[2+p1+p]{}]{}\
&=&[2H\_[inf]{}\^2\^2M\_p\^2c\_T]{} \^2([12(1+p)]{})(-ky)\^[p1+p]{}. This result is completely the same as Eq.(\[nT\]).
When $p\ll1$, we have \^2([12(1+p)]{})1+0.27p+[O]{}(p\^2) in Eq.(\[Ptilde\]) and $\tilde{\epsilon
}=-\frac{d\tilde{H}/d\tilde{t} }{\tilde{H}^2}\ll 1$. Thus with (\[Ptilde\]), we have P\_T=2[H]{}\^2/\^2 M\_P\^2, \[PT3\]i.e. Creminelli *et.al*’s result [@Creminelli:2014wna].
Actually, it is well known that the spectrum of GWs, as well as scalar perturbation, is independent of the disformal redefinition (\[gmunu1\]) of the metric [@Creminelli:2014wna][@Minamitsuji:2014waa]. An intuition argument for it is the comoving horizon of scalar perturbation = [1c\_T ]{}\~(-)\^[11+p]{}\~[1a H\_[inf]{}]{} i.e., the relation between the comoving wave number $k$ and the comoving sound horizon is not altered, where ${\tilde c}_s=1/c_T$ [@Creminelli:2014wna].
Conventionally, the superinflation breaks the NEC. Implementing the superinflation without the ghost instability is still a significant issue, e.g.[@Wang:2014kqa][@Cai:2014uka]. Here, we actually suggest such a superinflation scenario. It might be just a slow-roll inflation living in a disformal metric with $c_T$ gradually diminishing, however, if we see it with $c_T=1$, what we will feel is the superinflation. The violation of NEC in modified gravity does not necessarily mean ghost instability. Because the quadratic actions (\[paction\]) and (\[newaction\]) for the tensor (as well as those for scalar) are canonical, there is no ghost instability in both frames.
Cutoff of blue spectrum {#app-cutoff}
=======================
To avoid $P_T\sim 1$ at high frequency, we have to require that the diminishment of $c_T$ stop at a certain time $\tau_c$. Additionally, we assume that $c_T(t)$ will return to unity before the end of inflation, as in Sec. \[twoB\].
We assume that c\_T&=&(-H\_[inf]{})\^p\_c. We set $dy=c_Td\tau$. The solution of (\[eom4\]) is u\_2(y)=for $y>y_c$, and is $u_1(y)$ for $y<y_c$, which is actually (\[cq002\]), where $\nu=1+{1\over
2(1+p)}$, $y_c=c_{Tc}\tau_c$. When $-ky\ll1$, u\_2|C\_1-C\_2|. Thus the spectrum of primordial GWs is P\_T&=&[4k\^3\^2 M\_P\^2]{} [c\_T|u|\^2a\^2]{}= [2H\_[inf]{}\^2\^2M\_p\^2]{} f(p,y\_c,k), \[PT4\] where f(p,y\_c,k)&=&[4kc\_[Tc]{}]{}|C\_1-C\_2|\^2, and C\_1&=&-[i\^[3/2]{}16]{}, C\_2&=&[i\^[3/2]{}16]{}are set by the continuities of $u(y)$ and ${du/ dy}$ at $\tau_c$. We plot (\[PT4\]) in Fig.\[fig04\], and see that, although $P_T$ has a blue tilt, it is flat at a high frequency. We analytically calculate it as follows.
![$P_T/P_{T}^{inf}=f(p,y_c,k)$. The parameters of the magenta dashed and brown solid curves are $c_{Tc}=10^{-3}$ and $10^{-5}$, respectively, while we set $p=0.7$.[]{data-label="fig04"}](power-02.eps){width="55.00000%"}
When $-ky_c\ll1$, C\_1&=&-2\^[-[4+5p2(1+P)]{}]{}e\^[iky\_c]{} [1]{}(-ky\_c)\^[-[6+5p2(1+p)]{}]{} [(3+2p2(1+p))1+p]{}\
&&, C\_2&=&2\^[-[4+5p2(1+P)]{}]{}e\^[-iky\_c]{}[1]{} (-ky\_c)\^[-[6+5p2(1+p)]{}]{}[(3+2p2(1+p))1+p]{}\
&&. We have f(p,y\_c,k)&=&[4kc\_[Tc]{}]{}|C\_1-C\_2|\^2\
&&[2\^[-[p1+p]{}]{}9(1+p)\^2c\_[Tc]{}]{}\^2([3+2p2(1+p)]{}) (6+5p)\^2(-ky\_c)\^[[p1+p]{}]{}. Thus the tilt $n_T={p\over
1+p}$, which is the same as (\[nT\]).
When $-ky_c\gg1$, C\_1=e\^[[i4]{}-[i2]{}]{}, C\_2=e\^[[i4]{}-[i2]{}(+4ky\_c)]{}. We have f(p,y\_c,k) &&[1c\_[Tc]{}]{}. Thus the spectrum is flat.
From the above result, we can infer that if $c_T$ slowly diminishes to a value less than unity during inflation and then increases back to unity before the end of inflation, $\Omega_{gw}$ could be strongly boosted at the frequency band of Advanced LIGO/Virgo.
Discussion
==========
In the inflation scenario, obtaining a blue GWs spectrum ($n_T>0.1$) without the ghost instability while reserving a scale-invariant scalar spectrum with a slightly red tilt is still a challenge. We find that if the propagating speed of GWs gradually diminishes during inflation, the power spectrum of primordial GWs will be strongly blue, while that of the scalar perturbation may be unaffected.
It is well known that the blue-tilt GWs is the hallmark of superinflation [@Piao:2004tq][@Baldi:2005gk]. It may be implemented without ghost in G-inflation [@KYY], but it is difficult, however, to simultaneously give it a slightly red-tilt scalar spectrum [@Wang:2014kqa], see also [@Cai:2014uka]. Our scenario is actually a disformal dual to the superinflation, see Sec. \[app-disformal\]. In this duality, our background is actually a slow-roll inflation living in a disformal metric with $c_T$ gradually diminishing. However, if we see it with $c_T=1$, what we will feel is the superinflation, but there is no ghost instability. Thus our work might offer a far-sighted perspective on superinflation.
The blue tilt obtained is $0<n_T\lesssim 1$, which may significantly boost the stochastic GWs background at the frequency band of Advanced LIGO/Virgo, as well as the space-based detectors. This indicates that the primordial GWs recording the origin of the universe may be potentially measurable by the corresponding experiments.
To conclude, if a stochastic background of GWs is detected by Advanced LIGO/Virgo in the upcoming observing runs, it also possibly comes from the primordial inflation, and encodes the physics beyond GR at inflation scale.
**Acknowledgments**
This work is supported by NSFC, No. 11222546, 11575188, and the Strategic Priority Research Program of Chinese Academy of Sciences, No. XDA04000000. We thank Cong-Feng Qiao and Yun-Kau Lau for suggesting that we use the sensitivity curves of China’s Taiji program in space, which will appear in a full work report.
Scalar perturbation {#scalar}
===================
We work with the ADM metric $$\begin{aligned}
g_{\mu\nu}=\left(\begin{array}{cc}N_kN^k-N^2&N_j\\N_i& h_{ij}\end{array}\right),~~
g^{\mu\nu}=\left(\begin{array}{cc}-N^{-2}&\frac{N^j}{N^2}\\ \frac{N^i}{N^2}& h^{ij}-\frac{N^iN^j}{N^2}\end{array}\right)\,,\end{aligned}$$ where $h_{ij}=a^2e^{2\zeta}(e^{\gamma})_{ij}$, and $\gamma_{ii}=0=\partial_i\gamma_{ij}$. Generally, $N=1+\alpha$ and $N_i=\partial_i\beta$ are set for the scalar perturbations . It is convenient to define the normal vector of 3-dimensional hypersurface $n_\mu=n_0{dt/dx^\mu}=(n_0,0,0,0)$ and $n^{\mu}=g^{\mu\nu}n_\nu$. Using the normalization $n_\mu
n^\mu=-1$, one has $n_0=-N$, which suggests $n_\mu=(-N,0,0,0),
n^\mu=(\frac{1}{N},\frac{N^i}{N})$, and the 3-dimensional induced metric, orthogonal to the normal vector, i.e., $H_{\mu\nu}n^\nu=0$, can be defined to be $H_{\mu\nu}=g_{\mu\nu}+n_\mu n_\nu$, $$\begin{aligned}
H_{\mu\nu}=\left(\begin{array}{cc}N_kN^k&N_j\\N_i& h_{ij}\end{array}\right),~~
H^{\mu\nu}=\left(\begin{array}{cc}0&0\\ 0& h^{ij} \end{array}\right).\end{aligned}$$ The covariant derivative associated with $H_{\mu\nu}$ is $D_{\mu}$, which is applied to define the extrinsic curvature $K_{\mu\nu}$: K\_=[12N]{}(\_-D\_N\_-D\_N\_). We have & & K\_K\^-(K)\^2\
&=&[1(1+)\^2]{}{-6(-H)\^2 +4a\^[-2]{}e\^[-2]{}(-H)(\_i\_i+\_i\_i)\
&& +a\^[-4]{}e\^[-4]{}}, where $\delta K_{\mu\nu}=K_{\mu\nu}-H_{\mu\nu}H$.
Thus the quadratic action of scalar perturbation for (\[action1\]) and (\[action\]) is &&S\^[(2)]{}\_=dx\^4 M\_p\^2 { a\^3H\^2 \^2 - 27a\^3H\^2 \^2 + 9a\^3H\^2 \^2 - 18a\^3H\
&& +a( )\^2 - 2 a\_i\_i -[1c\_T\^2]{} }. \[scalar1\]The constraints can be solved as &=&[H]{},\
\_i\_i&=&[c\_T\^2H]{}(a\^2H-\_i\_i). Inserting them into (\[scalar1\]), S\^[(2)]{}\_=dx\^4 [M\_p\^2]{} a\^3 is obtained. Therefore, the scalar perturbation is not affected by the operator $\delta K_{\mu\nu}\delta
K^{\mu\nu}-(\delta K)^2$ at quadratic order.
[99]{}
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[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
[^4]: It is found that in the pre-big bang scenario (obtained in the context of string cosmology) the primordial GWs spectrum is blue [@Brustein:1995ah][@Gasperini:2002bn][@Gasperini:2007zz].
[^5]: In [@Giovannini:2015kfa], the author investigated the effect induced by the running of GWs’ refractive index $n(\tau)$, which is similar to that of $c_T$. But note that $c_T\neq 1/n$, as can be seen from the difference between Eq.(\[paction\]) here and the Eq.(2.8) in [@Giovannini:2015kfa].
[^6]: Here, $H_{Per}$ is defined as $H_{Per}={(z_T''/z_T)^{1/2}\over a c_T}$, where $z_T$ is given by Eq.(\[zt\]). For $c_T\sim (-\tau)^p$ and $a\sim(-\tau)^{-1}$, we have =\~[c\_TaH]{}.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this article, I have considered a real scalar field theory and able to show that under Bogoliubov transformation in infinite volume limit or thermodynamic limit the transformed Hamiltonian no longer invariant under U(1) action defined appropriately as it was before doing transformation. We also have checked this fact by looking at the correlation functions under the action of U(1) group. We suitably defined field operators that are associated with particle production phenomena then we can also show that correlation functions of such field operators also don’t follow U(1) invariance, shown in this article. This is a consequence of non-invariance of transformed Hamiltonian under U(1) action. Since, we know Bogoliubov transformation in curved spacetime is equivalent to doing a coordinate transformation, therefore this result directly shows the phenomena of particle production under the affect of gravity since changing coordinate is equivalent to turn on gravity according to Einstein’s equivalence principle in GR. I also show that particle production does not take place out of vacuum state but it can happen out of other many-particle states and vacuum state is not an eigenvector of Hamltonian operator in transformed Fock space and vacuum state does not remain vacuum state under time evolution.'
author:
- Susobhan Mandal
bibliography:
- 'bibtexfile.bib'
title: 'U(1) symmetry breaking under canonical transformation in real scalar field theory'
---
Introduction
============
We all know field theory describes a system containing infinitely many degrees of freedom and most of the time when we describe a field theoretical system we generally considered a infinite size system or system in thermodynamic limit. This limit is very crucial while we are doing canonical transformation that presereves commutation bracket. One such class is Bogoliubov transformations in scalar field theory which is often used in condensed matter physics [@casalbuoni2003lecture], [@timm2012theory], [@chalker2013quantum] and quantum field theory in curved spacetime [@birrell1980massive], [@biswas1995particle], [@jacobson2005introduction]. It can be shown and I will show that thermodynamic limit or infinite volume limit make this tranformation impossible and it created two inequivalent representations of two disjoint Fock space which is often used in quantum many body systems. Because of such inequivalent disjoint vector spaces, the operators both in original form and ones after transformation have their own seperate domain to act on states. Therefore, traditional ways of showing particle production [@parker2015creation] is not well-defined in thermodynamic limit. However, I actually show in this article that because of under such transformation Hamiltonian is no longer invariant under U(1) action defined in article therefore particle number of the system is not conserved which implies particle production in curved spacetime under gravity.
Massive scalar free-field theory in hamiltonian description
===========================================================
Massive scalar free-field theory in Minkowski spacetime
-------------------------------------------------------
Let’s Consider following action $$\begin{split}
S & =\int\sqrt{-\eta}d^{4}x\Big[\frac{1}{2}(\partial\phi(x))^{2}-\frac{1}{2}m^{2}\phi^{2}(x)\Big]\\
\eta & =\text{diag}(1,-1,-1,-1)
\end{split}$$ We can write field operators in following way $$\hat{\phi}(x) =\int\frac{d^{3}k}{\sqrt{(2\pi)^{3}2\omega_{\vec{k}}}}[\hat{a}_{\vec{k}}e^{-ik.x}+\hat{a}_{\vec{k}}^{\dagger}e^{ik.x}]$$ Here canonical conjugate momentum field operator is $\hat{\pi}=\dot{\hat{\phi}}$. And one can check from canonical commutation relation $$[\hat{\phi}(x),\hat{\pi}(y)]_{x^{0}=y^{0}} =i\delta^{(3)}(\vec{x}-\vec{y})$$ following algebra between creation-annihilation operators $$\begin{split}
[\hat{a}_{\vec{k}},\hat{a}_{\vec{k}'}] & =0=[\hat{a}_{\vec{k}}^{\dagger},\hat{a}_{\vec{k}'}^{\dagger}]\\
[\hat{a}_{\vec{k}},\hat{a}_{\vec{k}'}^{\dagger}] & =\delta^{(3)}(\vec{k}-\vec{k}')
\end{split}$$ And Hamiltonian operator for this system is following [@book:14817] $$\begin{split}
\hat{H} & =\int d^{3}x\Big[\frac{1}{2}\hat{\pi}^{2}(x)+\frac{1}{2}(\vec{\nabla}\phi(x))^{2}+\frac{1}{2}m^{2}\phi^{2}(x)\Big]\\
& =\int d^{3}k\frac{1}{2}\omega_{\vec{k}}[\hat{a}_{\vec{k}}\hat{a}_{\vec{k}}^{\dagger}+\hat{a}_{\vec{k}}^{\dagger}\hat{a}_{\vec{k}}]\\
& \equiv\int d^{3}k\omega_{\vec{k}}\hat{a}_{\vec{k}}^{\dagger}\hat{a}_{\vec{k}}, \ \omega_{\vec{k}}=\sqrt{\vec{k}^{2}+m^{2}}
\end{split}$$ The above expression can also be written using the definition of number operators $$\begin{split}
\hat{N}_{\vec{k}} & =\hat{a}_{\vec{k}}^{\dagger}\hat{a}_{\vec{k}}\\
\implies\hat{H} & =\int d^{3}k \ \omega_{\vec{k}}\hat{N}_{\vec{k}}
\end{split}$$ Note that Hamiltonian of this theory is invariant both global $U(1)$-group action which is following $$\begin{split}
\hat{a}_{\vec{k}} & \rightarrow e^{i\Theta}\hat{a}_{\vec{k}}\\
\hat{a}_{\vec{k}}^{\dagger} & \rightarrow e^{-i\Theta}\hat{a}_{\vec{k}}^{\dagger}
\end{split}$$ and local $U(1)$-group action which is following $$\begin{split}
\hat{a}_{\vec{k}} & \rightarrow e^{i\Theta_{\vec{k}}}\hat{a}_{\vec{k}}\\
\hat{a}_{\vec{k}}^{\dagger} & \rightarrow e^{-i\Theta_{\vec{k}}}\hat{a}_{\vec{k}}^{\dagger}
\end{split}$$ And vacuum of this theory is state $\ket{0}$ which is such that $$\hat{a}_{\vec{k}}\ket{0} =0, \ \forall\vec{k}$$ And all the single and multi-particle states can be constructed using operation of creation operators on the vacuum state.
Massive Scalar Free-Field Theory in different frame
---------------------------------------------------
Now we want to move to a different frame which is non-inertial and w.r.t an observer from this frame action can be written down using minimal prescription $$\begin{split}
S & =\int\sqrt{-g}d^{4}x\Big[\frac{1}{2}(\partial\phi(x))^{2}-\frac{1}{2}m^{2}\phi^{2}(x)\Big]\\
(\partial\phi(x))^{2} & =g^{\mu\nu}(x)\partial_{\mu}\phi(x)\partial_{\nu}\phi(x)
\end{split}$$ where the metric $g_{\mu\nu}(x)$ is non-trivial.\
Here we choose the mode functions to be solutions of Euler-Lagrange equation which is of following form $$\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\phi)+m^{2}\phi=0$$ Let $\{f_{n}(x)\}$ be complete set of such mode functions which solve above equation.\
We define an inner product between 2 such mode function in following way $$(f,g)\equiv i\int\sqrt{-g}d^{3}x \ f^{*}(x)[g^{0\nu}\overrightarrow{\partial}_{\nu}-g^{0\nu}\overleftarrow{\partial}_{\nu}]g(x)$$ And one can show that this inner product is time-translation invariant [@book:274311] $$\begin{split}
\partial_{0}(f,g) & = i\int d^{3}x \ \partial_{0}\Big[\sqrt{-g}f^{*}(x)[g^{0\nu}\overrightarrow{\partial}_{\nu}-g^{0\nu}\overleftarrow{\partial}_{\nu}]g(x)\Big]\\
& =i\int d^{3}x \ \partial_{\mu}\Big[\sqrt{-g}f^{*}(x)[g^{\mu\nu}\overrightarrow{\partial}_{\nu}-g^{\mu\nu}\overleftarrow{\partial}_{\nu}]g(x)\Big]\\
& -i\int d^{3}x \ \partial_{i}\Big[\sqrt{-g}f^{*}(x)[g^{i\nu}\overrightarrow{\partial}_{\nu}-g^{i\nu}\overleftarrow{\partial}_{\nu}]g(x)\Big]\\
=i\int d^{3}x \ & \Big[f^{*}(x)\partial_{\mu}[\sqrt{-g}g^{\mu\nu}\overrightarrow{\partial}_{\nu}]-\partial_{\mu}[\sqrt{-g}g^{\mu\nu}f^{*}(x)\overleftarrow{\partial}_{\nu}]g(x)\Big]\\
& -i\int d^{3}x \ \partial_{i}\Big[\sqrt{-g}f^{*}(x)[g^{i\nu}\overrightarrow{\partial}_{\nu}-g^{i\nu}\overleftarrow{\partial}_{\nu}]g(x)\Big]\\
=0
\end{split}$$ where we throw away the surface term because we have assumed that fields vanish at surface near spatial infinity and consider Euler-Lagrange equation followed by the modes.\
Similar definition is also there in first frame with only difference is the metric is Minkowski. So, we label the inner product in first and second frame to be 1 and 2 respectively.\
In this frame also we can write the field operators in following way $$\hat{\phi}'(y) = \sum_{n}[\hat{b}_{n}f_{n}(y)+\hat{b}_{n}^{\dagger}f_{n}^{*}(y)]$$ Now consider a point in spacetime which has different coordinates w.r.t two frames and let call them $x$ and $x'$ respectively in initial and final frame. Since, we are dealing with scalar field theory which has a nice property which is its tranformation property under general coordinate transformation $$\hat{\phi}'(x') =\hat{\phi}(x)$$ Using the definition of inner-product one can define a Bogoliubov type transformation which is defined in next subsection. And for convenience we choose momentum mode decomposition of fields from next section onwards by consiering spacetime which has spatial translational invariant for example spatially flat FRW spacetime [@book:274311].
Bogoliubov transformation
-------------------------
To show the Bogoliubov transformation [@jacobson2005introduction], [@sato1994bogoliubov], we consider a real scalar field with modes $\{\hat{a}_{\vec{k}}\}$. And they satisfy following algebra $$[\hat{a}_{\vec{k}},\hat{a}_{\vec{q}}^{\dagger}]=\delta^{(3)}(\vec{k}-\vec{q})$$ and rest of the commutator brackets are zero.\
With these one can construct Fock space $\mathcal{H}[a]$ with repeated applications of $\hat{a}_{\vec{k}}^{\dagger}$s on vacuum state denoted by $\ket{0}$ defined by $$\hat{a}_{\vec{k}}\ket{0}=0, \ \forall\vec{k}$$ Let us now consider the following Bogoliubov transformation(this is not most general transformation because most general transformation also can mix different momentum modes) $$\begin{split}
\hat{c}_{\vec{k}}(\theta) & =\cosh\theta_{\vec{k}}\hat{a}_{\vec{k}}-\sinh\theta_{\vec{k}}\hat{a}_{-\vec{k}}^{\dagger}
\end{split}$$ With these transformations in hand, one can check that new operators also satisfy $$[\hat{c}_{\vec{k}}(\theta),\hat{c}_{\vec{q}}^{\dagger}(\theta)]=\delta^{(3)}(\vec{k}-\vec{q})$$ and all other commutators vanishing.\
Now we consider vacuum relative to the operators $\{\hat{c}_{\vec{k}}(\theta)$, denoted by $\ket{0(\theta)}$ and defined by $$\hat{c}_{\vec{k}}(\theta)\ket{0(\theta)}=0, \ \forall\vec{k}$$ and we construct new Fock space representation $\mathcal{H}(c)$ by repeated application of $\{\hat{c}_{\vec{k}}^{\dagger}(\theta)$ on the vacuum state $\ket{0(\theta)}$.\
If now we assume the existence of an unitary operator $G(\theta)$ which generates the transformation $$U(\theta)\hat{a}_{\vec{k}}U^{-1}(\theta)=\hat{c}_{\vec{k}}(\theta)$$ where $U(\theta)=e^{iG(\theta)}$, then one can explicitly check that $$G(\theta)=i\int d^{3}k \ \theta_{\vec{k}}(\hat{a}_{\vec{k}}\hat{a}_{-\vec{k}}-\hat{a}_{-\vec{k}}^{\dagger}\hat{a}_{\vec{k}}^{\dagger})$$ And therefore, one can write the transformation operator in following way $$\begin{split}
U(\theta) & =e^{-\delta^{(3)}(0)\int d^{3}l\ln\cosh\theta_{\vec{k}}}e^{\int d^{3}k\tanh\theta_{\vec{k}}\hat{a}_{\vec{k}}^{\dagger}\hat{a}_{-\vec{k}}^{\dagger}}\\
& \times^{-\int d^{3}k\tanh\theta_{\vec{k}}\hat{a}_{-\vec{k}}\hat{a}_{\vec{k}}}
\end{split}$$ then we have $$\ket{0(\theta)}=e^{-\delta^{(3)}(0)\int d^{3}k\ln\cosh\theta_{\vec{k}}}e^{\int d^{3}k\tanh\theta_{\vec{k}}\hat{a}_{\vec{k}}^{\dagger}\hat{a}_{-\vec{k}}^{\dagger}}\ket{0}$$ where $\delta^{(3)}(0)$ in discrete limit is volume V. Therefore, unless $V<\infty$ and $\int d^{3}k\ln\cosh\theta_{\vec{k}}<\infty$, state $\ket{0(\theta)}$ does not belong to $\mathcal{H}(a)$ Fock space therefore, we can say that these two representations are inequivalent [@stepanian2013unitary].\
Note that transformation between operators are well-defined for any volume of the system, so once we defined the transformation we take the limit $V\rightarrow\infty$, for which transformation between operators are still well-defined but new and old Fock space becomes disjoint. Therefore, new and old creation and annihilation operators have seperate Hilbert space to act one or in otherwords theor domains become disjoint. This construction will be assumed from next section onwards.
Hamiltonian under Bogoliubov transformation
-------------------------------------------
Now let’s look at the Hamiltonian under Bogoliubov transformation but before that we need the inverse Bogoliubov transformation $$\begin{split}
\hat{c}_{\vec{k}}(\theta) & =\cosh\theta_{\vec{k}}\hat{a}_{\vec{k}}-\sinh\theta_{\vec{k}}\hat{a}_{-\vec{k}}^{\dagger}\\
\hat{c}_{\vec{k}}^{\dagger}(\theta) & =\cosh\theta_{\vec{k}}\hat{a}_{\vec{k}}^{\dagger}-\sinh\theta_{\vec{k}}\hat{a}_{-\vec{k}}\\
\implies\hat{a}_{\vec{k}} & =\cosh\theta_{\vec{k}}\hat{c}_{\vec{k}}+\sinh\theta_{\vec{k}}\hat{c}_{-\vec{k}}^{\dagger}\\
\hat{a}_{\vec{k}}^{\dagger} & =\cosh\theta_{\vec{k}}\hat{c}_{\vec{k}}^{\dagger}+\sinh\theta_{\vec{k}}\hat{c}_{-\vec{k}}
\end{split}$$ Using this we can write $$\begin{split}
\hat{H} & =\int d^{3}k \ \varepsilon_{\vec{k}}\hat{a}_{\vec{k}}^{\dagger}\hat{a}_{\vec{k}}\\
& =\int d^{3}k \ \varepsilon_{\vec{k}}[\cosh\theta_{\vec{k}}\hat{c}_{\vec{k}}^{\dagger}+\sinh\theta_{\vec{k}}\hat{c}_{-\vec{k}}]\\
& \times[\cosh\theta_{\vec{k}}\hat{c}_{\vec{k}}+\sinh\theta_{\vec{k}}\hat{c}_{-\vec{k}}^{\dagger}]\\
& =\int d^{3}k \ \varepsilon_{\vec{k}}[\cosh^{2}\theta_{\vec{k}}\hat{c}_{\vec{k}}^{\dagger}\hat{c}_{\vec{k}}+\sinh^{2}\theta_{\vec{k}}\hat{c}_{-\vec{k}}\hat{c}_{-\vec{k}}^{\dagger}\\
& +\sinh\theta_{\vec{k}}\cosh\theta_{\vec{k}}(\hat{c}_{-\vec{k}}\hat{c}_{\vec{k}}+\hat{c}_{\vec{k}}^{\dagger}\hat{c}_{-\vec{k}}^{\dagger})]\\
& \simeq\int d^{3}k \ \varepsilon_{\vec{k}}[(\cosh^{2}\theta_{\vec{k}}+\sinh^{2}\theta_{-\vec{k}})\hat{c}_{\vec{k}}^{\dagger}\hat{c}_{\vec{k}}\\
& +\sinh\theta_{\vec{k}}\cosh\theta_{\vec{k}}(\hat{c}_{-\vec{k}}\hat{c}_{\vec{k}}+\hat{c}_{\vec{k}}^{\dagger}\hat{c}_{-\vec{k}}^{\dagger})]
\end{split}$$ Note that under Bogoliubov transformation Hamiltonian breaks $U(1)$ invariance in new Fock space representation. Also note that this Hamiltonian looks similar to BCS superconductor Hamiltonian in mean-field approximation [@timm2012theory], [@casalbuoni2003lecture] but here we have non-trivial coefficients attached to the operators.\
Bow we are willing to write the Hamiltonian in matrix form and to do that we choose a representation w.r.t basis of the form $\begin{pmatrix}
\hat{c}_{\vec{k}}\\
\hat{c}_{-\vec{k}}^{\dagger}
\end{pmatrix}, \ \forall\vec{k}$. Using this representation we can write the Hamiltonian in following form $$\begin{split}
\hat{H}=\int d^{3}k\varepsilon_{\vec{k}} & \begin{pmatrix}
\hat{c}_{\vec{k}}^{\dagger} & \hat{c}_{-\vec{k}}
\end{pmatrix}\begin{bmatrix}
\cosh^{2}\theta_{\vec{k}} & \sinh\theta_{\vec{k}}\cosh\theta_{\vec{k}}\\
\sinh\theta_{\vec{k}}\cosh\theta_{\vec{k}} & \sinh^{2}\theta_{\vec{k}}
\end{bmatrix}\\
& \times\begin{pmatrix}
\hat{c}_{\vec{k}}\\
\hat{c}_{-\vec{k}}^{\dagger}
\end{pmatrix}
\end{split}$$ This representation will help us latter in finding off-diagonal components of 2-point correlation function or Green’s function.\
Before we proceed to next subsection where we start discussion of coherent states and partition function we want to make some comments
- First of all, note that new vacuum state that we got in new Fock space representation made out of combination of states of pair of $a-$ particles with opposite momentum, which can be seen directly from the mathematical definition of new vacuum state in terms of old vacuum state.
- Secondly, since Hamiltonian in new Fock space representation breaks the $U(1)$-invariance therefore we expect particle number is not conserved which will also reflect in correlation functions which we will show later.
Correlation functions
=====================
Description of Coherent states
------------------------------
Let’s strat with simple harmonic oscillator description first where we have as usual following algebra $$[\hat{a},\hat{a}^{\dagger}]=1, \ [\hat{a},\hat{a}]=0=[\hat{a}^{\dagger},\hat{a}^{\dagger}]$$ We define coherent states to be the states in Hilbert space which are eigenstates of annihilation operator $\hat{a}$ and defined in following way $$\ket{z}=e^{z\hat{a}^{\dagger}}\ket{0}$$ where $z$ is a complex number and $\ket{0}$ is vacuum state or ground state in this case. One can easily check that $$\hat{a}\ket{z}=\hat{a}e^{z\hat{a}^{\dagger}}\ket{0}=z\ket{z}$$ and similarly correspong dual state can be written as $$\bra{z}=\bra{0}e^{\bar{z}\hat{a}}$$ Similarly with little bit algebra one can show that $$\braket{z|z'}=e^{\bar{z}z'}$$ One can similarly show the resolution of identity $$\hat{I}=\int\frac{dz d\bar{z}}{2\pi i}e^{-z\bar{z}}\ket{z}\bra{z}$$ And for any normal ordered operator $A(\hat{a}^{\dagger},\hat{a})$ one can show easily that $$\bra{z}A(\hat{a}^{\dagger},\hat{a})\ket{z'}=A(\bar{z},z')e^{\bar{z}z'}$$
### Number operator as generator of phase
In above construction number operator is defined as $$\hat{N}=\hat{a}^{\dagger}\hat{a}$$ Now consider a coherent state $\ket{z}$ such that $\hat{a}\ket{z}=z\ket{z}$ where $z$ is some complex number. Then we look at the state $e^{-i\hat{N}\theta}\ket{z}$ $$\begin{split}
\hat{a}e^{-i\hat{N}\theta}\ket{z} & =e^{-i\hat{N}\theta}e^{i\hat{N}\theta}\hat{a}e^{-i\hat{N}\theta}\ket{z}\\
=e^{-i\hat{N}\theta} & \Big[\hat{a}+i\theta[\hat{N},\hat{a}]+\frac{(i\theta)^{2}}{2}[\hat{N},[\hat{N},\hat{a}]]+\ldots\Big]\ket{z}\\
=e^{-i\hat{N}\theta} & e^{-i\theta}\hat{a}\ket{z}=z e^{-i\theta}e^{-i\hat{N}\theta}\ket{z}\\
\implies e^{-i\hat{N}\theta}\ket{z} & \propto\ket{z e^{-i\theta}}
\end{split}$$ Therefore, we can see that number operator $\hat{N}$ is the generator of complex phase for coherent states. We will come to this point again later in this article.
Path integral using coherent states
-----------------------------------
We will want to compute the matrix elements of the evolution operator $\hat{U}$ defined by $$\hat{U}(t_{f},t_{i})=e^{-\frac{i}{\hbar}T\hat{H}(\hat{a}^{\dagger},\hat{a})}$$ where $T=(t_{f}-t_{i})$ and $\hat{H}(\hat{a}^{\dagger},\hat{a})$ is the Hamiltonian operator of the system in normal ordered form. Thus, if $\ket{i}$ and $\ket{f}$ denote two arbitrary initial and final states, we can write the matrix element of $\hat{U}(t_{f},t_{i})$ as $\bra{f}\hat{U}(t_{f},t_{i})\ket{i}$. Now we split the whole time interval into $N$-equal segments with $N\rightarrow\infty$, then each segment has length of $\epsilon\rightarrow0^{+}$, then using the first order approximation we can write $$\bra{f}\hat{U}(t_{f},t_{i})\ket{i}=\lim_{\epsilon\rightarrow0^{+}}\lim_{N\rightarrow\infty}\bra{f}\left(1-\frac{i}{\hbar}\epsilon\hat{H}(\hat{a}^{\dagger},\hat{a})\right)^{N}\ket{i}$$ Then using an overcomplete set $\{\ket{z_{j}}\}$ at each time $t_{j}$ where $j=1,\ldots,N$ and using the resolution of identity for coherent states one can show that $$\begin{split}
\lim_{\epsilon\rightarrow0^{+}}\lim_{N\rightarrow\infty}\bra{f} & \left(1-\frac{i}{\hbar}\epsilon\hat{H}(\hat{a}^{\dagger},\hat{a})\right)^{N}\ket{i}\\
=\int & \left(\prod_{j=1}^{N}\frac{dz_{j}d\bar{z}_{j}}{2\pi i}\right)e^{-\sum_{j}|z_{j}|^{2}}\\
\times\Big[\prod_{k=1}^{N-1}\bra{z_{k+1}} & \left(1-\frac{i}{\hbar}\epsilon\hat{H}(\hat{a}^{\dagger},\hat{a})\right)\ket{z_{k}}\Big]\\
\times\bra{f}\left(1-\frac{i}{\hbar}\epsilon\hat{H}(\hat{a}^{\dagger},\hat{a})\right)\ket{z_{N}} & \bra{z_{1}}\left(1-\frac{i}{\hbar}\epsilon\hat{H}(\hat{a}^{\dagger},\hat{a})\right)\ket{i}
\end{split}$$ In the limit $\epsilon\rightarrow0^{+}$ these matrix elements become $$\begin{split}
\bra{z_{k+1}}\left(1-\frac{i}{\hbar}\epsilon\hat{H}(\hat{a}^{\dagger},\hat{a})\right)\ket{z_{k}} & =\braket{z_{k+1}|z_{k}}\\
& \times\Big[1-\frac{i}{\hbar}\epsilon\hat{H}(\bar{z}_{k+1},z_{k})\Big]
\end{split}$$ And using the above information we can write $$\begin{split}
\bra{f}\hat{U}(t_{f},t_{i})\ket{i} & =\int\left(\prod_{j=1}^{N}\frac{dz_{j}d\bar{z}_{j}}{2\pi i}\right)e^{-\sum_{j}|z_{j}|^{2}}e^{\sum_{j=1}^{N-1}\bar{z}_{j+1}z_{j}}\\
\times & \prod_{k=1}^{N-1}\Big[1-\frac{i}{\hbar}\epsilon\hat{H}(\bar{z}_{k+1},z_{k})\Big]\braket{f|z_{N}}\braket{z_{1}|i}\\
& \times\Big[1-\frac{i\epsilon}{\hbar}\frac{\bra{f}\hat{H}\ket{z_{N}}}{\braket{f|z_{N}}}\Big]\Big[1-\frac{i\epsilon}{\hbar}\frac{\bra{z_{1}}\hat{H}\ket{i}}{\braket{z_{1}|i}}\Big]\\
=\int\mathcal{D}z\mathcal{D}\bar{z} & e^{\frac{i}{\hbar}\int_{t_{i}}^{t_{f}}dt\Big[\frac{\hbar}{2i}(z\partial_{t}\bar{z}-\bar{z}\partial_{t}z)-H(\bar{z},z)\Big]}\\
\times & e^{\frac{1}{2}(|z_{i}|^{2}+|z_{f}|^{2})}\bar{\psi}_{f}(z_{f})\psi_{i}(\bar{z}_{i})
\end{split}$$ where we have used the fact that $$\begin{split}
\bra{f} & =\int\frac{dz_{f}d\bar{z}_{f}}{2\pi i}e^{-|z_{f}|^{2}}\bar{\psi}_{f}(z_{f})\bra{z_{f}}\\
\ket{i} & =\int\frac{dz_{i}d\bar{z}_{i}}{2\pi i}e^{-|z_{i}|^{2}}\psi_{i}(\bar{z}_{i})\ket{z_{i}}
\end{split}$$
Extend the path integral to field theory
----------------------------------------
As we have seen in free-field theory we have harmonic oscillators with different momentum modes for which we have Hamiltonian in old Fock space as $$\hat{H}=\int d^{3}k \ \varepsilon_{\vec{k}}\hat{c}_{\vec{k}}^{\dagger}\hat{
c}_{\vec{k}}$$ Now we define states [@book:18638] $$\ket{\phi}=e^{\int d^{3}k\phi(\vec{k})\hat{c}_{\vec{k}}^{\dagger}}\ket{0}$$ and one can now check that $$\hat{c}_{\vec{l}}\ket{\phi}=\phi({\vec{l}})\ket{\phi}$$ as well as obeying the resolution of the identity in this space of states $$\hat{I}=\int\mathcal{D}\phi\mathcal{D}\bar{\phi}e^{-\int d^{3}k|\phi(\vec{k})|^{2}}\ket{\phi}\bra{\phi}$$ Therefore, using above extended quantities one can easily show that $$\begin{split}
\bra{f} & e^{-\frac{i}{\hbar}T\hat{H}}\ket{i}\\
=\int\mathcal{D}\phi\mathcal{D}\bar{\phi} & \bar{\Psi}_{f}(\phi(t_{f},\vec{k}))\Psi(\bar{\phi}(t_{i},\vec{k}))e^{\frac{1}{2}\int d^{3}k(|\phi(t_{i},\vec{k})|^{2}+|\phi(t_{f},\vec{k})|^{2})}\\
\times & e^{\frac{i}{\hbar}\int_{t_{i}}^{t_{f}}\int d^{3}k\Big[\frac{\hbar}{2i}(\phi(t,\vec{k})\partial_{t}\bar{\phi}(t,\vec{k})-\bar{\phi}(t,\vec{k})\partial_{t}\phi(t,\vec{k}))-\varepsilon_{\vec{k}}\bar{\phi}(t,\vec{k})\phi(t,\vec{k})\Big]}\\
=\int\mathcal{D}\phi\mathcal{D}\bar{\phi} & \ e^{\frac{i}{\hbar}\int_{t_{i}}^{t_{f}}\int d^{3}k\Big[i\hbar\bar{\phi}(t,\vec{k})\partial_{t}\phi(t,\vec{k})-\varepsilon_{\vec{k}}\bar{\phi}(t,\vec{k})\phi(t,\vec{k})\Big]}\\
\times & \bar{\Psi}_{f}(\phi(t_{f},\vec{k}))\Psi(\bar{\phi}(t_{i},\vec{k}))e^{\frac{1}{2}\int d^{3}k(|\phi(t_{i},\vec{k})|^{2}+|\phi(t_{f},\vec{k})|^{2})}
\end{split}$$ Now we consider partition function in grand canonical ensemble, defined by $$\mathcal{Z}=\text{Tr}e^{-\beta(\hat{H}-\mu\hat{N})}$$ which we want to evaluate using the path integral formalism where we extend the formalism in following way
- $\ket{i}=\ket{f}$ and arbitrary.
- summing over boundary states
- wick rotation to imaginary time $t\rightarrow-i\tau$ with time span $T\rightarrow-i\beta\hbar$ [@book:428978].
The result will be following [@laine2016basics], [@yang2011introduction] $$\mathcal{Z}=\int\mathcal{D}\phi\mathcal{D}\bar{\phi} \ e^{-S_{E}(\phi,\bar{\phi})}$$ where $$S_{E}(\phi,\bar{\phi})=\frac{1}{\hbar}\int_{0}^{\beta\hbar}d\tau\int d^{3}k\bar{\phi}(\tau,\vec{k})\Big[\hbar\partial_{\tau}-\xi_{\vec{k}}\Big]\phi(\tau,\vec{k})$$ with $\xi_{\vec{k}}=\varepsilon_{\vec{k}}-\mu$ and with periodic boundary condition $\phi(\tau,\vec{k})=\phi(\tau+\beta\hbar,\vec{k})$. This requiremnet suggests that we can decompose $\phi(\tau,\vec{k})$ in following way $$\phi(\tau,\vec{k})=\sum_{n}e^{i\omega_{n}\tau}\phi_{n}(\vec{k})$$ where $\omega_{n}=\frac{2\pi n}{\beta\hbar}$, which are known as matsubara frequencies.
Correlation function in curved spacetime under Bogoliubov transformation
------------------------------------------------------------------------
Recall that after doing Bogoliubov transformation new Hamiltonian for real scalar free-fiel theory was of the form $$\begin{split}
\hat{H} & =\int d^{3}k \ \varepsilon_{\vec{k}}[\cosh^{2}\theta_{\vec{k}}\hat{c}_{\vec{k}}^{\dagger}\hat{c}_{\vec{k}}+\sinh^{2}\theta_{\vec{k}}\hat{c}_{-\vec{k}}\hat{c}_{-\vec{k}}^{\dagger}\\
& +\sinh\theta_{\vec{k}}\cosh\theta_{\vec{k}}(\hat{c}_{-\vec{k}}\hat{c}_{\vec{k}}+\hat{c}_{\vec{k}}^{\dagger}\hat{c}_{-\vec{k}}^{\dagger})]\\
& \simeq\int d^{3}k \ \varepsilon_{\vec{k}}[(\cosh^{2}\theta_{\vec{k}}+\sinh^{2}\theta_{-\vec{k}})\hat{c}_{\vec{k}}^{\dagger}\hat{c}_{\vec{k}}\\
& +\sinh\theta_{\vec{k}}\cosh\theta_{\vec{k}}(\hat{c}_{-\vec{k}}\hat{c}_{\vec{k}}+\hat{c}_{\vec{k}}^{\dagger}\hat{c}_{-\vec{k}}^{\dagger})]
\end{split}$$ Therefore, for this Hamiltonian, the Euclidean action in the exponent of path integral description of partition function will be $$\begin{split}
S_{E}(\bar{\phi},\phi) & =\frac{1}{\hbar}\int_{0}^{\beta\hbar}d\tau\int d^{3}k\\
& \Big[\bar{\phi}(\tau,\vec{k})(\hbar\partial_{\tau}-(\cosh^{2}\theta_{\vec{k}}+\sinh^{2}\theta_{\vec{k}})\xi_{\vec{k}})\phi(\tau,-\vec{k})\\
-\xi_{\vec{k}}\sinh\theta_{\vec{k}} & \cosh\theta_{\vec{k}}(\phi(\tau,\vec{k})\phi(\tau,-\vec{k})+\bar{\phi}(\tau,\vec{k})\bar{\phi}(\tau,-\vec{k}))\Big]
\end{split}$$ which can be written in matrix representation in following way(denoting $\chi_{\vec{k}}=\cosh^{2}\theta_{\vec{k}}+\sinh^{2}\theta_{\vec{k}}, \ \eta_{\vec{k}}=\cosh\theta_{\vec{k}}\sinh\theta_{\vec{k}}$ and we assumed $\chi_{\vec{k}}=\chi_{-\vec{k}}, \ \eta_{\vec{k}}=\eta_{-\vec{k}}$) $$\begin{split}
S_{E}(\bar{\phi},\phi)=\frac{1}{\hbar}\int_{0}^{\beta\hbar}d\tau\int d^{3}k & \begin{pmatrix}
\bar{\phi}(\tau,\vec{k}) & \phi(\tau,-\vec{k})
\end{pmatrix}\\
\times\begin{bmatrix}
\frac{1}{2}(\hbar\partial_{\tau}-\xi_{\vec{k}}\chi_{\vec{k}}) & \xi_{\vec{k}}\eta_{\vec{k}}\\
\xi_{\vec{k}}\eta_{\vec{k}} & \frac{1}{2}(\hbar\overleftarrow{\partial}_{\tau}-\xi_{\vec{k}}\chi_{\vec{k}})
\end{bmatrix}
& \times\begin{pmatrix}
\phi(\tau,\vec{k})\\
\bar{\phi}(\tau,-\vec{k})
\end{pmatrix}\\
=\beta\sum_{n}\int d^{3}k \begin{pmatrix}
\bar{\phi}(\omega_{n},\vec{k}) & \phi(-\omega_{n},-\vec{k})
\end{pmatrix} & \times\\
\begin{bmatrix}
\frac{1}{2}(i\hbar\omega_{n}-\xi_{\vec{k}}\chi_{\vec{k}}) & \xi_{\vec{k}}\eta_{\vec{k}}\\
\xi_{\vec{k}}\eta_{\vec{k}} & \frac{1}{2}(-i\hbar\omega_{n}-\xi_{\vec{k}}\chi_{\vec{k}})
\end{bmatrix} & \times\begin{pmatrix}
\phi(\omega_{n},\vec{k})\\
\bar{\phi}(-\omega_{n},-\vec{k})
\end{pmatrix}
\end{split}$$ Now we define generating functional to get the correlation functions of any order. It is defined as(from now on we consider $\hbar=1$) $$\begin{split}
\mathcal{Z}[J,\bar{J}] & =\frac{1}{\mathcal{Z}}\int\mathcal{D}\phi \ \mathcal{D}\bar{\phi} \ e^{-S_{E}(\phi,\bar{\phi})}\\
& \times e^{\sum_{n}\int d^{3}k(\bar{J}(\omega_{n},\vec{k})\phi(\omega_{n},\vec{k})+J(\omega_{n},\vec{k})\bar{\phi}(\omega_{n},\vec{k}))}\\
& =e^{\bar{J}\mathcal{G}J}
\end{split}$$ where $\bar{J}\mathcal{G}J$ denotes a matrix multiplication with sum over modes and $\mathcal{G}$ is the propagator matrix or 2-point function matrix. Let’s write down $\bar{J}\mathcal{G}J$ explicitly $$\begin{split}
\bar{J}\mathcal{G}J & =\sum_{n}\int d^{3}k\begin{pmatrix}
\bar{J}(\omega_{n},\vec{k}) & J(-\omega_{n},-\vec{k})
\end{pmatrix}\\
& \times\frac{4}{\omega_{n}^{2}+\xi_{\vec{k}}^{2}(\chi_{\vec{k}}^{2}-4\eta_{\vec{k}}^{2})}\\
\times & \begin{bmatrix}
\frac{1}{2}(-i\omega_{n}-\xi_{\vec{k}}\chi_{\vec{k}}) & -\xi_{\vec{k}}\eta_{\vec{k}}\\
-\xi_{\vec{k}}\eta_{\vec{k}} & \frac{1}{2}(i\omega_{n}-\xi_{\vec{k}}\chi_{\vec{k}})
\end{bmatrix}\begin{pmatrix}
J(\omega_{n},\vec{k})\\
\bar{J}(-\omega_{n},-\vec{k})
\end{pmatrix}
\end{split}$$ Note that for $\theta_{\vec{k}}=0, \ \forall\vec{k}$ which is equivalent to saying we have not done Bogoliubov transformation in that case we can get back the known result that the 2-point correlation function is given by single quantity $<\bar{\phi}(\omega_{n},\vec{k})\phi(\omega_{n},\vec{k})>\propto\frac{1}{i\omega_{n}-\xi_{\vec{k}}}$.\
Now let’s do the matrix multiplication and write down the $\bar{J}\mathcal{G}J$ explicitly $$\begin{split}
\bar{J}\mathcal{G}J & =\sum_{n}\int d^{3}k\frac{4}{\omega_{n}^{2}+\xi_{\vec{k}}^{2}}\Big[\bar{J}(\omega_{n},\vec{k})(-i\omega_{n}-\xi_{\vec{k}}\chi_{\vec{k}})J(\omega_{n},\vec{k})\\
& -\xi_{\vec{k}}\eta_{\vec{k}}(\bar{J}(\omega_{n},\vec{k})\bar{J}(-\omega_{n},-\vec{k})+J(\omega_{n},\vec{k})J(-\omega_{n},-\vec{k}))\Big]
\end{split}$$ Note that for this case we found out that non-vanishing 2-point functions are $$\begin{split}
<\phi(\omega_{n},\vec{k})\phi(-\omega_{n},-\vec{k})> & =-\frac{8\xi_{\vec{k}}\eta_{\vec{k}}}{\omega_{n}^{2}+\xi_{\vec{k}}^{2}}\\
<\bar{\phi}(\omega_{n},\vec{k})\bar{\phi}(-\omega_{n},-\vec{k})> & =-\frac{8\xi_{\vec{k}}\eta_{\vec{k}}}{\omega_{n}^{2}+\xi_{\vec{k}}^{2}}\\
<\bar{\phi}(\omega_{n},\vec{k})\phi(\omega_{n},\vec{k})> & =\frac{4(-i\omega_{n}-\xi_{\vec{k}}\chi_{\vec{k}})}{\omega_{n}^{2}+\xi_{\vec{k}}^{2}}
\end{split}$$ Note that our result not only matches with known result in appropriate limit but also consistent with the fact that the first two result should be complex conjugate to each other, and here it’s trivially satisfied becuase of the fact that we consider $\eta_{\vec{k}},\chi_{\vec{k}}$ are real functions of $\vec{k}$ which we have assumed from the begining of the calculation for sake of convenience. Note also that non-zero value of the first two correlation function is a consequence of violation of particle number conservation.\
Evolution of number of particle in a state
------------------------------------------
To get to know that number of particles containing in states in new Fock space in free-field theory is not conserved one can also check following quantity which one can get from the Hamiltonian defined in new Fock space. Note that according to eq.(25) $$\begin{split}
\frac{d}{dt}(\hat{c}_{\vec{k}}^{\dagger}\hat{c}_{\vec{k}}) & =i[\hat{H},\hat{c}_{\vec{k}}^{\dagger}]\hat{c}_{\vec{k}}+i\hat{c}_{\vec{k}}^{\dagger}[\hat{H},\hat{c}_{\vec{k}}]\\
& =i\int d^{3}l \ \varepsilon_{\vec{l}}\Big[[(\chi_{\vec{l}}\hat{c}_{\vec{l}}^{\dagger}\hat{c}_{\vec{l}}+\eta_{\vec{l}}(\hat{c}_{-\vec{l}}\hat{c}_{\vec{l}}+\hat{c}_{\vec{l}}^{\dagger}\hat{c}_{-\vec{l}}^{\dagger})),\hat{c}_{\vec{k}}^{\dagger}]\hat{c}_{\vec{k}}\\
& +\hat{c}_{\vec{k}}^{\dagger}[(\chi_{\vec{l}}\hat{c}_{\vec{l}}^{\dagger}\hat{c}_{\vec{l}}+\eta_{\vec{l}}(\hat{c}_{-\vec{l}}\hat{c}_{\vec{l}}+\hat{c}_{\vec{l}}^{\dagger}\hat{c}_{-\vec{l}}^{\dagger})),\hat{c}_{\vec{k}}]\Big]\\
& =i\varepsilon_{\vec{k}}\Big[\chi_{\vec{k}}\hat{c}_{\vec{k}}^{\dagger}\hat{c}_{\vec{k}}+2\eta_{\vec{k}}\hat{c}_{-\vec{k}}\hat{c}_{\vec{k}}-\chi_{\vec{k}}\hat{c}_{\vec{k}}^{\dagger}\hat{c}_{\vec{k}}-2\eta_{\vec{k}}\hat{c}_{\vec{k}}^{\dagger}\hat{c}_{-\vec{k}}^{\dagger}\Big]\\
& =2i\eta_{\vec{k}}(\hat{c}_{-\vec{k}}\hat{c}_{\vec{k}}-\hat{c}_{\vec{k}}^{\dagger}\hat{c}_{-\vec{k}}^{\dagger})\neq0
\end{split}$$ Now if we consider a state say $\ket{\psi}$ in new Fock space then $$\begin{split}
\frac{d}{dt}\bra{\psi} & \hat{c}_{\vec{k}}^{\dagger}\hat{c}_{\vec{k}}\ket{\psi}\\
& =i\bra{\psi}[\hat{H},\hat{c}_{\vec{k}}^{\dagger}]\hat{c}_{\vec{k}}+i\hat{c}_{\vec{k}}^{\dagger}[\hat{H},\hat{c}_{\vec{k}}]\ket{\psi}\\
& =2i\eta_{\vec{k}}\bra{\psi}(\hat{c}_{-\vec{k}}\hat{c}_{\vec{k}}-\hat{c}_{\vec{k}}^{\dagger}\hat{c}_{-\vec{k}}^{\dagger})\ket{\psi}\neq 0, \ \text{in general}\\
& =2i\eta_{\vec{k}}[\bra{\psi}\hat{c}_{-\vec{k}}\hat{c}_{\vec{k}}\ket{\psi}-\bra{\psi}\hat{c}_{-\vec{k}}\hat{c}_{\vec{k}}\ket{\psi}^{*}]\\
& =-4\eta_{\vec{k}}\text{Im}[\bra{\psi}\hat{c}_{-\vec{k}}\hat{c}_{\vec{k}}\ket{\psi}]\\
\implies\frac{d}{dt}\bra{\psi} & \sum_{\vec{k}}\hat{c}_{\vec{k}}^{\dagger}\hat{c}_{\vec{k}}\ket{\psi}=\frac{d}{dt}\bra{\psi}\hat{N}\ket{\psi}\neq0
\end{split}$$ which will be clear if one use completeness relation of occupation number basis of new Fock space. This clearly shows violation of particle number conservation.\
Recall I have mentioned earlier that number operator is a generator of phase in coherent states, in the above calculation we can alearly see that number operator does not commute with the Hamiltonian under Bogoliubov transformation $[\hat{H},\hat{N}]\neq0$, therefore phase generator or the charge corresponfing to U(1) symmetry is broken. Because of such non-commutativity Hamiltonian and number operator don’t have simultaneous eigenstates. For example we can clearly see that vacuum state which is defined to be a state that is annihilated by annihilation operator is no longer an eigenstate of Hamiltonian operator although it conatains zero number of particles since vacuum expectation value of Hamiltonian is zero. And we can also check that there is no particle production happen in vacuum state since $$\begin{split}
\frac{d}{dt}\bra{0(\theta)} & \hat{c}_{\vec{k}}^{\dagger}\hat{c}_{\vec{k}}\ket{0(\theta)}\\
=2i\eta_{\vec{k}}\bra{0(\theta)} & (\hat{c}_{-\vec{k}}\hat{c}_{\vec{k}}-\hat{c}_{\vec{k}}^{\dagger}\hat{c}_{-\vec{k}}^{\dagger})\ket{0(\theta)}=0\\
\implies\frac{d}{dt}\bra{0(\theta)} & \hat{N}\ket{0(\theta)}=0
\end{split}$$ Note that saying particle creation out of vacuum by showing that old vacuum state in old Fock space representation is not true vacuum in new Fock space(which is mostly done [@degner2010cosmological], [@Ford:1997hb], [@biswas1995particle], [@book:274311], [@hossenfelder2003particle], [@0305-4470-8-4-022], [@ford1987gravitational], [@birrell1980massive], [@winitzki2005cosmological], [@frieman1989particle]) is not equivalent of saying violation of particle number conservation because old vacuum state and multi-particle states in old Fock space do not belong to new Fock space in field theory because of infinite volume limit(even is some cases people consider amplitude of particle propagation under time evolution in path integral formalism by considering a initial and final states to be vacuum states at different time [@chitre1977path], [@duru1986particle] which is also not a correct description). Therefore, one should carefully choose action of operactors according to their domain.\
Now let’s check whether or not the vacuum state remains vacuum states under infinitesimal time evolution $$\begin{split}
\hat{c}_{\vec{k}}e^{-i\hat{H}\epsilon}\ket{0(\theta)} & =e^{-i\hat{H}\epsilon}e^{i\hat{H}t}\hat{c}_{\vec{k}}e^{-i\hat{H}\epsilon}\ket{0(\theta)}\\
=e^{-i\hat{H}\epsilon} & \Big[\hat{c}_{\vec{k}}+i\epsilon[\hat{H},\hat{c}_{\vec{k}}]+\mathcal{O}(\epsilon^{2})\Big]\ket{0(\theta)}\\
=e^{-i\hat{H}t} & \Big[i\epsilon[\hat{H},\hat{c}_{\vec{k}}]+\mathcal{O}(\epsilon^{2})\Big]\ket{0(\theta)}\\
=e^{-i\hat{H}t} & [(-\chi_{\vec{k}}\hat{c}_{\vec{k}}-2\eta_{\vec{k}}\hat{c}_{-\vec{k}}^{\dagger})+\mathcal{O}(\epsilon^{2})]\ket{0(\theta)}\neq\ket{0(\theta)}
\end{split}$$ So, we can see that even under infinitesimal time evolution vacuum state is no longer vacuum state of new transformed Fock space.\
Note also that in this whole setup we have not considered about action functional of this theory. From the action one can easily notice that since it is a real scalar field theory there is no breaking of $U(1)$-symmetry at all but once we write down the Hamiltonian and define what would be proper action of U(1) to check not charge conservation but rather particle number conservation because charge conservation may not violate because from vacuum state one can produce particle-antiparticle pairs but since for a real scalar field charge is zero we don’t have to worry about charge conservation.
Non-invariance under U(1) action in an example of 2-particle scattering interaction
-----------------------------------------------------------------------------------
Now consider that system is interacting with the following interaction term in old Fock space represenatation $$\hat{H}_{\text{int}}=\sum_{\vec{q},\vec{k},\vec{l}}v(\vec{q})\hat{a}_{\vec{k}+\vec{q}}^{\dagger}\hat{a}_{\vec{l}-\vec{q}}^{\dagger}\hat{a}_{\vec{l}}\hat{a}_{\vec{k}}$$ where $v(\vec{q})$ is the interaction strength which depends on the exchanged momentum 2 particle scattering process at tree-level.\
If we do Bogolibuov transformation(then taking thermodynamic limit since we are considering field theory) which is equivalent to switch on gravity then in new Fock space representation we will get following interaction term $$\begin{split}
\hat{H}_{\text{int}}=\sum_{\vec{q},\vec{k},\vec{l}}v(\vec{q}) & \Big[(\cosh\theta_{\vec{k}+\vec{q}}\hat{c}_{\vec{k}+\vec{q}}^{\dagger}+\sinh\theta_{\vec{k}+\vec{q}}\hat{c}_{-\vec{k}-\vec{q}})\\
& \times(\cosh\theta_{\vec{l}-\vec{q}}\hat{c}_{\vec{l}-\vec{q}}^{\dagger}+\sinh\theta_{\vec{l}-\vec{q}}\hat{c}_{-\vec{l}+\vec{q}})\\
\times(\cosh\theta_{\vec{l}}\hat{c}_{\vec{l}}+\sinh\theta_{\vec{l}} & \hat{c}_{-\vec{l}}^{\dagger})\times(\cosh\theta_{\vec{k}}\hat{c}_{\vec{k}}+\sinh\theta_{\vec{k}}\hat{c}_{-\vec{k}}^{\dagger})\Big]\\
=\cosh\theta_{\vec{k}+\vec{q}}\cosh\theta_{\vec{l}-\vec{q}} & \cosh\theta_{\vec{l}}\cosh\theta_{\vec{k}}\hat{c}_{\vec{k}+\vec{q}}^{\dagger}\hat{c}_{\vec{l}-\vec{q}}^{\dagger}\hat{c}_{\vec{l}}\hat{c}_{\vec{k}}\\
+\cosh\theta_{\vec{k}+\vec{q}}\cosh\theta_{\vec{l}-\vec{q}} & \cosh\theta_{\vec{l}}\sinh\theta_{\vec{k}}\hat{c}_{\vec{k}+\vec{q}}^{\dagger}\hat{c}_{\vec{l}-\vec{q}}^{\dagger}\hat{c}_{\vec{l}}\hat{c}_{-\vec{k}}^{\dagger}\\
+\cosh\theta_{\vec{k}+\vec{q}}\cosh\theta_{\vec{l}-\vec{q}} & \sinh\theta_{\vec{l}}\cosh\theta_{\vec{k}}\hat{c}_{\vec{k}+\vec{q}}^{\dagger}\hat{c}_{\vec{l}-\vec{q}}^{\dagger}\hat{c}_{-\vec{l}}^{\dagger}\hat{c}_{\vec{k}}\\
+\cosh\theta_{\vec{k}+\vec{q}}\cosh\theta_{\vec{l}-\vec{q}} & \sinh\theta_{\vec{l}}\sinh\theta_{\vec{k}}\hat{c}_{\vec{k}+\vec{q}}^{\dagger}\hat{c}_{\vec{l}-\vec{q}}^{\dagger}\hat{c}_{-\vec{l}}^{\dagger}\hat{c}_{-\vec{k}}^{\dagger}\\
+\cosh\theta_{\vec{k}+\vec{q}}\sinh\theta_{\vec{l}-\vec{q}} & \cosh\theta_{\vec{l}}\cosh\theta_{\vec{k}}\hat{c}_{\vec{k}+\vec{q}}^{\dagger}\hat{c}_{-\vec{l}+\vec{q}}\hat{c}_{\vec{l}}\hat{c}_{\vec{k}}\\
+\cosh\theta_{\vec{k}+\vec{q}}\sinh\theta_{\vec{l}-\vec{q}} & \cosh\theta_{\vec{l}}\sinh\theta_{\vec{k}}\hat{c}_{\vec{k}+\vec{q}}^{\dagger}\hat{c}_{-\vec{l}+\vec{q}}\hat{c}_{\vec{l}}\hat{c}_{-\vec{k}}^{\dagger}\\
+\cosh\theta_{\vec{k}+\vec{q}}\sinh\theta_{\vec{l}-\vec{q}} & \sinh\theta_{\vec{l}}\cosh\theta_{\vec{k}}\hat{c}_{\vec{k}+\vec{q}}^{\dagger}\hat{c}_{-\vec{l}+\vec{q}}\hat{c}_{-\vec{l}}^{\dagger}\hat{c}_{\vec{k}}\\
+\cosh\theta_{\vec{k}+\vec{q}}\sinh\theta_{\vec{l}-\vec{q}} & \sinh\theta_{\vec{l}}\sinh\theta_{\vec{k}}\hat{c}_{\vec{k}+\vec{q}}^{\dagger}\hat{c}_{-\vec{l}+\vec{q}}\hat{c}_{-\vec{l}}^{\dagger}\hat{c}_{-\vec{k}}^{\dagger}\\
+\sinh\theta_{\vec{k}+\vec{q}}\cosh\theta_{\vec{l}-\vec{q}} & \cosh\theta_{\vec{l}}\cosh\theta_{\vec{k}}\hat{c}_{-\vec{k}-\vec{q}}\hat{c}_{\vec{l}-\vec{q}}^{\dagger}\hat{c}_{\vec{l}}\hat{c}_{\vec{k}}\\
+\sinh\theta_{\vec{k}+\vec{q}}\cosh\theta_{\vec{l}-\vec{q}} & \cosh\theta_{\vec{l}}\sinh\theta_{\vec{k}}\hat{c}_{-\vec{k}-\vec{q}}\hat{c}_{\vec{l}-\vec{q}}^{\dagger}\hat{c}_{\vec{l}}\hat{c}_{-\vec{k}}^{\dagger}\\
+\sinh\theta_{\vec{k}+\vec{q}}\cosh\theta_{\vec{l}-\vec{q}} & \sinh\theta_{\vec{l}}\cosh\theta_{\vec{k}}\hat{c}_{-\vec{k}-\vec{q}}\hat{c}_{\vec{l}-\vec{q}}^{\dagger}\hat{c}_{-\vec{l}}^{\dagger}\hat{c}_{\vec{k}}\\
+\cosh\theta_{-\vec{k}-\vec{q}}\cosh\theta_{\vec{l}-\vec{q}} & \sinh\theta_{\vec{l}}\sinh\theta_{\vec{k}}\hat{c}_{-\vec{k}-\vec{q}}\hat{c}_{\vec{l}-\vec{q}}^{\dagger}\hat{c}_{-\vec{l}}^{\dagger}\hat{c}_{-\vec{k}}^{\dagger}\\
+\sinh\theta_{\vec{k}+\vec{q}}\sinh\theta_{\vec{l}-\vec{q}} & \cosh\theta_{\vec{l}}\cosh\theta_{\vec{k}}\hat{c}_{-\vec{k}-\vec{q}}\hat{c}_{-\vec{l}+\vec{q}}\hat{c}_{\vec{l}}\hat{c}_{\vec{k}}\\
+\sinh\theta_{\vec{k}+\vec{q}}\sinh\theta_{\vec{l}-\vec{q}} & \cosh\theta_{\vec{l}}\sinh\theta_{\vec{k}}\hat{c}_{-\vec{k}-\vec{q}}\hat{c}_{-\vec{l}+\vec{q}}\hat{c}_{\vec{l}}\hat{c}_{-\vec{k}}^{\dagger}\\
\end{split}$$ $$\begin{aligned}
\begin{split}
+\sinh\theta_{\vec{k}+\vec{q}}\sinh\theta_{\vec{l}-\vec{q}} & \sinh\theta_{\vec{l}}\cosh\theta_{\vec{k}}\hat{c}_{-\vec{k}-\vec{q}}\hat{c}_{-\vec{l}+\vec{q}}\hat{c}_{-\vec{l}}^{\dagger}\hat{c}_{\vec{k}}\\
+\sinh\theta_{\vec{k}+\vec{q}}\sinh\theta_{\vec{l}-\vec{q}} & \sinh\theta_{\vec{l}}\sinh\theta_{\vec{k}}\hat{c}_{-\vec{k}-\vec{q}}\hat{c}_{-\vec{l}+\vec{q}}\hat{c}_{-\vec{l}}^{\dagger}\hat{c}_{-\vec{k}}^{\dagger}
\end{split}\end{aligned}$$ Note that even the interaction term gets modified in curved spacetime such way that it violates particle number conservation because of the fact interaction is not invariant under the action of global $U(1)$ group but momentum conservation still holds. Remember we have chosen a curved spacetime where notion momentum modes are well-defined which means that spacetime line-element or metric is invariant under spatial translations.\
According to the non-vanishing 2-point function that we have found all the interaction terms in the interaction Hamiltonian in new Fock space representation contribute in 4-point and higher order correlation function in interacting theory which is easy to see if one follows perturbative approach.\
Conclusion
----------
In the beginning of this article I emphasized on the fact that in thermodynamic limit although we have 2 disjoint vector spaces but still the we can do the canonical transformation. And we also restrict ourself to new Fock space because after taking infinite volume limit we can’t get back to the old Fock space. In this article I am able to show how to look at the particle production phenomena under change in coordinate transformation or under frame change which is equivalent of doing Bogoliubov transformation in field theory in thermodynamic limit. We have shown how does change in frame breaks both global and local U(1) invariance which is suitably defined. We have also seen that there is no particle production happen out of vacuum state in new transformed Fock space under time evolution but it can happen out of other many-particle states and vacuum state is not an eigenvector of Hamltonian operator in transformed Fock space and vacuum state does not remain vacuum state under time evolution.
Acknowledgement
===============
Author wants to thank Dr. Golam Mortuza Hossain, Gopal Sardar for helpful discussion regarding the subject matter and their comments on the idea of this paper. Author would also like to thank CSIR to support this work through JRF fellowship.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We prove ballistic behaviour as well as an annealed functional central limit theorem for random walks in mixing random environments (RWRE). The ballistic hypothesis will be an effective polynomial condition as the one introduced by Berger, Drewitz, and Ramírez (*Comm. Pure Appl. Math,* [**67**]{}, (2014) 1947–1973). The novel idea therein was the construction of several simultaneous renormalization steps, providing more flexibility for seed estimates. For our proof, we indeed follow a similar path, and introduce a new mixing effective criterion which will be implied by the polynomial condition. This allows us to prove, in a mixing framework, the RWRE conjecture concerning the equivalence between each condition $(T^\gamma)|\ell$, for $\gamma\in (0,1)$ and $\ell \in \mathbb S^{d-1}$. This work complements the previous work of Guerra (*Ann. Probab.* [**47**]{} (2019) 3003–3054) and completes the answer about the meaning of condition $(T'')|\ell$ in a mixing setting, an open question posed by Comets and Zeitouni (*Ann. Probab.* [**32**]{} (2004) 880–914).'
address:
- 'Pontificia Universidad Católica de Chile\'
- 'Universidade Federal do Rio de Janeiro\'
- 'Universidade Federal do Rio de Janeiro\'
author:
-
-
-
title: On an Effective Class of Ballistic Random Walks in Mixing Random Environments
---
Introduction {#secIntro}
============
Random walk in random environment (RWRE) is a widely-studied stochastic model. However, its asymptotic behaviour is still poorly understood when the underlying dimension $d$ of the walk is larger than one, especially for the non-i.i.d. random environment case. The article [@Gue17] arose while the authors were holding weekly seminars and opened a path to study ballistic properties of the random walk process under weaker assumptions than the usual Kalikow’s condition. This condition intrinsically implies ballistic nature (cf. Lemma 2.2 in[@SZ99]) for the walk and was the standing assumption in previous works as [@CZ01], [@CZ02] and [@RA03], among others. The i.i.d. random environment case has shown that Kalikow’s condition is not the appropriate criterion in order to detect ballistic behaviour, indeed is stronger.
The main issue to establish a precise relation connecting local conditions on the environment law and a given asymptotic behaviour (which might be considered the main objective in this field) comes from the fact that under the annealed measure the random walk process is not Markovian. The non-Markovian character, already even when the environment is endowed with an i.i.d. structure, makes harder to use ergodic devices to handle its asymptotic laws in terms of conditions on the random environment distribution. In turn, the loss of Markovian property is associated to the very definition of the annealed law as the semidirect product between quenched and environment laws. Indeed this does not inherit the Markov property coming from its quenched factor. As a result of this increasing complexity, when the environment has a mixing-type of structure, the random walk process loses further properties. For instance, one can show that the multidimensional i.i.d. renewal structure of [@SZ99] fails to be a regeneration in mixing environments.
Our main objectives in this article are firstly to provide an effective condition under which ballistic behaviour is fulfilled. Secondly we derive a proof which extends the i.i.d. conjecture in RWRE, regarding the equivalence between all of the ballisticity conditions $(T^\gamma)|\ell$, for $\gamma \in (0,1)$ and $\ell\in\mathbb S^{d-1}$, where as a matter of definition $$\mathbb S^{d-1}:=\{x\in\mathbb R^d:\, \sum_{i=1}^d x_i^2=1\}.$$ The former aim will be essentially accomplished using diffusive controls on the orthogonal fluctuation to the *quasi-asymptotic direction* of the walk process; these controls are provided under stretched exponential upper bounds of the unlikely walk exit from large boxes. Renormalization procedures under those diffusive controls yield sharp inequalities for the environmental probability of unlikely quenched events. Following Sznitman’s construction [@Sz00] in a mixing setting (cf. Section 5 in [@Gue17]), these suffice to get appropriate estimates for renewal time tails. The proof will be concluded in these terms after an application of the functional CLT of [@CZ02], which in turn, once the integrability conditions are fulfilled, depends only on the theory of Markov chains with infinite connections [@IG80].
For the second aim we will strongly follow the proof argument given in [@BDR14], where the authors proved the so-called effective criterion starting from a class of polynomial condition. Indeed a first target for us shall be to construct a *mixing effective criterion*, which will imply condition $(T')|\ell$. Afterwards, an exhaustive use of renormalization schemes shall produce the decay required by our *mixing effective criterion*, under an effective polynomial condition introduced in Section \[secpoly\]. Alongside, the big picture behind the proof strategy in [@BDR14] is right for our purposes, nevertheless we must be apart of their proof argument owed to minor problems related to split quasi independent environmental events. We present an appropriate version of that argument within the framework of mixing random environments as it shall be explained in the corresponding section. We start with formal developments in the next paragraph.
A $d$-dimensional random walk in a random environment (RWRE) evolves on the lattice $\mathbb Z^d$, where the environmnet is specified by random transitions to nearest neighbors at each lattice site. More precisely, let $d\geq2$ (case $d=1$ is well understood and we recommend the survey [@Ze04]) be the underlying dimension, $\kappa\in (0, 1/(4d))$ and consider the $(2d-1)$-dimensional simplex $\mathcal P_\kappa$ defined by $$\label{simplex}
\mathcal P_\kappa:=\left\{x\in\mathbb R^{2d}: \, \sum_{i=1}^{2d}x_i=1,\, x_i\geq 2\kappa, \,\,\mbox{ for }i\in[1,2d] \right\}.$$ Consider the product space $\Omega:=\mathcal P_\kappa^{\mathbb Z^d}$ which denotes the set of environments, endowed with the canonical product $\sigma$-algebra $\mathfrak{F}_\Omega$ and fix an ergodic probability measure $\mathbb P$ on $\mathfrak{F}_\Omega$. We use the notation $|\,\cdot\,|_1$ and $|\,\cdot\,|_2$ to denote the $\ell_1$ and $\ell_2$-distance on $\mathbb R^d$ respectively; and furthermore, for $A, B\subset \mathbb Z^d$, $i\in \{1,2\}$, the notation $d_i(A,B)$ stands for the canonical $\ell_i$-distance between sets $A,\, B$, i.e. $d_i(A,B):=\inf\{|x-y|_i,\, x\in A, y\in B\}$. For a given environment $\omega:=(\omega(y, e))_{y\in \mathbb Z^d, e\in \mathbb Z^d, |e|_1=1} \in \Omega$, $x\in \mathbb Z^d$, the *quenched law* $P_{x,\omega}$ is defined as the law of the canonical Markov chain $(X_n)_{n\geq 0}$ with state space in $\mathbb Z^d$, starting from $x\in \mathbb Z^d$ and stationary transition probabilities given by the environment $\omega$, i.e. $$\begin{gathered}
P_{x,\omega}[X_0=x]=1 \\
P_{x,\omega}[X_{n+1}=X_n+e|X_n]=\omega(X_n,e),\,\, |e|_1=1.\end{gathered}$$ The *annealed law* $P_x$ is defined as the semidirect product probability measure $\mathbb P\otimes P_{x,\omega}$ on the product space $\Omega\times (\mathbb Z^d)^{\mathbb N}$. With some abuse of notation we call as well $P_x$ the marginal law under this measure of the random walk process $(X_n)_{n\geq0}$. Before introducing the class of mixing assumptions that will be considered, we recall the definition of $r$-Markovian field:
For $r\geq 1$, let $\partial^r V=\{z\in \mathbb Z\setminus V: d_1(z, V)\leq r\}$ be the $r-$boundary of the set $V \subset \mathbb Z$. A random environment $(\mathbb P, \mathfrak{F}_{\Omega})$ on $\mathbb Z^d$ is called $r$-Markovian if for any finite $V\subset \mathbb Z^d$, $$\mathbb P[(\omega_{x})_{x\in V}\in \cdot|\mathfrak{F}_{V^c}]=\mathbb P[(\omega_x)_{x\in V}\in \cdot|\mathfrak{F}_{\partial ^r V}], \,\, \mathbb P-a.s.,$$ where $\mathfrak{F}_{\Lambda}=\sigma(\omega_x, \, x\in \Lambda)$.
We can now introduce two types of randomness on the environment to be used throughout this article as standing assumptions:
Let $C$ and $g$ be positive real numbers. We will say that an $r$-Markovian field $(\mathbb P, \mathfrak{F}_{\Omega} )$ satisfies the strong mixing condition **(SM)**$_{C,g}$ if for all finite subsets $\Delta\subset V \subset \mathbb Z^d$ with $d_1(\Delta, V^c)\geq r$, and $A\subset V^c$, $$\label{sma}
\frac{d\mathbb P[(\omega_x )_{x\in \Delta}\in \cdot | \eta]}{d \mathbb P[(\omega_x )_{x\in \Delta}\in \cdot | \eta']}\leq \exp\left( C \sum_{x\in \partial^r \Delta, y \in \partial^r A}e^{-g|x-y|_1}\right)$$ for $\mathbb P-$a.s. all pairs of configurations $\eta, \,\eta'\in \mathcal P_{\kappa}^{\mathbb Z^d} $ which agree over the set $V^c \backslash A$. Here we have used the notation $$\mathbb P[(\omega_x )_{x\in \Delta}\in \cdot | \eta]=\mathbb P[(\omega_x )_{x\in \Delta}\in \cdot |\mathfrak{F}_{V^c}]|_{(\omega_x)_{x\in V^c}=\eta}.$$
Let $C$ and $g$ be positive real numbers. We say an $r$-Markovian field $(\mathbb P, \mathfrak{F}_{\Omega} )$ satisfies Guo’s strong mixing condition **(SMG)**$_{C,g}$ if for all finite subsets $\Delta\subset V \subset \mathbb Z^d$ with $d_1(\Delta, V^c)\geq r$, and $A\subset V^c$, $$\label{smg}
\frac{d\mathbb P[(\omega_x )_{x\in \Delta}\in \cdot | \eta]}{d \mathbb P[(\omega_x )_{x\in \Delta}\in \cdot | \eta']}\leq \exp\left( C \sum_{x\in \Delta, y \in A}e^{-g |x-y|_1}\right) \,$$ for $\mathbb P-$a.s. all pairs of configurations $\eta, \,\eta'\in \mathcal P_{\kappa}^{\mathbb Z^d} $ which agree over the set $V^c \backslash A$.
The first condition **(SM)**$_{C,g}$ is in the spirit of the one introduced by Dobrushin and Shlosman in [@DS85]; the second one **(SMG)**$_{C,g}$ is motivated by the work of Guo [@Guo14].
It is straightforward to see that the random walk process $(X_n)_{n\geq 0}$ under the annealed law $P_0$ is not Markovian (think about the first return time to the initial position). In this framework, worst things happen under either **(SM)**$_{C,g}$ or **(SM)**$_{C,g}$. For instance it is direct to see that the renewal structure introduced in [@SZ99] fails to be a regeneration if $g$ is finite, indeed it is unknown if any regeneration structure exists at all.
For a subset $A\subset\mathbb Z^d$ we introduce its boundary $\partial A:=\partial^1 A$ and the random variable $T_A$ to denote: $$\begin{gathered}
%H_A:=\inf\{n\geq 0:\, X_n\in A\}\\
T_A:=\inf\{n\geq 0:\, X_n\notin A\}.%=H_{\mathbb Z^d\setminus A}.\end{gathered}$$ Following the original formulations in [@Sz02] for i.i.d. random environments and as a result of Lemma \[lemmaTgamma\], for $\gamma\in (0,1]$ and $\ell\in \mathbb S^{d-1}$ we can and do define the so-called ballisticity conditions as follows.
\[deftgammaandtprime\] We say that condition $(T^\gamma)|\ell$ holds, if for any positive number $b$, there exists some neighborhood $U\subset \mathbb S^{d-1}$ of $\ell$, such that $$\limsup_{\substack{L\rightarrow\infty}} L^{-\gamma}\ln\left(P_0\left[\widetilde{T}_{-bL}^{\ell'}<T_{L}^{\ell'}\right]\right)<0$$ holds, for each $\ell'\in U$, where we have used the standard notation (in the RWRE literature): for $c\in \mathbb R$ and $u\in \mathbb S^{d-1}$, $$\begin{gathered}
\label{extimslab}
T_{c}^u:=\inf\{n\geq 0: X_n\cdot u\geq c\},\,\, \mbox{along with} \\
\widetilde{T}_{c}^{u}:=\inf\{n\geq 0: X_n\cdot u\leq c \}.\end{gathered}$$ One then defines $(T')|\ell$ as the requirement that $(T^\gamma)|\ell$ holds, for all $\gamma\in (0,1)$.
Conditions that require weaker decays than the previous ones are useful in this work.
\[defpolasympandec\] For $M>0$ we say that condition $(P)_M|\ell$ holds, if for all $b>0$ there exists some neighborhood $U\subset \mathbb S^{d-1}$ of $\ell$ such that $$\lim_{\substack{L\rightarrow\infty}}L^{M}P_0\left[\widetilde{T}_{-bL}^{\ell'}<T_{L}^{\ell'}\right]=0,$$ for all $\ell'\in U$. Furthermore, for a given rotation $R$ of $\mathbb R^d$ with $R(e_1)=\ell$, real numbers $L, L', \widetilde{L}>3\sqrt{d}$ we denote a *box specifications* by $\mathcal B:=\mathcal B(R, L, L', \widetilde L)$ to which we attach the box $B_{\mathcal B}$ defined by: $$B_{\mathcal B}:=R\left((-L,L')\times (-\widetilde L, \widetilde L)^{d-1}\right)$$ along with its *frontal boundary part* defined by $$\partial^+B_{\mathcal B}:=\partial B_{\mathcal B}\cap\{z: z\cdot \ell\geq L', |z\cdot R(e_j)|<\widetilde L\, \forall j\in [2,d]\}.$$ We further introduce and attach to $\mathcal B$ random variables: $$\begin{gathered}
q_{\mathcal B}:=P_{0,\omega}\left[X_{T_{B_{\mathcal B}}}\notin \partial^+B_{\mathcal B}\right]=1-p_{\mathcal B}\,\, \mbox{ and} \\
\rho_{\mathcal B}:=\frac{q_{\mathcal B}}{p_{\mathcal B}}.\end{gathered}$$ We then say that the *mixing effective criterion* in direction $\ell$ holds and denoted this by $(EC)|\ell$ holds, if $$\inf_{\substack{\mathcal B, a}}c_{11}\widetilde L^{d-1}L^{4(d-1)+1}\mathbb E\left[\rho_{\mathcal B}^{a}\right]<1,$$ where the infimum runs over $a\in (0,1]$, all the box specifications $\mathcal B(R,L-2,L+2, \widetilde L )$ with $R(e_1)=\ell$, $L^4>\widetilde L>L$, $L>c_{10}$, and where in turn $c_{10},c_{11}>3\sqrt d$ are certain prescribed dimensional constants. As it was anticipated one of the advantage of this criterion is its effective character, i.e. it can be checked by inspection on a finite box.
- Notice that in Definition \[defpolasympandec\], the polynomial decay required therein and the condition itself is not effective. That condition is only instrumental and will imply our effective polynomial condition of Definition \[defpoly\]. We would rather define this asymptotic condition instead of the effective one because is simpler and displays better the connection with conditions $(T^\gamma)|\ell$.
- We observe that in our *mixing effective criterion* we allow to orthogonal dimensions be of order $L^4$ and not $L^3$ as usual. That flexibility was disposed and will be proven (cf. Section \[sectionce\]) in this form to simplify the proof of Theorem \[mainth1\].
Our first result is a proof of the RWRE conjecture of Sznitman [@Sz02] in the present mixing setting.
\[mainth1\] Let $M>9d$, $\gamma\in (0,1)$, $C,g>0$ and $\ell\in \mathbb S^{d-1}$. Then the following statements are equivalent under either **(SMG)**$_{C,g}$ or **(SM)**$_{C,g}$:
- $(T^\gamma)|\ell$ holds.
- $(T')|\ell$ holds.
- $(P)_M|\ell$ holds.
- $(EC)|\ell$ holds.
[ *Comment.*]{} The question concerning the validity of extended RWRE conjecture on the equivalence between condition $(T)|\ell$ and any of the above ballisticity conditions, proved in [@GR18] for i.i.d. environments, remains unanswered in the mixing framework.
We now state the result regarding the annealed invariance principle.
\[mainth2\] Let $C,g,>0$, $M>9d$, $\ell\in \mathbb S^{d-1}$ with $g>2\ln(1/\kappa)$ and assume that the RWRE satisfies conditions: $(P)_M|\ell$, and either: **(SMG)**$_{C,g}$ or **(SM)**$_{C,g}$ hold. Then there exist a deterministic non-degenerate covariance matrix $R$ and a deterministic vector $v$ with $v\cdot \ell>0$, such that under $P_0$; denoting by $$S_n(t):=\frac{X_{[nt]}-vt}{\sqrt{n}},$$ the path $S_n(t)$ taking values in the space of right continuous functions possessing left limits endowed with the supremum norm, converges in law to a standard Brownian motion with covariance matrix $R$.
Notice that Theorem \[mainth1\] requires an assumption concerning the mixing strength: $g>2\ln(1/\kappa)$. Of course $g$ can be taken arbitrary large under the assumption of i.i.d. environment. To some extent the proof will display the minimality of this requirement if we assume the step involved in Proposition \[protransfluctuation\] is an unescapable step into the proof. On the other hand, we remark that at the same time of our definition of scales and good boxes was made, a lesser requirement for the polynomial condition was needed. Indeed a detailed analysis of the proof makes us see that it actually requires a little more than $M=9d-1$.
Let us explain the structure of this work. Section \[secPrel\] introduces primary properties about $(T^\gamma)|\ell$ conditions, in particular there we prove that a box version is equivalent to the slab version given here. It is introduced as well in Section \[secPrel\] the approximate renewal structure of Comets and Zeitouni [@CZ01]. Furthermore, at the end of the section we lay out the schedule to follow for the main proof. Section \[secbasicpropTgamma\] shows that stretched exponential moments are fulfilled under corresponding stretched exponential decays of unlikely exit events. Section \[sectionce\] will provide a construction of a mixing effective criterion implying condition $(T')|\ell$. Section \[secpoly\] has the purpose of introducing a polynomial effective condition, under which the effective criterion holds, besides a proof for Theorem \[mainth1\] is disposed at the end of that section. Finally in Section \[secproofmainth\] is developed appropriate machinery to get estimates for the approximate renewal time tails, and also is proven therein Theorem \[mainth2\].
Preliminaries: Basic fact about $(T^\gamma)$ and Approximate Renewal Structure in Mixing Environments {#secPrel}
=====================================================================================================
On equivalent formulations for $(T^\gamma)$
-------------------------------------------
The target of this subsection will be to provide different formulations of the condition $(T^\gamma)|l$, for parameters $l\in \mathbb S^{d-1}$ and $\gamma\in (0,1)$. This section follows a similar analysis to Section 2.1 in [@Gue17]. We begin with the definition of directed systems of slabs.
We say that $l_0,l_1, \ldots ,l_k\, \in \mathbb S^{d-1}$, $a_0=1, a_1, a_2, \ldots, a_k\,>0$, $b_0,b_1, \ldots,b_k>0 $ generate an $l_0$-directed system of slabs of order $\gamma$, when
- $l_0, l_1\ldots, l_k$ generate $\mathbb R^d$
- $\mathcal D=\{x\in \mathbb R^d: x\cdot l_0\in [-b_0,1], l_i\cdot x\geq -b_i, i\in[0,k]\}\,\subset \{x\in \mathbb R^d: l_i\cdot x<a_i, i\in[1,d]\}$
- $\limsup_{\substack{L\rightarrow \infty}}L^{-\gamma}\,\ln P_0\left[\widetilde{T}_{-b_iL}^{l_i}<T_{a_iL}^{l_i}\right]<0$, for $i\in [0,k]$.
For positive numbers $L$ and $L'$, we introduce the notation $B_{L,L',l}(x)$ to denote the box $$\label{generalboxes}
B_{L,L',l}(x):=
x+R\left(\left(-L,L\right)\times\left(-L',L'\right)^{d-1}\right)
\cap\mathbb{Z}^d,$$ where $R$ is a rotation on $\mathbb R^d$, $x\in \mathbb Z^d$ and $l\in\mathbb S^{d-1}$ so that $R$ satisfies $R(e_1)=l$ (the specific form of such a rotation is immaterial for our purposes).
Let $\gamma\in(0,1)$, we can prove then
\[lemmaTgamma\] The following assertions are equivalents (the proof is given in the Appendix):
- The data $l_0,l_1, \ldots ,l_k\, \in \mathbb S^{d-1}$, $a_0=1, a_1, a_2, \ldots, a_k\,>0$, $b_0,b_1, \ldots,b_k>0 $ generate an $l_0$-directed system of slabs of order $\gamma$.
- For some positive constants $b$ and $\hat{r}$, and large $L$, there are finite subsets $\Delta_L\subset \mathbb Z^d$, with $0\in \Delta_L\subset \{x\in \mathbb Z^d: x\cdot l_0\geq -bL\}\cap \{x\in \mathbb R^d: |x|_2\leq \hat{r}L\}$ and $$\limsup_{\substack{L\rightarrow \infty}}L^{-\gamma}\, \ln P_0\left[X_{T_{\Delta_L}}\notin \partial^+\Delta_L\right]<0,$$ where $\partial^+\Delta_L=\partial \Delta \cap \{x\in \mathbb R^d: x\cdot l\geq L \}$.
- For some $c>0$, one has $$\label{Tgammasquare}
\limsup_{\substack{L\rightarrow\infty}}\, L^{-\gamma}\ln P_0\left[X_{T_{B_{L, cL, l_0}(0)}}\notin \partial^+ B_{L, cl, l_0}(0)\right]<0.$$
Furthermore, as easily seen any of these conditions is equivalent to the previously defined condition $(T^\gamma)|_{l_0}$ in Definition \[deftgammaandtprime\].
Approximate Renewal Structure {#Apre}
-----------------------------
We recall the approximate renewal structure of F. Comets and O. Zeitouni (cf. [@CZ01]). We state results without proofs, the great majority of them can be found in Section 2.2 of [@Gue17].
Throughout this section we assume that condition $(T^\gamma)|\ell$ holds, for some $\gamma\in (0,1)$ and a fixed $\ell\in \mathbb S^{d-1}$. Observe that one can assume direction $\ell$ is such that there exists number $h\in (0, \infty)$ with $$\label{rul}
l:=h\ell\in \mathbb Z^d.$$ As it was argued in [@Gue17], this is not a further restriction. Denoting the canonical orthonormal basis of $\mathbb R^d$ by $\{e_i, \, i\in[1,d]\}$, we consider the probability measure $\overline P_0$ be defined by $$\overline P_0:=\mathbb P\otimes Q\otimes P_{\omega, \varepsilon}^0\,\,\,\, \mbox{on}\,\,\, \Omega\times (\mathcal W)^{\mathbb N} \times (\mathbb Z^d)^{\mathbb N},$$ where $\mathcal{W}=\{z:\, z=\pm e_i, \, \mbox{ for some }\, i\in
[1,d]\}\cup\{0\}$, which is defined as follows: $Q$ is a product probability measure such that with each sequence $\varepsilon=(\varepsilon_1, \varepsilon_2, \,\ldots)\in (\mathcal
W)^{\mathbb N}$, for $i\in [1,d]$ we have $Q[\varepsilon_1=\pm
e_i]=\kappa$ and $Q[\varepsilon_1=0]=1-2d \kappa$. Then for fixed random elements $\varepsilon\in (\mathcal W)^{\mathbb N}$ and $\omega \in \Omega$, we define $P_{\omega, \varepsilon}^0$ as the law of the Markov chain $(X_n)_{n\geq 0}$ with state space in $\mathbb Z^d$, starting from $0\in \mathbb R^d$ and transition probabilities $$P_{\omega, \varepsilon}^0[X_{n+1}=X_n+e| X_n]=\mathds{1}_{\{\varepsilon_{n+1}=e\}}+\frac{\mathds{1}_{\{\varepsilon_{n+1}=0\}}}{1-2d\kappa}\left(\omega(X_n,e)-\kappa\right),$$ where $e$ is an element of the unit sphere in the lattice: $\{y\in \mathbb Z^d:\,
|y|_2=1\}$. An important property of this auxiliary probability space stems from the easy fact to verify fact that the law of $(X_n)_{n\geq0}$ under $Q\otimes P_{\omega, \varepsilon}^0$ coincides with the law under $P_{0,\omega}$, while the law under $\mathbb P \otimes
P_{\omega, \varepsilon}^0$ coincides with $P_0$.
Define now the sequence $\bar{\varepsilon}$ of length $|l|_1\in \mathbb N$ in the following form: $\bar{\varepsilon}_1=\bar{\varepsilon}_2=\ldots=\bar{\varepsilon}_{|l_1|}=\mbox{sign}(l_1)e_1$, $\bar{\varepsilon}_{|l_1|+1}=\bar{\varepsilon}_{|l_1|+2}=\ldots=\bar{\varepsilon}_{|l_1|+|l_2|}=\mbox{sign}(l_2)e_2$, $\ldots , \,
,\bar{\varepsilon}_{|l|_1-|l_d|+1}=\ldots=\bar{\varepsilon}_{|l|_1}=\mbox{sign}(l_d)e_d$. Notice that $l_i:=\ell\cdot e_i$ for $i\in[1,d]$.
For $\zeta>0$ small, $x\in \mathbb Z^d$, the cone $C(x,l,\zeta)\subset \mathbb Z^d$ will be given by $$\label{cone}
C(x,l,\zeta):=\{y\in \mathbb Z^d: (y-x)\cdot l \geq \zeta |l|_2
|y-x|_2 \}.$$ We will assume that $\zeta$ is small enough in order to satisfy the following requirement: $$\bar{\varepsilon}_1,\bar{\varepsilon}_1+\bar{\varepsilon}_2,\, \ldots\, , \bar{\varepsilon}_1+\bar{\varepsilon}_2+\ldots+\bar{\varepsilon}_{|l|_1}\in C(0, l, \zeta).$$ For $L\in |l|_1\mathbb N$ we will denote by $\bar{\varepsilon}^{(L)}$ the vector $$\bar{\varepsilon}^{(L)}=\overbrace{(\bar{\varepsilon}, \bar{\varepsilon},\, \ldots\, ,\bar{\varepsilon},\bar{\varepsilon})}^{L/|l|_1-\mbox{times}}$$ of length equal to $L$. Setting $$D':=\inf\{n\geq0:\, X_n\notin C(X_0, l , \zeta)\},$$ we have:
\[Dundert\] Assume condition $(T^\gamma)|\ell$ holds for some $\gamma \in (0,1)$, and fix a constant $\mathfrak{r}$ and a rotation $R$ as in item $iii)$ of Lemma \[lemmaTgamma\]. Then there exists $c_1>0$ such that if $\zeta
<\min\left\{\frac{1}{9d}, \frac{1}{3d\mathfrak{r}}\right\}$, then $$P_0[D'=\infty]\geq c_1.$$
The proof is given in Lemma 2.3 of [@Gue17] which uses a previous result, Proposition 5.1 in [@GR17].
We choose $\zeta>0$ satisfying the hypotheses of Lemma \[Dundert\]. For each $L\in |l|_1\mathbb N$, we define $S_0=0$, and denoting by $\theta$ the canonical time shift, we set $$\begin{gathered}
S_1=\inf\{n\geq L:\, X_{n-L}\cdot l >\max_{0\leq j<n-L}\{X_j\cdot
l\}, \,
(\varepsilon_{n-L},\ldots,\varepsilon_{n-1})=\bar{\varepsilon}^{(L)}\},\\
R_1=D'\circ \theta_{S_1}+S_1,\end{gathered}$$ and for $n>1$ $$\begin{gathered}
S_n=\inf\{n>R_{n-1}:\, X_{n-L}\cdot l >\max_{0\leq j<n-L}\{X_j\cdot
l\}, \,
(\varepsilon_{n-L},\ldots,\varepsilon_{n-1})=\bar{\varepsilon}^{(L)}\},\\
R_n=D'\circ \theta_{S_n}+S_n.\end{gathered}$$ For given $L$ as above, these random variables are stopping times for the canonical filtration of the pair $(X_n,
\varepsilon_n)_{n\geq 0}$. Notice also that the chain of inequalities $$S_0=0<S_1\leq R_1\leq \ldots \leq S_n\leq R_n \ldots \leq \infty$$ is satisfied, with strict inequality if the left member is finite. Setting $$K:=\inf\{n\geq 1: S_n<\infty, R_n=\infty\},$$ one defines the first time of asymptotic regeneration $\tau_1:=\tau_1^{(L)}=S_K \leq \infty$. We get rid the dependence on $L$ from $\tau_1$ when there is not risk of confusion, notice also that $\tau_1$ depends on direction $\ell$. A qualitative characterization of the time $\tau_1=n$ is as follows: the first time $n$ that the walk takes a strict record level in direction $l$ at time $n-L$, after this the walk is pushed through direction $l$ by unit steps on the lattice $\mathbb Z^d$ just owed to the action of $\bar{\varepsilon}^{(L)}$ sequence in the probability space $(Q,
(\mathcal W)^{\mathbb N})$, independently on the environment, and finally for any future $j>n$ the walk remains forever inside the cone $C(X_n, l, \zeta)$.
The next lemma shows that the construction makes sense.
\[tranunderttau\] Assume $(T^{\gamma})|\ell$ holds for some $\gamma\in (0,1)$. Then $P_0-$a.s. (recall (\[rul\])) $$\label{TTRANSIENT}
\lim_{\substack n\rightarrow \infty}\, X_n\cdot l=\infty.$$ and there exists a deterministic $L_0>0$, so that for each $L\geq L_0$, with $L\in \,|l|_1\mathbb N$, one has $\overline P_0-$a.s. $$\label{tau1finite}
\tau_1^{(L)}<\infty.$$
The claim (\[tau1finite\]) is a straightforward application of Lemma \[Dundert\], see Lemma 6.2 of [@GR17] for details. As for the proof of claim (\[TTRANSIENT\]), see Lemma 6.1 of [@GR17] and page 517 of [@Sz02].
Choosing $L$ and $\zeta$ as prescribed by Lemmas \[Dundert\]-\[tranunderttau\], one has that $\overline P_0-$a.s. $\{R_k<\infty\}\,=\,\{S_{k+1}<\infty\}$ and $S_1<\infty$ by (\[TTRANSIENT\]).
Let us now define the iterated regeneration times of $\tau_1$ via: $$\tau_n=\tau_1\circ \theta_{\tau_{n-1}}+\tau_{n-1}$$ for $n>1$ (by convention $\tau_0=0$). It is routine to verify that for any $k\in \mathbb N$, $\overline P_0-$a.s. $\tau_k<\infty$.
On the Almost Renewal Structure for Random Walks in Strong Mixing Environments {#section2}
------------------------------------------------------------------------------
Our mixing assumptions provide an approximate renewal structure when one considers the increments of the $\tau_1$ iterates. More precisely, we define the $\sigma$-algebra $\mathcal
G_1$ by $$\sigma\left(\omega(y, \cdot): y\cdot l< X_{\tau_1}\cdot l- (L|l|_2)/(|l|_1),\, (\varepsilon_i)_{0\leq i\leq \tau_1},\,(X_i)_{0 \leq i\leq \tau_1}\right),$$ and for $x\mathbb Z^d$ and $L\in|l|_1 \mathbb N$ we introduce the $\sigma-$algebra $$\label{sigmafrak}
\mathfrak{F}_{x,L}:=\sigma(\omega(y,\cdot): (y-x)\cdot l\leq -(L|l|_2)/|l|_1 ).$$ An important technical fact is the content of the next proposition whose proof is in Proposition 3.1 of [@Gue17].
\[propare\] For each $t\in (0,1)$ there exists $L_0:=L_0(C, g, \kappa, l,
d,r)$, such that $\overline P_0-$a.s. $$\begin{aligned}
\label{approxre1}
&\exp\left(-e^{-g\, tL}\right)\overline{P}_0[(X_{n}-X_{0})_{n\geq0}\in \cdot\,|\, D'=\infty] \\
\nonumber
&\leq \overline P_0[(X_{\tau_1+n}-X_{\tau_1})_{n\geq0}\in \cdot \,|\,\mathcal G_1]\\
\nonumber
&\leq \exp\left(e^{-g\, tL}\right)\overline{P}_0[(X_{n}-X_{0})_{n\geq0}\in \cdot\,|\, D'=\infty]\end{aligned}$$ holds, for all $L\geq L_0$ with $L\in|l|_1 \mathbb N$.
We close this subsection with a straightforward extension of the previous proposition which will be stated in the next corollary, for reference purposes. As a natural extension to $\mathcal G_1$, we define the sigma-algebra $\mathcal G_i$, where $i\in \mathbb N$, by $$\mathcal G_i=\sigma\left(\omega(y, \cdot): y\cdot l< X_{\tau_i}\cdot l- (L|l|_2)/(|l|_1),\, (\varepsilon_i)_{0\leq j\leq \tau_i},\,(X_j)_{0 \leq j\leq \tau_i}\right),$$ then an induction argument makes us conclude:
\[corren\] Assume either: **(SM)**$_{C, g}$ or **(SMG)**$_{C, g}$ and let $j\in\mathbb N, \, t\in (0,1)$. Then there exists $L_0=L_0(C,g,
\kappa, l, d,r)$ such that $\overline P_0-$a.s. $$\begin{aligned}
&\exp\left(-e^{-g\,tL}\right)\overline{P}_0[(X_{n}-X_{0})_{n\geq0}\in \cdot\,|\, D'=\infty] \\ &\leq\overline{P}_0[(X_{\tau_j+n}-X_{\tau_j})_{n\geq0}\in \cdot \,|\,\mathcal G_j]\\
&\leq \exp\left(e^{-g\, tL}\right)\overline{P}_0[(X_{n}-X_{0})_{n\geq0}\in \cdot\,|\, D'=\infty]\end{aligned}$$ holds, for all $L\geq L_0$ with $L\in |l|_1\mathbb N$.
Overview of the Proof
---------------------
We shall explain the general strategy to prove Theorems \[mainth1\] and \[mainth2\]. This subsection has an informal character and we refer to corresponding sections for mathematical developments, nevertheless we describe here some auxiliary results needed which might help to understand the general idea behind our proof and avoid to focus on cumbersome computations. The rough plan is easy to explain:
1. \[1\]$(P)_M|\ell$ implies an effective polynomial effective condition $(P_M)|\ell$.
2. \[2\]Using condition $(P_M)|\ell$ we prove an effective criterion $(EC)|\ell$ in mixing random environments.
3. \[3\]The mixing effective criterion $(EC)|\ell$ implies condition $(T')$. On the other hand, from the very definitions is straightforward the converse implication: $(T')|\ell$ implies $(P)_M|\ell$. This proves Theorem \[mainth1\].
4. \[4\]Using condition $(T')|\ell$ we obtain several controls that at the end they will provide suitable estimates for tails of the approximate regeneration time of Subsection \[section2\]. The result is proven after an application of the CLT given in [@CZ02].
We shall explain quickly each point at the outline above.
- The statement in (\[1\]) is a geometric fact. Indeed proceeding with similar analysis as in proof of Lemma \[lemmaTgamma\], the very definition of condition $(P)_M|\ell$ in (\[defpolasympandec\]) implies an analogous statement in terms of boxes with *linear dimensions* at each coordinate, and this provides the proof (cf. end of Section \[secpoly\] for further details or [@Li16] for an alternative argument).
- The strategy to prove $(P_M)|\ell \, \Rightarrow \, (EC)|\ell$ is split into prove in turn Lemma \[lemmakpolybad\] and Proposition \[propquenpol\]. Combination of this two results makes us see $\mathbb E[\rho_{\mathcal B}^{a}]$ (recall notation in (\[defpolasympandec\])) decays faster than any polynomial function on $L$, with a box specification $\mathcal B:=\mathcal B(R, L-2,L+2,4L^3)$ and $a\in (0,1)$ depends on $L$. This is the content of Section \[secpoly\].
- The effective criterion defined in (\[defpolasympandec\]) by a similar analysis as the one in [@Sz02] will imply condition $(T')|\ell$. Several steps will be extended in order to separate sites where transitions depends, because in this mixing setting we need more than disjointness for those sets of sites. We shall develop the formal procedure in Section \[sectionce\].
- Under $(T')|\ell$ we prove diffusive type of controls on the orthogonal fluctuation to the approximate asymptotic direction of the walk path in Proposition \[protransfluctuation\]. Renormalization schemes using as seed estimate the one displayed in Proposition \[propatyquenest\] dispose us to prove a sharp-inequality of super-exponential character concerning the estimation of the $\mathbb P-$ probability of an atypical quenched event. That is Proposition \[propatyquenest\], which combined with Lemma \[decomtail1\] proves finite moments for arbitrary finite powers of the approximate regeneration time $\tau_1^{(L)}$. This will be the content of Section \[secproofmainth\].
Further estimates under $(T^\gamma)$ in mixing environments {#secbasicpropTgamma}
===========================================================
Throughout this section, we let $\gamma \in (0,1)$ along with $\ell\in \mathbb S^{d-1}$. The subject of this section is to establish that under $(T^\gamma)|\ell$ one obtains the existence of some finite stretched exponential moments for the regeneration position.
Following argument in Section 4 of [@Gue17], we first pick $h\in (0,\infty)$ so that (\[rul\]) is satisfied. Thus denoting $l=h\ell$, it is possible to construct regeneration times $\tau_1^{(L)}$ along vector $l$, which depend on $L\in|l|_1\mathbb N$ and we further assume that $L\geq L_0$, where $L_0$ is as in Proposition \[propare\]. For the construction of $\tau_1^{(L)}$, it is also needed a cone through $l$ (cf. \[cone\]), which in turn is determined by a cone angle $\zeta>0$ and it shall be convenient to take that angle as any positive number satisfying: $$\label{zeta}
\zeta<\min\left\{\frac{1}{9d},\, \frac{1}{3d\mathfrak{r}},\,\cos\left(\frac{\pi}{2}-\arctan(3\mathfrak{r})\right)\right\}.$$ Furthermore, we choose a constant $\mathfrak{r}>0$ such that (\[Tgammasquare\]) in item $iii)$ of Lemma \[lemmaTgamma\] is fulfilled.
\[expmpr\] Assume that $(T^\gamma)|\ell$ and either: **(SM)**$_{C,g}$ or **(SMG)**$_{C,g}$ hold. Then there exist positive constants $c_2$, $c_3$ and $L_0$, such that for all $L\geq L_0$, with $L\in |l|_1 \mathbb N$ we have that
$$\label{expmompos}
\overline{E}_0[\exp\left(c_2\left(\kappa^L X_{\tau_1}\cdot l\right)^\gamma\right)]<c_3$$
holds.
Under either mixing assumption: **(SM)**$_{C,g}$ or **(SMG)**$_{C,g}$, the procedure to derive Proposition 4.1 in [@Gue17] leads us to the existence of $\widetilde c:=\widetilde c(C,g)>0$ and $L_0>0$ such that for any $c>0$, $\gamma>0$ (notice that $\widetilde c$ and $L_0$ do not depend on $c,\gamma$) and $L\in |l|_1\mathbb N$, with $L\geq L_0$ one has: $$\begin{gathered}
\label{xtaucdotl}
\overline{E}_0\left[\exp\left(c\left(\kappa^L X_{\tau_1}\cdot l\right)^\gamma\right)\right]\leq\\
\nonumber
\sum_{k \geq 1}\overline{E}_0\left[\exp\left( c\left(\kappa^L X_{S_k}\cdot l\right)^\gamma\right), S_k<\infty\right]\left(\exp\left(e^{-\widetilde c L}\right)P_0\left[D'=\infty\right]\right).\end{gathered}$$ For $k\geq 0$ we define random variables $M_k$ along with $\overline M$ as follows $$M_0:=0, \hspace{2.5ex} \mbox{for $k\geq1$}\hspace{1.5ex} M_k:=\sup_{\substack{0\leq n\leq R_k}} \left\{X_n \cdot l\right\} \hspace{2.5ex}\mbox{and}\hspace{2ex}\overline M:=\sup_{\substack{0\leq n \leq D'}}\left\{X_n\cdot l\right\} .$$ As was proven in [@Gue17], it is possible to get a further decomposition for the series term in (\[xtaucdotl\]). More precisely, we use inequality $(a+b)^\gamma\leq a^\gamma+b^\gamma$ for $a,b>0$, to conclude that there exist constants $\widetilde{c}_{2}$, $\widetilde L_0$ and a further constant $\overline c:=\overline c(\widetilde{c}_2, \widetilde{L}_0)>0$, such that for $L\geq \widetilde L_0$ and $c\leq \widetilde c_2$ we have (by convention $R_0:=0$) $$\begin{aligned}
\label{splittingrp}
&\sum_{k \geq 1}\overline E_0\left[\exp\left( c\left(\kappa^L X_{S_k}\cdot l\right)^\gamma\right), S_k<\infty\right]\\
\nonumber
&\leq \overline E_0\left[\exp\left( c\left(\kappa^L X_{S_1}\cdot l\right)^\gamma\right), S_1<\infty\right]\\
\nonumber
&+\overline c \times \sum_{k\geq 2} E_0\left[\exp\left(c\left(\kappa^L M_{k-1}\right)^\gamma\right),\, R_{k-1}<\infty \right].\end{aligned}$$ We claim that for $k\geq 1$ $$\begin{gathered}
\nonumber
\overline E_0\left[\exp\left(c\left(\kappa^L M_{k}\right)^\gamma\right),\, R_k<\infty \right]\leq \overline E_0\left[\exp\left(c\left(\kappa^LX_{S_{k}}\cdot l\right)^\gamma\right), S_k<\infty\right] \\
\label{claimMk-1}
\times\left(\exp\left(e^{-\widetilde c L}\right)E_0\left[ \exp\left(c\left(\kappa^L \overline M\right)^\gamma\right)\, ,D'<\infty\right]\right),\end{gathered}$$ and we claim as well that: $$\label{claims1}
\overline{E}_0\left[\exp\left( c\left(\kappa^L X_{S_1}\cdot l\right)^\gamma\right), S_1<\infty\right]\leq \overline c.$$ Indeed, using once again the inequality $(a+b)^\gamma\leq a^\gamma+b^\gamma$ for $a,b>0$, we get (with the hopeful clear notation) $$\begin{aligned}
&\overline E_0\left[\exp\left(c\left(\kappa^L M_{k}\right)^\gamma\right),\, R_k<\infty \right]\leq\\
&\overline E_0\left[\exp\left(c\left(\kappa^LX_{S_k}\cdot l\right)^\gamma\right), \,S_k<\infty;\right.\\ &\left.\times\exp\left(c\left(\kappa^L(M_k-X_{S_k}\cdot l)\right)^\gamma\right)\, ,D'\circ\theta_{S_k}<\infty \right].\end{aligned}$$ On the other hand, observe that on the event $\left\{S_k=n, \, X_{S_k}=x\right\}$ for given $n\in \mathbb N$ and $x\in \mathbb Z^d$, the random variables: $$\begin{gathered}
\label{s1nx}
\mathcal S_{1,n,x}:=E_{Q\otimes P_{\epsilon,\omega}^0}\left[\exp\left(c\left(\kappa^L X_{S_k}\cdot l\right)^\gamma\right), \,S_k=n,\,X_n=x\right]\mbox{ and}\\
\label{s2nx}
\mathcal S_{2,n,x}:=E_{Q\otimes P_{theta_n\epsilon, \theta_x \omega}^0}\left[\exp\left(c\left(\kappa^L \overline M\right)^\gamma\right), \,D'\circ \theta_n<\infty\right]\end{gathered}$$ are $$\begin{gathered}
\sigma\left(\epsilon_i,i<n,\,\, \omega(y,\cdot), y\cdot l<x\dot l-L|l|_2/|l|_1\in\right) \mbox{ and,} \\
\sigma\left(\epsilon_i, i\geq n, \,\, \omega(z,\cdot), z\in C(x,l,\zeta)\right)\end{gathered}$$ measurable respectively. Furthermore, the two terms (\[s1nx\])-(\[s2nx\]) satisfy $$\begin{gathered}
E_{Q\otimes P_{\epsilon,\omega}^0}\left[\exp\left(c\left(\kappa^LX_{S_k}\cdot l\right)^\gamma\right), \,S_k<\infty;\right.\\ \left.\exp\left(c\left(\kappa^L(M_k-X_{S_k}\cdot l)\right)^\gamma\right)\, ,D'\circ\theta_{S_k}<\infty\right]\\
=\mathcal S_{1,n,x}\, \mathcal S_{2,n,x}.\end{gathered}$$ Hence after decomposing the event $\{S_k<\infty\}$, one can and do apply the mixing assumptions **(SM)**$_{C,g}$ or **(SMG)**$_C,g$ to find that $$\begin{gathered}
\overline E_0\left[\exp\left(c\left(\kappa^L M_{k}\right)^\gamma\right),\, R_k<\infty \right]\leq
\overline E_0\left[\exp\left(c\left(\kappa^LX_{S_k}\cdot l\right)^\gamma\right),\, S_k<\infty\right]\times\\
\exp\left(e^{-\widetilde c L}\right)\overline E_0\left[\exp\left(c\left(\kappa^L \overline M\right)^\gamma\right),\, D'=\infty\right],\end{gathered}$$ which ends the proof of the first claim in (\[claimMk-1\]).
The second part of the claim can be derived with same type of argument as above (4.11) in [@Gue17], therefore the proof will be omitted.
In view of claims (\[claimMk-1\]) and (\[claims1\]), an induction argument makes us conclude that for $k\geq 1$ $$\begin{gathered}
\overline{E}_0\left[\exp\left( c\left(\kappa^L X_{S_{k}}\cdot l\right)^\gamma\right), S_{k}<\infty\right]\leq\\
\overline c \left(\exp\left(e^{-\widetilde c L}\right)\left(E_0\left[\overline c\exp\left(c\left(\kappa^L \overline M\right)^\gamma\right)\,,D'<\infty\right]\right)^{k-1}\right),\end{gathered}$$ and hence going back to (\[splittingrp\]) and appealing to (\[xtaucdotl\]), we get the inequality: $$\begin{gathered}
\nonumber
\overline E_0\left[\exp\left(c\left(\kappa^LX_{\tau_1}\cdot l\right)^\gamma\right)\right]\leq\left(\exp\left(e^{-\widetilde c L}\right)P_0\left[D'=\infty\right]\right)\times\\
\label{finalxtaul}
\sum_{k\geq 1}\overline c \left(\overline c E_0\left[\exp\left(c\left(\kappa^L \overline M\right)^\gamma\right),\,D'<\infty\right]\right)^{k-1}.\end{gathered}$$ In virtue of Lemma \[Dundert\], the next lemma ends the proof Proposition \[expmpr\].
\[expintM\] There exist constants $c_4, c_5>0$, such that $$\label{expbM}
E_0\left[\exp \left(c_4\left(\overline{M}\right)^\gamma\right),\, D'<\infty\right]<c_5.$$
It will be sufficient to prove that for some $c>0$, there exists finite $c'>0$ such that $$E_0\left[\exp \left(c\left(M'\right)^\gamma\right), D'<\infty\right]<c',$$ where as a matter of definition (recall (\[rul\])): $$M':=\sup_{\substack{0\leq n \leq D'}}\{(X_n-X_0)\cdot \ell\} .$$ Notice that $$\begin{gathered}
\nonumber
E_0[\exp\left(c\left(M'\right)^\gamma\right), D'< \infty ]\leq e^cP_0[D'< \infty ]+\\
\sum_{m\geq0} \exp\left(c2^{(m+1)\gamma}\right)P_0[2^m\leq M'<2^{m+1},D'< \infty ].\end{gathered}$$ As a result of the above inequality, it suffices to get $\gamma-$stretched exponential controls for large $m$ on the probability: $$P_0[2^m\leq M'<2^{m+1},\,D'< \infty ].$$ To this end, it will be convenient to introduce the following stopping time for the canonical filtration of the walk: $$\label{D0}
D'(0):=\inf\{n\geq0: X_n\notin C(0,l,\zeta)\}$$ Plainly, using the notation of (\[extimslab\]) and recalling that we chose $\mathfrak r>0$ satisfying (\[Tgammasquare\]), we have $$\begin{aligned}
\label{decompositionM}
&P_0[2^m\leq M'<2^{m+1},D'< \infty ]\\
\nonumber
&\leq P_0[T^\ell _{2^m} \leq D'<\infty, T^\ell_{2^{m+1}}\circ\theta_{T^\ell_{2^m}}
> D'(0)\circ\theta_{T^\ell_{2^m}}]\\
\nonumber
&\leq P_0[ X_{T^\ell_{2^m }}\not \in \partial^+B_{2^m,\mathfrak{r}2^m,\ell}(0), T^\ell_{2^m } \leq D'<\infty ]\\
\nonumber
&+P_0[X_{T^\ell_{2^m}}\in
\partial^+B_{2^m,\mathfrak{r}2^m,\ell}(0), T^\ell_{2^{m+1}}\circ\theta_{T^\ell_{2^m}}> D'(0)\circ\theta_{T^\ell_{2^m}}].\end{aligned}$$ Notice that on the event of the first probability on the right most expression in (\[decompositionM\]), $P_0-$a.s. one has $$X_{T_{B_{2^m,\mathfrak{r}2^m,\ell}(0)}} \notin \partial^+B_{2^m,\mathfrak{r}2^m,\ell}(0).$$ Therefore, with the help of item $iii)$ in Lemma \[lemmaTgamma\] we get for large $m$, $$\begin{aligned}
\nonumber
&P_0[ X_{T^\ell_{2^m }}\not \in
\partial^+B_{2^m,\mathfrak r2^m,\ell}(0), T^\ell_{2^m } \leq
D'<\infty ]\\
\label{expb1}
&\leq\exp\left(-\mathfrak{c}2^{m\gamma}\right)\end{aligned}$$ for some suitable $\mathfrak{c}>0$. As for the second term on the right most expression of (\[decompositionM\]), for $m\in \mathbb N$ we introduce the box frontal boundary $F_{m}$ via: $$F_{m}=\partial^+B_{2^m,\mathfrak r 2^m,\ell}(0).$$ Applying the strong Markov property we find that $$\begin{gathered}
\nonumber
P_0[X_{T^\ell _{2^m}}\in
\partial^+B_{2^m,\mathfrak r 2^m,u}(0), T^\ell
_{2^{m+1}}\circ\theta_{T^\ell_{2^m}}>
D'(0)\circ\theta_{T^\ell_{2^m}}]\leq\\
\label{expb2}
\sum_{y\in F_{m}}P_y[T^\ell _{2^{m+1} }>D'(0)].\end{gathered}$$ In order to find un upper bound for the last probability entering at (\[expb2\]), we will bound from below the probability of its complement. We define for $x\in \mathbb Z^d$, the box $$B_x:=B_{2^{m-1},\mathfrak{r}2^{m-1},\ell}(x),$$ and notice that under assumption (\[zeta\]) $$\mathfrak{r}\left(2^m+2^{m-1}\right)\leq \tan \left( \frac{\pi}{2}-\arccos(\zeta)\right)\,2^{m-1}.$$ Thus, for any $y\in F_{m}$ and $z\in \partial^+ B_y$ both boxes $B_y$ and $B_z$ are inside of the cone $C(0,l,\zeta)$ (see Figure \[fig\]).
![The boxes $B_y$ and $B_z$ are inside of $C(0,l,\zeta)$.[]{data-label="fig"}](figure1.pdf "fig:"){width="8cm"}\
Consequently for $y \in F_m$, we get: $$\begin{gathered}
\nonumber
P_y[T^\ell_{2^{m+1}} < D'(0)]\geq\\
\label{Pyc2}
\sum_{z \in \partial^+ B_y}\mathbb{E}[P_{y,\omega}[X_{T_{B_y}}\in
\partial^+ B_y, X_{T_{B_y}}=z,
(X_{T_{B_z}}\in\partial^+ B_z)\circ\theta_{T_{B_y}}]].\end{gathered}$$ It will be now convenient to introduce for $m\in \mathbb N$ a *second boundary set*, denoted by $\bar F_{m}$ and given by $$\bar F_{m}:=\partial[\cup_{y\in F_{m}}B_y]\cap R([ 2^{m-1}+2^m,\infty)\times \mathbb{R}^{d-1}),$$ In turn, we introduce *good environment events* $G_{\bar F_{m}}$ via: $$G_{\bar F_{m}}:=\left\{\omega\in \Omega : \sup_{\substack{z\in\bar F_m} } P_{z,\omega}[X_{T_{B_z}}\in
\partial^+B_z]> 1-\exp(-\mathrm{c} 2^{(m-1)\gamma}), \right\},$$ where the constant $\mathrm{c}>0$ will be chosen later on. We use the Markov property to get that the right most hand probability in inequality (\[Pyc2\]) is greater or equal to $$\label{expression}
\left(1-\exp(-\mathrm{c} 2^{\gamma(m-1)})\right)\left(P_y[X_{T_{B_y}}\in \partial^+
B_y]- P_y[(G_{\bar F_{m}})^c]\right),$$ where $(E)^c$ denotes the complement of set $E$.
Furthermore, using stationarity of $P_0$ and (\[Tgammasquare\]) we have that for $x\in\mathbb R^d$ and large $m$ $$\begin{gathered}
\nonumber
P_{x}[X_{T_{B_x}}\not\in \partial^+B_x]=P_{0}[X_{T_{B_0}}\not\in
\partial^+B_0]\leq \\
\label{k1}
\exp\left(-\mathfrak{w}2^{(m-1)\gamma}\right),\end{gathered}$$ for certain constant $\mathfrak{w}>0$.
Thus we see that (\[expression\]) is greater than $$\label{in1}
\left(1-\exp\left(-\mathrm{c} 2^{(m-1)\gamma}\right)\right)\left(1-\exp\left(-\mathfrak{w}2^{(m-1)\gamma}\right)-P_y[(G_{\bar F_{m}})^c]\right).$$
We choose $\mathrm{c}=\mathfrak{w}/2$ to get as a result of (\[k1\]) and Chevyshev’s inequality $$\begin{gathered}
P_y[(G_{\bar F_{m}})^c]\leq\\
\nonumber
|\bar F_{m}|\exp\left(\mathrm{c}2^{(m-1)\gamma}\right)\sup_{x\in
\bar F_{m}}P_{x}[ X_{T_{B_x}}\not\in
\partial^+B_x]\leq\exp\left(-\mathfrak{t}2^{(m-1)\gamma}\right),
\label{estimatek1}\end{gathered}$$ for suitable $\mathfrak{t}>0$, where we have used that $$\max\left\{|\bar F_{m}|, |F_m| \right\}\leq \left(6\mathfrak{r}2^{m}\right)^{d-1}.$$ Consequently, for large $m$ we can find a further positive constant $\widetilde{c}$ such that: $$\label{Destimate}
P_y[T^\ell_{2^{m+1}} \leq D'(0)]\geq 1-\exp\left(-\widetilde{c}2^{m\gamma}\right)$$ for all $y\in \bar{F}_m$.
In view of (\[expb1\]), (\[expb2\]) and (\[Destimate\]), the claim (\[expbM\]) follows.
The next proposition will be used to prove Proposition \[protransfluctuation\]. We first define the random variable $$\label{supt1}
Y:=\sup_{\substack{0\leq n \leq \tau_1}}|X_n|_2.$$ We close this section with the following reinforcement to Theorem \[expmpr\]:
\[corexp\] Assume $(T^\gamma)|\ell$ and either: **(SM)**$_{C,g}$ or **(SMG)**$_{C,g}$. Then there exist positive constants $c_{6}$, $c_{7}$ and $L_0$ such that $$\label{esupt1}
\overline{E}_0[e^{c_{6}\left(\kappa^L Y\right)^\gamma}]\leq c_{7}.$$ for all $L\geq L_0$ with $L\in |l|_1\mathbb N$.
We have for large $u$ $$\begin{aligned}
\nonumber
&\overline{P}_0\left[Y\geq u\right]=\overline{P}_0\left[\sup_{0\leq n\leq \tau_1}|X_n|_2\geq u\right] \\
\nonumber
&\leq \overline{P}_0\left[T_{\Delta_{\frac{u}{2\hat{r}}}}<\tau_1\right] \\
\nonumber
&\leq \overline{P}_0\left[X_{\tau_1}\cdot l\geq \frac{u}{2\hat{r}} \right]+\overline{P}_0\left[X_{\tau_1}\cdot l<\frac{u}{2\hat{r}}, \, T_{\Delta_{\frac{u}{2\hat{r}}}}<\tau_1\right]\\
\nonumber
&\exp\left(-c_2\left(\kappa^L\,\frac{u}{2r}\right)^\gamma\right)\bar{E}_0\left[\exp\left(c_2\left(\kappa^L\,X_{\tau_1}\cdot l\right)^\gamma\right)\right]+P_0\left[X_{T_{\Delta_{\frac{u}{2\hat{r}}}}}\notin \partial^+\Delta_{\frac{u}{2\hat{r}}}\right],\end{aligned}$$ where in the last step we have used that for integer $m\geq 0$ by definition $X_m\cdot
l<X_{\tau_1}\cdot l$, when $0\leq m <\tau_1 $. Keeping in mind the *layer cake decomposition* (cf. [@Ru87], Chapter8, Theorem 8.16), the claim of the corollary follows after applying Lemma \[lemmaTgamma\] and Proposition \[expmpr\].
An effective Criterion for Mixing Environments {#sectionce}
==============================================
Let us introduce in this section a *mixing effective criterion*, extending the effective criterion of [@Sz02]. This criterion will be equivalent to $(T')|\ell$ and it is effective because it can be checked by inspection at the environment on a finite box, as opposed to the asymptotic character of the definition for $(T')|\ell$. Our method shall fallow a closer analysis to the argument presented by A-S. Sznitman in Section 2 in [@Sz02]. For reference purposes we recall the notation introduced in Definition \[defpolasympandec\].
Let $\ell\in \mathbb S^{d-1}$ be a direction and consider boxes transversal to that direction encoded by box specifications $\mathcal{B}(R, L, L', \widetilde{L})$, where $R$ is a rotation of $\mathbb R^d$ with $R(e_1)=\ell$ and $L, L', \widetilde{L}$ are positive numbers. Then, the box attached to this box specification is: $$\label{boxB}
B:=R\left((-L, L')\times(-\widetilde L, \widetilde L)^{d-1}\right)\cap \mathbb Z^d$$ and its positive boundary will be defined by: $$\label{partialB}
\partial^+B:=\partial B\cap \{x\in \mathbb Z^d: \, x\cdot \ell \geq L', \, |x\cdot R(e_i)|< \widetilde L, i\in [2,d]\}$$ It will be also attached to $\mathcal B(R, L, L', \widetilde{L})$, random variables: $$\begin{gathered}
\nonumber
\rho_\mathcal B(\omega):=\frac{q_\mathcal B(\omega)}{p_\mathcal B(\omega)}\in [0,\infty], \, \,\mbox{ where}\\
\label{ravaB}
q_\mathcal B(\omega):=P_{0,\omega}\left[X_{T_B}\notin\partial^+B\right]=1-p_\mathcal B(\omega).\end{gathered}$$ Consider now a rotation $R$ with $R(e_1)=\ell$, positive numbers $L_0, L_1,\widetilde L_0$ and $\widetilde L_1$ satisfying: $$\label{lolonenum}
3\sqrt{d}<L_0<L_1,\hspace{3ex}3\sqrt{d}<\widetilde L_0<\widetilde L_1,$$ and the box specification $\mathcal B_0(R,L_0-1,L_0+1, \widetilde L_0 )$ and $\mathcal B_1(R,L_0-1,L_0+1,\widetilde L_0)$. Whenever there is not risk of confusion, we shall drop the dependence on the environment of the random variables attached to this box specifications (cf. \[ravaB\]). Let us add a basic remark which will be used throughout this and the next sections.
Note that by Cauchy-Schwarz inequality, for a sequence of real numbers $(x_i)_{1\leq i \leq d}$ we have $$\sum_{i=1}^d|x_i|\leq \sqrt{d} \sqrt{\sum_{i=1}^d x_i^2}=:c_1 \sqrt{\sum_{i=1}^d x_i^2}.$$ Therefore defining $c_1:= \sqrt{d}$, we have that for any pair of points $x, y\in \mathbb R^d$ such that $|x-y|_2\leq \eta$, there exists a self voiding and nearest neighbour path $[[x,y]]$ of length less or equal to $c_1\eta$.
The next proposition provides an appropriate mixing estimate for moments of the random variable $\rho_1$ in terms of random variable $\rho_0$ moments.
\[proprecurce\] There exist $c_2>3\sqrt{d}$, $c_3(d), c_4(d), c_5(d), c_6(d)>1$, such that when $N_0:=L_1/L_0\geq 3$, $L_0\geq c_2$, $\widetilde L_1\geq 48 N_0 \widetilde L_0$, one has for any $a\in (0,1]$: $$\begin{gathered}
\nonumber
\mathbb E\left[\rho_1^{a/4}\right]\leq c_3 \left\{(2\kappa)^{-10 c_1 L_1}\left(c_4\widetilde L_1^{d-2}\frac{L_1^3}{L_0^2}\widetilde L_0\mathcal F_{SM}\mathbb E\left[q_0\right]\right)^{\frac{\widetilde L_1}{16 N_0\widetilde L_0}}\right. \\
\label{critrenor}
\left.+\sum_{0\leq k\leq N_0+1}\left(F_{SM}\, c_4\widetilde L_1^{d-1}\mathbb E\left[\rho_0^a\right]\right)^{\frac{N_0+m-1}{4}}\right\},\end{gathered}$$ where $\mathcal F_{SM}$ and $F_{SM}$ are positive functions that depends on $L_0$, $\widetilde L_0$, $L_1$ and $\widetilde L_1$. In fact they are bounded from above as follows: $$\begin{aligned}
\mathcal F_{SM} &\leq \exp\left(c_5(\widetilde L_1)^2 (L_1)^2 (\widetilde L_1)^{2(d-2)}\exp\left(-2gN_0\widetilde L_0\right)\right) \mbox{ and}\\
F_{SM} &\leq \exp\left( c_6 (\widetilde L_1)^{2{d-1}}(L_0)^2N_0\exp\left(-\frac{3gL_0}{2}\right)\right).\end{aligned}$$
It will be convenient to introduce thin slabs transversal to direction $\ell$. More precisely, for $i\in \mathbb Z^d$, we define the set $\mathcal H_i$ by: $$\label{Hi}
\mathcal H_i:=\left\{y\in \mathbb Z^d: \, \exists x\in \mathbb Z^d, \, |x-y|_1=1,\,(x\cdot\ell -iL_0)(y-iL_0)\leq 0 \right\},$$ and the function $I:\mathbb Z^d\rightarrow\mathbb Z$ so that: $$\label{I}
I(z)=i\in \mathbb Z,\,\, \mbox{on} z\in \left\{y\in \mathbb Z^d: y\cdot \ell \in [iL_0-L_0/2, iL_0+L_0/2)\right\}.$$ We note that $I(\mathcal H_i)=i$ since $L_0>2$. We then the successive visit times to the slabs $\mathcal H_i$ are defined through the recursion: $$\begin{gathered}
\nonumber
V_0=0,\,\, V_1=\inf\{n\geq 0,\,X_n\in\mathcal H_{I(X_0)+1}\cup \mathcal H_{I(X_0)-1} \}, \,\, \mbox{ and}\\
\label{visitHi}
V_i=V_1\circ \theta_{V_{i-1}}+V_{i-1},\,\, \mbox{ for $i>1$.}\end{gathered}$$ For a given environment $\omega \in \Omega$ and $x\in \mathbb Z^d$, let us introduce random variables $q_{x,\omega}$ and $p_{x,\omega}$ as $$\label{slabqp}
\widehat q_{x,\omega}=P_{x,\omega}\left[X_{V_1}\in \mathcal H_{I(X_0)-1}\right]=1-\widehat p_{x,\omega},$$ together with the random variable: $$\label{rhohat}
\widehat \rho_{i,\omega}:=\sup\left\{\frac{\widehat q_{x,\omega}}{\widehat p_{x,\omega}},\,\,x\in \mathcal H_i,\,\, |x\cdot R(e_j)|<\widetilde L_1, \, j\in[2,d]\right\},$$ for $i\in \mathbb Z$. We denote by $n_0:=[N_0]$ and consider the nonnegative function $f$ on $\{n_0+2,n_0+1,\ldots \}\times\Omega$ defined by: $$\begin{gathered}
\nonumber
f(n_0+2,\omega)=0, \,\, \mbox{ and} \\
\label{ffunc}
f(i,\omega)=\sum_{i\leq j \leq n_0+1}\, \prod_{j< m\leq n_0+1}(\widehat \rho_{m,\omega})^{-1},\,\, \mbox{ for $i\geq n_0+1$}.\end{gathered}$$ We shall drop the $\omega$-dependence from these random variables when it is fixed in the estimates. Introducing the stopping time $\widetilde T$ via: $$\label{ttilderen}
\widetilde T:=\inf\{n\geq 0,\,\, |X_n\cdot R(e_j)|\geq \widetilde L_1,\,\, \mbox{ for some }j\in [2,d] \}$$ one has the quenched estimate: $$\label{quenf}
P_{0,\omega}\left[\widetilde{T}_{-L_1+1}^\ell<\widetilde T\wedge T_{L_1+1}^\ell\right]\leq \frac{f(0)}{f(-n_0+1)}.$$ We will quickly prove claim (\[quenf\]) only for later reference purposes in the next section. Indeed for the original proof see [@Sz02], pages 523-524. We will first prove that the quenched random function $$\eta(m):=E_{0,\omega}\left[f(I(X_{V_{m\wedge \tau}})), \, V_{m\wedge \tau}\leq \widetilde T\right]$$ is decreasing in $m$. In fact, we have: $$\begin{aligned}
\eta(m+1)\leq&E_{0,\omega}\left[f(I(X_{V_{(m)\wedge \tau}})), \, V_{(m+1)\wedge \tau}\leq \widetilde T,\tau\leq m\right]\\
& +E_{0,\omega}\left[f(I(X_{V_{(m+1)\wedge \tau}})), \, V_{(m+1)\wedge \tau}\leq \widetilde T,\tau> m\right]\\
\leq & E_{0,\omega}\left[f(I(X_{V_{(m)\wedge \tau}})), \, V_{m\wedge \tau}\leq \widetilde T,\tau\leq m\right]\\
&+E_{0,\omega}\left[\tau>m,\, V_{m\wedge\tau}\leq \widetilde T,\,E_{X_{V_m},\omega}\left[f(I(X_{V_1}))\right]\right].\end{aligned}$$ where we have applied the strong Markov property in the last step. On the other hand, notice that $P_{0,\omega}-$a.s. on the event $\{\tau>m, \, V_{m\wedge\tau}\leq \widetilde T\}$ we have $$\begin{aligned}
E_{X_{V_m},\omega}\left[f(I(X_{V_1}))\right]=&f(I_{X_{V_m}})+\widehat p_{X_{V_m},\omega}\left(f(I(X_{V_m})+1)-f(I_{X_{V_m}})\right)\\
&+q_{X_{V_m},\omega}\left(f(I(X_{V_m})-1)-f(I_{X_{V_m}})\right).\end{aligned}$$ However, the expression $$\widehat p_{X_{V_m},\omega}\left(f(I(X_{V_m})+1)-f(I_{X_{V_m}})\right)
+q_{X_{V_m},\omega}\left(f(I(X_{V_m})-1)-f(I_{X_{V_m}})\right).$$ is less than $0$ since $P_{0,\omega}-$a.s. $X_{V_m}\in\mathcal H_{I(X_{V_m})}$ and definitions (\[rhohat\])-(\[ffunc\]). Therefore $\eta(m)$ is decreasing and then applying Fatou’s lemma we get the proof of claim (\[quenf\]).
The next step shall be derive a mixing estimate for the annealed probability $P_0[\widetilde T< \widetilde T_{-L_1+1}^\ell \wedge T_{L_1+1}^\ell]$. For this end, we introduce for $z\in \mathbb R^d$ the semi-norm $$|z|_{\perp}:=\sup\{z\cdot R(e_j), j\in [2,d]\}$$ as well as the stopping times for $j\in[2,d]$, $u\in \mathbb R$ $$\begin{gathered}
\nonumber
\sigma_{u}^{j,+}=\inf\left\{n\geq 0, \,\, X_n\cdot R(e_j)\geq u\right\}\,\, \mbox{ and}\\
\label{sigmatimes}
\sigma_{u}^{j,-}=\inf\left\{n\geq 0, \,\, X_n\cdot R(e_j)\leq -u\right\}.\end{gathered}$$ We set $$\label{escalejlo}
J=\left[\frac{\widetilde N_0}{2(n_0+1)}\right], \,\, \mbox{with }\,\, \widetilde{N}_0=\frac{\widetilde L_1}{\widetilde L_0},\,\, \mbox{ and }\,\,\overline L_0=2(n_0+1)\widetilde L_0.$$ By virtue of our hypotheses, one has that $J\geq [24 N_0/(n_0+1)]\geq 18$. Since $J\overline L_0<\widetilde L_1$, on the event $\{\widetilde T\leq\widetilde T_{-L_1+1}^\ell\wedge T_{L_1+1}^\ell \}$, one gets $P_0$-a.s. there exists some $n\in \mathbb N$, so that $|X_n\cdot R(e_j)|\geq J\overline L_0$ for some $j\in {2,d}$ and for all $i<n$ the walk is inside of the $B_1$-box. Thus: $$\label{pttilde}
P_0[\widetilde T\leq \widetilde T_{-L_1+1}^\ell\wedge T_{L_1+1}^\ell]\leq \sum_{2\leq j\leq d}P_0[\sigma_{J\overline L_0}^{+,j}\leq T_{B_1}]+P_0[\sigma_{J\overline L_0}^{-, j}\leq T_{B_1}].$$
Let us simply write $\sigma_u$ by $\sigma_{u}^{+,2}$ and prove an estimate for the probability $P_0[\sigma _{J\overline L_0}\leq T_{B_1}]$, the other terms entering at (\[pttilde\]) can be treated in a similar fashion with analogous bounds. We let $k\in \mathbb Z$ and define the cylinder $$\label{cylk}
c_\perp(k)=\{w\in\mathbb R^d, w\cdot R(e_2)\in \overline L_0[k,k+1), \,\, |w\cdot R(e_i)|<\widetilde L_1,\, \forall i\in [3,d]\}$$ together with a discrete cylinder $\overline{c}(k)$ made from $c_\perp (k) $, via: $$\label{trcylk}
\overline c(k)=\{x\in \mathbb Z^d, \, \inf_{w\in c_\perp(k)}|x-w|\leq (2n_0+1)\widetilde L_0,\,x\cdot l\in (-L_1+1, L_1+1) \}.$$ As an application of the strong Markov property on the *quenched probabilty measure* $P_{0,\omega}$, we have $$\label{sigmdes}
P_0[\sigma_{J\overline L_0}\leq T_{B_1}]\leq E_0[\sigma_{(J-2)\overline L_0}<T_{B_1}, P_{X_{\sigma_{(J-2)\overline L_0}},\omega}[\sigma_{J\overline L_0}\leq T_{B_1}]].$$ As a result of the choices (\[escalejlo\]) and the definitions (\[cylk\])-(\[trcylk\]), on the event $\{\sigma_{(J-2)\overline L_0}<T_{B_1}\}$, one has $P_0-$a.s the random variable $X_{\sigma_{(J-2)\overline L_0}}$ is in $\overline c(J-2)$, since $2n_0+1<2(n_0+1)$. Denoting by $H^i$, with $i\geq 0$ the iterates of the stopping time $H^1= T_{B_1}\wedge T_{X_0+B_0}$ and defining the stopping time $$\begin{gathered}
S=\inf\{k\geq 0, (X_k-X_0)\ell \leq -L_0+1 \,\,\mbox{or}\,\, |(X_k-X_0)\cdot R(e_j))|\geq \widetilde L_0, \\ \mbox{for some}\,\, j\in [2,d] \},\end{gathered}$$ we note that when $y\in c_\perp(J-2)\cap B_1$, one has for $\omega \in\Omega$ $$P_{y,\omega}\mbox{-a.s.}\,\, \bigcap_{i=0}^{2n_0+1}\theta_{H^i}^{-1}\{H^1<S\}\rightarrow\{T_{B_1}<\sigma_{J\overline L_0}\}$$ and the path $X_\cdot$ described for the event of the left above $P_{y,\omega}$-a.s. remains into $\overline c(J-2)$.
Therefore, using the notations attached to the box specification $\mathcal B_0$ as well as denoting by $t$ the space shift on the environment $\Omega$, for $\omega in \Omega $, $y\in \overline c(J-2)$ one has $P_{y,\omega}$-a.s. for any $i\in [0,2n_0+1]$ the inequality $P_{X_{H^i},\omega}[H^1<S]\geq\inf_{x\in \overline c(J-2)} p_{\mathcal B_0}\circ t_x(\omega)$. Thus an application of the strong Markov property gives $$\begin{gathered}
P_{0}\mbox{-a.s. on the event }\{\sigma_{(J-2)\overline L_0}<T_{B_1}\},\\
P_{X_{\sigma_{(J-2)\overline L_0}},\omega}[T_{B_1}<\sigma_{J\overline L_0}]\geq\left\{\inf_{x\in \overline{c}(J-2)}p_{B_0}\circ t_x(\omega)\right\}^{2(n_0+1)}\stackrel{def}=\varphi(J-2,\omega).\end{gathered}$$ Inserting the previous estimate into (\[sigmdes\]), we have $$P_0[\sigma_{J\overline L_0}\leq T_{B_1}]\leq \mathbb E[P_{0,\omega}[\sigma_{(J-4)\overline L_0}<T_{B_1}](1-\varphi(J-2),\omega)].$$ A crucial point in our mixing case stems from the fact that the random variable $P_{0,\omega}[\sigma_{(J-4)\overline L_0}<T_{B_1}]$ is $$\sigma \left(\omega(y,\cdot),\, y \cdot R(e_2)< (J-4)\overline L_0, \,y\in B_1(R, L_1-1, L_1+1,\widetilde L_1)\right)$$ measurable, while the random variable $(1-\varphi(J-2),\omega)$ is measurable according to the $\sigma$-algebra given by $$\begin{gathered}
\sigma\left(\omega(y,\cdot),\, (J-3)\overline L_0 \leq y \cdot R(e_2)< J \overline L_0, y\cdot \ell\in \left(-(L_1+L_0)+2, \right.\right.\\
\left.\left.(L_1+L_0)+2\right), \mbox{ and for all }\, i\in [3,d]\, |y \cdot R(e_i)|<\widetilde L_1+ \overline L_0\right).\end{gathered}$$ Writing as $A, B\subset \mathbb Z^d$ the sets of sites on which the transitions need to be known to determined the measurability of $P_{0,\omega}[\sigma_{(J-4)\overline L_0}<T_{B_1}]$ and $(1-\varphi(J-2),\omega)$ respectively, when $L_0>3r$ we have (recall the definitions of constants $C$ and $g$ in (\[sma\])-(\[smg\])) $$\begin{gathered}
\sum_{x\in A, y\in B} C\exp(-g|x-y|_1)\leq C|A||B|\exp\left(-g\overline L_0\right)\\
\leq c(d)(J\overline L_0)^2 (L_1)^2 (\widetilde L_1)^{2(d-2)}\exp\left(-g\overline L_0\right)\end{gathered}$$ for some suitable $d$-dependent constant $c(d)$. Analogously one gets $$\sum_{x\in \partial^r A, y\in \partial^r B} C\exp(-g|x-y|_1)\leq c(d)(J\overline L_0)^2 (L_1)^2 (\widetilde L_1)^{2(d-2)}\exp\left(-g\overline L_0\right)$$ We choose the function $\mathcal F_{SM}$ as $$\max\left\{e^{\sum_{x\in \partial^r A, y\in \partial^r B} C\exp(-g|x-y|_1)},e^{\sum_{x\in A, y\in B} C\exp(-g|x-y|_1)}\right\}.$$ Thus, applying either of our mixing assumptions: **(SMG)**$_{C,g}$ or **(SM)**$_{C,g}$ we get $$\begin{gathered}
P_0[\sigma_{J\overline L_0}\leq T_{B_1}]\leq \mathbb P_0[\sigma_{(J-4)\overline L_0}<T_{B_1}]\\
\mathbb E[(1-\varphi(J-2),\omega)]\times \mathcal F_{SM}.\end{gathered}$$ On the other hand, it can be seen that the next upper bound (cf. [@Sz02] page 526): $$\mathbb E[(1-\varphi(J-2,\omega))]\leq \widetilde c(d)\widetilde L_1^{(d-2)}\frac{L_1^3}{L_0^2}\widetilde L_0\mathbb E[q_0]$$ holds, for a suitable constant $\widetilde c(d)>0$. Therefore, going back to (\[sigmdes\]) we get $$\begin{gathered}
P_0[\sigma_{J\overline L_0}< T_{B_1}]\leq P_0[\sigma_{(J-4)\overline L_0}<T_{B_1}]\\
\mathcal F_{SM}\, \widetilde c(d)\widetilde L_1^{(d-2)}\frac{L_1^3}{L_0^2}\widetilde L_0\mathbb E[q_0]\end{gathered}$$ and furthermore, an induction argument lead us to $$\begin{gathered}
P_0[\sigma_{J\overline L_0}< T_{B_1}]\leq\\
\left(\mathcal F_{SM}\widetilde c(d)\widetilde L_1^{(d-2)}\frac{L_1^3}{L_0^2}\widetilde L_0\mathbb E[q_0]\right)^{m},\end{gathered}$$ for any $0\leq m \leq [J/4]$. Since $[J/4]\geq \widetilde L_1/(16 N_0\widetilde L_0)$ by hypothesis and using the remark above \[cylk\] about similar estimates for the other dimensions, we find that $$\begin{gathered}
\nonumber
P_0[\widetilde T<\widetilde T_{-L_1+1}^\ell \wedge T_{L_1+1}^\ell]\leq 2(d-1)\times\\
\label{finesttransv}
\left(\mathcal F_{SM}\widetilde c(d)\widetilde L_1^{(d-2)}\frac{L_1^3}{L_0^2}\widetilde L_0\mathbb E[q_0]\right)^{\frac{\widetilde L_1}{16 N_0\widetilde L_0}}.\end{gathered}$$ In virtue of our choice for the constant $\kappa$ in (\[simplex\]), with the notation of (\[ravaB\]), we have $\rho_1\leq (2\kappa)^{-2c_1L_1}$. We denote by $T$ the random variable $$P_{0,\omega}[\widetilde T_{-L_1+1}^\ell<\widetilde T\wedge T_{L_1+1}^\ell]+P_{0,\omega}[\widetilde T<\widetilde T_{-L_1+1}^\ell \wedge T_{L_1+1}^\ell]$$ and now consider the event $$\mathcal G=\{\omega\in \Omega:\, P_{0,\omega}[\widetilde T<\widetilde T_{-L_1+1}^\ell \wedge T_{L_1+1}^\ell]\leq (2\kappa)^{9c_1 L_1} \}.$$ We then note that on that event we find $$\label{ongpho1}
\rho_1\leq \frac{T}{(1-T)_+}\stackrel{(\ref{quenf})}\leq \frac{f(0)+f(-n_0+1)(2\kappa)^{9c_1L_1}}{\left(f(-n_0+1)-f(0)-f(-n_0+1)(2\kappa)^{9c_1L_1}\right)_+}.$$ Besides switching the constants $\kappa$ by $2\kappa$, the same argument as the one given in [@Sz02], page 527 makes us conclude that on the event $\mathcal G$ $$\label{quenestonG}
\rho_1(\omega)\leq 2\sum _{0\leq m\leq n_0+1}\prod_{-n_0+1<j\leq m}\widehat \rho_{j,\omega},$$ provided $L_0\geq c_2$, where $c_2$ is a suitable dimensional dependent positive constant. Notice that when $i\in \mathbb Z$, we have for any $x\in \mathcal H_i$ the inequality (see (\[ravaB\])-(\[slabqp\]) for notation): $$\frac{\widehat q_{x,\omega}}{p_{x,\omega}}\leq \rho_0\circ t_x.$$ As a result for $i\in \mathbb Z^d$ recalling the definition (\[rhohat\]) and writing $\mathcal Y_i$ the set $\{x: \forall \, j\in [2,d]\,|x\cdot R(e_j)|<\widetilde L_1\}\cap \mathcal H_i$, we obtain $$\label{rhodotilde}
\widehat \rho_{i}\leq \sup_{z\in \mathcal Y_i}\rho_0\circ t_z\stackrel{def}=\overline{\rho}_{i,\omega}.$$ A further step in the proof will be to insert the above estimate $\ref{rhodotilde}$ into (\[quenestonG\]). Once we have done that, using the inequality $(a+b)^\gamma\leq a^\gamma+b^\gamma$ for $a,b\geq0$ and $\gamma\in(0,1)$ we obtain $$\label{estonG1}
\mathbb E\left[(\rho_1)^{\frac{a}{4}}(\omega),\mathcal G\right]\leq 2\sum _{0\leq m\leq n_0+1}\mathbb E\left[\prod_{-n_0+1<j\leq m}(\overline \rho_{j,\omega})^{\frac{a}{4}}\right].$$ We now split each product entering at (\[estonG1\]) into four groups depending on the residues modulo $4$. For integer $j\in[0,3]$ we call $\mathcal M_j=\{i\in \mathbb Z: i=j\,\, (\mbox{mod }4)\}$ and integer $m\in[0,n_0+1]$, applying Bunyakovsky-Cauchy-Shuarz inequality twice we find that: $$\begin{aligned}
\nonumber
\mathbb E\left[\prod_{-n_0+1<j\leq m}(\overline \rho_{j,\omega})^{\frac{a}{4}}\right]&\leq
\mathbb E^{\frac{1}{2}}\left[\prod_{\substack{-n_0+1<j\leq m\\ j\in \mathcal M_0 \cup \mathcal M_1}}(\overline \rho_{j,\omega})^{\frac{a}{2}}\right]\,
\mathbb E^{\frac{1}{2}}\left[\prod_{\substack{-n_0+1<j\leq m\\ j\in \mathcal M_2 \cup \mathcal M_3}}(\overline \rho_{j,\omega})^{\frac{a}{2}}\right]\\
\label{estonG2}
&\leq \prod_{0\leq i\leq 3}\mathbb E^{\frac{1}{4}}\left[\prod_{\substack{-n_0+1<j\leq m\\ j\in \mathcal M_i}}(\overline \rho_{j,\omega})^a\right].\end{aligned}$$ Observe that for given $i\in [0,3]$, when $j_1, j_2\in \mathcal M_i$ with $j_1< j_2$ using either **(SM)**$_{C,g}$ or **(SMG)**$_{C,g}$ of our mixing assumptions, if $L_0\geq 3r$, arguing with a closer analysis as the one before (\[finesttransv\]), we obtain $$\label{decoupling}
\mathbb E\left[(\overline \rho_{j_1,\omega})^a\, (\overline \rho_{j_2,\omega})^a\right]\leq \mathbb E\left[(\overline \rho_{j_1,\omega})^a\right]\left(F_{SM}\mathbb E\left[(\overline \rho_{j_2,\omega})^a\right]\right),$$ provided we first set $j=\max\{k: \, k\in \mathcal M_i\cap (-n_0+1,n_0+1]\}$ and define sets $A_{i,j_1}$, $B_{i}$ (recall we have fixed the integer $i\in[0,3]$) as follows $$\begin{aligned}
\nonumber
A_{i,j_1}:=&\{z\in \mathbb Z^d: \forall i\in [2,d]\, |z\cdot R(e_i)|<\widetilde L_1+\widetilde L_0,\, z\cdot \ell\in \\
\nonumber
&(-L_0+j_1L_0, L_0+3+j_1L_0) \},\\
\nonumber
B_I:=&\bigcup_{j' \in \mathcal M_i\cap(-n_0+1,n_0+1]\setminus\{j\}}A_{i,j'}\end{aligned}$$ and then in these terms the mixing factor $F_{SM}$ is given by: $$\begin{aligned}
\label{mix2}
F_{SM}:=&\max\left\{e^{\sum_{x\in A_{i,j}, y \in B_i}\, C\exp\left(-g|x-y|_1\right)}, e^{\sum_{x\in \partial^r A_{i,j}, y \in \partial^r B_i}\, C\exp\left(-g|x-y|_1\right)}\,\right\}\\
\nonumber
\leq& \exp\left(\mathfrak c (d)(\widetilde L_1)^{2{d-1}}(L_0)^2N_0\exp\left(-\frac{3gL_0}{2}\right)\right)\end{aligned}$$ for certain dimensional constant $\mathfrak c>0$.
Therefore for $i\in [0,3]$ and $m\in (-n_0+1,n_0+1]$, successive conditioning in each term on the right of (\[estonG2\]) from the biggest $j$ *afterwards along direction* $\ell$, turns out $$\mathbb E^{\frac{1}{4}}\left[\prod_{\substack{-n_0+1<j\leq m\\ j\in \mathcal M_i}}(\overline \rho_{j,\omega})^a\right]\leq \prod_{\substack{-n_0+1<j\leq m\\ j\in \mathcal M_i}}\left(F_{SM}\mathbb E[(\overline \rho_{j,\omega})^{a}]\right)^{\frac{1}{4}}$$ Combining this last inequality in (\[estonG2\]) and the bound in (\[finesttransv\]) with the help of Chevyshev’s inequality we get $$\begin{gathered}
\nonumber
\mathbb E[(\rho_1)^{\frac{a}{4}}]\leq (2\kappa)^{-ac_1L_1}\mathbb P[\mathcal G^c]+2\sum_{0\leq m\leq n_0+1}\prod_{-n_0+1<j\leq m}\left(F_{SM}\,\mathbb E[(\overline \rho_{j,\omega})^a]\right)^{\frac{1}{4}}\\
\nonumber
\stackrel{(\ref{finesttransv})} \leq (2\kappa)^{-10c_1L_1}\left(\widetilde c\widetilde L_1^{d-2}\frac{L_1^3}{L_0^2}\widetilde L_0\mathcal F_{SM}\,\mathbb E[q_0]\right)^{\frac{\widetilde L_1}{16 N_0\widetilde L_0}}+\\
\label{endpropec}
2\sum_{0\leq m\leq n_0+1}\prod_{-n_0+1<j\leq m}\left(F_{SM}\mathrm c \widetilde L_1^{d-1}\mathbb E[\rho_0^{a}]\right)^{\frac{1}{4}},\end{gathered}$$ for some $\mathrm c>0$. The claim of the proposition follows in virtue of the right most term in (\[endpropec\]).
The precedent proposition will be instrumental to apply recursively a procedure along fast growing scales. We need to introduce some further defintions in order to explain that procedure. We let $$\label{u0valpha}
u_0\in (0,1),\hspace{7ex}v=16,\hspace{7ex}\alpha=320$$ and consider sequences of positive numbers $(L_k)_{k\geq 0}$ together with $(\widetilde L_k)_{k\geq 0}$ encoding box specifications $\mathcal B_k(L_k-1,L_k+1, \widetilde L_k)$ and satisfying: $$\begin{gathered}
\nonumber
L_0\geq c_2,\,\hspace{4ex}L_0\leq \widetilde L_0 \leq L_0^4,\, \hspace{4ex} \mbox{and when }k\geq 0\\
\label{escalesL_k}
L_{k+1}=N_k L_k\, \hspace{4ex}\mbox{with }N_k=\frac{\alpha c_1}{u_0}v^k\\
\nonumber
\widetilde L_{k+1}=N_k^4\widetilde L_k.\end{gathered}$$ Notice that from the definitions (\[escalesL\_k\]), one has for $k\geq 0$ $$\label{expescaleL_k}
L_k=\left(\frac{\alpha c_1}{u_0}\right)^k\, v^{\frac{k(k-1)}{2}}\hspace{0.7ex}\mbox{and }\hspace{0.7ex}\widetilde L_k=\left(\frac{L_k}{L_0}\right)^4\widetilde L_0.$$ Let us obtain appropriate upper bounds under scales of display (\[escalesL\_k\]) for both functions: $\mathcal F_{SM}$ and $F_{SM}$, provided that $L_0\geq \mathfrak c$ for a suitable constant $\mathfrak c>0$ to be determined. Observe that as a result of both expressions in (\[expescaleL\_k\]), letting $k$ play the role of $0$ for the box specifications involved in Proposition \[proprecurce\], it is clear that for some constant $c_7(d, C,g),c_8(d,C,g)>0$ and $\mathfrak{c}_9(d,C,g)>0$, when $L_0\geq \mathfrak{c}_9$ for all $k\geq 1$: $$c_4\mathcal F_{SM}(k)\leq c_7,\, \hspace{0.7ex}\mbox{and }\hspace{0.7ex}c_4 F_{SM}(k)\leq c_8,$$ where we have written $\mathcal F_{SM}(k)$ and $F_{SM}(k)$ to stress that these functions depend on scales $(L_k)_{k\geq 0}$ and $(\widetilde L_k)_{k\geq 0}$. Furthermore, notice that for $k=0$ we have $$\mathcal F_{SM}(0)\leq \exp\left(c_5\left(\frac{\alpha c_1 L_0}{u_0}\right)^{8d+2}e^{-2g\left(\frac{\alpha c_1}{u_0}\right)L_0}\right)\leq c_7,$$ whenever $L_0\geq \mathrm{c}_9$ for a suitable constant $\mathrm{c}_9>0$. On the other hand, an upper bound for the function $F_{SM}(0)$ can be obtained as follows $$\begin{gathered}
F_{SM}\leq \exp\left(c_6\left(\frac{\alpha c_1}{u_0}\right)^{8d-3}L_0^{8d-2}e^{-\frac{3gL_0}{2}}\right)\\
\leq \exp\left(u_0^{-(8d-3)}e^{-gL_0}\right)\leq c_8,\end{gathered}$$ provided that $L_0\geq \widetilde c(d)$ and $u_0\in [e^{-\frac{gL_0}{8d-3}},1]$. Thus we can and do define the constant $$\mathfrak c=\max\left\{\mathfrak c_9,\, \mathrm c_9,\, \widetilde c\right\}.$$ As a result, under the choice of scales given in (\[escalesL\_k\]), Proposition \[proprecurce\] can be reformulated to get rid the mixing terms :$\mathcal F_{SM}$ and $F_{SM}$. More precisely, when $L_0\geq \mathfrak c$ and $k\geq 0$, $$\begin{gathered}
\nonumber
\mathbb E\left[\rho_{k+1}^{a/4}\right]\leq c_3 \left\{(2\kappa)^{-10 c_1 L_{k+1}}\left(c_7\widetilde L_{k+1}^{d-2}\frac{L_{k+1}^3}{L_k^2}\widetilde L_k\mathbb E\left[q_k\right]\right)^{\frac{\widetilde L_{k+1}}{16 N_k\widetilde L_k}}\right. \\
\label{critrenor1}
\left.+\sum_{0\leq k\leq N_k+1}\left(c_8\widetilde L_{k+1}^{d-1}\mathbb E\left[\rho_k^a\right]\right)^{\frac{N_k+m-1}{4}}\right\}.\end{gathered}$$ The next lemma provides a recursion to obtain controls of stretched exponential type on certain moments of $\rho_k, \, k\geq 0$. The main assumption will be the seed estimate, as we shall see soon that estimate is what we will call effective criterion. Keeping in mind scales $(L_k)_{k\geq 0}$, $(\widetilde L_k)_{k\geq 0}$ satisfying (\[escalesL\_k\]) we have
\[lemmacriterion\] There exists a positive constant $c_9(d)\geq\max\{\mathfrak c, c_2\}$, so that whenever $L_0\geq c_9$ along with $ L_0\leq \widetilde L_0\leq L_0^4$ and for some $a_0\in (0,1]$, $u_0\in [\max\{e^{-\frac{gL_0}{8d-3}},(2\kappa)^{\frac{7L_0}{4d-1}}\},1]$, the inequality $$\label{seedce}
\varphi_0\stackrel{\mbox{def}}=(c_7\vee c_8)\widetilde L_1^{(d-1)}L_0\mathbb E[\rho_0^{a_0}]\leq(2\kappa)^{u_0L_0}$$ holds, then for all $k\geq 0$ one has that $$\label{kestimce}
\varphi_k\stackrel{\mbox{def}}=(c_7\vee c_8)\widetilde L_{k+1}^{(d-1)}L_k\mathbb E[\rho_k^{a_k}]\leq(2\kappa)^{u_kL_k},$$ where $$\label{akuk}
a_k=a_04^{-k},\,\hspace{7ex }\mbox{and }\hspace{7ex}u_k=u_0v^{-k}.$$
We argue by induction following a similar procedure of [@Sz02], Lemma2.2. Consider the set $$A:=\{k\geq 0: \varphi_k>(2\kappa)^{u_k L_k}\}$$ and notice that the claim of the lemma follows once we have proven that for some constant $c_9$, $L_0\geq c_9$ implies $A=\varnothing$. Assume that $A\neq \varnothing$ and we will derive a contradiction. By hypothesis $0\notin A$ and thus denoting by $k+1$ the minimal natural number in $A$, we apply inequality (\[critrenor1\]) to get $$\label{criterecurk}
\varphi_{k+1}\leq c_3 (c_7\vee c_8)\widetilde L_{k+2}^{(d-1)}L_{k+1}\left\{(2\kappa)^{-10c_1L_{k+1}}\varphi_k^{\frac{N_k^3}{16}}+\sum_{0\leq m\leq N_k+1}\varphi_k^{\frac{[N_k+m-1]}{4}}\right\}.$$ Since $k\notin A$ we have: $$(2\kappa)^{-10c_1L_{k+1}}\varphi_k^{\frac{N_k^3}{32}}\leq
(2\kappa)^{-10c_1L_{k+1}}(2\kappa)^{\frac{u_kL_{k+1}N_k^2}{32}}\stackrel{(\ref{u0valpha})-(\ref{escalesL_k})}\leq1.$$ Using $[N_k]-1\geq N_k/2$ and $k\notin A$ once again, we have $$\begin{gathered}
\nonumber
\varphi_{k+1}\leq c_3(c_7\vee c_8)\widetilde L_{k+2}^{(d-1)}L_{k+1}\left\{\varphi_k^{\frac{N_k^3}{32}}+L_{k+1}\varphi_k^{\frac{N_k}{8}}\right\}\\
\label{almfinishlemre}
\leq 2c_3 (c_7\vee c_8)\widetilde L_{k+2}^{(d-1)}L_{k+1}^2\varphi_k^{\frac{N_k}{16}}\,\,(2\kappa)^{u_{k+1}L_{k+1}}.\end{gathered}$$ The rest of the proof will consist in finding a constant $c_9>0$ so that whenever $L_0\geq c_9$ one has $$\label{claim}
2c_3 (c_7\vee c_8)\widetilde L_{k+2}^{(d-1)}L_{k+1}^2\varphi_k^{\frac{N_k}{16}}\leq 1 ,$$ which produces the contradiction. For this end, we observe that in view of (\[escalesL\_k\]), after performing some basic estimations using $\widetilde L_0\leq L_0^4$, we get $$2c_3 (c_7\vee c_8)\widetilde L_{k+2}^{(d-1)}L_{k+1}^2\varphi_k^{\frac{N_k}{16}}\leq2c_3 (c_7\vee c_8)L_k^{4d-2}v^{4(d-1)}N_k^{4d-1}(2\kappa)^{20c_1 L_k}$$ Since $\kappa \leq 1/4$, it is clear that by inspection at (\[expescaleL\_k\]) one can find a constant $\widetilde c(d)$ so that whenever $L_0\geq \widetilde c$, $$2c_3(c_7\vee c_8)L_k^{4d-2}v^{4(d-1)}(2\kappa)^{c_1L_k}\leq 1.$$ As a result, $$\begin{gathered}
\nonumber
2c_3 (c_7\vee c_8)\widetilde L_{k+2}^{(d-1)}L_{k+1}^2\varphi_k^{\frac{N_k}{16}}\leq N_k^{4d-1}\kappa^{19c_1 L_k}\\
\label{ineceimp}
=\left(\frac{\alpha c_1}{u_0}\right)^{4d-1}v^{(4d-1)k}(2\kappa)^{19c_1(\frac{\alpha c_1}{u_0})^kv^{\frac{k(k-1)}{2}}}.\end{gathered}$$ In order to finish the proof we observe that one can choose $\widehat{c}(d)$ such that when $L_0\geq \widehat c$, the right most expression in (\[ineceimp\]) is smaller than $1$ for all $k\geq 1$. On the other hand, notice that for $k=0$, since $u_0\in[(2\kappa)^{\frac{7L_0}{4d-1}},1]$ by choosing a further positive constant $\overline c(d)$, for $L_0\geq \overline c$ one has $$\left(\frac{\alpha c_1}{u_0}\right)^{4d-1}(2\kappa)^{19c_1(\frac{\alpha c_1}{u_0})^k v^{\frac{k(k-1)}{2}}}\leq u_0^{-4d+1}(2\kappa)^{7L_0}\leq 1.$$ As was mentioned the claim (\[claim\]) turns out a contradiction with the assumption $A\neq \varnothing$ and ends the proof.
A crucial point shall be to prove that under an appropriate version of a seed condition as in (\[seedce\]), condition $(T')|\ell$ holds, however we shall first introduce some further notations and remarks. We are looking for a nice expression for $\varphi_0$ of Lemma \[lemmacriterion\]. Notice that in virtue of (\[escalesL\_k\]), $\varphi_0$ in (\[seedce\]) equals: $$(c_7\vee c_8)\left(\frac{\alpha c_1}{u_0}\right)^{4(d-1)}\widetilde L_0^{d-1}L_0.$$ It will be also convenient to consider the function $$\lambda:=[\max\{e^{-\frac{gL_0}{8d-3}},(2\kappa)^{\frac{7L_0}{4d-1}}\},1]\rightarrow[0,\infty],\,\, \lambda(u)=u^{4(d-1)}(2\kappa)^{uL_0}$$ which has its maximum value: $$\left(\frac{4(d-1)}{e\ln\left(\frac{1}{2\kappa}L_0\right)}\right)^{4(d-1)},$$ at point $u_0=\frac{4(d-1)}{L_0\ln(1/(2\kappa))}$, with $1>u_0>\max\{e^{-\frac{gL_0}{8d-3}},(2\kappa)^{\frac{7L_0}{4d-1}}\}$ provided that $L_0\geq \mathfrak c_{10}$ for a suitable constant $\mathfrak c_{10}(d,g,C)>0$.
We define the constant $c_{10}:=c_{10}(d, C,g)>0$ via: $$\label{const10}
c_{10}=\max\left\{\mathfrak c_{10}, c_9\right\}$$ as well as the constant $c_{11}:=c_{11}(d, C,g)>0$ $$\label{const10}
c_{11}=2^{d-1}\left(\frac{e\ln(\frac{1}{2\kappa})}{4(d-1)}\right)^{4(d-1)}(c_7\vee c_8)\alpha c_1$$ then, we proceed to state the main theorem of this section:
\[theocrite-tprime\] Assume that: $$\label{effectivecriterion}
\inf_{\mathcal B(R, L-2, L+2,\widetilde L), a\in [0,1]}\left(c_{11}\widetilde L^{d-1}L^{4(d-1)+1}\mathbb E\left[\rho_{\mathcal B}^{a}\right]\right)<1,$$ where the infimum runs over all the box specifications $\mathcal B(R, L-2, L+2,\widetilde L)$ where $R$ is a rotation with $R(e_1)=\ell$, $L\geq c_{10}$ and $L \leq \widetilde L< L^4$. Then condition $(T')|\ell$ is satisfied.
In virtue of hypothesis (\[effectivecriterion\]), setting $\widetilde{L'}=(\widetilde L+1)\wedge L^4\,\,(>\widetilde L)$, there exist some $a\in [0,1]$ and a box specification $\mathcal B(R, L-2, L+2,\widetilde {L'})$ such that $$\label{efcri}
c_{11}2^{-(d-1)}\widetilde{L'}^{d-1}L^{4(d-1)+1}\mathbb E\left[\rho_{\mathcal B(R, L-2, L+2,\widetilde {L'})}^{a}\right]<1,$$ since $$\mathbb E\left[\rho_{\mathcal B(R, L-2, L+2,\widetilde {L'})}^a\right]\leq \mathbb E\left[\rho_{\mathcal B(R, L-2, L+2,\widetilde L)}^a\right],$$ for $a\in [0,1]$. We take a rotation $R'$ near enough $R$ so that $R'(e_1)=\ell'$ and (cf. (\[ravaB\]) for notation) $$\label{inROT}
p_{\mathcal B(R,L-2,L+2,\widetilde{L'})}\leq p_{\mathcal B'}$$ where we denote by $\mathcal B'$ the box specification $\mathcal B(R', L-1,L+1, \widetilde {L'})$. Hence using inequality (\[inROT\]) one gets $$c_{11}2^{-(d-1)}\widetilde{L'}^{d-1}L^{4(d-1)+1}\mathbb E\left[\rho_{\mathcal B'}^{a}\right]<1,$$ which is in the spirit as the expression for $\varphi_0$ of Lemma \[lemmacriterion\], provided that we replace $L_0$ by $L$, $\widetilde L_0$ by $\widetilde{L'}$, $\ell$ by $R'(e_1)$ and $u_0= 4(d-1)/(\ln(1/(2\kappa))L)$. Therefore letting $L$ play the role of $L_0$, we can apply Lemma \[lemmacriterion\] under scales given in (\[escalesL\_k\])-(\[expescaleL\_k\]). For this end, for large $M$, $b,\,\, \widetilde b>0$ we take $k\geq 0$ so that $L_k< \widetilde b M\leq L_{k+1}$ and notice that defining the cylinder set $C$ by
$$C=\left\{x\in \mathbb Z^d:\,\,|x|_\perp\leq\frac{bM}{L_k},\, x\cdot \ell'\in(-\widetilde{b}M, bM)\right\},$$ as a result of applying first Chevyshev’s inequality and then $\mathbb E[q_k]\leq \mathbb E [\rho_k^{a_k}]$, we have $$\mathbb P[\mathcal H]\leq |C|\kappa^{\frac{u_kL_k}{2}},$$ where in the notations of Lemma \[lemmacriterion\] we have set $$\mathcal H=\left\{\omega\in \Omega, \,\, \exists x\in C:\,q_k\circ t_x\geq \kappa^{\frac{u_kL_k}{2}} \right\}.$$ On the other hand, the strong Markov property implies that on the event $\Omega\setminus\mathcal H$, $$P_0[T_{bM}^{\ell'}<\widetilde{T}_{-\widetilde b M}^{\ell'}]\geq \left(1-\kappa^{\frac{u_kL_k}{2}}\right)^{\left[\frac{bM}{L_k}\right]+1}$$ Therefore we can find some suitable constant $c>0$, so that for large $M$: $$P_0[T_{bM}^{\ell'}>\widetilde{T}_{-\widetilde b M}^{\ell'}]\leq \exp\left(-\widetilde b L\, e^{-c\sqrt{\ln(\widetilde b L)}}\right)$$ holds. It is clear now using Lemma \[lemmaTgamma\] the required claim follows.
It is possible to prove within the framework of this new effective criterion a decay nearer exponential than the previous one, however we will not need such improvement here (see [@GR15] for details).
Polynomial Mixing Condition: Proof of Theorem \[mainth1\] {#secpoly}
=========================================================
It is our main concern here to prove that a polynomial condition to be introduced in Definition \[defpoly\], implies the mixing effective criterion previously introduced in Section \[sectionce\]. As mentioned in the introduction, we follow a similar analysis as the one in [@BDR14], with a slight modification of their Definition 3.6 and a different choice of growth for the scales. This will be clearer after Remark \[remarkgoodpoly\].
We start with the choice of scales and then we shall introduce our *mixing polynomial condition*. We set $v:=44$ and consider for given $k\in \mathbb N$, positive numbers $N_k$ satisfying for $k\geq 0$: $$\label{scalespoly}
N_{k+1}=\left(\left[\frac{15c_1 N_0 \ln(1/\kappa)}{2\ln(N_0)}\right]+1\right)v^{k+1} N_k, \mbox{ with } N_0\geq 3\sqrt d,$$ later on further restriction on $N_0$ will be required.
We let $k$ be a nonnegative integer, $\ell\in\mathbb S^{d-1}$ be fixed and we fix as well, a rotation denoted $R$ of $\mathbb R^{d}$ such that $R(e_1)=\ell$. For $z\in N_k\mathbb Z\times N_k^3\mathbb Z^{d-1}$ we define boxes $\widetilde B_{1,k}(z),\, \dot{B_{1,k}}(z),\,B_{2,k}(z),$ along with the frontal boundary part $\partial^+B_{2,k}(z)$ of the last box, as follows $$\begin{aligned}
\label{boxespolynomial}
\widetilde B_{1,k}(z):= &R\left(z+[0,N_k]\times[0,N_k^{3}]^{d-1}\right)\cap \mathbb Z^d \\
\nonumber
B_{2,k}(z) := &R\left(z+\left(-\frac{N_k}{11}, N_k+\frac{N_k}{11}\right)\times\left(-\frac{N_k^3}{10}, N_k^3+\frac{N_k^3}{10}\right)^{d-1}\right)\cap \mathbb Z^{d},\\
\nonumber
\dot B_{1,k}(z):=&R\left(z+(0,N_k)\times (0,N_k^3)^{d-1}\right)\cap \mathbb Z^d \mbox{ and,}\\
\nonumber
\partial^+B_{2,k}(z):=&\partial B_2(z)\cap\left\{y\in\mathbb Z^d:\,(y-z)\cdot \ell \geq N_k, \right.\\
\nonumber
&\left.\forall i\in[2,d]\,\,-\frac{N_k^3}{10}<(y-z)R(e_i)<\frac{11N_k^3}{10} \right\}.\end{aligned}$$ Although the choice of numbers above seems in part arbitrary, we actually need for instance in the $\ell-$direction add a (here add means add $a$ times $N_k$ to the corresponding length of the box $B_{2,k}(0)$ in relation to the same length of $\widetilde B_{1,k}(0)$) number $a\in (0,1)$ such that the relation: $N_k-(4aN_k)=$ something proportional to $N_k$ holds (cf. Remark \[remarkgoodpoly\] for a further explanation). Any number less than $1/4$ does this, we have chosen $a=1/11$, therefore along direction $\ell$ points out the box dimensions at level $k$ are $N_k+(1/11)N_k$.
For integer $k\geq0$, let us denote by $\mathfrak B_k$ the set of boxes of scale $k$: $$\label{setofboxesk}
\mathfrak{B}_k:=\{B_{2,k}(z),\, z\in N_k\mathbb Z\times N_k^3\mathbb Z^{d-1} \},$$ and we also denote by $\mathfrak L_k$ the subset of $\mathbb R^d$: $$\label{setposboxpoly}
\mathfrak L_k:=N_k\mathbb Z \times N_k^3\mathbb Z^{d-1}.$$
The upcoming renormalization procedure requires the following: for each box $B_{2,k}(x)\in \mathfrak B_k$ where $k\geq 1$ and $x\in \mathfrak L_k$, the boxes $\dot B_{1,k-1}(z)$, $z\in \mathfrak L_{k-1}$ such that $\dot B_{1,k-1}(z)\subset B_{2,k}(x)$ form a *quasi-covering* of $B_{2,k}(x)$, in the following sense: $$\label{unionscalespoly}
B_{2,k}(x)\subset \bigcup_{\substack{t \in N_{k-1} \mathbb Z\times N_{k-1}^3\mathbb Z^{d-1}\\
\dot B_{1,k-1}(t)\subset B_{2,k}(x) }}\widetilde B_{1,k-1}(t).$$ This is satisfied if $N_0$ is such that for any $k\geq 1$ we have $$\label{requpolno}
N_{k}/N_{k-1}\in 110\mathbb N.$$ It will be convenient to assume that $N_0$ is a fixed number satisfying the previous requirement. We do introduce now our *mixing polynomial condition*.
\[defpoly\] For $M>0$ and $\ell\in \mathbb S^{d-1}$ we say that the mixing polynomial condition $(P_M)|\ell$ holds if for some $N_0$ large enough and satisfying (\[requpolno\]) we have $$\sup_{\substack x\in \widetilde B_{1,0}(0)}P_x\left[X_{T_{B_{2,0}(0)}}\notin \partial^+ B_{2,0}(0)\right]<\frac{1}{N_0^{M}}$$ holds.
Roughly speaking, the main idea behind the next renormalization procedure is: consider for large $L$ a box of *order* $B_{2,L}(0)$, and consider as well the first $k$ such that $N_k\leq L<N_{k+1}$. Then we will approximate with $k-$multiple renormalization procedures the unlikely walk exit probability from $B_{2, L}(0)$ and not only one as usual. In formal terms we need to introduce the notion of suitable into this polynomial framework.
\[defgoodboxpoly\] Consider $z\in \mathfrak L_k$, then for $k=0$ we say that box $B_{2,0}(z)\in \mathfrak B_0$ is $N_0-$*Good* if $$\label{goodboxpoly0}
\inf_{\substack{x\in \widetilde B_{1,0}(z)}}P_{x,\omega} [X_{T_{B_{2,0}(z)}}\in \partial^+B_{2,0}(z)]>1-\frac{1}{N_0^5},$$ otherwise we say that box $B_{2,0}(z)$ is $N_0-$*Bad*.
Inductively, for $k\geq 1$ and $z\in \mathfrak L_k$ we say that box $B_{2,k}(z)\in\mathfrak B_k$ is $N_k-$*Good* if there exists $t\in \mathfrak L_{k-1}$ with $\dot B_{2,k-1}(t)\in \mathfrak B_{k-1}$, $\dot B_{1,k-1}(t)\subset B_{2,k}(z)$ such that for each another $y\in \mathfrak L_{k-1}$, with $B_{2,k-1}(y)\in\mathfrak B_{k-1}$, $\dot B_{1, k-1}(y)\subset B_{2,k}(z)$ and $B_{2, k-1}(y)\cap B_{2,k-1}(t)=\varnothing$, implies that the box $B_{2, k-1}(y)$ is $N_{k-1}-$*Good*.
Otherwise, we say that box $B_{2,k}(z)$ is $N_k-$*Bad*.
Several remarks are needed for reference purposes later on.
\[remarkgoodpoly\]
- Informally for a given $k\geq 1$, a box $B_{2,k}(0)$ is good if there exists *at most one bad box* $B_{2,k-1}(y)$, $y\in \mathfrak L_{k-1}$ of scale $k-1$ inside of $B_{2,k}(0)$. We actually might have more than only one, indeed all of those boxes intersecting $B_{2,k-1}(y)$ might be bad, but not more. We neither have all the boxes inside of scale $k-1$ inside of $B_{2,k}(0)$, we have rather all of the their *frontal parts:* $\dot B_{1,k-1}$ inside of $B_{2,k}(0)$.
- Notice that the property of being “Good” for a box $B_{2,k}(x)$, with $x\in \mathfrak L_k, \, k\geq 0$ under this definition, depends at most on transitions at sites of the set $\mathcal{B}_{k,x}$ defined by: $$\begin{aligned}
\label{attachedboxpoly}
\mathcal B_{k,k}:=&\left\{z\in\mathbb Z^d z\in R\left(x+(\sum_{i=0}^{k}\frac{-N_i}{11},(1+1/11)N_{k}+\sum_{i=0}^{k-1}\frac{N_i}{11})\times \right.\right.\\
\nonumber
&\left.\left.(-\sum_{i=0}^{k}\frac{N_i^3}{10}, (1+1/10)N_k^3 + \sum_{i=0}^{k-1}\frac{N_i^3}{10})^{d-1}\right) \right\},
\end{aligned}$$ which is straightforward to prove by induction. We further observe that $$\sum_{i=0}^{k-1}\frac{N_i^3}{10}\leq \frac{N_k^3}{10} \mbox{ and }\sum_{i=0}^{k-1}\frac{N_i}{11}\leq \frac{N_k}{11}.$$
- We note as well that a given box $B_{2,k}(z)$, with fixed $z\in \mathfrak L_k$ and $k\geq 0$, the number of boxes in $\mathfrak B_k$ intersecting $B_{2,k}(z)$ along a given fixed direction $\pm R(e_i),\, i\in[1,d]$ is at most three (if we include the box itself). Furthermore, along a given direction we have nonconsecutive boxes are disjoint, indeed they are separated at least a distance $9N_k/11$ in terms of the $\ell_1-$norm. Consequently along a fixed direction $\pm R(e_i),\, i\in[1,d]$ disjoint boxes have attached boxes of the type (\[attachedboxpoly\]) separated in $\ell_1-$norm $7N_0/11$ and thus the present choice of scales makes capable to apply mixing conditions on renormalization schemes.
The next objective is to get *doubly exponential upper bounds* in $k$ for the probability of a given box in $\mathfrak B_k$ to be $N_k-$*Bad*, we begin with case $k=0$.
\[lemmaseedpolybad\] Assume $(P_M)|\ell$ to be fulfilled for given $M>0$ and $\ell \in \mathbb S^{d-1}$. Then for any $z\in\mathfrak L_0$ $$\label{estimabad0}
\mathbb P[B_{2,0}(z) \mbox{ is } N_0-\mbox{\textit{Bad}} ]\leq \frac{1}{N_0^{M-3(d+1)}}$$ holds.
Let $M$, $\ell$ and $z$ be as in the statement of lemma. Then from the very Definition \[defgoodboxpoly\] we have $$\begin{aligned}
\mathbb P[B_{2,0}(z) \mbox{ is } N_0-\mbox{\textit{Bad}}]&= \mathbb P [\sup_{\substack x\in \widetilde B_{1,0}(z)}P_{x,\omega}[X_{T_{B_{2,0}(z)}}\notin \partial^+B_{2,0}(z)]\geq\frac{1}{N_0^5}]\\
&\leq N_0^5 \sum_{\substack{x\in \widetilde B_{1,0}(z)}}P_x[X_{T_{B_{2,0}(z)}}\notin \partial^+B_{2,0}(z)]\\
&\leq N_0^5|\widetilde B_{1,0}(z)|\sup_{\substack{x\in \widetilde B_{1,0}(z)}}P_x[X_{T_{B_{2,0}(z)}}\notin \partial^+B_{2,0}(z)]\\
&\leq N_0^5|\widetilde B_{1,0}(z)|\frac{1}{N_0^M}\leq\frac{1}{N_0^{M-5-3d+2}},\end{aligned}$$ where we have used Chevyshev’s inequality in the first inequality and the hypothesis $(P_M)|\ell$ in the penultimate inequality above, the proof is complete.
We continue with the estimates for general $k\geq1$ in the following:
\[lemmakpolybad\] Assume $(P_M)|\ell$ to be fulfilled for $M>9d$ and given $\ell\in \mathbb S^{d-1}$. For $k\geq 0$ we let $z\in \mathfrak L_k$ be fixed, then there exists positive constants $\eta_1:=\eta_1(d)$ and $\eta_2:=\eta_2(\eta_1, d)$ such that whenever $N_0\geq \eta_1$ and $N_0$ satisfies (\[requpolno\]) we have $$\label{estimabadk}
\mathbb P[B_{2,k}(z) \mbox{ is } N_k-\mbox{\textit{Bad}}]\leq \exp\left(-\eta_2 2^{k}\right)$$ holds.
Let $M$, $\ell$ and $z$ be as in the statement of this proposition. Consider a sequence of positive constants $(c_j)_{j\geq0}$ defined by (recall Definition \[sma\] and \[smg\] for notation) $$\begin{aligned}
\label{ckpoly}
c_0=&(M-3(d+1))\ln(N_0), \mbox{ and for $k\geq 0$ }\\
\nonumber
c_{k+1}=&c_{k}-\frac{\ln \left(\frac{12}{10}\left(\left[\frac{15c_1 N_0 \ln(1/\kappa)}{2\ln(N_0)}\right]+1\right)v^{k+1}\right)^{6d-4}}{2^{k+1}}-\\
\nonumber
&\frac{\ln\left(2C|B_{2,k}|^2e^{-\frac{gN_k}{4}}\right)}{2^{k+1}}.\end{aligned}$$ We shall prove by induction the following claim: for any $k\geq 0$, the inequality $$\label{claimbadboxk}
\mathbb P[B_{2,k}(z) \mbox{ is } N_k-\mbox{\textit{Bad}}]\leq \exp\left(-c_k2^{k}\right)$$ holds. Later on we shall prove that $\inf_{\substack{k\geq 0}}c_k\geq1$ for some choice of $N_0$ large enough, and thus we will eventually finish the proof.
We first focus on the case $k=0$ in (\[estimabadk\]). Using Lemma \[lemmaseedpolybad\] one can rewrite (\[estimabad0\]) as $$\mathbb P[B_{2,0}(z) \mbox{ is } N_0-\mbox{\textit{Bad}} ]\leq \exp\left((-M+3d+3)\ln(N_0)\right).$$ Hence, it suffices to prove the induction step $k$ to $k+1$. For this end, we now assume inequality (\[claimbadboxk\]) holds for some $k\geq0$, we shall prove that claim (\[claimbadboxk\]) holds when $k$ is replaced by $k+1$. Indeed, let $z\in \mathfrak L_{k+1}$ be fixed and consider the environmental event: $z-$Bad:=“$B_{2,k+1}(z)$ is $N_{k+1}-$*Bad*”. Using Definition \[defgoodboxpoly\] one sees $z-$Bad is a subset of the event: “there exist two disjoint boxes $B_{2,k}(t_1),B_{2,t_2}(t_2)$ of scale $k$ such that $\dot B_{1,k}(t_1),\dot B_{1,t_2}(t_2)$ are contained in $B_{2,k+1}(z)$ and $B_{2,k}(t_1),B_{2,t_2}(t_2)$ are $N_k-$*Bad*”. Therefore, introducing for $k\geq0$, the set: $$\begin{aligned}
\Lambda_k:=&\{(t_1, t_2)\in \mathfrak L_k\times \mathfrak L_k:\,B_{2,k}(t_1)\cap B_{2,k}(t_2)=\varnothing,\\
&\dot B_{1,k}(t_1), \dot B_{1,k}(t_2)\subset B_{2,k+1}(z) \},\end{aligned}$$ we have $$\begin{aligned}
\nonumber
\mathbb P[z-\mbox{Bad}]\leq& \mathbb P[\exists (t_1, t_2)\in \Lambda_k: B_{2,k}(t_1), B_{2,k}(t_2) \mbox{ are }N_k-\mbox{\textit{Bad}} ]\\
\label{inebadz}
\leq &\sum_{\substack{(t_1,t_2)\in \Lambda_k}}\mathbb P[B_{2,k}(t_1), B_{2,k}(t_2) \mbox{ are }N_k-\mbox{Bad}].\end{aligned}$$ Observe that by the third item in Remark \[remarkgoodpoly\] disjoint boxes of same scale $k$ depends on transition prescribed by sets separated in $\ell_1-$norm at least $7N_k/11$. As a result, using either: **(SMG)**$_{C,g}$ or **(SM)**$_{C,g}$ the last probability inside the sum in (\[inebadz\]) splits into a product of two factors up to a mixing correction. More precisely, we introduce for $k\geq 0$ and $(t_1,t_2) \in \Lambda_k$ the mixing factor (under notation of Remark \[remarkgoodpoly\]): $$\begin{aligned}
\Gamma_k:=\max&\left\{\exp\left(\sum_{x\in \mathcal B_{k,t_1}, y\in \mathcal B_{k,t_2} }Ce^{-g|x-y|_1}\right),\right.\\
&\left.\exp\left(\sum_{x\in \partial^r \mathcal B_{k,t_1}, y\in \partial^r \mathcal B_{k,t_2 }}Ce^{-g|x-y|_1}\right)\right\},\end{aligned}$$ then under the previous notation and either: **(SMG)**$_{C,g}$ or **(SM)**$_{C,g}$ we have $$\begin{aligned}
&\mathbb P[B_{2,k}(t_1), B_{2,k}(t_2) \mbox{ are }N_k-\mbox{Bad}]\leq\\
&\Gamma_k \times \mathbb P[B_{2,k}(t_1)\mbox{ is }N_k-\mbox{\textit{Bad}}]\times
\mathbb P[B_{2,k}(t_2)\mbox{ is }N_k-\mbox{\textit{Bad}}].\end{aligned}$$ Going back to (\[inebadz\]) together with induction hypothesis (\[claimbadboxk\]) we have the probability $\mathbb P[z-\mbox{Bad}]$ is less than $$\begin{aligned}
\nonumber
& |(t_1,t_2)\in \Lambda_k|\times \Gamma_k\times e^{-c_k2{-(k+1)}}\\
\nonumber
\leq& \left(\frac{12}{10}\left(\left[\frac{15c_1 N_0 \ln(1/\kappa)}{2\ln(N_0)}\right]+1\right)v^{k+1}\right)^{6d-4}\\
\nonumber
&\times\exp\left(-\left(c_k-\frac{\ln(\Gamma_k)}{2^{k+1}}\right)2^{k+1}\right)\\
\nonumber
\leq& \exp\left(-2^{k+1} \left(c_k-\frac{\ln \left(\frac{12}{10}\left(\left[\frac{15c_1 N_0 \ln(1/\kappa)}{2\ln(N_0)}\right]+1\right)v^{k+1}\right)^{6d-4}}{2^{k+1}}-\right.\right.\\
\label{inepoly1}
&\left.\left.\frac{\ln\left(2C|B_{2,k}|^2e^{-\frac{gN_k}{4}}\right)}{2^{k+1}}\right)\right).\end{aligned}$$ We have used in the previous chain of inequalities (\[inepoly1\]) a non-sharp estimate: $$\Gamma_k\leq e^{2C|B_{2,k}|^2e^{-\frac{gN_k}{4}}},$$ together with the very construction prescribed by (\[scalespoly\]), recall also the discussion after (\[unionscalespoly\]) and in (\[requpolno\]).
We now use definitions given in (\[ckpoly\]) to get that $$\mathbb P[z-\mbox{Bad}]\leq \exp\left(-c_{k+1}2^{k+1}\right)$$ provided $N_0\geq \eta$. The induction is finished, therefore as was mentioned it suffices to prove at this point the claim: there exists constant $\eta'(d)\geq \eta$ such that the inequality: $$\inf_{\substack{j\geq 0}}c_j\geq 1$$ holds, provided that $N_0\geq \eta'$. However a rough counting argument gives $$\lim_{k\rightarrow\infty}c_k-c_0$$ is smaller than $$\begin{aligned}
&\sum_{\substack{j\geq 0}}\frac{\ln \left(\frac{12}{10}\left(\left[\frac{15c_1 N_0 \ln(1/\kappa)}{2\ln(N_0)}\right]+1\right)v^{j+1}\right)^{6d-4}}{2^{j+1}}+
\frac{\ln\left(2C|B_{2,j}|^2e^{-\frac{gN_j}{4}}\right)}{2^{j+1}} \\
\leq&(6d-4)\ln\left(\frac{12}{10}\left(\left[\frac{15c_1 N_0 \ln(1/\kappa)}{2\ln(N_0)}\right]+1\right)\right)+(6d-4)2\ln(v)+1.\end{aligned}$$
It is now straightforward to verify $\inf_{k\geq0} c_k\geq 1$ using that $c_0=(M-3(d+1))\ln(N_0)$ provided $N_0$ be large but fixed, which ends the proof.
We now take a further step into the proof of Theorem \[mainth1\]. Similarly as the argument given in [@BDR14], quenched exponential bounds for the unlikely exit event from a *Good* box will be needed. Notice that we will have a weaker decay in the quenched estimate than in the similar result stated in Proposition 3.9 of [@BDR14]. It is mostly owed to the present choice of scales (\[scalespoly\]) and definition of good boxes in Definition \[defgoodboxpoly\], which were introduced in that form to clearly avoid any intersection problem. Roughly speaking under our scaling construction, for a given box of scale $k$ we have less boxes of scale $k-1$ intersecting it. The formal statement is as follows in the next proposition.
\[propquenpol\] For integer $k\geq 0$ we let $z\in \mathfrak L_k$ and $B_{2,k}(z)\in \mathfrak B_k$ be $N_k-$*Good*. Then there exist positive constants $\eta_3:=\eta_3(d)$ and $\eta_4:=\eta_4(\eta_3,d)$ such that whenever $N_0\geq \eta_3$ and $N_0$ satisfies (\[requpolno\]) we have $$\label{Quenchedexpbo}
\sup_{x\in \widetilde B_{1,k}(z)}P_{x,\omega}\left[X_{T_{B_{2,k}(z)}}\notin \partial^+B_{2,k}(z)\right]\leq e^{-\frac{\eta_4 N_k}{v^{k+1}}}$$ holds.
By stationarity under spatial shifts of the probability measure $\mathbb P$, it is enough to prove the proposition for general $k\geq 0$ and $z=0$. Henceforth we assume that $z=0$ along with $B_{2,k}(0)$ is $N_k-$*Good* for given $k\geq 0$.
We will first prove by induction the statement: for each $k\geq 0$ there exists a sequence of positive numbers $(c_k)_{k\geq0}$ such that whenever $N_0\geq \zeta_1$ $$\label{quenchestgoodp}
\sup_{x\in \widetilde B_{1,k}}P_{x,\omega}\left[X_{T_{B_{2,k}}}\notin \partial^+B_{2,k}\right]\leq e^{-c_k N_k}$$ holds.
Where the sequence $(c_k)_{k\geq0}$ can be derived along the proof of the induction, however we keep the convention as in the previous proposition of defining them here: $$\begin{aligned}
\label{ckquepoly}
&c_0:=\frac{5\ln N_0}{N_0}, \mbox{ and by for $k\geq1$}\\
\nonumber
&c_k:=\frac{5\ln N_0}{(44)^k N_0}.\end{aligned}$$
Our method to prove inequality (\[quenchestgoodp\]) will follow a similar analysis as the one given in Section \[sectionce\], whose argument was originally owed to Sznitman [@Sz02].
Let us start with $k=0$, in this case we observe that whenever $B_{k,2}$ is $N_k-$*Good* by Definition \[defgoodboxpoly\] we have $$\label{0induction}
\sup_{\substack{x\in \widetilde B_1}}P_{x}[X_{T_{B_{2,0}}}\notin\partial^+B_{2,0}]\leq \frac{1}{N_0^5}=e^{-5\frac{\ln{N_0}}{N_0}N_0}.$$ Hence we have (\[quenchestgoodp\]) holds from the very definition of $c_0$ in (\[ckquepoly\]).
We now proceed to prove the induction step for $k\geq1$. it amounts to prove the statement of (\[quenchestgoodp\]) for $k$ assuming that (\[quenchestgoodp\]) is satisfied for $k$ replaced by $k-1$.
For this purpose, we introduce for $k\geq 0$, $u\in\mathbb R$ and $i\in [2,d]$ the $\mathcal F_n-$stopping times $\sigma_u^{+i}$ and $\sigma_u^{-i}$ defined by $$\begin{aligned}
\sigma_u^{+i}:=&\inf\{n\geq 0 : (X_n-X_0)\cdot R(e_i)\geq u\} \mbox{ and}\\
\sigma_u^{-i}:=&\inf\{n\geq 0 : (X_n-X_0)\cdot R(e_i)\leq u\}.\end{aligned}$$ Let $i\in [2,d]$ and denote by $\vartheta_k^{+i}$ and $\vartheta_k^{-i}$ the stopping times $$\begin{aligned}
\sigma_{\frac{N_k^3}{10}}^{+i} \mbox{ and }\sigma_{-\frac{N_k^3}{10}}^{-i}\end{aligned}$$ respectively. Before formal developments, notice that *levels* $\pm N_k^3$ entering at the $\sigma$’s stopping times above, we say $\vartheta_k^{+2}$ leave out the event where walk starts from any point in $\widetilde B_{1,k}(0)$ exits box $B_{2,k}(0)$ before that stopping time when the walk exit is known through direction $+R(e_2)$. It will be useful to introduce as well, the path event $\mathfrak I_k$ for $k\geq 1$ $$\mathfrak I_k:=\{\exists i\in [2,d]: \vartheta_{k-1}^{+i}\leq T_{B_{2,k}(0)} \ \mbox{ or } \ \vartheta_{k-1}^{-i}\leq B_{2,k}(0)\}.$$ In order to simplify notation, we shall drop the dependence of $0\in \mathbb R^d$ from the boxes $\widetilde B_{1,k}:=\widetilde B_{1,k}(0)$ and $B_{2,k}:=B_{2,k}(0)$.
Plainly letting $x\in \widetilde B_{1,k}$ be arbitrary and observing that the following decomposition $$\begin{aligned}
\label{decomquenpoly}
P_{x,\omega}\left[X_{T_{B_{2,k}}}\notin \partial^+B_{2,k}\right]\leq &P_{x,\omega}[\mathfrak I_{k}]+ \\
\nonumber
&P_{x,\omega}\left[(\mathbb Z^d)^{\mathbb N}\setminus \mathfrak I_{k}, X_{T_{B_{2,k}}}\cdot \ell\leq -N_{k}(1+1/11)\right]\end{aligned}$$ holds. In virtue of this last inequality, we will get upper bounds for the terms on the right most hand of inequality (\[decomquenpoly\]). Let us start with bounding from above the first one of them.
Thus, we further decompose the first term on the right hand side of inequality (\[decomquenpoly\]), indeed we have $$\label{ordecpoly}
P_{x,\omega}\left[\mathfrak I_k\right]\leq \sum_{i=2}^{d}\left(P_{x,\omega}\left[\vartheta_{k-1}^{+i}\leq T_{B_{2,k}}\right]+ P_{x,\omega}\left[\vartheta_{k-1}^{-i}\leq T_{B_{2,k}}\right]\right).$$ As a result it suffices to prove an upper bound in the spirit of the one in (\[quenchestgoodp\]) for a single stopping time of the $\vartheta-$type, we say stopping time $\vartheta_{k-1}^{+2}$, the others cases are analogous and the general procedure to be displayed below will provide the same upper bound for them.
Define the *orthogonal norm* $|\,\cdot\,|_\perp$ by $$|z|_\perp:=\sup_{\substack{i\in[2,d]}}|z\cdot R(e_i)|$$ for $z\in \mathbb R^d$ (this semi-norm was already introduced, we repeat here for reader convenience).
For easy of notation we set $$n_{k-1}:=\frac{1}{2}\left(\frac{13N_k}{11N_{k-1}}+1\right)\in \mathbb N \mbox{ (by construction),}$$ and note $2n_{k-1}-1$ is the number of consecutive boxes $B_{2,k-1}(t), \, t\in\mathfrak L_{k-1}$ of scale $k-1$ along direction $R(e_1)=\ell$ such that $\dot B_{1,k-1}(t) \subset B_{2,k}$ (recall comments after (\[unionscalespoly\])).
In these terms, it will be convenient to define for the fixed integer $k\geq1$, numbers $J_{k-1}$ and $\overline L_{k-1}$ by (recall $[\,\cdot\,]$ denotes the integer part function) $$\begin{aligned}
\label{scalesorthofluc}
\nonumber
J_{k-1}:=& \frac{10N_{k}^{3}}{220N_{k-1}^3n_{k-1}} \mbox{ and} \\
\overline{L_{k-1}}:=& \frac{22n_{k-1}N_{k-1}^3}{10}.\end{aligned}$$
Note that for any $x\in \widetilde B_{1,k}$ we have $P_{x,\omega}-$a.s. $$\left\{\vartheta_k^{+2}\geq\sigma_{J_{k-1}\overline L_k}^{+2}\right\}.$$
It will be useful to introduce sets $c_\perp(j)$ for integer $j$ to determine levels along direction $R(e_2)$ as follows: $$\begin{aligned}
c_\perp(j):=&\left\{ z\in \mathbb Z^d:\,z\cdot R(e_2)\in \overline L_{k-1}[j,j+1),\right. \\
&\left.-1/10 N_k^3 < z \cdot R(e_i)< 11/10 N_k^3 \right\}\end{aligned}$$ together with the discrete truncated cylinder $\overline c(j)$ defined by $$\begin{aligned}
\overline c(j):=&\left\{z\in \mathbb Z^d:\,\inf_{w\in c_\perp(z)}|z-w|_\perp\leq 2n_{k-1},\right.\\
&\left. z\cdot \ell\in (-(1/11)N_k,(12/11)N_k ) \right\}.\end{aligned}$$ Let us further introduce for technical matters to be clarify soon, the stopping time $\sigma_{u}^{2x}$ depending on $u\in\mathbb R$ and $x\in \widetilde B_{1,k}$, defined by $$\sigma_{u}^{2x}:=\inf\{n\geq0:\, (X_n-x)R(e_2)\geq u\}.$$ We now observe that using the strong Markov property for arbitrary $x\in \widetilde B_{1,k}$, one gets $$\begin{aligned}
\label{decompsigma}
&P_{x,\omega}\left[\vartheta_{k-1}^{+2}\leq T_{B_{2,k}}\right]\leq P_{x,\omega}\left[\sigma_{J_{k-1}\overline L_{k-1}}^{+2}\leq T_{B_{2,k}}\right]=\\
\nonumber
& E_{x,\omega}\left[\sigma_{(J_{k-1}-2)\overline L_{k-1}}^{+2}<T_{B_{2,k}},P_{X_{\sigma_{(J_{k-1}-2)\overline L_{k-1}}^{+2}},\omega}\left[\sigma_{J_{k-1}\overline L_{k-1}}^{2x}\leq T_{B_{2,k}}\right]\right].\end{aligned}$$ Plainly we have $P_{x,\omega}-$a.s. on the event $\{\sigma_{(J_{k-1}-2)\overline L_{k-1}}^{+2}<T_{B_{2,k}}\}$, the random variable $$X_{\sigma_{(J_{k-1}-2)\overline L_{k-1}}^{+2}}$$ belongs to $c_\perp(J_{k-1}-2)$. The strategy to prescribe on the walk path is going to bound from below the probability of the complementary event $\{\sigma_{J_{k-1}\overline L_{k-1}}^{2x}\leq T_{B_{2,k}}\}$, for paths starting from arbitrary $y\in c_\perp(J_{k-1}-2)$. Indeed, the strategy consists in pushing the walk successively $(2n_{k-1}-1)-$times to exit boxes of scale $k-1$ inside of $B_{2,k}$ by their boundary side where $\ell$ points out. This will be fulfilled with relatively high probability since $B_{2,k}$ is $N_k-\textit{Good}$.
Formally to prescribe the strategy we introduce a sequence of stopping times $(H^i)_{i\geq0}$ along with two sequences of successive random positions $(Y_i)_{i\geq 0}$, $(Z_i)_{i\geq0}$, which are recursively defined via: $$\begin{aligned}
&H^0=0,\,Y_0=X_0,\, Z_0\in\{z\in \mathfrak L_{k-1}:\, Y_0 \in \widetilde B_{1,k-1}(z)\}, \, , \\
&H^1=T_{B_{2,k}}\wedge T_{ B_{2,k-1}(Z_0)},\ \mbox{ and for $i>1$,} \\
&Y_{i-1}=X_{H^{i-1}}, \,\ Z_{i-1}\in \{z\in \mathfrak L_{k-1}:\,Y_{i-1} \in \widetilde B_{1,k-1}(z)\},\\
&H^i=H^{i-1}+H^1\circ \theta_{H^{i-1}}.\end{aligned}$$ Notice that the construction of sequence $(Z_i)_{i\geq0}$ makes use of finite arbitrary choices.
One also defines the stopping time $S$ to indicate a *wrong* exit from box $B_{2,k-1}$, defined by $$S:=\inf\left \{n\geq 0: X_n \in \partial B_{2,k-1}(Z_0)\setminus \partial^+ B_{2,k-1}(Z_0)\right\}.$$ For any $y\in c_\perp(J_{k-1}-2)$, since (\[unionscalespoly\]) one has $P_{y,\omega}-$a.s. on the event $$\bigcap_{i=0}^{2n_{k-1}-2}\theta_{H^i}^{-1}\{S> H^1\}$$ the walk exits $B_{2,k}$ before time $\sigma_{J_{k-1}\overline L_{k-1}}^{2x}$. We need to introduce some further definitions in order to get estimates under this strategy.
Let us first define for $i\in[0,2n_{k-1}-2]$ and $j\in \mathbb Z$ the set: $$\begin{aligned}
\Theta_{i,j-2}:=&\{z\in \mathbb Z^d:\, \exists w\in \mathfrak L_{k-1}, z\in \widetilde B_{1,k-1}(w), \\
&w\cdot \ell=-\left(\frac{1}{11}N_k-iN_{k-1}\right), B_{2,k-1}(w)\subset \overline c\left(j-2\right)\}\end{aligned}$$ and then we introduce the random variable $$\psi_{i,j-2}:=\inf_{h\in \Theta_{i,j-2}}P_{h,\omega}[S>H_1].$$ As a result of applying the strong Markov property successively many times, we get $$\begin{aligned}
\label{inequenortho}
P_{y,\omega}[T_{B_{2,k}}<\sigma_{J_{k-1}\overline L_{k-1}}^{2x}]\geq \prod_{i=0}^{2n_{k-1}-2}\psi_{i,J_{k-1}-2}=:\varphi(J_{k-1}-2).\end{aligned}$$ Hence in virtue of (\[decompsigma\]), for any $x\in \widetilde B_{1,k}$ we have got the estimate $$\begin{aligned}
\label{ineinothop}
&P_{x,\omega}[\sigma_{J_{k-1}\overline L_{k-1}}^{+2}\leq T_{B_{2,k}}]\leq\\
\nonumber
& P_{x,\omega}\left[\sigma_{(J_{k-1}-2)\overline L_{k-1}}^{+2}\leq T_{B_{2,k}}\right]\left(1-\varphi(J_{k-1}-2)\right).\end{aligned}$$ In turn, let us further decompose part of the most right hand term in (\[ineinothop\]), in order to iterate the given process up to this point, to be precise: $$\begin{aligned}
\label{ineindorthpoly}
&P_{x,\omega}\left[\sigma_{(J_{k-1}-2)\overline L_{k-1}}^{+2}\leq T_{B_{2,k}}\right]\leq\\
\nonumber
&E_{x,\omega}\left[\sigma_{(J_{k-1}-5)\overline L_{k-1}}^{+2}\leq T_{B_{2,k}},P_{X_{\sigma_{(J_{k-1}-5)\overline L_{k-1}}^{+2}}}\left[\sigma_{(J_{k-1}-3)\overline L_{k-1}}^{2x}\right]\right].\end{aligned}$$ We observe that using (\[ineindorthpoly\]), the same procedure applied at *level*: $$X_{\sigma_{(J_{k-1}-5)\overline L_{k-1}}^{+2}}$$ instead of $$X_{\sigma_{(J_{k-1}-2)\overline L_{k-1}}^{+2}}$$ will turn out a strategy for any $y$ in $c_\perp((J_{k-1}-5)\overline L_{k-1})$ which will force the walk successively $(2n_0-1)-$times through direction $\ell$, to exit successive boxes $B_{2,k-1}(t'), t'\in \mathfrak L_{k-1}$, such that $\dot B_{1,k-1}(t')\subset B_{2,k}$ by their boundary side $\partial^+B_{2,k-1}(t')$.
Thus for any $y\in \overline c(J_{k-1}-5)$ we have $P_{y,\omega}-$a.s. on event outlying above the walk exit $B_{2,k}$ before getting level $J_{k-1}-3)\overline L_{k-1}$ along direction $R(e_2)$, i.e.: $$\begin{aligned}
\label{strategyortho}
&\bigcap_{i=0}^{2n_{k-1}-2}\theta_{H^i}^{-1}\{S> H^1\}\subset\\
\nonumber
&\{\sigma_{(J_{k-1}-3)\overline L_{k-1}}^{2x}> T_{B_{2,k}}\}.\end{aligned}$$ It will be useful stress that the event on the left hand of (\[strategyortho\]) in this case, makes use of disjoint boxes of the ones previously used. Therefore for arbitrary $x\in \widetilde B_{1,k}$ one obtains by induction $$\label{inducorthpoly}
P_{x,\omega}\left[\vartheta_{k-1}^{+2}\leq T_{B_{2,k}}\right]\leq \prod_{i=1}^{[J_{k-1}/3-2/3]}\left(1-\varphi(J_{k-1}-3i-2)\right).$$ Moreover, by assumption $B_{2,k}$ is $N_k-$*Good*, hence one sees in virtue of remark above about the disjointness of the boxes involved in the events into product terms in (\[inducorthpoly\]), we have that there exist at most three $N_{k-1}-$*Bad* boxes in at most one of the events involved in $\varphi(J_{k-1}-3i)$, for (we have used here as well, Remark \[remarkgoodpoly\] about number of intersection of boxes along a fixed direction).
Thus for any $x\in \widetilde B_{1,k}$ we will bound from above by one the eventual level $i\in [1,[(J_{k-1}-2)/3]]$ containing the bad box, recalling (\[decompsigma\]) and applying the induction hypothesis to each besides one term inside of product (\[inducorthpoly\]), we see $$\begin{aligned}
P_{x,\omega}\left[\vartheta_{k-1}^{+2}\leq T_{2,k}\right] \leq&
\left(1-(1-e^{-c_{k-1}N_{k-1}})^{2n_{k-1}-1}\right)^{[(J_{k-1}-2)/3]-1}\\
\leq&\left((2n_{k-1}-1)e^{-c_{k-1}N_{k-1}}\right)^{[(J_{k-1}-2)/3]-1}\\
\leq&e^{-c_{k-1}N_{k-1}J_{k-1}/6+\ln(2n_{k-1}-1)J_{k-1}/6}\end{aligned}$$ which is satisfied provided that $N_0\geq \eta$ for certain positive constant $\eta$. As was mentioned, similar bounds satisfy for others bounds in (\[ordecpoly\]), hence for arbitrary $x\in\widetilde B_{1,k}$ $$\label{orthpolyest}
P_{x,\omega}\left[\mathfrak I_k\right]\leq e^{-c_{k-1}N_{k-1}J_{k-1}/6+\ln(2n_{k-1}-1)J_{k-1}/6+\ln(2d-2)}.$$ We now treat the second term on the right most hand of inequality (\[decomquenpoly\]). Our procedure will be similar as the one given in the previous section, in fact we try to emulate one dimensional computation with the use of *quasi-martingales*. We define for $i\in \mathbb Z$, the strip $\mathcal H_i$ via $$\mathcal H_i:=\left\{z\in \mathbb Z^d:\ \exists z'\in \mathbb Z,\ |z-z'|_1=1, (z\cdot \ell- iN_{k-1})\cdot(z'\cdot\ell -i N_{k-1} )\leq 0\right\}.$$ Recall that we have fixed $x\in\widetilde B_{1,k}$, which in turn amount to fix the integer $k\geq1$. For later purposes, we further define the set $\mathcal{\widehat H}_{0}$ by $$\mathcal{\widehat H}_{0}:=\mathcal H_0\cap \{z\in\mathbb R^d:\ |(z-x)\cdot R(e_j)|<\frac{N_k^3}{10}\},$$ and throughout this part of the proof we are going to let $x_1\in \mathcal{\widehat H}_{0}$ be arbitrary. Furthermore, we also define the function $I(\cdot)$ on $\mathbb R^{d}$ such that $I(z)=i$ whenever $z\cdot\ell\in[iL_{k-1}-\frac{L_{k-1}}{2}, iL_{k-1}+\frac{L_{k-1}}{2}) $. If $N_0\geq \eta'$ for certain constant, then $I(\mathcal H_i)=i$ for each $i\in \mathbb Z$. We introduce a sequence $(V_j)_{j\geq0}$ the successive times of visits to different strips $\mathcal H_i,\, i\in \mathbb Z$ recursively via: $$\begin{aligned}
\label{Vtimes}
&V_0=0, \, V_1=\inf\{n\geq0:\, X_n\in \mathcal H_{I(X_0)-1}\cup \mathcal H_{I(X_0)+1}\}, \\
\nonumber
&\mbox{and for $n\geq 1$ } V_n=V_{n-1}+V_1\circ \theta_{V_{i-1}}.\end{aligned}$$ Let us also define, for $\omega \in \Omega$ and $y\in \mathbb Z^d$ random variables $q(y,\omega)$, $p(y,\omega)$ and $\rho(y,\omega)$, as follows: $$\begin{aligned}
q(y,\omega):=&P_{y,\omega}\left[X_{V_1}\in \mathcal H_{I(X_0)-1}\right]=:1-p(y,\omega), \\
\rho(y,\omega):=&\frac{q(y,\omega)}{p(y,\omega)}.\end{aligned}$$ In these terms for $i\in \mathbb Z$ we further define the random variable $\rho(i):=\rho_\omega (i)$ given by $$\rho(i):=\sup_{y\in \mathcal{\widetilde H}_i}\frac{q(y,\omega)}{p(y,\omega)},$$ where in turn for $i\in \mathbb Z$, the truncated strip $\mathcal{ \widetilde H}_i$ is defined by $$\mathcal{ \widetilde H}_i=\mathcal H_i\cap B_{2,k}.$$ It will be useful to introduce an environmental random function $f_{\omega}:\mathbb Z \rightharpoonup\mathbb R$ defined as $$\begin{aligned}
f_\omega(j)=0, \mbox{ for $j\geq \frac{12N_{k}}{11N_{k-1}}+1=:m_{k-1}\in \mathbb N$,} \\
f_\omega(j)=\sum_{j\leq m\leq m_{k-1}-1}\prod_{m<i\leq m_{k-1}-1}\rho^{-1}(i) \mbox{ for }j<m_{k-1}.\end{aligned}$$ For simplicity we drop the freezing variable $\omega$ when there is not risk of confusion. We set $$w_{k-1}:=\frac{N_k}{11N_{k-1}}\mbox{($\in \mathbb N$ by construction cf. (\ref{requpolno})),}$$ and then we assert that claim (recall $x_1\in\mathcal{\widehat H}_{0}$) $$\label{quenchestleftpoly}
P_{x_1,\omega}\left[X_{T_{B_{2,k}}}\cdot \ell\leq \frac{-N_{k}}{11} , (\mathbb Z^{d})^{\mathbb N}\setminus \mathfrak I_k\right]\leq \frac{f(0)}{f(-w_{k-1})}$$ holds.
Indeed the proof of claim (\[quenchestleftpoly\]) is similar as the one after (\[quenf\]), therefore we omit it. We continue with estimating the right most term in (\[quenchestleftpoly\]) as follows $$\label{inefpoly}
\frac{f(0)}{f(-w_{k-1})}\leq\frac{\sum_{0\leq m\leq m_{k-1}-1}\prod_{\substack{m<j\leq m_{k-1}-1}}\rho^{-1}(j)}{\prod_{\substack{-w_{k-1}<j\leq m_{k-1}-1}}\rho^{-1}(j)}.$$ On the other hand, recall that by Remark \[remarkgoodpoly\], for a $N_k-$ *Good* box $B_{2,k}$ the maximum number of bad boxes *inside* along direction $\ell$, is three. We also observe that when $z\in \mathcal{\widetilde{H}}_i \cap B_{2,k}$, for some $i\in \mathbb Z$, there exists a point $y:=y(z)\in \widetilde B_{1,k-1}(t)$ for some $t\in \mathfrak L$ such that $|z-y|_1=1$, together with $y\cdot \ell\geq iN_{k-1}$. As a result of uniform ellipticity (\[simplex\]) we have $$\rho(i)\leq \sup_{z\in \bigcup_{\substack{t\in \mathfrak L_{k-1}}}\widetilde B_{1,k-1}(t)\cap\mathcal{ \widetilde H }_i} \frac{\frac{1}{\kappa}P_{z,\omega}\left[X_{T_{B_{2,k}}}\notin \partial^+ B_{2,k}\right]}{1-\frac{1}{\kappa}P\left[X_{T_{B_{2,k}}}\notin \partial^+ B_{2,k}\right]}.$$ Using the last estimate, uniform ellipticity again for those eventually bad boxes and applying the induction hypothesis into (\[inefpoly\]), we get that $$\begin{aligned}
\label{estimateleftpoly}
&P_{x,\omega}\left[X_{T_{B_{2,k}}}\cdot \ell\leq \frac{-N_{k}}{11} , (\mathbb Z^{d})^{\mathbb N}\setminus \mathfrak I_k\right]\leq\\
\nonumber
&e^{3c_1\ln(1/\kappa)N_{k-1}}\sum_{i=w_{k-1}}^{m_{k-1}-1} \left(e^{-c_{k-1} N_{k-1}+\ln(1/\kappa)-\ln\left(1-(1/\kappa)e^{-c_{k-1}N_{k-1}}\right)}\right)^{i-3}.\end{aligned}$$ It was assumed that $x_1\in \mathcal{\widehat H}_{0}$ to obtain (\[estimateleftpoly\]), however for $z\in \widetilde B_{1,k}$ one has $P_{z,\omega}-$a.s. on the event $$\{X_{T_{B_{2,k}}}\cdot \ell\leq \frac{-N_{k}}{11} , (\mathbb Z^{d})^{\mathbb N}\setminus \mathfrak I_k\},$$ the random time $\widehat H:=\inf\{n\geq 0:\, X_n\in \mathcal{\widehat H}_{0}\}$ is finite and $\widehat H\leq T_{B_{2,k}}$. Thus applying the Markov property $$\begin{aligned}
&P_{z,\omega}\left[X_{T_{B_{2,k}}}\cdot \ell\leq \frac{-N_{k}}{11} , (\mathbb Z^{d})^{\mathbb N}\setminus \mathfrak I_k\right]\\
\leq&\sum_{x_1\in \mathcal{\widehat H}_{0}}P_{z,\omega}[X_{\widehat H}=x_1]P_{x_1,\omega}[X_{T_{B_{2,k}}}\cdot \ell\leq \frac{-N_{k}}{11} , (\mathbb Z^{d})^{\mathbb N}\setminus \mathfrak I_k]\\
\leq&\sup_{x_1\in \mathcal{\widehat H}_{0}}P_{x_1,\omega}[X_{T_{B_{2,k}}}\cdot \ell\leq \frac{-N_{k}}{11} , (\mathbb Z^{d})^{\mathbb N}\setminus \mathfrak I_k].\end{aligned}$$ which proves that estimate (\[estimateleftpoly\]) holds for any $x\in \widetilde B_{1,k}$.
In view of (\[decomquenpoly\]), estimate (\[orthpolyest\]) and inequality (\[estimateleftpoly\]), for arbitrary $x\in \widetilde B_{1,k}$ under $B_{2,k}$ is $N_k-$*Good* we have that $P_{x,\omega}[X_{T_{B_{2,k}}}\notin \partial^+B_{2,k}] $ is less than: $$\begin{aligned}
\label{lastpolyine}
&2e^{3c_1\ln(1/(2\kappa))N_{k-1}} \\
\nonumber
&\times \sum_{i=w_{k-1}}^{m_{k-1}-1} \left(e^{-c_{k-1} N_{k-1}+\ln(1/(2\kappa))-\ln(1-(1/(2\kappa))e^{-c_{k-1}N_{k-1}})}\right)^{i-3} \\
\nonumber
\leq& \frac{2e^{3c_1\ln(1/2\kappa)N_{k-1}}}{1-e^{-c_{k-1} N_{k-1}+\ln(1/(2\kappa)-\ln\left(1-(1/(2\kappa))e^{-c_{k-1}N_{k-1}}\right)}}\\
\nonumber
&\times e^{-c_{k-1} N_{k-1}(w_{k-1}-3)+\ln(1/(2\kappa))(w_{k-1}-3)-\ln(1-(1/(2\kappa))e^{-c_{k-1}N_{k-1}})(w_{k-1}-3)}.\\
\nonumber\end{aligned}$$ Using $w_{k-1}-3\leq w_{k-1}/2$ for large but fixed $N_0$, we get in turn that the last expression in inequality (\[lastpolyine\]) is less than $$\begin{aligned}
\label{finalquenchespoly}
\leq& e^{6c_1\ln(1/(2\kappa))N_{k-1}-c_{k-1} N_{k-1}(\frac{1}{2}w_{k-1})+2\ln(1/(2\kappa))(\frac{1}{2}w_{k-1})}\\
\nonumber
\leq & e^{6c_1\ln(1/(2\kappa))N_{k-1}-\frac{c_{k-1}N_{k-1} N_k}{33N_{k-1}}}\\
\nonumber
\leq & e^{-\frac{c_{k-1}N_{k-1}N_k}{44N_{k-1}}} \mbox{ (cf. definition (\ref{ckquepoly}))}\\
\nonumber
\leq & e^{-c_{k}N_k},\end{aligned}$$ provided that $N_0\geq \zeta$, for some constant $\zeta$. Thus (\[finalquenchespoly\]) completes the induction and the proof as well.
The previous Lemma \[lemmakpolybad\] along with Proposition \[propquenpol\] makes us able to prove a stronger decay for the probability of an unlikely exit. In order to develop the formal setting to state the next proposition we introduce the condition representing a stronger decay as the one represented by Definition \[defpoly\].
\[deftgamaofn\] Recall that we have fixed $\ell\in \mathbb S^{d-1}$ and a rotation $R$ of $\mathbb R^d$ such that $R(e_1)=\ell$. We define the box $\widehat{B}_N$ for $N\geq 3\sqrt d$ by $$\widehat B_N:= R\left(\left(-N,N+1 \right)\times\left(-\frac{21N^3}{10}, \frac{21N^3}{10}\right)^{d-1}\right)\cap \mathbb Z^d$$ together with its *frontal boundary part* $\partial^+\widehat B_N$ defined by $$\partial^+ \widehat B_N:= \partial \widehat B \cap\{z\in \mathbb Z^d:\, z\cdot \ell \geq 10 N\}.$$ Setting $\Gamma: 3\sqrt d\rightharpoonup (0,1)$, so that $N \rightharpoonup 1/(\ln(N))^{\frac{1}{2}}$.
We let $N\geq 3\sqrt d$ , $\ell\in \mathbb S^{d-1}$ and $R$ be a rotation as above. We say that condition $(T^{\Gamma(N)})|\ell$ holds, if there exist $N_0\geq 3\sqrt d$ and a constant $c>0$ such that for all $N\geq N_0$ one has $$P_0\left[X_{T_{\widehat B_N}}\notin \partial ^+ \widehat B_N\right]\leq e^{-cN^{\frac{c}{(\ln N)^{1/2}}}}.$$
Throughout the remaining of this section we let $\ell\in\mathbb S^{d-1}$, $R$ a rotation of $R\mathbb R^d$ such that $R(e_1)=\ell$ , $\Gamma$ be the function of Definition \[deftgamaofn\] and $M>9d$.
\[proptgampoly\] Assume condition $(P_M)|\ell$, then condition $(T^{\Gamma(N)})|\ell$ holds.
Since assumption $(P_M)|\ell$ holds, we can and do consider the construction of scales and boxes of (\[scalespoly\]). We also use Lemma \[lemmakpolybad\] together with Proposition \[propquenpol\] along this proof. Thus we consider for large $N$ the first natural $k$ such that $N_k\leq N<N_{k+1}$. We let $\mathcal G$ be the environmental event defined by $$G:=\{\omega\in \Omega:\, B_{2,k}(z)\in \mathfrak B_k, z\in \mathfrak L_k \mbox{ is } N_{k}-\mbox{Good, if }B_{2,k}\cap\widehat B_N\neq\varnothing \}.$$ We next decompose the underlying probability through the event $\mathcal G$ as follows (under notation as in Definition \[deftgamaofn\]): $$\begin{aligned}
P_0\left[X_{T_{\widehat B_N}}\notin \partial ^+ \widehat B_N\right]\leq&\mathbb E\left[\mathds 1_{G}P_{0, \omega}[X_{T_{\widehat B_N}}\notin \partial ^+ \widehat B_N]\right]\\
&+\mathbb E\left[\mathds 1_{\Omega\setminus G}P_{0,\omega}[X_{T_{\widehat B_N}}\notin \partial ^+ \widehat B_N]\right].\end{aligned}$$ On the other hand, using Lemma \[lemmakpolybad\] and a rough counting argument, we have $$\begin{aligned}
\label{ngoodtgamma}
\mathbb E\left[\mathds 1_{\Omega\setminus G}P_{0,\omega}[X_{T_{\widehat B_N}}\notin \right.&\left. \partial ^+ \widehat B_N]\right]\\
\nonumber
&\leq \mathbb E\left[\Omega\setminus G\right] \\
\nonumber
&\stackrel{(\ref{lemmakpolybad})}\leq \left(\frac{13N_{k+1}}{11N_k}+2\right)\times\left(\frac{12N_{k+1}^3}{10N_k^3}+2\right)^{d-1}e^{-2^k}\\
\nonumber
&\leq c(d)e^{-2^{k}+(3d-2)(k+1)\ln v}.\end{aligned}$$ Following the proof method developed to prove Proposition \[propquenpol\] and under notation therein, we define a sequence of stopping times $(H^i)_{i\geq 0}$ as well as two sequences of random positions $(Y_i)_{i\geq0}$ and $(Z_i)_{i\geq0}$ given by: $$\begin{aligned}
&H^0:=0, \, Y_0=X_0, \, Z_0\in\{z\in \mathfrak L_{k}:\, Y_0\in \widetilde B_{1,k}(z)\}\\
&H^1=T_{\widehat B_N}\wedge T_{B_{2,k}(z)}, \mbox{ and for $i>1$}\\
&Y_{i-1}=X_{H^{i-1}},\, Z_{i-1}\in\{z\in\mathfrak L_k:\, Y_{i-1}\in \widetilde B_{1,k}(z)\}, \, H^i=H_{i-1}+H^1\circ \theta_{H^{i-1}}.\end{aligned}$$ We stress again that the construction of sequence $(Z_i)_{i\geq0}$ makes use of finite arbitrary choices in virtue of (\[scalespoly\]) and (\[unionscalespoly\]). We also define the stopping time $S$ by $$S:=\inf\left \{n\geq 0: X_n \in \partial B_{2,k-1}(Z_0)\setminus \partial^+ B_{2,k-1}(Z_0)\right\}$$ Thus we can see by a similar argument as the one established in the proof of Proposition \[propquenpol\], $P_{0, \omega}-$a.s. $$\bigcap_{\substack{i=0}}^{[N/N_k]+1} \theta_{H^i}^{-1}\{S>H^1\}\subset \left\{X_{T_{B_{2,k+1}(0)}}\in \partial^+B_{2,k+1}(0)\right\}.$$ It is now straightforward to verify that the following chain of inequalities: $$\begin{aligned}
\label{goodtgamma}
\mathbb E\left[\mathds 1_{G}P_{0,\omega}[X_{T_{\widehat B_N}}\notin \partial ^+ \widehat B_N]\right]&\leq 1-\left(1-e^{-\frac{c_0 N_k}{v^{k}}}\right)^{[N/N_k]+1}\\
&\leq (N/N_k+1)e^{-\frac{c_0 N_k}{v^{k}}}\end{aligned}$$ holds, where $c_0$ is as in (\[ckquepoly\]). We combine the last estimates (\[goodtgamma\])-(\[ngoodtgamma\]), to get the existence of a constant $\widetilde c>0 $ such that for large $N$, $$\begin{aligned}
P_0\left[X_{T_{\widehat B_N}}\notin \partial ^+ \widehat B_N\right]&\leq 2 c(d)e^{-2^{k}+(3d-2)(k+1)\ln v}\leq \exp\left\{-\widetilde c N^{\frac{\widetilde c}{(\ln N)^{1/2}}}\right\} .\end{aligned}$$ This finishes the proof.
The next technical tool needed to prove the effective criterion under our polynomial condition, is an estimate for the $\mathbb P-$probability of a quenched atypical event. Notice that the result will be weaker than similar estimates, for instance the one to be established in Proposition \[propatyquenest\].
We keep notation as in Definition \[deftgamaofn\]. We observe that condition $(T^{\Gamma(N)})|\ell$ is weaker than the natural extension of the definition in the spirit of Lemma \[lemmaTgamma\] for $\Gamma(N)-$stretched exponential decay. The crucial point here is that in order to $(T^\gamma)|\ell$ holds, for $\gamma\in (0,1]$ is required (as Lemma \[lemmaTgamma\] precisely says) to have *linear growth* for the underlying box along the orthogonal space to direction $\ell$. Indeed, it is straightforward using Lemma \[lemmaTgamma\] to see that $(T^{\Gamma(N)})|\ell$ is implied by the corresponding extension of $(T^\gamma)$ in direction $\ell$ as in Lemma \[lemmaTgamma\], but there is not direct proof to derive the converse implication. We will follow a proof argument close to [@BDR14] in order to overcome this issue.
\[propoweakaqe\] Let $(T^{\Gamma(N)})|\ell$ be fulfilled. We let $\mathcal U_N\subset \mathbb Z^d$ be a box defined by $$\mathcal U_N:=R\left(N-2, N+2)\times (-N^3+1, N^3-1)^{d-1}\right)\cap\mathbb Z^d$$ and we also define their frontal boundary part $\partial^+\mathcal U_N$ by $$\partial^+\mathcal U_N:= \partial \mathcal U_N\cap\{z\in \mathbb Z^d:\, z\cdot\ell\geq N+2\}.$$ We set the function $\epsilon:[3\sqrt{d},\infty)\rightharpoonup [0,1]$ defined by $\epsilon(N)=\frac{1}{(\ln N)^{\frac{3}{4}}}$.
Then for each function $\beta:[3\sqrt d, \infty)\rightharpoonup[0,1]$ such that $\lim_{M\rightarrow\infty}\epsilon(M)/\beta(M)<1$, we have for large $N$, $$\label{weakaqe}
\mathbb P\left[ P_{0,\omega}\left[X_{T_{\mathcal U_N}}\in \partial^+\mathcal U_N \right]\leq \frac{e^{-2c_1\ln(1/\kappa)N^{\beta}}}{2}\right]\leq \frac{4e\times 6^{d-1}}{\left(\frac{N^{\beta(N)}}{4\times 6^{d-1}}\right)!}.$$
Observe that $\mathcal U_N$ is within the class of boxes entering in the infimum of (\[effectivecriterion\]).
Under notation of the statement of this proposition we assume $(T^{\gamma(N})|\ell$ holds and let $\beta$ be any function with the prescribed assumptions. We need to establish a one-step renormalization scheme which turns out the prescribed decay by (\[weakaqe\]). To this end, we consider $N_0:=N^{\epsilon(N)}$ for large $N\geq 3\sqrt d$ and define the set $\mathfrak L_{N_0}$ via: $$\mathfrak L_{N_0}:=N_0\mathbb Z\times N_0^{3}\mathbb Z^{d-1}.$$ We introduce for $z\in \mathfrak L_{N_0}$ boxes $widetilde B_1(z)$, $B_2(z)$ as well as its *frontal boundary part* $\partial^+ B_2(z)$ defined by: $$\begin{aligned}
\widetilde B_1(z) &:=R\left(z+[0,N_0]\times [0, N_0^3]^{d-1} \right)\cap \mathbb Z^d, \\
B_2(z)&:= R\left(z+(-N_0,N_0+1)\times (-\frac{21N_0}{10}, \frac{31N_0^3}{10}) \right)\cap \mathbb Z^d \mbox{ and }\\
\partial^+ B_2(z)&\partial B_2(z)\cap\{y\in \mathbb Z^d:\, (y-z)\cdot\ell\geq N_0+1\}\end{aligned}$$ It is then convenient to define *Good boxes*: for $z\in \mathfrak L_{N_0}$ box $B_2(z)$ is $N_0-$*Good* if $$\inf_{\substack{y\in\widetilde B_1(z)}}P_{y,\omega}\left[X_{T_{B_2(z)}}\in \partial^+ B_2(z)\right]>1-\frac{1}{N^{\epsilon^{-1}}}.$$ We say that box $B_2(z)$ is $N_0-$*Bad* otherwise.
We are going to bound from above the environmental probability of a box $B_2(z)$ to be $N_0-$*Bad*. We observe that for arbitrary $z\in \mathfrak L_{N_0}$ using Proposition \[proptgampoly\], we have that $$\begin{aligned}
\nonumber
\mathbb P\left[B_{2}(z)\mbox{ is }N_0-\mbox{\textit{Bad}}\right]&=\mathbb P\left[\sup_{\substack{y\in \widetilde
B_1(z)}}P_{y,\omega}[X_{T_{B_2(z)}}\notin \partial^+B_2(z)]\geq \frac{1}{N^{\epsilon^{-1}}}\right]\\
\nonumber
&\leq N^{\epsilon^{-1}} |\widetilde B_1(z)| e^{-\widetilde c N_{0}^{\frac{\widetilde c}{(\ln N_0)^{1/2}}}}\\
\nonumber
&\leq e^{-\widetilde c N^{\frac{\widetilde c(N)}{(\ln N)^{7/8}}}+(\epsilon^{-1}(N)+3d\epsilon(N))\ln(N)}\\
\label{estimabadwaqe}
&\leq e^{-\widehat c N^{\frac{\widehat c}{(\ln N)^{7/8}}}}\end{aligned}$$ holds, for certain positive dimensional constants $\widetilde c$ and $\widehat c$.
We need to introduce the already well-known strategy to exit from box $\mathcal U_N$ starting at $0\in \mathbb R^d$. Further definitions will be required so as to describe the environmental event where the strategy fails. We define the set of boxes involved in the forthcoming strategy $\mathcal G_{N}$ by $$\begin{aligned}
\mathcal G_N:=&\{B_{2}(z),\, z\in \mathfrak L_{N_0},\, z\cdot e_1=j N_0,\, j\in[0,[N/N_0]+1],\,\mbox{and for $i\in[2,d]$ }\\
&z\cdot e_i=jN_0^3, j\in [-[N/N_0]-1, ([N/N_0]+1)]\}.\end{aligned}$$ Define as well, the environmental event $\mathfrak G_{\beta(N)}$ via $$\mathfrak G_{\beta(N)}:=\left\{\sum_{B_{2}(z)\in \mathcal G_N, z\in \mathfrak L_{N_0}}\mathds{1}_{\{ B_2(z) \mbox{ is } N_0-\mbox{\textit{Bad} } \}}\leq N^{\beta}\right\}.$$ Analogously as in cases of Proposition \[propquenpol\] and Proposition \[proptgampoly\] we introduce the strategy of successive exits from boxes of type $B_2(z),$ $z\in\mathfrak L_{N_0}$ by their frontal boundary part, afterwards we shall see that on event $\mathfrak G_{\beta(N)}$, the strategy shall imply the complementary event involved in the probability (\[weakaqe\]), and then prove $\mathfrak G_{\beta(N)}$ has high $\mathbb P-$ probability. Despite overcharged notation, with the purpose of completeness we introduce again sequences of random positions $(Y_i)_{i\geq0}$, $(Z_i)_{i\geq0}$ together with stopping times $(H^i)_{i\geq 0}$ defined as follows: $$\begin{aligned}
\nonumber
H^0&=0,\,Y_0=X_0, \, Z_0\in \{z\in \mathfrak L_{N_0}:\, Y_0\in\widetilde B_1(z) \}\\
\nonumber
H^1&=T_{\mathcal U_N}\wedge T_{B_2(Z_0)}, \,\mbox{ and for $i>1$}\\
\nonumber
Y_{i-1}&=X_{H^{i-1}},\, Z_{i-1}\in \{z\in \mathfrak L_{N_0}:\, Y_{i-1}\in \widetilde B_1(z)\}, \, H^i=H^{i-1}+H^1\circ \theta_{H^{i-1}}.\end{aligned}$$
We define as well the stopping time $S$ via: $$S=\inf\{n\geq 0: \, X_n\in \partial B_2(Z_0)\setminus \partial^+B_2(z)\}.$$
It is routine using uniform ellipticity (\[simplex\]) to see that for large $N$ on the event $\mathfrak G_{\beta(N)}$ one has $\mathbb P-$a.s. $$\begin{aligned}
\nonumber
P_{0,\omega}\left[X_{T_{\mathcal U_N}}\in \partial^+\mathcal U_N \right]&>P_{0,\omega}\left[\bigcap_{0\leq i\leq \frac{N}{N_0}}\theta_{H^i}^{-1}\left\{S>H^1\right\}\right]\\
\nonumber
&\geq (2\kappa)^{L^{\epsilon(N)+\beta(N)}}\left(1-\frac{1}{N^{\epsilon^{-1}(N)}}\right)^{[N/N_0]+1}\\
\label{goodestpoly}
&\geq \frac{1}{2}e^{2\ln(2\kappa)\beta (N)}.\end{aligned}$$ It is our purpose now, to decompose the set $\mathcal G_N$ into smaller sets which would have boxes elements with appropriate disjointness so as to apply mixing conditions. Keeping that purpose in mind, we introduce for integer $i\in [0,3]$ subsets $\mathcal G_{N,i, e_1}$ of $\mathcal G_N$ defined by $$\mathcal G_{N,i, e_1}:=\{B_{2}(z)\in \mathcal G_N :\, z\in \mathfrak L_{N_0},\, \frac{z\cdot e_1}{N_0} =i \mod 4\}.$$ Next, we further decompose each one of the four $\mathcal G_{N,i, e_1}$, $i\in [0,3]$ along the *orthogonal space to* $\ell$. For this end, fix the integer $i\in [0,3]$, and define for $j_2,\ldots, j_{d}\in [0,5]$ subsets $\mathcal G_{N, i,j_2,\ldots,j_{d}}$ of $\mathcal G_{N, i,e_1}$ given by: $$\mathcal G_{N, i,j_2,\ldots,j_{d}}:=\{B_{2}(z)\in \mathcal G_{N, i,e_1}: \, z\in \mathfrak L_{N_0},\, \frac{z\cdot e_k}{N_0^3}=j_k\mod 6\, \forall k\in[2,d] \}.$$ Thus we have constructed $4\times 6^{d-1}$ subsets $\mathcal G_{N, i,j_1,\ldots,j_{d-1}}$ of $\mathcal G_N$, where $i\in [0,3]$ and $j_2,\ldots, j_d\in [0,5]$ such that each pair of boxes $B_2(z_1), B_2(z_2)\in \mathcal G_{N, i,j_2,\ldots,j_{d}}$ with $z_1,z_2\in\mathfrak L_{N_0} $ are at least $2N_0=2N^{\epsilon(N)}$ separated in terms of $\ell^1-$distance. We plainly have as well, $$\bigcup_{\substack{i,j_2,\ldots,j_d\in[0,3]\times[0,5]^{d-1}}} \mathcal G_{N, i,j_2,\ldots,j_{d}}=\mathcal G_N.$$ Under these terms, we can rewrite event $\mathfrak G_{\beta(N)}$ in the form: $$\begin{aligned}
&\mathfrak G_{\beta(N)}=\\
&\left\{\sum_{\substack{i,j_2,\ldots,j_d\in[0,3]\times[0,5]^{d-1}}}\sum_{\substack{B_2(z)\in \mathcal G_{N, i,j_2,\ldots,j_{d}},\\
z\in \mathfrak L_{N_0} }}\mathds 1_{\{B_2(z) \mbox{ is }N_0-\mbox{\textit{Bad}} \}}>N^{\beta(N)} \right\}\end{aligned}$$ and then it is clear that $$\begin{aligned}
\label{Gcom}
&\Omega \setminus \mathfrak G_{\beta(N)}\\
\nonumber
&\subset \bigcup_{\substack{i,j_2,\ldots,j_d\in[0,3]\times[0,5]^{d-1}}} \left\{\sum_{\substack{B_2(z)\in \mathcal G_{N, i,j_2,\ldots,j_{d}}\\z\in \mathfrak L_{N_0} }}\mathds 1_{\{B_2(z) \mbox{ is }N_0-\mbox{\textit{Bad}} \}}>\frac{N^{\beta(N)}}{4\times 6^{d-1}}\right\}.\end{aligned}$$ On the other hand notice that using remark above, for given integers $i, j_2,\ldots, j_d$ and any integer number $$k\in \left[0, \frac{([N/N_0]+1)\times (2([N/N_0]+1))^{d-1}}{4\times 6^{d-1}}\right]=:[0, N']$$ (the last number represents an upper bound for the amount of elements in $\mathcal G_{N, i,j_2,\ldots,j_{d}}$ ), using either **(SM)**$_{C,g}$ or **(SMG)**$_{C,g}$, and a counting argument we have for large $N$, $$\begin{aligned}
&\mathbb P\left[B_2(z_1), B_2(z_2),\ldots,B_2(z_k)\in \mathcal G_{N, i,j_2,\ldots,j_{d}} \mbox{ are (and not more) } N_0-\mbox{\textit{Bad}} \right]\stackrel{(\ref{estimabadwaqe})}\leq\\
&\exp\left(\sum_{j=0}^{k-1}(2Cj)\times 3^{d}(2N_0)(6N_0^3)^{d-1}e^{-2gN_0}\right)\exp\left(- \widehat c \, k \, e^{\widehat c(\ln N)^{1/8}}\right).\end{aligned}$$ Hence for large $N$ there exists a dimensional dependent constant $\breve{c}>0$, such that $$\begin{aligned}
&\mathbb P\left[\sum_{\substack{B_2(z)\in \mathcal G_{N, i,j_2,\ldots,j_{d}}\\z\in \mathfrak L_{N_0}}}\mathds 1_{\{B_2(z)\mbox{ is }N_0-\mbox{\textit{Bad}}\}}=k\right] \leq \binom{N'}{k}\exp\left(-\breve c \, k\, e^{ \breve c (\ln N)^{1/8}}\right).\end{aligned}$$ holds. As a result of combining this last estimate with (\[Gcom\]), together with some basic estimates one sees $$\begin{aligned}
\nonumber
&\mathbb P[\mathfrak G_{\beta(N)}]\\
\nonumber
&\leq \sum_{\substack{i,j_2,\ldots,j_d\in[0,3]\times[0,5]^{d-1}}}\mathbb P\left[\mathds 1_{\{B_{2}(z)\mathcal G_{N, i, i,j_2,\ldots,j_d}, B_2(z)\mbox{ is }N_0-\mbox{\textit{Bad}}\}}>\frac{N^{\beta(N)}}{4\times 6^{d-1}}\right]\\
\nonumber
&\leq 4\times 6^{d-1}\sum_{j\geq \frac{N^{\beta(N)}}{4\times 6^{d-1}}}\binom{N'}{j}\exp\left(-\breve c \, k\, e^{ \breve c (\ln N)^{1/8}}\right)\\
\label{lastestquenchpoly}
&\leq 4\times 6^{d-1}\sum_{j\geq \frac{N^{\beta(N)}}{4\times 6^{d-1}}}\binom{N'}{j}\frac{1}{(N')^{j}}\leq 4\times 6^{d-1}\frac{e}{\left(\frac{N^{\beta(N)}}{4\times 6^{d-1}}\right)!}.\end{aligned}$$ The claim involved in the statement of this proposition is proven in virtue of (\[goodestpoly\]) and (\[lastestquenchpoly\]).
The last step required in order to get the effective criterion of Section \[sectionce\] starting from a polynomial condition as in Definition \[defpoly\], will be to integrate the random variable $\rho$ given in (\[ravaB\]). The formal statement comes in the next:
Assume $(P_M)|\ell$ be fulfilled for $M>9d$ and $\ell\in \mathbb S^{d-1}$ (for some $N_0$ large enough but fixed, see Definition \[defpoly\]). Then $(EC)|\ell$ holds (cf. Definition \[defpolasympandec\]).
Assume condition $(P_M)|\ell$ be fulfilled for some $N_0$ large enough. Consider large $N$ and recall the notation introduced for a box specification $\mathcal B:=\mathcal B(R,N-2, N+2, 4N^3 )$ in Section \[sectionce\], where the rotation $R$ satisfies $R(e_1)=\ell$ and we set $B$ for the box attached to $\mathcal B$. In virtue of Proposition \[proptgampoly\] there exists a constant $\widetilde c>0$ such that for large $N$ (see (\[ravaB\]) for notation) $$\label{est1poly}
\mathbb E\left[p_{\mathbb B}(\omega)\right]\leq e^{-\widetilde cN^{\frac{\widetilde c}{(\ln N)^{1/2}}}}.$$ In order to apply inequality (\[est1poly\]) and Proposition \[propoweakaqe\], we define parameters: $$\begin{aligned}
\nonumber
\beta_1:=\frac{\widetilde c}{2(\ln N)^{1/2}}\\
\nonumber
\alpha:=\frac{\widetilde c}{3(\ln N)^{1/2}}\\
\label{parampolyfinal}
a:=\frac{1}{N^{\alpha}}\end{aligned}$$ Following the proof argument of Section 2.2 in [@BDR14] we split the expectation $\mathbb E\left[\rho_{\mathcal B}^{a}\right]$ into $L$ terms, where $$L:=\left[\frac{2(1-\beta_1)}{\beta_1}\right]+1.$$ Indeed, the decomposition split the underlying denominator in the expectation into terms given by $$\begin{aligned}
\mathcal E_0&:=\mathbb E\left[\rho_{\mathcal B}^a, P_{0,\omega}\left[X_{T_B}\in \partial^+B\right]>\frac{e^{-2c_1\ln(1/(2\kappa))N^{\beta_1}}}{2}\right], \\\end{aligned}$$ for $j\in\{1,\ldots, L-1\}$ we define: $$\begin{aligned}
\mathcal E_j&:=\mathbb E\left[\rho_{\mathcal B}^a,\frac{e^{-2c_1\ln(1/(2\kappa))N^{\beta_{j+1}}}}{2} <P_{0,\omega}\left[X_{T_B}\in \partial^+B\right]\leq\frac{e^{-2c_1\ln(1/(2\kappa))N^{\beta_j}}}{2}\right] \\\end{aligned}$$ and the last term is $$\begin{aligned}
\mathcal E_L&:=\mathbb E\left[\rho_{\mathcal B}^a, P_{0,\omega}\left[X_{T_{B}}\in \partial^+B\right]\leq \frac{e^{-2c_1\ln(1/(2\kappa))N^{\beta_L}}}{2}\right],\end{aligned}$$ where in turn the numbers $\beta_j$ for $j\in \{1,\ldots,L\}$ are prescribed by $$\beta_j:=\beta_1+(j-1)\frac{\beta_1}{2}.$$ Notice that using uniform ellipticity (\[simplex\]) one sees that $\mathcal E_L=0$ since $\beta_L\geq 1$ and $\mathbb P-$a.s. one has $$P_{0,\omega}\left[X_{T_B}\in \partial^+B\right]>e^{-2c_1\ln(1/(2\kappa))N}.$$ On the other hand an application of Jensen’s inequality first and then (\[est1poly\]) we have $$\begin{aligned}
\nonumber
\mathcal E_0 &\leq 2^{a}e^{2c_1\ln(1/(2\kappa))N^{\beta_1-\alpha}}e^{-\widetilde c N^{-\alpha+\widetilde c(\ln N)^{-1/2}}}\\
\label{estterjply}
&\leq 2^{a}e^{2c_1\ln(1/(2\kappa))N^{\frac{\widetilde c}{6(\ln N)^{1/2}}}}e^{-\widetilde cN^{\frac{2\widetilde c}{3(\ln N)^{1/2}}}}.\end{aligned}$$ Furthermore, we use the atypical quenched estimate provided by Proposition \[propoweakaqe\] to get that for $j\in\{1,\ldots,L-1\}$ there exist positive constants $\widehat c, \breve c$ such that $$\begin{aligned}
\nonumber
\mathcal E_j\leq&2\exp(N^{\beta_{j+1}-\alpha})\mathbb P\left[P_{0,\omega}[X_{T_B}\in\partial^+B]\leq \frac{e^{2c_1\ln(1/(2\kappa))N^{\beta_j}}}{2}\right]\\
\nonumber
\leq&\widehat c\, 2\exp\left(2c_1\, \ln(1/(2\kappa))N^{\beta_{j+1}-\alpha}\right)\exp\left(-\breve c N^{\beta_j}\ln(N^{\beta_j})\right)\\
\label{esttermj2poly}
\leq&\widehat c\, 2\exp\left(2c_1\, \ln(1/(2\kappa))N^{\beta_j+(\beta_1/2)-2\beta_1/3}\right)\exp\left(-\breve c N^{\beta_j}\ln(N^{\beta_j})\right).\end{aligned}$$ Combining inequalities (\[estterjply\]) and (\[esttermj2poly\]) we see that $\mathbb E[\rho_{\mathcal B}^a]$ is less than any polynomial function in $N$, therefore the proof is complete.
It is now straightforward to prove Theorem \[mainth1\].
It is a geometric fact to prove the implication: $(P)_M|\ell \,\Rightarrow \, (P_M)|\ell$, we provide a proof here. We let boxes $\widetilde B_{1,0}$ and $B_{2,0}$ be defined as in (\[boxespolynomial\]) for large $N_0$. Consider Definition \[defpolasympandec\] for the polynomial *asymptotic* condition; then setting $b=1/13$ we can find a neighborhood $\mathcal U_\ell\subset \mathbb S^{d-1}$ such that $\ell\in \mathcal U_\ell$ and $$\lim_{N\rightarrow\infty}N^MP_0\left[\widetilde T_{-bN}^{\ell'}<T_N^{\ell}\right]=0$$ for each $\ell'\in \mathcal U_\ell$. Thus in particular taking $\epsilon=\frac{1}{2(d-1)}$ there exist a large $L_0$ together with a neighborhood $\mathcal U_\ell\subset \mathbb S^{d-1}$ of $\ell$ such that $$\label{polydecayasymp}
P_0\left[\widetilde T_{-bL_0}^{\ell'}<T_{L_0}^{\ell}\right]<\frac{1}{2(d-1)L_0^M},$$ for all $\ell'\in \mathcal U_\ell$. We take $$N_0=\frac{11L_0}{12.5},$$ and we keep in mind the box involved in Definition \[defpoly\]. Since $\mathcal U_\ell$ is open containing $\ell$ there exists a strictly positive number $\alpha$ such that the following requirements are satisfied:
- the vectors $\ell_i^{\pm}:=\ell\pm \alpha R(e_i)/|\ell\pm \alpha R(e_i)|_2 \,\in \mathcal U_\ell$ for each $i\in [2,d]$. We stress that there are $2(d-1)$ vectors $\ell_i^{\pm}$, i.e. the symbol $\pm$ is an abbreviation.
- $\alpha$ is small enough such that $$\cos(\arctan(\alpha))\leq \min\left\{\frac{12N_0}{11L_0},\frac{11L_0}{13N_0}\right\}$$ holds.
Consider now the set $\mathcal D\subset\mathbb Z^d$ defined by: $$\mathcal D:=\{x\in\mathbb R^d:\, x\cdot \ell\in (-\frac{N_0}{11}, \frac{12N_0}{11}), \,\forall i\in[2,d] x\cdot \ell_i^{\pm}>-\frac{L_0}{13}\}.$$ Observe that for any $x\in \widetilde B_{1,0}(0)$ we have $$x+\mathcal D\cap \mathbb Z^d \subset B_{2,0}(0).$$ For a set $A\subset \mathbb R^d$ we define its boundary $\partial A$ in $\mathbb Z^d$ by $$\partial A:=\{z\in \mathbb Z^d:\, z\notin A, \, \ \exists z'\in A\cap\mathbb Z^d \, \ |z-z'|_1=1\}.$$ We then introduce the frontal part of the boundary for set $\mathcal D$, denoted by $\partial^+\mathcal D$ and defined as follows: $$\partial^+\mathcal D:=\{z\in\mathbb Z^d:\, z\in \partial \mathcal D,\,z\cdot \forall i\in[2,d] x\cdot \ell_i^{\pm}>-\frac{L_0}{13} \},$$ and therefore we note that for any $x\in\widetilde B_{1,0}(0)$ we have $P_x-$a.s. one has $$T_{ X_0+\partial\mathcal D\setminus\partial^+\mathcal D}\leq T_{B_{2,0}(0)}.$$ By stationarity of the probability space $(\Omega, \mathfrak F_{\Omega}, \mathbb P)$ and using (\[polydecayasymp\]), we get for arbitrary $x\in\widetilde B_{1,0}(0)$ $$\begin{aligned}
P_x[X_{T_{B_{2,0}(0)}}\notin \partial^+B_{2,0}(0)]&\leq P_{x}[X_{H_{x+\partial \mathcal D}}\notin x+\partial^+\mathcal D]\\
&\leq P_0[X_{H_{\partial \mathcal D}}\notin \partial^+\mathcal D]\\
&\leq \sum_{i=2}^{d}P_0[\widetilde T_{-bL_0}^{\ell_i^+}<T_{L_0}^{\ell_i^+}]+P_0[\widetilde T_{-bL_0}^{\ell_i^-}<T_{L_0}^{\ell_i^-}]\\
&\stackrel{(\ref{polydecayasymp})}<\frac{1}{N_0^{M}},\end{aligned}$$ which ends the proof.
On the other hand, the previous theorem in conjunction with Theorem \[theocrite-tprime\] proves that $(P_M)|\ell$ implies $(T')|\ell$. Finally, it is clear using any of the equivalent definitions contained in statement of Lemma \[lemmaTgamma\], the implication $(T^\gamma)|\ell\, \Rightarrow\,(P_M)|\ell$ holds, for any $\gamma>0$.
Estimates for tails of regeneration times: Proof of Theorem \[mainth2\] {#secproofmainth}
=======================================================================
We shall prove the main Theorem \[mainth2\] in this section. Roughly speaking, the objective will be to start assuming condition $(P^M)|\ell$ and as a result of Theorem \[mainth1\] we have that condition $(T')|\ell$ holds. Next, assuming condition $(T')|\ell$ we provide several auxiliary results leading up to bound from above the annealed tails of the regeneration times in such a form that the random variable $\tau_1$ will possess arbitrary finite order moments. We end the proof by applying the annealed central limit theorem of F. Comets and O. Zeitouni [@CZ02].
Henceforward, we will assume $(T')|\ell$ with $h\ell=l\in \mathbb Z^d$ for some $h>0$ (cf. (\[rul\])), either of the mixing conditions: **(SM)**$_{C,g}$ or **(SMG)**$_{C,g}$ and $g>2\ln(1/\kappa)$ (cf. Definitions \[sma\]-\[smg\]), where $C, g>0$ and $2\kappa>0$ is the uniform elliptic constant prescribed by (\[simplex\]). Furthermore, we introduce for $L\in |l|_1\mathbb N$ the approximate regeneration time $\tau_1^{(L)}$ as in Section \[section2\], as well as the approximate asymptotic direction $\hat v_L:=E_0[X_{\tau_1^{(L)}}|D'=\infty]/|E_0[X_{\tau_1^{(L)}}|D'=\infty]|_2$. It is convenient to define the orthogonal projector $\Pi_{\hat v_L}: \,\mathbb Z^d\,\rightarrow\,\mathbb Z^d$ which projects vectors onto the orthogonal subspace to direction $\hat v_L$ by: $$\label{projvl}
\Pi_{\hat v_L}(z)=z-(z\cdot v_L)v_L.$$ We introduce for $M>0$ the time $\mathcal L_M$ of last visit to the half space $\mathcal H_M=\{z\in \mathbb Z^d: z\cdot l\leq M\}$ by: $$\label{lastvistM}
\sup\{n\geq0: X_n\cdot l\leq M \}.$$ Furthermore, we let $$t=\frac{\frac{2\ln\left(\frac{1}{\kappa}\right)}{g}+1}{2}.
\label{defintransfluct}$$ The next proposition will be fundamental to apply renormalization schemes, indeed the given estimate will be used in order to get seed control.
\[protransfluctuation\] Let $\eta>0$ and $$\rho\in \left(\frac{gt+2\ln\left(\frac{1}{\kappa}\right)}{2gt},1\right),$$ noting that the interval above is nonempty under assumption **(R)**$_{C,g}$. For large $M>0$, we let $L\in |l|_1\mathbb N$ be the least integer of the set $$A=\left\{\overline L\in |l|_1\mathbb N: \, e^{-gt\overline L}\leq M^{2\rho-2-\epsilon}\right\},$$ where $$\label{epsilonchoise}
\epsilon:=\frac{(2\ln(1/\kappa)(2-2\rho))}{(gt-2\ln(1/\kappa))}>0.$$ Then there exists $c_{12}>0$ such that for large $M$ $$\begin{gathered}
\nonumber
P_0\left[\sup_{0\leq n\leq \mathcal L_M}\,|\Pi_{\hat{v}_L}(X_n)|_2>\eta \,M^\rho\right]\leq\\
\label{claimproptranv}
\exp\left(-c_{12}M^{(2\rho-1)-\epsilon}\right)\end{gathered}$$ holds.
Observe that under such a $\rho$ above, we have that $(2\rho-1)-\epsilon>0$ holds. Moreover, it is a matter of taking limits when $g\rightarrow\infty$ to see that we recover Sznitman’s result for i.i.d. random environments. The proof is based on a combination of Theorems A.1-A.2 of [@Sz02] and Proposition 4.5 of [@Gue17].
We let $\rho$ and $\eta>0$ as in the statement of the proposition. Notice that since $2\rho-1<\rho$ together with $(T')|\ell$ holds, we can and do choose $\gamma\in (0,1)$, with $2\rho-1<\gamma \rho$ such that $(T^\gamma)|\ell$ holds. We take $M>0$ large enough so that the least integer $L$ of the set (recall definition in (\[epsilonchoise\])) $$A=\left\{\overline L\in |l|_1 \mathbb N:\, e^{-gt\overline L}\leq M^{2\rho-2-\epsilon} \right\}$$ fulfills the following requirements: $$\begin{gathered}
L \geq c_{10}\,\, \hspace{1.5ex}\mbox{(cf. Proposition \ref{expmpr} for notation) and}\\
\label{Limp}
L \geq 16 \frac{|l|_1}{|l|_2 3}\left(2(c_{11}/c_{10}^2)+1\right).\end{gathered}$$
Observe now that in order to prove claim (\[claimproptranv\]) we can and do replace $$\sup_{0\leq n\leq \mathcal L_M}|\Pi(X_n)|_2$$ by $\sup_{0\leq n\leq\mathcal L_M}X_n\cdot w$, where $w\in \mathbb S^{d-1}$ and $w\cdot \hat v_L=0$. We consider the regeneration time $\tau_1^{(L)}$ constructed as in Section \[section2\] along direction $l$ and define for $n\geq0$ (recall convention $\tau_0^{(L)}:=0$): $$K_n=\sup\{k\geq 0:\,\tau_k^{(L)}\leq n \}$$ Setting $c_l=|l|_1/|l|_2$, we see by the very definition of $\tau_1^{(L)}$ that $\overline P_0$-a.s. on the event $0\leq n \leq \mathcal L_M$ one has $K_n\leq (c_lM)/L$. As a result of that upper bound for $K_n$, writing $X_n\cdot w=(X_n-X_{\tau_{K_n}})\cdot w + X_{\tau_{K_n}}\cdot w$, recalling notation (\[supt1\]), we set $Y=X*$ to get $$\begin{gathered}
\nonumber
P_0\left[\sup_{0 \leq n \leq \mathcal L_M}\,X_n\cdot w>\eta M^{\rho}\right]\leq \sum_{0\leq k\leq\frac{c_l M}{L} }\overline P_0\left[X*\circ \theta_{\tau_k}>\frac{\eta M^\rho}{3}\right]+\\
\label{decomptranflucv}
\overline P_0\left[X_{\tau_1}\cdot w> \frac{\eta M^\rho}{3}\right]+\sum_{2\leq k\leq \frac{c_l M}{L}}\overline P_0\left[(X_{\tau_k}-X_{\tau_1})\cdot w>\frac{\eta M^\rho}{3}\right].\end{gathered}$$ The first two terms on the right hand side of inequality (\[decomptranflucv\]) can be bounded from above using condition $(T^\gamma)|\ell$. More precisely, we first introduce constants $b$ and $\phi$ via: $$\begin{gathered}
\nonumber
b=\frac{2\ln(1/\kappa)}{gt-2\ln(1/\kappa)} \,\, \mbox{ and }\\
\label{constantphiandb}
\phi=e^{-\frac{bgt|l|_1}{2(b+1)}},\end{gathered}$$ then applying for integer $k\in [0, c_lM/L]$ Corollary \[corren\] in combination with Chevyshev’s inequality to get $$\begin{gathered}
\overline P_0\left[X*\circ \theta_{\tau_k}>\frac{\eta M^\rho}{3}\right]=\overline P_0\left[c_{10}(\phi M^{-\epsilon}X*\circ \theta_{\tau_k})^\gamma>\frac{c_{10}(\eta \phi)^\gamma M^{\gamma(\rho-\epsilon)}}{3}\right]\\
\leq \exp \left(-\frac{c_{10}(\eta \phi)^\gamma M^{\gamma (\rho-\epsilon)}}{3^\gamma}\right)\overline E_0\left[\exp\left(c_{10}(\phi M^{-\epsilon}X*\circ \theta_{\tau_k})^\gamma\right)\right]\\
\leq e^{\exp\left(-gtL\right)}\exp \left(-\frac{c_{10}(\eta\phi)^\gamma M^{\gamma (\rho-\epsilon)}}{3^\gamma}\right)\overline E_0\left[\exp\left(c_{10}(\phi M^{-\epsilon}X*)^\gamma\right)|D'=\infty\right].\end{gathered}$$ Plainly, the same upper bound holds for the second term on the right most hand of inequality (\[decomptranflucv\]). Thus $$\begin{gathered}
\nonumber
\sum_{0\leq k\leq\frac{c_l M}{L} }\overline P_0\left[X*\circ \theta_{\tau_k}>\frac{\eta M^\rho}{3}\right]+\overline P_0\left[X_{\tau_1}\cdot w> \frac{\eta M^\rho}{3}\right]\\
\label{boundfirsttwo}
\leq \left(\frac{4Mc_l}{L}\right)\exp \left(-\frac{c_{10}(\eta\phi)^\gamma M^{\gamma (\rho-\epsilon)}}{3^\gamma}\right)\overline E_0\left[\exp\left(c_{10}(\phi M^{-\epsilon}X*)^\gamma\right)|D'=\infty\right].\end{gathered}$$ Let us now proceed by examining the order of the last term on right most expression inside of the sum in (\[decomptranflucv\]). For this end, we let $2\leq k\leq (c_lM)/L$ and notice that defining $\mathcal H_{k,M}:=\{\exists j\in [2, k]:\, |(X_{\tau_{j}}-X_{\tau_{j-1}})\cdot w| \geq \delta M^{\frac{2\rho-1}{\gamma}}\}$ for $\delta>0$ to be chosen later on, we have $$\begin{gathered}
\nonumber
\overline P_0\left[(X_{\tau_k}-X_{\tau_1})\cdot w>\frac{\eta M^\rho}{3}\right]\leq \\
\label{decotran2}
\overline P_0\left[ \mathcal H_{k,M}\right]+\overline P_0\left[(X_{\tau_k}-X_{\tau_1})\cdot w>\frac{\eta M^\rho}{3}, (\mathcal H_{k,M})^c \right].\end{gathered}$$ We first bound from above the left term on the right most expression entering at inequality (\[decotran2\]). Observe that a close argument to the given to get estimate in (\[boundfirsttwo\]) turns out: $$\begin{gathered}
\nonumber
\overline P_0\left[ \mathcal H_{k,M}\right]\leq \sum_{2\leq j\leq k}\overline P_0\left[|(X_{\tau_j}-X_{\tau_{j-1}})\cdot w|\geq\delta M^{\frac{M^{2\rho-1}}{\gamma}}\right]\\
\nonumber
\leq \sum_{2\leq j\leq k}2\exp\left(-(\delta\phi)^\gamma c_{10}M^{2\rho-1-\gamma\epsilon}\right)\overline E_0\left[ \exp\left(c_{10}(\phi M^{-\epsilon}X*)^\gamma\right)|D'=\infty\right]\\
\label{boundintwhofirst}
\leq 2k\exp\left(-c_{10}(\delta\phi)^\gamma M^{2\rho-1-\gamma\epsilon}\right)\overline E_0\left[ \exp\left(c_{10}(\phi M^{-\epsilon}X*)^\gamma\right)|D'=\infty\right].\end{gathered}$$ We now turn to bound the remaining and hardest to handle term, however we must keep in mind the estimate in (\[boundintwhofirst\]) and continue fixing $k\in[2, (c_lM)/L]$. For that purpose, we fix some $\zeta\in (1/2,1]$ and apply exponential Chevyshev’s inequality to the second term on right most hand of inequality (\[decotran2\]) to get $$\begin{gathered}
\label{decomhard}
\overline P_0\left[(X_{\tau_k}-X_{\tau_1})\cdot w>\frac{\eta M^\rho}{3}, (\mathcal H_{k,M})^c \right]\leq
e^{-\zeta\frac{\eta M^\rho}{M^{(1-\rho)+\epsilon}}}\times\\
\nonumber
\left(\overline E_0\left[e^{\zeta\frac{\sum_{2\leq j\leq k}(X_{\tau_j}-X_{\tau_{j-1}})\cdot w}{M^{(1-\rho)+\epsilon}}}, \forall i\in[2,k],\,|(X_{\tau_i}-X_{\tau_{i-1}})\cdot w|<\delta M^{\frac{2\rho-1}{\gamma}} \right]\right).\end{gathered}$$ On the other hand, either of our mixing hypotheses: **(SM)**$_{C,g}$ or **(SMG)**$_{C,g}$ along with Lemma \[propare\] and successive conditioning lead us to: $$\begin{gathered}
\left(\overline E_0\left[e^{\zeta\frac{\sum_{2\leq j\leq k}(X_{\tau_j}-X_{\tau_{j-1}})\cdot w}{M^{(1-\rho)+\epsilon}}}, \forall i\in[2,k],\,|(X_{\tau_i}-X_{\tau_{i-1}})\cdot w|<\delta M^{\frac{2\rho-1}{\gamma}} \right]\right)\\
\leq \left(\exp\left(e^{-gtL}\right)\overline E_0\left[\exp\left(\frac{\zeta X_{\tau_1}\cdot w}{M^{1-\rho+\epsilon}}\right), |X_{\tau_1}\cdot w| <\delta M^{\frac{2\rho-1}{\gamma}}|D'=\infty\right]\right)^{\frac{c_l M}{L}},\end{gathered}$$ for $2 \leq k\leq (c_lM)/L$. Using that for $|u|\leq 1$ one can find some $\nu\in (1/2,1]$ so that $$|e^{u}-1-u|<\nu u^2,$$ and choosing $\alpha>\epsilon/2$, with $\alpha<(2\rho-1)/\gamma\wedge (1-\rho+\epsilon)$, $$\begin{gathered}
\nonumber
\overline E_0\left[\exp\left(\frac{\zeta X_{\tau_1}\cdot w}{M^{1-\rho+\epsilon}}\right), |X_{\tau_1}\cdot w| <\delta M^{\frac{2\rho-1}{\gamma}}|D'=\infty\right]\\
\nonumber
\leq \overline E_0\left[1 +\frac{\zeta X_{\tau_1}\cdot w}{M^{(1-\rho)+\epsilon}}+\nu \frac{\zeta^2 (X_{\tau_1}\cdot w)^2}{M^{(2-2\rho)+2\epsilon}},\,|X_{\tau_1}\cdot w|\leq M^{\alpha}|D'=\infty\right]\\
\nonumber
+\overline E_0\left[\exp\left(\frac{\zeta X_{\tau_1}\cdot w}{M^{1-\rho +\epsilon}}\right), M^\alpha<|X_{\tau_1}\cdot w|<\delta M^{\frac{2\rho-1}{\gamma}}|D'=\infty\right]\\
\nonumber
\leq 1+ \nu \frac{\zeta^2}{M^{2-2\rho+\epsilon}}\overline E_0\left[\left(M^{-\frac{\epsilon}{2}}X_{\tau_1}\cdot w\right)^2|D'=\infty\right]-\frac{\zeta}{M^{1-\rho+(\epsilon/2)}}\times\\
\nonumber
\overline E_0\left[M^{-\frac{\epsilon}{2}}X_{\tau_1}\cdot w,\, |X_{\tau_1}\cdot w|> M^{\alpha}|D'=\infty\right]\\
\label{hh}
+\overline E_0\left[\exp\left(\frac{\zeta X_{\tau_1}\cdot w}{M^{1-\rho +\epsilon}}\right), M^\alpha<|X_{\tau_1}\cdot w|<\delta M^{\frac{2\rho-1}{\gamma}}|D'=\infty\right].\end{gathered}$$ Applying Cauchy-Schwarz inequality together with assumption $(T')|\ell$, we have
$$\begin{gathered}
\nonumber
\overline E_0\left[M^{-\frac{\epsilon}{2}}X_{\tau_1}\cdot w, |X_{\tau_1}\cdot w|> M^\alpha|D'=\infty\right]\leq\\
\nonumber
\overline E_0\left[\left(M^{-\frac{\epsilon}{2}}X_{\tau_1}\cdot w\right)^2|D'=\infty\right]^{\frac{1}{2}}\left(\overline P_0\left[|X_{\tau_1}\cdot w|> M^\alpha|D'=\infty\right]\right)\\
\nonumber
\leq E_0\left[\left(M^{-\frac{\epsilon}{2}}X_{\tau_1}\cdot w\right)^2|D'=\infty\right]^{\frac{1}{2}}\\
\label{bound3}
\times \exp(-c_{10}\phi^\gamma M^{\gamma(\alpha-\epsilon/2})\overline E_0\left[\exp\left(c_{10}(\phi M^{-\frac{\epsilon}{2}}X*)^\gamma\right)|D'=\infty\right],\end{gathered}$$
where we have proceeded as in (\[boundfirsttwo\])-(\[boundintwhofirst\])-(\[bound3\]). At this point we are going to leave the main proof subject to prove that the expectations entering at inequalities (\[boundfirsttwo\])-(\[boundintwhofirst\]) are bounded. From the very definition of $L$ one has $$e^{-\frac{bgt|l|_1}{2(b+1)}}M^{-\frac{\epsilon}{2}}=e^{-\frac{bgt|l|_1}{2(b+1)}}M^{-\frac{b(2-2\rho)}{2}}<e^{-\frac{bgtL}{2(b+1)}}=\kappa^L.$$ As a result, by (\[boundfirsttwo\])-(\[boundintwhofirst\])-(\[bound3\]) we can rewrite them as $$\begin{gathered}
\nonumber
\sum_{0\leq k\leq\frac{c_l M}{L} }\overline P_0\left[X*\circ \theta_{\tau_k}>\frac{\eta M^\rho}{3}\right]+\overline P_0\left[X_{\tau_1}\cdot w> \frac{\eta M^\rho}{3}\right]\\
\nonumber
\leq \left(\frac{4Mc_l}{L}\right)\exp \left(-\frac{c_{10}(\eta\phi)^\gamma M^{\gamma (\rho-\epsilon)}}{3^\gamma}\right)c_{11},\\
\nonumber
\overline P_0\left[ \mathcal H_{k,M}\right]\leq \sum_{2\leq j\leq k}\overline P_0\left[|(X_{\tau_j}-X_{\tau_{j-1}})\cdot w|\geq\delta M^{\frac{M^{2\rho-1}}{\gamma}}\right]\\
\nonumber
\leq 2k\exp\left(-c_{10}(\delta\phi)^\gamma M^{2\rho-1-\gamma\epsilon}\right)c_{11}\,\, \mbox{ and}\\
\nonumber
\overline E_0\left[M^{-\frac{\epsilon}{2}}X_{\tau_1}\cdot w, |X_{\tau_1}\cdot w|> M^\alpha|D'=\infty\right]\leq\\
\label{recastbounds}
\exp(-c_{10}\phi^\gamma M^{\gamma(\alpha-\epsilon/2})(c_{11}/c_{10}\phi).\end{gathered}$$ We now go back to the last expression of the right most hand in inequality (\[hh\]), notice that $$\begin{gathered}
\overline E_0\left[\exp\left(\frac{\zeta X_{\tau_1}\cdot w}{M^{1-\rho +\epsilon}}\right), M^\alpha<|X_{\tau_1}\cdot w|<\delta M^{\frac{2\rho-1}{\gamma}}|D'=\infty\right]\\
\leq \exp\left(-\xi M^{\gamma \alpha}\right)+\frac{\zeta}{M^{1-\rho+(\epsilon/2)}}\times\\
\int_{M^{\alpha-(\epsilon/2)}}^{\delta M^{\frac{2\rho-1}{\gamma}-(\epsilon /2)}}\exp\left(\zeta\frac{u}{M^{1-\rho+(\epsilon/2)}}\right)\overline P_0[M^{-\frac{\epsilon}{2}}X_{\tau_1}\cdot w>u|D'=\infty]du\end{gathered}$$
$$\label{layercake1}
\leq \exp\left(-\xi M^{\gamma \alpha}\right)+\frac{\zeta'}{M^{1-\rho+(\epsilon/2)}}\int_{M^{\alpha-(\epsilon/2)}}^{\delta M^{\frac{2\rho-1}{\gamma}-(\epsilon /2)}}\exp\left(\zeta\frac{u}{M^{1-\rho+(\epsilon/2)}}-\xi u^\gamma\right)du,$$
where $\xi=c_{10}\phi^\gamma$ and $\zeta'=\zeta \overline E_0\left[e^{c_{10}(\phi M^{-\frac{\epsilon}{2}}X_{\tau_1}\cdot w)^\gamma}\right]$. Observe now that $(2\rho-1)/\gamma+(\rho-1)-\epsilon<\gamma \rho$, since $2\rho-1<\gamma \rho$. Using this we get $$\frac{\zeta'}{M^{1-\rho+(\epsilon/2)}}\int_{M^{\alpha-(\epsilon/2)}}^{\infty}\exp\left(-\frac{\xi u^\gamma}{2}\right)du\leq \exp\left(-\frac{\xi M^{\gamma(\alpha-(\epsilon/2))}}{3}\right)$$ Therefore, in virtue of this last inequality and going back to (\[hh\]), for large $M$ we have found $$\begin{gathered}
\overline E_0\left[\exp\left(\frac{\zeta X_{\tau_1}\cdot w}{M^{1-\rho+\epsilon}}\right), |X_{\tau_1}\cdot w| <\delta M^{\frac{2\rho-1}{\gamma}}|D'=\infty\right]\\
\leq \exp(2\nu \zeta^2 (c_{11}/c_{10}^{2})M^{2\rho-2-\epsilon}).\end{gathered}$$ Hence, under our choice of $L\in |l|_1\mathbb N$ given by the property (\[Limp\]) one has that $$-\zeta \mu +\left(2\nu \zeta^2 c_{11}/(c_{10})^2 +1\right)\frac{c_l}{L}<-(\zeta\mu)/4.$$ As a result of applying this inequality into (\[decomhard\]) and using that in (\[decotran2\]), we have $$\label{ineq1tran}
\sum_{2\leq k\leq c_lM/L}\overline P_0\left[(X_{\tau_k}-X_{\tau_{1}})\cdot w>\frac{\eta M^{\rho}}{3}\right]\leq \exp\left(-\frac{\zeta\eta M^{2\rho-1-\epsilon}}{8}\right).$$ Thus, we have ended the proof provided that we combine (\[recastbounds\])-(\[ineq1tran\]) with (\[decomptranflucv\]).
In order to bound tails for $L\in |l|_1\mathbb N$ of the random variable $\tau_1(L)$ under assumption $(T')|\ell$, we shall follow Section 5 of [@Gue17] (see also Section **III** of [@Sz00] for the original argument in i.i.d. terms). The next lemma is a first connection between condition $(T')|\ell$ and tails of regeneration times. We fix a rotation $R$ of $\mathbb R^d$ with $R(e_1)=\ell$ and as in Section \[secPrel\] we choose $\mathfrak r$ so that (\[Tgammasquare\]) is satisfied. We define for $M>0$ the hypercube (cf. Lemma \[lemmaTgamma\] which uses the underline rotation $R$): $$\label{hypercube}
C_M:=B_{M.\mathfrak rM,\ell}(0)$$ and we then prove:
\[decomtail1\] There exist $c_{16}=c_{16}(d,\kappa,g)>0$ and $L_0>0$ with $L_0\in |l|_1 \mathbb N$ so that for each $L\geq L_0$ with $L\geq\in|l|_1 \mathbb N$ and any function $M:\mathbb R^+ \,\mapsto \, \mathbb R^+$, satisfying $\lim_{u\rightarrow\infty}\,M(u)=\infty$ we have that for large $u$ and $\gamma\in(0,1)$ $$\label{integratingtau1}
\overline P_0\left[\tau_1>u\right]\leq P_0\left[T_{C_M}=T_{M(u)}^\ell>u\right]+e^{-c_{16}\left(\kappa^LM(u)\right)^\gamma}.$$
For large $u$, using Proposition \[expmpr\] we have $$\begin{gathered}
\overline P_0\left[\tau_1>u\right]\leq\\
\overline P_0\left[\tau_1>u,\, X_{\tau_1}\cdot l <|l|_2 M(u)\right]+\overline P_0\left[X_{\tau_1}\cdot l\geq |l|_2M(u)\right]\leq\\
\overline P_0\left[\tau_1>u,\, X_{\tau_1}\cdot l <|l|_2 M(u)\right]+e^{-\frac{c_2|l|_2^\gamma\left(\kappa^L M(u)\right)^\gamma}{2}}.\end{gathered}$$ On the other hand, using that $\tau_1=T_{X_{\tau_1}\cdot l}^l$ we find that: $$\overline P_0\left[\tau_1>u,\, X_{\tau_1}\cdot l <|l|_2 M(u)\right]\leq P_0\left[T_{M(u)}^\ell>u\right].$$ We now decompose the previous last probability to get: $$\begin{gathered}
P_0\left[T_{M(u)}^\ell>u\right]\leq\\
P_0\left[T_{M(u)}^\ell=T_{C_{M(u)}}>u \right]+ P_{0}\left[X_{T_{C_{M(u)}}}\notin \partial^+ C_{M(u)}\right].\end{gathered}$$ Using $(T')|\ell$ we obtain $$\begin{gathered}
P_{0}\left[X_{T_{C_{M(u)}}}\notin \partial^+ C_{M(u)}\right]\leq e^{-\widetilde c (M(u))^\gamma}\end{gathered}$$ where $\widetilde c>0$ is certain constant. Thus the requirement (\[integratingtau1\]) follows.
We continue with a version of the so-called atypical quenched estimate. Its construction strongly depends on Proposition \[protransfluctuation\]. Furthermore, the next result shall be the cornerstone to get the final estimate in combination with Lemma \[decomtail1\]. We first introduce the set $$\label{slabUl}
U_{M}=\left\{x\in \mathbb Z^d:\, |x\cdot \ell|<M\right\}$$ for $M>0$. Then we have (under the notation introduced in (\[defintransfluct\])):
\[propatyquenest\] For $\beta \in (0,1)$ and $c>0$ $$\label{atyquenest}
\limsup_{\substack{M\rightarrow\infty}}M^{-\chi}\ln \mathbb P\left[P_{0,\omega}\left[X_{T_{U_M}}\cdot \ell>0\right]\leq e^{-cM^\beta}\right]<0,$$ where either $\chi\in (0,1)$ or $\chi < d\left(\frac{3gt}{gt-2\ln(1/\kappa)}\beta-\frac{2gt}{gt-2\ln(1/\kappa)}\right)$ if $\beta>\frac{2gt}{3gt-2\ln(1\kappa)}$.
We observe that for any $\beta\in (0,1)$ and $c>0$ using condition $(T')|\ell$ one can take $\chi:=\gamma\in(0,1)$ such that $(T^\gamma)|\ell$ holds. Thus as an application of Chevyshev’s inequality we can find $\widetilde c>0$ such that $$\mathbb P\left[P_{0,\omega}\left[X_{T_{U_M}}\cdot \ell>0\right]\leq e^{-cM^\beta}\right]\leq \frac{e^{-\widetilde c M^\gamma}}{1-e^{-cM^\beta}}$$ holds. As a result independently of $\beta\in (0,1)$ and $c>0$ we have that (\[atyquenest\]) holds with $\chi\in (0,1)$. Hence we restrict ourself to the case $\beta\in (0,1)$ satisfying $$\label{betbig}
\beta >\frac{2gt}{3gt-2\ln(1\kappa)}..$$ Notice that if $\beta$ satisfies (\[betbig\]), then $\beta > \frac{(2d+1)gt-2\ln(1/\kappa)}{3dgt}$ and therefore $$d\left(\frac{3gt}{gt-2\ln(1/\kappa)}\beta-\frac{2gt}{gt-2\ln(1/\kappa)}\right)>1.$$ We shall now follow closely the proof argument of Proposition 5.2 in [@Gue17] with the help of Proposition \[protransfluctuation\]. For large $M$, we shall construct strategies which will happen with high probability on the environment law ensuring that the walks escapes from slab $U_M$ by the boundary side $$\partial^+U_M:=\partial U_M\cap\left\{z:\,z\cdot \ell \geq M\right\}$$ with quenched probability bigger than $e^{-cM^\beta}$. Recall definitions (\[defintransfluct\])-(\[epsilonchoise\]) and let $\gamma$ be a real number in the interval: $$\left(\frac{gt+2\ln(1/\kappa)}{2gt},1\right).$$ We pick $M_0>3\sqrt d$ so that the natural $L\in|l|_1\mathbb N$ defined by $$L:=\min\left\{\mathfrak L:\, e^{-gt\mathfrak L}\leq M^{2\gamma-2-\epsilon}\right\}$$ is such that $L\geq L_0$ (where $L_0$ is the maximum between the ones of Corollary \[corren\] and Corollary \[corexp\]) and with the purpose of using Proposition \[protransfluctuation\], we ask $$L\geq\frac{16|l|_1(2c_{11}+c_{10}^2)}{3|l|_2c_{10}^2}$$ as well. For that given $L$ one then choose a rotation $\widehat R$ of $\mathbb R^d$ with $$\widehat R(e_1)=\widehat v_L=\frac{\overline E_0\left[X_{\tau_1}|\, D'=\infty\right]}{|\overline E_0\left[X_{\tau_1}\,|D'=\infty\right]|_2}\hspace{2ex} \mbox{as was introduced before Proposition \ref{protransfluctuation}.}$$ We now introduce for $z\in M_0\,\mathbb Z^d$ the following blocks: $$\begin{gathered}
\widetilde{B}_1(z):=\widehat R\left(z+(0,M_0)^d\right)\cap \mathbb Z^d \hspace{2ex}\mbox{and}\\
\label{blocksaqe}
\widetilde{B}_2(z):=\widehat R\left(z+(-M_0^\gamma, M_0+M_0^\gamma)^d\right)\cap \mathbb Z^d.\end{gathered}$$ We further define the frontal part of the boundary of $\widetilde B_2(z)$ by $$\label{boundaryb2aqe}
\partial^+\widetilde B_2(z):=\partial \widetilde B_2(z)\cap \left\{y:\, y\cdot \widetilde v\geq z\cdot \widetilde v+M_0+M_0^\gamma\right\}$$ The aforementioned strategies involve the definition of *good* and *bad* boxes. We say that site $z\in M_0\,\mathbb Z^d$ is $M_0$-good if $$\label{goodbox}
\sup_{\substack{x\in \widetilde B_1(z)}}P_{x,\omega}\left[X_{T_{B_2(z)}}\geq \partial^+B_2(z)\right]\geq \frac{1}{2}$$ and $M_0$-bad otherwise.
\[lemmagoodbox\] Let $\gamma\in \left(\frac{gt+2\ln(1/\kappa)}{2gt},1\right)$. Then one has that $$\label{estbadbox}
\limsup_{\substack{M_0\rightarrow\infty}}M^{-(2\gamma-1-\epsilon)}\,\sup_{\substack{z\in M_0\, \mathbb Z^d}} \ln \mathbb P\left[z\,\mbox{ is }\,M_0-\mbox{bad}\right]<0.$$
For $z\in M_0\, \mathbb Z^d$, $$\mathbb P\left[z\,\mbox{ is }\,M_0-\mbox{bad}\right]\leq 2|\widetilde B_1(z)|\sup_{\substack{x\in \widetilde B_1(z)}}P_x\left[X_{T_{\widetilde B_2(z)}}\notin \partial^+\widetilde B_2(z)\right].$$ Observe now that for arbitrary $x\in \widetilde B_1(z)$, the block $\widetilde B_{2}(z)$ is included in a ball of radius $3\sqrt dM_0$ centered at $x$. Thus one has that $P_x$-a.s $$T_{\widetilde B_2(z)}\leq T_{x\cdot \ell +3\sqrt d M_0}^\ell,$$ and moreover, notice that on the event $\left\{X_{T_{\widetilde B_2(z)}}\notin \partial^+\widetilde B_2(z)\right\}$ we have $P_x$-a.s. either: $$\left(X_{T_{\widetilde B_2(z)}}-x\right)\leq -\frac{M_0^\gamma}{2}\hspace{2ex}\mbox{or}\hspace{2ex}\left|\Pi_{\widehat v}\left(X_{T_{\widetilde B_2(z)}}-x\right)\right|_2\geq \frac{M_0^\gamma}{2},$$ where the notations as in the beginning of this section. Consequently as in the proof of Lemma 5.3 of [@Gue17] for a suitable constant $c(d)$ we find $$\begin{gathered}
\label{lastinbadboxest}
\mathbb P\left[z\,\mbox{ is }\,M_0-\mbox{bad}\right]\leq \\
\nonumber
c(d)M_0^d \left(P_0\left[\sup_{\substack{0\leq n\leq T_{3\sqrt d M_0}^\ell}}\Pi_{\widehat v}\left(X_n\right)\geq \frac{\widehat v\cdot \ell M_0^\gamma }{4}\right]+P_0\left[\widetilde T_{-\frac{\widehat v\cdot \ell M_0^\gamma }{4}}^\ell<\infty\right]\right).\end{gathered}$$ We construct the random variable $\tau_1:=\tau_1(L_0)$ along direction $\ell$ with $L_0$ as in Proposition \[expmompos\], and using that $$P_0\left[\widetilde T_{-\frac{\widehat v\cdot \ell M_0^\gamma }{4}}^\ell<\infty\right]\leq \overline P_0\left[X_{\tau_1}\cdot \ell \geq \frac{\widehat v\cdot \ell M_0^\gamma }{4}\right],$$ the assertion (\[estbadbox\]) follows since the existence of non-dependent on $L$ constants $k_1>0$ and $k_2>0$ (with same argument as in Remark 4.4 in [@Gue17]) so that $$k_1\leq \widehat v_L\cdot \ell \leq k_2$$ together with applying Proposition \[protransfluctuation\].
It is convenient to introduce some further terminology. Consider $M>0$ and $M_0$ as above, and attach to each site $z\in M_0\, \mathbb Z^d$ the column $$\label{columnz}
Col(z):=\left\{z' \in M_0\,\mathbb Z^d: \exists j \in [0,J] \, \mbox{ with }\, z'=z + j M_0 e_1 \right\},$$ where $J$ is the smallest integer satisfying $JM_0\widehat v_L\cdot \ell\geq 3M $. We choose $M_1>0$ an integer multiple of $M_0$ and we also define the tube attached to site $z\in M_0\,\mathbb Z^d$, $$\label{tubez}
Tube(z):=\left\{z'\in M_0\,\mathbb Z^d: \exists j_1,\ldots, j_d \in \left[0,\frac{M_1}{M_0}\right],\, z'=z+\sum_{i=1}^d j_iM_0e_i\right\}.$$ The crucial point of the strategy is that one way for the walk starting from $0$ to escape from $U_M$ by $\partial^+U_M$ is to get to one of the bottom block in $Tube(0)$ containing the largest amount of good boxes, and then moving along this column up to its top. Thus, defining: $$\label{Topz}
Top(z):=\bigcup_{\substack{z'\in Tube(z)}}\partial^+\widetilde B_2(z'+JM_0e_1),$$ together with the neighbourhood of a tube, $$\label{neighbouz}
V(z):=\left\{x\in \mathbb Z^d:\exists y\in \bigcup_{\substack{z'\in Tube(z)\\ 0\leq j \leq J}}\widetilde B_1(z'+jM_0e_1), |x-y|_1\leq 3dM_1\right\}.$$ One has the following:
\[lemmacompaqe\] For $z\in M_0\,\mathbb Z^d$, we let $n(z,w)$ be the random variable: $$\label{minbadblocks}
\min_{\substack{z'\in Tube(z)}}\left\{\sum_{j=0}^J \mathds{1}_{z'+jM_0e_1\hspace{0.5ex}\mbox{is }M_0-\mbox{bad}}\right\}.$$ There exists $c_{10}>0$ such that for any $z\in M_0\, \mathbb Z^d$ and any $x\in D(z)$ where $$D(z):=\bigcup_{\substack{z'\in Tube(z)\\ 0\leq j\leq J}}\widetilde B_1(z'+jM_0e_1),$$ we have $$\label{eqlemcomaqe}
P_{x,\omega}\left[H_{Top(z)}<T_{V(z)}\right]> (2\kappa)^{c_{10}\left(M_1+JM_0^\gamma+n(z,\omega)M_0\right)}\frac{1}{2^{J+1}}.$$
It is easy to see that replacing $\kappa$ by $2\kappa$ in virtue of (\[simplex\]) the proof of Lemma 3.3 in [@Sz00] provides the claim (\[eqlemcomaqe\]).
Keeping in mind Lemma \[lemmagoodbox\], we choose $\gamma\in ((gt+2\ln(1/\kappa))/2gt,1)$ so that $$\label{xidef}
\xi:=\frac{1-\beta}{1-\gamma}<\beta<1,$$ Let us note that the choice of $\gamma$ is possible under assumption (\[betbig\]), since: $$\beta>\frac{2gt}{3gt-2\ln(1/\kappa)}\,\Leftrightarrow\,\frac{2\beta-1}{\beta}>\frac{gt+2\ln(1/\kappa)}{2gt}.$$ We then choose $$\nu>1-\gamma,$$ along with for large $M>0$: $$M_0=\rho_1 M^\xi,\hspace{2ex}M_1=\left[\rho_2M^{\beta-\xi}\right]M_0,\hspace{2ex}N_0=\left[\rho_3M^{\beta-\xi}\right],$$ where the constants $\rho_1,\,\rho_2,\,\rho_3$ possibly depend on the constant of the model and $c$ in (\[eqlemcomaqe\]). They are chosen so that for large $M$: $$\begin{gathered}
\label{constrain1aqe}
(2\kappa)^{c_{10}JM_0^\gamma}, \, (2\kappa)^{c_{10}M_1}, \, (2\kappa)^{c_{10}N_0M_0}, \,\left(\frac{1}{2}\right)^{J+1}>\exp\left(-\frac{c}{5}M^\beta\right).\\
\label{constrain2aqe}
\frac{N_0}{3}>(J+1)\frac{(e^2-1)}{M_0^\nu}, \, \, \mbox{ and}\\
\label{constrain3aqe}
\mbox{any nearest neighbor path within $V(0)$, between $0$ and $Top(0)$,} \\
\nonumber
\mbox{first exits $U_M$ through $\partial^+ U_M$.}\end{gathered}$$ That choice is possible because in order to satisfy \[constrain1aqe\] and \[constrain2aqe\] it is sufficient to take $\rho_1$ large enough and then $\rho_2=\rho_3=c(10 c_{10}\rho_1\ln\frac{1}{2\kappa})^{-1}$. We also note that (\[constrain3aqe\]) is fulfilled since $\beta<1$ implies $\beta -\xi>1-(1+\nu)\xi$. We now borrow the last arguments in proof of Proposition 5.2 [@Gue17] to conclude that $$\limsup_{M}\, M^{d(\beta-\xi)}\,\ln \mathbb P\left[P_{0,\omega}\left[X_{T_{U_M}}\cdot \frac{l}{|l|_2}\geq M\right]\leq e^{-cL^\beta}\right]<0,$$ which ends the proof of claim (\[atyquenest\]) provided we vary $\gamma$ according to (\[xidef\]).
We finally finish the procedure to bound tails of regenerations times $\tau_1(L)$, for $L\in |l|_1\mathbb N$. Roughly speaking, it is a Markov chain argument which links Proposition \[propatyquenest\] and Lemma \[decomtail1\] with the upper bound. More precisely, recalling definition (\[defintransfluct\]) one has:
\[thboundtails\] Let $$\sigma=\min\left\{\frac{gt-2\ln(1/\kappa)}{3gt-2\ln(1/\kappa)},\, \frac{(d-1)gt+2\ln(1/\kappa)}{(3d+1)gt-2\ln(1/\kappa)}\right\},$$ and notice that $\sigma>0$. There exist positive constants $c_{11}$, $c_{12}$ and $L_0\in |l|_1\,\mathbb N$ such that for each $L\geq L_0$ with $L\in |l|_1\,\mathbb N$ and $\alpha\in(1,1+\sigma)$, $$\label{tailtau1}
\overline P_0\left[\tau_1^{(L)}>u\right]\leq e^{-c_{11}\kappa^L\left(\ln(u)\right)^\alpha}+e^{-c_{12}\left(\ln(u)\right)^\alpha}.$$
Let $\alpha\in (1,1+\sigma)$ and for large $u$ define $$\begin{gathered}
\nonumber
\Delta(u)=\frac{1}{10\sqrt{d}}\, \frac{\ln(u)}{\ln\left(\frac{1}{\kappa}\right)}\,\, \mbox{ and }\,\,M(u)= N(u)\Delta(u), \\
\label{scalestail}
\hspace{15ex}\mbox{where } N(u)=\left[(\ln(u))^{\alpha-1}\right].\end{gathered}$$ For the rest of the proof we shall drop $u$ in $M, \,\Delta$ and $N$. We apply Lemma \[decomtail1\] with the function $M$ above to get $$\overline P_0\left[\tau_1^{(L)}>u\right]\leq e^{-c_{11}\kappa^L\left(\ln(u)\right)^{\gamma \alpha}}+P_0\left[T_{C_M}>u\right].$$ Since $\gamma$ can be done close to $1$ and the upper bound for $\alpha$ is not reached, it suffices to prove $$\label{claimlemmatail}
\limsup_{\substack{M\rightarrow\infty}}\ln(u)^{-\alpha}\ln\left(P_0\left[T_{C_M}>u\right]\right)<0.$$ We decompose as follows $$\begin{gathered}
\nonumber
P_0\left[T_{C_M}>u\right]\leq \mathbb E\left[P_{0,\omega}\left[T_{C_M}>u\right],\, \forall x\in C_M\, P_{x,\omega}\left[T_{C_M}> \frac{u}{(\ln(u))^\alpha}\right]\leq \frac{1}{2}\right]+\\
\label{decomtauitail}
\mathbb E\left[P_{0,\omega}\left[T_{C_M}>u\right],\, \exists x_1\in C_M\, P_{x_1,\omega}\left[T_{C_M}>\frac{u}{(\ln(u))^\alpha}\right]>\frac{1}{2}\right].\end{gathered}$$ The first term on the right most expression in (\[decomtauitail\]), as a result of the strong Markov property is smaller than $$\left(\frac{1}{2}\right)^{[\ln(u)^\alpha]}.$$ We turn to bound the second term on the right most hand in (\[decomtauitail\]). It was shown in the proof of Proposition 5.5 [@Gue17] the assertion: $$\begin{gathered}
\nonumber
\left\{\omega\in \Omega:\exists x_1\in C_M\, P_{x_1,\omega}\left[T_{C_M}>\frac{u}{(\ln(u))^\alpha}\right]>\frac{1}{2} \right\}\subset\\
\label{contained}
\left\{\omega\in \Omega:\, \exists x_2 \in C_M\, P_{x_2,\omega}\left[\widetilde H_{x_2}>T_{C_M}\right]\leq \frac{2|C_M|(\ln(u))^\alpha}{u}\right\},\end{gathered}$$ holds, where as a matter of definition, for $x\in \mathbb Z^d$ $$\widetilde H_{x}:=\inf\{n\geq 1:\, X_n=x \}.$$ Moreover, observe that on the environment event on the right in (\[contained\]), we take $y:=x_2\in C_M$ satisfying $$\label{ysiteatypical}
P_{y,\omega}\left[\widetilde H_{y}>T_{C_M}\right]\leq \frac{2|C_M|(\ln(u))^\alpha}{u}$$ and choose $x\in \mathbb Z^d$ a closest lattice point to $y+\ell K \in\mathbb R^d$, where $$K\leq\left[\frac{\ln{u}}{3\ln(1/(2\kappa))}\right].$$ Thus we can take a nearest neighbours and self avoiding path on $\mathbb Z^d$ starting from $y$ and ending at $x$ of length at most $\left[\frac{\ln{u}}{3\ln(1/(2\kappa))}\right]$. In virtue of uniform ellipticity (cf. (\[simplex\])), we get $$\label{yatypical2}
P_{y,\omega}\left[\widetilde H_{y}>T_{C_M}\right]\geq u^{-\frac{1}{3}}P_{x,\omega}\left[H_y>T_{C_M}\right],$$ and notice that $x\in C_M$ by (\[ysiteatypical\]). Therefore, considering for $i\in \mathbb Z$ *strips* $$\mathcal G_i:=\partial \{z:\, z\cdot \ell< \Delta i\},$$ then for a given $y\in C_M$ satisfying (\[ysiteatypical\]), since $\left[\frac{\ln{u}}{3\ln(1/(2\kappa))}\right]>2\Delta+d$ move $y$ to $x$ along direction $\ell$ and inside of $\{z:(i-2)\Delta\leq z\dot\ell\leq i\Delta\}$, for some $i\in [-N+2, N-1]$ requires less than $\frac{\ln{u}}{3\ln(1/(2\kappa))}$ steps, we can find a further site $x\in C_M\cap \mathcal G_i$, for some integer $i$ such that for large $u$: $$\begin{gathered}
P_{x,\omega}\left[\widetilde T_{(i-1)\Delta}^\ell>C_M\right]\leq P_{x,\omega}\left[H_{y}>T_{C_M}\right]\stackrel{(\ref{ysiteatypical})-(\ref{yatypical2})}\leq \frac{1}{\sqrt u}.\end{gathered}$$ As a result, for large $u$ $$\begin{gathered}
\nonumber
\mathcal R:=\left\{\omega\in \Omega:\,\exists x_1\in C_M\, P_{x_1,\omega}\left[T_{C_M}>\frac{u}{(\ln(u))^\alpha}\right]>\frac{1}{2} \right\}\subset\\
\label{contained2}
\left\{\omega\in \Omega: \exists x, i:\, i\in\in \mathbb [-N+2,N-1], x\in C_M\cap \mathcal G_i\mbox{ with }\right.\\
\nonumber
\left.P_{x,\omega}\left[\widetilde T_{(i-1)\Delta}^\ell>T_{C_M}\right]\leq \frac{1}{\sqrt u}\right\}.\end{gathered}$$ For $i\in \mathbb Z$, we define random variables $$X_i:=\left\{
\begin{array}{ll}
-\ln\left( \inf_{x\in C_M \cap \mathcal G_i }
P_{x,\omega}\left[\widetilde{T}_{(i-1)\Delta}^\ell>T_{(i+1)\Delta}^\ell\right]\right)
&\quad{\rm if}\quad C_M\cap \mathcal G_i\neq\emptyset,\\ 0&\quad{\rm
if}\quad C_M\cap \mathcal G_i=\emptyset.
\end{array}
\right.$$ For $i\in [-N+1, N]$ and $x\in \mathcal G_i$, a standard application of the strong Markov property provides us with $$P_{x,\omega}\left[\widetilde T_{(i-1)\Delta}>T_{C_M}\right]\geq \exp\left(-\sum_{j=i}^N X_i\right).$$ Therefore, $$\label{probrarevent}
\mathbb P\left[\mathcal R\right]\leq \mathbb P\left[\sum_{i=-N+1}^N X_i\geq \frac{\ln (u)}{2}\right]\leq 2N \sup_{\substack{i\in [-N+1, N]}}\mathbb P\left[X_i\geq \frac{\ln(u)}{4N}\right].$$ On the other hand, observe that for arbitrary $i\in \mathbb Z$ and $\vartheta>0$ we have (see \[slabUl\] for notation) $$\label{atypquen}
\mathbb P\left[X_i>\vartheta\right]\leq |C_M|\mathbb P\left[P_{0,\omega}\left[X_{T_{U_\delta}}\cdot\ell \geq \Delta\right]\leq e^{-\vartheta}\right].$$ Hence, we apply (\[atypquen\]) into inequalities (\[probrarevent\]) and then we use Proposition (\[propatyquenest\]) to conclude that for large $u$ there exists a suitable constant $\widetilde c$ such that (cf. (\[defintransfluct\]) for notation) $$\mathbb P\left[\mathcal R\right]\leq \exp\left(-\widetilde c \ln(u)^\chi\right),$$ where $\chi<d\left( \frac{3gt}{gt-2\ln(1/\kappa)}(2-\alpha)-\frac{2gt}{gt-2\ln(1/\kappa)}\right)$ and whenever: $$\label{req1}
2-\alpha>\frac{2gt}{3gt-2\ln(1/\kappa)}\Leftrightarrow \alpha<1+\frac{gt-2\ln(1/\kappa)}{3gt-2\ln(1/\kappa)}.$$ In turn, we require that $$\begin{gathered}
\nonumber
\alpha<d\left( \frac{3gt}{gt-2\ln(1/\kappa)}(2-\alpha)-\frac{2gt}{gt-2\ln(1/\kappa)}\right)\Leftrightarrow\\
\label{req2}
\alpha<1+\frac{(d-1)gt +2\ln(1/\kappa)}{(3d+1)gt-2\ln(1/\kappa)}.\end{gathered}$$ The claim of the theorem follows since requirements (\[req1\])-(\[req2\])
As a result of Theorem \[thboundtails\] and the standard analysis result (cf. [@Ru87], Chapter 8, Theorem 8.16), for $L\geq L_0$ with $L\in |l|_1\mathbb N$ there exists a constant $M=M(L)$ such that $$\label{tdmomenttau1}
\mathbb P\left[\frac{\overline E_0\left[\left(\kappa^L\tau_1(L)\right)^3, \,D'=\infty|\,\mathfrak L_{0,L}\right]}{P_0\left[D'=\infty|\,\mathfrak L_{0,L}\right]}>M\right]=0.$$ The result of Theorem \[mainth2\] follows from equation (\[tdmomenttau1\]) after applying Theorem 2 in [@CZ02].
Appendix {#appendix .unnumbered}
========
We give a proof of Lemma \[lemmaTgamma\], which is very similar to that in [@Gue17], subsection 6.1. However in the last reference was missing a proof for the last equivalence, thus by completeness we provide a proof here for case $\gamma<1$ and the equivalence with the statement of Definition \[deftgammaandtprime\].
The proof of $(i)\Rightarrow(ii)$ can be found in [@Sz02], pages 516-517. Therefore, we turn to prove $(ii)\Rightarrow(iii)$. By $(ii)$, there exist $b, \hat{r}>0$ , so that for large $L$ there are finite subsets $\Delta_L$ with $0\in\Delta _L \subset \{x\in \mathbb Z^d: x\cdot l_0\geq -bL\}\cap \{x\in \mathbb R^d: |x|_2\leq \hat{r}L\}$ and $$\limsup_{\substack{L\rightarrow \infty}}L^{-\gamma}\, \ln P_0\left[X_{T_{\Delta_L}}\notin \partial^+\Delta_L\right]<0.$$ Therefore, one can find a positive constant $\widetilde{c}$ so that for large $L$: $$P_0\left[X_{T_{\Delta_L}}\notin \partial^+\Delta_L\right]<e^{-\widetilde{c}L^\gamma}.$$ Furthermore, by taking the intersection of the set $\Delta_L$ with $\{x\in \mathbb Z^d: \,x\cdot l_0<L\}$, without loss of generality we can and do assume that $\Delta_L\subset \{x\in \mathbb Z^d: \, x\cdot l_0<L \}$. Consider box $B_{L, \hat{r},b, l_0}(0)$ defined by $$B_{L, \hat r, b, l_0}(0):=\widetilde{R}\left((-bL,L)\times(-\hat{r}L, \hat{r}L)^{d-1}\right),$$ where $\widetilde{R}$ is a rotation on $\mathbb R^d$ with $\widetilde {R}(l_0)=e_1$. We also define its frontal boundary part by $$\partial ^+B_{L, \hat r, b, l_0}(0):=\partial B_{L, \hat r, b, l_0}(0)\cap \{z\in \mathbb Z^d:\, z\cdot l_0\geq L\}.$$ We then have that $\Delta_L \subset \widetilde{B}_{L, \hat{r}, b, l_0}(0)$, and consequently for large $L$, $$P_0\left[X_{T_{B_{L, \hat{r}, b, l_0}(0)}}\in \partial^+B_{L, \hat{r}, b, l_0}(0)\right]\geq P_0\left[X_{T_{\Delta_L}}\in \partial^+\Delta_L\right]> 1-e^{-\widetilde{c}L^\gamma}.$$ Notice that if $b\leq1$, we choose $c$ in $(iii)$ as $\hat r$, and we finish the proof. Otherwise, we can proceed as follows: we take $N=bL$ and consider now box $$B_{N, \hat{r}([b]+1)N, l_0}(0)$$ defined according (\[generalboxes\]). We introduce for integer $i\in [1,[b]]$ a sequence $(T_i)_{1 \leq i \leq [b]}$ of $(\mathcal F_n)_{n\geq0}-$stopping times via $$\begin{aligned}
\nonumber
T_1&=T_{\widetilde{B}_{L, \hat{r}, b, l_0}(X_0)}, \,\, \mbox{ and for $i>1$ }\\
\label{stoppinti}
T_i&=T_1\circ\theta_{T_{i-1}}+T_{i-1}.\end{aligned}$$ We also introduce the stopping time $S$ which codifies the unlikely walk exit from box $\widetilde{B}_{L, \hat{r}, b, l_0}(X_0)$ and is defined by: $$S:=\inf_{n\geq 0}\{n\geq0:\, X_n\in\partial \widetilde{B}_{L, \hat{r}, b, l_0}(X_0)\setminus \partial^+\widetilde{B}_{L, \hat{r}, b, l_0}(X_0)\}.$$ As a result of terminology above, we have $$\begin{gathered}
\nonumber
P_0\left[X_{T_{B_{N, \hat{r}([b]+1)N, l_0}(0)}}\in \partial^+B_{N, \hat{r}([b]+1)N, l_0}(0)\right]\geq \\
\label{caixas}
P_0\left[T_1<S, (T_1<S)\circ\theta_{T_1},\ldots, (T_1<S)\circ \theta_{T_{[b]}} \right].\end{gathered}$$ It is convenient at this point to introduce *boundary sets* $F_i$, $i\in [1, [b]]$ as follows $$\begin{aligned}
F_1&=\partial^+ B_{L, \hat{r}, b, l_0}(0)\, \, \mbox{ and for $i>1$}\\
F_i&=\bigcup_{y\in F_{i-1}}\partial^+B_{L, \hat{r}, b, l_0}(y),\end{aligned}$$ where $B_{L, \hat{r}, b, l_0}(y)=y+ B_{L, \hat{r}, b, l_0}(0)$. We also define for $i\in[1,[b]]$, *good environmental events* $G_i$ via $$G_i=\left\{\omega\in \Omega:\, P_{y,\omega}\left[T_1<S\right]\geq 1 -e^{-\frac{\widetilde{c}L^\gamma}{2}},\,\forall y\in F_i \right\}.$$ Observe that the left hand of inequality (\[caixas\]) is greater than $$\begin{gathered}
\mathbb E\left[P_0,\omega\left[T_1<S, (T_1<S)\circ\theta_{T_1},\ldots, (T_1<S)\circ \theta_{T_{[b]}}\right] \mathds{1}_{G_{[b]}}\right].\end{gathered}$$ In turn, writing for simplicity $B_{L, \hat{r}, b, l_0}(y)$ as $B(y)$ for $y\in \mathbb Z^d$, the last expression equals $$\begin{aligned}
&\sum_{y\in F_{[b]}}\mathbb E\left[P_{0,\omega}\left[T_1<S,\ldots ,X_{T_{[b]}}=y\right]P_{y,\omega}\left[X_{T_{B(y)}}\in \partial^+B(y)\right]\mathds{1}_{G_{[b]}}\right]\\
& \geq \left(1-e^{-\frac{\widetilde{c}}{2}L^{\gamma}}\right)\left(P_0\left[T_1<S,\ldots,(T_1<S)\circ \theta_{T_{[b]-1}} \right]-\mathbb P[(G_{[b]})^c]\right),\end{aligned}$$ where we have made use of the Markov property to obtain the last inequality. We iterate this process to see $$\begin{aligned}
\nonumber
&P_0\left[X_{T_{B_{N, \hat{r}([b]+1)N, l_0}(0)}}\in \partial^+B_{N, \hat{r}([b]+1)N, l_0}(0)\right]\\
\label{finalcaixa}
&\geq \left(1-e^{-\frac{\widetilde{c}L^\gamma}{2}}\right)^{[b]+1}-\sum_{i=1}^{[b]}\left(1-e^{-\frac{\widetilde{c}L^\gamma}{2}}\right)^{[b]-i}\mathbb P[G_i^c].\end{aligned}$$ Notice that using Chebysev’s inequality, we have for $i\in [1,[b]]$ and large $L$, $$\label{Gi}
\mathbb P[G_i^c]\leq\sum_{y\in F_i}\mathbb P\left[P_{y,\omega}\left[X_{T_{B(y)}}\notin \partial^+B(y)\right]>e^{-\frac{\widetilde{c}L^\gamma}{2}}\right]\leq e^{-\frac{\widetilde{c}L^\gamma}{4}}.$$ From (\[finalcaixa\]), the fact that $b$ is finite and independent of $L$ and the estimate (\[Gi\]); there exists a constant $\eta>0$, so that for large $N$ $$P_0\left[X_{T_{B_{N, \hat{r}([b]+1)N, l_0}(0)}}\in \partial^+B_{N, \hat{r}([b]+1)N, l_0}(0)\right]\geq 1-e^{-\eta N^\gamma}$$ and this ends the proof of the required implication.
To prove the implication $(iii)\Rightarrow(i)$, we fix a rotation $R$ on $\mathbb R^d$, with $R(e_1)=l_0$ and such that $R$ is the underlying rotation of hypothesis in $(iii)$. For small $\alpha$ we define $2(d-1)$-*directions* $l_{+i}$ and $l_{-i}$, $i\in [2,d]$ $$\begin{gathered}
l_{+i}=\frac{l_0+\alpha R(e_i)}{|l_0+\alpha R(e_i)|_2} \,\, \mbox{ and }
l_{-i}=\frac{l_0-\alpha R(e_i)}{|l_0-\alpha R(e_i)|_2}.\end{gathered}$$ Following a similar argument as the one in [@GR17], Proposition 4.2, pages 13-15; but using $\gamma-$stretched exponential decay instead of polynomial one; we conclude that there exists a small and positive $\alpha$, so that for each $i\in [2,d]$ there are some $r_i>0$, with $$\label{piexp}
\limsup_{\substack{L\rightarrow}\infty}\,L^{-\gamma} \ln P_0\left[X_{T_{B_{L, r_i L, l_{\pm i}}(0)}}\notin \partial^+B_{L, r_i L, l_{\pm i}}(0)\right]<0,$$ Observe that the requirements prescribed in (\[piexp\]) finish the proof of $(iii)$ implies $(i)$ by taking $$\begin{gathered}
a_0=1,\,a_1=a_2=\ldots=a_{2(d-1)}=\frac{1}{2},\\
b_0=b_1=\ldots=b_{2(d-1)}=1 ,\\
l_0,\, l_1=l_{+1},l_2=l_{-1},\ldots, l_{2(d-1)-1}=l_{+(d-1)}, l_{2(d-1)}=l_{-(d-1)},\end{gathered}$$ and then observing that for $i\in[0,2(d-1)]$ we have $$P_0\left[\widetilde{T}_{-b_iL}^l<T_{a_iL}^l\right]\leq P_0\left[X_{T_{B_{L, r_i L,l_{i} }(0)}}\notin \partial^+B_{L, r_i L,l_{i} }(0)\right].$$ Finally, it is straightforward by the same argument to see that $(iii)$ implies Definition \[deftgammaandtprime\], indeed for $\alpha$ as above, the set $$\mathcal A_{\alpha}:=\{\ell\in\mathbb S^{d-1}:\ \ell=\frac{l_0+\alpha' R(e_i)}{|l_0+\alpha' R(e_i)|_2},\ \alpha'\in(-\alpha,\alpha),\ i\in[2,d]\}$$ is open in $\mathbb S^{d-1}$ and contains $l_0$. Furthermore, for any positive number $b$ we have $$\limsup_{\substack{L\rightarrow\infty}}L^{-\gamma}\ln P_0[\widetilde T_{-bL}^\ell<T_{L}^\ell]<0,$$ for each direction $\ell\in \mathcal A_{\alpha}$, an thus we prove the statement in Definition \[deftgammaandtprime\]. Conversely, assume that statement of Definition \[deftgammaandtprime\] holds for direction $l_0$, then there exits a positive number $\varepsilon$ such that $$\limsup_{\substack{L\rightarrow\infty}}L^{-\gamma}\ln P_0[\widetilde T_{-bL}^\ell<T_{L}^\ell]<0,$$ for each direction $\ell\in \left(l_0+\{x\in\mathbb R^d: |x|_2<\varepsilon\}\right)\cap\mathbb S^{d-1}=:\widetilde B(\varepsilon)$. Since the set $\widetilde B(\varepsilon)$ is not plane and in fact has curvature!, any $d-$ different elements $l_1$, $l_2$, $\ldots,$ $l_d$ of $\widetilde B(\varepsilon)$ are linearly independents and thus they span $\mathbb R^d$. Take data: $l_0,$ $l_1,$ $\ldots,$ $l_d$, $a_0=1,$ $a_1=1,$ $\ldots,$ $a_d=0$ and arbitrary positive numbers $b_0,$ $b_1,$ $\ldots,$ $b_d$ and it is clear that they generate an $l_0-$ directed system of slabs of order $\gamma$, which ends the proof.
Acknowledgments {#acknowledgments .unnumbered}
===============
Enrique Guerra thanks his wife: Stephanie Alfaro, his sister: Edith Guerra and his mother: Patricia Aguilar, for taking care of him during a treatment for a disease that he had between May and August of 2019.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
This article serves as a continuation for the discussion in [@EHT], we analyze the invariance properties of the gravity path-integral measure derived from canonical framework, and discuss which path-integral formula may be employed in the concrete computation e.g. constructing a spin-foam model, so that the final model can be interpreted as a physical inner product in the canonical theory.
The present article is divided into two parts, the first part is concerning the gauge invariance of the canonical path-integral measure for gravity from the reduced phase space quantization. We show that the path-integral measure is invariant under all the gauge transformations generated by all the constraints. These gauge transformations are the local symmetries of the gravity action, which is implemented without anomaly at the quantum level by the invariant path-integral measure. However, these gauge transformations coincide with the spacetime diffeomorphisms only when the equations of motion is imposed. But the path-integral measure is not invariant under spacetime diffeomorphisms, i.e. the local symmetry of spacetime diffeomorphisms become anomalous in the reduced phase space path-integral quantization.
In the second part, we present a path-integral formula, which formally solves all the quantum constraint equations of gravity, and further results in a rigging map in the sense of refined algebraic quantization (RAQ). Then we give a formal path-integral expression of the physical inner product in loop quantum gravity (LQG). This path-integral expression is simpler than the one from reduced phase space quantization, since all the gauge fixing conditions are removed except the time-gauge. The resulting path-integral measure is different from the product Lebesgue measure up to a local measure factor containing both the spacetime volume element and the spatial volume element. This formal path-integral expression of the physical inner product can be a starting point for constructing a spin-foam model.
author:
- |
[Muxin Han]{}[^1]\
\
[MPI f. Gravitationsphysik, Albert-Einstein-Institut,]{}\
[Am Mühlenberg 1, 14476 Potsdam, Germany]{}\
\
[Institut f. Theoretische Physik III, Universität Erlangen-Nürnberg]{}\
[Staudtstra[ß]{}e 7, 91058 Erlangen, Germany]{}
title: |
Canonical Path-Integral Measures for Holst and Plebanski Gravity:\
II. Gauge Invariance and Physical Inner Product
---
Introduction
============
The path-integral measure plays an important role in the path-integral quantization of a dynamical system. However at first look, the choice of path-integral measure is ambiguous. Even for a simple system like a free particle on $\mathbb{R}^3$, all the different measures on the space of paths can play the role as a path-integral measure, and lead to different quantum theories. Usually the way to select a right measure for path-integral is to establish the equivalence with the canonical quantization of the same system. For the simple systems, e.g. a free particle on $\mathbb{R}^3$ and the free fields on the Minkowski spacetime, the correct path-integral measures are simply the formal Lebesgue measures. But for general systems more complicated than these, in particular a constrained system, the path-integral measure is unlikely to be always a Lebesgue meausure. It is suggested by the literatures e.g. [@QuantizationGauge] that for a general constrained system, one should use a measure derived from the Liouville measure on the reduced phase space. The reason is that the reduced phase space Liouville measure has direct relation with the canonical quantization in the reduced phase space of the constrained system. The path-integral formulation is of interest in the quantization of general relativity (GR), a theory where space-time covariance plays a key role. The spacetime diffeomorphism of GR results in that GR formulate gravity as a dynamical system with a number of first-class constraints with some certain complications. In quantizing such a complicated dynamical system as GR, one has to be careful about the choice of path-integral measure.
Currently loop quantum gravity (LQG) is a mathematically rigorous quantization of general relativity that preserves background independence — for reviews, see [@book; @rev]. The spin-foam model can be thought of as a path-integral framework for loop quantum gravity, directly motivated from the ideas of path-integral adapted to reparametrization-invariant theories [@sfrevs; @spinfoam2]. So far, however only the kinematical structure of LQG is used in motivating the spin-foam framework, while the canonical formulation of the dynamics in LQG doesn’t contribute to the current spin-foam models. Instead the resulting framework is remarkably close to the path-integral quantization of so-called BF theory, a topological field theory whose quantization is exactly known [@bf]. GR can in fact be formulated as a contrained BF theory, yielding the Plebanski formulation [@plebanski], and for this reason the Plebanski formulation is usually the starting point for deriving the dynamics of spin-foams. However, the consistency between spin-foam model and canonical framework of LQG hasn’t been well understood. And the path-integral measure consistent with canonical theory hasn’t been incorporated in the current spin-foam models yet.
In [@EHT], we derive two path-integral formulations for both the Holst and Plebanski-Holst actions from the reduced phase space quantization. Thus we show that their path-integral measures are consistent with the canonical theory, so they are candidates for the spin-foam construction. However, there are immediately two questions concerning the resulting path-integral formulae:
1. The first question is a conceptual question: The resulting path-integral measure[^2] is not a formal Lebesgue measure, but with a so called, local measure factor of the shape ${\mathcal V}^nV_s^m$, where ${\mathcal V}$ is the spacetime volume element and $V_s$ is the spatial volume element. And the powers $m,n$ are different between the cases of the Holst action and the Plebanski-Holst action. The appearance of spatial volume element breaks the manifest spacetime diffeomorphism invariance of the path-integral measure, which leads to the first question: What is the implication of this diffeomorphism non-invariant path-integral measure? Does it mean that this path-integral quantization of gravity breaks the spacetime diffeomorphism invariance?
2. The second one is a practical question: In the path-integral formulae in [@EHT] given by reduced phase space quantization, the integrands contain several gauge fixing conditions, one for each first-class constraints. This fact reflects that we are considering the quantization of a gauge system. For the conventional computation of the path-integral amplitude, one often need to introduce the ghost fields and write down an effective action. However, if we consider the background independent quantization for GR such as the spin-foam models, the gauge fixing terms are too complicated to be implemented. Then the question is: Can we find some ways to circumvent the gauge fixing conditions, in order to make path-integrals in [@EHT] computable?
Our answer for the first question is: If our path-integral quantization is consistent with the canonical theory of gravity, then the local symmetry of spacetime diffeomorphism is broken in the quantum level. The reason is the following: It turns out in [@wald] that the local symmetry group Diff(M) corresponding to the spacetime diffeomorphisms is not projectable under the projection map from the space of metric to the phase space. There doesn’t exist any canonical generator on the phase space generating spacetime diffeomorphisms for the phase space variables. Thus Diff(M) is not the group of gauge symmetries in the canonical GR. A direct consequence is that the first-class constraints (the spatial diffeomorphism constraint and the Hamiltonian constraint) generate a constraint algebra which is not a Lie algebra, i.e. the structure functions appears. Therefore the gauge transformations doesn’t form a group in the canonical theory of gravity, since they are generated by the first-class constraints. The collection of the gauge transformations is at most an enveloping algebra, whose generic element is a product of infinitesimal gauge transformations. We refer to this enveloping algebra as the “Bergmann-Komar group” BK(M) [@BK]. It coincides with Diff(M) only when the equation of motion is imposed. This Bergmann-Komar group essentially determines the dynamical symmetry of canonical GR, while Diff(M) is only the kinematical symmetry of the theory. Here by kinematical we mean that the symmetry group is insensitive to the form of the Lagrangian. This point can be illustrated by comparing the Einstein-Hilbert action with the high-derivative action [d]{}\^4x R\_(x) R\^(x). Both the actions are spacetime diffeomorphism invariant, but their dynamics are dramatically different, which can be seen from their constraints.
Therefore we can see that a path-integral quantization of GR with the local symmetries of Diff(M) cannot be consistent with the canonical theory with the gauge symmetries of BK(M). They are consistent at most in the semiclassical limit. Then an immediate question is whether the dynamical symmetry of the Bergmann-Komar group BK(M) is implemented in our path-integral quantization in [@EHT]. The answer is positive. We will show in the present paper that the Bergmann-Komar group BK(M), which is also a collection of local symmetry of the gravity action, is implemented anomaly-freely in the path-integral quantization derived from reduced phase space quantization. Therefore all the informations of the dynamical gauge symmetry have been incorporated in this path-integral quantization. Moreover, if we approximate the path-integral around a classical solution of the equation of motion, the semiclassical limit recovers the spacetime diffeomorphism invariance.
Since our path-integral measure is derived from canonical framework of GR, instead of asking it to be diffeomorphism invariant, one should rather ask whether it can solve all the quantum constraint equations. More precisely, given the kinematical Hilbert space ${\mathcal H}_{Kin}$ of GR (which can be realized by the kinematical Hilbert space in LQG), we represent the classical constraints $C_I$ to be operators $\hat{C}_I$ on the kinematical Hilbert space. Then the quantum constraint equations are $\hat{C}_I\Psi=0$. A correct path-integral formula should gives a rigging map for the refined algebraic quantization (RAQ) [@RAQ], mapping the kinematical states in a dense domain of ${\mathcal H}_{Kin}$ to the space of solutions of the quantum constraint equations. In the present paper, we will show that the path-integral formula derived from reduced phase space quantization does give the desired rigging map (will be denoted by $\eta_{\omega}$), which formally solves all the (Abelianized) constraints of GR quantum mechanically. Finally we can write down formally a physical inner product of LQG in terms of this path-integral formula. This result means that this path-integral formula correctly represents the quantum dynamics of GR.
Our resulting path-integral expression of the physical inner product also effectively answer the second question above. It turns out that all the gauge fixing conditions are removed in the path-integral representing the physical inner product, except the so called, time-gauge. For example, we will show that the physical inner product can be formally represented by a path-integral of the Plebanski-Holst action \_ø(f’)|\_ø(f)\_[Phys]{}&=&\
Z\_T(f,f’)&=&\_[II]{}[D]{}ø\_\^[IJ]{}[D]{}B\_\^[IJ]{} \_[xM]{} [V]{}\^[13/2]{} V\^[9]{}\_s \^[20]{}(\_[IJKL]{} B\_\^[IJ]{} B\_\^[KL]{}-[V]{}\_) \^3(T\_c)\
&&\[i\_M B\^[IJ]{}(F-\* F)\] f’(A\_a\^i)\_[t\_i]{}\[1\] where ${\mathcal V}$ and $V_s$ are spacetime volume element and spatial volume element respectively. The time-gauge $T_c$ has to be there since our analysis starts from the Ashtekar-Barbero-Immirzi Hamiltonian formulation [@ABI], which is the starting point of LQG. Since the gauge fixing conditions disappear, this physical inner product represented by the path-integral is ready for the concrete computation, by employing the technique of the spin-form model. The resulting spin-foam model will possess a direct canonical interpretation as a physical inner product.
One can see from Eq.(\[1\]) that, as we expected, the local measure factor ${\mathcal V}^{13/2} V^{9}_s$ appears in the physical inner product, it actually reflects the fact that we are considering a dynamical system with constraints. The quantum effect of this type of local measure factor has been discussed in the literature since 1960s (see for instance [@FV; @GT]) in the formalism of geometrodynamics and its [*background-dependent*]{} quantizations (stationary phase approximation). The outcome from the earlier investigations appears that in the background-dependent quantization, this local measure factor only contributes to a divergent part of the loop-order amplitude, thus their meanings essentially depend on the regularization scheme. Then it turns out that one can always choose certain regularization schemes such that, either the local measure factor never contributes to the transition amplitude (e.g. dimensional regularization), or it is canceled by the divergence from the action [@FV; @GT]. Thus in the end, the effect from the local measure factor may be ignored in the practical computation of background-dependent quantization.
In the formalism of connection-dynamics for GR, however, when we perform [*background-independent*]{} quantization like the spin-foam models, in principle the local measure factor should not be simply ignored, because the regularization arguments in background-dependent quantization are not motivated in the background-independent context anymore. For example, the spin-foam models are defined on a triangulation of the spacetime manifold with a finite number of vertices, where at each vertex the value of the local measure factor is finite, and the action also doesn’t show any divergence. Thus in principle one has to consider the quantum effect implied by this local measure factor in the context of spin-foam model, if one wants to relate the spin-foam model to the canonical theory. An immediate consequence is that the crossing symmetry in spin-foam model may be broken by the spatial volume measure factor, which is consistent with the canonical LQG in terms of Hamiltonian constraint operator and master constraint operator [@QSD].
The present article is organized as follows:
In section \[generalinv\], we first review the reduced phase space quantization for a general constrained system, and perform the derivation for its path-integral formulation. And give a general argument about in which circumstance the path-integral measure is invariant under the infinitesimal gauge transformations generated by the first-class constraints.
In section \[GRinv\], the general consideration is applied to the case of GR. We analyze the path-integral measures for both the ADM formalism and the Holst action. We show that the path-integral measures is invariant under the Bergmann-Komar group, which is also a collection of the local symmetries. However, the spacetime diffeomorphism symmetries become anomalous in this path-integral quantization.
In section \[PIRAQ\], we first briefly review the general programme of refined algebraic quantization. Then for a general time-reparametrization invariant constrained system, we give a general formal expression of the rigging map by using the path-integral from its reduced phase space quantization. After that we apply the general expression to the case of gravity coupling with 4 real massless scalar field. In the end, we obtain the physical inner product of GR formally represented by a path-integral formula of the Holst action or the Plebanski-Holst action.
A general dynamical system with both first-class and second-class constraints {#generalinv}
=============================================================================
Reduce phase space quantizations of the constraint system {#reduce}
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We first consider a general reparametrization-invariant dynamical system, whose Hamiltonian is a linear combination of constraints. We will employ the relational framework to construct Dirac observables of the system [@observable] and then perform the canonical quantization in reduced phase space. The advantages of this approach are that: (1) We will obtain a direct interpretation for the path-integral amplitude with boundary *kinematical state* as the physical inner poduct between *physical states* in the canonical reduced phase space quantization in terms of relational framework; (2) This approach will also help us in considering the rigging map in refined algebraic quantization, which will be proposed in the section \[PIRAQ\].
We first briefly recall the relational framework of constructing Dirac observables for general covariant systems (see [@observable] for details, see also [@tina]). First of all, we consider the phase space for a dynamical system $({\mathcal M},{\omega})$ with a collection of the first-class constraints $C_I$, $I\in{\mathcal I}$ which is an arbitrary index set. These first-class constraints in the most general case close under the Poisson bracket from ${\omega}$: {C\_I,C\_J}=f\_[IJ]{}\^[ K]{}C\_K where $f_{IJ}^{K}$ in general is a structure function. If the dynamical system also has a number of second-class constraints $\Phi_i$, we denote by ${\mathcal M}$ the constraint surface where all the second-class constraints vanish, and ${\omega}$ is the Dirac symplectic structure on ${\mathcal M}$. We choose a collection of the gauge variant functions $T^I$ (clock functions), $I\in{\mathcal I}$, providing a local coordinatization of the gauge orbit $[m]$ of any point $m$ in the phase space, at least in a neighborhood of the constraint surface $\overline{{\mathcal M}}:=\{m\in{\mathcal M}\ |\ C_I(m)=0,\ \forall I\in{\mathcal I}\}$. It follows that the matrix $A_{I}^J:=\{C_I,T^J\}$ must be locally invertible. Consider the equivalent set of constraints C’\_I:=\_J \[A\^[-1]{}\]\^J\_IC\_J the new set of constraints has the properties that $\{C'_I,T^J\}\approx\delta^J_I$ and their Hamiltonian vector fields $X_I:=\chi_{C'_I}$ are weakly commuting. For any real numbers ${\beta}^I$ X\_:=\_I\^I X\_I. Then the gauge transformation can be expressed in terms of formal Taylor series for any function $f$ on the phase space \_(f):=(X\_)f=\_[n=0]{}\^X\_\^nf by the mutually weak commutativity of $X_I$. The idea of relational framework is that by using the gauge variant clock variables $T^I$, we construct a (at least weakly) gauge invariant Dirac observable from each gauge variant phase space function. The resulting Dirac oberservables can separate the points on the reduce phase space. It allows one to coordinatize the reduced phase space by the Dirac observables so constructed. Given a collection of the phase space constants ${\tau}^I$, the weakly gauge invariant Dirac observable associated with the partial observables $f$ and $T^I$ are defined by [@observable] O\_f():=\[\_(f)\]\_[\_(T\^I)=\^I]{}.\[Of\] One can check that ${\alpha}_{\beta}(T^I)\approx T^I+{\beta}^I$. Notice that after equating ${\beta}^I$ with ${\tau}^I-T^I$, the previously phase space independent quantities ${\beta}$ become phase space dependent, therefore it is important in Eq.(\[Of\]) to first compute the action of $X_{\beta}$ with ${\beta}^I$ treated as phase space independent and only then to set it equal to ${\tau}^I-T^I$. Therefore on the constraint surface $O_f({\tau})$ can be expressed as formal series O\_f()\_[\^I=\^I-T\^I]{}=\_[{k\_I}=0]{}\^\_[I]{}\_I(X\_I)\^[k\_I]{}f A significant consequence of the above construction is that the map $O({\tau}): f\to O_f({\tau})$ is a weak Poisson homomorphism (homomorphism only on $\overline{{\mathcal M}}$) from the Poisson algebra of the functions on the constraint surface defined by $C_I$ ,$T^J$ with respect to the Dirac bracket $\{,\}_D$ to the Poisson algebra of the weak Dirac observables $O_f({\tau})$ [@observable]. To write the relation explicitly, &&O\_f()+O\_[f’]{}()=O\_[f+f’]{}(), O\_f()O\_[f’]{}()O\_[ff’]{}(),\
&&{O\_f(),O\_[f’]{}()}{O\_f(),O\_[f’]{}()}\_DO\_[{f,f’}\_D]{}() where the Dirac bracket is explicitly given by {f,f’}\_D={f,f’}-{f,C\_I}\[A\^[-1]{}\]\^I\_J{T\^J,f’}+{f’,C\_I}\[A\^[-1]{}\]\^I\_J{T\^J,f}.
Suppose that we can choose the canonical coordinates, such that the clock functions $T^I$ are some of the canonical coordinates, we write down the complete canonical pairs $(q^a,p_a)$ and $(T^I,P_I)$ where $P_I$ is the conjugate momentum of $T^I$. And at least locally one can write the constraints $C_I$ in an equivalent form: \_I=P\_I+h\_I(q\^a,p\_a,T\^J)\[solve\] thus one can solve the constraints by setting $P_I=-h_I(q^a,p_a,T^J)$. On the constraint surface, the Dirac observable associated with the clocks $T^I$ O\_[T\^I]{}():=\[\_(T\^I)\]\_[\_(T\^I)=\^I]{}=\^I is simply a constant on the phase space. We also define the Dirac observables associated with the canonical pairs $(q^a,p_a)$ Q\^a():=O\_[q\^a]{}() P\_a():=O\_[p\_a]{}() and the “equal-time" Poisson bracket: {P\_a(),Q\^b()}\^b\_a {P\_a(),P\_b()}{Q\^a(),Q\^b()}0 Thus we can see that for each ${\tau}$ the pairs $(Q^a({\tau}),P_a({\tau}))$ form the canonical coordinates of the reduced phase space.
On the other hand, the collection of the constraints $\tilde{C}_I=P_I+h_I(q^a,p_a,T^J)$ forms a strongly Abelean constraint algebra. The reason is the following: $\tilde{C}_I$’s are first class, i.e. $\{\tilde{C}_I, \tilde{C}_J\}=\tilde{f}_{IJ}^{\ \ K}\tilde{C}_K$ for some new structure function $\tilde{f}$. the left hand side is independent of $P_I$, so must be the right hand side. Then the right hand side can be evaluated at any value of $P_I$. So we set $P_I=-h_I$. If we write $C'_I=\sum_JK_{IJ}\tilde{C}_J$ for a regular matrix $K$ and since $\{C'_I,T^J\}\approx\delta^J_I=\{\tilde{C}_I,T^J\}$, we obtain that $K_{IJ}\approx\delta^J_I$ and $C'_I=\tilde{C}_I+O(C^2)$, which means that $C'_I$ and $\tilde{C}_I$ is different by the terms quadratic in the constraints. It follows that the Hamiltonian vector fields $X_I$ and $\tilde{X}_I$ of $C'_I$ and $\tilde{C}_I$ are weakly commuting. We now set $H_I({\tau})=H_I(Q^a({\tau}),P_a({\tau}),{\tau}):=O_{h_I}({\tau})\approx h_I(Q^a({\tau}),P_a({\tau}),{\tau})$. For any function $f$ depending only on $p^a$ and $q_a$, we have the “equal-time"commutator: {H\_I(), O\_f()}&& O\_[{h\_I,f}\_D]{}() = O\_[{h\_I,f}]{}() = O\_[{\_I,f}]{}() = \_[{k}]{}\_[J]{}\_JX\_J\^[k\_J]{}\_If\
&&\_[{k}]{}\_[J]{} \_I\_JX\_J\^[k\_J]{}f \_[{k}]{}\_[J]{} [X]{}\_I\_JX\_J\^[k\_J]{}f\
&=&O\_f()\[Hamiltonian\] which means that the Dirac observable $H_I({\tau})$ is a “time-dependent" generator for the gauge flow on the constraint surface. Thus we call $H_I({\tau})$ the (time-dependent) physical Hamiltonian if we are dealing with a general reparametrization-invariant system with vanishing Hamiltonian. Moreover, the algebra of the physical Hamiltonians is weakly Abelean, because the flows ${\alpha}^{\tau}: O_f({\tau}_0)\mapsto O_f({\tau}+{\tau}_0)$ forms a Abelean group of weak automorphisms.
Then we come to the quantization on the reduced phase space, where all the classical constraint is solved and all the classical elementary observables are invariant under gauge transformation. We start our quantization in Heisenberg picture. On the kinematical level, the quantum algebra ${\mathfrak{A}}$ is generated by the gauge invariant observables $\hat{Q}^a({\tau})$ and $\hat{P}_a({\tau})$ with the “equal-time" canonical commutation relation for any ${\tau}$ (in particular ${\tau}=0$) &&\[\_a(),\^b()\]=-i{P\_a(),Q\^b()}-i\^b\_a\
&&\[\_a(),\_b()\]=-i{P\_a(),P\_b()}0\
&&\[\^a(),\^b()\]=-i{Q\^a(),Q\^b()}0.\[CCR\] Given the quantum algebra ${\mathfrak{A}}$, one can find the representation Hilbert space ${\mathcal H}$ of ${\mathfrak{A}}$ via GNS construction by any positive linear functional on ${\mathfrak{A}}$. Note that we can call ${\mathcal H}$ the physical Hilbert space because all the constraints have been solved in the classical level. Furthermore,
We say that the quantum dynamics exists for the present system provided that there exists a representation ${\mathcal H}$ such that for each ${\tau}$ all the physical Hamiltonians $H_I(Q^a({\tau}),P_a({\tau}),{\tau})$ are represented as the densely defined self-adjoint operators on ${\mathcal H}$. We say that the quantum dynamics is anomaly-free provided that all physical Hamiltonians $H_I(Q^a({\tau}),P_a({\tau}),{\tau})$ form a commutative algebra for each ${\tau}$.
If the quantum dynamics is anomaly-free, it means that all the classical gauge symmetries are manifestly reproduced in the quantum theory. Suppose we have a representation of ${\mathfrak{A}}$ such that an anomaly-free quantum dynamics exists, on this representation, we have the Heisenberg picture “equation of motion" from Eq.(\[Hamiltonian\]): =-i{H\_I(Q\^a(),P\_a(),), O\_f()}-i\_f()\[quantumHamiltonian\] thus the operators $\hat{H}_I({\tau})\equiv H_I(\hat{Q}^a({\tau}),\hat{P}_a({\tau}),{\tau})$ are the physical Hamiltonian operators of the quantum system, which generate the multi-finger evolutions of the Heisenberg operators. On the other hand, The time-dependent operators $\hat{H}_I({\tau}):=H_I(\hat{Q}^a(0),\hat{P}_a(0),{\tau})$ generate the multi-finger evolution of quantum states in Schödinger picture. For each $I$, a unitary propagator $U_I({\tau}_I,{\tau}_I')$ is defined by a formal Dyson expansion: U\_I(\_I,\_I’):=1+\_[n=1]{}\^(-i)\^[n]{}\_[\_I’]{}\^[\_I]{}[d]{}\_[I,n]{}\_[\_I’]{}\^[\_[I,n]{}]{}[d]{}\_[I,n-1]{}\_[\_I’]{}\^[\_[I,2]{}]{}[d]{}\_[I,1]{} \_I(\_[I,1]{})\_I(\_[I,n]{}) which at least formally solves the Schödinger equation iU\_I(\_I,\_I’)=\_I(\_I)U\_I(\_I,\_I’). Since the representation is anomaly-free, we define the multi-finger evolution unitary propagator $U({\tau},{\tau}')$ by U(, ’):=\_[I]{}U\_I(\_I,\_I’) Given two “multi-finger time" ${\tau}_N=\{{\tau}_{I,N}\}_I,{\tau}_0=\{{\tau}_{I,0}\}_I$ and two physical state (Heisenberg states) $\Psi,\Psi'\in{\mathcal H}$, their corresponding Schödinger states are denoted by $\Psi({\tau}_N),\Psi'({\tau}_0)$. The physical multi-finger transition amplitude is defined by the physical inner product between these two physical Heisenberg states ${\langle}\Psi|\Psi'{\rangle}$. We then perform the standard skeletonization procedure [@brown] for this physical transition amplitude: &&|’ = (\_N)|U(\_N, \_0)|’(\_0)\
&=&[d]{}Q\^a(\_N)[d]{}Q\^a(\_[N-1]{})[d]{}Q\^a(\_1)[d]{}Q\^a(\_0)\
&&|Q\^a(\_N)Q\^a(\_N)|Q\^a(\_[N-1]{})Q\^a(\_1)|Q\^a(\_[0]{})Q\^a(\_0)|’\[skeleton\] where $|Q^a({\tau}) {\rangle}{\langle}Q^a({\tau})|$ is the projection valued measure associated with $\hat{Q}^a({\tau})$ (we assume $\hat{Q}^a({\tau})$ is represented as a self-adjoint operator for each ${\tau}$). From Eq.(\[skeleton\]), we see that a (discrete) path $c$ in the space of ${\tau}$ is selected for this skeletonization precedure. We denote by ${\mathcal T}$ the space of ${\tau}$ and by $c:\mathbb{R}\to{\mathcal T}$ the path parametrized by the parameter $t$, and $c(t_n)={\tau}_n$. We will call this parameter $t$ the “external time parameter". However the value of ${\langle}\Psi|\Psi'{\rangle}$ is manifestly independent of the choice of the path $c$ (external-time reparametrization) by the anomaly-freeness. This fact is a reflection of the general covariance of the system.
Following the way in [@brown], we arrive at a formal path-integral formula of the physical inner product of Heisenberg states: &&|’\
&=& [’]{}(Q\^a(c(t\_0)),c(t\_0))\
&&i\_[n=1]{}\^NFormally take the continuous limit $N\to\infty$, we obtain a formal Hamiltonian path-integral expression: &&|’\
&=&\_[t]{} e\^[i\_[t\_i]{}\^[t\_f]{}[d]{}t\[P\_a(c(t))\_a(c(t))-[H]{}\_I(c(t))\^I(t)\]]{} [’]{}(Q\^a(c(t\_i)),c(t\_i))\
&=&\_[t]{} \_[t]{}(P\_I(t)+H\_I(c(t))) (T\^I(t)-c\^I(t)) e\^[i\_[t\_i]{}\^[t\_f]{}[d]{}t\[P\_a(c(t))\_a(c(t))+[P]{}\_I(t)\^I(t)\]]{}\
&& [’]{}(Q\^a(c(t\_i)),T\^I(t\_i)) Because of the $\delta$-functions $\delta\Big(T^I(t)-c^I(t)\Big)$, each $Q^a({\tau})\approx[{\alpha}_{\beta}(q^a)]_{{\beta}^I={\tau}^I-T^I}$ reduces to $q^a$ and the same for the momenta $P^a({\tau})$. Therefore &&|’\
&=&\_[t]{} \_[t]{}(P\_I(t)+H\_I(t)) (T\^I(t)-c\^I(t)) e\^[i\_[t\_i]{}\^[t\_f]{}[d]{}t\[p\_a(t)\_a(t)+[P]{}\_I(t)\^I(t)\]]{}\
&& [’]{}(q\^a(t\_i),T\^I(t\_i))\[RPI1\] Eq.(\[RPI1\]) can also be shortly written as &&|’\
&=&[D]{}p\_a[D]{}q\^a[D]{}P\_I[D]{}T\^I \_[t,I]{} e\^[ i\_[t\_i]{}\^[t\_f]{}[d]{}t]{} [’]{}(q\^a\_i,T\^I\_i)\[RPI2\] This result implies that the path-integral amplitude with boundary *kinematical states* ${\Psi(q^a_f,T^I_f)}$ and ${\Psi'}(q^a_i,T^I_i)$ can be interpreted as the physical inner product between the corresponding *physical states* in the relational framework (we will come back and discuss more about this point in section \[PIARM\]). On the other hand, it is also a Faddeev-Popov path-integral formula with the gauge fixing functions $T^I-{\tau}^I$ and unit Faddeev-Popov determinant. We can also obtain the path-integral with the original constraint $C_I$ and general gauge fixing condition $\xi^I$ (two sets of the equations $\xi^I=0$ and $T^I={\tau}^I$ have to share the same solutions) by the relation: \_I&&({C\_I,\^J}) \_I = ({P\_K,T\^L}) () ()\_I\
&=&({P\_I,T\^J}) \_Iwhere $\sqrt{|D_1|}$ denotes the Faddeev-Popov determinant. Recall that if the dynamical system also has a number of second-class constraints $\Phi_i$, the formal measure ${\mathcal D}p_a{\mathcal D}q^a{\mathcal D}P_I{\mathcal D}T^I$ is the Liouville measure on the constraint surface where all second-class constraints vanish. Then it turns out that (the proof will be shown shortly later) [D]{}p\_a[D]{}q\^a[D]{}P\_I[D]{}T\^I=[D]{}x\^A(t) \_[t]{} (\_i\[x\^A(t)\]) where ${\mathcal D}x^A(t) \prod_{t}\sqrt{\det {\omega}[x^A(t)]}$ is the Liouville measure on the full phase space, $\Phi_{\alpha}$ are the second-class constraints, and $|D_2|=\det\left(\{\Phi_{\alpha}, \Phi_{\beta}\}\right)$ is the Dirac determinant. Inserting these relations into Eq.(\[RPI2\]) &&|’\
&=&[D]{}x\^A(t)\_[t]{}\_[t]{}\_[t]{}\
&&( iS\[x\^A(t)\]) [’]{}\[x\^A(t\_i)\] \[RPI3\] where $S[x^{A}(t)]$ denotes the action of the system. It is remarkable that this path-integral Eq.(\[RPI3\]) is independent of the reparametrization of the external time $t$, since the physical inner product (multi-finger physical transition amplitude) is manifestly independent of the choice of path $c$. This fact reflects that we are dealing with a general reparametrization-invariant system, whose Hamiltonian is a linear combination of constraints.
Finally we write down the partition function from relational framework: [Z]{}\_[Relational]{} &=&[D]{}x\^A(t)\_[t]{}\_[t]{}\
&&\_[t]{} ( iS\[x\^A(t)\]).\[RPI4\]
I will show in the follows that the path-integral formula Eq.(\[RPI4\]) coincides with the reduced phase space path-integral quantization using Liouville measure [@QuantizationGauge]. The following discussion also (1) includes the case that the Hamiltonian is non-vanishing on the constraint surface; (2) manifestly shows that different canonical formulations of the same constrained system result in the equivalent path-integral quantizations.
We consider a general regular dynamical system associated with a $2n$-dimensional phase space $({\Gamma},{\Omega})$ with symplectic coordinates $(p_a,q^a)$ ($a=1,\cdots,n$). There are $m$ first-class constraints $C_I$ ($I=1,\cdots,m$) and $2k$ second-class constraint $\Psi_i$ ($i=1,\cdots,2k$) for this dynamical system, and we suppose $m+k<n$ in order that there are still some unconstrained dynamical degree of freedom. Note that since the constraints $C_I$ are first-class, they close under Poisson bracket determined by ${\Omega}$ modulo second-class constraints, and weakly Poisson commute with the second-class constraints $\Psi_i$, i.e. {C\_I,C\_J}=f\_[IJ]{}\^[ K]{}C\_K+g\_[IJ]{}\^[ k]{}\_k {C\_I,\_i}=u\_[Ii]{}\^[ J]{}C\_J+v\_[Ii]{}\^[ j]{}\_j where $f_{IJ}^{\ \ K}$, $g_{IJ}^{\ \ k}$, $u_{Ii}^{\ \ J}$ and $v_{Ii}^{\ \ j}$ are in general the structure functions depending on the phase space coordinates.
Now the total Hamiltonian $H_{tot}$ is written as: H\_[tot]{}=H+f\^i\_i+ł\^IC\_I where ${\lambda}^I$ are the free Lagrangian multipliers but $f^i$ are determined as functions on the constraint surface by the consistency of all the constraints. $H$ is the non-vanishing Hamiltonian on the reduced phase space, which is a function weakly commute with the first class constraints, i.e {H,C\_I}=U\_I\^JC\_J+V\_I\^j\_j where $U_I^J$ and $V_I^j$ are in general the functions depending on the phase space coordinates.
In order to have a unique equation of motion without the free Lagrangian multipliers, one can introduce $m$ gauge fixing functions $\chi^I$ ($C_I$ and $\chi^I$ shouldn’t be weakly Poisson commute) to locally cut the gauge obits generated by the first class constraints (we ignore the potential problems about Gribov copies). The implementation of the local gauge fixing conditions $\chi^I\approx0$ reduce the original first-second-class-mixed constrained system into a purely second-class constrained system. Thus the total Hamiltonian is re-defined by H\_[tot]{}=H+f\^i\_i+ł\^IC\_I+\_I\^I By the consistency of $C_I$ and $\chi^I$, their coefficients ${\lambda}^I$ and ${\beta}_I$ are determined as some certain functions on the constraint surface defined by $\Psi_i=C_I=\chi^I=0$. This constraint surface ${\Gamma}_R$ combined with the Dirac symplectic structure ${\Omega}_R$ determined by the constraints $C_I$, $\chi^I$, and $\Psi_i$ is (at least locally) symplectic isomorphic to the reduced phase space for the constrained system (see [@QuantizationGauge] for proof).
Then we consider the path-integral quantization of a general regular[^3] dynamical system with both the first-class and the second-class constraints. We have shown that this kind of constrained system can always be reduced into a purely second-class constrained system by introducing a certain number of gauge fixing conditions, so it is enough to consider the path-integral quantization for a general regular dynamical system only associated with the second-class constraints [@QuantizationGauge]. We denote its $2n$-dimensional phase space by $(\Gamma,\Omega)$ with a general coordinates $x^I$ $I=1,\cdots,2n$. Suppose there is a irreducible set of $2m$ regular second-class constraints $\chi_\alpha$ $\alpha=1,\cdots,2m$, then the reduced phase space $(\Gamma_R,\Omega_R)$ is defined by the sub-manifold ${\Gamma}_R=\{x\in\Gamma:\chi_\alpha(x)=0,\ \forall\alpha=1,\cdots,2m\}$ with the Dirac symplectic structure $\omega_R$ determined by $\chi_{\alpha}$. On the reduced phase space $(\Gamma_R,\Omega_R)$, all the degree of freedom is free of constraint, thus it is straight-forward to (heuristically) define the path-integral partition function. We introduce $2(n-m)$ coordinates $y^i$ $i=1,\cdots,2(n-m)$ on ${\Gamma}_R$, then the partition function of the system is defined by a path-integral with respect to an infinite product of Liouville measure associated to $\omega_R$: $$\begin{aligned}
Z:=\int\prod_{t\in[t_1,t_2]}\left[{\mathrm d}y^{i}(t)\sqrt{\left|\det\omega_R[y^i(t)]\right|}\ \right]\exp(iS[y^i(t)])\end{aligned}$$ where $S[y^{i}(t)]$ denotes the action of the system.
In order to rewrite the path-integral formula in terms of the original phase space coordinates $x^I$. We make a coordinate transformation on $\Gamma$ from $\{x^I\}_{I=1}^{2n}$ to $\{y^i\}_{i=1}^{2(n-m)}$ and $\{\chi_\alpha\}_{\alpha=1}^{2m}$, where $\{y^{i}\}_{i=1}^{2(n-m)}$ are the coordinates on ${\Gamma}_R$ such that $\{y^i,\chi_\alpha\}\approx0$ (it is always possible, see Theorem 2.5 of [@QuantizationGauge]). Then the simplectic structure on $\Gamma$ can be written as $$\begin{aligned}
\omega=(\omega_\chi)^{\alpha\beta}\mathrm{d}\chi_\alpha\wedge\mathrm{d}\chi_\beta+(\omega_R)_{ij}\mathrm{d}y^i\wedge\mathrm{d}y^j.\end{aligned}$$ In fact, $(\omega_\chi)^{\alpha\beta}$ is the inverse of the Dirac matrix $\Delta_{\alpha\beta}=\{\chi_\alpha,\chi_\beta\}$ Thus we obtain a relation of the integration measure by the invariance of the Liouville measure under the coordinate transformation: $$\begin{aligned}
\sqrt{\left|\det\omega[x^I]\right|}\prod_I\mathrm{d}x^I=\sqrt{\left|\frac{\det\omega_R[y^i]}{\det\Delta}\right|}\prod_\alpha\mathrm{d}\chi_\alpha\prod_i\mathrm{d}y^i.\nonumber\end{aligned}$$ Therefore we obtain the desired path-integral formula in terms of $x^I$ $$\begin{aligned}
Z=\int\prod_{t\in[t_1,t_2]}\left[{\mathrm d}x^{I}(t)\sqrt{\left|\det\omega[x^I(t)]\right|}\ \right]\prod_{t\in[t_1,t_2]}\left[\sqrt{\left|\det\Delta[x^I(t)]\right|}\prod_\alpha\delta(\chi_\alpha[x^I(t)])\right]\exp(iS[x^I(t)])\label{generalZ}\end{aligned}$$ Note that since this path-integral formula is a quantization on the reduced phase space, the partition function $Z$ is independent of the choice of the gauge fixing function $\chi_{\alpha}$.
We apply the path-integral formula Eq.(\[generalZ\]) to our general first-second-class-mixed constraint system. The phase space coordinates $x^I$ are chosen to be the symplectic coordinates $(p_a,q^a)$. The second-class constraints $\chi_{\alpha}$ are chosen to be the first-class constraints $C_I$, the gauge fixing functions $\chi^I$, and the second-class constraints $\Psi_i$. On the constraint surface, the Dirac matrix $\Delta$ takes the following form:
$\Delta_{{\alpha}{\beta}}$ $C_I$ $\Psi_i$ $\chi^I$
---------------------------- ---------------- ------------ ---------------
$C_J$ 0 0 $\Delta_{FP}$
$\Psi_j$ 0 $\Delta_D$ $\Theta$
$\chi^J$ $-\Delta_{FP}$ $-\Theta$ $\Xi$
where $(\Delta_D)_{ij}:=\{\Psi_i,\Psi_j\}$ is the Dirac matrix for the second-class constraint $\Psi_i$, and $(\Delta_{FP})_{IJ}:=\{C_I,\chi^J\}$ is the Faddeev-Popov matrix for the gauge fixing functions $\chi^I$. The absolute value for the determinant of $\Delta$ is ||=|\_D||\_[FP]{}|\^2 Therefore, the partition function of the constrained system is written as Z&=&\_[t]{}\_[t]{}\
&&\_[t]{}(iS) which precisely coincides with the path-integral formula Eq.(\[RPI4\]) from the relational framework and the canonical quantization on the reduced phase space, when the Hamiltonian $H$ vanishes. Here we have split the integrand into four different factors in order to clarify the different physical meanings.
1. The first factor \_[t]{}is essentially an infinite product of the Liouville measures on the constraint surface ${\Gamma}_\Psi:=\left\{x\in{\Gamma}\ |\ \Psi_i\left[x\right]=0\right\}$ equipped with the Dirac symplectic structure ${\Omega}_\Psi$. This product Liouville measure is invariant under the canonical transformations on $({\Gamma}_\Psi,{\Omega}_\Psi$.
2. The second factor \_[t]{}is a product of the $\delta$-functions of the first-class constraints $C_I$, which will be exponentiated and contribute the total Hamiltonian, after we use the Fourier decompositions of the $\delta$-functions. The third factor \_[t]{}is a typical Faddeev-Popov term which always appears in the gauge fixed path-integral.
3. And the last factor is the exponentiated action where the action $S\left[p_a(t), q^a(t)\right]$ reads: S=\_[t\_1]{}\^[t\_2]{}[d]{}t.
The next step is to employ the Fourier decomposition of the $\delta$-functions so that we write the partition function as Z&=&\_[t]{}\_[t]{}\
&&\_[t]{}\
&&i\_[t\_1]{}\^[t\_2]{}[d]{}t where we have exponentiated the first-class constraints $C_I$, and replaced the second factor by a formal product measure of the Lagrangian multipliers.
The gauge invariance of path-integral measure
---------------------------------------------
In this subsection, we will consider the invariance of the total measure factor D:=\_[t]{}\_[t]{}\[Dmu\] under the gauge transformations generated by the first-class constraints.
We first consider the first-class constraints reduced on the second-class constraint surface $({\Gamma}_\Psi,{\Omega}_\Psi)$, we denote the restricted first-class constraints $C_I\big|_{G_{\Psi}}$ also by $C_I$, which won’t result in any mis-understanding, because we will restrict all the following discussion on the phase space $({\Gamma}_\Psi,{\Omega}_\Psi)$. On the phase space $({\Gamma}_\Psi,{\Omega}_\Psi)$, the constraint algebra is the Poisson algebra generated by $C_I$ under the Dirac bracket $\{,\}_\Psi$ determined by ${\Omega}_\Psi$ {C\_I,C\_J}\_&=&{C\_I,C\_J}-{C\_I,\_i}(\_D\^[-1]{})\^[ik]{}{\_k,C\_J}\
&=&f\_[IJ]{}\^[ K]{}C\_K+g\_[IJ]{}\^[ k]{}\_k+(\_D\^[-1]{})\^[ik]{}\
&[\^=]{}&C\_K\
&[\^=]{}&F\_[IJ]{}\^[ K]{}C\_K where “ $^\Psi\!\!\!\!\!\!=$ " means the equality on $({\Gamma}_\Psi,{\Omega}_\Psi)$ and $F_{IJ}^{\ \ K}\equiv\left[f_{IJ}^{\ \ K}+u_{Ii}^{\ \ L}C_L\left(\Delta_D^{-1}\right)^{ik}u_{Jk}^{\ \ K}\right]$ is the structure function for the first-class constraint algebra on $({\Gamma}_\Psi,{\Omega}_\Psi)$. For the Dirac bracket between $C_I$ and the Hamiltonian $H$, we have {H,C\_I}\_&=&{H,C\_I}-{H,\_i}(\_D\^[-1]{})\^[ik]{}{\_k,C\_I}\
&=&U\_I\^JC\_J+V\_I\^j\_j-{H,\_i}(\_D\^[-1]{})\^[ik]{}\
&[\^=]{}&C\_J\
&[\^=]{}&W\_I\^JC\_J Now we consider the infinitesimal gauge transformation generated by a first-class constraint $C_J$. For any function $f$ on the phase space $({\Gamma}_\Psi,{\Omega}_\Psi)$, it is defined by f:=f+{f,C\_J}\_In the path-integral measure Eq.(\[Dmu\]), there is a factor of the product measure of the Lagrangian multipliers $\left[\prod_{I=1}^m {\mathrm d}{\lambda}^I(t)\right]$ whose gauge transformation is not obvious so far, because the constraints $C_I$ are the phase space functions and their Hamiltonian vector fields don’t have action on the Lagrangian multipliers. However the gauge transformations for the Lagrangian multipliers ${\lambda}^I$ is obtained if we ask the action is invariant under gauge transformation generated by the first-class constraints, i.e. the canonical variables and Lagrangian multipliers should transform at the same time and form the local symmetries of the action.
Apply this infinitesimal gauge transformation generated by $C_J$ to the total action $S_{tot}$ (also assume the change of ${\lambda}^I$), we obtain the transformed total action by S\_[tot]{}&=&\_[t\_1]{}\^[t\_2]{}[d]{}t\_[\_]{}\
&=&\_[t\_1]{}\^[t\_2]{}[d]{}t\_[\_]{}+o(\^2) where we have used the fact that the kinetic term is unchanged under canonical transformation and ł\^IC\_I:=\^IC\_I+ł\_I+o(\^2). Obviously, if at the same time we transform the Lagrangian multiplier by ł\^K\^K:=ł\^K-the total action $S_{tot}$ is unchanged up to the order of ${\epsilon}^2$ S\_[tot]{}=S\_[tot]{}+o(\^2) or in another word: \_[J,]{} S\_[tot]{}:=\_[0]{}=0 Therefore we give a name for this simultaneous transformation
The simultaneous infinitesimal transformations for both the phase space functions and the Lagrangian multipliers induced by the first-class constraint $C_J$ f:=f+{f,C\_J}\_ [and]{} ł\^K\^K:=ł\^K-which results in $\delta_{J,{\epsilon}} S_{tot}=0$ is called the **infinitesimal local symmetry** generated by $C_J$.
There are two remarks:
- It is well known that the local symmetries of the action in general result in the constraints in the canonical framework. Given an classical action, there is a projection map from the space of field configurations to its phase space. This projection map usually projects the infinitesimal local symmetry transformations of the action to the same number of infinitesimal phase space gauge transformations generated by the constraints [@wald]. With respect to this, the construction of infinitesimal gauge transformations for action from constraints is its reverse procedure. But this reverse procedure sometimes cannot capture all the local symmetry transformations, espectially in general relativity, because some local symmetry transformations may not be projectable, e.g. the field-independent spacetime diffeomorphisms.
- The reason for only considering infinitesimal transformations is the following: Here we are discussing the most general cases in which the first-class constraint algebra is not a Lie algebra (there are some structure functions). Therefore the collection of gauge transformations doesn’t have a group structure and actually, is at most an enveloping algebra of the first-class constraint algebra. A generic element of the enveloping algebra is a finite product of infinitesimal gauge transformations.
An infinitesimal local symmetry generated by a first-class constraint is said to be implemented quantum mechanically without anomaly if the total path-integral measure D:=\_[t]{}\_[t]{} is invariant (up to an overall constant) under this local symmetry transformation.
Under this gauge transformation, the transformation behavior of the total measure is given by &&\_[t]{}\_[t]{}\
&&\_[t]{}\_[t]{}\
&=&\_[t]{}\_[t]{}(1- F\_[IJ]{}\^[ I]{}) where we have used the fact that the first factor of the total measure is an infinite product of the Liouville measures on the constraint surface ${\Gamma}_\Psi$ equipped with the Dirac symplectic structure ${\Omega}_\Psi$. This product Liouville measure is invariant under the canonical transformations on $({\Gamma}_\Psi,{\Omega}_\Psi$. Therefore we have proven the following result:
\[invariancecondition\] The local symmetry generated by the first-class constraint $C_J$ is implemented without anomaly in reduced phase space quantization if and only if the trace of the structure function, $F_{IJ}^{\ \ I}$, is a phase space constant.
Note that we only need the invariance of the measure $D\mu$ up to an overall constant because essentially the quantities we are computing is the physical inner product ${\langle}\Psi|\Psi'{\rangle}$ (see Eq.(\[RPI3\])). More precisely, we may choice a reference vector in the physical Hilbert space ${\mathcal H}$, the meaningful quantity is the ratio |’\_Ø= which is invariant under a re-scaling of the path-integral measure.
The gauge invariance of the path-integral measure of gravity {#GRinv}
============================================================
The path-integral measure in ADM formalism
------------------------------------------
We would like to apply our general consideration to gravity. First of all we consider the ADM formalism of the 4-dimensional canonical general relativity. The ADM formalism formulates canonical general relativity as a purely first-class constrained system, whose total Hamiltonian is a linear combination of first-constraints: H\_[tot]{}:=\_[d]{}\^[3]{}x \[N\^a(x) H\_a(x)+N(x) H(x)\]where $H_a$ and $H$ are spatial diffeomorphism constraint and Hamiltonian constraint respectively. They are expressed as: H\_a&=&-q\_[ac]{}D\_b P\^[bc]{}\
H&=&-R-P\^[ab]{}P\^[cd]{} The first-class constraint algebra is given by {H\_a(N\^a),H\_b(M\^b)}&=&H\_a(-[L]{}\_N\^a)\
{H(N),H\_a(N\^a)}&=&H(-[L]{}\_N)\
{H(N),H(M)}&=&H\_a(q\^[ab]{}\[N\_bM-M\_bN\]) It is not hard to see that the constraint algebra is not a Lie algebra since there is a structure function appearing in the commutator between two Hamiltonian constraints. Therefore the gauge transformations generated by these constraints don’t form a group but only can form a enveloping algebra, to which we refer as the Bargmann-Komar Group BK(M) [@BK]. This enveloping algebra obtains a group structure when the equations of motion is imposed, and only in this case BK(M)=Diff(M). We will come back to this point in Section \[BKvsDiff\].
We apply our previous general consideration to ADM formalism and write down the path-integral partition function in terms of canonical variables $(P^{ab}, q_{ab})$ Z\_[ADM]{}&=&\_[xM]{} \_[xM]{} i[d]{}t[d]{}\^3x P\^[ab]{}q\_[ab]{}\
&&\_[xM]{}\
&=&\_[xM]{} \_[xM]{} i[d]{}t[d]{}\^3x \[P\^[ab]{}q\_[ab]{}+N\^a(x) H\_a(x)+N(x)H(x)\]\
&&\_[xM]{}where the path-integral measure and the ADM-action read D\_[ADM]{}&=&\_[xM]{} \_[xM]{}\
S\_[ADM]{}\[P\^[ab]{},q\_[ab]{},N\^a,N\]&=&[d]{}t[d]{}\^3x \[P\^[ab]{}q\_[ab]{}+N\^a(x) H\_a(x)+N(x)H(x)\]In the following we need to write down the local symmetries of $S_{ADM}$ generated by spatial diffeomorphism constraint and Hamiltonian constraint. These transformations have to transform not only the phase space variables but also the Lagrangian multipliers which are shift vector and lapse function, in order to keep the ADM-action invariant.
First of all, we consider the infinitesimal gauge transformation $T^{\epsilon}_{Diff}$ generated by the spatial diffeomorphism constraint $H_a({\epsilon}^a)$. For a phase space function $f[P^{ab},q_{ab}]$, the transformation is given by T\^\_[Diff]{}f:=f+{f,H\_a(\^a)} Then we look at the transformation of the action &&S\_[ADM]{}=[d]{}t[d]{}\^3x \[P\^[ab]{}q\_[ab]{}+N\^a(x) H\_a(x)+N(x)H(x)\]\
&& S\_[ADM]{}\
&&=[d]{}t[d]{}\^3x+o(\^2) where we have used the fact that the kinetic term is unchanged under the canonical transformations, and T\^\_[Diff]{}H\_a(N\^a)&=&H\_a(T\^\_[Diff]{}N\^a)+(T\^\_[Diff]{}H\_a)(N\^a)+o(\^2)\
T\^\_[Diff]{}H(N)&=&H(T\^\_[Diff]{}N)+(T\^\_[Diff]{}H)(N)+o(\^2) If we suppose the following transformations assigned to the lapse function and the shift vector correspondingly: T\^\_[Diff]{}N\^a&=&N\^a+[L]{}\_N\^a\
T\^\_[Diff]{}N&=&N+[L]{}\_N\[ADMDiff\] the action $S_{ADM}\left[P^{ab},q_{ab},N^a,N\right]$ is invariant up to $o({\epsilon}^2)$ under the transformation generated by the spatial diffeomorphism constraint $T^{{\epsilon}}_{Diff}:\left(P^{ab},q_{ab},N^a,N\right)\mapsto\left(T^{{\epsilon}}_{Diff}P^{ab},T^{{\epsilon}}_{Diff}q_{ab},T^{{\epsilon}}_{Diff}N^a,T^{{\epsilon}}_{Diff}N\right)$. Therefore \_[[Diff]{},]{}S\_[ADM]{}:=\_[||||0]{}=0 If the smearing fields ${\epsilon}^a(x)$ are the compact support vector fields, the norm $||\vec{{\epsilon}}||$ can be chosen as $\sup_{x\in{\Sigma}}{\left}|{\epsilon}^a(x)\ {\epsilon}^a(x){\right}|^{1/2}$. Thus we have shown that $T^{{\epsilon}}_{Diff}:\left(P^{ab},q_{ab},N^a,N\right)\mapsto\left(T^{{\epsilon}}_{Diff}P^{ab},T^{{\epsilon}}_{Diff}q_{ab},T^{{\epsilon}}_{Diff}N^a,T^{{\epsilon}}_{Diff}N\right)$ is an infinitesimal local symmetry generated by the spatial diffeomorphism constraint. Furthermore it is manifest that the transformation Jacobian in Eq.(\[ADMDiff\]) is independent of all the fields $\left(P^{ab},q_{ab},N^a,N\right)$, so it is obvious that the path-integral measure D\_[ADM]{}&=&\_[xM]{} \_[xM]{}is invariant under $T^{{\epsilon}}_{Diff}$ up to an overall constant.
Then we consider the infinitesimal gauge transformation $T^{{\epsilon}}_H$ generated by the Hamiltonian constraint. For any phase space function $f[P^{ab},q_{ab}]$, the transformation is given by T\^\_[H]{}f:=f+{f,H()} Then we look at the transformation of the action &&S\_[ADM]{}=[d]{}t[d]{}\^3x \[P\^[ab]{}q\_[ab]{}+N\^a(x) H\_a(x)+N(x)H(x)\]\
&& S\_[ADM]{}\
&&=[d]{}t[d]{}\^3x+o(\^2) where we have used the fact that the kinetic term is unchanged under canonical transformations, and T\^\_[H]{}H\_a(N\^a)&=&H\_a(T\^\_[H]{}N\^a)+(T\^\_[H]{}H\_a)(N\^a)+o(\^2)\
T\^\_[H]{}H(N)&=&H(T\^\_[H]{}N)+(T\^\_[H]{}H)(N)+o(\^2) If we suppose the following transformations assigned to the lapse function and shift vector correspondingly: T\^\_[H]{}N\^a&=&N\^a-q\^[ab]{}(N\_b-\_bN)\
T\^\_[H]{}N&=&N-[L]{}\_ = N-N\^a\_a\[ADMH\] the action $S_{ADM}\left[P^{ab},q_{ab},N^a,N\right]$ then is invariant up to $o({\epsilon}^2)$ under the gauge transformation generated by the Hamitonian constraint $T^{{\epsilon}}_{H}:\left(P^{ab},q_{ab},N^a,N\right)\mapsto\left(T^{{\epsilon}}_{H}P^{ab},T^{{\epsilon}}_{H}q_{ab},T^{{\epsilon}}_{H}N^a,T^{{\epsilon}}_{H}N\right)$. Therefore \_[[H]{},]{}S\_[ADM]{}:=\_[||||0]{}=0 If the smearing fields ${\epsilon}(x)$ are compact support functions, the norm $||{{\epsilon}}||$ can be chosen as $\sup_{x\in{\Sigma}}{\left}|{\epsilon}(x){\right}|$. Thus we have shown that $T^{{\epsilon}}_{H}:\left(P^{ab},q_{ab},N^a,N\right)\mapsto\left(T^{{\epsilon}}_{H}P^{ab},T^{{\epsilon}}_{H}q_{ab},T^{{\epsilon}}_{H}N^a,T^{{\epsilon}}_{H}N\right)$ is an infinitesimal local symmetry generated by the Hamiltonian constraint. Furthermore the transformation Jacobian $\partial \left(T^{\epsilon}_{H}N^a,T^{\epsilon}_{H}N\right)/\partial\left(N^a,N\right)$ in Eq.(\[ADMH\]) is expressed as
$\frac{\partial \left(T^{\epsilon}_{H}N^a,T^{\epsilon}_{H}N\right)}{\partial\left(N^a,N\right)}$ $T^{\epsilon}_{H}N^a$ $T^{\epsilon}_{H}N$
-------------------------------------------------------------------------------------------------- ---------------------------------------------------------------- -------------------------
$N^a$ $1$ $-\partial_a{\epsilon}$
$N$ $q^{ab}\left({\epsilon}\partial_b-\partial_b{\epsilon}\right)$ $1$
$\ =\ I+{\mathcal E}$
where the perturbation matrix ${\mathcal E}$ is traceless. Thus the Jacobian determinant is $\det{\left}(I+{\mathcal E}{\right})=1+\mathrm{Tr}{\mathcal E}=1$. As a result the path-integral measure D\_[ADM]{}&=&\_[xM]{} \_[xM]{}is invariant under $T^{{\epsilon}}_{H}$.
So far we assumed the infinitesimal transformation parameter ${\epsilon}^\mu=({\epsilon},\vec{{\epsilon}})$ doesn’t depend on the time-parameter $t$. However if we let ${\epsilon}^\mu={\epsilon}^\mu(t,\vec{x})$, the Eqs.(\[ADMDiff\]) and (\[ADMH\]) have generalized expressions. Now we write Eqs.(\[ADMDiff\]) and (\[ADMH\]) in a uniform expression and add certain time-derivative terms T\^[\^]{}N\^a&=&N\^a+[L]{}\_N\^a+q\^[ab]{}(\_bN-N\_b)+\^a\
T\^[\^]{}N&=&N+\^a\_aN-N\^a\_a+\[ADMgen\] Firstly, combined with the gauge transformations for phase space variables, one can check that Eq.(\[ADMgen\]) leaves both the action $S_{ADM}$ and the path-integral measure $D\mu_{ADM}$ invariant (the consistency conditions $\dot{H}=\dot{H}_a=0$ should be used to show the invariance of the action). Secondly the generalization Eq.(\[ADMgen\]) is motivated by the spacetime diffeomorphisms (see section \[BKvsDiff\]) and can be derived systematically [@Pons]. In order to derive it, the phase space has to be enlarged to include the lapse $N$ and the shift $N^a$ as well as their conjugated momenta $\Pi$ and $\Pi_a$. The set of the first-class constraints should also be extended to include $\Pi\approx0$ and $\Pi_a\approx0$. Then the extended gauge transformation generator H()+H\_a(\^a)+[d]{}\^3x\[\_a([L]{}\_N\^a+q\^[ab]{}(\_bN-N\_b)+\^a)+(\^a\_aN-N\^a\_a+)\]generates the gauge transformations on all ten components of the spacetime metric, which coincide with spacetime diffeomorphism if the equations of motion are imposed.
Let’s summarize all above results:
\[ADMinvmeasure\] There exists a collection of infinitesimal transformations $T^{{\epsilon}^\mu}$ generated by the spatial diffeomorphism constraint $H_a$, and the Hamiltonian constraint $H$: f&& T\^[\^]{}f:=f+{f,H()+H\_a(\^[a]{})}\
N\^a&&T\^[\^]{}N\^a:=N\^a+[L]{}\_N\^a+q\^[ab]{}(\_bN-N\_b)+\^a\
N&&T\^[\^]{}N:=N+\^a\_aN-N\^a\_a+ These transformations are the infinitesimal local symmetries of the action, i.e. \_[[\^]{}]{}S\_[ADM]{}=\_[||\^||0]{}=0 for ${\epsilon}^{\mu}$ compact support and $||{\epsilon}^\mu||=\sup_{(t,\vec{x})\in M}|{\epsilon}^\mu(t,\vec{x}){\epsilon}^\mu(t,\vec{x})|^{1/2}$. And the total path-integral measure D\_[ADM]{}&=&\_[xM]{} \_[xM]{}is invariant under these transformations (up to an overall constant).
The Bergmann-Komar group BK(M) is the enveloping algebra generated by the collection of $T^{{\epsilon}^\mu}$ as the transformations on the space of metric $g_{{\alpha}{\beta}}=(q_{ab},N,N^a)$, i.e. :=[F]{}({T\^[\^]{}}\_[\^]{})/\~$\{T^{{\epsilon}^\mu}\}_{{\epsilon}^\mu}$ is the free algebra generated by $T^{{\epsilon}^\mu}$ whose generic element read T\^[\_1\^]{}T\^[\_2\^]{}T\^[\_n\^]{} and “$\sim$" denotes the equivalence relations from the linearity and the commutation relations of $T^{{\epsilon}^\mu}$.
The Bergmann-Komar group BK(M) is a collection of the infinitesimal local symmetries of classical Einstein-Hilbert action, which are implemented without anomaly in the reduced phase space path-integral quantization.
The field-independent spacetime diffeomorphism group {#BKvsDiff}
----------------------------------------------------
There exists another collection of the local symmetries of the Einstein-Hilbert action, which is the set of the *field-independent* spacetime diffeomorphisms Diff(M). The generators of Diff(M) are Lie-derivatives ${\mathcal L}_u$ along the 4-vector field $u^\mu$. The commutator between two Lie-derivatives \[[L]{}\_u,[L]{}\_[u’]{}\]=[L]{}\_[\[u,u’\]]{} endows the collection of generators a Lie algebra structure thus gives Diff(M) a group structure. In order to compare the the *field-independent* spacetime diffeomorphisms and the local symmetries in Bergamnn-Komar group and check if the local symmetries in Diff(M) can be implemented without anomaly in the above path-integral quantization, we have to show how the diffeomorphisms in Diff(M) act on the components $(q_{ab},N,N^a)$ (for details see [@GHTW] and the references therein).
The components $(q_{ab},N,N^a)$ can be solved from 4-metric $g_{{\alpha}{\beta}}$ by the following relations N\^a=g\^[ab]{}g\_[tb]{}, N\^2=-g\_[tt]{}+g\^[ab]{}g\_[ta]{}g\_[tb]{}, q\_[ab]{}=g\_[ab]{} where $g^{ab}$ is the inverse of the spatial metric $g_{ab}$. The infinitesimal change of the 4-metric under the spacetime diffeomorphism is given by the Lie-derivative \_ug\_:=[L]{}\_ug\_=u\^\_g\_+2g\_[(]{}\_[)]{}u\^ To derive the induced transformation for lapse and shift, one should use the relation $u^\mu={\epsilon}n^\mu+X^\mu_{,a}{\epsilon}^a$ or u\^t=, u\^a=\^a- After some calculus, the infinitesimal changes of lapse and shift under spacetime diffeomorphism is obtained explicitly \_uN\^a&=&[L]{}\_N\^a+q\^[ab]{}(\_bN-N\_b)+\^a\
\_uN&=&\^a\_aN-N\^a\_a+ which coincides with Eq.(\[ADMgen\]) and actually is the motivation for adding the time-derivative terms in Eq.(\[ADMgen\]). On the other hand, the infinitesimal changes of spatial metric $q_{ab}$ and its momentum $P^{ab}$ are represented as the Lie-derivatives of the spatial fields \_uq\_[ab]{}:=[L]{}\_uq\_[ab]{}, \_uP\^[ab]{}:=[L]{}\_uP\^[ab]{} However, only when equations of motion is imposed, [L]{}\_uq\_[ab]{}(x)={q\_[ab]{}(x),H()+H\_a(\^[a]{})}, [L]{}\_uP\^[ab]{}(x)={P\^[ab]{}(x),H()+H\_a(\^[a]{})} Thus we see that the relation BK(M)=Diff(M) holds only when the equations of motion are imposed.
Then the question is whether the local symmetries in Diff(M) can be implemented quantum mechanically without anomaly. It turns out in [@wald; @Pons] that all the infinitesimal field-independent non-spatial diffeomorphisms cannot be taken from the space of metric to the phase space by the projection map in [@wald]. In another word, there is no notion of the local symmetries on the phase space corresponding to the non-spatial diffeomorphisms. Thus any non-spatial diffeomorphism belonging to Diff(M) cannot be written as a canonical transformation (unless on shell) and doesn’t leave the Liouville measure $\prod_{x\in M}\left[{\mathrm d}P^{ab}(x)\ {\mathrm d}q_{ab}(x)\right]$ invariant. In conclusion, the classical local symmetries in the spacetime diffeomorphism group Diff(M) is anomalous in the reduce phase space path-integral quantization of general relativity.
The path-integral measure from the Holst action
-----------------------------------------------
We apply the general procedure to the case of Holst action, whose canonical framework is studied in [@Barros]. The total Hamiltonian is a linear combination of constraints (the physical Hamiltonian on reduced phase space vanishes). the expressions of the first-class constraints $G^{IJ}$, $H_a$, $H$, and second-class constraint $C^{ab}$, $D^{ab}$ is given by (we follow the notation of [@Barros]) G\_[IJ]{}&=&\_a(-\*)\^a\_[IJ]{}+A\_[aI]{}\^[ K]{}(-\*)\^a\_[JK]{}-A\_[aJ]{}\^[ K]{}(-\*)\^a\_[IK]{}\
H\_a&=&(F-\*F)\_[ab]{}\^[IJ]{}\[A\] \^b\_[IJ]{}-A\^[IJ]{}\_aG\_[IJ]{}\
H&=&(F-\*F)\_[ab]{}\^[IJ]{}\[A\] \^a\_[IK]{} \^b\_[JL]{} \^[KL]{}+ł\_[ab]{}(A,)C\^[ab]{}\
C\^[ab]{}&=& \^[IJKL]{}\^a\_[IJ]{}\^b\_[KL]{}\
D\^[ab]{}&=&\*\^[c]{}\_[IJ]{}(\^[aIK]{}D\_c\^[bJL]{}+\^[bIK]{}D\_c\^[aJL]{})\_[KL]{} where $D^{ab}$ is the secondary constraint with $\{H(x),C^{ab}(x')\}=D^{ab}(x)\delta(x,x')$, ${\lambda}_{ab}(A,\pi)$ is the Lagrangian multiplier determined by the consistency of $D^{ab}$. The constraint algebra is given by {G\_[IJ]{}(Ł\^[IJ]{}),G\_[KL]{}(Ø\^[KL]{})}&=&4G\_[IJ]{}(Ł\^[IK]{}Ø\^[JL]{}\_[KL]{})\
{G\_[IJ]{}(Ł\^[IJ]{}),H\_a(N\^a)}&=&G\_[IJ]{}(-[L]{}\_Ł\^[IJ]{})\
{G\_[IJ]{}(Ł\^[IJ]{}),H(N)}&=&0\
{G\_[IJ]{}(Ł\^[IJ]{}),C\^[ab]{}(c\_[ab]{})}&=&0\
{G\_[IJ]{}(Ł\^[IJ]{}),D\^[ab]{}(d\_[ab]{})}&=&0\
{H\_a(N\^a),H\_b(M\^b)}&=&H\_a(-[L]{}\_N\^a)\
{H(N),H\_a(N\^a)}&=&H(-[L]{}\_N) = -H(N\^a\_aN-N\_aN\^a)\
{C\^[ab]{}(c\_[ab]{}),H\_c(N\^c)}&=&C\^[ab]{}(-[L]{}\_c\_[ab]{})\
{D\^[ab]{}(d\_[ab]{}),H\_c(N\^c)}&=&D\^[ab]{}(-[L]{}\_d\_[ab]{})\
{H(N),H(M)}&=&H\_a(\[q\]q\^[ab]{}\[N\_bM-M\_bN\])+C\^[ab]{}()+D\^[ab]{}()\
{H(N),C\^[ab]{}(c\_[ab]{})}&=&D\^[ab]{}(Nc\_[ab]{})+C\^[ab]{}()\
{H(N),D\^[ab]{}(d\_[ab]{})}&=&D\^[ab]{}()+C\^[ab]{}()\
{C\^[ab]{}(c\_[ab]{}),D\^[ab]{}(d\_[ab]{})}&=&C\^[ab]{}()+4\^2q\^[ab]{}q\^[cd]{}(c\_[ac]{}d\_[bd]{}-c\_[ab]{}d\_[cd]{}) where we have skipped some unimportant structure functions related to the second-class constraints. Note that the smearing functions $N$ for the Hamiltonian constraint $H$ and $c_{ab}$ for $C^{ab}$ have density weight $-1$, and the smearing function $d_{ab}$ for $D^{ab}$ has density weight $-2$, they are all assumed to be independent of phase space variables.
In contrast to the ADM formalism, the canonical formulation of Holst action results in a non-regular constrained system (the rank of Dirac matrix is not a constant). There are five disjoint sectors of solutions for the second-class constraint equation $C^{ab}\approx0$, i.e. there exists a non-degenerated triad field $e_a^i$ and an additional 1-form field $e^0_a$ such that $$\begin{aligned}
\nonumber
(I \pm) && \pi^a_{IJ}= \pm {\epsilon}^{abc} e_b^I e_c^J \nonumber\\
(II \pm) && \pi^a_{IJ} = \pm \frac{1}{2} {\epsilon}^{abc} e_b^K e_c^L{\epsilon}_{IJKL}\nonumber\\
(\text{Degenerated}) && \pi^a_{IJ} = 0\end{aligned}$$ where the degenerated sector results in a degenerated Dirac matrix while other sectors lead to non-degenerated Dirac matrix. In order to proceed the path-integral quantization in section \[reduce\], we have to exclude the degenerated sector. For simplicity we only consider a single sector, say, II+. The analysis for all other non-degenerated sectors is essentially the same.
The path-integral partition function can be written down immediately Z&=&\_[xM]{} \_[xM]{}\
&&\_[xM]{} i[d]{}t[d]{}\^3x\
&=&\_[xM]{} \_[xM]{}\
&&\_[xM]{}\
&&i[d]{}t[d]{}\^3x\[pi\] where the first factor in square bracket is the Liouville measure on the constraint surface $({\Gamma}_\Psi,{\Omega}_\Psi)$ defined by $C^{ab}=D^{ab}=0$. We denote by $D\mu$ the total path-integral measure: D:=\_[xM]{} \_[xM]{}$\Delta_D$ is the determinant of the Dirac matrix for the second-class constraint
&{C\^[ab]{}(x),C\^[cd]{}(x’)}&, {C\^[ab]{}(x),D\^[cd]{}(x’)} \
&{D\^[ab]{}(x),C\^[cd]{}(x’)}&, {D\^[ab]{}(x),D\^[cd]{}(x’)}
=
&0&, {C\^[ab]{}(x),D\^[cd]{}(x’)} \
&{D\^[ab]{}(x),C\^[cd]{}(x’)}&, {D\^[ab]{}(x),D\^[cd]{}(x’)}
Therefore $|\det\Delta_D|=[\det G]^2$ where $G$ is the matrix determined by G\^[ab,cd]{}(x,x’)={C\^[ab]{}(x),D\^[cd]{}(x’)}\^[2]{} \^3(x,x’)\[G\] its determinant is $\det G=[\det q]^8$ up to overall constant.
As we did before, the first-class constraint algebra on the phase space $({\Gamma}_\Psi,{\Omega}_\Psi)$ is obtained by computing the Dirac bracket with respect to the second-class constraint {G\_[IJ]{}(Ł\^[IJ]{}),G\_[KL]{}(Ø\^[KL]{})}\_&[\^=]{}&G\_[IJ]{}(4Ł\^[IK]{}Ø\^[JL]{}\_[KL]{})\
{G\_[IJ]{}(Ł\^[IJ]{}),H\_a(N\^a)}\_&[\^=]{}&G\_[IJ]{}(-[L]{}\_Ł\^[IJ]{})\
{G\_[IJ]{}(Ł\^[IJ]{}),H(N)}\_&[\^=]{}&0\
{H\_a(N\^a),H\_b(M\^b)}\_&[\^=]{}&H\_a(-[L]{}\_N\^a)\
{H(N),H\_a(N\^a)}\_&[\^=]{}&H(-[L]{}\_N) = -H(N\^a\_aN-N\_aN\^a)\
{H(N),H(M)}\_&[\^=]{}&H\_a(\[q\]q\^[ab]{}\[N\_bM-M\_bN\]) By using this constraint algebra, we are going to derive the local symmetries of the Holst action generated by the first-class constraints, and to check the invariance of total measure $D\mu$ under these local symmetries. So we will see if these classical local symmetries is implemented without anomaly in the reduced phase path-integral quantization. First of all we consider the infinitesimal $SO(\eta)$ gauge transformation $T^{\epsilon}_{G}$ generated by $G_{IJ}({\epsilon}^{IJ})$, where ${\epsilon}^{IJ}={\epsilon}^{IJ}(t,\vec{x})$ is a spacetime $so(\eta)$-function fT\^\_[G]{}f:=f+{f,G\_[IJ]{}(\^[IJ]{})}\_\[Gtrans\] which results in the transformation of Holst action: &&S=[d]{}t[d]{}\^3x\
&& S\
&&=[d]{}t[d]{}\^3x If we suppose the Lagrangian multipliers obey the following transformation, \_t\^[IJ]{}&&T\^\_[G]{}[A]{}\_t\^[IJ]{}:=A\_t\^[IJ]{}-4A\_t\^[\[I|K|]{}\^[J\]L]{}\_[KL]{}-2\_t\^[IJ]{}\
N\^a&&T\^\_[G]{}N\^a:=N\^a\
N&&T\^\_[G]{}N:=N\[Gtrans1\] the Holst action stays unchanged: $S\left[T^{\epsilon}_{G}{A}_a^{IJ},T^{\epsilon}_{G}{\pi}^a_{IJ},T^{\epsilon}_{G}{A}_t^{IJ},T^{\epsilon}_{G}{N}^a,T^{\epsilon}_{G}{N}\right]=S\left[A_a^{IJ},\pi^a_{IJ},A_t^{IJ},N^a,N\right]$ by using the consistency condition $\partial_t{G}^{IJ}=0$ and the gauge transformation of $A_a^{IJ}$ \_a\^[IJ]{}&&T\^\_[G]{}[A]{}\_a\^[IJ]{}=A\_a\^[IJ]{}+{A\_a\^[IJ]{}(x),G\_[IJ]{}(\^[IJ]{})}\_=A\_a\^[IJ]{}-4A\_a\^[\[I|K|]{}\^[J\]L]{}\_[KL]{}-2\_a\^[IJ]{}. Thus we obtain the SO($\eta$) local symmetry of the Holst action generated by Gauss constraint. It is easy to see that we have recovered the SO($\eta$) local gauge transformation of the spacetime connection field, i.e. \_\^[IJ]{}&&T\^\_[G]{}[A]{}\_\^[IJ]{}=A\_\^[IJ]{}-4A\_\^[\[I|K|]{}\^[J\]L]{}\_[KL]{}-2\_\^[IJ]{}. Since the Jacobian matrix from Eq.(\[Gtrans1\]) is independent of the phase space variables, the total path-integral measure $D\mu$ is invariant (up to an overall constant) under $T^{\epsilon}_G$.
Next we analyze the infinitesimal gauge transformation $T^{\epsilon}_{Diff}$ generated by the spatial diffeomorphism constraint $H_{a}({\epsilon}^{a})$, ${\epsilon}^a={\epsilon}^a(t,\vec{x})$. For any phase space function, the infinitesimal gauge transformation generated by $H_a({\epsilon}^a)$ reads fT\^\_[Diff]{}f:=f+{f,H\_[a]{}(\^[a]{})}\_\[Difftrans\] which results in the transformation of Holst action: &&S=[d]{}t[d]{}\^3x\
&& S\
&&=[d]{}t[d]{}\^3x Obviously, in order that $\delta_{Diff,{\epsilon}}S=0$, the infinitesimal transformation of Lagrangian multipliers takes the following form: \_t\^[IJ]{}&&T\^\_[Diff]{}[A]{}\_t\^[IJ]{}:=A\_t\^[IJ]{}+[L]{}\_A\_t\^[IJ]{}-A\_a\^[IJ]{}\_t\^a=A\_t\^[IJ]{}+\^a\_aA\_t\^[IJ]{}-A\_a\^[IJ]{}\_t\^a\
N\^a&&T\^\_[Diff]{}N\^a:=N\^a+[L]{}\_N\^a+\_t\^a=N\^a+\^b\_bN\^a-N\^b\_b\^a+\_t\^a\
N&&T\^\_[Diff]{}N:=N+[L]{}\_N=N+\^a\_aN-N\_a\^a\[Difftrans1\] note that here the lapse $N$ has density-weight -1. Thus we obtain the local symmetry of the Holst action generated by spatial diffeomorphism constraint. Again since the Jacobian matrix from Eq.(\[Difftrans1\]) is independent of phase space variables, the total path-integral measure $D\mu$ is invariant (up to an overall constant) under $T^{\epsilon}_{Diff}$.
The last task is to consider the infinitesimal gauge transformation $T^{\epsilon}_{H}$ generated by the Hamiltonian constraint $H({\epsilon})$ with ${\epsilon}={\epsilon}(t,\vec{x})$. For any phase space function, the infinitesimal gauge transformation generated by $H({\epsilon})$ reads fT\^\_[H]{}f:=f+{f,H()}\_\[Htrans\] which results in the transformation of the Holst action: &&S=[d]{}t[d]{}\^3x\
&& S\
&&=[d]{}t[d]{}\^3x\] in order that the action is invariant $\delta_{H,{\epsilon}}S=0$, the infinitesimal transformations of the Lagrangian multipliers takes the following form, in order to obtain the local symmetries of the Holst action generated by the Hamiltonian constraint $H$ \_t\^[IJ]{}&&T\^\_[H]{}[A]{}\_t\^[IJ]{}:=A\_t\^[IJ]{}+\[q\]q\^[ab]{}(N\_b-\_bN)A\_a\^[IJ]{}-N\^a{A\_a\^[IJ]{},H()}\_\
N\^a&&T\^\_[H]{}N\^a:=N\^a-\[q\]q\^[ab]{}(N\_b-\_bN)\
N&&T\^\_[H]{}N:=N-[L]{}\_+\_t=N-(N\^a\_a-\_aN\^a)+\_t\[Htrans1\] note that here the lapse $N$ has density-weight -1. the Jacobian matrix $\partial \left(T^{\epsilon}_{H}{A}_t^{IJ},T^{\epsilon}_{H}N^a,T^{\epsilon}_{H}N\right)/\partial\left({A}_t^{IJ},N^a,N\right)$ for the transformation Eq.(\[Htrans1\]) is expressed as
$\frac{\partial \left(T^{\epsilon}_{H}{A}_t^{IJ},T^{\epsilon}_{H}N^a,T^{\epsilon}_{H}N\right)}{\partial\left({A}_t^{IJ},N^a,N\right)}$ $T^{\epsilon}_{H}{A}_t^{IJ}$ $T^{\epsilon}_{H}N^a$ $T^{\epsilon}_{H}N$
---------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------- ------------------------------------------------------------------------ -----------------------------------------------------------
$A_t^{IJ}$ $1$ $0$ $0$
$N^a$ $\left\{A_a^{IJ},H({\epsilon})\right\}_\Psi$ $1$ $-\left(\partial_a{\epsilon}-{\epsilon}\partial_a\right)$
$N$ $-[\det q]q^{ab}A_a^{IJ}\left({\epsilon}\partial_b-\partial_b{\epsilon}\right)$ $[\det q]q^{ab}\left({\epsilon}\partial_b-\partial_b{\epsilon}\right)$ $1$
which can be written as an identity matrix with the addition of a perturbation, i.e. =I+
0 & 0 & 0\
{A\_a\^[IJ]{},H()}\_& 0 & -(\_a-\_a)\
-\[q\]q\^[ab]{}A\_a\^[IJ]{}(\_b-\_b) & \[q\]q\^[ab]{}(\_b-\_b) & 0
I+[E]{}The perturbation ${\mathcal E}$ is again traceless, so the determinant of the Jacobian matrix equals $1+\mathrm{Tr}{\mathcal E}=1$, which means that the local symmetries of the Holst action generated by $H$, Eqs.(\[Htrans\]) and (\[Htrans1\]), leave the total path-integral measure $D\mu$ invariant.
The result of this section is summarized in the following theorem:
\[invmeasure\] (1) The infinitesimal transformations induced by $SO(\eta)$ Gauss constraint $G^{IJ}$ f&& T\^\_[G]{}f:=f+{f,G\_[IJ]{}(\^[IJ]{})}\_\
[A]{}\_t\^[IJ]{}&&T\^\_[G]{}[A]{}\_t\^[IJ]{}:=A\_t\^[IJ]{}-4A\_t\^[\[I|K|]{}\^[J\]L]{}\_[KL]{}+2\_t\^[IJ]{}\
N\^a&&T\^\_[G]{}N\^a:=N\^a\
N&&T\^\_[G]{}N:=N form a group of SO($\eta$) local symmetries of the Holst action and leave the path-integral measure $D\mu$ invariant (up to an overall constant). So the classical SO($\eta$) local symmetries of the Holst action is implemented without anomaly in reduced phase space path-integral quantization.
\(2) There exist infinitesimal transformations $T^{{\epsilon}^\mu}$ (${\epsilon}^\mu={\epsilon}^\mu(t,\vec{x})$) induced by the spatial diffeomorphism constraint $H_a$ and the Hamiltonian constraint $H$ f&& T\^[\^]{}f:=f+{f,H()+H\_a(\^[a]{})}\_\
[A]{}\_t\^[IJ]{}&&T\^[\^]{}[A]{}\_t\^[IJ]{}:=A\_t\^[IJ]{}+\[q\]q\^[ab]{}(N\_b-\_bN)A\_a\^[IJ]{}-N\^a{A\_a\^[IJ]{},H()}\_+[L]{}\_A\_t\^[IJ]{}-A\_a\^[IJ]{}\_t\^a\
N\^a&&T\^[\^]{}N\^a:=N\^a-\[q\]q\^[ab]{}(N\_b-\_bN)+[L]{}\_N\^a+\_t\^a\
N&&T\^[\^]{}N:=N+[L]{}\_N-[L]{}\_+\_tThe enveloping algebra generated by $\{T^{{\epsilon}^\mu}\}_{{\epsilon}^\mu}$ is the Bergmann-Komar group BK(M) represented on the space of connections and tetrads. Each element of BK(M) is an infinitesimal local symmetry of the Holst action $\delta_{{\epsilon}^\mu}S=0$. And the total path-integral measure $D\mu$ is invariant (up to an overall constant) under these transformations. Thus the Bargmann-Komar group BK(M) as a collection of classical local symmetries is implemented without anomaly in the reduced phase space path-integral quantization.
Path-integral and refined algebraic quantization {#PIRAQ}
================================================
Since the dynamics of GR is completely determined by the constraints, the criterion for the correctness of a path-integral is that it should solve all the constraints of GR quantum mechanically. In the following analysis, we will discuss this point in the framework of refined algebraic quantization (RAQ).
Refined algebraic quantization
------------------------------
Consider a general constrained dynamical system with a number of the first-class constraints $C_I$, we denote the full phase space by $({\mathcal M},{\omega})$ and the canonical coordinate on it by $(p_a,q^a)$. In the procedure of canonical quantization of the constrained system, it is practically simpler to first perform the quantization on the phase space $({\mathcal M},{\omega})$, when one doesn’t have enough knowledges about the reduced phase space variables. Suppose we have done this quantization and obtained a kinematical Hilbert space ${\mathcal H}_{Kin}$, which is a space of the $L^2$-functions of configuration variables $q^a$. After that, one should represent the constraints as the operators $\hat{C}_I$ on ${\mathcal H}_{Kin}$, and impose the quantum constraint equation $\hat{C}_I\Psi=0$. In general the solution of the constraint equations doesn’t belong to the kinematical Hilbert space, but will contained by the algebraic dual ${\mathfrak{D}}^\star_{Kin}$ of a dense domain ${\mathfrak{D}}_{Kin}\subset{\mathcal H}_{Kin}$, which is supposed to be invariant under all the $\hat{C}_I$ and $\hat{C}_I^\dagger$. So what we are looking for is the state $\Psi\in{\mathfrak{D}}^\star$ such that: \[\^\_If\]:=’\_I\[f\]=0, fThe space of solutions is denoted by ${\mathfrak{D}}^\star_{Phys}$. The physical Hilbert space will be a subspace of ${\mathfrak{D}}^\star_{Phys}$. And ${\mathfrak{D}}^\star_{Phys}$ will be the algebraic dual of a dense domain ${\mathfrak{D}}_{Phys}\in{\mathcal H}_{Phys}$, which is the invariant domain of the algebra of operators corresponding to the Dirac observables. Hence we obtain a Gel’fand triple: \_[Phys]{}[H]{}\_[Phys]{}\^\_[Phys]{}
A systematic construction of the physical Hilbert space is available if we have an anti-linear rigging map: :\_[Kin]{}\_[Phys]{}\^; f(f) such that (1.) $\eta(f')[f]$ is a positive semi-definite sesquilinear form on ${\mathfrak{D}}_{Kin}$. (2.) For all the Dirac observables $\hat{O}$ on the kinematical Hilbert space, we have $\hat{O}'\eta(f)=\eta(\hat{O}f)$. If such a rigging map $\eta$ exists, we define the physical inner product by (f)|(f’)\_[Phys]{}:=(f’)\[f\], f,f’\_[Kin]{}. Then a null space ${\mathfrak{N}}\subset{\mathfrak{D}}_{Phys}^\star$ is defined by ${\left}\{\eta(f)\in{\mathfrak{D}}_{Phys}^\star\ \big|\ ||\eta(f)||_{Phys}=0\ {\right}\}$. Therefore \_[Phys]{}:=(\_[Kin]{})/And the physical Hilbert space ${\mathcal H}_{Phys}$ is defined by the completion of ${\mathfrak{D}}_{Phys}$ with respect to the physical inner product.
The above general procedure is called the Refined Algebraic Quantization (RAQ) [@RAQ], which follows the Dirac quantization procedure of a first-class constrained system, and independent from the reduced phase space quantization described in section \[reduce\].
The path-integral as a rigging map {#PIARM}
----------------------------------
The discussion in section \[reduce\] suggests a anti-linear map $\eta^{{\tau}_f,{\tau}_i}_{\omega}$ from ${\mathfrak{D}}_{Kin}$ to ${\mathfrak{D}}^\star_{Kin}$, by a choice of the dense domain ${\mathfrak{D}}_{Kin}\subset{\mathcal H}_{Kin}$, two multi-finger clock values ${\tau}_f, {\tau}_i$, and a reference vector ${\omega}\in{\mathfrak{D}}_{Kin}$: &&\^[\_f,\_i]{}\_ø: \_[Kin]{}\_[Kin]{}\^; f\^[\_f,\_i]{}\_ø(f)\
&&\^[\_f,\_i]{}\_ø(f)\[f’\]:= for all $f,f'\in{\mathfrak{D}}_{Kin}$. As it is mentioned in section \[reduce\], the path-integral definition of $\eta^{{\tau}_f,{\tau}_i}_{\omega}(f)[f']$ doesn’t depend on the choice of path $c$ in the ${\mathcal T}$-space. However, for general kinematical state $f,f'$, $\eta^{{\tau}_f,{\tau}_i}_{\omega}(f)[f']$ depends on the choice of the initial and final points of $c$, i.e. it depends on the value of ${\tau}_i=c(t_i)$ and ${\tau}_f=c(t_f)$. The reason is the following: given a specific ${\tau}_0$ and a kinematical state $f\in{\mathfrak{D}}_{Kin}$, we restrict the function $f(q^a,T^I)$ on the surface defined by $T^I-{\tau}^I_0=0$ and keep in mind that $q^a=Q^a(0)\equiv Q^a$. Then we obtain a wave-packet $f(Q^a,{\tau}_0)$ at the multi-finger time ${\tau}_0$. This wave-packet belongs to the Hilbert space ${\mathcal H}$ from reduced phase space quantization, and serves as the initial wave-packet for multi-finger time evolution. That is, we obtain a wave function $F_{{\tau}_0}(Q^a,{\tau})$ by solving the Schödinger equation for each $I$: iF\_[\_0]{}(Q\^a,)=\_I()F\_[\_0]{}(Q\^a,) with the initial condition: F\_[\_0]{}(Q\^a,\_0)=f(Q\^a,\_0) The multi-finger history of the solution $F_{{\tau}_0}(Q^a,{\tau})$ corresponds to a Heisenberg state $|F_{{\tau}_0}{\rangle}\in{\mathcal H}$ (we use the lower case letter to denote the kinematical state in ${\mathcal H}_{Kin}$, while the corresponding state in ${\mathcal H}$ is denoted by the corresponding capital letter). Note that the above map from kinematical states to the Heisenberg states in ${\mathcal H}$ is not injective for a given ${\tau}_0$, since different kinematical functions can have the same restriction on the surface $T^I-{\tau}^I_0=0$. As a result, by reversing the calculation in section \[reduce\], we see that \^[\_f,\_i]{}\_ø(f)\[f’\]= which shows the dependence on ${\tau}_f$ and ${\tau}_i$. However, for the purpose of the following analysis, we remove this dependence by integral over all possible ${\tau}_f$ and ${\tau}_i$, and define a new anti-linear map $\eta_{\omega}$, which is the candidate for a formal rigging map: &&\_ø: \_[Kin]{}\_[Kin]{}\^; f\_ø(f)\
&&\_ø(f)\[f’\]:==\
&=& where we use the fact that ${\langle}F_{{\tau}_f}|F'_{{\tau}_i}{\rangle}$ is independent of the path $c(t)$ except its initial and final points, so the integrals $\int\prod_{t=[t_i,t_f]}{\mathrm d}c(t)$ are canceled from the denominator and numerator except $\int{\mathrm d}{\tau}_f{\mathrm d}{\tau}_i$. Suppose $\eta_{\omega}(f)[f']$ is finite for all $f,f'\in{\mathfrak{D}}_{Kin}$, $\eta_{{\omega}}(f)$ then is an element in ${\mathfrak{D}}^\star_{Kin}$. If $\eta_{\omega}$ is a rigging map, the image of $\eta_{\omega}$ should be in ${\mathfrak{D}}_{Phys}^\star$, we have to first check if $\eta_{\omega}(f)$ solves all the quantum constraint equations.
We suppose all the constraints $C_I$ have been Abelianized as $C'_I=P_I+h_I(p_a,q^a,T^I)$. The corresponding operators is defined by ’\_If(q\^a,T\^I):=\[-i+h\_I(-i,q\^a,T\^I)\] f(q\^a,T\^I) Then we compute $\eta_{\omega}(f)[\hat{C}'_If']$ for any two kinematical states $f,f'\in{\mathfrak{D}}_{Kin}$ &&[D]{}p\_a(t)[D]{}q\^a(t)[D]{}P\_I(t)[D]{}T\^I(t) \_[t,I]{}(P\_I+h\_I) e\^[ i\_[t\_i]{}\^[t\_f]{}[d]{}t]{} ’\_If’(q\^a\_i,T\^I\_i)\
&=&[D]{}p\_a(t)[D]{}q\^a(t)[D]{}T\^I(t) e\^[ i\_[t\_i]{}\^[t\_f]{}[d]{}t]{} \[-i+h\_I(-i,q\_i\^a,T\_i\^I)\] f’(q\^a\_i,T\^I\_i) We now consider the collection of $T^I$ as the multi-finger time parameters, then the constraint equations ’\_IF(q\^a,T\^I):=\[-i+h\_I(-i,q\^a,T\^I)\] F(q\^a,T\^I)=0 can be consider as the multi-finger time evolution equations. Similar to the case with ${\tau}$-parameter, given a kinematical state $f$, a specific $T_0$ and the initial condition $F(q^a,T_0^I)=f(q^a,T_0^I)$, a solution $F_{T_0}(q^a,T^I)$ can be determined in principle and corresponds to a state (a Schödinger state $|F_{T_0}(T){\rangle}$ or a Heisenberg state $|F_{T_0}{\rangle}$ ) in ${\mathcal H}$. And an unitary propagator $U(T_f,T_i)$ is defined on ${\mathcal H}$ in the same way as it was in section \[reduce\]. Therefore, we have &&[D]{}p\_a(t)[D]{}q\^a(t)[D]{}P\_I(t)[D]{}T\^I(t) \_[t,I]{}(P\_I+h\_I) e\^[ i\_[t\_i]{}\^[t\_f]{}[d]{}t]{} ’\_If’(q\^a\_i,T\^I\_i)\
&=&[D]{}T\^I(t)F\_[T\_f]{}(T\_f)|U(T\_f,T\_i)\[-i+(T\_i)\]|F’\_[T\_i]{}(T\_i)\
&=&[D]{}T\^I(t)(iF\_[T\_f]{}(T\_f)|U(T\_f,T\_i)) |F’\_[T\_i]{}(T\_i)+F\_[T\_f]{}(T\_f)|U(T\_f,T\_i) (T\_i)|F’\_[T\_i]{}(T\_i)\
&=&0 by some certain boundary conditions of $f(q^a_i,T^I_i)$ and $f'(q^a_i,T^I_i)$, note that ${\mathcal D}T^I(t):=\prod_{t\in[t_i,t_f]}{\mathrm d}T^I(t)$. The above manipulation shows that \_ø(f)\[’\_If’\]=0. So the map $\eta_{\omega}$ is from ${\mathfrak{D}}_{Kin}$ to the space of solutions ${\mathfrak{D}}_{Phys}^\star$ of the quantum constraint equations.
Moreover, it is clear that $\eta_{\omega}(f)[f']$ is a positive semi-definite sesquilinear form on ${\mathfrak{D}}_{Kin}$ because by definition \_ø(f)\[f’\]== Therefore we define formally the physical inner product via the map $\eta_{\omega}$ \_ø(f’)|\_ø(f)\_[Phys]{}&:=&\_ø(f)\[f’\]\
&=&\
&&\[rigging\] On the other hand, the physical inner product Eq.(\[rigging\]) can also be equivalently expressed by the group averaging of the Abelianized constraints, i.e. if the Abelianized constraint ${C}'_I$ are represented as commutative self-adjoint operators $\hat{C}'_I$, \_ø(f)\[f’\]= The formal proof for this equivalence is shown in [@muxin]. From this equation, it is clear that for any operator $\hat{O}$ such that its adjoint $\hat{O}^\dagger$ commutating with all the constraints $\hat{C}_I'$ ’\_ø(f)\[f’\]=\_ø(f)\[\^f’\]=\_ø(f)\[f’\] Therefore $\eta_{\omega}$ satisfies all the requirements, thus is qualified as a rigging map.
The physical Hilbert space ${\mathcal H}_{Phys}$ is defined by the image of $\eta_{{\omega}}$ modulo a null space of zero-norm states, while the detailed structure of the null space and physical Hilbert space ${\mathcal H}_{Phys}$ should be studied case by case.
Before we come to the next section, it is remarkable that Eq.(\[rigging\]) removes all the gauge fixing conditions from the original path-integral formula Eq.(\[RPI3\]). It will greatly simplify the path-integral of gravity, since the gauge fixings are often hard to implement in the case of quantum gravity.
The case of gravity with scalar fields
--------------------------------------
We now apply our general consideration to the case of gravity with four massless real scalar fields $T^I$ $(I=0,\cdots,3)$, where the action for the scalar fields reads S\_[KG]{}\[T\^I,g\_\]=-\_[I=0]{}\^3[d]{}\^4x g\^\_T\^I \_T\^I\[dustaction\] where ${\alpha}$ is the coupling constraint.
One can perform the Legendre transformation of Eq.(\[dustaction\]) according to the 3+1 decomposition of the spacetime manifold $M\simeq\mathbb{R}\times{\Sigma}$. The resulting total Hamiltonian of the system is a linear combination of first class constraints $C^{tot}$ and $C^{tot}_a$: C\^[tot]{}&=&C+C\^[KG]{}, C\^[KG]{} = \[q\^[ab]{}\_aT\^I\_bT\^I+P\_I\^2\]\
C\^[tot]{}\_a&=&C\_a+C\^[KG]{}\_a, C\_a\^D = P\_I\_aT\^I\[constraint\] here $C$ and $C_a$ are respectively the standard Hamiltonian constraint and diffeomorphism constraint for gravity. The symplectic structure here is different from the standard case of gravity only by adding the scalar variables $T^I$ and their conjugate momenta $P_I$.
When the local Jacobian matrix = (
[cc]{} &\
\_aT\^0 & \_a T\^j
) is non-degenerated, the constraint Eqs.(\[constraint\]) can be locally solved into the equivalent form as Eq.(\[solve\]) \^[tot]{}&=&P+h, h = h(P\^[ab]{},q\^[ab]{},T\^I)\
\^[tot]{}\_j&=&P\_j+h\_j, h\_j = h\_j(P\^[ab]{},q\^[ab]{},T\^I).\[csolve\] Since $C^{tot}$ is quadratic to $P$, we have to restrict $P$ to take value only in half real-line in order to obtain $\tilde{C}^{tot}$. The constraint Eqs.(\[csolve\]) form a strongly Abelean constraint algebra.
Following the general rule in section \[PIARM\], we can immediately write down the formal rigging map. We first consider the canonical GR in the ADM formalism &&\_ø: \_[Kin]{}\_[Phys]{}\^; f\_ø(f), \_ø(f)\[f’\]:=\
&& \[ADMrigging\] Note that the integral of $P$ is only over a half of the real-line.
The formal rigging map Eq.(\[ADMrigging\]) can be equivalently re-expressed in terms of the connection variables, by add the SU(2) Gauss constraint $G_i=\partial_aE^a_i+\epsilon_{ij}^{\ \ k}A_a^jE^a_k$ and the corresponding gauge fixing condition $\xi_i$. The purpose for such a re-expression is to relating the kinematical framework of LQG, which is formulated mathematical-rigorously and better understood than the ADM case &&\_ø: \_[Kin]{}\_[Phys]{}\^; f\_ø(f), \_ø(f)\[f’\]:=\
&&\[LQGrigging\] where $\Delta_{FP}$ is the Faddeev-Popov determinant. In the case that all the kinematical states we considered are SU(2) gauge invariant, we can remove the gauge fixing term $\Delta_{FP}\ \delta\Big(\xi_j\Big)$ from Eq.(\[LQGrigging\]) without changing anything. The reason is shown by the standard Faddeev-Popov trick, here we briefly outline the method (see also e.g.[@Weinberg]):
Suppose a integral formula is written as Z=[D]{} f\[X\] \_[FP]{}\[X\] (\_I\[X\]) where $\xi_I$ are gauge fixing functions. We replace the variable $X$ everwhere by $X_\L$, which is an finite gauge transformation of $X$ generated by the first-class constraints. Then Z=[D]{} f\[X\_Ł\] \_[FP]{}\[X\_Ł\] (\_I\[X\_Ł\]) We are assuming that the first-class constraint algebra is a Lie algebra so that the gauge transformations form a group. Since the infinitesimal parameter $\L$ is arbitrary, $Z$ cannot dependent on $\L$. Therefore we integral over $\L$ with a certain suitable weight-function $\rho[\L]$ Z[D]{}Ł\^I =[D]{}Ł\^I [D]{} f\[X\_Ł\] \_[FP]{}\[X\_Ł\] (\_I\[X\_Ł\]) If we also assume that both the path-integral measure ${\mathcal D}\mu[X]$ and the function $f[X]$ are invariant under the gauge transformations, we will obtain Z[D]{}Ł\^I =[D]{} f\[X\] [D]{}Ł\^I \_[FP]{}\[X\_Ł\] (\_I\[X\_Ł\]) By using an explicit expression of Faddeev-Popov determinant $\Delta_{PF}$. We know that \_[PF]{}\[X\_Ł\]=(|\_[ł=0]{}) We consider the result of performing the gauge transformation with parameters $\L^I$ followed by the gauge transformation with parameters ${\lambda}^I$ as a *product* infinitesimal gauge transformation with parameters $\Theta^I(\L,{\lambda})$. Then |\_[ł=0]{}=|\_[=Ł]{}|\_[ł=0]{}= |\_[ł=0]{} Therefore if we choose the weight-function =\^[-1]{} We then have [D]{}Ł\^I \_[FP]{}\[X\_Ł\] (\_I\[X\_Ł\])=[D]{}Ł\^I () (\_I\[X\_Ł\])=1 As a result Z=\^[-1]{}[D]{} f\[X\] which means that $Z$ is expressed as a simpler integral $\int{\mathcal D}\mu[X]\ f[X]$ divided by a infinite gauge obit volume. Note that if one can define a Haar measure on the gauge group, the weight-function can be chosen as 1.
The above derivation only depends on two non-trivial assumptions: (1.) Both the measure ${\mathcal D}\mu$ and the function $f$ are gauge invariant; (2.) The gauge transformations form a Lie group. In our case of Eq.(\[LQGrigging\]), both integrals in the numerator and denominator fulfill the assumptions. Firstly, in the same way as we discussed in previous sections, the SU(2) gauge transformations are local symmetries of the action and anomaly-free in the path-integral (leave the measure invariant). And we have chosen all the kinematical states $f,f',{\omega}$ are SU(2) gauge invariant. Secondly, the SU(2) gauge transformations form a Lie group. As a result, &&\_ø: \_[Kin]{}\_[Phys]{}\^; f\_ø(f), \_ø(f)\[f’\]:=\
&&\[LQGrigging1\] where the gauge fixing terms are removed, and the overall factors of infinite gauge volumes in both numerator and denominator are canceled with each other.
Another step is to transform the delta functions and express Eq.(\[LQGrigging1\]) in terms of original constraints $C^{tot}$ and $C^{tot}_a$ &&\_ø: \_[Kin]{}\_[Phys]{}\^; f\_ø(f), \_ø(f)\[f’\]:=\
&&\[LQGrigging2\] where R=||= |(
[cc]{} [P\_0]{} & [P\_j]{}\
\_aT\^0 & \_a T\^j
)| which depends on both the variables of gravity and the variables of dust. As it was shown generally in section \[PIARM\], $\eta_{\omega}(f)$ formally solves the quantum constraint equations defined by the Abelianized constraints Eq.(\[csolve\]). And \_ø(f’)|\_ø(f)\_[Phys]{}:=\_ø(f)\[f’\] is qualified to be a physical inner product.
There is a remarkable feature of Eq.(\[LQGrigging2\]): when we derive Eq.(\[LQGrigging2\]), we restrict the kinematical states $f,f',{\omega}$ to be SU(2) gauge invariant. However, even if we apply some gauge non-invariant states to Eq.(\[LQGrigging2\]), it actually only depends on the equivalence class of the kinematical states $f,f'$, in which different states are related by SU(2) gauge transformations. We can see the reason by a change of variables in the integral, we replace the variables $A_a^i,E^a_i$ by the SU(2) gauge transformed ones $T^\L_GA_a^i,T^\L_GE^a_i$ everywhere (which doesn’t change anything), with an arbitrary time-space-dependent parameter $\L(t,\vec{x})$. If we choose the reference vector ${\omega}$ to be gauge invariant, by the fact that $SU(2)$ gauge transformations leaves the measure, delta functions and the kinetic term invariant up to field independent constants, we then obtain that &&\
&=& where $\L_i$ and $\L_f$ are relatively independent. This observation shows that for a given kinematical state $f$, $\eta_{\omega}(f)$ solves ALL the quantum constraints, including Gauss constraint, Diffeomorphism constraint and Hamiltonian constraint.
The path-integral rigging map in terms of spacetime covariant field variables
-----------------------------------------------------------------------------
It is more convenient to perform the computation with the path-integral formula in terms of original spacetime covariant field variables and the Lagrangian in a manifestly covariant form. Thus it is better for us to transform Eq.(\[LQGrigging2\]) into the path-integral in terms of original spacetime covariant field variables, so that we can evaluate the physical inner product by using the techniques of spin-foam model.
It is shown in the appendix of [@EHT] that the canonical path-integral formula from the Ashtekar-Barbero-Immirzi Hamiltonian (the one appeared in Eq.(\[LQGrigging2\])) is equivalent to the canonical path-integral for the Holst action Eq.(\[pi\]) after imposing the time-gauge. We incorporate this result to the physical inner product (we first ignore the scalar field contribution, but will add it back afterward): \_ø(f’)|\_ø(f)\_[Phys]{}&=&\
Z\_T(f,f’)&=&\_[xM]{} \_[xM]{}\
&&\_[xM]{} i\_[t\_i]{}\^[t\_f]{}[d]{}t\_[\_t]{}[d]{}\^3x f’(A\_a\^i)\_[t\_i]{}\[HP\] where the time-gauge condition $T_c:={\epsilon}^{ijk}\pi^a_{0i}\pi^{b}_{jk}{\epsilon}_{abc}=0$, $M$ is the spacetime region bounded by initial and final slices ${\Sigma}_{t_i}$ and ${\Sigma}_{t_f}$.
In order to let Eq.(\[HP\]) consistent with Eq.(\[LQGrigging\]), one need to be careful for the following reasons: Recall that the simplicity constraint $$C^{ab} = {\epsilon}^{IJKL}\pi^a_{IJ}\pi^b_{KL} \approx 0$$ has five disjoint sectors of solutions, where $\pi^a_{IJ}$ takes one of the five forms $$\begin{aligned}
(I \pm) && \pi^a_{IJ} = \pm {\epsilon}^{abc} e_b^I e_c^J \nonumber\\
(II \pm) && \pi^a_{IJ} = \pm \frac{1}{2} {\epsilon}^{abc} e_b^K e_c^L{\epsilon}_{IJKL}\nonumber\\
(Deg) && \pi^a_{IJ} = 0.\end{aligned}$$ If we define $\pi^a_i := \frac{1}{2} \pi^a_{0i}$ and $\tilde{\pi}^a_i:= \frac{1}{4} \epsilon_i{}^{jk} \pi^a_{jk}$. Then we have the following different properties for different sectors $$\begin{aligned}
(I+) &\Rightarrow& \det \pi^a_i = 0\text{ and }(\det \tilde{\pi}^a_i)(\det e^i_a) > 0, \nonumber\\
(I-) &\Rightarrow& \det \pi^a_i = 0\text{ and }(\det \tilde{\pi}^a_i)(\det e^i_a) < 0, \nonumber\\
(II+) &\Rightarrow& \det \tilde{\pi}^a_i = 0\text{ and }(\det \pi^a_i)(\det e^i_a) > 0, \nonumber\\
(II-) &\Rightarrow& \det \tilde{\pi}^a_i = 0\text{ and }(\det \pi^a_i)(\det e^i_a) < 0 .\end{aligned}$$ where we can also see that the four sectors $(I \pm)$ and $(II \pm)$ are disjoint. Because of the appearance of these five sectors of solutions, we have to clarify which sector is contained in the integral of Eq.(\[HP\]) in order to be consistent with Eq.(\[LQGrigging\]). It turns out in the appendix of [@EHT] that the integral of Eq.(\[HP\]) only contain two sectors, either $(I\pm)$ or $(II\pm)$. Thus we restrict ourselves in the sectors $(II\pm)$.
The next step is to perform the Henneaux-Slavnov trick [@BHNR] to eliminate the secondary second-class constraint $D^{ab}$ in the path-integral formula. We express the delta functions $\delta(H)$ and $\delta(D^{ab})$ by their Fourier decompositions, Z\_T(f,f’)&=&\_[II]{} \_[xM]{} \_[xM]{}\
&&\_[xM]{} i\_[t\_i]{}\^[t\_f]{}[d]{}t\_[\_t]{}[d]{}\^3x f’(A\_a\^i)\_[t\_i]{} Then We consider a change of variables which is also a canonical transformation for the canonical fields on different spatial slices ${\Sigma}_t$. It is generated by the functional F(t)&:=&-\_[\_t]{}[d]{}\^3x \_[ab]{}(t,) C\^[ab]{}/N(t,)\
\_[ab]{}(t,) &=&[d]{}\_[ab]{}(t,) t\
&=&0 t={t\_i,t\_f} so this change of variable doesn’t affect the kinematical state $f,f'$ on the boundary. The integral measure $\left[{\mathrm d}A_a^{IJ}(x)\ {\mathrm d}\pi_{IJ}^a(x)\right]$ is the Liouville measure on the phase space thus is invariant under this canonical transformation, $\sqrt{|\det\Delta_D(x)|}$ is also invariant under the canonical transformation generated by $F$ since $\left\{C^{ab}(x), G^{cd,ef}(x',x'')\right\}=0$, and the product of $\delta$-functions for both time-gauge and first-class constraints are invariant under this canonical transformation because {G\_[IJ]{}(Ł\^[IJ]{}), C\^[ab]{}(c\_[ab]{})}&=&0\
{C\^[ab]{}(c\_[ab]{}),C\^[cd]{}(d\_[cd]{})}&=&0\
{H\_a(N\^a),C\^[bc]{}(c\_[bc]{})}&=&C\^[ab]{}([L]{}\_c\_[ab]{}) Under the canonical transformation generated by $F$, the change of kinetic term $\delta\int{\mathrm d}t\int{\mathrm d}^3x\ \pi_{IJ}^a\partial_t\left(A_a^{IJ}-\frac{1}{\gamma}*A_a^{IJ}\right)$ is proportional to $\int{\mathrm d}t\int{\mathrm d}^3x\ C^{ab}\partial_t(d_{ab}/N)$ which also vanishes by the delta functions $\delta(C^{ab})$ in front of the exponential. So $H$ and $D^{ab}$ are the only terms variant in this canonical transformation. Moreover because $\left\{H(x),C^{ab}(x')\right\}=D^{ab}(x)\delta(x,x')$ modulo the terms proportional to $C^{ab}$ and $\left\{C^{ab}(x),D^{cd}(x')\right\}=G^{ab,cd}(x,x')$ we can obtain explicitly the transformation behavior of $H(N)$ and $D^{cd}(d_{cd})$ in the time period $[t_i+{\epsilon},t_f-{\epsilon}]$ modulo the terms vanishing on the constraint surface defined by $C^{ab}=0$ e\^[\_[-F]{}]{}[H]{}(N)&&\_[n=0]{}\^{F,H(N)}\_[(n)]{} = [d]{}\^3x N(x)H(x)+[d]{}\^3x d\_[ab]{}(x)D\^[ab]{}(x)-[d]{}\^3y[d]{}\^3zd\_[ab]{}(y)d\_[cd]{}(z)G\^[ab,cd]{}(y,z)\
e\^[\_[-F]{}]{}[D]{}\^[cd]{}(d\_[cd]{})&&\_[n=0]{}\^{F,D\^[cd]{}(d\_[cd]{})}\_[(n)]{} = [d]{}\^3x d\_[cd]{}(x)D\^[cd]{}(x)-[d]{}\^3x[d]{}\^3y d\_[cd]{}(x)d\_[ab]{}(y)G\^[ab,cd]{}(x,y) here $\chi_{-F}$ is the Hamiltonian vector field associated by the phase space function $-F$, and the series terminated because of {C\^[ab]{}(x), G\^[cd,ef]{}(x’,x”)}=0. When we take ${\epsilon}\to0$, the integral on the exponential becomes [d]{}t[d]{}\^3xThen we perform the integral over $d_{ab}$, we obtain &&Z\_T(f,f’)\
&=&\_[II]{} \_[xM]{}\
&& f’(A\_a\^i)\_[t\_i]{}\[HP1\] We include the scalar degrees of freedom, since the scalar field contributions all commute with $F(t)$, we get &&Z\_T(f,f’)\
&=&\_[II]{} \[[D]{}P\_I[D]{}T\^I\]\_[xM]{} R\[P\_I,T\^I,q\^[ab]{}\]\
&&i\_[t\_i]{}\^[t\_f]{}[d]{}t\_[\_t]{}[d]{}\^3x\
&&f’(A\_a\^i,T\^I)\_[t\_i]{}\[HP2\] We then consider the scalar field contributions of the total action on the exponential: S\_[KG]{}&=&\_[t\_i]{}\^[t\_f]{}[d]{}t\_[\_t]{}[d]{}\^3x\
&=&\_[t\_i]{}\^[t\_f]{}[d]{}t\_[\_t]{}[d]{}\^3x\
&=&\_[t\_i]{}\^[t\_f]{}[d]{}t\_[\_t]{}[d]{}\^3x\
&=&-\_[t\_i]{}\^[t\_f]{}[d]{}t\_[\_t]{}[d]{}\^3x\^2-[d]{}\^4x g\^\_T\^I \_T\^I where the second term is the original covariant action of the scalar fields, and the first term contributes the integral of $P_I$. Let’s extract the integral of $P_I$ out of the path-itnegral formula: &&[D]{}P\_I R e\^[-i\_[t\_i]{}\^[t\_f]{}[d]{}t\_[\_t]{}[d]{}\^3x\[P\_I-( n\^[\_]{}T\^I)\]\^2]{}\
&=&[D]{}P\_I |(
[cc]{} [P\_0]{} & [P\_j]{}\
\_aT\^0 & \_a T\^j
)| e\^[-i\_[t\_i]{}\^[t\_f]{}[d]{}t\_[\_t]{}[d]{}\^3x\[P\_I-( n\^[\_]{}T\^I)\]\^2]{}\
&=&()\^[5/2]{}[D]{}X\_I |(
[cc]{} [X\_0+ n\^[\_]{}T\^0]{} & [X\_j+ n\^[\_]{}T\^j]{}\
\_aT\^0 & \_a T\^j
)| e\^[-i\_[t\_i]{}\^[t\_f]{}[d]{}t\_[\_t]{}[d]{}\^3x X\_I\^2]{}\
&&2\^[5/2]{}\^[3/2]{}V\_s\^[3/2]{}N\^[-5/2]{}[J]{}\_[KG]{}\[,T\^I,q\^[ab]{}\]\[JKG\]
After the above manipulation for the scalar field degrees of freedom, we follow the same way in [@EHT] to solve the simplicity constraint $C^{ab}$, and obtain a path-integral of the Holst action coupling to the scalar fields Z\_T(f,f’)&=&\_[II]{} [D]{}A\_\^[IJ]{} [D]{}e\^I\_[D]{}T\^I \_[xM]{}|[V]{}\^[1/2]{} V\_s\^6| [J]{}\_[KG]{}\[,T\^I,q\^[ab]{}\] \^3(e\^0\_a) \[\_M e\^Ie\^J(\* F\_[IJ]{}-F\_[IJ]{})\]\
&&\[-\_M[d]{}\^4x g\^\_T\^I \_T\^I\]f’(A\_a\^i,T\^I)\_[t\_i]{}\[HSSM\] It is the path-integral representation of the physical inner product for GR coupling to the scalar fields. However, the Jacobian ${\mathcal J}_{KG}$ involves integral expression Eq.(\[JKG\]) thus is hard to practically compute. But we can approximate this formula Eq.(\[HSSM\]) by the pure gravity contribution, if we assume (1.) $f,f',{\omega}$ depend on $A_a^i$ only; (2.) the coupling constant ${\alpha}$ is small and negligible. The reason is that if ${\alpha}$ is negligible, ${\mathcal J}_{KG}$ then becomes independent of $q^{ab}$, then the integrals of scalar field variables are factored out from Eq.(\[HSSM\]) and canceled between the denominator and the numerator of the physical inner product: \_ø(f’)|\_ø(f)\_[Phys]{}&=& Under this approximation, we write down the path-integral representation of the physical inner product Z\_T(f,f’)=\_[II]{} [D]{}A\_\^[IJ]{} [D]{}e\^I\_\_[xM]{}|[V]{}\^[1/2]{} V\_s\^6| \^3(e\^0\_a) \[\_M e\^Ie\^J(\* F\_[IJ]{}-F\_[IJ]{})\] f’(A\_a\^i)\_[t\_i]{}\[HSM\] Moreover, Eq.(\[HSM\]) is equivalent to the path-integral of a Plebanski-Holst action [@EHT] Z\_T(f,f’)&=&\_[II]{}[D]{}A\_\^[IJ]{}[D]{}B\_\^[IJ]{} \_[xM]{} |[V]{}\^[13/2]{} V\^[9]{}\_s| \^[20]{}(\_[IJKL]{} B\_\^[IJ]{} B\_\^[KL]{}-[V]{}\_) \^3(T\_c)\
&&\[i\_M B\^[IJ]{}(F-\* F)\] f’(A\_a\^i)\_[t\_i]{}.\[PSM\]
Conclusion and discussion
=========================
The aim of the present paper has been to analyze the gauge invariance of the path-integral measure consistent with the canonical settings, also to give a path-integral formula appropriate for both conceptual and practical purposes. Our previous discussions show that the path-integral measure we obtained is invariant under all the gauge transformations (local symmetries) of GR generated by the first-class constraints. These gauge transformations form the Bergmann-Komar “group" (enveloping algebra) which is the collection of the dynamical symmetries. We also obtain the desired path-integral formula which formally solves all the constraints of GR quantum mechanically. And the pure gravity part of physical inner product is formally represented by these path-integral formula Eqs.(\[HSM\]) or (\[PSM\]), which is ready for the spin-foam model construction.
There are several remarks concerning our result Eqs.(\[HSM\]) and (\[PSM\]):
- In Eqs.(\[HSM\]) and (\[PSM\]) most of the gauge fixing conditions, which often appear in the quantization of gauge system, disappear in our case. Since it is hard to implement gauge fixing conditions in the spin-foam quantization, the Eqs.(\[HSM\]) and (\[PSM\]) simplify the further construction and computation at this point. However, the appearance of time-gauge will have non-trivial contribution to the construction of a spin-foam model.
- Our construction formally explains that if we are working toward formally a rigging map and physical inner product consistent with the Ashtekar-Barbero-Immirzi canonical formulation [@ABI], we have to include both two sectors $(II\pm)$ in constructing the spin-foam models from Plebanski path-integral. It simplifies the construction because in constructing a spin-foam model, we would like to first remove the delta functions of the simplicity constraint $
{\epsilon}_{IJKL}\ B_{{\alpha}{\beta}}^{IJ}\ B_{{\gamma}\delta}^{KL}=\frac{1}{4!}{\mathcal V}{\epsilon}_{{\alpha}{\beta}{\gamma}\delta}
$ and consider a pure BF-theory, and implement the simplicity constraint afterwards. In considering BF theory, we need the integral for each component of $B_{{\alpha}{\beta}}^{IJ}$ to be over the full real-line in order to obtain a product of $\delta{\left}(F{\right})$ (more precisely the delta functions of holonomies). However if it was restricted to only one single sector $(II+)$ or $(II-)$, the integral for each component of $B_{{\alpha}{\beta}}^{IJ}$ would be only over a half real-line.
- The local measure factors (${\mathcal V}^{1/2} V_s^6$ for Holst and ${\mathcal V}^{13/2} V^{9}_s$ for Plebanski-Holst) appear in our result Eqs.(\[HSM\]) and (\[PSM\]). It means that in order to interpret the path-integral amplitude (or spin-foam amplitude) as a physical inner product in the canonical theory, one has to properly implement these local measure factors into the spin-foam model. The detailed implementation and construction will appear in the future publication.
- In order to construct the physical Hilbert space ${\mathcal H}_{Phys}:=\eta_{{\omega}}({\mathfrak{D}}_{Kin})/{\mathfrak{N}}$, it is needed to clarify the null space ={\_ø(f)\_[Phys]{}\^ | ||\_ø(f)||\_[Phys]{}=0 } implied by the physical inner product defined by Eqs.(\[HSM\]) or (\[PSM\]). Therefore the further research is necessary regarding the implication from the path-integral formula Eqs.(\[HSM\]) or (\[PSM\]).
Acknowledgments {#acknowledgments .unnumbered}
===============
M.H. is grateful for the advises from Thomas Thiemann and the fruitful discussion with Aristide Baratin, Bianca Dittrich, and Jonathan Engle. M.H. also would like to gratefully acknowledge the support by International Max Planck Research School and the partial support by NSFC Nos. 10675019 and 10975017.
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[^1]: [ [email protected]]{}
[^2]: In a path-integral formula $\int{\mathcal D}\mu\ e^{iS}$, some author use the term “a path-integral measure” referring to ${\mathcal D}\mu$, but some others use it referring to ${\mathcal D}\mu\ e^{iS}$. We are following the first convention. But there is no different between these two convention when we are consider the invariance of the path-integral measure under the symmetries of the action $S$.
[^3]: A regular constrained system means that its Dirac matrix has a constant rank on the phase space.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Robert P. Kirshner'
title: Foundations of supernova cosmology
---
Foundations of supernova cosmology
==================================
Supernovae and the discovery of the expanding universe
------------------------------------------------------
Supernovae have been firmly woven into the fabric of cosmology from the very beginning of modern understanding of the expanding, and now accelerating universe. Today’s evidence for cosmic acceleration is just the perfection of a long quest that goes right back to the foundations of cosmology. In the legendary Curtis-Shapley debate on the nature of the nebulae, the bright novae that had been observed in nebulae suggested to Shapley (1921) (see Trimble, 1995) that the systems containing them must be nearby. Otherwise, he reasoned, they would have unheard-of luminosities, corresponding to M = -16 or more. Curtis (1921) countered concluding that “the dispersion of the novae in spirals and in our galaxy may reach ten magnitudes...a division into two classes is not impossible.” Curtis missed the opportunity to name the supernovae, but he saw that they must exist if the galaxies are distant. Once the distances to the nearby galaxies were firmly established by the observation of Cepheid variables (Hubble, 1925), the separation of ordinary novae and their extraordinary, and much more luminous super cousins, became clear.
A physical explanation for the supernovae was attempted by Baade and Zwicky (1934). Their speculation that supernova energy comes from the collapse to a neutron star is often cited, and it is a prescient suggestion for the fate of massive stars, but not the correct explanation for the supernovae that Zwicky and Baade studied systematically in the 1930s. In fact, the spectra of all the supernovae that they discovered and followed up in those early investigations were of the distinct, but spectroscopically mysterious, hydrogen-free type that today we call SN Ia. They are not powered by core collapse, but by a thermonuclear flame. Baade (1938) showed that the luminosities of the supernovae in their program were more uniform than those of galactic novae, with a dispersion of their peak luminosities near 1.1 mag, making them suitable as extragalactic distance indicators. Right from the beginning, supernovae were thought of as tools for measuring the universe.
Nature has more than one way to explode a star. This was revealed clearly by Minkowski (1941) who observed a distinct spectrum for some supernovae, different from those obtained for the objects studied by Baade. SN 1940B had strong hydrogen lines in its spectrum. These are the stars whose energy source we now attribute to core collapse in massive stars. At the time, it seemed sensible to call Baade’s original group Type I (SN I) and the new class Type II (SN II). The small dispersion in luminosity for Baade’s sample resulted from his good luck in having Zwicky discover a string of supernovae that were all of a single type. SN I are generally less luminous than the galaxies in which they occur. (Introductory texts, and introductory remarks in colloquia concerning supernovae usually get this basic fact wrong.) The SN II are, generally speaking, fainter than SN I and have a larger dispersion in their luminosity. Separating the supernovae, on the basis of their spectra, into distinct physical classes is one way they have become more precise as distance indicators. By the late sixties, Kowal (1968) was able to make a Hubble diagram for 19 SN I. The scatter about the Hubble line for this sample, which reached out to the Coma Cluster of galaxies at a redshift of 7000 km/s was about 0.6 magnitudes. These were photographic magnitudes, obtained with the non-linear detectors of the time, and they contained no correction for absorption by dust in the host galaxies, which we now know is an important source of scatter in the observed samples. But this was a promising step forward.
{width="85.00000%"}
\[\]
In 1968, there was plenty of room for improvement in the precision of SN I measurements and in extending the redshift range over which they were studied. As Kowal forecast: “These supernovae could be exceedingly useful indicators of distance. It should be possible to obtain average supernova magnitudes to an accuracy of 5% to 10% in the distances.” He also predicted the future use of supernovae to determine cosmic acceleration: “It may even be possible to determine the second-order term in the redshift-magnitude relation when light curves become available for very distant supernovae.” The “second-order term” would be the one that indicated cosmic acceleration or deceleration. Along with the Hubble constant (which would require reliable distances from Cepheid variables), this deceleration term was expected to provide an account of cosmic kinematics, and, in the context of General Relativity, for the dynamics of the Universe, as sketched for astronomers in the classic paper by Sandage (1961).
On the last page of this paper, Sandage worked out the observational consequences of the exponential expansion that would be produced by a cosmological constant. He explicitly shows that you cannot decide between an accelerating universe of this type and the steady-state model (they would both have q$_{0}$ = -1). Yet, in 1968, the measurement of deceleration was presented by Sandage (1968) as a decisive test between the steady-state model, which predicted acceleration, and Friedmann cosmologies where matter would produce deceleration. It is possible that, if cosmic acceleration had been discovered earlier, it might have been taken as evidence in favor of the steady-state model. It was the richer physical context of cosmological information, such as the cosmic microwave background, that led to a much different conclusion in 1998.
### Classifying supernovae
In 1968, there was ample room for technical improvement in the measurements themselves, a need for a proper account for the effects of dust, and just as important, well into the 1980s the classification scheme for SN I was still incomplete. Core-collaspe supernovae were mixed in among the thermonuclear explosions that make up most of the Type I supernovae. As described by Zwicky (1965) and later by Oke and Searle (1974) the definition of a SN I was empirical: it meant that the spectrum resembled the bright supernova SN 1937C as extensively studied by Minkowski (1939). The bright supernova SN 1972E, observed with a new generation of spectrophotometric instruments by Kirshner et al. (1973a) in the infrared (Kirshner et al. 1973b) and at late times (Kirshner et al. 1975) provided a rich template for redefining the spectra of Type Ia supernovae. The distinctive feature in Type I supernova spectra is a broad and deep absorption observed at about 6150 Angstroms, attributed by Pskovskii (1968) to absorption by Si II. However, there were a handful of SN I, usually dubbed “peculiar” SN I, whose spectra resembled the other SN I in other respects, but which lacked this distinctive absorption line at maximum light. We now understand that this is not just a minor detail: the SN Ib (and their more extreme cousins, the SN Ic) are completely different physical events, ascribed to core-collapse in massive stars that have lost their hydrogen envelopes in late stages of stellar evolution (Branch and Doggett, 1985; Uomoto and Kirshner, 1985; Wheeler and Levreault, 1985; Wheeler and Harkness, 1990; Filippenko, 1997). The notation SN Ia was introduced to refer to the original class of supernovae, like SN 1937C and SN 1972E, that has no hydrogen or helium lines in the spectrum and the strong Si II feature.
Once the SN Ib were distinguished from the SN Ia, the homogeneity of the SN Ia improved, with the scatter about the Hubble line decreasing to 0.65-0.36 mag, depending on which objects were selected and which photometric bands were used (Tammann and Leibundgut, 1990; Branch and Miller, 1990, 1993; Della Valle and Panagia, 1992). This work rested on the assumption that the SN Ia were identical, so that a single underlying template for the light curve (Leibundgut, 1988) could be used to interpolate between the observations of any individual object to determine its apparent brightness at maximum light in the B-band, and put all the objects on a common scale.
#### SN II as cosmological distance indicators
The idea that supernovae could be used to measure cosmological parameters had more than one component. Another line of work employed Type II supernovae. As pointed out by Kirshner and Kwan (1974), the expanding photospheres of these hydrogen-rich supernovae provide the possibility to measure distances without reference to any other astronomically determined distance. The idea of the Expanding Photosphere Method (EPM) is that the atmosphere was not too far from a blackbody, so the temperature could be determined from the observed energy distribution. If you measure the flux and temperature, that determines the angular size of the photosphere. Since you can measure the temperature and flux many times during the first weeks after the explosion, an observer can establish the angular expansion rate of a supernova. At the same time, absorption lines formed in the expanding atmosphere, from hydrogen and from weaker lines that more closely trace the expansion of the photosphere, give the expansion velocity. If you know the angular rate of expansion from the temperature and flux and the linear rate from the shape of the absorption lines, you can solve for the distance to a Type II supernova. The combination of the supernova’s redshift and distance allows for a measurement of the Hubble constant that does not depend on any other astronomically-determined distance. The departure of the energy distribution for a supernova atmosphere from a blackbody could be computed, as done by Schmidt, Eastman, and Kirshner (1992), and this held out the prospect of making more precise distance measurements to SN II than had been achieved for SN Ia.
Wagoner (1977) noted that this approach could be extended to high redshift to measure the effects of cosmic deceleration, and also pointed out that the EPM provided an internal test of its own validity: if the distance determined remained the same, while the temperature and the velocity of the atmosphere changed, this was a powerful sign that the measurement was consistent. This was an important point, since the prospects for using galaxies as the principal tracer of cosmic expansion were dimming, due to evidence that the luminosity of a galaxy could easily change over time due to stellar evolution and galaxy mergers. Even sign of this change was not known for certain. Galaxies might grow brighter over time due to mergers, and they might grow dimmer due to stellar evolution. In either case, unless the effect was carefully calibrated, it could easily swamp the small changes in apparent magnitude with redshift that hold the information on the history of cosmic expansion. Supernovae, though fainter than galaxies, were discrete events that would not have the same set of changes over cosmic time. The use of SN II for cosmology has recently been revived and it promises to provide an independent path to measuring expansion and perhaps even acceleration (Poznanski et al., 2008).
For SN II, the expanding photospheres provide a route to distances that can accommodate a range of intrinsic luminosities and still provide accurate distances, because the atmospheres have hydrogen and behave like those of other stars. For SN Ia, the atmospheres are more difficult to analyze, but the hope was simpler: that the physics underlying the explosion of a SN Ia would determine its luminosity. The idea that SN Ia were identical explosions has a theoretical underpinning. In the earliest pictures, the SN Ia were imagined to come from the ignition of a carbon-oxygen white dwarf at the Chandrasekhar mass (Hoyle and Fowler, 1960; Colgate and McKee, 1969). In models of this type, a supersonic shock wave travels through the star, burning it thoroughly into iron-peak isotopes, especially Ni$^{56}$. Such a standard explosion of a uniform mass would lead to a homogenous light curve and uniform luminosity, making SN Ia into perfect standard candles. The exponential light curves that suggest an energy input from radioactivity and the late-time spectra of SN Ia, which are made up of blended iron emission lines were broadly consistent with this picture. Though the simple theoretical idea that SN Ia are white dwarfs that ignite near the Chandrasekhar mass has been repeated many times as evidence that SN Ia must be perfect standard candles, nature disagrees. Observations show that there is a factor of three range in luminosity from the most luminous SN Ia (resembling SN 1991T) to the least luminous (resembling SN 1991-bg). Despite the facts, many popular (and professional!) accounts of SN Ia assert that SN Ia are standard candles because they explode when they reach the Chandrasekhar limit. This is wishful thinking.
#### Searching for SN Ia for cosmology
Nevertheless, the hope that SN Ia might prove to be good standard candles began to replace the idea that brightest cluster galaxies were the standard candles best suited to measuring the deceleration of the universe. As a coda to his pioneering automated supernova search, Stirling Colgate imagined the way in which a similar search with the Hubble Space Telescope might find distant supernovae (Colgate, 1979). A more sober analysis of the problem by Gustav Tammann estimated the sample size that would be needed to make a significant detection of deceleration using HST (Tammann, 1979). The result was encouraging: depending on the dispersion of the SN Ia, he found between 6 and 25 SN Ia at z$ \sim $0.5 would be needed to give a 3$\sigma$ signal of cosmic deceleration. Tammann got the quantities right– it was only the sign of the effect that was wrong.
Unwilling to wait for the advent of the Hubble Space Telescope, a pioneering group from Denmark began a program of supernova observations using the Danish 1.5 meter telescope at ESO (Hansen, Jorgensen, and Norgaard-Nielsen, 1987; Hansen et al., 1989). Their goal was to find distant supernovae, measure their apparent magnitudes and redshifts, and, on the assumption the SN Ia were standard candles, fit for q$_{0}$ from the Hubble diagram. This method is described with precision in the chapter in this book by Pilar Ruiz-Lapuente. The difference between q$_{0}$ = 0.1 and q$_{0}$ of 0.5 is only 0.13 mag at redshift of 0.3. At the time they began their work, there was hope that the intrinsic scatter for SN Ia might be as small as 0.3 mag. To beat the errors down by root-N statistics to make a 3 sigma distinction would take dozens of well-observed supernovae at z $\sim$ 0.3.
The Danish group used the search rhythm developed over the decades by Zwicky and his collaborators for finding supernovae. Since the time for a Type Ia supernova to rise to maximum and fall back by a factor of 2 is roughly one month, monthly observations in the dark of the moon are the best way to maximize discoveries. Observations made toward the beginning of each dark run were most useful, since that allowed time to follow up each discovery with spectroscopy and photometry. This is the pattern Zwicky established with the Palomar 18-inch Schmidt and which was used for many years by Sargent and Kowal with the 48-inch Schmidt at Palomar (Kowal, Sargent, and Zwicky, 1970). It is the pattern used by the Danish group, and all the subsequent supernova search teams until the introduction of dedicated searches like that of Kare et al. (1988) and the rolling search led by John Tonry (Barris et al., 2004) that became the model for the recent ESSENCE and SNLS searches.
But there was something new in the Danish search. Photographic plates, which are large but non-linear in their response to light, were replaced by a Charge Coupled Device (CCD). The advantages were that the CCD was much more sensitive to light (by a factor of $\sim$ 100!) and that the digital images were both linear and immediately available for manipulation in a computer. Fresh data taken at the telescope could be processed in real time to search for new stars, presumably supernovae, in the images of galaxy clusters. The new image needed to be registered to a reference image taken earlier, the two images appropriately scaled to take account of variations in sky brightness, the better of the two images blurred to match the seeing of the inferior image, and then subtracted. The Danish team implemented these algorithms and demonstrated their success with SN 1998U, a SN Ia in a galaxy at redshift 0.31 (Norgaard-Nielsen et al., 1989). Although this group developed the methods for finding distant supernovae in digital data, the rate at which they were able to find supernovae was disappointingly low. Instead of making steady progress toward a cosmologically-significant sample at a rate of, say, one object per month, they only found one supernova per year. At this rate, it would take 10 years to beat down the measuring uncertainty and to begin to learn about the contents of the universe. And that was in the optimistic case where the intrinsic scatter of SN Ia was assumed to be small. Instead, the observational evidence was pointing in the opposite direction, of larger dispersion among the SN Ia. Another early effort, carried out by the Lawrence Berkeley Laboratory at the 4m Anglo-Australian Telescope had even less luck. Despite building a special-purpose prime focus CCD camera to find supernovae, they reported none (Couch et al., 1991).
### SN Ia as standard candles– not!
Starting in 1986, careful observations made with CCD detectors showed ever more clearly that the luminosity and the light curve shapes for SN Ia were not uniform (Phillips et al., 1987). In 1991, two supernovae at opposite extremes of the luminosity scale showed for certain that this variety was real, and needed to be dealt with in order to make SN Ia into effective distance measuring tools. SN 1991bg (Leibundgut et al., 1993; Filippenko et al., 1992) was extremely faint and SN 1991T (Phillips et al., 1992) was extremely bright. Despite hope for a different result, and a theoretical argument why their luminosities should lie in a narrow range, Type Ia supernovae simply are not standard candles: they are known to vary over a factor of three in their intrinsic luminosity. The size of the sample needed to make a cosmological measurement scales as the square of the scatter, so, in 1991, the truly productive thing to harness supernovae for cosmology was not to find more distant supernovae, but to learn better how to reduce the uncertainty in the distance for each object.
Using a set of well-sampled SN Ia light curves with precise optical photometry and accurate relative distances, Phillips (1993) demonstrated a correlation between the shape of a SN Ia light curve and the supernova’s luminosity. Supernovae with the steepest declines are the least luminous. More interestingly, even among the supernovae that do not lie at the extremes of the distribution marked by SN 1991T and SN 1991bg, the relation between luminosity and light curve shape provides an effective way to decrease the scatter in the Hubble diagram for SN Ia. Phillips used this correlation to decrease the observed scatter about the Hubble line to about 0.3 mag.
This made the path forward a little clearer. What was needed was a well-run supernova search for relatively nearby supernovae that could guarantee accurate follow-up observations. Mark Phillips, Mario Hamuy, Nick Suntzeff and their colleagues at Cerro Tololo Inter-American Observatory and at the University of Chile’s Cerro Calán observatory worked together to conduct such a search, the Calán-Tololo Supernova Search (Hamuy et al., 1993). The technology was a hybrid of the past and the future– photographic plates were used on the venerable Curtis Schmidt telescope (named in honor of Heber D. Curtis, of the debate cited earlier) at Cerro Tololo to search a wide field (25 square degrees) in each exposure. Despite the drawbacks of photographic plates as detectors, this large field of view made this the most effective search for nearby supernovae. The plates were developed on the mountain, shipped by bus to Santiago, and then painstakingly scanned by eye with a blink comparator to find the variable objects. The modern part was the follow-up. Since the search area was large enough to guarantee that there would be objects found each month, CTIO scheduled time in advance on the appropriate telescopes for thorough photometric and spectroscopic follow-up with CCD detectors. The steady weather at Cerro Tololo and the dedicated work at Cerro Calán led to a stream of supernova discoveries and a rich collection of excellent supernova light curves. For example, in 1996, the Calán-Tololo group published light curves of 29 supernovae obtained on 302 nights in 4 colors (Hamuy et al., 1996a). This is what was needed to develop reliable ways to use the supernova light curves to determine the intrinsic luminosity of SN Ia, and to measure the luminosity distance to each object (Hamuy et al., 1996b). The Calán-Tololo Search was restricted to redshifts below 0.1, so it did not, by itself, contain information on the cosmology. However, it provided the data needed to understand how to measure distances with supernovae, and, when used in combination with high-z supernovae, it had the potential to help determine the cosmology.
### Dust or cosmology?
However, the accuracy of the distance measurements was compromised by the uncertain amount of dust absorption in the each supernova host galaxy. Two parallel approaches were developed. One, led by Mark Phillips and his colleagues, used the observational coincidence, first noted by Paulina Lira, that the evolution in the color B-V had a very small dispersion at ages from 30 to 90 days after maximum (Phillips et al., 1999). By measuring the observed color at those times, the absorption could be inferred and the true distance measured. The other, based on the same data set, and then later extended through observations at the Whipple Observatory of the Center for Astrophysics, used an empirical method to find that intrinsically faint supernovae are also intrinsically redder. Since the light curve shape, which was the strongest clue to supernova luminosity, was not greatly affected by absorption, it was possible to determine both the distance and the absorption by dust to each supernova. A formal treatment of the extinction using Bayes’ theorem was used to determine the best values and their uncertainty (Riess, Press, and Kirshner, 1996a). This MLCS (Multi-color Light Curve Shape) approach was also used to examine whether the dust in other galaxies was the same as dust in the Milky Way (Riess, Press, and Kirshner, 1996b). While the early indications were that the dust in other galaxies had optical properties that were consistent with those found in the Galaxy, as the samples of supernovae have grown larger and the precision of the measurements has improved, this simple picture is no longer tenable. These early workers recognized that measuring the extinction to individual supernovae was an essential step in deriving reliable information on the cosmology. After all, the dimming due to an accelerating cosmology at redshift 0.5 is only of order 0.2 magnitudes. If instead this dimming were produced by dust like the dust of the Milky Way, the additional reddening would be only 0.07 mag in the B-V color, so good photometry in multiple bands was essential to make reliable inferences on the presence or absence of cosmic acceleration.
### Early results
The earliest observations of the Supernova Cosmology Project (SCP) did not take account of these requirements. Their observations of SN 1992bi at z= 0.458 were made in only one filter, making it impossible, even in principle, to determine the reddening (Perlmutter et al., 1995). No spectrum for this object was obtained, but it was completely consistent with being a SN Ia. This was a striking demonstration that the search techniques used by the SCP, which resembled those of the Danish team, could reliably detect transient events in galaxies at the redshifts needed to make a cosmologically interesting measurement. The search was carried out with a 2048 by 2048 pixel CCD camera at the 2.5 m Isaac Newton Telescope, whose increased speed over the Danish system made it plausible that a supernova could be found in each month’s observing. As with the Calán-Tololo search being carried out at low redshift, it was reasonable for the SCP to schedule follow-up observations. The SCP developed the “stretch” method for accounting for the connection between luminosity and light curve shape in the B and V bands. This works very well, but does not, by itself, account for the effects of dust extinction (Goldhaber et al., 2001).
The High-Z Supernova Team (HZT) was formed in 1995 by cooperation between members of the Calán-Tololo group and supernova workers at the Harvard-Smithsonian Center for Astrophysics and ESO. The goal was to apply the new methods for determining the intrinsic luminosity and reddening of a supernova, developed from the low-redshift samples, to objects at cosmologically interesting distances. This required mastering the techniques of digital image subtraction. The first object found by the High-Z Team was SN 1995K, at a redshift of 0.479, which, at that time was the highest yet published (Leibundgut et al., 1996). Observations were obtained in two colors, and the supernova’s spectrum showed it was a genuine Type Ia. Leibundgut et al. used the observations to show that the light curve for SN 1995K was stretched in time by a factor of (1 + z), just as expected in an expanding universe.
The time-dilation effect had been discussed in 1939 by Olin Wilson (1939), sought in nearby data by Rust (1974), and by Leibundgut (1990). Publications by Goldhaber and his colleagues of the SCP (Goldhaber et al., 1996, 2001) show this effect in their data, though the degeneracy between the light curve shape as analyzed by the “stretch” method and time dilation requires some (quite plausible) constraints on changes in the supernova population with redshift to draw a firm conclusion. Another approach to the same problem uses the evolution of the spectra of SN Ia to show in an independent way that the clocks governing distant supernovae appear to run slower by the factor (1 + z) (Foley et al., 2005; Blondin et al., 2008).
In the mid-1990s, important technical developments improved the ability to discover distant supernovae. At the National Optical Astronomy Observatories, new 2K x 2K CCD systems were implemented at the 4-meter telescopes at Kitt Peak and at Cerro Tololo. In 1997, the Big Throughput Camera (Wittman et al., 1998) became available for general use at the 4 meter telescope at Cerro Tololo. This 16 Megapixel camera set the standard for distant supernova searches and was employed by both SCP and HZT as they developed the samples that led to the discovery of cosmic acceleration.
But the path to cosmic acceleration was not smooth or straight. In July 1997, based on 7 objects, the SCP published the first cosmological analysis based on supernovae (Perlmutter et al., 1997). Comparing their data from z $\sim$0.4, most of which was obtained through just one filter, to the nearby sample from Calán-Tololo (Hamuy et al., 1996a) they found a best value for $\Omega_M$ of 0.88, and concluded that their results were “inconsistent with Lambda-dominated, low-density, flat cosmologies.”
Some theorists had begun to speculate that $\Lambda$ was the missing ingredient to reconcile the observations of a large value for the Hubble Constant (Freedman, Madore, and Kennicutt, 1997), the ages of globular clusters, and a low value for $\Omega_M$ in a flat cosmology (Ostriker and Steinhardt, 1995; Krauss and Turner, 1995). If the universe was flat with a total $\Omega$ of 1, and had $\Omega_M$ of 0.3, then subtraction pointed to a value for $\Lambda$ of 0.7 and you could match the ages of the globular clusters even if the Hubble constant was significantly larger than previously thought. But the initial results of the SCP pointed in the opposite direction, and their evidence for deceleration threw the cold water of data on these artfully-constructed arguments.
The situation began to change rapidly late in 1997. Both teams used the Hubble Space Telescope to observe supernovae that had been found from the ground. The precision of the HST photometry was very good, with the supernova well resolved from the host galaxy thanks to the unique angular resolution of HST. Once the difficult task of accurately connecting the HST photometry to the ground-based work was complete, the observations could be combined to provide additional constraints at the beginning of 1998. For the SCP, there was one additional object from HST, at a record redshift of 0.83. When combined with a subset of the data previously published in July, the analysis gave a qualitatively different answer. In their January 1998 Nature paper (submitted on October 7, 1997), the SCP now found that “these new measurements suggest that we may live in a low-mass-density universe.” There was no observational evidence presented in this paper for cosmic acceleration (Perlmutter et al., 1998). For the High-Z Team, the HST-based sample was larger, with 3 objects, including one at the unprecedented redshift of 0.97 (Garnavich et al. 1998). Although the HZT additional sample of ground-based high-redshift observations was meager (just 1995K), using the same MLCS and template-fitting techniques on both the high-z and low-z samples, and augmenting the public low-z sample from Calán-Tololo with data from the CfA improved the precision of the overall result. Taken at face value, the analysis in this paper, submitted on October 14, 1997 and published on January 14, 1998, showed the tame result that matter alone was insufficient to produce a flat universe, and, more provocatively, if you insisted that $\Lambda$ was zero, and the universe was flat, then the best fit to the data had $\Omega_M$ less than 0. This was a very tentative whisper of what, with hindsight, we can now see was the signal of cosmic acceleration.
An accelerating Universe
------------------------
### First results
Both teams had larger samples under analysis during the last months of 1997, and it was not long before the first analyses were published. The High- Z Team, after announcing their results at the Dark Matter meeting in February 1998 (Filippenko and Riess, 1998), submitted a long article entitled “Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant” to the Astronomical Journal on March 13, 1998. This appeared in the September 1998 issue (Riess et al., 1998). It used a sample of 16 high-z and 34 nearby objects obtained by High-Z Team members, along with the methods developed at the CfA and by the Calán-Tololo group to determine distances, absorptions, and their uncertainties for each of these objects. The data clearly pointed to cosmic acceleration, with luminosity distances in the high-z sample 10-15% larger than expected in a low-mass density universe without $\Lambda$. The High-Z Team also published a long methods paper (Schmidt et al., 1998) and an analysis of this data set in terms of the dark energy equation of state (Garnavich et al., 1998).
The SCP, after showing their data at the January 1998 AAS meeting, cautiously warned that systematic uncertainties, principally the possible role of dust absorption, made it premature to conclude the universe was accelerating. They prepared a long paper for publication that showed the evidence from 42 high redshift objects and 18 low-redshift objects from the Calán-Tololo work. This was submitted to the Astrophysical Journal on September 8, 1998 and appeared in June 1999 (Perlmutter et al., 1999). Although the SCP had no method of their own for determining the reddening and absorption to individual supernovae, they showed that the color distributions of their high-z sample and the objects they selected from the Calán-Tololo sample had similar distributions of restframe color, an indication that the extinction could not be very different in the two samples. They also applied the method of Riess, Press, and Kirshner (1996a) to determine the absorption in the cases where they had the required data. The analysis showed, with about the same statistical power as the High-Z Team paper, that the luminosity distances to supernovae clearly favored a picture in which the universe was accelerating.
### Room for doubt?
Two important questions about these results soon surfaced.
One was whether the results of the two groups were independent. Some of the machinery for analyzing the data sets, for example, the K-corrections to take account of the way supernova redshifts affect the flux in fixed photometric bands, were based on the same slender database of supernova spectra. Similarly, the low-redshift sample used by the SCP was made up entirely of objects observed by the HZT. The two teams cooperated on observing a few of the high redshift objects and both teams used the data for those objects. A small number of co-authors showed up on both the High-Z Team and the Supernova Cosmology Project publications. But the analysis was done independently, most of the high-redshift samples were disjoint, and the astronomical community generally took the agreement of two competing teams to imply that this result was real. But it was the integrity of the results, not the friction of the personalities, that made this work credible.
Another question about the initial results was whether the measured effect– a small, but significant dimming of the distant supernovae relative to nearby ones, was due to cosmology, to some form of dust, or to evolution in the properties of SN Ia with redshift (Aguirre, 1999a,b; Aguirre and Haiman, 2000; Drell, Loredo, and Wasserman, 2000). Aguirre explored the notion that there might be “grey dust” that would cause dimming without reddening. Theoretical difficulties included the limit imposed by using all the available solids, distributing them uniformly, and staying under the limit imposed on the thermal emission from these particles by observations in the far infrared. A direct approach to the possible contribution of dust came from measurements of supernovae over a wider wavelength range– the dust could not be perfectly grey, and a wider range of observations, made with infrared detectors, would reveal its properties more clearly. The earliest application of this was by the HZT (Riess et al., 2000), who observed a supernova at z = 0.46 in the rest-frame I band, with the goal of constraining the properties of Galactic dust or of the hypothetical gray dust. They concluded that the observed dimming of the high-z sample was unlikely to be the result of either type of dust. Much later, this approach was employed by the SCP (Nobili et al., 2005). Dust obscuration, and the relation of absorption to reddening, remains the most difficult problem in using supernova luminosity distances for high-precision cosmology, but the evidence is strong that dust is not responsible for the $\sim $0.25 mag dimming observed at z $\sim $0.5.
A second route to excluding grey dust was to extend observations of SN Ia to higher redshift. If the dimming were due to uniformly distributed dust, there would be more of it along the line-of-sight to a more distant supernova. Due to the discovery of a supernova in a repeat observation of the Hubble Deep Field (Gilliland, Nugent, and Phillips, 1999) and unconscious follow-up with the NICMOS program in that field, Adam Riess and his collaborators were able to construct observations of SN 1997ff at the extraordinary redshift of z $\sim $1.7 (Riess et al., 2001). In a flat universe with $\Omega_{\Lambda} \sim 2/3$ and $\Omega_{M} \sim 1/3$, there is a change in the sign of the expected effect on supernova apparent brightness. Since the matter density would have been higher at this early epoch by a factor (1+z)$^{3}$, the universe would have been decelerating at that time, if the acceleration is due to something that acts like the cosmological constant. The simplest cosmological models predict that a supernova at z $\sim$1.7 will appear brighter than you would otherwise expect. Dust cannot reverse the sign of its effect, so these measurements of the light curve of a SN Ia at z $\sim$1.7 provided a powerful qualitative test of that idea. While the data were imperfect, the evidence, even from this single object, was inconsistent with the grey dust that would be needed to mimic the effect of cosmic acceleration at lower redshift.
Another way of solidifying the early result was to show that the spectra of the nearby supernova of Type Ia, the supernovae at z $\sim $0.5 that gave the strongest signal for acceleration, and spectra of the most distant objects beyond z of 1 give no sign of evolution. While the absence of systematic changes in the spectra with epoch isn’t proof that the luminosities do not evolve, it is a test which the supernova could have failed. They do not fail this test. The early HZT results by Coil et al. (2000) show that, within the observational uncertainties, the spectra of nearby and the distant supernovae are indistinguishable. This approach was explored much later by the SCP (Hook et al., 2005) with consistent results.
### After the beginning
By the year 2000, the context for analyzing the supernova results, which give a strong constraint on the combination ($\Omega_{\Lambda} - \Omega_{M}$) soon included strong evidence for a flat universe with ($\Omega_{\Lambda} + \Omega_{M} = 1$) from the power spectrum of the CMB (de Bernardis et al., 2002) and stronger evidence for the low value of $\Omega_M$ from galaxy clustering surveys (Folkes et al., 1999). The concordance of these results swiftly altered the conventional wisdom in cosmology to a flat $\Lambda$CDM picture. But the concordance of these various methods does not mean that they should lean on each other for support like a trio of drunkards. Instead, practitioners of each approach need to assess its present weaknesses and work to remedy those. For supernovae, the opportunities included building the high-z sample, which was still only a few handfuls, extending its range to higher redshift, augmenting the low-z sample, identifying the systematic errors in the samples, and developing new, less vulnerable methods for measuring distances to supernovae.
#### Building the High- z sample
The High-Z Team published additional data in 2003 that augmented the High-Z sample and extended its range to z = 1.2 (Tonry et al., 2003). Using the 12K CCD detector at the Canada-France-Hawaii Telescope and the Suprime-Cam at Subaru 8.2m telescope, the HZT then executed a “rolling” search of repeated observations with a suitable sampling interval of 1-3 weeks for 5 months (Barris et al., 2004). This enabled the High-Z Team to double the world’s sample of published objects with z $>$ 0.7, to place stronger constrains on the possibility of grey dust, and improve knowledge of the dark energy equation-of-state. The publication by the SCP of 11 SN Ia with 0.36 $<$ z $<$ 0.86 included high-precision HST observations of the light curves and full extinction corrections for each object (Knop et al., 2003).
By this point, in 2003, the phenomenon of cosmic acceleration was well established and the interpretation as the effect of a negative pressure component of the universe fit well into the concordance picture that now included results from WMAP (Spergel et al., 2003). But what was not so clear was the nature of the dark energy. Increasing the sample near z $\sim$0.5 was the best route to improving the constraints on dark energy. One way to describe the dark energy is through the equation of state index $w=p/\rho$. For a cosmological constant, 1 +w = 0. Back-of-the-envelope calculations showed that samples of a few hundred high-z supernovae would be sufficient to constrain w to a precision of $10\%$. As before, two teams undertook parallel investigations. The Supernova Legacy Survey (SNLS), carried out at the Canada-France-Hawaii telescope, included many of the SCP team. The ESSENCE program (Equation of State: SupErNovae trace Cosmic Expansion) carried out at Cerro Tololo included many of the High-Z Team. This phase of constraining dark energy is thoroughly described in the chapter in this book by Michael Wood-Vasey,
The SNLS observing program was assigned 474 nights over 5 years at CFHT. They employed the one-degree imager, Megacam, to search for supernovae and to construct their light curves in a rolling search, with a 4 day cadence, starting in August of 2003. In 2006, they presented their first cosmological results, based on 71 SN Ia, that gave a value of 1+w = -0.023 with a statistical error of 0.09, consistent with a cosmological constant (Astier et al., 2006).
The ESSENCE program used the MOSAIC II imager at the prime focus of the 4m Blanco telescope. They observed with this 64 Megapixel camera every other night for half the night during the dark of the moon in the months of October, November, and December for 6 years, starting in 2002. The survey is described by Miknaitis et al. (2007) and cosmological results from the first 3 years of data were presented in 2006 (Wood-Vasey et al., 2007). The ESSENCE analysis of 60 SN Ia gave a best value for 1+ w = -0.05, with a statistical error of 0.13, consistent with a cosmological constant and with the SNLS results. Combining the SNLS and ESSENCE results gave a joint constraint of 1 + w = -0.07 with a statistical error of 0.09.
We can expect further results from these programs, but the easy part is over. Bigger samples of distant supernovae do not assure improved knowledge of dark energy because systematic errors are now the most important source of uncertainty. These include photometric errors and uncertainties in the light curve fitting methods, but also more subtle matters such as the way dust absorption affects the nearby and distant samples. Collecting large samples is still desirable, especially if the photometric errors are small, but tightening the constraints on the nature of dark energy will also demand improved understanding of supernovae and the dust that dims and reddens them.
![[*Top panels:*]{} Hubble diagram and residuals for MLCS17. The new CfA3 points are shown as rhombs and the OLD and High–z points as crosses. [*Bottom panel:*]{} Hubble diagram of the CfA3 and OLD nearby SN Ia (from Hicken et al., 2009a).](MLCS17_hubble_residuals.ps "fig:"){width="95.00000%"} ![[*Top panels:*]{} Hubble diagram and residuals for MLCS17. The new CfA3 points are shown as rhombs and the OLD and High–z points as crosses. [*Bottom panel:*]{} Hubble diagram of the CfA3 and OLD nearby SN Ia (from Hicken et al., 2009a).](MLCS17_LCpaper_hubblediagram.ps "fig:"){width="100.00000%"}
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#### Extending its range
While the work of Tonry et al. (2003) and Barris et al. (2004) showed that it was possible, with great effort, to make observations from the ground of supernovae beyond a redshift of 1, the installation of the Advanced Camera for Surveys (ACS) on HST provided a unique opportunity to search for and follow these extremely high-redshift objects (Blakeslee et al., 2003). By enlisting the cooperation of the GOODS survey, and breaking its deep exposures of extragalactic fields into repeated visits that formed a rolling search, the Higher-Z Team, led by Adam Riess, developed effective methods for identifying transients, selecting the SN Ia from their colors, obtaining light curves, determining the reddening from IR observations with NICMOS, and measuring the spectra with the grism disperser that could be inserted into the ACS (Riess et al., 2004a,b, 2007). This program has provided a sample of 21 objects with z $>$ 1, and demonstrated directly the change in acceleration, the “cosmic jerk” , that is the signature of a mixed dark matter and dark energy universe. The demise of the ACS brought this program to a halt. It is possible that the planned servicing mission can restore HST to this rich line of investigation.
![[*Left panels:*]{} Today’s best constraints from the Constitution data set on $\Omega_{M}$ and $\Omega_{\Lambda}$. The lower panel shows the combination of the SN contours with the BAO prior. [*Right panels:*]{} Same for $w$ versus $\Omega_{M}$ in a flat Universe (Hicken et al., 2009b).](SALT.OMOL.ps "fig:"){width="47.50000%"} ![[*Left panels:*]{} Today’s best constraints from the Constitution data set on $\Omega_{M}$ and $\Omega_{\Lambda}$. The lower panel shows the combination of the SN contours with the BAO prior. [*Right panels:*]{} Same for $w$ versus $\Omega_{M}$ in a flat Universe (Hicken et al., 2009b).](SALT.OMw.ps "fig:"){width="47.50000%"}
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#### Augmenting the low-z sample
Both the High-Z Team (which included members of the Calán-Tololo supernova program) and the SCP depended on low redshift observations of supernovae to establish the reality of cosmic acceleration. The samples at high redshift were assembled, at great effort, and high cost in observing time at the world’s largest telescopes because it was clear that these data could shift our view of the universe. The low-z samples require persistence, careful attention to systematic effects, and promised no shift in world view. They have been slower to develop. Two early steps forward were the publication by Riess of 22 BVRI light curves from his Ph.D. thesis at Harvard (Riess et al., 1999), and the publication of 44 UBVRI light curves from the thesis work of Jha (Jha, Riess, and Kirshner, 2007). The U-band observations in Jha’s work were especially helpful in analyzing the HST observations of the Higher-Z program, since, for the highest redshift objects observed with HST, most of the observations correspond to ultraviolet emission in the supernova’s rest frame. Jha also revised and retrained the MLCS distance estimator that Riess had developed, using this larger data set, and dubbed it MLCS2k2. Recently, Kowalski compiled the “Union” data sample (Kowalski et al., 2008). His work assessed the uncertainties in combining data from diverse sources, and, by applying stringent cuts to the data, provided a set of 57 low-redshift and 250 high-redshift supernovae to derive constraints on dark energy properties. Kowalski noted the imbalance of the low-z and high-z samples and emphasized the opportunity to make a noticeable improvement in the constraints on dark energy by increasing the sample size for the nearby events.
A third Ph.D. thesis at Harvard, by Malcolm Hicken, has just been completed that finally brings the low-redshift sample out of the statistical limit created by our slow accumulation of nearby objects and begins to encounter the systematic limit imposed by imperfect distance estimators. Hicken analyzed the data for 185 SN Ia in 11500 observations made at the Center for Astrophysics over the period from 2001 to 2008. This large and homogenous data set improves on the Union data set compiled by Kowalski to form the (more perfect) Constitution data set (Hicken et al., 2009a,b). When Hicken uses the same distance fitter used by Kowalski to derive the expansion history and fits to a constant dark energy, he derives 1 + w = 0.013 with a statistical error of about 0.07 and a systematic error that he estimates at 0.11. As discussed below, one important contribution to the systematic error that was not considered by Kowalski is the range of results that is produced by employing different light curve fitters such as SALT, SALT2, and MLCS2k2 which handle the properties of dust in different ways.
This CfA work is a follow-up program that exploits the supernova discovery efforts carried out at the Lick Observatory by Alex Filippenko, Weidong Li, and their many collaborators (Filippenko et al., 2001) as well as a growing pace of supernova discoveries by well-equipped and highly motivated amateur astronomers. Since the selection of the Constitution supernova sample is not homogeneous, information extracted from this sample concerning supernova parent populations and host galaxy properties needs to be handled with caution, but it suggests that even after light curve fitting, the SN Ia in Scd, Sd, or Irregular galaxy hosts are intrinsically fainter than those in Elliptical or S0 hosts, as reported earlier by Sullivan, based on the SCP sample (Sullivan, 2003). The idea of constructing a single fitting procedure for supernovae in all galaxy types has proved effective, but it may be missing a useful clue to distinct populations of SN Ia in galaxies that are and are not currently forming stars. There may be a variety of evolutionary paths to becoming a SN Ia that produce distinct populations of SN Ia in star-forming galaxies that are not exactly the same as the SN Ia in galaxies where star formation ceased long ago (Mannucci et al., 2005; Sullivan et al., 2006; Scannapieco and Bildsten, 2005). Constructing separate samples and deriving distinct light curve fitting methods for these stellar populations may prove useful once the samples are large enough.
A step in this direction comes from the work at La Palma, building up the sample at the sparsely-sampled redshift range near z = 0.2. (Altavilla et al., 2009). A comprehensive approach to sampling has been taken by the Sloan Supernova Survey (Frieman et al., 2008). By repeatedly scanning a 300 deg$^2$ region along the celestial equator, the survey identified transient objects for spectroscopic follow-up with excellent reliability and has constructed ugriz light curves for over 300 spectroscopically confirmed SN Ia. With excellent photometric stability, little bias in the supernova selection, and a large sample in the redshift range 0.05 $<$ z $<$ 0.35, this data set will be a powerful tool for testing light curve fitting techniques, provide a low-redshift anchor to the Hubble diagram, and should result in a more certain knowledge of dark energy properties.
In the coming years, comprehensive results from the SCP’s SN Factory (Aldering et al., 2002), the Carnegie Supernova Program (Hamuy et al., 2006), and the analysis of the extensive KAIT archive (Filippenko et al., 2001) should change the balance of the world’s sample from one that is just barely sufficient to make statistical errors smaller than systematic errors, to one that provides ample opportunity to explore the ways that sample selection might decrease those systematic errors.
Shifting to the infrared
------------------------
Coping with the effects of dust absorption was an important contribution of the early work by Phillips et al. (1999) and by Riess, Press, and Kirshner (1996a), the later work by Knop et al. (2003) and Jha, Riess, and Kirshner (2007) and it continues to be the most difficult and interesting systematic problem in supernova cosmology. The formulations that worked sufficiently well to measure 10% effects will not be adequate for the high precision measurements that are required for future dark energy studies. The analysis of the low-redshift supernova data by Conley et al. (2007) showed that either the ratio of reddening to extinction in the supernova hosts was distinctly different from that of the Milky Way (R$_{V}$ = 1.7 instead of the conventional value of 3.1) or there was a “Hubble Bubble”– a zone in which the local expansion rate departed from the global value. As discussed by Hicken (2009a), today’s larger sample does not show evidence for the Hubble Bubble, but the value of R$_{V}$ that performs best for MLCS2k2 and for SALT is significantly smaller than 3.1. It seems plausible that the sampling for earlier work was inhomogeneous, with highly reddened objects present only in the nearby region. If the correction for reddening in these cases was not carried out accurately, they could contribute to the illusion of a Hubble Bubble. But the evidence for a small effective value of R$_{V}$ has not gone away. It seems logical to separate the contribution due to reddening from the contribution that might result from an intrinsic relation between supernova colors and supernova luminosity, as done in MLCS2k2, but the approach of lumping these together, as done by the fitting techniques dubbed SALT and SALT2, also works well empirically (Guy et al., 2005, 2007). In the ESSENCE analysis, the effects of extinction on the properties of the observed sample were carefully considered, and found to affect the cosmological conclusions. Getting this problem right will be an important part of preparing for higher precision cosmological measurements with future surveys.
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Fortunately, there is a very promising route to learning more about dust, avoiding its pernicious effects on supernova distances, and deriving reliable and precise measures of dark energy properties. That route is to measure the properties of the supernovae in the rest frame infrared. As shown in the pioneering work of Krisciunas, Phillips, and Suntzeff (2004), nearby SN Ia in the Hubble flow behave as very good standard candles when measured in near infrared bands (NIR), typically J, H, and K$_s$. This work has recently been extended by Wood-Vasey et al. who used the PAIRITEL system (a refurbished and automated version of the 2MASS telescope) at Mount Hopkins to obtain near infrared light curves that double the world’s sample (Wood-Vasey et al., 2008) . Even with no correction for light curve shape or dust absorption, the NIR light curves for SN Ia exhibit a scatter about the Hubble line that is typically 0.15 mag. This is comparable to the scatter that is achieved by the output of the elaborate light curve fitters now in use for optical data that correct for the width of the light curve’s peak and use the optical colors to infer dust corrections. This means that the SN Ia actually do behave like standard candles– but in the NIR! What’s more, the effects of dust absorption generally scale as $1/\lambda$, so the effects of extinction on the infrared measurements should be 4 times smaller than at the B band. When combined with optical data, the infrared observations can be used to determine the properties of the dust, and to measure even more accurate luminosity distances. Early steps toward these goals are underway (Friedman et al., 2009).
The next ten years
------------------
Goals for the coming decade are to improve the constraints on the nature of dark energy by improving the web of evidence on the expansion history of the universe and on the growth of structure through gravitation (Albrecht et al., 2006, 2009; Frieman, Turner, and Huterer, 2008; Ruiz-Lapuente, 2007). Supernovae have an important role to play because they have been demonstrated to produce results. Precise photometry from homogeneous data, dust absorption determined with near-IR measurements, and constructing useful subsamples in galaxies with differing star formation histories are all areas where we know improvement in the precision of the distance measurements is possible. More speculative, but plausible, would be the use of supernova spectra in a systematic way to improve the distance estimates. Implementation of statistically sound ways to use the light curves (and possibly spectra) to determine distances should make the results more reliable and robust. What is missing is a level of theoretical understanding for the supernova explosions themselves that could help guide the empirical work, and provide confidence that stellar evolution is not subtly undermining the cosmological inferences (Hoeflich, Wheeler, and Thielemann, 1998; Ruiz–Lapuente, 2004). Large samples from Pan-STARRS, the Dark Energy Survey, and, if we live long enough, from JDEM and LSST will eventually be available. The chapter in this book by Alex Kim makes a persuasive case for the effectiveness of a thorough space-based study of supernovae. Our ability to use these heroic efforts effectively depends on improving our understanding of supernovae as astronomical objects in the context of galaxy formation, stellar evolution, and the physics of explosions. Then we can employ the results with confidence to confirm, or, better yet, to rule out some of the weedy garden of theoretical ideas for the dark energy described in other chapters of this book!
Acknowledgements
----------------
Supernova research at Harvard is supported by the US National Science Foundation through grant AST06-06772. I am grateful for the long series of excellent postdocs and students who have contributed so much to this work. As postdocs, Alan Uomoto, Bruno Leibundgut, Pilar Ruiz-Lapuente, Eric Schlegel, Peter Hoeflich, David Jeffery, Peter Garnavich, Tom Matheson, Stéphane Blondin, and Michael Wood-Vasey. And as graduate students Ron Eastman, Chris Smith, Brian Schmidt, Jason Pun, Adam Riess, Saurabh Jha, Marayam Modjaz, Malcolm Hicken, Andy Friedman, and Kaisey Mandel.
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{
"pile_set_name": "ArXiv"
}
|
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abstract: 'Hybrid analog-digital (AD) beamforming structure is a very attractive solution to build low cost massive multiple-input multiple-output (MIMO) systems. Typically these systems use a set of fixed beams for transmission and reception to avoid the need to obtain channel state information at transmitter (CSIT) for each antenna element individually. However, such a method can not fully exploit the potential of hybrid AD beamforming systems. Alternatively, CSIT can be estimated by assuming a model for the propagation channel, whereas this model is only validated in millimeter-wave (mmWave) band thanks to its poor scattering nature. In this paper, we focus on time division duplex (TDD) systems with hybrid beamforming structure and propose a reciprocity calibration scheme that allows to acquire full CSIT. Different to existing CSIT acquisition methods, our approach does not require any assumption on the channel model and can, in theory, estimate the CSIT up to an arbitrary small error.'
author:
- 'Xiwen ˜JIANG, ˜ and ˜Florian ˜Kaltenberger, ˜ [^1] [^2]'
bibliography:
- 'refs.bib'
title: Channel reciprocity calibration in TDD hybrid beamforming massive MIMO systems
---
Channel reciprocity calibration, channel state information at transmitter (CSIT), hybrid analog-digital beamforming, massive MIMO, time division duplex (TDD).
Introduction {#sec:intro}
============
Massive multiple-input multiple-output (MIMO) is considered as a key enabler of the next generation of wireless communication networks, as it has the potential to dramatically increase the network capacity [@marzetta2010noncooperative]. To bring this concept to practice, it is essential to reduce the cost of building up such complex systems. Among the most promising solutions, hybrid analog-digital (AD) beamforming structure has achieved great attention. By introducing phase shifters and reducing the number of expensive components on digital and RF chains, such as digital-to-analog/analog-to-digital converters (DACs/ADCs) as well as signal mixers, hybrid beamforming structure opens up possibilities to build relatively low cost massive MIMO systems.
A common way of enabling hybrid beamforming is to pre-define a set of fixed beams in the downlink (DL) on which pilots are transmitted to a user equipment (UE) who then simply selects the best beam and sends the index back to the base station (BS), which will use it directly for data transmission [@kim2013multi; @han2015large]. Such systems have also been specified for LTE-Advanced Pro, in the so-called full dimension (FD) MIMO system[@ji2017overview], but are clearly suboptimal compared to the case where full channel state information at the transmitter (CSIT) is available [@flordelis2017massive]. Under the assumption of full CSIT, a hybrid massive MIMO system can achieve the same performance of any fully digital beamforming scheme, as long as the number of RF chains is at least twice the number of data schemes [@sohrabi2016hybrid2]. However, acquisition of CSIT in a hybrid massive MIMO system is a non-trivial matter, both for frequency division duplex (FDD) and time division duplex (TDD) systems.
The problem was studied in the millimeter-wave (mmWave) band in [@alkhateeb2014channel], where the channel can be considered to have only a few number of dominant rays because of the poor scattering nature of the channel. While this method works out well for mmWave, it can hardly be generalized to an arbitrary channel, especially when hybrid beamforming massive MIMO systems are used in a sub-6GHz band. In this paper, we propose to perform the DL CSIT acquisition based on the channel reciprocity property in a time division duplex (TDD) system. In fact, as long as the DL and uplink (UL) transmission happens within the channel coherence time, the physical channel is reciprocal. This property was used when massive MIMO was introduced in [@marzetta2010noncooperative] to avoid large channel feedback to the BS in UL. The only problem is that the transmit and receive radio frequency (RF) chains in transceivers (hardware from DAC to antenna at the transmit path and that from antenna to ADC at the receive path) are not reciprocal, thus calibration is needed to compensate the hardware asymmetry.
{width="1.6\columnwidth"}
In a fully digital TDD system, numerous calibration methods have been proposed. Reciprocity calibration using “over-the-air" signal processing was introduced for classical MIMO systems in [@guillaud2005practical; @kaltenberger2010relative], where the BS exchanges pilots with the UE to estimate bi-directional channel. A total least squares (TLS) problem can then be formulated to estimate the calibration coefficients. Such methods, however, can not be directly applied to massive MIMO systems since all UEs need to feed back their measured DL CSI during the calibration procedure. In [@shi2011efficient], authors propose to choose a reference UE to assist the BS to calibrate in order to reduce the UL feedback. BS internal calibration was then introduced in [@shepard2012argos] for the Argos massive MIMO testbed, where the reference UE is replaced by a reference antenna, so that BS can perform internal calibration without the involvement of UE. However, the Argos calibration method is quite sensitive to the placement of the reference antenna, thus is not easy for real environment deployment and not suitable for antennas in a distributed topology. In order to take up this challenge, methods based on bi-directional transmissions between antenna pairs are proposed in [@rogalin2014scalable; @vieira2014reciprocity]. These methods were initially designed for distributed massive MIMO systems but show good performance in co-localized systems as well. Other methods such as [@papadopoulos2014avalanche; @luo2015robust; @wei2016mutual] address different aspects in the calibration problem, including speeding up the whole calibration procedure, reducing the number of transmission needed for calibration and calibration dedicated to maximum ratio transmission (MRT). In [@vieira2017reciprocity], the authors propose a maximum likelihood (ML) estimator to enhance the accuracy of reciprocity calibration.
Regardless the variety of calibration methods, none of them can be directly used in a hybrid AD beamforming structure [@han2015large]. This is the main reason why TDD reciprocity based methods have been left behind in this type of massive MIMO systems. In this paper, we introduce a reciprocity calibration method which allows us to avoid beam training or selection and acquire CSIT without any assumption on the channel. The main contributions of this paper are as follows:
- We propose a reciprocity calibration method for TDD hybrid beamforming massive MIMO systems. This problem was never addressed before, although calibration methods for fully digital systems were introduced more than a decade ago.
- Based on reciprocity calibration, we illustrate that TDD hybrid beamforming systems have the potential to acquire the CSIT up to an arbitrary small error, thus can fully release its beamforming potential. This provides novel ways to operate hybrid beamforming systems rather than performing beam training using a fix set of pre-determined beams.
The notation adopted in this paper conforms to the following convention. Vectors and matrices are denoted in lowercase bold and uppercase bold respectively: $\mathbf{a}$ and $\mathbf{A}$. $(\cdot)^*$, $(\cdot)^T$, $(\cdot)^H$, $(\cdot)^{-1}$, $(\cdot)^{-T}$ denote element-wise complex conjugate, transpose, Hermitian transpose, inverse, and transpose together with inverse, respectively. $\otimes$ denotes the Kronecker product operator. $\mbox{diag}\{a_1, a_2, \dots, a_M\}$ denotes a diagonal matrix with its diagonal composed of $a_1, a_2, \dots, a_M$, $\mbox{rank}(\mathbf{A})$ represents the rank of matrix $\mathbf{A}$, whereas $\mbox{vec}(\mathbf{A})$ denotes the vectorization of the matrix $\mathbf{A}$. $\mathbb{C}$ denotes the set of complex numbers. Finally, the Frobenius norm is denoted by $\|\cdot\|_F$.
System model
============
Hybrid Structure {#subsec:hybrid_structure}
----------------
The structure of a TDD hybrid beamforming transceiver is shown in Fig. \[fig:hybrid\_architecture\] [@li2016mmwave] where the digital beamformer is connected to $N_{RF}$ RF chains, which, through an analog beamforming network, are connected with power amplifiers (PAs)/low noise amplifiers (LNAs) and $N_{ant}$ antennas, where $N_{ant} \geq N_{RF}$. Note that it is also possible to place PAs and LNAs in the RF chains before the analog beamformer so that the number of amplifiers are less. However, in that case, each amplifier needs more power since it amplifies signal for multiple antennas. Additionally, in the transmission mode, the insertion loss of analog precoder working in the high power region makes the transceiver less efficient in terms of power consumption. In the reception mode, the fact of having phase shifters before LNAs also results in a higher noise figure in the receiver. It is thus a better choice to have PAs and LNAs close to antennas. To this reason, we stick our study in this paper to the structure in Fig. \[fig:hybrid\_architecture\]. The discussion in this paper, however, can also be applied to the case where the PAs/LNAs are placed before the analog beamformer.
The analog beamformer is interpreted as analog precoder and combiner in the transmit and receive path, respectively. Two types of architecture can be found in literature [@sohrabi2016hybrid; @han2015large]:
- **Subarray architecture:** Each RF chain is connected to $N_{ant}/N_{RF}$ phase shifters as shown in Fig. \[fig:analog\_per\_rf\]. Such a structure can be found in [@huang2010hybrid; @guo2012hybrid; @kim2013multi; @roh2014millimeter]
- **Fully connected architecture:** $N_{ant}$ phase shifters are connected to each RF chain. Phase shifters with the same index are then summed up to be connected to the corresponding antenna, as shown in Fig. \[fig:analog\_full\_str\]. This structure can be found in [@nsenga2010mixed; @el2012low; @alkhateeb2013hybrid; @alkhateeb2014channel].
In terms of CSIT acquisition, since the BS is not fully digital, assigning orthogonal pilots to different antennas for DL channel estimation per antenna can not be used. Additionally, even assuming that we can have perfect channel estimation for all antennas at the UE, it is unfeasible to feed this information back to the BS, because in a massive MIMO system, the UL overhead will be so heavy that at the time the BS gets the whole CSIT, the information has already outdated.
In order to address the problem, we make use of the inherent reciprocity property in TDD systems. We firstly show how this is possible for “subarray architecture” by enabling reciprocity calibration. We then provide some ideas to calibrate a fully connected hybrid architecture.
[.49]{} ![Two types of analog beamforming structure.[]{data-label="fig:analog_bf_2types"}](analog_type1.png "fig:"){width="0.99\columnwidth"}
[.49]{} ![Two types of analog beamforming structure.[]{data-label="fig:analog_bf_2types"}](analog_type2.png "fig:"){width="\columnwidth"}
{width="1.6\columnwidth"}
System Model
------------
Consider a subarray hybrid beamforming system with a single user shown in Fig. \[fig:hybrid\_str\], where a BS with $N_{ant}^{BS}$ antennas communicates $N_s$ data streams in the DL to a UE with $N_{ant}^{UE}$ antennas. $N_{RF}^{BS}$ and $N_{RF}^{UE}$ are the number of RF chains at the BS and the UE, respectively, such that $N_s \leq N_{RF}^{BS} \leq N_{ant}^{BS}$ and $N_s \leq N_{RF}^{UE} \leq N_{ant}^{UE}$. In Fig. \[fig:hybrid\_str\], we use ${\mathbf{V}}_{BB}^{BS} \in \mathbb{C}^{N_{RF}^{BS}\times N_s}$ and ${\mathbf{W}}_{BB}^{UE} \in \mathbb{C}^{N_s \times N_{RF}^{UE}}$ to represent the baseband digital beamforming matrix at the BS and at the UE, respectively. ${\mathbf{V}}_{RF}^{BS} \in \mathbb{C}^{N_{ant}^{BS} \times N_{RF}^{BS}}$ and ${\mathbf{W}}_{RF}^{UE} \in \mathbb{C}^{N_{RF}^{UE} \times N_{ant}^{UE}}$ are the analog beamforming precoders and combiners. We use ${\mathbf{T}}_1^{BS} \in \mathbb{C}^{N_{RF}^{BS}\times N_{RF}^{BS}}$, ${\mathbf{T}}_2^{BS} \in \mathbb{C}^{N_{ant}^{BS}\times N_{ant}^{BS}}$, ${\mathbf{R}}_1^{UE} \in \mathbb{C}^{{N_{RF}^{UE}\times N_{RF}^{UE}}}$ and ${\mathbf{R}}_2^{UE} \in \mathbb{C}^{N_{ant}^{UE}\times N_{ant}^{UE}}$ to represent the transfer functions of the corresponding hardwares. The diagonal elements of ${\mathbf{T}}_1^{BS}$ and ${\mathbf{R}}_1^{UE}$ capture the hardware characteristics of the $N_{RF}^{BS}$ and $N_{RF}^{UE}$ RF chains including the DACs/ADCs, signal mixers and some other components around, whereas, their off-diagonal elements represent the RF crosstalk. Similarly, the diagonal of ${\mathbf{T}}_2^{BS}$ and ${\mathbf{R}}_2^{UE}$ are used to represent the properties of amplifiers as well as some surrounding components after phase shifter on each branch and their off-diagonal elements represent RF crosstalk and antenna mutual coupling [@balanis2016antenna]. If we transmit a signal ${\mathbf{s}}$ through a channel ${\mathbf{C}}\in \mathbb{C}^{N_{ant}^{UE}\times N_{ant}^{BS}}$, at the output of the UE’s digital combiner, we have $$\label{eqn:sig_model}
{\mathbf{y}}= {\mathbf{W}}_{BB}^{UE}{\mathbf{R}}_1^{UE}{\mathbf{W}}_{RF}^{UE}{\mathbf{R}}_2^{UE}{\mathbf{C}}{\mathbf{T}}_2^{BS}{\mathbf{V}}_{RF}^{BS}{\mathbf{T}}_1^{BS}{\mathbf{V}}_{BB}^{BS}{\mathbf{s}}+ {\mathbf{n}},$$ where ${\mathbf{y}}$ is the $N_s\times 1$ received signal vector and ${\mathbf{n}}$ following the circularly symmetric complex Gaussian distribution $\mathcal{CN}(0,\sigma_n^2{\mathbf{I}})$ is the noise vector.
In a TDD system, the physical channel is reciprocal within the channel coherence time, i.e., in the reverse transmission, the UL physical channel from UE to BS can be represented by ${\mathbf{C}}^T$. However, apart from special cases in mmWave, where the channel enjoys limited scattering property, channel estimation, usually performed in the digital domain, does not directly provide ${\mathbf{C}}$ but a composite channel including ${\mathbf{C}}$ and transceivers’ hardware properties, which are not reciprocal. This will be detailed in Section \[subsec:csit\_aquis\].
CSIT acquisition based on TDD reciprocity calibration
=====================================================
In this section, we describe how to acquire accurate CSIT based on TDD channel reciprocity. Especially, as transmit and receive RF chains break down the inherent reciprocity, we introduce our calibration method to compensate the hardware asymmetry.
Equivalent System Model
-----------------------
In order to acquire CSIT and calibrate TDD systems, let us firstly introduce an equivalent system model which simplifies the signal model in , where we observe that the hardware blocks are mixed up with digital and analog beamforming matrices. Note that ${\mathbf{T}}_1^{BS}$ (similar for ${\mathbf{R}}_1^{UE}$) represents the hardware properties on the $N_{RF}^{BS}$ RF chains, where the $N_{RF}^{BS}$ diagonal elements mainly capture the random phases generated by the corresponding RF chains and the off-diagonal elements represent the RF crosstalk, i.e., the RF leakage from one RF chain to the others. Proper RF circuit design usually ensures very small RF crosstalk with regard to the diagonal values, which is also proven by the measurement results in [@Jiang2015], leading to the fact that, in reality, ${\mathbf{T}}_1^{BS}$ and ${\mathbf{R}}_1^{UE}$ can be considered to be diagonal. Since ${\mathbf{V}}_{RF}^{BS}$ and ${\mathbf{W}}_{RF}^{UE}$, representing the analog beamformers for each RF chain, have block diagonal structures, the matrix multiplication is commutative if we introduce a Kronecker product such as ${\mathbf{V}}_{RF}^{BS}{\mathbf{T}}_1^{BS} = ({\mathbf{T}}_1^{BS}\otimes {\mathbf{I}}_{BS}){\mathbf{V}}_{RF}^{BS}$ and ${\mathbf{R}}_1^{UE}{\mathbf{W}}_{RF}^{UE} = {\mathbf{W}}_{RF}^{UE}({\mathbf{R}}_1^{UE}\otimes {\mathbf{I}}_{UE})$, where ${\mathbf{I}}_{BS}$ and ${\mathbf{I}}_{UE}$ are identity matrices of size $N_{ant}^{BS}/N_{RF}^{BS}$ and $N_{ant}^{UE}/N_{RF}^{UE}$, respectively. The signal model in thus has an equivalent representation as $$\begin{aligned}
{\mathbf{y}}= & \underbrace{{\mathbf{W}}_{BB}^{UE}{\mathbf{W}}_{RF}^{UE}}_{{{\mathbf{W}}}_{UE}}
\underbrace{({\mathbf{R}}_1^{UE}\otimes {\mathbf{I}}_{UE}){\mathbf{R}}_2^{UE}}_{{{\mathbf{R}}}_{UE}} {\mathbf{C}}\\
& \underbrace{{\mathbf{T}}_2^{BS}({\mathbf{T}}_1^{BS}\otimes {\mathbf{I}}_{BS})}_{{{\mathbf{T}}}_{BS}}
\underbrace{{\mathbf{V}}_{RF}^{BS}{\mathbf{V}}_{BB}^{BS}}_{{{\mathbf{V}}}_{BS}}{\mathbf{s}}+ {\mathbf{n}},
\label{eqn:sig_equ_model}
\end{aligned}$$ where we group up the digital and analog transmit and receive beamforming matrices into ${\mathbf{V}}_{BS}$ and ${\mathbf{W}}_{UE}$. The hardware transfer functions are merged to ${\mathbf{T}}_{BS}$ and ${\mathbf{R}}_{UE}$.
An intuitive understanding of this alternative representation on the BS transmit part is shown in Fig. \[fig:hybrid\_str3\], where we 1) replace all shared hardware components (mixers, filters) on RF chain by its replicas on each branch with a phase shifter; 2) change the order of hardware components such that all components in ${\mathbf{T}}_{BS}$ go to the front end near the antennas.
Note that this equivalent model is general for different hardware implementation, i.e., no matter how hardware impairments are distributed on the hybrid structure, we can always use these two steps to create an equivalent system model. For example, if there’s any hardware impairment within the phase shifter or in DAC, they can also be extracted out and put into ${\mathbf{T}}_{BS}$ using the same methodology.
![Equivalent hybrid structure.[]{data-label="fig:hybrid_str3"}](hybrid_structure_equivalent.png){width="0.7\columnwidth"}
Full CSIT Acquisition Based on Reciprocity Calibration {#subsec:csit_aquis}
------------------------------------------------------
Let us look at the DL and UL transmission between the BS and the UE using TDD mode, the bi-directional transmission represented in the equivalent signal model is given by: $$\left\{
\begin{aligned}
{\mathbf{y}}_{BS{\rightarrow}UE} &= {\mathbf{W}}_{UE}\underbrace{{\mathbf{R}}_{UE}{\mathbf{C}}{\mathbf{T}}_{BS}}_{{\mathbf{H}}_{BS{\rightarrow}UE}}{\mathbf{V}}_{BS}{{\mathbf{s}}}_{BS} +{\mathbf{n}}_{UE},\\
{\mathbf{y}}_{UE{\rightarrow}BS} &= {\mathbf{W}}_{BS}\underbrace{{\mathbf{R}}_{BS}{\mathbf{C}}^T{\mathbf{T}}_{UE}}_{{\mathbf{H}}_{UE{\rightarrow}BS}}{\mathbf{V}}_{UE}{{\mathbf{s}}}_{UE} + {\mathbf{n}}_{BS},\\
\end{aligned}
\right.$$ where ${\mathbf{C}}$ and ${\mathbf{C}}^T$ are the reciprocal DL and UL air propagation channel. From the point of view of digital signal processing, the channel does not only include the physical channel ${\mathbf{C}}$ but also the hardware transfer functions of the radio front ends, thus we define the effective channel ${\mathbf{H}}_{BS{\rightarrow}UE} = {\mathbf{R}}_{UE}{\mathbf{C}}{\mathbf{T}}_{BS}$ and ${\mathbf{H}}_{UE{\rightarrow}BS}={\mathbf{R}}_{BS}{\mathbf{C}}^T{\mathbf{T}}_{UE}$. Since the transmit and receive RF chains use different hardware components, it is clear that ${\mathbf{T}}_{BS} \neq {\mathbf{R}}_{BS}^T$ and ${\mathbf{T}}_{UE} \neq {\mathbf{R}}_{UE}^T$. Thus the hardware radio front ends break the TDD channel reciprocity, i.e. ${\mathbf{H}}_{BS{\rightarrow}UE}\neq {\mathbf{H}}_{UE{\rightarrow}BS}$.
In order to compensate the hardware asymmetry and to achieve the reciprocity, we establish the relationship between the DL and UL effective channels as follows $${\mathbf{H}}_{BS{\rightarrow}UE} ={\mathbf{R}}_{UE}{\mathbf{T}}_{UE}^{-T}{\mathbf{H}}_{UE{\rightarrow}BS}^T{\mathbf{R}}_{BS}^{-T}{\mathbf{T}}_{BS}.$$ Defining ${\mathbf{F}}= {\mathbf{R}}^{-T}{\mathbf{T}}$ for both BS and UE, we have $${\mathbf{H}}_{BS{\rightarrow}UE} ={\mathbf{F}}_{UE}^{-T}{\mathbf{H}}_{UE{\rightarrow}BS}^T{\mathbf{F}}_{BS}.
\label{eqn:H_recip}$$ We observe that the DL CSIT can be represented as the UL CSI ${\mathbf{H}}_{UE{\rightarrow}BS}$ tuned with two matrices only dependent on the transceivers’ hardware, denoted as ${\mathbf{F}}_{BS}$ and ${\mathbf{F}}_{UE}$, which are named as calibration matrices at the BS and the UE, respectively. As long as we have the three matrices in , we can estimate the DL CSIT.
Note that if the UE has only one antenna, ${\mathbf{F}}_{UE}$ becomes a scalar and can be ignored, since the ambiguity of a complex scalar value on the obtained CSIT will not change the final created beam pattern [@shepard2012argos]. According to [@R1-091752; @R1-091794; @R1-094622], even if the UE has more than one antenna (but significantly less than the eNB), the UE calibration error has little effect to the performance of reciprocity. Especially phase calibration errors at the UE have no effect on the performance, and relative amplitude calibration mismatch at UE side can have some impact. Thus, when the antenna number at the UE is limited, the UE side calibration is not necessarily needed from the point of view of DL beamforming. Taking ${\mathbf{F}}_{UE}$ as the identity matrix does not impact much the performance[^3].
It is worth noting that both ${\mathbf{F}}_{BS}$ and ${\mathbf{F}}_{UE}$ represent hardware properties, which are independent to the propagation channel ${\mathbf{C}}$, leading to the fact that they are quite stable during the time. Measurements in [@shepard2012argos] show that the variation of calibration coefficients deviates from the mean angle with an average of less than 2.6% (maximum 6.7%), and from the mean amplitude less than 0.7% (maximum 1.4%), over a period of 4 hours. This implies that calibration does not have to be performed very frequently. In the sequel, we firstly describe the effective channel estimation method for ${\mathbf{H}}_{UE{\rightarrow}BS}$ estimation. We then present an internal reciprocity calibration scheme, where BS (or UE if needed) can estimate its own calibration matrices internally.
Effective Channel Estimation {#subsec:eff_ch_est}
----------------------------
In order to obtain the UL CSI, we need to estimate the effective channel based on pilot transmission. This is also needed for internal calibration at the BS and UE side, thus in order to make the description general, we drop the subscript BS and UE and use ${\mathbf{H}}={\mathbf{R}}{\mathbf{C}}{\mathbf{T}}$ to denote the effective channel, where ${\mathbf{T}}$ and ${\mathbf{R}}$ are $N_{ant}^t \times N_{RF}^t$ and $N_{RF}^r \times N_{ant}^r$ matrices.
Consider sending pilots (${\mathbf{s}}={\mathbf{p}}$) using $K$ transmit precoders combined with $L$ different receive combiners, we can totally accumulate $KL$ measurements: $$\underbrace{[{\mathbf{y}}_{l,k}]}_{{\mathbf{Y}}} =
\underbrace{[{\mathbf{W}}_1^T, \dots, {\mathbf{W}}_L^T]^T}_{\tilde{{\mathbf{W}}}}{\mathbf{H}}\underbrace{[{\mathbf{V}}_1{\mathbf{p}}_1, \dots, {\mathbf{V}}_K{\mathbf{p}}_K]}_{\tilde{{\mathbf{P}}}}
+ \underbrace{[{\mathbf{n}}_{l,k}]}_{{\mathbf{N}}}.$$ where ${\mathbf{y}}_{l,k}$ is the block element of ${\mathbf{Y}}$ on the $l^{th}$ row and $k^{th}$ column. $\tilde{{\mathbf{W}}}$ and $\tilde{{\mathbf{P}}}$ are matrices of size $N_sL\times N_{ant}^r$ and $N_{ant}^t \times K$, respectively. To obtain the channel estimation, we vectorize the receive vector as $$\mbox{vec}({\mathbf{Y}}) =\underbrace{\tilde{{\mathbf{P}}}^T\otimes\tilde{{\mathbf{W}}}}_{{\mathbf{D}}}\cdot\mbox{vec}({\mathbf{H}}) + \mbox{vec}({\mathbf{N}}),$$ where we define ${\mathbf{D}}=\tilde{{\mathbf{P}}}^T\otimes\tilde{{\mathbf{W}}}$. The least squares (LS) channel estimator is $$\mbox{vec}({\mathbf{H}}) = ({\mathbf{D}}^H{\mathbf{D}})^{-1}{\mathbf{D}}^H\cdot\mbox{vec}({\mathbf{Y}}).
\label{eqn:ch_est}$$ In order to guarantee that the estimation problem is over determined, we should have $\mbox{rank}({\mathbf{D}}) \geq N_{ant}^t\times N_{ant}^r$, where $\mbox{rank}({\mathbf{D}}) = \mbox{rank}(\tilde{{\mathbf{P}}}^T)\mbox{rank}(\tilde{{\mathbf{W}}})$ according to Kronecker product’s property on matrix rank. Noting that $\mbox{rank}(\tilde{{\mathbf{P}}}^T) \leq \min (N_{ant}^t, K)$ and $\mbox{rank}(\tilde{{\mathbf{W}}}) \leq \min (N_sL, N_{ant}^r)$, thus, in order to meet the sufficient condition of over determination on the estimation problem, we should have $K \geq N_{ant}^t$ and $L \geq N_{ant}^r/N_s$. Note that since the objective here is to estimate the effective channel, digital precoder and combiner are not necessarily needed, i.e. pilots for channel estimation can be inserted after the digital precoder. In this case $N_s=N_{RF}$ and $L \geq N_{ant}^r/N_{RF}$. Additionally, in a multi-carrier system, where, for example, orthogonal frequency division multiplexing (OFDM) modulation is used, it is possible to allocate different carriers to the pilots of different RF chains. Assuming $\beta$ the number of frequency multiplexing factor on transmit RF chains, the number of the needed transmit precoder $K \geq N_{ant}^t/\beta$.
The effective channel estimation can be used to obtain UL channel estimation but will also be served to estimate calibration matrices as will be presented hereafter.
Internal Reciprocity Calibration {#subsec:intern_calib}
--------------------------------
One basic idea in estimating calibration matrix consists in accumulating extensively pairs of channel measurement $\hat{{\mathbf{H}}}_{BS{\rightarrow}UE}$ and $\hat{{\mathbf{H}}}_{UE{\rightarrow}BS}$, based on which ${\mathbf{F}}_{BS}$ and ${\mathbf{F}}_{UE}$ can be estimated. However, such a method implies that we have to exchange the estimated channel information between BS and UE during the calibration, which introduces an extra cost. It is thus more reasonable to perform calibration internally within the antenna array, at the BS and the UE, independently.
Internal calibration means that the pilot-based channel estimation happens between different antennas of the same transceiver. Let us equally partition the total $N_{ant}$ antennas into two groups ${\mathcal{A}}$ and ${\mathcal{B}}$, e.g., ${\mathcal{A}}= \{1,2,...,\frac{N_{ant}}{2}\}$ and ${\mathcal{B}}= \{\frac{N_{ant}}{2}+1,...,N_{ant}\}$, as shown in Fig. \[fig:inter\_calib\]. When the antennas in group ${\mathcal{A}}$ are connected to the transmit path of $\frac{N_{RF}}{2}$ RF chains, the antennas in group B are connected to the receive path of the rest $\frac{N_{RF}}{2}$ RF chains. We firstly perform an intra-array transmission from ${\mathcal{A}}$ to ${\mathcal{B}}$, and within the channel coherence time, we switch the roles of group ${\mathcal{A}}$ and ${\mathcal{B}}$ in order to transmit signal from ${\mathcal{B}}$ to ${\mathcal{A}}$. The bi-directional received signals are given by $$\left\{
\begin{aligned}
{\mathbf{y}}_{{\mathcal{A}}{\rightarrow}{\mathcal{B}}} &= {\mathbf{W}}_{\mathcal{B}}{\mathbf{R}}_{\mathcal{B}}{\mathbf{C}}{\mathbf{T}}_{\mathcal{A}}{\mathbf{V}}_{\mathcal{A}}{\mathbf{p}}_{\mathcal{A}}+{\mathbf{n}}_{{\mathcal{A}}{\rightarrow}{\mathcal{B}}},\\
{\mathbf{y}}_{{\mathcal{B}}{\rightarrow}{\mathcal{A}}} &= {\mathbf{W}}_{\mathcal{A}}{\mathbf{R}}_{\mathcal{A}}{\mathbf{C}}^T{\mathbf{T}}_{\mathcal{B}}{\mathbf{V}}_{\mathcal{B}}{\mathbf{p}}_{\mathcal{B}}+ {\mathbf{n}}_{{\mathcal{B}}{\rightarrow}{\mathcal{A}}},
\end{aligned}
\right.
\label{eqn:bi_trans}$$ where ${\mathbf{p}}_{\mathcal{A}}$ and ${\mathbf{p}}_{\mathcal{B}}$ are transmitted pilots, ${\mathbf{C}}$ is the reciprocal intra-array channel whereas ${\mathbf{n}}_{{\mathcal{A}}{\rightarrow}{\mathcal{B}}}$ and ${\mathbf{n}}_{{\mathcal{B}}{\rightarrow}{\mathcal{A}}}$ are noise.
If we use ${\mathbf{H}}_{{\mathcal{A}}{\rightarrow}{\mathcal{B}}}={\mathbf{R}}_{\mathcal{B}}{\mathbf{C}}{\mathbf{T}}_{\mathcal{A}}$ and ${\mathbf{H}}_{{\mathcal{B}}{\rightarrow}{\mathcal{A}}}={\mathbf{R}}_{\mathcal{A}}{\mathbf{C}}^T{\mathbf{T}}_{\mathcal{B}}$ to represent the bi-directional effective channels between group ${\mathcal{A}}$ and ${\mathcal{B}}$, including the physical channel in the air as well as transceiver’s hardware, similar to , we have $$\label{eqn:recip}
{\mathbf{H}}_{{\mathcal{A}}{\rightarrow}{\mathcal{B}}} ={{\mathbf{F}}_{\mathcal{B}}}^{-T}{\mathbf{H}}_{{\mathcal{B}}{\rightarrow}{\mathcal{A}}}^T{{\mathbf{F}}_{\mathcal{A}}},$$ where ${\mathbf{F}}_{\mathcal{A}}= {\mathbf{R}}_{\mathcal{A}}^{-T}{\mathbf{T}}_{\mathcal{A}}$ and ${\mathbf{F}}_{\mathcal{B}}= {\mathbf{R}}_{\mathcal{B}}^{-T}{\mathbf{T}}_{\mathcal{B}}$ are the calibration matrices. In a practical system, the off-diagonal elements ${\mathbf{F}}_{\mathcal{A}}$ and ${\mathbf{F}}_{\mathcal{B}}$ representing the RF crosstalk and antenna mutual coupling are much smaller than the diagonal elements representing the main calibration coefficients. In fact, in-depth theoretical modeling on the calibration matrix in [@petermann2013multi], system measurements from experiment such as in [@Jiang2015], as well as practical experience in fully digital testbeds such as in [@shepard2012argos; @vieira2017reciprocity] all indicate tha ${\mathbf{F}}_{\mathcal{A}}$ and ${\mathbf{F}}_{\mathcal{B}}$ can be considered to be diagonal. The calibration estimation problem will thus be much simplified. Besides if we use ${\mathbf{F}}$ to denote the whole calibration matrix, we have ${\mathbf{F}}= \mbox{diag}\{{\mathbf{F}}_{\mathcal{A}}, {\mathbf{F}}_{\mathcal{B}}\} = \mbox{diag}\{f_1,\ldots,f_N\}$, where $\mbox{diag}\{\cdot\}$ represents the operation to construct a diagonal matrix with given elements on its diagonal.
![Internal calibration where the whole antenna array is partitioned into group ${\mathcal{A}}$ and group ${\mathcal{B}}$. We then perform intra-array measurement between the two groups.[]{data-label="fig:inter_calib"}](hybrid_int_cal_sub_array.png){width="\columnwidth"}
Internal reciprocity calibration consists in estimating ${\mathbf{F}}$ based on the intra-array channel measurement $\hat{{\mathbf{H}}}_{{\mathcal{A}}{\rightarrow}{\mathcal{B}}}$ and $\hat{{\mathbf{H}}}_{{\mathcal{B}}{\rightarrow}{\mathcal{A}}}$, without any involvement of other transceivers. Since the calibration coefficients stay quite stable during a relatively long time, once they are estimated, we can use them together with instantaneously estimated UL channel estimation $\hat{{\mathbf{H}}}_{UL}$ to obtain CSIT.
Let us denote the antenna index in group ${\mathcal{A}}$ and ${\mathcal{B}}$ by $i$ and $j$, respectively, since ${\mathbf{F}}$ is a diagonal matrix, we have $$\begin{split}
&h_{i{\rightarrow}j} = f_{j}^{-1}h_{j{\rightarrow}i}f_{i}, \\
\mbox{where,} \;\; i\in \{1,2,& ...,\frac{N_{ant}}{2}\}, j\in \{\frac{N_{ant}}{2}+1,...,N_{ant}\}.
\end{split}$$ The problem then becomes very similar to that in [@rogalin2014scalable]. Let us use $J$ to denote the cost function of a LS problem: $$J(f_1, f_2,...,f_{ant}) = \sum_{i\in A, j\in B}|f_j h_{i{\rightarrow}j} -f_i h_{j{\rightarrow}i}|^2.
\label{eqn:LS_cost_fun}$$
Estimating the calibration coefficients consists in minimizing $J$ subject to a constraint, e.g., assuming a unit norm or the first calibration coefficient to be known. We adopt here the unit norm constraint, such as $\|{\mathbf{f}}\| = 1$, where ${\mathbf{f}}$ is the diagonal vector of ${\mathbf{F}}$. The Lagrangian function of the constrained LS problem is given by $$L({\mathbf{f}}, \lambda) = J({\mathbf{f}}) - \lambda(\|{\mathbf{f}}\|^2-1),$$ where $\lambda$ is the Lagrangian multiplier. By setting the partial derivatives of $L({\mathbf{f}}, \lambda)$ with regard to $f_i^*$ and $f_j^*$ to zeros, respectively, where $f_i^*$ and $f_i$ are treated as if they were independent variable [@hjorungnes2007complex], we obtain $$\left\{
\begin{split}
\frac{\partial L({\mathbf{f}}, \lambda)}{\partial f_i^*} &= \Sigma_{j\in B} (f_i|h_{j{\rightarrow}i}|^2 - f_j h_{j{\rightarrow}i}^*h_{i{\rightarrow}j}) -\lambda f_i= 0,\\
\frac{\partial L({\mathbf{f}}, \lambda)}{\partial f_j^*} &= \Sigma_{i\in A} (f_j|h_{i{\rightarrow}j}|^2 - f_i h_{i{\rightarrow}j}^*h_{j{\rightarrow}i}) -\lambda f_j= 0.
\end{split}
\right.
\label{eqn:partial}$$ The matrix representation of is ${\mathbf{Q}}{\mathbf{f}}=\lambda{\mathbf{f}}$, where ${\mathbf{Q}}\in\mathbb{C}^{{N_{ant}}\times N_{ant}}$, whose element on its $i$-th row and $m$-th column is $$Q_{i,m} = \left\{
\begin{split}
&\Sigma_{j\in {\mathcal{B}}}|h_{j {\rightarrow}i}|^2 \; &\mbox{for} \;\; &m = i,\\
&-h_{m {\rightarrow}i}^*h_{i{\rightarrow}m} \;&\mbox{for}\;\; &m \in {\mathcal{B}},\end{split}
\right.$$ and the element on the $j$-th row and $m$-th column is given by $$Q_{j,m} = \left\{
\begin{split}
&\Sigma_{i\in {\mathcal{A}}}|h_{i {\rightarrow}j}|^2 \; &\mbox{for} \;\; &m = j,\\
&-h_{m {\rightarrow}j}^*h_{j{\rightarrow}m} \;&\mbox{for}\;\; &m \in {\mathcal{A}}.\end{split}
\right.$$ whereas all other elements are $0$. The solution is given by the eigenvector of $Q$ corresponding to the eigenvalue having smallest magnitude.
Calibration for fully connected structure {#subsec:calib_full}
-----------------------------------------
Until now, we have concentrated the reciprocity based CSIT acquisition method under the subarray structure. In this section, we give some ideas on how to calibrate a fully connected architecture for CSIT acquisition. Consider a system with BS and UE both using fully connected hybrid beamforming structure as in Fig. \[fig:hybrid\_sys\_full\_str\].
{width="1.8\columnwidth"}
We use ${\mathbf{U}}^t_{BS} \in \mathbb{C}^{N_{ant}^{BS}\times N_{ant}^{BS}N_{RF}^{BS}}$ and ${\mathbf{U}}^r_{UE} \in \mathbb{C}^{N_{RF}^{UE}N_{ant}^{UE}\times N_{ant}^{UE}}$ to denote the summation array between the PA and the antennas at the BS and the corresponding summation operation between the antennas and LNAs at the UE, respectively. The signal model can be written as $$\label{eqn:sig_model_full}
{\mathbf{y}}= {\mathbf{W}}_{BB}^{UE}{\mathbf{R}}_1^{UE}{\mathbf{W}}_{RF}^{UE}{\mathbf{U}}^r_{UE}{\mathbf{R}}_2^{UE}{\mathbf{C}}{\mathbf{T}}_2^{BS}{\mathbf{U}}^t_{BS}{\mathbf{V}}_{RF}^{BS}{\mathbf{T}}_1^{BS}{\mathbf{V}}_{BB}^{BS}{\mathbf{s}}+ {\mathbf{n}},$$ An example of the summation array ${\mathbf{U}}^t_{BS}$ for $N_{ant}^{BS} =4 $ and $N_{RF}^{BS}=2 $ (i.e. 8 phase shifters) has the following structure: $${\mathbf{U}}^t_{BS} =
\begin{bmatrix}
1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\
\end{bmatrix}
\label{eqn:sum_matrix_example}$$
As ${\mathbf{U}}^t_{BS}$ can be viewed as a block row vector composed of $N_{RF}^{BS}$ identity matrix ${\mathbf{I}}_{N_{ant}^{BS}}$, i.e. $\begin{bmatrix}{\mathbf{U}}^t_{BS} = {\mathbf{I}}_{N_{ant}^{BS}} & {\mathbf{I}}_{N_{ant}^{BS}} & \cdots & {\mathbf{I}}_{N_{ant}^{BS}} \end{bmatrix}$, we can use a Kronecker product to commute ${\mathbf{T}}_2^{BS}{\mathbf{U}}^t_{BS}$ such as ${\mathbf{T}}_2^{BS}{\mathbf{U}}^t_{BS} = {\mathbf{U}}^t_{BS}({\mathbf{I}}_{N_{RF}^{BS}}\otimes{\mathbf{T}}_2^{BS})$. This is equivalent to move the replicas of the PAs (as well as other components) near the transmit antennas onto each branch before the summation operation. A similar approach can be adopted for the UE, we can thus get an equivalent system model of : $$\label{eqn:sig_model_full_equ}
\begin{aligned}
{\mathbf{y}}= &\underbrace{{\mathbf{W}}_{BB}^{UE}{\mathbf{W}}_{RF}^{UE}}_{{\mathbf{W}}_{UE}}
\underbrace{({\mathbf{R}}_1^{UE}\otimes{\mathbf{I}}_{N_{ant}^{UE}})({\mathbf{I}}_{N_{RF}^{UE}}\otimes{\mathbf{R}}_2^{UE})}_{{\mathbf{R}}_{UE}}
\underbrace{{\mathbf{U}}^r_{UE}{\mathbf{C}}{\mathbf{U}}^t_{BS}}_{\tilde{{\mathbf{C}}}}\\
&\underbrace{({\mathbf{I}}_{N_{RF}^{BS}}\otimes{\mathbf{T}}_2^{BS})({\mathbf{T}}_1^{BS}\otimes{\mathbf{I}}_{N_{ant}^{BS}})}_{{\mathbf{T}}_{BS}}
\underbrace{{\mathbf{V}}_{RF}^{BS}{\mathbf{V}}_{BB}^{BS}}_{{\mathbf{V}}_{BS}}{\mathbf{s}}+ {\mathbf{n}},
\end{aligned}$$ where ${\mathbf{I}}_{N_{ant}^{BS}}$ and ${\mathbf{I}}_{N_{RF}^{UE}}$ are identity matrices. If we consider ${\mathbf{U}}^r_{UE}{\mathbf{C}}{\mathbf{U}}^t_{BS}$ as a composite propagation channel $\tilde{{\mathbf{C}}}$, the equivalent signal model is similar to .
When the system is in the UL transmission, the switches at the BS are connected to receive paths whereas those at the UE are connected to transmit paths. Thus, the UL composite channel can be written as ${\mathbf{U}}^r_{BS}{\mathbf{C}}^T{\mathbf{U}}^t_{UE}$, which can be verified as $\tilde{{\mathbf{C}}}^T$, implying that reciprocity is maintained for the composite propagation channel. Note that if there exists some hardware impairment in the summation operation, we can represent ${\mathbf{U}}^t$ and ${\mathbf{U}}^r$ by ${\mathbf{E}}^t{\mathbf{U}}_0^t$ or ${\mathbf{U}}_0^r{\mathbf{E}}^r$ where ${\mathbf{U}}_0$ is the ideal summation matrix as in , ${\mathbf{E}}^t$ and ${\mathbf{E}}^r$ are impairment matrices which can be absorbed into ${\mathbf{T}}_2^{BS}$ or ${\mathbf{R}}_2^{UE}$.
For a fully connected architecture, internal reciprocity calibration is not feasible since it is not possible to partition the whole antenna array into transmit and receive antenna groups. To enable TDD reciprocity calibration, a reference UE with a good enough channel should be selected to assist the BS to calibrate, such as [@shi2011efficient] proposed for a fully digital system. In this case, the bi-directional transmission no longer happens between two partitioned antenna groups ${\mathcal{A}}$ and ${\mathcal{B}}$ but between the BS and the UE. The selected reference UE needs to feed back its measured DL channel to the BS during the calibration procedure. Methods in Section \[subsec:intern\_calib\] can still be used to estimate the calibration matrices for both BS and UE. Note that although UE feedback is heavy, the calibration does not have to be done very frequently, thus such a method is still feasible. Another possible way is to use a dedicated device at the BS to assist the antenna array for calibration, e.g., using a reference antenna as in [@shepard2012argos]. Using this method, DL channel measurements feedback from UE can be avoided, but a dedicated digital chain needs to be allocated to the assistant device, introducing an extra cost.
Simulation Results
==================
As a proof-of-concept, we perform internal calibration simulation for a subarray hybrid transceiver with 64 antennas and 8 RF chains. To the extent of our knowledge, signal mixers and amplifiers are the main sources of hardware asymmetry. For different RF chains, signal mixers introduce random phases when multiplying the baseband signal with the carrier, whereas the gain imbalance between different amplifiers can cause their output signal having different amplitudes. Other components can also have some minor impacts, e.g., the non-accuracy in the phase shifter can add a further random factor to the phase. In this simulation, we capture the main effects of these hardware properties introduced by signal mixers and amplifiers, though the calibration method is not limited to this simplified case. We assume that the random phases introduced by the signal mixers in ${\mathbf{T}}_1^{BS}$ and ${\mathbf{R}}_1^{BS}$ are uniformly distributed between $-\pi$ and $\pi$ whereas the amplitudes in ${\mathbf{T}}_2^{BS}$ and ${\mathbf{R}}_2^{BS}$ are independent variables uniformly distributed in $[1-\epsilon \;1+\epsilon]$, with $\epsilon$ chosen such that the standard deviation of the squared-magnitude is 0.1.
The intra-array channel model between antenna elements strongly depends on the antenna arrangement in the array, antenna installation, as well as the frequency band. In the simulation, we focus on a sub-6GHz scenario and adopt the experiment based intra-array radio channel in [@vieira2017reciprocity], where the physical channel $c_{i,j}$ between two antenna elements $i$ and $j$ in the same planar antenna array is modeled as $$c_{i,j} = |\bar{c}_{i,j}|\mbox{exp}(j2\pi\phi_{i,j})+\tilde{c}_{i,j}.
\label{eqn:intra_array_ch_model}$$ In , $\bar{c}_{i,j}$ is the near field path[^4] between two antenna elements and $\tilde{c}_{i,j}$ absorbs all other multi-path contributions due to reflections from obstacles around the antenna array. For simplicity reasons, we assume the 64 antennas follows a co-polarized linear arrangement with an antenna space of half of the wavelength. According to the measurements in [@vieira2017reciprocity], the magnitude for two half-wavelength spaced antennas are $-15$dB and at each distance increase of half of the wavelength, $|\bar{c}_{i,j}|$ decreases by 3.5dB. $\phi_{i,j}$ is modeled as uniformly distributed in $[0, 1)$ since a clear dependence with distance was not found. The multi-path component is modeled by an i.i.d zero-mean circularly symmetric complex Gaussian random variable with variance $\sigma^2=0.001$.
For the internal calibration, different antenna partition strategies are possible, where the optimal solution is yet to be discovered. In our simulation, we choose two different antenna partition scenarios: “two sides partition" and “interleaved partition", as shown in Fig. \[fig:antenna\_partition\]. The “two sides partition" separates the whole antenna array to group ${\mathcal{A}}$ and ${\mathcal{B}}$ on the left and right sides whereas the “interleaved partition" assigns every 8 antennas to ${\mathcal{A}}$ and ${\mathcal{B}}$ alternatively.
![(a) “two sides partition" where group ${\mathcal{A}}$ and ${\mathcal{B}}$ contain 32 antennas on the left and right sides of the linear antenna array, respectively; (b) “interleaved partition" where every 8 antennas are assigned to group ${\mathcal{A}}$ and ${\mathcal{B}}$, alternatively.[]{data-label="fig:antenna_partition"}](two-sides-partition.png){width="\columnwidth"}
![(a) “two sides partition" where group ${\mathcal{A}}$ and ${\mathcal{B}}$ contain 32 antennas on the left and right sides of the linear antenna array, respectively; (b) “interleaved partition" where every 8 antennas are assigned to group ${\mathcal{A}}$ and ${\mathcal{B}}$, alternatively.[]{data-label="fig:antenna_partition"}](interleaved-partition.png){width="\columnwidth"}
In the first simulation, we would like to verify the feasibility to calibrate a hybrid beamforming transceiver using internal calibration. For this purpose, we use the “two sides partition" scenario and assume no noise in the bi-directional transmission between group ${\mathcal{A}}$ and ${\mathcal{B}}$. We use 8 randomly generated independent QPSK symbols as pilots after the baseband digital beamforming and only apply analog precoding whose weights have a unit amplitude, with their phases uniformly distributed in $[-\pi\;\pi)$. Using $K=32$ and $L=5$ such randomly generated transmit and receive analog beam weights to accumulate 160 measurements[^5] and applying the method \[subsec:intern\_calib\] on the accumulated signal, we can obtain the estimated calibration coefficients. For the purpose of illustration, we eliminate the complex scalar ambiguity, the results are shown in Fig. \[fig:F\_est\].
![Estimated calibration matrix vs. real calibration matrix. The blue circles are predefined calibration coefficients and the red stars are estimated values after elimination of the complex scalar ambiguity.[]{data-label="fig:F_est"}](F_est){width="\columnwidth"}
We observe that the calibration matrix are partitioned in 8 groups, corresponding to 8 RF chains each with its own signal mixer. On each angle, elements have different amplitudes, which mainly correspond to the gain imbalance of independent amplifiers on each branch. We also observe that the estimated calibration parameters perfectly match the predefined values, implying that we can recover the coefficients using the proposed method. In a practical system, as no real value of ${\mathbf{F}}$ is known, all estimated coefficients have an ambiguity up to a common complex scalar value as explained in Section \[subsec:intern\_calib\]. In the next simulation, we study the calibration performance with regard to the number of intra-array channel measurements. Since the measurements are within the antenna array, noise from both transmit and receive hardware can impact the received signal’s quality. For antennas near each other, the main noise source comes from the transmit signal, usually measured in error vector magnitude (EVM). Assuming a transmitter with an EVM of $-20$dB, the SNR of the transmit signal is $40$dB. For antennas far away from each other, noise at the receiver is the main limiting factor. Assuming that the system bandwidth is $5$MHz, the thermal noise at room temperature would be $-107$dBm at the receiving antenna. Using a radio chain with a noise figure of $10$dB and a total receive gain equaling to $0$dB, the noise received in the digital domain would be around $-97$dBm. We assume a 0dBm transmission power per antenna and use the intra-array channel model as in . The calibrated coefficients are measured in its normalized mean square error (NMSE), such as $$\mbox{NMSE}_{{\mathbf{F}}} = \frac{\|\hat{{\mathbf{F}}}-{\mathbf{F}}\|^2}{\|{\mathbf{F}}\|^2}.$$
{width="0.97\columnwidth"}
{width="0.97\columnwidth"}
{width="0.97\columnwidth"}
{width="0.97\columnwidth"}
The results are shown in Fig. \[fig:MSE\_F\] for “two sides partition" and “interleaved partition". We observe in both cases that, when $K < 32$, the estimation of ${\mathbf{F}}$ can not converge, since the intra-array channel estimation problem is under-determined, as explained in Section \[subsec:eff\_ch\_est\]. As long as $K \geq 32$ and $L \geq 8$, it is possible to estimate ${\mathbf{F}}$ up to an accuracy with an NMSE below $10^{-2}$. The “interleaved partition" has a better performance than the “two sides partition" when the minimum $K$ and $L$ requirements are met. This can be explained by the fact that the received signals in the “interleaved partition" have more balanced amplitudes than in the “two sides partition", where, the bi-directional transmission between far away antenna elements have very little impact on the estimation of ${\mathbf{F}}$ since the received signal are small. Note that different sets of transmit and receive analog precoding weights can lead to different performance in the estimation of ${\mathbf{F}}$, with the best set left to be discovered. In our simulation, we randomly choose a set of weights and use it for both the “two sides partition" and the “interleaved partition". For comparison purpose, the set of weights for given $K$ and $L$ values (e.g $K=32, L=8$) is a subset for the weights used when $K$ and $L$ are bigger (e.g $K=33, L=9$).
Since we simulate the intra-array transmission, both the transmit and receive noise have been taken into account. In order to understand the impact from the two noise sources, let us simulate for them independently under both antenna partition scenarios. Fig. \[fig:mse\_trxN\] illustrates the NMSE of ${\mathbf{F}}$ with independently considered noise for “two sides partition" and “interleaved partition". It is obvious that, in both cases, the noise at the transmit side is dominant and limits the accuracy of the estimated ${\mathbf{F}}$ whereas if only the receiver’s thermal noise is considered, NMSE of ${\mathbf{F}}$ becomes negligible. In fact, if we look back at , it is the errors present in the bi-directional channel estimation $h_i$ and $h_j$ with the highest amplitudes (i.e. internal channels between nearby antenna elements) that dominate the cost function. For a receiving antenna near the transmitting element, the received transmit noise is much higher than the thermal noise generated at the receiving antenna itself.
When the system has accomplished internal calibration, it can use the estimated calibration matrix together with instantaneously estimated UL channel to assess the DL CSIT in order to create a beam for data transmission. The accuracy of the estimated DL CSIT depends on both the UL CSI and the estimated calibration matrices. In order to study the impact of both factors, we assume a simple scenario where a subarray hybrid structure BS performs beamforming towards a single antenna UE, such as in [@jiang2016accurately]. In this case, the DL channel ${\mathbf{h}}_{BS {\rightarrow}UE}^T$ (we use transpose since the DL channel is a row vector) can be estimated by $\hat{f}_{UE}^{-1}\hat{{\mathbf{h}}}_{UE{\rightarrow}BS}^T\hat{{\mathbf{F}}}_{BS}$, where $\hat{{\mathbf{h}}}_{UE{\rightarrow}BS}$ is the estimated UL channel. $\hat{{\mathbf{h}}}_{UE{\rightarrow}BS} = {\mathbf{h}}_{UE{\rightarrow}BS} + \Delta {\mathbf{h}}_{UE{\rightarrow}BS}$, where $\Delta {\mathbf{h}}_{UE{\rightarrow}BS}$ is the UL channel estimation error, ${\mathbf{h}}_{UE{\rightarrow}BS} = {\mathbf{R}}_{BS}\mathbf{c} t_{UE}$, with the UL physical channel vector $\mathbf{c}$ modeled as a standard Rayleigh fading channel. In our case, the calibration coefficients at the BS and the UE can be combined such as ${\mathbf{F}}= f_{UE}^{-1}{\mathbf{F}}_{BS}$. Its estimation $\hat{{\mathbf{F}}}$ can be represented by $\hat{{\mathbf{F}}} = {\mathbf{F}}+\Delta {\mathbf{F}}$ with $\Delta {\mathbf{F}}$ denoting the estimation error. The estimation errors in $\Delta {\mathbf{h}}_{UE}$ and $\Delta {\mathbf{F}}$ are assumed to be i.i.d Gaussian random variables with zero mean and $\sigma_{n,UL}^2$, $\sigma_{{\mathbf{F}}}^2$ as their variance, respectively. $\mbox{NMSE}_{{\mathbf{F}}}$ can be calculated as $N_{ant}^{BS}\sigma_{{\mathbf{F}}}^2/\|{\mathbf{F}}\|^2$. Without considering the complex scalar ambiguity, which does not harm the finally created beam, we can calculate the NMSE of the DL CSI as $$\label{eqn:mse_F_err}
\begin{aligned}
\mbox{NMSE}_{DL} =&
\frac{1}{N_{ant}^{BS}}\mathbb{E}{\left[\|\hat{{\mathbf{h}}}_{UE{\rightarrow}BS}^T\hat{{\mathbf{F}}}-{\mathbf{h}}_{BS{\rightarrow}UE}^T\|^2\right]} \\
=& \frac{1}{N_{ant}^{BS}}\mathbb{E}{\left[\|{\mathbf{h}}_{UE {\rightarrow}BS}^T\Delta {\mathbf{F}}+ \Delta {\mathbf{h}}_{UE {\rightarrow}BS}^T\hat{{\mathbf{F}}}\|^2\right]} \\
=&\frac{1}{N_{ant}^{BS}}\mbox{Tr}\left\{\Delta{\mathbf{F}}^H\mathbf{\Omega}^*\Delta{\mathbf{F}}+ \sigma_{n, UL}^2\hat{{\mathbf{F}}}^H\hat{{\mathbf{F}}}\right\}
\end{aligned}$$ where $\mathbf{\Omega}$ is the covariance matrix of the UL channel, i.e. $\mathbf{\Omega} = \mathbb{E}[{\mathbf{h}}_{UE {\rightarrow}BS}{\mathbf{h}}_{UE {\rightarrow}BS}^H]$.
The NMSE of the calibrated CSIT as a function of different $\mbox{NMSE}_{\mathbf{F}}$ and $\mbox{NMSE}_{UL}$[^6] is shown in Fig. \[fig:MSE\_vs\_ULMSE\]. We observe that when the accuracy of the UL CSI is low, it is the main accuracy limiting factor on the calibrated DL CSIT. As the UL CSI accuracy increases, the accuracy on $\hat{{\mathbf{F}}}$ begins to influence the DL CSIT. In a calibrated system where $\mbox{NMSE}_{\mathbf{F}}= 10^{-2}$ and $\mbox{NMSE}_{UL}=10^{-2}$, it is possible to obtain DL CSIT with an NMSE under $10^{-1}$.
![The accuracy of acquired CSIT as a function of NMSE of the reciprocity calibration matrix and instantaneously measured UL CSI.[]{data-label="fig:MSE_vs_ULMSE"}](MSE_vs_ULMSE){width="\columnwidth"}
Conclusion
==========
In this paper, we presented a CSIT acquisition method based on reciprocity calibration in a TDD hybrid beamforming massive MIMO system. Compared to state-of-the-art methods which assume a certain structure in the channel such as the limited scattering property validated only in mmWave, this method can be used for all frequency bands and arbitrary channels. Once the TDD system is calibrated, accurate CSIT can be directly obtained from the reverse channel estimation, without any beam training or selection. It thus offers a new way to operate hybrid AD beamforming systems.
[^1]: This work was supported in part by Huawei Mathematical and Algorithmic Sciences Lab in Paris through the project of “Modeling, Calibrating and Exploiting Channel Reciprocity for Massive MIMO". This work was also supported in part by the French Government (National Research Agency, ANR) through the “Investments for the Future” Program \#ANR-11-LABX-0031-01.
[^2]: The authors are with Communication Systems Department, EURECOM, Campus SophiaTech, 06410 Biot, France (e-mail: [email protected]; [email protected]).
[^3]: In a multi-user scenario, the impact from UE side calibration might increase with the number of served UEs. In this case, each UE can feed back its calibration coefficients back to the BS.
[^4]: This term is called “antenna mutual coupling" in [@vieira2017reciprocity], which is slightly different from the classical mutual coupling defined in [@balanis2016antenna] where two nearby antennas are both transmitting or receiving. We thus call this term “near field path" describing the main signal propagation from one antenna to its neighbor element.
[^5]: Note that in a practical multi-carrier system, the channel estimation on different RF chains can be performed on different frequencies as explained in Section \[subsec:eff\_ch\_est\], the needed K can then be much less.
[^6]: $\mbox{NMSE}_{UL}=\frac{1}{N_{ant}^{BS}}\mathbb{E}\left[\|\Delta {\mathbf{h}}_{UE {\rightarrow}BS}\|^2\right] = \sigma_{n,UL}^2$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We explore Jaeger’s state model for the HOMFLYPT polynomial. We reformulate this model in the language of Gauss diagrams and use it to obtain Gauss diagram formulas for a two-parameter family of Vassiliev invariants coming from the HOMFLYPT polynomial. These formulas are new already for invariants of degree 3.'
address: 'The Ohio State University, Mansfield, 1680 University Drive, Mansfield, OH 44906. [[email protected]]{}Department of mathematics, Technion, Haifa 32000, Israel. [[email protected]]{}'
author:
- SERGEI CHMUTOV and MICHAEL POLYAK
title: Elementary combinatorics of the HOMFLYPT polynomial
---
Introduction {#s:intro .unnumbered}
============
The [*HOMFLYPT polynomial*]{} $P(L)$ is an invariant of oriented link $L$. It is defined as the Laurent polynomial in two variables $a$ and $z$ with integer coefficients satisfying the following skein relation and the initial condition: $$\label{eq:skein}
aP({{\raisebox}{-4.2mm}{{\includegraphics}[width=10mm]{lrints.eps}}}) - a^{-1}P({{\raisebox}{-4.2mm}{{\includegraphics}[width=10mm]{rlints.eps}}}) = zP({{\raisebox}{-4.2mm}{{\includegraphics}[width=10mm]{twoup.eps}}})\ ;\qquad P({{\raisebox}{-4.2mm}{{\includegraphics}[width=10mm]{unkn.eps}}})\quad
= \quad 1\,.$$ If $L$ is an unlink with $m$ components then $P(L)=\Bigl(\frac{a-a^{-1}}{z}\Bigr)^{m-1}$. The proof of the existence of such an invariant is long and cumbersome. It was established simultaneously and independently by five groups of authors [@HOM; @PT].
This paper is devoted to Gauss diagram formulas for Vassiliev invariants coming from the HOMFLYPT polynomial. It is known [@GPV] that any Vassiliev knot invariant may be presented by a Gauss diagram formula. This type of formulas is the simplest for computation purposes; however, the algorithm for producing them is complicated and until recently only few lower degree cases were described explicitly. The first description of such formulas for an infinite family of Vassiliev invariants was given in [@CKR], where the coefficients of the Conway polynomial were considered. This paper generalizes the result of [@CKR] to the HOMFLYPT polynomial.
We use a non-standard change of variables (used formely in [@G2]), leaving $z$ alone and plugging in $a=e^h$ to obtain a power series $\sum_{k,l}p_{k,l}h^kz^l$. The coefficients $p_{k,l}$ are Vassiliev invariants of degree $\leqslant k+l$, see [@G2]. We give the Gauss diagram formulas for $p_{k,l}$ for arbitrary $k,l$. These formulas are new already for invariants of degree 3.
The paper is organized in the following way. In Section \[s:homfly\] we start from the scheme of [@H; @LM; @PT], extracting from it an explicit state model for the HOMFLYPT following [@Ja] in Section \[s:jaeger\]. We then briefly review the notions of Gauss diagrams in Section \[s:gaus-diagr\] and reformulate the state model in these terms in Section \[s:jaeger-ga\]. The expansion of $P(L)$ into power series in $h$ and $z$ is considered in Section \[s:vas-from-homfly\]. In the same section we remind the definition of the Gauss diagram formulas for Vassiliev invariants. Finally, we describe the Gauss diagram formulas for $p_{k,l}$ in Section \[s:result\]. In the last Section \[s:example\] we analyze low degree cases in details.
Note that using instead of the skein relation for the two-variable Kauffman polynomial, one gets a similar state model. We plan to consider the resulting Gauss diagram formulas in a forthcoming paper.
We are grateful to O. Viro, L. Traldi, and to the anonymous referee for numerous corrections to the first version of the paper and useful remarks. This work has been done when both authors were visiting the Max-Plank-Institut für Mathematik in Bonn, which we would like to thank for excellent work conditions and hospitality. The second author was supported by a grant 3-3577 of the Israel Ministry of Science and ISF grant 1261/05.
HOMFLYPT and descending diagrams {#s:homfly}
================================
The skein relation allows one to calculate the HOMFLYPT polynomial of a link. Following [@H; @LM; @PT], this can be done by ordering a link diagram and then transforming it into a descending diagram. We call a diagram $D$ [*ordered*]{}, if its components $D_1$, $D_2$,…,$D_m$ are ordered and on every component a (generic) base point is chosen. An ordered diagram is [*descending*]{}, if $D_i$ is above $D_j$ for all $i<j$ and if for every $i$ as we go along $D_i$ starting from its base point along the orientation we pass each self-crossing first on the overpass and then on the underpass.
An elementary step of the algorithm computing $P(L)$ consists of the following procedure. Suppose that $D$ is an ordered diagram and that the subdiagram $D_1,\dots,D_{i-1}$ is already descending. We go along $D_i$ (starting from the base point) looking for the first crossing which fails to be descending. At such a crossing $x$ we change it using the skein relation. Namely, depending on the sign $\e$ (the local writhe) of the crossing, we express $P(D)$ as $$\label{eq:descend}\begin{array}{ccl}
P({{\raisebox}{-4.2mm}{{\includegraphics}[width=10mm]{lrints.eps}}})&=&a^{-2}P({{\raisebox}{-4.2mm}{{\includegraphics}[width=10mm]{rlints.eps}}}) + a^{-1}zP({{\raisebox}{-4.2mm}{{\includegraphics}[width=10mm]{twoup.eps}}})\vspace{5pt}\\
P({{\raisebox}{-4.2mm}{{\includegraphics}[width=10mm]{rlints.eps}}})&=&a^{2}P({{\raisebox}{-4.2mm}{{\includegraphics}[width=10mm]{lrints.eps}}}) - azP({{\raisebox}{-4.2mm}{{\includegraphics}[width=10mm]{twoup.eps}}})
\end{array}$$ Denote the corresponding diagrams $D^\e$, $D^{-\e}$, $D^0$.
The ordering of $D=D^\e$ induces an ordering of $D^{-\e}$ (in an obvious way); the ordering of $D^0$ requires some explanation. If $x$ was a crossing of $D_i$ with $D_j$, $j>i$, then these two components merge into a single component $D^0_i$ of $D^0$, with a base point being the base point of $D_i$. If $x$ was a self-crossing of $D_i$, then $D_i$ splits into two components: $D^0_i$, which contains the base point of $D_i$, and $D^0_{i+1}$, where we choose the base point in a neighborhood of $x$. In both cases the order of remaining components shifts accordingly.
The diagrams $D^{-\e}$, $D^0$ are “more” descending than $D^\e$. At the next step we apply the same procedure to each of them.
\[ex:1a\] For the trefoil $3_1$ the algorithm consists of two steps, illustrated in the figure below. The diagram $D^+$ appearing in the first step is already descending; the diagram $D^0$ is not, so the second step is needed to transform it.
$${{\raisebox}{-15pt}[90pt][25pt]{\begin{picture}(160,15)(0,0)
\put(0,0){{\includegraphics}[width=160pt]{31alg.eps}} \put(20,60){\mbox{$D^-$}}
\put(-60,85){\mbox{Step 1:}}\put(-60,20){\mbox{Step 2:}}
\put(85,60){\mbox{$D^+$}}\put(150,60){\mbox{$D^0$}}
\put(25,-5){\mbox{$D^-$}}\put(98,-5){\mbox{$D^+$}}
\put(140,-5){\mbox{$D^0$}}
\end{picture}}}$$ Hence $P(3_1)=a^2\cdot 1 - az\Bigl(
a^2\cdot\frac{a-a^{-1}}{z}-az\cdot 1\Bigr) = (2a^2-a^4)+a^2z^2$.
State model reformulation {#s:jaeger}
=========================
The state model of [@Ja] for the HOFMLYPT polynomial is a convenient reformulation of the algorithm of Section \[s:homfly\].
A [*state*]{} $S$ on a link diagram $D$ is a subset of its crossings. The HOMFLYPT polynomial is going to be a sum over the states. Let $D(S)$ be the link diagram obtained by smoothing every crossing in $S$ according to orientation and $c(S)$ be the number of its components. We will not use the topology of $D(S)$, however its combinatorics will determine the contribution of the state $S$ to the state sum. The contribution will be a product of a global weight of the state as a whole, $\bigl(\frac{a-a^{-1}}{z}\bigr)^{c(S)-1}$ and local weights of crossings of the diagram.
The ordering of $D$ induces an ordering of $D(S)$ (in the way explained in Section \[s:homfly\] above) and thus determines a tracing of the link $D(S)$. The local weight $\lw$ of a crossing $x$ of $D$ depends on the first passage of a neighborhood of $x$ in the tracing and on the sign $\e$ of $x$. Namely, if $x$ is in $S$ and we approach $x$ first time on an overpass of $D$ then $\lw=0$ (since such a situation does not occur in the above algorithm). If we approach $x$ on an underpass of $D$ then $\lw=\e a^{-\e}z$ (i.e., the coefficient of $D^0$ in ). In the case if $x$ does not belong to $S$ and we approach $x$ first time on an overpass then $\lw=1$ (since in the above algorithm we do not apply the skein relation to $x$). If we approach $x$ on an underpass then $\lw=a^{-2\e}$ (i.e., the coefficient of $D^{-\e}$ in ). These assignments can be summarized in the following figure. $$\begin{array}{c||c|c|c|c}
{\raisebox}{5pt}{First
passage:}&{{\raisebox}{10pt}[0pt][0pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{br2.eps}}
\end{picture}}}&{{\raisebox}{0pt}[20pt][3pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{br1.eps}}
\end{picture}}}&
{{\raisebox}{0pt}[0pt][0pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{br4.eps}}
\end{picture}}}&{{\raisebox}{0pt}[0pt][0pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{br3.eps}}
\end{picture}}} \\ \hline\hline
{{\raisebox}{-8pt}[15pt][10pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{cr_p.eps}}
\end{picture}}} & 0 & a^{-1}z & 1 & a^{-2}\\ \hline
{{\raisebox}{-8pt}[15pt][10pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{cr_m.eps}}
\end{picture}}} & -az & 0 & a^2 & 1
\end{array}$$ Denote by $\plw:=\prod_x \lw$ the product of local weights of all crossings. For a link $L$ with a diagram $D$ we have [@Ja Proposition 2]: $$P(L)=\sum_S\ \ \plw\cdot \left(\frac{a-a^{-1}}{z}\right)^{c(S)-1}$$
Consider a based trefoil diagram $D$ and a state $S$ consisting of one crossing $\{x_1\}$. $$D={{\raisebox}{-15pt}[20pt][20pt]{\begin{picture}(40,15)(0,0)
\put(0,0){{\includegraphics}[width=40pt]{31.eps}}
\put(17,-3){\mbox{\scriptsize $x_1$}}
\put(34,23){\mbox{\scriptsize $x_2$}}
\put(0,26){\mbox{\scriptsize $x_3$}}
\end{picture}}} \hspace{2cm}
D(S)={{\raisebox}{-20pt}[20pt][0pt]{\begin{picture}(40,15)(0,0)
\put(0,0){{\includegraphics}[width=40pt]{conwlin.eps}}
\end{picture}}}$$ The tracing of $D(S)$ first approaches the crossing $x_1$ on the strand which was an underpass in $D$. So its weight will be $\lvw{x_1}{D}=-az$. Similarly the weights of the other two crossings are $\lvw{x_2}{D}=a^2$ and $\lvw{x_3}{D}=1$. So the total contribution from this state will be equal to $-a^3z\bigl(\frac{a-a^{-1}}{z}\bigr) = a^2-a^4$. The next table shows the contributions from all eight states. Non-zero weights come from states corresponding to descending diagrams appearing in the end of the algorithm, see Example \[ex:1a\]. $$\begin{array}{c|c|c|c|c|c|c|c}
\emptyset & \{x_1\} & \{x_2\} & \{x_3\} & \{x_1,x_2\} & \{x_1,x_3\}
&
\{x_2,x_3\} & \{x_1,x_2,x_3\} \\ \hline \makebox(0,12){}
a^2& a^2-a^4 & 0 & 0 & a^2z^2 & 0 & 0 & 0
\end{array}$$ So we recover the result of Example \[ex:1a\]: $P(3_1)=(2a^2-a^4)+
z^2a^2$.
Smoothing a crossing from a state $S$ changes the number of components by one. Hence the cardinality $|S|$ and the difference $m-c(S)$ (where $m$ is the number of components of $D$) are congruent modulo 2. Therefore the HOMFLYPT polynomial $P(L)$ is even in each of the variables $a$ and $z$ if $m$ is odd, and it is an odd polynomial if $m$ is even.
The negative powers of $z$ come from the factors $\bigl(\frac{a-a^{-1}}{z}\bigr)^{c(S)-1}$. A smoothing of a crossing $x\in S$ may increase $c(S)$ by one, however this increment will be compensated by a local weight $\lw$. As a consequence we have that the lowest power of $z$ in the HOMFLYPT polynomial of $L$ is at least $-m+1$. In particular the HOMFLYPT polynomial $P(K)$ of a knot $K$ is a genuine polynomial in $z$, i.e., does not contain terms with negative powers of $z$.
Gauss diagrams {#s:gaus-diagr}
==============
Gauss diagrams provide an alternative and more combinatorial way to present links. For a link diagram $D$ consider a collection of (counterclockwise) oriented circles parameterizing it. Two preimages of a crossing of $D$ we unite in a pair and connect them by an arrow pointing from the overpassing preimage to the underpassing one. To each arrow we assign a sign $\pm1$ of the corresponding crossing. The result is called the [*Gauss diagram*]{} $G_D$ of the link diagram $D$. A link can be uniquely reconstructed from the corresponding Gauss diagram [@GPV].
For example, a Gauss diagram of the trefoil looks as follows. $$D=\ {{\raisebox}{-15pt}[20pt][20pt]{\begin{picture}(40,15)(0,0)
\put(0,0){{\includegraphics}[width=40pt]{31.eps}}
\end{picture}}} \hspace{3cm}
G_{D}=\ {{\raisebox}{-15pt}[27pt][35pt]{\begin{picture}(40,15)(0,0)
\put(0,0){{\includegraphics}[width=40pt]{gd-31bp.eps}}
\put(-135,-15){\mbox{\tt A knot and its Gauss diagram}}
\end{picture}}}\label{d3-1}$$
Not every diagram with arrows is realizable as a Gauss diagram of a classical link. For example, ${{\raisebox}{-5pt}[14pt][8pt]{\begin{picture}(15,15)(0,0)
\put(0,0){{\includegraphics}[width=15pt]{gauss-nr.eps}}
\end{picture}}}\ $ is not realizable regardless of signs of its arrows. An [*abstract Gauss diagram*]{}, or an [*arrow diagram*]{} is a generalization of a notion of Gauss diagram, in which we forget about realizability. In other words, an arrow diagram consists of a number of oriented circles with several arrows connecting pairs of distinct points on them. The arrows are equipped with signs $\pm1$. We consider these diagrams up to orientation preserving diffeomorphisms of the circles.
We are going to work with [*ordered Gauss diagrams*]{}, i.e. Gauss diagrams with ordered circles and a base point $\bp_1, \bp_2, \dots,
\bp_m$ on each circle corresponding to an ordering of $D$. Similarly, an [*ordered arrow diagram*]{} is an arrow diagram equipped with an ordering of the circles and a base point (different from the end points of the arrows) on each of them.
Two Gauss diagrams represent isotopic links if and only if they are related by a finite number of Reidemeister moves (see, for example, [@GPV; @Oll; @CDbook]). $$\Omega_1:\ {{\raisebox}{-15pt}[18pt][15pt]{\begin{picture}(100,15)(0,0)
\put(0,0){{\includegraphics}[width=100pt]{virrI.eps}}
\put(-2,5){\mbox{$\scriptstyle \e$}}
\put(85,5){\mbox{$\scriptstyle \e$}}
\end{picture}}} \hspace{2cm}
\Omega_2:\ {{\raisebox}{-15pt}[0pt][0pt]{\begin{picture}(95,15)(0,0)
\put(0,0){{\includegraphics}[width=95pt]{virrII1.eps}}
\put(6,18){\mbox{$\scriptstyle \e$}}
\put(22,18){\mbox{$\scriptstyle -\e$}}
\end{picture}}}$$ $$\Omega_3: {{\raisebox}{-18pt}[20pt][22pt]{\begin{picture}(120,15)(0,0)
\put(0,0){{\includegraphics}[width=120pt]{virrIII.eps}}
\end{picture}}}\ .$$ Note that the segments involved in $\Omega_2$ or $\Omega_3$ may lie on the different components of the link. So the order in which they are traced along the link may be arbitrary.
An [*ordered link*]{} is an equivalence class of ordered Gauss diagrams modulo Reidemeister moves which do not involve base points. For $m=1$ this notion is equivalent to the notion of long knots defined as embeddings of ${\mathbb{R}}$ into ${\mathbb{R}}^3$ which coincide with a standard embedding (say, an $x$-axis) outside a compact. It is well known that for classical knots the theories of long and closed knots coincide.
State models on Gauss diagrams {#s:jaeger-ga}
==============================
All notions and constructions of Section \[s:jaeger\] have a straightforward translation to the language of Gauss diagrams.
A [*state*]{} $S$ on an abstract Gauss diagram $G$ is a subset of its arrows. Let $G(S)$ be the abstract Gauss diagram obtained by doubling every arrow in $S$ as in the figure $${{\raisebox}{-6pt}[0pt][5pt]{\begin{picture}(30,15)(0,0)
\put(0,0){{\includegraphics}[width=30pt]{arrow.eps}}
\end{picture}}}\qquad
{{\raisebox}{-2pt}[0pt][0pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{totor.eps}}
\end{picture}}}\qquad{{\raisebox}{-6pt}[0pt][0pt]{\begin{picture}(30,15)(0,0)
\put(0,0){{\includegraphics}[width=30pt]{darrow.eps}}
\end{picture}}}\ ,$$ and let $c(S)$ be the number of its circles. The ordering of $G$ induces an ordering of $G(S)$. The local weight $\lwg$ of an arrow $\a$ of $G$ in general depends on whether $\a$ belongs to $S$, on the first passage in a neighborhood of $\a$ in the tracing of $G(S)$, and on the sign $\e$ of $\a$. Given a table of such local weights, we denote by $\plwg:=\prod_\a \lwg$ the product of local weights of all arrows and define a polynomial $P(G)$ by $$\label{eq:PG}
P(G):=\sum_S\ \ \plwg\cdot \left(\frac{a-a^{-1}}{z}\right)^{c(S)-1}$$
The table of local weights for the HOMFLYPT state model (readily taken from Section \[s:jaeger\]) is shown below. $$\label{eq:lwg}
\begin{array}{c||c|c|c|c}
{\raisebox}{7pt}{First passage:}&
{{\raisebox}{0pt}[25pt][3pt]{\begin{picture}(35,15)(0,0)
\put(0,0){{\includegraphics}[width=35pt]{fp-b.eps}}
\end{picture}}}&{{\raisebox}{0pt}[0pt][0pt]{\begin{picture}(35,15)(0,0)
\put(0,0){{\includegraphics}[width=35pt]{fp-t.eps}}
\end{picture}}}&
{{\raisebox}{0pt}[0pt][0pt]{\begin{picture}(35,15)(0,0)
\put(0,0){{\includegraphics}[width=35pt]{fp-l.eps}}
\end{picture}}}&{{\raisebox}{0pt}[0pt][0pt]{\begin{picture}(35,15)(0,0)
\put(0,0){{\includegraphics}[width=35pt]{fp-r.eps}}
\end{picture}}} \\ \hline\hline
{{\raisebox}{-8pt}[22pt][15pt]{\begin{picture}(40,15)(0,0)
\put(0,0){{\includegraphics}[width=40pt]{cr_p-gd.eps}}
\end{picture}}} & a^{-1}z & 0 & a^{-2} & 1 \\
\hline {{\raisebox}{-8pt}[22pt][15pt]{\begin{picture}(40,15)(0,0)
\put(0,0){{\includegraphics}[width=40pt]{cr_m-gd.eps}}
\end{picture}}} & -az & 0 & a^2 & 1
\end{array}$$
For the Gauss diagram of the trefoil the states with non-zero weights are the following. $$\begin{array}{r||c|c|c}
\mbox{States of\quad } {{\raisebox}{-10pt}[20pt][18pt]{\begin{picture}(30,15)(0,0)
\put(0,0){{\includegraphics}[width=30pt]{gd-31bp.eps}}
\end{picture}}}\ :&
{{\raisebox}{-10pt}[0pt][0pt]{\begin{picture}(30,15)(0,0)
\put(0,0){{\includegraphics}[width=30pt]{s31-0.eps}}
\end{picture}}} & {{\raisebox}{-10pt}[0pt][0pt]{\begin{picture}(33,15)(0,0)
\put(0,0){{\includegraphics}[width=33pt]{s31-1.eps}}
\end{picture}}} &
{{\raisebox}{-10pt}[0pt][0pt]{\begin{picture}(32,15)(0,0)
\put(0,0){{\includegraphics}[width=32pt]{s31-12.eps}}
\end{picture}}} \\
\mbox{Weights}\ : & 1\cdot a^2\cdot 1&
1\cdot (-az)\cdot a^2\cdot \Bigl(\frac{a-a^{-1}}{z}\Bigr)&
1\cdot (-az)\cdot (-az)
\end{array}$$ Hence, $P(G)=(2a^2-a^4)+ z^2a^2$.
The HOMFLYPT polynomial defined by this state model may be called the [*descending*]{} HOMFLYPT polynomial. An [*ascending*]{} HOMFLYPT polynomial may be defined in a similar way, interchanging the values of the first two columns and the last two columns in the table (\[eq:lwg\]) of local weights. For classical links these two polynomials coincide.
Vassiliev invariants coming from the HOMFLYPT polynomial {#s:vas-from-homfly}
========================================================
HOMFLYPT power series
---------------------
A standard way [@BN; @BL] to relate Vassiliev invariants to the HOMFLYPT polynomial is to make a substitution $a=e^{Nh}$, $z=e^h-e^{-h}$ and then take the Taylor expansion of $P(L)$ in the variable $h$. The coefficient at $h^n$ turns out to be a Vassiliev invariant of order $\leqslant n$ which depends on a parameter $N$.
In this paper we are working in a different way, following [@G2]. Namely, we substitute $a=e^h$ and take the Taylor expansion in $h$. The result will be a Laurent polynomial in $z$ and a power series in $h$. Let $p_{k,l}(L)$ be its coefficient at $h^kz^l$. It is not difficult to see that for any link $L$ the total degree $k+l$ is not negative. (It also follows from the Jaeger model in Section \[s:jaeger\].)
$p_{k,l}(L)$ is a Vassiliev invariant of order $\leqslant k+l$.
Indeed, plugging $a=e^h$ into the skein relation we get $$P({{\raisebox}{-4.2mm}{{\includegraphics}[width=10mm]{lrints.eps}}})\ -\ P({{\raisebox}{-4.2mm}{{\includegraphics}[width=10mm]{rlints.eps}}})\ = \ zP({{\raisebox}{-4.2mm}{{\includegraphics}[width=10mm]{twoup.eps}}}) + h(\mbox{some terms})\ .$$ Since all terms of the HOMFLYPT polynomial have non-negative total degree in $z$ and $h$, the terms of the right hand side has degree at least $1$. Therefore, if we change $n+1$ crossings in different places then the alternating sum of the $2^n$ polynomials will have the degree of its monomials $\geqslant n+1$. Hence the coefficient at any degree $n$ term will be zero.
After substitution $a=e^h$ and the Taylor expansion in $h$ the factor $\frac{a-a^{-1}}{z}$ becomes $ \frac{2h + \dots}{z}$. In other words its total degree in $h$ and $z$ is not negative. Therefore, the total degree $k+l$ of the monomial $h^kz^l$ of $P(L)$ is not negative, however the exponent $l$ of $z$ may be as negative as $-k+1$.
Our next goal is to describe the Gauss diagram formulas for $p_{k,l}(L)$. Note that the case $k=0$ corresponds to the substitution $a=1$ into the HOMFLYPT polynomial, i.e. to the Conway polynomial. Thus $p_{0,l}(L)$ are coefficients of the Conway polynomial for which the Gauss diagram formulas were found in [@CKR]. Thus our work may be considered as a generalization of [@CKR].
Gauss diagram formulas for Vassiliev invariants {#s:arrow-diagr}
-----------------------------------------------
Let $\A$ be a free ${\mathbb{Z}}$-module generated by ordered arrow diagrams with $m$ circles. Define a map $I:\A\to\A$ by $I(G):=\sum_{A\subseteq G} A$ for any (abstract, ordered) Gauss diagram $G$, and extend it to $\A$ by linearity. Here $A\subseteq G$ means the arrow subdiagram $A$ containing the same circles as the whole diagram $G$ but only a subset of arrows of $G$ with their signs. A natural scalar product on $\A$ is given by $(A,B):=0$ if $A$ is not equal to $B$, and $(A,B):=1$ if $A=B$ for a pair of arrow diagrams $A$ and $B$. Let us define a pairing $\scp{A}{G}:=(A,I(G))$.
Let $A$ be a fixed element of $\A$. By [*a Gauss diagram formula*]{} we mean a function $\I_A$ on abstract Gauss diagrams defined by $\I_A: G\mapsto \scp{A}{G}$.
If $A$ is chosen at random then $\I_A(G)$ usually changes under Reidemeister moves and thus does not define any link invariant. However, for some special choice of $A$ it might be a link invariant. According to [@GPV] any Vassiliev invariant of long knots can be expressed by a Gauss diagram formula. In the following sections we describe an algorithm for finding such formulas for invariants $p_{k,l}(L)$ coming from the HOMFLYPT polynomial.
For shortness of notation, further we will use [*unsigned arrow diagrams*]{}, understanding by that a linear combination of arrow diagrams with all possible choices of signs and appearing with a coefficients $\pm1$ depending on whether even or odd number of negative signs were chosen.
If $m=2$ and $$A\ =\ {{\raisebox}{-12pt}[15pt][12pt]{\begin{picture}(70,15)(0,0)
\put(0,0){{\includegraphics}[width=70pt]{adln.eps}}
\end{picture}}}\ :=\
{{\raisebox}{-12pt}[0pt][0pt]{\begin{picture}(70,15)(0,0)
\put(0,0){{\includegraphics}[width=70pt]{adln-p.eps}}
\end{picture}}}\ -\ {{\raisebox}{-12pt}[0pt][0pt]{\begin{picture}(70,15)(0,0)
\put(0,0){{\includegraphics}[width=70pt]{adln-m.eps}}
\end{picture}}}\ ,$$ then $\I_A(G)$ is equal to the linking number of components.
If $m=1$ and $$A\ =\ {{\raisebox}{-10pt}[0pt][12pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{gd2.eps}}
\end{picture}}}\ :=\ {{\raisebox}{-10pt}[0pt][0pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{gd2-pp.eps}}
\end{picture}}}\
-\ {{\raisebox}{-10pt}[0pt][0pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{gd2-pm.eps}}
\end{picture}}}\ -\ {{\raisebox}{-10pt}[0pt][0pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{gd2-mp.eps}}
\end{picture}}}\
+\ {{\raisebox}{-10pt}[0pt][0pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{gd2-mm.eps}}
\end{picture}}}\ ,$$ then $\I_A(G)$ is equal to the second coefficient of the Conway polynomial, $p_{0,2}(G)$ (see [@PV]).
Gauss diagram formulas for HOMFLYPT coefficients {#s:result}
================================================
Our aim is to figure out contributions of various arrow subdiagrams to $p_{k,l}$, using the state model from Section \[s:jaeger-ga\].
Consider a state model on an arrow diagram $A$ with the following table of local weights $\lwa$: $$\label{eq:lwa}
\begin{array}{c||c|c|c|c}
{\raisebox}{7pt}{First passage:}&
{{\raisebox}{0pt}[25pt][3pt]{\begin{picture}(35,15)(0,0)
\put(0,0){{\includegraphics}[width=35pt]{fp-b.eps}}
\end{picture}}}&{{\raisebox}{0pt}[0pt][0pt]{\begin{picture}(35,15)(0,0)
\put(0,0){{\includegraphics}[width=35pt]{fp-t.eps}}
\end{picture}}}&
{{\raisebox}{0pt}[0pt][0pt]{\begin{picture}(35,15)(0,0)
\put(0,0){{\includegraphics}[width=35pt]{fp-l.eps}}
\end{picture}}}&{{\raisebox}{0pt}[0pt][0pt]{\begin{picture}(35,15)(0,0)
\put(0,0){{\includegraphics}[width=35pt]{fp-r.eps}}
\end{picture}}} \\ \hline\hline
{{\raisebox}{-8pt}[22pt][15pt]{\begin{picture}(40,15)(0,0)
\put(0,0){{\includegraphics}[width=40pt]{cr_p-gd.eps}}
\end{picture}}} & e^{-h}z & 0 & e^{-2h}-1 & 0 \\
\hline {{\raisebox}{-8pt}[22pt][15pt]{\begin{picture}(40,15)(0,0)
\put(0,0){{\includegraphics}[width=40pt]{cr_m-gd.eps}}
\end{picture}}} & -e^hz & 0 & e^{2h}-1 & 0
\end{array}$$
Let $\plwa=\prod_{\a\in A}\lwa$ and define a power series in $h$ and $z$ by $$\label{e:coeff}
W(A)=\sum_S \plwa \left(\frac{e^h-e^{-h}}{z}\right)^{c(S)-1}$$ Denote by $w_{k,l}$ the coefficient of $h^kz^l$ in $W(A)$, so that $W(A)=\sum_{k,l} w_{k,l}(A) h^k z^l$.
Now the linear combination $A_{k,l}\in\A$ can be defined as follows. $$\displaystyle A_{k,l} := \sum\ w_{k,l}(A)\cdot A$$
\[th:main\] Let $G$ be a Gauss diagram of an ordered link $L$. Then $$p_{k,l}(L)=\I_{A_{k,l}}(G)=\scp{A_{k,l}}{G}\ .$$
According to \[s:jaeger-ga\], the HOMFLYPT is equal to $$P(G)=\sum_{S\subset G}\ \ \plwg\cdot \left(\frac{a-a^{-1}}{z}\right)^{c(S)-1}.$$ We have $$\begin{gathered}
\plwg=\prod_{\a\in G} \lwg=\prod_{\a\in S} \lwg \prod_{\a\in G\sminus S} \lwg=\\
=\prod_{\a\in S} \lwg \sum_{A\supset S}\left(\prod_{\a\in A\sminus S} (\lwg-1) \prod_{\a\in G\sminus A} 1\right)=\\
=\sum_{A\supset S}\left(\prod_{\a\in S} \lwg\prod_{\a\in A\sminus S}
(\lwg-1)\right).\end{gathered}$$ Therefore $$\begin{gathered}
P(G)=\sum_{S\subset G}\ \ \sum_{A\supset S}\left(\prod_{\a\in S} \lwg\prod_{\a\in A\sminus S} (\lwg-1)\right)\cdot \left(\frac{a-a^{-1}}{z}\right)^{c(S)-1}=\\
=\sum_{A\subset G}\ \ \sum_{S\subset A}\left(\prod_{\a\in S}
\lwg\prod_{\a\in A\sminus S} (\lwg-1)\right)\cdot
\left(\frac{a-a^{-1}}{z}\right)^{c(S)-1}\end{gathered}$$ Comparing tables and of local weights, we get $$\prod_{\a\in S} \lwg\prod_{\a\in A\sminus S} (\lwg-1)=\prod_{\a\in A} \lwa=\plwa$$ Thus $$P(G)=\sum_{A\subset G}\ \ \sum_{S\subset A}\plwa\cdot
\left(\frac{a-a^{-1}}{z}\right)^{c(S)-1}$$ And the theorem follows.
Contributions of various diagrams to $A_{k,l}$ {#sub:contrib}
----------------------------------------------
A state $S$ of an arrow diagram $A$ is called [*ascending*]{}, if in the tracing of $A(S)$ we approach a neighborhood of every arrow (not only the ones in $S$) first at the arrow head. As easy to see from the weight table, only ascending states contribute to $W(A)$. In particular, the first end point of an arrow in $A$ (as we move from the base point along the orientation) must be an arrow head.
Note that since $e^{\pm 2h}-1=\pm 2h + \mbox{(higher degree terms)}$ and $\pm e^{\mp h}z= \pm z + \mbox{(higher degree terms)}$, the power series $W(A)$ starts with terms of degree at least $|A|$, the number of arrows of $A$. Moreover, the $z$-power of $\plwa
\left(\frac{e^h-e^{-h}}{z}\right)^{c(S)-1}$ is equal to $|S|-c(S)+1$. Therefore, for fixed $k$ and $l$, the weight $w_{k,l}(A)$ of an arrow diagram may be non-zero only if $A$ satisfies the following conditions:
- $|A|$ is at most $k+l$;
- there is an ascending state $S$ such that $c(S)=|S|+1-l$.\[con-iii\]
For diagrams of the highest degree $|A|=k+l$, the contribution of an ascending state $S$ to $w_{k,l}(A)$ is equal to $(-1)^{|A|-|S|}2^k\e(A)$, where $\e(A)$ is the product of signs of all arrows in $A$. If two such arrow diagrams $A$ and $A'$ with $|A|=k+l$ differ only by signs of arrows, their contributions to $A_{k,l}$ differ by the sign $\e(A)\e(A')$. Thus all such diagrams may be combined to the unsigned diagram $A$, appearing in $A_{k,l}$ with the coefficient $\sum_S(-1)^{|A|-|S|}2^k$ (where the summation is over all ascending states of $A$ with $c(S)=|S|+1-l$).
Arrow diagrams with isolated arrows do not contribute to $A_{k,l}$. Indeed, consider an arrow diagram $A\cup a$ with an isolated arrow $a$. Every state $S$ of $A$ corresponds to two states of $A\cup
a$: $S$ and $S\cup a$. Depending on the orientation of $\a$ their weights will be either both $0$, or $(e^{-2\e h}-1)\plwa$ and $\e
e^{-\e h}z\frac{e^h-e^{-h}}{z}\plwa$. In both cases they sum up to $0$, since $(e^{-2\e h}-1)+\e e^{-\e h}(e^h-e^{-h})=0$.
Coefficients of the Conway polynomial
-------------------------------------
The Conway polynomial is obtained from the HOMFLYPT polynomial by setting $h=0$. So our formulas for $A_{0,l}$ are the Gauss diagram formulas for coefficients of the Conway polynomial, discovered earlier by Michael Khoury and Alfred Rossi [@CKR]. Indeed, only states with $|S|=|A|$ and $c(S)=1$ contribute to $w_{0,l}(A)$. Since these are diagrams of the highest degree, according to \[sub:contrib\] they may be combined into unsigned ascending diagrams which appear with coefficients 1.
For example, in the case $m=1$ of long knots, states with $c(S)=1$ exist only for even number $l$ of arrows. For $l=2$ and $l=4$ the resulting linear combinations $A_{0,l}$ are
$$\begin{array}{rcl}
A_{0,2} &=& {{\raisebox}{-12pt}[15pt][20pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{cd22arw.eps}}
\end{picture}}}\ ;\hspace{1cm}
A_{0,3} = 0\ ;\\
A_{0,4} &=& \ard{cd4-01arw}\
+\ \ard{cd4-07arw1} + \ard{cd4-07arw2} + \ard{cd4-07arw3} + \ard{cd4-07arw4}+ \\
&&\hspace{-8pt}
+ \ard{cd4-05arw1} + \ard{cd4-05arw2} + \ard{cd4-05arw3} + \ard{cd4-05arw4}
+ \ard{cd4-05arw5} + \ard{cd4-05arw6} + \ard{cd4-05arw7} + \ard{cd4-05arw8} + \\
&&\hspace{-8pt}
+ \ard{cd4-06arw1} + \ard{cd4-06arw2} + \ard{cd4-06arw3} + \ard{cd4-06arw4}
+ \ard{cd4-06arw5} + \ard{cd4-06arw6} + \ard{cd4-06arw7} + \ard{cd4-06arw8}\ .
\end{array}$$
Low degree examples {#s:example}
===================
Let us describe the corresponding formulas for degree 2 and 3 invariants of knots, i.e. $k+l=2,3$, $m=1$. The case $A_{0,2}$ was described above. A direct check shows that $A_{2,0}=0$. Let us explicitly find the formula for $A_{1,2}$. The maximal number of arrows is equal to 3. To get $z^2$ in $W(A)$ we need ascending states with either $|S|=2$ and $c(S)=1$, or $|S|=3$ and $c(S)=2$. In the first case the equation $c(S)=1$ means that the two arrows of $S$ must intersect. In the second case the equation $c(S)=2$ does not add any restrictions on the relative position of arrows. In cases $|S|=|A|=2$ or $|S|=|A|=3$, since $S$ is ascending, $A$ itself must be ascending as well.
For diagrams of the highest degree $|A|=1+2=3$, we should count ascending states of unsigned arrow diagrams with the coefficient $(-1)^{3-|S|}2$, i.e. $-2$ for $|S|=2$ and $+2$ for $|S|=3$. There are only four types of (unsigned) 3-arrow diagrams with no isolated arrows: \#1
[-13pt]{}\[18pt\]\[10pt\]
(25,15)(0,0) (0,0)[\[width=25pt\][\#1.eps]{}]{}
\#1
[-20pt]{}\[12pt\]\[25pt\]
(25,15)(0,0) (0,0)[\[width=25pt\][\#1.eps]{}]{}
\#1
[-12pt]{}\[15pt\]\[15pt\]
(25,15)(0,0) (0,0)[\[width=25pt\][\#1.eps]{}]{}
$$\lhd{bcd35-1}; \qquad {{\raisebox}{-10pt}[0pt][0pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{bcd34-3.eps}}
\end{picture}}}\ \ , \qquad \lhd{bcd34-1},\qquad {{\raisebox}{-10pt}[0pt][0pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{bcd34-2.eps}}
\end{picture}}}\ \ .$$ Diagrams of the same type differ by directions of arrows.
For the first type, recall that the first arrow should be oriented towards the base point; this leaves 4 possibilities for directions of the remaining two arrows. One of them, namely $\lhd{aiv-n}$ does not have ascending states with $|S|=2,3$. The remaining possibilities, together with their ascending states, are shown in the table: $$\begin{array}[t]{||c|c|c|c||}\hline\hline
\fhd{aiv-3} & \fhd{aiv-2} & \fhd{aiv-1} & \fhd{aiv-1} \\
\shd{civ-3} & \shd{civ-2} & \shd{civ-1} & \shd{civ-4} \\
\hline\hline\end{array}$$ The final contribution of this type of 3-arrow diagrams to $A_{1,2}$ is equal to\
$$-2\ {{\raisebox}{-12pt}[0pt][15pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{aiv-3.eps}}
\end{picture}}}\
-2\ {{\raisebox}{-12pt}[0pt][0pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{aiv-2.eps}}
\end{picture}}}\ .$$
The remaining three types of 3-arrow diagrams differ by the location of the base point. A similar consideration shows that 5 out of the total of 12 arrow diagrams of these types, namely $${{\raisebox}{-10pt}[15pt][15pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{aiii-n1.eps}}
\end{picture}}}\ ,\quad
{{\raisebox}{-10pt}[0pt][0pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{aiii-n2.eps}}
\end{picture}}}\ ,\qquad
{{\raisebox}{-13pt}[0pt][0pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{ai-n.eps}}
\end{picture}}}\ ,\qquad
{{\raisebox}{-10pt}[0pt][0pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{aii-n1.eps}}
\end{picture}}}\ ,\quad
{{\raisebox}{-10pt}[0pt][0pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{aii-n2.eps}}
\end{picture}}}$$ do not have ascending states with $|S|=2,3$. The remaining possibilities, together with their ascending states, are shown in the table: $$\begin{array}[t]{||c|c|c||c|c|c||c|c|c||}\hline\hline
{{\raisebox}{-17pt}[10pt][25pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{aiii-1.eps}}
\end{picture}}}& {{\raisebox}{-17pt}[0pt][0pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{aiii-2.eps}}
\end{picture}}}
& {{\raisebox}{-17pt}[0pt][0pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{aiii-2.eps}}
\end{picture}}}
& \fhd{ai-1} & \fhd{ai-2} & \fhd{ai-3} &
{{\raisebox}{-17pt}[0pt][0pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{aii-2.eps}}
\end{picture}}}& {{\raisebox}{-17pt}[0pt][0pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{aii-1.eps}}
\end{picture}}}
& {{\raisebox}{-17pt}[0pt][0pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{aii-3.eps}}
\end{picture}}} \\
{{\raisebox}{-9pt}[15pt][15pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{ciii-1.eps}}
\end{picture}}}& {{\raisebox}{-9pt}[0pt][0pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{ciii-2.eps}}
\end{picture}}}
& {{\raisebox}{-9pt}[0pt][0pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{ciii-3.eps}}
\end{picture}}}
& \shd{ci-1} & \shd{ci-2} & \shd{ci-3} &
{{\raisebox}{-9pt}[0pt][0pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{cii-2.eps}}
\end{picture}}}& {{\raisebox}{-9pt}[0pt][0pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{cii-1.eps}}
\end{picture}}}
& {{\raisebox}{-9pt}[0pt][0pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{cii-3.eps}}
\end{picture}}} \\
\hline\hline\end{array}$$
The final contribution of this type of 3-arrow diagrams to $A_{1,2}$ is equal to $$-2\ {{\raisebox}{-10pt}[13pt][15pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{aiii-1.eps}}
\end{picture}}}
-2\ {{\raisebox}{-13pt}[0pt][0pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{ai-1.eps}}
\end{picture}}}
-2\ {{\raisebox}{-13pt}[0pt][0pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{ai-2.eps}}
\end{picture}}}
+2\ {{\raisebox}{-13pt}[0pt][0pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{ai-3.eps}}
\end{picture}}}
-2\ {{\raisebox}{-10pt}[0pt][0pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{aii-2.eps}}
\end{picture}}}\ .$$
Besides 3-arrow diagrams, some 2-arrow diagrams contribute to $A_{1,2}$ as well. Since $|A|=2<k+l=3$, contributions of 2-arrow diagrams depend also on their signs. Such diagrams must be ascending (since $|S|=|A|=2$) and should not have isolated arrows. There are four such diagrams, looking like $\lhd{ad2}$, but with different signs $\e_1$, $\e_2$ of arrows. For each of them $\plwa=\e_1\e_2 e^{-(\e_1+\e_2)h}z^2$. If $\e_1=-\e_2$, then $\plwa=-z^2$, so the coefficient of $hz^2$ vanishes and such diagrams do not occur in $A_{1,2}$. For two remaining diagrams with $\e_1=\e_2=\pm1$, coefficients of $hz^2$ in $\plwa$ are equal to $\mp2$ respectively.
Combining all the above contributions, we finally get $$A_{1,2} = -2\Bigl( {{\raisebox}{-12pt}[0pt][15pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{aiv-3.eps}}
\end{picture}}}+
{{\raisebox}{-12pt}[0pt][0pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{aiv-2.eps}}
\end{picture}}}+{{\raisebox}{-10pt}[13pt][15pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{aiii-1.eps}}
\end{picture}}}
+{{\raisebox}{-13pt}[0pt][0pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{ai-1.eps}}
\end{picture}}}+{{\raisebox}{-13pt}[0pt][0pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{ai-2.eps}}
\end{picture}}}
-{{\raisebox}{-13pt}[0pt][0pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{ai-3.eps}}
\end{picture}}}+{{\raisebox}{-10pt}[0pt][0pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{aii-2.eps}}
\end{picture}}}
+ \shd{ad2pp} - \shd{ad2mm}\Bigr)\ .$$
The invariant $\I_{A_{1,2}}=\scp{A_{1,2}}{\cdot}$ can be simplified further. Note that for any classical Gauss diagram $G$, $\scp{
{{\raisebox}{-13pt}[17pt][13pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{ai-2.eps}}
\end{picture}}}}{G}=
\scp{ {{\raisebox}{-13pt}[0pt][0pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{ai-3.eps}}
\end{picture}}} }{G}$. This follows from the symmetry of the linking number. Indeed, supposed we have matched two vertical arrows (which are the same in both diagrams) with two arrows of $G$. Let us consider the orientation preserving smoothings of the corresponding two crossings of the link diagram $D$ associated with $G$. The smoothened diagram $\widetilde{D}$ will have three components. Matchings of the horizontal arrow of our arrow diagrams with an arrow of $G$ both measure the linking number between the first and the third components of $\widetilde{D}$, using crossings when the first component overpasses (underpasses, respectively) the third one. Thus, as functions on classical Gauss diagrams, ${{\raisebox}{-13pt}[17pt][5pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{ai-2.eps}}
\end{picture}}}$ is equal to ${{\raisebox}{-13pt}[17pt][5pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{ai-3.eps}}
\end{picture}}}$ and we have $$p_{1,2}(G) = -2\langle {{\raisebox}{-12pt}[0pt][15pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{aiv-3.eps}}
\end{picture}}}+
{{\raisebox}{-12pt}[0pt][0pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{aiv-2.eps}}
\end{picture}}}+{{\raisebox}{-10pt}[13pt][15pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{aiii-1.eps}}
\end{picture}}}
+{{\raisebox}{-13pt}[0pt][0pt]{\begin{picture}(25,15)(0,0)
\put(0,0){{\includegraphics}[width=25pt]{ai-1.eps}}
\end{picture}}}+{{\raisebox}{-10pt}[0pt][0pt]{\begin{picture}(27.5,15)(0,0)
\put(0,0){{\includegraphics}[width=27.5pt]{aii-2.eps}}
\end{picture}}}
+ \shd{ad2pp} - \shd{ad2mm}\ , G\rangle\ .$$
In a similar way one may check that $A_{3,0}=-4 A_{1,2}$.
Let us compute the coefficients of $hz^2$ and $h^3$ of the HOMFLYPT polynomial on the trefoil from Section \[d3-1\], see page . $$\scp{A_{1,2}}{G}= 2\scp{\shd{ad2mm}}{G}=2\qquad
\mbox{and}\qquad \scp{A_{3,0}}{G}= -8\ ,$$ It is easy to verify these coefficients in the Taylor expansion of $P(3_1)=(2e^{2h}-e^{4h})+ e^{2h}z^2$.
[ABC]{}
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`http://www.math.toronto.edu/~drorbn/Goussarov/`.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present the first broadband $\lambda = 1\rm mm$ spectrum toward the $z=2.56$ Cloverleaf Quasar, obtained with Z-Spec, a 1-mm grating spectrograph on the 10.4-meter Caltech Submillimeter Observatory. The 190–305 GHz observation band corresponds to rest-frame 272 to 444 [$\rm \mu m$]{}, and we measure the dust continuum as well as all four transitions of carbon monoxide (CO) lying in this range. The power-law dust emission, $F_\nu = 14\rm mJy \left({\nu}/{240 GHz}\right)^{3.9}$ is consistent with the published continuum measurements. The CO [[$J{\!=\!}6{\!\rightarrow\!}5$]{}]{}, [[$J{\!=\!}8{\!\rightarrow\!}7$]{}]{}, and [[$J{\!=\!}9{\!\rightarrow\!}8$]{}]{} measurements are the first, and now provide the highest-J CO information in this source. Our measured CO intensities are very close to the previously-published interferometric measurements of [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{}, and we use all available transitions and our $^{13}$CO upper limits to constrain the physical conditions in the Cloverleaf molecular gas disk. We find a large mass (2–50$\times10^{9}\,\rm M_{\odot}$) of highly-excited gas with thermal pressure $\rm nT > 10^6\,\rm K cm^{-3}$. The ratio of the total CO cooling to the far-IR dust emission exceeds that in the local dusty galaxies, and we investigate the potential heating sources for this bulk of warm molecular gas. We conclude that both UV photons and X-rays likely contribute, and discuss implications for a top-heavy stellar initial mass function arising in the X-ray-irradiated starburst. Finally we present tentative identifications of other species in the spectrum, including a possible detection of the [$\rm{H_2O}$]{} [$\rm 2_{0,2}{\!\rightarrow\!}1_{1,1}$]{} transition at $\rm \lambda_{rest}$ = 303 [$\rm \mu m$]{}.'
author:
- 'C.M. Bradford, J.E. Aguirre, R. Aikin, J.J. Bock, L. Earle, J. Glenn, H. Inami, P.R. Maloney, H. Matsuhara, B.J. Naylor, H.T. Nguyen, J. Zmuidzinas'
bibliography:
- 'bradford\_master.bib'
title: The Warm Molecular Gas Around the Cloverleaf Quasar
---
Introduction
============
Since the first measurements of CO in the powerful IRAS galaxy FSC10214 at z=2.3 [@bvb92] nearly two decades ago, the field of molecular line measurements in early-universe galaxies has grown steadily (see review by @svb05). Several tens of sources are now known, drawn from a wide variety of parent populations including IRAS galaxies, optically-selected quasars, submillimeter galaxies, and now Spitzer-selected galaxies (e.g. [@frayer08]). As with the submillimeter and millimeter-wave continuum measurements, observations of high-z molecular lines benefit from a negative K-correction: the spectral lines generally carry more power at higher frequency. In particular for carbon monoxide, the run of the CO line luminosity with J typically increases up to the mid-J transitions (J$\sim$5–10) for actively starforming galaxies. This CO rotational spectrum when measured in energy units thus peaks in the 200–500 [$\rm \mu m$]{} regime, making millimeter-wave spectroscopy a powerful tool for studying molecular gas in the early universe. Measurements of the CO spectrum across its peak constrains the temperature, density and total mass of molecular gas. While observations of mid-J CO transitions in the local universe are hampered by poor transmission in narrow atmospheric windows, a millimeter-wave systems can have approximately uniform sensitivity across the peak of the CO SED for $z>$1.
All high-$z$ millimeter-wave observations to date have employed heterodyne receivers, originally with single-dish telescopes but soon after with large interferometer arrays. SIS receiver sensitivities, $T_{rec}\sim \rm 30~K$, are close to the photon background limit at mountaintop sites, and are an excellent match to the large-collecting area arrays. However, their instantaneous fractional bandwidths are at most $\sim10\%$, even with the new wideband backends (e.g. 8 GHz for ALMA). They thus require a separate tuning for each spectral line, and searching for unknown lines and/or unknown redshifts is time-consuming. A more comprehensive approach to finding redshifts and probing molecular gas contents of distant galaxies can be obtained with complete spectral coverage, to the extent possible given the telluric windows. CO is but one of the important coolants, and it is generally accepted that other species, notably water, become important as the molecular gas becomes more excited. Z-Spec is the first spectrograph designed specifically for this type of measurement at wavelengths longward of the mid-IR; a first-order grating with simultaneous coverage over the entire 1 mm atmospheric window.
After a brief introduction to the Cloverleaf quasar, Section \[sec:obs\] provides a description of Z-Spec and our observations at the CSO. The line and continuum parameters extracted from our spectrum are presented in Section \[sec:results\], and the findings are discussed in Section \[sec:discussion\].
{width="16cm"}
The Cloverleaf
--------------
We have observed the lensed $\rm z=2.56$ broad-absorption-line quasar system H1413+1143, also known as the Cloverleaf. The source was originally detected in an optical spectroscopic survey [@Hazard_84], but a warm dust component also emits some $\sim 7\times 10^{13}$[L$_{\odot}$]{} (intrinsic) in the rest-frame mid-IR [Lutz\_07]{}, ascribed to reradiated energy from the accretion zone. The SED also suggests a second distinct dust emission component dominating the rest-frame far-IR and submillimeter, but with a total bolometric luminosity $\sim$5–10% that of the AGN dust component [@Barvainis_92; @Barvainis_95; @Weiss_05]. The measurement of PAH features with intensities consistent with a starburst in this far-IR / submm component leads to the picture that the Cloverleaf is a composite object: an usually powerful partially-embedded AGN accompanied by a weaker but still tremendous starbust which alone would be similar to a submillimeter galaxy [@Lutz_07]. These large luminosities, combined with a substantial lensing magnification factor ($\sim11$) makes the Cloverleaf an excellent laboratory for studying the AGN / host interaction in what is likely the era of peak activity in galaxies. In particular, the molecular gas reservoir is the raw material for both the star formation and ultimately the nuclear accretion, and the impact of UV and X-ray photons from the stars or AGN on the molecular gas is a key aspect of the starburst / AGN interaction.
Some of the earliest high-$z$ molecular gas measurements were detections of the CO [[$J{\!=\!}3{\!\rightarrow\!}2$]{}]{} transition redshifted to the 3 mm band [@Barvainis_94; @Barvainis_97]. These measurements confirmed the presence of a large gas reservoir ($\sim 10^{10}$ [M$_{\odot}$]{}) inferred from the dust SED. Subsequent observations of the [[$J{\!=\!}3{\!\rightarrow\!}2$]{}]{} and [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{} transitions with the BIMA, OVRO and IRAM interferometers have steadily improved [@Wilner_95; @Yun_97], culminating in an IRAM [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{} map which fully resolves the four Cloverleaf components (@Alloin_97 \[hereafter A97\], @Kneib_98a). @Venturini_03 used the [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{} image to model the CO [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{} source and lens and find (after correcting to the modern cosmology) that the intrinsic source is a disk with radius of 650 pc, inclined by 30$^\circ$. The lens is formed with two galaxies of comparable mass at 0.25[$^{\prime\prime}$]{} and 0.71[$^{\prime\prime}$]{} from the line of sight. Our Z-Spec measurement complements this spatial information by providing a survey of several CO transitions spanning the peak of the spectrum. The uniform calibration allows us to anchor the total molecular gas conditions and energetics. The result provides constraints on the gas heating sources: stars or the active nucleus, and on the impact of the active nucleus on the putative host starburst.
Z-Spec Instrument and Observations {#sec:obs}
==================================
Z-Spec is a single-beam grating spectrometer which disperses the 190–308 GHz band across an linear array of 160 bolometers. The grating approach is novel: a curved grating operates in a parallel plate waveguide which is fed by a single-mode corrugated feedhorn. The instrument coupling to the CSO telescope is thus well-approximated with a Gaussian-beam approach, and the measured efficiencies and beam sizes are consistent with this. Details of the grating design and testing can be found in @Naylor_03 [@Bradford_04; @Earle_06; @Inami_08]. The detector spacing increases from $\sim1$ part in 400 at the low-frequency end ($\Delta\nu = 500$ MHz), to $\sim1$ part in 250 at the high-frequency end of the band ($\Delta\nu =1200$ MHz), while the spectrometer resolving power runs from $\sim1$ part in 300 at low frequencies to $\sim1$ part in 250 at the high frequency end of the band. Thus the system is marginally under-sampled, especially at the high-frequencies.
Spectral profiles for all 160 channels have been measured with a long-path Fourier-transform spectrometer, with channel center frequencies adjusted slightly per observations of multiple transitions in the spectral standard IRC+10216. Refinement of the Z-Spec frequency scale is ongoing as we incorporate line measurements from an increasing library of astronomical sources; for the data presented here, we are confident that the channel frequencies are known to better than 200 MHz across the band, or approximately one fifth of a channel width. We expect to improve this further in the near future.
The entire structure is cooled with an adiabatic demagnetization refrigerator and operates at temperatures between 60 and 85 mK to facilitate photon-background-limited detection. The bolometers developed at the JPL Microdevices Laboratory are individually-mounted silicon-nitride micro-mesh absorbers with quarter-wave backshorts, read-out with neutron-transmutation-doped Germanium thermistors. With an operational optical loading is $\sim1-3 \times 10^{-13}\rm W$, and phonon NEPs of $4\times10^{-18} \rm W Hz^{-1/2}$, these detectors are the most-sensitive, lowest-background bolometers fielded to date for astrophysics.
Z-Spec observes in a traditional chop-and-nod mode, with the secondary chopping between 1 and 2 Hz, and a nod period of 20 seconds. Because it is not possible to modulate the spectral response of the instrument relative to the bolometer array, and the spectral resolution elements are not oversampled, it is critical to both insure excellent array yield, and carefully calibrate each detector’s response. The situation is complicated by the fact that the both the bolometer loading and the bath temperature are changing throughout a typical observing night, while primary calibration sources are relatively scarce. To address this, we have built a library of planetary observations in varying conditions, and we fit the dependence of each bolometer’s response on its operating voltage, a proxy for the combination of bath temperature and optical loading. This relationship provides a calibration correction which is used to bridge intervals between astronomical calibration observations. Based on the self-consistency obtained with this scheme on planets and quasar calibrators, we estimate that the channel-to-channel calibration uncertainties are less than 10%, except at the lowest frequencies which are degraded by the wing of the 186 GHz atmospheric water line.
The Cloverleaf was observed on 2008 April 8 and April 15, with a total observing time (including chopping) of 7.9 hours, equally split between the two nights. On April 8, the weather was excellent, with [$\tau_{\rm 225 GHz}$]{} between 0.04 and 0.06, while April 16 had higher opacity: [$\tau_{\rm 225 GHz}$]{}$\sim$ 0.15–0.2. Each channel’s signal is summed individually on a nod-by-nod basis, with each nod individually calibrated and weighted by the inverse square of the detector noise (in Janskys) measured around the chop frequency. No modifications are made to the overall spectral shape at any point. Z-Spec’s sensitivity is weather dependent: on April 8, the sensitivity per spectral channel was 300–500 $\rm mJy\,\sqrt{sec}$, but degraded to 500–700 $\rm mJy\,\sqrt{sec}$ on the 15th, with greater degradation at the band edges, where the opacity increase is larger. The measured sensitivities are consistent with a simple model in which photon noise from the sky and telescope is the dominant contribution to the system noise. The final RMS uncertainty in each channel is shown with the error bars in Figure 1. It ranges from 2.5 to 4 mJy for most channels, with a median of 3.3 mJy.
Results, Continuum and Line Flux Extractions {#sec:results}
============================================
The Z-Spec spectrum toward the Cloverleaf system is shown in Figure \[fig:spec\]. To extract line fluxes, we perform a simultaneous fit to a single component power-law continuum and the multiple lines. Each channel’s measured spectral response profile is used in the fitting since the spectrometer is not critically sampled, and the grating has spectral sidelobes at the $\sim$1% level. All line widths are fixed, since Z-Spec is not sensitive to widths below $\sim$1000 [$\,\rm km\,s^{-1}$]{}. The PdB measurements of the CO [[$J{\!=\!}3{\!\rightarrow\!}2$]{}]{} shows a width of 420 [$\,\rm km\,s^{-1}$]{}(W03), while the @Kneib_98a spectrum of [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{} has a Gaussian fit to some of the profile giving FWHM$\sim$450 [$\,\rm km\,s^{-1}$]{}, though the profile is non-Gaussian and the true RMS width of the profile is larger than this. It is conceivable that the line width increases with $J$—this would be expected if warmer material lies preferentially closer to the point-like nucleus. We adopt 500 [$\,\rm km\,s^{-1}$]{} for the Z-Spec fits. This choice is not crucial; we have found that the fits for the integrated line fluxes are not strongly sensitive to the adopted line width for values below $\sim$800 [$\,\rm km\,s^{-1}$]{}. A total of six spectral lines are fit, the results are presented in Table \[tab:lines\].
The [\[CI\]]{} [[$J{\!=\!}2{\!\rightarrow\!}1$]{}]{} ($\nu=227.5~\rm GHz$) and the CO [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{} ($\nu=226.7~\rm GHz$) transitions are separated by only 1000 [$\,\rm km\,s^{-1}$]{}, or about one Z-Spec channel, so blending is a problem. To address this, we adopt an iterative approach to make appropriate use of prior information: we first fit the continuum plus the three unblended CO lines ([[$J{\!=\!}6{\!\rightarrow\!}5$]{}]{}, [[$J{\!=\!}8{\!\rightarrow\!}7$]{}]{}, and [[$J{\!=\!}9{\!\rightarrow\!}8$]{}]{}) to determine their fluxes and a CO redshift. We find $z=2.5585 \pm 0.0015$, a value in good agreement with previous measurements (our error derives from the accuracy of our frequency calibration, and can be improved in the future). With the redshift determined, we then fit fluxes to the CO [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{} and [\[CI\]]{} [[$J{\!=\!}2{\!\rightarrow\!}1$]{}]{} lines, finding $45.3\pm6.4$ and $8.5\pm7.4$ [$\rm Jy\,km\,s^{-1}$]{}, respectively (see Table \[tab:lines\]). Our current frequency scale uncertainty corresponds to interchanging $\sim$4 [$\rm Jy\,km\,s^{-1}$]{} of flux between the two transitions about these fit values, with the sum of the two conserved, a value which is less than but comparable to the statistical errors. Our results are consistent with the various [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{} integrated intensity measurements from the Plateau de Bure. B97 and A97 quote $43.7\pm2.2$ and $43.3\pm2.4$ [$\rm Jy\,km\,s^{-1}$]{}, respectively, using a fit to a Gaussian profile with $\Delta V_{\rm FWHM} =376$[$\,\rm km\,s^{-1}$]{} (adopted based on the lower-$J$ measurements). However, A97 also report 50.1$\pm2.8$ [$\rm Jy\,km\,s^{-1}$]{} when the linewidth is fit as a free parameter (finding $\Delta V_{\rm FWHM} = 480$ [$\,\rm km\,s^{-1}$]{}). Given the linewidth discussion above, we adopt the larger of these as our input to the analysis, but adopt a 15% systematic uncertainty to accommodate the range of measurements. As the CO temperature spectrum plotted in Figure \[fig:sed\] shows, this is in good agreement with the [[$J{\!=\!}6{\!\rightarrow\!}5$]{}]{} and [[$J{\!=\!}8{\!\rightarrow\!}7$]{}]{} measurements, giving confidence to the overall calibration and noise estimates. Our [\[CI\]]{} flux is also formally consistent with the W05 measurement ($5.2\pm0.3$ [$\rm Jy\,km\,s^{-1}$]{}). An emission feature at 277.6 GHz is also fit, tentatively identified as water and discussed in Section \[sec:water\]. Finally, we have obtained upper limits to six other transitions, discussed briefly in Section \[sec:limits\]. The potential emission feature at 208 GHz is unidentified at present.
[cccccccc]{} CO [[$J{\!=\!}3{\!\rightarrow\!}2$]{}]{} & 345.79 & 33.2 & 97.2 & 13.2 & 1.7 & 0.55 & W03\
CO [[$J{\!=\!}3{\!\rightarrow\!}2$]{}]{} & 345.79 & 33.2 & 97.2 & 9.9 & 0.6 & 0.41 & B97\
[\[CI\]]{} [[$J{\!=\!}1{\!\rightarrow\!}0$]{}]{}& 492.16& 23.6 &138.3 &3.6&0.4 & 0.21 &W05\
CO [[$J{\!=\!}4{\!\rightarrow\!}3$]{}]{}& 461.04 & 55.3 & 129.58 & 21.1 & 0.8 & 1.17 & B97\
CO [[$J{\!=\!}5{\!\rightarrow\!}4$]{}]{}& 576.27 & 83.0& 161.96 & 24.0 & 1.7 & 1.66 & B97\
CO [[$J{\!=\!}6{\!\rightarrow\!}5$]{}]{}& 691.47 & 116 &194.3 & 37.0& 8.1 & 3.1 & this work\
CO [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{}& 806.65& 155 & 226.7& 50.1 &2.2 & 4.9& B97, A97\
CO [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{}& 806.65 & 155 & 226.7 & 45.3 & 6.3 & 4.4 & this work\
[\[CI\]]{} [[$J{\!=\!}2{\!\rightarrow\!}1$]{}]{}& 809.34 & 37 & 226.6 & 5.2 & 0.3 & 0.50 & W05\
[\[CI\]]{} [[$J{\!=\!}2{\!\rightarrow\!}1$]{}]{}& 809.34 & 37 & 226.6 & 8.5 & 7.4 & 0.83 & this work\
CO [[$J{\!=\!}8{\!\rightarrow\!}7$]{}]{}& 921.80& 199 & 259.0 & 51.4 & 4.7 & 5.7 & this work\
CO [[$J{\!=\!}9{\!\rightarrow\!}8$]{}]{}& 1036.91 & 249& 291.4& 41.8 & 5.8 & 5.2 & this work\
far-IR dust & & 5.4$\times 10^4$ & W03, L07\
\
\
[$\rm{H_2O}$]{} [$\rm 2_{0,2}{\!\rightarrow\!}1_{1,1}$]{}& 987.93 & 101 & 277.6 & 20.3 & 6.1 & 2.4\
[$\rm{H_2O}$]{} [$\rm 2_{1,1}{\!\rightarrow\!}2_{0,2}$]{}& 752.03 & 137 & 211.4 & $<$14.2 & 5.7 & $<$1.3\
Absorption feature & & & 288.3 & -17.4 & 5.8\
LiH [[$J{\!=\!}2{\!\rightarrow\!}1$]{}]{}& 887.29 & 21.3 & 249.3 & $<$12.3 & 4.9 & $<$1.9\
CH$^+$ [[$J{\!=\!}1{\!\rightarrow\!}0$]{}]{}& 835.07 & 43.1 & 234.7 & $<$14.0 & 5.6 & $<$1.6\
$^{13}$CO [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{}& 771.19 & 148 & 216.81 & $<$11.7 & 4.7 & $<$1.1\
$^{13}$CO [[$J{\!=\!}8{\!\rightarrow\!}7$]{}]{}& 881.27 & 190 & 247.76 & $<$14.8 & 5.9 & $<$1.6\
\[tab:lines\]
Discussion {#sec:discussion}
==========
Dust Continuum {#sec:continuum}
--------------
The best fit continuum across the Z-Spec band is $F_{\nu} = 14.1\pm 0.4\, \rm mJy \left({\nu}/{240 GHz}\right)^{3.91\pm0.17}$. This is consistent with previous measurements, for which the best compilation is in W03 (their Figure 3); we show the latest measurements in our band in Figure \[fig:spec\]. The MAMBO bolometer measurement lies very close to the Z-Spec spectrum; the PdB continuum measurement at 224 GHz is slightly lower, but we note that this data point lies below the best fit of W03 including all the higher-frequency data. Indeed, the various measurements and limits from 97 GHz to 400 GHz do not all lie on a single greybody curve, but our best-fit exponent of 3.91 from 190 to 300 GHz is consistent with the W03 best fit: dust at 50 K with $\beta=2$ and with unit emissivity between 50–300 [$\rm \mu m$]{}.
![Cloverleaf CO spectral line energy distribution (SLED) using all available transitions. The top panel is in bolometric energy units, the bottom in brightness temperature units, referred to the VS03 source size assuming $m=11$. We have adoped a 20% systematic uncertainty for the measurements other than those from Z-Spec, shown in the temperature plot. A model corresponding to conditions near the peak in the pressure likelihood ($\rm T=56\,K, n_{H_2}=2.8\times 10^4, cm^{-3}$, $\rm N_{CO} / dv = 1.6\times10^6\,\rm cm^{-2}\,km^{-1}\,s$, normalized to match the observed [[$J{\!=\!}8{\!\rightarrow\!}7$]{}]{} flux) is overplotted as a solid line in both scales. Dashed lines in the temperature scale show themalized blackbody emission for temperatures of 25, 50 and 100 K (higher T is shallower slope), arbitrarily normalized to the model value of 39 K at [[$J{\!=\!}6{\!\rightarrow\!}5$]{}]{}. To produce 78% of the total observed intensities, the area filling factors relative to the VS03 disk would have to be 3.3, 1.1, and 0.46, respectively. \[fig:sed\]](f2top.eps "fig:"){width="8"}\
![Cloverleaf CO spectral line energy distribution (SLED) using all available transitions. The top panel is in bolometric energy units, the bottom in brightness temperature units, referred to the VS03 source size assuming $m=11$. We have adoped a 20% systematic uncertainty for the measurements other than those from Z-Spec, shown in the temperature plot. A model corresponding to conditions near the peak in the pressure likelihood ($\rm T=56\,K, n_{H_2}=2.8\times 10^4, cm^{-3}$, $\rm N_{CO} / dv = 1.6\times10^6\,\rm cm^{-2}\,km^{-1}\,s$, normalized to match the observed [[$J{\!=\!}8{\!\rightarrow\!}7$]{}]{} flux) is overplotted as a solid line in both scales. Dashed lines in the temperature scale show themalized blackbody emission for temperatures of 25, 50 and 100 K (higher T is shallower slope), arbitrarily normalized to the model value of 39 K at [[$J{\!=\!}6{\!\rightarrow\!}5$]{}]{}. To produce 78% of the total observed intensities, the area filling factors relative to the VS03 disk would have to be 3.3, 1.1, and 0.46, respectively. \[fig:sed\]](f2bottom.eps "fig:"){width="8"}
CO Excitation and Radiative Transfer Modeling {#sec:rad}
---------------------------------------------
The run of CO line luminosity with $J$ for all published measurements is shown in Figure \[fig:sed\]. The luminosity scale shows the quoted statistical uncertainties for the various measurements. Systematic errors are clearly present judging from the published [[$J{\!=\!}3{\!\rightarrow\!}2$]{}]{} through [[$J{\!=\!}5{\!\rightarrow\!}4$]{}]{} measurements, and we have assumed that systematic effects produce a normally-distributed uncertainty of $\sigma=20\%$ of the measured flux for the non-Z-Spec measurements. This statistical uncertainty is shown in the temperature scale.
Our approach is to consider the properties of the gas which is confined to the physical size inferred by VS03, a 650 pc-radius disk inclined by $i=30^\circ$ to present a 559 pc semi-minor axis. VS03 resolve 78% of the total [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{} flux in this source, and we adopt this fraction for other mid-J transitions, in particular for [[$J{\!=\!}8{\!\rightarrow\!}7$]{}]{}, our adopted reference transitions for bolometric calculations given its clean measurement with Z-Spec. The large observed line luminosities arising from the VS03 disk immediately imply that the molecular gas in the disk is very warm. If the disk radiates isotropically with a velocity FWHM of 500 [$\,\rm km\,s^{-1}$]{},the brightness temperatures must be $\rm T_{RJ}=$ 37, 40, 31, 20 K, for the [[$J{\!=\!}6{\!\rightarrow\!}5$]{}]{}, [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{}, [[$J{\!=\!}8{\!\rightarrow\!}7$]{}]{}, and [[$J{\!=\!}9{\!\rightarrow\!}8$]{}]{} lines, respectively. The corresponding physical temperatures assuming optically-thick, thermalized emission are even higher: 50, 56, 49, 39 K. An area filling factor or optical depth less than unity would correspond to even higher physical temperatures. Adopting a velocity width smaller than the 500 [$\,\rm km\,s^{-1}$]{} that we assume would also correspond to even higher physical temperatures.
For a more quantitative analysis, we begin with a variant of the RADEX code [@VanderTak_07] to model the CO excitation and radiative transfer. RADEX is a non-LTE code that uses an iterative, escape probability formalism to treat the line radiative transfer. Three independent physical variables provide the input into RADEX: the gas density ($n_{\rm H_2}$), the kinetic temperature ($T$), and the CO column density per unit linewidth $N_{\rm CO}/\Delta v$, which sets the optical depth scale. RADEX calculates the excitation temperatures, line optical depths, and line surface brightnesses in the CO lines. Because of the uncertainties in the magnification and the beam filling factor (assumed to be the same for all transitions), we compare the line ratios (with respect to the [[$J{\!=\!}8{\!\rightarrow\!}7$]{}]{} line), rather than the absolute line fluxes. We have used RADEX in this way to generate the CO line intensities on a large grid in density (10$^2$–10$^8$ cm$^{-3}$), temperature (0–300 K), and CO column per unit linewidth (10$^{14}$–10$^{20}\,\rm cm^{-2}\,km^{-1}\,s$).
The CO column density per unit linewidth is not strongly constrained by the $^{12}$CO line ratios alone, though very small optical depths are not allowed because some degree of radiative trapping is required to populate the levels up to $J=9$ (T= 249 K). To address this, we also compare the upper limits for the $^{13}$CO [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{} and [[$J{\!=\!}8{\!\rightarrow\!}7$]{}]{} transitions in our band with the output of a RADEX model for this species, assuming that its fractional abundance relative to $^{12}$CO is 1/40, a value derived in a multi-level study of the NGC 253 @Henkel_93.
We then follow the methodology of @Ward_03 to generate likelihood distributions for $n_{\rm H_2}$, $T$, and $N_{\rm
CO}/\Delta v$ by comparing the RADEX results with the observed line ratios. We assume uniform priors in the logarithm of all physical parameters, except for two prior constraints. First, the total mass of gas producing the CO lines in the disk cannot exceed the dynamical mass implied by the observed velocity spread and the size of the disk. This eliminates very large CO optical depths according to: $$\frac{N_{\rm CO} }{dv} < \frac{M_{\rm dyn} X_{\rm CO}}{1.4 m_{\rm H_2}} \frac{1}{\pi R^2_{\rm d} \cos{i}} \frac{1}{\Delta v}= 1.3\times10^{18}\,\rm cm^{-2}\, km^{-1}\, s.$$ Here the dynamical mass in the disk is based on the 650 pc modeled size, the measured aspect ratio implying $i=30^\circ$, and a circular velocity of 375[$\,\rm km\,s^{-1}$]{}/$\sin{30^\circ}$, per the B97 HWZI profile width (and confirmed by W03): $\rm M_{dyn}=8.5\times10^{10}\,M_{\odot}$. $\rm X_{CO}$ is the CO abundance relative to [H$_2$]{} (taken at $2\times10^{-4}$) and $\mu=1.4$ is the mean molecular weight in units of $\rm M_{H_2}$.
The second constraint requires a minimum temperature sufficient to produce the observed luminosity in the finite-sized disk, described above. Figure \[fig:nt1\] shows the effect on the likelihood distributions of adding this constraint: with it, the physical temperature is required to be above $\sim$50 K, without it, the temperature likelihood suggests T$\sim$30–60 K. The constrained-temperature likelihood suggests somewhat lower densities, and provides a better-defined total thermal pressure than the unconstrained likelihood, peaking between 0.8–3$\times10^{6}\rm \,K\,cm^{-3}$. We note that the derived likelihoods depend on the assumed linewidth only through this second constraint. In its absence, the likelihoods depend only on the integrated intensities. As described above, the adoption of 500 [$\,\rm km\,s^{-1}$]{} linewidth value is actually a conservative value in this context: smaller linewidths corresponds to even higher temperatures, which would provide a more stringent constraint on the final likelihoods.
![Likelihood distributions for the physical conditions in the Cloverleaf host, with (red) and without (black) a prior applied to insure that the temperature is sufficient to produce the observed luminosity in region with size given by the modeled disk. For both, the constraint in column density per linewidth is applied, eliminating N$_{CO} / \Delta v$ values much above $10^{18} \rm cm^{-2}\,km^{-1}s$. Otherwise, uniform priors are adopted in the logarithm of n, T and N$_{CO} / \Delta v$. All transitions are included. \[fig:nt1\]](f3top.eps "fig:"){width="8.7cm"}\
![Likelihood distributions for the physical conditions in the Cloverleaf host, with (red) and without (black) a prior applied to insure that the temperature is sufficient to produce the observed luminosity in region with size given by the modeled disk. For both, the constraint in column density per linewidth is applied, eliminating N$_{CO} / \Delta v$ values much above $10^{18} \rm cm^{-2}\,km^{-1}s$. Otherwise, uniform priors are adopted in the logarithm of n, T and N$_{CO} / \Delta v$. All transitions are included. \[fig:nt1\]](f3bottom.eps "fig:"){width="8.7cm"}
While the disk dominates the mid-J line emission, it may not dominate the integrated large-beam low-J (e.g. [[$J{\!=\!}1{\!\rightarrow\!}0$]{}]{} through [[$J{\!=\!}3{\!\rightarrow\!}2$]{}]{}) fluxes. We explore the degree to which a single component model can account for all the transitions by both including and excluding the [[$J{\!=\!}3{\!\rightarrow\!}2$]{}]{} measurement (the lowest-J transition available). The results are shown in Figure \[fig:nt2\]. Not surprisingly, higher excitation conditions are favored when the [[$J{\!=\!}3{\!\rightarrow\!}2$]{}]{} is excluded, but the effect is small, and indicates that a single component is a suitable model for all the observed transitions. Our adopted likelihoods for the physical conditions are thus shown with the black curves in both Figures \[fig:nt1\] and \[fig:nt2\], yielding a temperature of 50–100 K and a density n$_{\rm H_2} > 3\times10^3\,\rm cm^{-3}$. As an example, the spectrum in Figure \[fig:sed\] is overlaid with the fluxes predicted by a model near the peak of the pressure likelihood with $\rm T = 56\,\rm K$, $n_{\rm H_2}=2.8\times10^4\,\rm cm^{-3}$, and $N_{\rm CO}/\Delta v= 1.6\times10^{16}\,\rm cm^{-2}\,km^{-1}\,s$. We note that for our peak likelihood conditions, the gas density and the optical depth parameter are consistent with the gas having at least enough velocity dispersion to correspond to virialized motion under its own self-gravity. Rearranging Equation 2 in @Papadopoulos_07 for our units gives: $$\rm K_{\rm vir} = \frac{19.0}{\sqrt{\alpha}}\left[ \frac{N_{\rm CO}/ dv}{10^{17}}\right]^{-1}\,\left[\frac{n_{H_2}}{10^3 cm^{-3}}\right]^{1/2}\,\left[\frac{X_{CO}}{4\times10^{-4}}\right] > 1$$ where $\rm N_{CO} / dv$ is in units of $\rm cm^{-2}\,km^{-1}\,s$, $\rm X_{CO}$ is the CO abundance relative to [H$_2$]{}, $\alpha$ is a parameter ranging from 1–2.5 per the cloud density profile, and the inequality accounts for the fact that the gas can of course be subject to more than its own gravity. For a density of $10^4\,\rm cm^{-3}$, the inequality is satisfied as long as $\rm N_{CO} / dv$ is less than $3.8\times10^{18}\,\rm cm^{-2}\,km^{-1}\,s$. Our derived likelihoods thus imply velocity dispersions which exceed the virial requirement by about an order of magnitude, perhaps due to large-scale turbulence and/or the influence of additional mass (of at most $\sim$3 times that in the gas itself).
![Likelihood distributions for the physical conditions in the Cloverleaf host, both with (black) and without (red) the [[$J{\!=\!}3{\!\rightarrow\!}2$]{}]{} measurements included. The temperature and optical depth constraints are applied, but otherwise uniform priors are assumed in the logarithms of n, T and N$_{CO} / \Delta v$. \[fig:nt2\]](f4top.eps "fig:"){width="8.7cm"}\
![Likelihood distributions for the physical conditions in the Cloverleaf host, both with (black) and without (red) the [[$J{\!=\!}3{\!\rightarrow\!}2$]{}]{} measurements included. The temperature and optical depth constraints are applied, but otherwise uniform priors are assumed in the logarithms of n, T and N$_{CO} / \Delta v$. \[fig:nt2\]](f4bottom.eps "fig:"){width="8.7cm"}
Mass of Molecular Gas
---------------------
Our results are broadly consistent with the analysis of B97; cf. their Figure 2. As in that paper, we can directly compute (a likelihood for) the mass by scaling from the observed line luminosities and using the implicit information on the line optical depth from RADEX, instead of relying on an uncertain CO X-factor The calculation depends somewhat on the cloud geometry, as it determines the ratio of mass to emitting area; we have assumed spherical clouds, for which the working relationship is $$\frac{M}{L}=\frac{c}{12\pi I}\frac{N_{\rm CO}/\Delta v}{X_{CO}}\,\mu\, m_{H_2} = 69\, N_{17}\left(\frac{X_{CO}}{10^{-4}}\right)^{-1}I^{-1} \,\frac{M_\odot}{L_\odot},$$ where $N_{17}=[N/\Delta v]/[10^{17}\rm cm^{-2}/(km\,s^{-1})]$, the intensity $I$ is in cgs integrated intensity units (erg s$^{-1}$ cm$^{-2}$ sr$^{-1}$). This conversion results in a similar mass when applied to the various mid-J transitions, and Figure \[fig:mass\] shows the likelihood distribution for the total gas mass, again with and without [[$J{\!=\!}3{\!\rightarrow\!}2$]{}]{}. Our likelihood corresponds to a true molecular gas mass in the source of $M\sim 0.2-30 \times 10^{10} M_\odot$, consistent with previous gas mass estimates derived from CO, dust, and neutral carbon which range from 1.3–1.6$\times10^{10}$ [M$_{\odot}$]{} (W05). We note for completeness the consistency around the peak values in the mass and optical depth likelihoods. The gas column density is the estimated $\rm M=6\times10^{9}\, M_\odot$ distributed over the (projected) VS03 disk: $\rm N_{H}\sim4.6\times10^{23}\,cm^{-2}$. The CO optical depth parameter should be the corresponding CO column ($\rm N_{CO} = 2\times10^{-4}\, N_{H}\sim4.6\times10^{19}\,cm^{-2}$) divided by the linewidth of the source (400–500 [$\,\rm km\,s^{-1}$]{}), giving $\rm N_{CO} / dv \sim 0.9-1.2\times10^{17}\,cm^{-2}\,km^{-1}\,s$.
Origin of the Warm Molecular Gas
--------------------------------
The CO emission observed in the Cloverleaf is unusually intense, even if the entire far-IR luminosity is produced by a starburst with $L_{\rm SB} = L_{\rm 40-120 \mu m} \approx 5.5\times10^{12}\,$[L$_{\odot}$]{} as suggested by @Lutz_07, W03. The most luminous CO lines, the [[$J{\!=\!}8{\!\rightarrow\!}7$]{}]{}, [[$J{\!=\!}9{\!\rightarrow\!}8$]{}]{} each emit a fraction $\sim 10^{-4}$ of this total starburst luminosity (see Table \[tab:lines\]), and we measure the total CO luminosity through the likelihood formalism, finding that it is well-constrained at 4.6–6.8 $\times$ the [[$J{\!=\!}8{\!\rightarrow\!}7$]{}]{} luminosity (Figure \[fig:mass\]). We therefore have $$L_{\rm CO}\approx 3.3\times 10^{9}\rm\, L_{_\odot} \approx 6.1\times 10^{-4}\, L_{far-IR},$$ assuming that the same value of $m$ applies to both the gas and dust. This is more extreme than for the nuclei of the nearby starburst galaxies. In NGC 253, for example, the CO SED (in energy units) in the central 180 pc appears to peak at [[$J{\!=\!}6{\!\rightarrow\!}5$]{}]{} or [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{}; and each of these lines carry 2–3$\times10^{-5}$ times the far-IR luminosity, so the total CO luminosity fraction is $\sim$1–2$\times10^{-4}$ [@Bradford_03; @Hailey-Dunsheath_08; @Guesten_06]. Moreover, recent mid-J CO observations with the ZEUS spectrometer [@Stacey_07] in a sample of local-universe LIRGS and ULIRGS show mid-J ([[$J{\!=\!}6{\!\rightarrow\!}5$]{}]{} and/or [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{}) CO fractional luminosities comparable to or less than those in NGC 253 (T. Nikola, personal communication). Thus molecular gas cooling relative to the far-IR dust emission in the Cloverleaf host demonstrably exceeds that in even the extreme local-universe systems. We now consider potential heating sources for this bulk of warm molecular material.
![LEFT: Likelihood for the total molecular gas mass traced with CO in the Cloverleaf host, assuming a magnification factor $m$ of 11. RIGHT: Likelihood distribution for the total luminosity in the all the CO transitions. Color coding is as in Figure \[fig:nt2\][]{data-label="fig:mass"}](f5.eps){width="8.7cm"}
### Stellar Ultraviolet (PDRs) {#sec:pdr}
Recent PDR model calculations (@Kaufman_06, hereafter K06) compute CO line intensities which can be compared directly with the observations. These models parametrize the space of physical conditions with two quantities: the incident far-UV flux ($\rm G_0$), and the molecular hydrogen density of the nascent molecular cloud ($n_{\rm H_2}$). They indicate that for relatively high densities ($n > \rm few\times10^4$) and modest UV fields ($1<G_0<10^4$), the CO [[$J{\!=\!}6{\!\rightarrow\!}5$]{}]{} and [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{} transitions can have intensities more than $10^{-4}$ times the total far-IR luminosity. We compare two measured ratios with model calculations available online as part of the K06 model: 1) the measured intensity ratio of the CO [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{} to the total far-IR ($=8.8\times10^{-5}$, assuming that they come from the same physical region), and 2) the CO [[$J{\!=\!}6{\!\rightarrow\!}5$]{}]{} to [[$J{\!=\!}2{\!\rightarrow\!}1$]{}]{} (=23, assuming [[$J{\!=\!}2{\!\rightarrow\!}1$]{}]{} is thermalized at the same temperature a s[[$J{\!=\!}3{\!\rightarrow\!}2$]{}]{}). We find that these ratios can be consistent with the K06 PDR model with G$_0$=1–5$\times 10^3$ and $n_{\rm H_2} =$1–4$ \times 10^5\,\rm cm^{-3}$. The density estimate, when combined with our likelihood analysis of the thermal pressure, suggests gas temperatures in the range of 10–300 K, reasonable given the range of temperatures expected in the PDR.
However, the neutral carbon intensities reported thus far are are not consistent with this picture. For the PDR parameters discussed above, the K06 model predicts that the [\[CI\]]{}[[$J{\!=\!}1{\!\rightarrow\!}0$]{}]{} intensity relative to the CO [[$J{\!=\!}3{\!\rightarrow\!}2$]{}]{} and [[$J{\!=\!}4{\!\rightarrow\!}3$]{}]{} intensities should be 0.2–0.4 and 0.1–0.3, respectively; ratios which are close to those observed. But the [\[CI\]]{} [[$J{\!=\!}2{\!\rightarrow\!}1$]{}]{} to [[$J{\!=\!}1{\!\rightarrow\!}0$]{}]{} intensity ratio is predicted to be $\sim$6, while it is observed at only 2–2.5. The PDR model is predicting warmer neutral carbon than is observed, a fact which is generally difficult to reconcile with the conditions derived from the CO spectrum and which is discussed further in \[sec:xdr\] below.
For both total far-IR flux, and CO [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{}, the observed intensity emerging from the disk is $\sim$50 times greater than predicted by the single PDR model which fits the ratios. If UV photons are responsible for the heating of the gas, then on average in the Cloverleaf disk, we are seeing $\sim$50 PDR surfaces superposed along the line of sight. Correspondingly, the column density corresponding to a visual extinction ($\rm A_{\rm V}$) of 10 magnitudes is $\sim$1/50 of the total gas column estimated in Section \[sec:rad\], if it is averaged over the 650-pc disk. The derived far-UV field of $G_0 \sim $1–5$\times 10^3$ is modest, suggesting that the Cloverleaf PDR is not directly adjacent to an OB star as in an ionization-bounded HII region, but that the PDRs are typically 0.1 to 0.5 pc from the OB stars. This is quite reasonable, if the $5.5\times10^{12}$ [L$_{\odot}$]{} attributed to the starburst arises from 100–10000 [L$_{\odot}$]{} stars in a region of size given by the VS03 model, then the average interstellar separation is 0.2–1 pc.
### Hard X-Rays from the Active Nucleus {#sec:xdr}
The hard X-ray continuum of the active nucleus itself could power the observed CO emission. Hard (E$>$1 keV) X-rays can penetrate a large gas column (N$_{\rm H}>$10$^{22}$ cm$^{-2}$) forming an X-ray Dissociation Region (XDR: @Maloney_96). The key quantity for understanding the XDR heating and cooling is the ratio of the X-ray energy deposition rate per particle (set by the X-ray luminosity and attenuating column) to the gas density which sets the recombination and cooling rates. We do not have a direct view of the AGN in the Cloverleaf. The observed X-ray emission is believed to be scattered [@Chartas_00], and even this scattered emission is subject to an uncertain absorption in the gas along the line of sight. Thus the true X-ray flux must be inferred or modeled based on the observed spectrum. At the bolometric luminosity estimated for the Cloverleaf AGN ($\rm L_{\rm bol}\approx 7\times
10^{13}$[L$_{\odot}$]{}[@Lutz_07]), the $1-20$ keV X-rays typically carry $\sim 5\%$ of $L_{\rm bol}$ [@Mushotzky_93] (though with a scatter of a factor of $\sim 3$ in this relation). This implies that $\rm L_x\approx 3\times 10^{12}$[L$_{\odot}$]{}. @Chartas_07 present a model (their Figure 9) explaining the Cloverleaf X-ray spectra from Chandra and XMM-Newton in which the total 2–10 keV luminosity of the central region is $2\times10^{46}\,\rm erg\,s^{-1}$ or 5$\times 10^{12}$[L$_{\odot}$]{}(instrinsic, with $m\sim10$), based on an observed flux which is a factor of $\sim$200 lower.
While these inferences clearly have large uncertainties, it is very likely that the hard X-ray luminosity approaches the 40–120[$\rm \mu m$]{} far-infrared luminosity, and $${L_{\rm CO}\over L_{\rm X}}\approx 10^{-3}.$$ The large column densities derived in the CO-line analysis mean that much of this hard X-ray emission will be absorbed. Unlike in a PDR, the X-rays input a much greater fraction their energy ($\sim$0.1–1) into the gas than the dust, and since the CO lines will be an important coolant, this ratio immediately indicates that the AGN could readily power the CO emission we observe.
To explore this possibility, we have generated XDR models for parameters appropriate to the Cloverleaf system. The input parameters are the total gas density and the incident X-ray flux, which is set (for fixed $L_{\rm X}$) by the distance to the AGN and the column density of absorbing gas between the hard X-ray source and the modeled region. The results also depend on the optical depth in the coolings lines; we have used a total hydrogen column per unit linewidth of $10^{21}\rm cm^{-2}/km\,s^{-1}$, as suggested by the CO line analysis.
![CO abundance (relative to H nuclei) in the distance-gas density space of the XDR model. At small distances and low densities, gas is largely atomic. The blue contours show gas temperature. \[fig:X1\]](f6.eps){width="8.5cm"}
Figure \[fig:X1\] shows the predicted CO abundance as a function of gas density and cloud distance (denoted R) from the nucleus. The assumed X-ray luminosity is conservatively adopted to be 10$^{46}\,\rm erg\,s^{-1}$ and the total attenuating column is $\rm N_{H, att}=3\times 10^{23} cm^{-2}$, of order but less than the estimated total column from the CO analysis. This results in a factor of 22 attenuation and the resulting X-ray flux at 600 pc distance is thus 10 $\rm erg\,s^{-1}\,cm^{-2}$. Larger values of L$_{\rm X}$ or smaller values of $N_{att}$ shift the contours up and to the right. We have adopted solar abundances for elements in the gas-phase. At small R and relatively low density (lower left corner of the plot), the CO abundance is small, as the gas is warm and atomic. However, over most of the plotted range, the CO abundance is large (more than $\sim$a few $\times 10^{-5}$), reaching nearly $3\times 10^{-4}$ at the largest R and densities, for which all the gas-phase carbon is in CO.
![CO rotational cooling for X-ray heated clouds in the XDR model space. The contours show surface brightness of all the CO lines in units of erg / s / cm$^2$. Values of 0.2 and 0.6 correspond to the observed surface brightness assuming filling factors between 1/3 and 1 in the VS03 disk. Dashed blue contours show thermal pressure, values denoted with $\log{\rm nT}$ \[fig:X2\]](f7.eps){width="8.5cm"}
As noted above, the surface brightness of the CO emission in the Cloverleaf is very large. The estimated total CO rotational line luminosity (equation \[2\]), distributed over the VS03 disk with unit area filling factor implies a surface flux of approximately 0.2 ergs cm$^{-2}$ s$^{-1}$. If the area filing factor is less than unity, then even larger local surface brightness is required. We plot this local CO surface brightness predicted by our XDR models in Figure \[fig:X2\], and conclude that for densities above $3\times10^4$ cm$^{-3}$, and R from 500–1500 pc, our model is consistent with the observed CO cooling, and suggests area filling factors in the range of 0.3–1. Of course, even if the area filling factor is unity, we note that the [*volume*]{} filling factors can still be quite small. Our results are consistent with the XDR model of @Meijerink_07, who present the CO [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{} to [[$J{\!=\!}3{\!\rightarrow\!}2$]{}]{} intensity ratio as a function on incident X-ray flux and density (their Figure 6). The observed ratio is 8.9–12.3, depending which[[$J{\!=\!}3{\!\rightarrow\!}2$]{}]{} measurement is adopted, suggesting an X-ray flux of 5–20 $\rm erg\,s^{-1}\,cm^{-2}$, which compares well with our working value of $\sim$10 at $\rm d=600\,pc$.
![Ratio of the total CO cooling to the [\[CI\]]{} [[$J{\!=\!}1{\!\rightarrow\!}0$]{}]{} cooling in the XDR model. The heavy contour corresponds to the observed value (130). \[fig:ci\]](f8.eps){width="8.5cm"}
More generally, the XDR models indicate that CO rotational line emission can be $\sim
0.1$ of the absorbed X-ray energy, so only $\sim 1\%$ of the hard X-ray luminosity of the AGN needs to be absorbed in a CO-luminous XDR to produce the CO emission. However, we emphasize that while the XDR can easily heat the gas, it is not likely to appreciably heat the co-extant dust the way that the PDR front would. In an XDR, the ratio of total CO rotational line flux to the [*locally-generated*]{} far-infrared flux from grains (due to absorption of emitted line photons and to UV photons produced by excitation and re-combination) can easily exceed $10^{-3}$, and can be as large as $\sim 0.1$. (This ratio depends on the cloud column density as well as the column density per unit linewidth, since the far-infrared dust emission depends on the former.) Since the observed ratio is $\sim 5\times 10^{-4}$, the observed far-IR / submillimeter dust emission can not be powered by the XDR processes. There must be additional source(s) of local grain heating, e.g., star formation, or re-radiated continuum from the AGN.
As with the PDR, the XDR framework does not perfectly account for the observed [\[CI\]]{} intensities, though it may provide a better match in terms of total cooling. The XDR model generally predicts lower [\[CI\]]{} relative to CO than the PDR, and the [\[CI\]]{} [[$J{\!=\!}1{\!\rightarrow\!}0$]{}]{} line is well-matched with the XDR model. The ratio of the total CO emergent intensity including all transitions to the [\[CI\]]{} [[$J{\!=\!}1{\!\rightarrow\!}0$]{}]{} intensity is plotted in Figure \[fig:ci\]. The observed value of 130 is highlighted, as are ratios a factor of two above and below this value to allow for systematic uncertainties in both. As with the analysis described above, the allowed region corresponds to sub-kpc distances and densities greater than 10$^5\,\rm cm^{-3}$. However, it remains true that the [\[CI\]]{} temperature is far too low; the [\[CI\]]{} [[$J{\!=\!}2{\!\rightarrow\!}1$]{}]{} to [[$J{\!=\!}1{\!\rightarrow\!}0$]{}]{} ratio (R = 2.3 in power units per W03, W05) suggests a physical temperature of only $\sim$30 K, a value generally inconsistent with both the the XDR and the PDR models. W05 suggest that the excess [\[CI\]]{} [[$J{\!=\!}1{\!\rightarrow\!}0$]{}]{} emission may arise in an (additional) cool gas component exterior to the material traced with the CO [[$J{\!=\!}7{\!\rightarrow\!}6$]{}]{}. Such a scenario may be consistent with our findings, as we allow for a second component of low-excitation material outside the disk. However, this would mean that the [\[CI\]]{} [[$J{\!=\!}1{\!\rightarrow\!}0$]{}]{} which is associated with the mid-J CO would be less, corresponding to higher ratios in Figure \[fig:ci\], requiring higher densities and lower distances from the central source. A large allocation of the [\[CI\]]{} [[$J{\!=\!}1{\!\rightarrow\!}0$]{}]{} to this external component is difficult to reconcile with the overall XDR picture. Another possibility is that the measurements of the relatively weak [\[CI\]]{} intensities are subject to systematic errors. We do note that our measurement favors a slightly higher flux for the [\[CI\]]{} [[$J{\!=\!}2{\!\rightarrow\!}1$]{}]{} than W05 (giving $R\sim3.8$, for T$\sim$40 K), though with low significance. We do note that the W05 measurement of [\[CI\]]{} jtwo puts the [\[CI\]]{} ratio well below that of M82 (R = 4.3, per [@Stutzki_97]), the only external galaxy for which the [[$J{\!=\!}2{\!\rightarrow\!}1$]{}]{} measurement is published.
In the context of the XDR model, the above analyses suggest that the gas density is high: $\rm n_{H_2} \ge 5 \times 10^4 cm^{-3} $, for the bulk of the material. This density is higher than required by the CO intensity ratios alone (Figure \[fig:nt2\]); it derives in essence from requiring the total observed CO luminosity to be generated in a volume constrained by the VS03 disk model. High densities have been suggested given the measurement of high-dipole-moment species, beginning with HCN [[$J{\!=\!}1{\!\rightarrow\!}0$]{}]{} (@Solomon_03), and now with HCO$^+$ [[$J{\!=\!}1{\!\rightarrow\!}0$]{}]{} [@Riechers_06]. The two transitions are comparable: $\rm L_{HCO^+} = 0.8 \pm 0.2 L_{HCN}$, and the fractional intensity relative to L$_{\rm FIR}$ nicely matches the linear correlation found by @Gao_04 for the local-universe LIRGS and ULIRGS. While mid-IR pumping schemes could potentially account for these lines, @Riechers_06 conclude that the match between the HCN and HCO$^+$ lines is best explained by both species producing optically-thick thermalized emission (at least for the [[$J{\!=\!}1{\!\rightarrow\!}0$]{}]{} transition), requiring $\rm n_{H_2} \ge 10^5$. This is fully consistent with the both the XDR and the high-density PDR scenarios.
### Turbulence, Cosmic Rays {#sec:turb}
Finally, we note that there are other potential energy sources which will preferentially heat the gas relative to the dust. Cosmic-ray-ionization heating has been proposed to explain the warm molecular material in the nuclei of M 82 and NGC 253 [@Hailey-Dunsheath_08; @Bradford_03; @Suchkov_93]. Cosmic Rays are generated in supernovae associated with massive star formation, and like X-rays, they penetrate throughout the bulk of the molecular material. Simple dissipation of mechanical energy is also a potential heating source. @Bradford_05 apply the shock models of @Draine_83 [@Draine_84; @Roberge_90] to the molecular gas around SGR A$^*$ and find that low-velocity ($v\sim$10–20 [$\,\rm km\,s^{-1}$]{}) magnetohydrodynamic (MHD) shocks produced by cloud-cloud collisions can produce warm material which cools primarily via the mid-J CO rotational transition, again without appreciable heating of the dust. The arguments presented in these studies would apply to the Cloverleaf host, and could also boost the mid-$J$ CO emission relative to that which arises in the PDR alone.
### Prospects for Atomic Gas Tracers
Further distinction between the PDR scenario and the X-ray and other various bulk heating scenarios could be provided by the atomic fine-structure transitions. In particular, the \[OI\] $63\mu$m and \[CII\] $158\mu$m will be important. The ratio of these fine-structure line intensities to the total CO cooling is a powerful discriminant: in the PDR the fine structure lines carry an order of magnitude more power than the CO since most of the UV heating is deposited in the atomic gas, while in the XDR, the energy is more uniformly distributed between the atomic and molecular components. Indeed, our models suggest that the \[OI\] $63\mu$m intensity and the total CO rotational cooling are comparable for the XDR. With the Cloverleaf’s unusual luminosity and lensing factor, these fine-structure lines may be accessible with the spectrometers on Herschel.
### Conclusion – A Composite Solution {#sec:composite}
We conclude that there are several potential gas heating sources, each of which could contribute substantially to powering the CO emission in the Cloverleaf host. The far-IR / submillimeter continuum component appears distinct from the mid-IR component, and the PAH fractional luminosities relative to this far-IR dust component are similar to those observed in starburst systems. These facts are difficult to explain without a massive starburst. On the other hand, since the fractional energy carried in the molecular gas exceeds that of the local starburst analogs by factors of 2–5, and there is a powerful X-ray source which is demonstrably capable of contributing 10–100% of the required heating, it is difficult to exclude the X-rays as a heating source. We conclude that the molecular material around the Cloverleaf QSO is most likely heated by both UV photons from young stars and and X-rays from the accretion zone, with neither dominant by more than $\sim$1 order of magnitude.
X-rays which penetrate and heat the bulk of the molecular gas will impact the properties of the star formation, particularly the stellar mass function. Simple thermodynamic arguments along the lines of the Jeans mass or Bonnor-Ebert mass show that the characteristic mass required for gravitational collapse increases as the ’minimum temperature’ of the gas increases. This is precisely the effect that a bulk heating mechanism would produce. Neglecting the effects of magnetic fields and rotation, this mass is given by $$M_{BE} = \frac{C_{BE} \,v_T^4}{P^{1/2}\, G^{3/2}} \propto \frac{T^{3/2}}{n^{1/2}} \propto \frac{T^2}{P^{1/2}}, \label{eq:bonner}$$ where $T$, $n$, and $P$ are temperature, number density, and thermal pressure, respectively of the environment from which the stars must form, and $C_{BE}$ is a numerical constant. Detailed theoretical approaches confirm the sense of this relation, finding a scaling between the characteristic mass scale $M_*$ in a stellar IMF, and the minimum temperature to which molecular gas can cool—$M_{*} \propto T_{min}^{\gamma}$—with exponent $\gamma$ ranging from 1.7 (obtained in numerical experiments [@Jappsen_05]) to 3.35 (obtained in an analytic treatment [@Larson_85]). In the Galaxy, $M_*\sim0.5$ [M$_{\odot}$]{} and $T_{min}\sim8$ K, values numerically consistent with Equation \[eq:bonner\]. While our analysis does not directly measure $T_{min}$, it must be true that $T_{min}$ is increased relative to the Galaxy. A plausible assertion is that $T_{min}$ scales as the typical temperature derived from fits to CO line ratios: at least $\sim$50K per our analysis of the Cloverleaf fluxes, compared with $\sim$22 K in the inner Galaxy per the COBE FIRAS measurements [@Fixsen_99]. This simple scaling would suggest that $M_*$ in the Cloverleaf starburst is 4–15 times the Galactic value: some 2–5 [M$_{\odot}$]{}. As @Larson_98 and others have pointed out, such a top-heavy IMF converts a given mass of gas into a greater total stellar (and thus far-IR) luminosity than with the Salpeter IMF. This is the leading scenario proposed to explain the factor of $\sim$3–5 discrepancy between the observed stellar mass buildup and the star-formation history in the $4<z<1$ era [@Perez-Gonzalez_08; @Dave_08; @Hopkins_06]. While the Cloverleaf may be somewhat more extreme than the typical star-forming system, the increased prevalence of AGN in this early epoch may generally result in more massive stars than those formed in the Galaxy today.
Additional Features in the Spectrum
-----------------------------------
While it has yet to be explored completely in a extragalactic source, the short submillimeter band is expected to host low-lying transitions of light molecules and ions other than CO, some of which may be bright. We now discuss some tentative identifications which could be followed up with the large collecting area and spectral resolution of the interferometers.
### Water in the Cloverleaf {#sec:water}
The feature at 277.6 GHz is fit with a Gaussian line of $\nu_{\rm rest} = 987.4 \pm 0.55\rm GHz$, adopting the systemic redshift of the Cloverleaf. This might be identified with the lowest-lying transition of ortho-hydronium $o-\rm H_3O^+$ ($0^-_0\rightarrow 1^+_0$, $\nu_{\rm rest} = 984\rm\,GHz$), but unlike the other low-lying $\rm H_3O^+$ transitions ($\nu_{\rm rest} = 396, 388, 364\, \rm GHz$), it is not expected to be bright given the lack of radiative pumping pathways to excite the upper level [@Phillips_92]. Moreover, the abundance of $\rm H_3O^+$ even in extreme Arp 220-like nuclei is believed to be less than $1/10$ that of water [@VanderTak_08]. A better match to the fit, and a more convincing scenario is that this feature is the 987.9 GHz ([$\rm 2_{0,2}{\!\rightarrow\!}1_{1,1}$]{}) transition connecting the first excited level of $p$-[$\rm{H_2O}$]{} at 54 K to the second excited level at 101 K. Attempts to detect high-redshift rotational water transitions have been made. [@Encrenaz_93; @Casoli_94] report tentative detections of the [$\rm 2_{1,1}{\!\rightarrow\!}2_{0,2}$]{} transition ($\nu_{\rm rest}=752.0\,\rm GHz$) in IRAS F10214+4724 ($\rm z=2.3$). Remarkably, the reported intensity (0.65 [$\,\mathrm{K\,km\,s^{-1}}$]{} at 30-m) is only a factor of 0.45 times that of the nearby CO [[$J{\!=\!}6{\!\rightarrow\!}5$]{}]{} transition (1.4 [$\,\mathrm{K\,km\,s^{-1}}$]{} 30-m, per @Solomon_92b). More recently, [@Riechers_06a] report an upper limit on the $3_{1,3}\rightarrow2_{2,0}$ (183 GHz) transition in MG 0751+2716 at $\rm z=3.2$.
The a priori interpretation of water spectra is difficult even with multiple line detections, as evidenced by the work analyzing the ISO spectra toward SGR B2 [@Neufeld_95; @Cernicharo_06] and the Orion outflow [@Harwit_98; @Cernicharo_06a]. Morever, since most of the low-lying water lines are not accessible at zero redshift from the ground, there are no good Galactic templates with which to compare the [$\rm 2_{0,2}{\!\rightarrow\!}1_{1,1}$]{} measurement. Surveys have been conducted only in the ground-state [$\rm 1_{1,0}{\!\rightarrow\!}1_{0,1}$]{} (538 [$\rm \mu m$]{}) line with the KAO, SWAS, and ODIN (e.g. [@Ashby_00; @Neufeld_03; @Snell_00]), finding low abundances ($<10^{-8}$) or upper limits in GMCs and cloud cores. Similarly, @Wilson_07 derive upper limits to the water abundance of $<10^{-8}$ from non-detections in nearby starburst galaxies, albeit with the large (2.1) ODIN beam. However, much larger water abundances (more than $10^{-5}$) are inferred in warm cores and outflows, notably the Orion outflow.
In spite of the ODIN ground-state non-detections in local starbursts, the ISO far-IR spectrum of the Arp 220 nucleus [@Gonzalez-Alfonso_04] demonstrates that water can be abundant and produce powerful features even in a large-beam (i.e. average) extragalactic spectrum, at least in extreme sources. The 179 [$\rm \mu m$]{} absorption from the nucleus shows an equivalent width of $\sim$400[$\,\rm km\,s^{-1}$]{}(in absorption). These investigators quote [$\rm{H_2O}$]{} column densities of 2–10$\times10^{17}\,\rm cm^{-2}$ toward the Arp 220 nucleus, corresponding to abundances of 1–5$\times10^{-8}$ for N$_{\rm H2}\sim 10^{25}\,\rm cm^{-2}$. We measure an equivalent width of $\sim$580[$\,\rm km\,s^{-1}$]{}, which is of the same order, albeit in emission. We do note that the models of @Cernicharo_06a suggest that even in the SGR B2 geometry, the [$\rm 2_{0,2}{\!\rightarrow\!}1_{1,1}$]{} transition in particular could be brought into emission (in contrast with most of the other transitions) due to a favorable pumping / cascade network in the para levels. While there is no straightforward detailed reconciliation of the tentative detection with published models, we note that water emission is more easy to understand in the context of an XDR than in a traditional PDR. Our XDR model suggests a water abundance of order $10^{-7}$, greater than that derived in Arp 220, and more than would be expected in the bulk of molecular material of a PDR.
### Upper limits, absorption identification {#sec:limits}
Before concluding, we briefly note for completeness upper limits for two other transitions, marked in the spectrum in Figure \[fig:spec\]. The radical CH$^+$ has its ground state transition ([[$J{\!=\!}1{\!\rightarrow\!}0$]{}]{}) at $\nu_{\rm rest} = 806\rm GHz$ ($\nu_{\rm obs}= 234.7\rm GHz$). The spectrum shows this channel is above the continuum at the 1.5$\sigma$ level; a formal 2.5$\sigma$ upper limit to the flux is 14 Jy km/s. The LiH [[$J{\!=\!}2{\!\rightarrow\!}1$]{}]{} transition at $\nu_{\rm rest} = 887\rm GHz$ ($\nu_{\rm obs}= 249.4\rm GHz$) is also above the continuum at low significance, with a formal upper limit of 12.3 Jy [$\,\rm km\,s^{-1}$]{}.
The absorption feature at 287 GHz is significant, at 3$\sigma$, Interestingly, this frequency corresponds to the 557 GHz ground state of o-[$\rm{H_2O}$]{} at z=0.93, perfectly within the redshift range expected for the Cloverleaf lens [@Kneib_98b] ($\bar{z} = 0.9 \pm 0.1$). The equivalent width of the absorption is $\sim$600 km/s, which would imply that there must be nearly complete absorption over a several hundred km/s velocity range. A galaxy perfectly positioned along the line of sight could potentially do this. For $\tau$ to be unity over 600 km/s requires a molecular hydrogen column of $6\times 10^{23}\,\rm cm^{-2}$, assuming an ortho-to-para ratio of unity, a water abundance of 10$^{-8}$, and that there is negligible population in the upper levels. The lens is located 0.6 arcsec from the projected Cloverleaf images, thus the light passes within $<$5 kpc of the lens, so the geometry is not implausible. Further investigation of this intriguing possibility will require higher spectral and spatial resolution measurements.
Conclusions
===========
We have built and are now using the first broadband spectrograph for the millimeter / submillimeter, providing a new observational approach for studying galaxies at all redshifts. Among the first experiments is the measurements of the rest-frame 272–444 [$\rm \mu m$]{} spectrum of the Cloverleaf system which contains the bolometric peak in the CO rotational spectrum. In our analysis of the CO intensities and other published data we find the following:
1. [The intensity ratios of the [[$J{\!=\!}3{\!\rightarrow\!}2$]{}]{} through [[$J{\!=\!}9{\!\rightarrow\!}8$]{}]{} CO transitions can be produced with a single gas component. The excitation is high, with thermal pressure $> 10^6\,\rm K\,cm^{-3}$. The 0.2–5$\times 10^{10}$ [M$_{\odot}$]{} of excited molecular gas in the Cloverleaf host must fill a projected size of at least half the 650-pc-radius @Venturini_03 disk. A more compact distribution is not physical based on the absolute line intensities.]{}
2. [As @Lutz_07 conclude, the match between the far-IR dust component and the PAH emission suggests that far-IR-emitting dust component represents a massive starburst. However, the molecular gas cooling in the CO transitions alone is a large fraction ($6\times10^{-4}$) of the bolometric luminosity of this starburst. This is a factor of a few larger than in the local-universe starburst galaxies for which mid-J CO intensities are available.]{}
3. [Given the powerful AGN of the Cloverleaf system, X-rays are likely to be an important energy source for molecular gas even several hundred parsecs from the nucleus. Our XDR model show that $\sim$5% of the bolometric luminosity of the AGN in hard X-rays can easily provide the energy input required to match the observed CO cooling. The X-rays do not appreciably contribute to the far-IR dust continuum.]{}
Our interpretation is that the Cloverleaf host is indeed a undergoing a massive starburst, but that it has additional energy input into the ISM via hard X-rays originating in the accretion zone. As a result, the stellar IMF is likely biased toward much higher masses than in the Galaxy, with a typical stellar mass $M_*$ of several [M$_{\odot}$]{}. With this characteristic mass replacing the Salpeter value of $\sim$0.5 [M$_{\odot}$]{}, the gas consumption rate at the observed luminosity is several times lower than that given by applying the @Kennicutt_98 prescription. The starburst we are witnessing in the Cloverleaf may therefore extend much longer than the 30 Myr estimated by @Lutz_07; it could last for a few hundred Myr. A similar scenario, though perhaps less extreme, may be typical of the star-forming galaxies in the first half of the Universe’s history when energy release peaked.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
The Bethe-Salpeter formalism represents the most accurate method available nowadays for computing neutral excitation energies and optical spectra of crystalline systems from first principles. Bethe-Salpeter calculations yield very good agreement with experiment but are notoriously difficult to converge with respect to the sampling of the electronic wavevectors. Well-converged spectra therefore require significant computational and memory resources, even by today’s standards. These bottlenecks hinder the investigation of systems of great technological interest. They are also barriers to the study of derived quantities like piezoreflectance, thermoreflectance or resonant Raman intensities.
We present a new methodology that decreases the workload needed to reach a given accuracy. It is based on a double-grid on-the-fly interpolation within the Brillouin zone, combined with the Lanczos algorithm. It achieves significant speed-up and reduction of memory requirements. The technique is benchmarked in terms of accuracy on silicon, gallium arsenide and lithium fluoride. The scaling of the performance of the method as a function of the Brillouin Zone point density is much better than a conventional implementation. We also compare our method with other similar techniques proposed in the literature.
address: ' European Theoretical Spectroscopy Facility, Institute of Condensed Matter and Nanosciences, Université catholique de Louvain, Chemin des étoiles 8, bte L7.03.01, 1348 Louvain-la-Neuve, Belgium'
author:
- Yannick Gillet
- Matteo Giantomassi
- Xavier Gonze
bibliography:
- 'interp.bib'
title: 'Efficient On-the-Fly Interpolation Technique for Bethe-Salpeter Calculations of Optical Spectra'
---
Bethe-Salpeter Equation; Lanczos algorithm; 78.20.Bh; 71.15.Dx
Introduction
============
The calculation of optical properties from first principles can be achieved with different levels of approximation and different computational requirements. The Bethe-Salpeter Equation (BSE) in the framework of Many-Body Perturbation Theory is the most precise and sophisticated approach to compute the macroscopic dielectric function including the attractive electron-hole interaction [@Onida2002]. This formalism has been used since 1998 [@Albrecht1998] for the first-principles computation of the optical spectra of semiconductors and insulators. Even though its application to simple materials is reasonably frequent nowadays, the calculation of optical properties of complex materials with more than two dozen atoms in the unit cells is still challenging (see e.g. the work of Kresse *et al.* [@Kresse2012] and Rinke *et al.* [@Rinke2012]). Algorithmic improvements [@Rohlfing2000; @Paier2008; @Gruning2011; @Sander2015], and new theoretical developments to introduce temperature dependence [@Marini2008], or to compute resonant Raman intensities from derivatives of optical response [@Gillet2013] are active domains of research.
Independently of the approximations used, the precise description of the dielectric properties usually requires a large number of wavevectors to sample the Brillouin Zone (BZ). Each wavevector of the BZ, indeed, gives contributions at different transition energies and small changes in the mesh density can induce oscillations in the dielectric properties [@Gillet2013]. The construction of the BSE Hamiltonian requires the computation of matrix elements connecting different points in the wavevector mesh. This computation is the most time-consuming part of the BSE flow and its cost renders well-converged results difficult to achieve. The extraction of the macroscopic dielectric function from the inversion of the Hamiltonian matrix constitutes the last step of the BSE flow. Two different approaches are commonly used nowadays: direct diagonalization and Lanczos algorithms following the seminal work of Haydock [@Haydock1980]. The Lanczos approach was employed for the first time for the solution of the BSE by Benedict *et al.* [@Benedict1998a; @Benedict1999]. It is based on the construction of a chain of vectors obtained by performing simple matrix-vector operations.
Since 1998, different numerical methods have been proposed to help reduce the computational cost. Rohlfing and Louie [@Rohlfing2000] (abbreviated RL) proposed a double-grid technique in which the kernel is interpolated starting from a homogeneous coarse mesh. This approach is used, for example, in the BerkeleyGW code [@Deslippe2012]. Another technique by Fuchs *et al.* [@Fuchs2008], uses an inhomogeneous mesh and refines the sampling to extract specific information for bound excitons. The large Hamiltonian matrices obtained with these methods are then treated either by direct diagonalization or with the Lanczos algorithm. Alternatively, one can perform the average of optical properties using several independent shifted coarse grids, as introduced by Paier *et al.* [@Paier2008] and, later, by Gillet *et al.* [@Gillet2013].
A completely different approach to the BSE problem has been proposed recently by Kammerländer *et al.* [@Kammerlander2012]. The authors focus on a single frequency and avoid the setup of the entire matrix and the direct diagonalization by using an iterative technique that takes advantage of a double grid to solve the Dyson equation.
In the present work, we propose a new method that combines the RL interpolation with the Lanczos-Haydock algorithm without requiring the storage of the full matrix. We also generalize the RL approach to include multi-linear interpolation, and we reformulate the algorithm to render it more scalable and less memory demanding. We present different levels of interpolation, with different computational loads.
This article is organized as follows. In Section \[sec2\], we describe the main equations and the iterative approach used to solve the BSE. Section \[sec3\] presents the interpolation methodology while the technical details of the implementation are discussed in Section \[sec4\]. Finally, in Section \[sec5\], we apply our technique with different interpolation levels to three different crystalline systems: bulk silicon, gallium arsenide and lithium fluoride. We conclude with a comparison between our method and other similar techniques proposed by Paier *et al.* [@Paier2008] and Gillet *et al.* [@Gillet2013], and the work of Kammerländer *et al.* [@Kammerlander2012].
The Bethe-Salpeter Equation and the Lanczos recursion algorithm \[sec2\]
========================================================================
In the so-called Tamm-Dancoff Approximation (TDA) [@Gruning2009], the matrix elements of the BSE Hamiltonian in the transition space, i.e. products of valence and conduction bands, are given by $$\begin{aligned}
H_{vc\boldsymbol{k},v'c'\boldsymbol{k}'} = \left( \varepsilon_{c\boldsymbol{k}}
- \varepsilon_{v\boldsymbol{k}} \right) \delta_{\boldsymbol{k} \boldsymbol{k}'}
\delta_{vv'} \delta_{cc'} + K_{vc\boldsymbol{k},v'c'\boldsymbol{k}'},
\label{bse1}\end{aligned}$$ where the kernel $K$ is defined as $$\begin{aligned}
K_{vc\boldsymbol{k},v'c'\boldsymbol{k}'} =
2 {\ensuremath{\left\langle vc\boldsymbol{k} \left| \bar{v} \right| v'c'\boldsymbol{k}' \right\rangle}} -
{\ensuremath{\left\langle vc\boldsymbol{k} \left| W \right| v'c'\boldsymbol{k}' \right\rangle}},
\label{bse2}\end{aligned}$$ with $$\begin{aligned}
{\ensuremath{\left\langle vc\textbf{k} \left| \bar{v} \right| v'c'\textbf{k} \right\rangle}} &= \int \int
\psi_{v\textbf{k}}(\textbf{r}) \psi_{c\textbf{k}}^*(\textbf{r})
\bar{v}(\textbf{r} - \textbf{r}') \psi_{v'\textbf{k}'}^*(\textbf{r}')
\psi_{c'\textbf{k}'}(\textbf{r}') d\textbf{r}' d\textbf{r} \label{bse3a} \\
{\ensuremath{\left\langle vc\textbf{k} \left| W \right| v'c'\textbf{k}' \right\rangle}} &= \int \int
\psi_{v\textbf{k}}(\textbf{r}) \psi_{v'\textbf{k}'}^*(\textbf{r})
W(\textbf{r},\textbf{r}') \psi_{c\textbf{k}}^*(\textbf{r}')
\psi_{c'\textbf{k}'}(\textbf{r}') d\textbf{r}' d\textbf{r}.
\label{bse3b}\end{aligned}$$
In the above expressions, $v$ and $c$ stands for valence and conduction band indices, $\textbf{k}$ is a wavevector in the BZ, $\varepsilon_{n\textbf{k}}$ and $\psi_{n\textbf{k}}$ are the energies and wavefunctions of band $n$ at point $\textbf{k}$. Equation represents the so-called exchange term and takes into account local-fields effects. $\bar{v}$ is a modified bare Coulomb potential obtained from the bare potential $v(\boldsymbol{G})$ by setting the $\boldsymbol{G} = 0$ component to zero. The expression in Eq. is usually referred to as the direct term and takes into account the static screened Coulomb interaction, $W$, through the inverse dielectric function $\epsilon^{-1}(\textbf{r},\textbf{r}')$ via $$\begin{aligned}
W(\textbf{r},\textbf{r}') = \int d\textbf{r}''
\epsilon^{-1}(\textbf{r},\textbf{r}'') v(\textbf{r}''-\textbf{r}'),
\label{eq:Wcoul}\end{aligned}$$
The wavefunctions and eigenenergies are usually obtained from a standard Kohn-Sham calculation [@Hohenberg1964; @Kohn1965] and a scissors operator may be employed to mimic the opening of the gap introduced by the GW approximation [@Onida2002]. For BSE applications, it is common to compute the direct term with a screened interaction obtained within the Random-Phase Approximation (RPA) [@Adler1962; @Wiser1963]. Alternatively, one can employ the much cheaper model dielectric function proposed by Cappellini in Ref. [@Cappellini1993].
Finally, the macroscopic dielectric function $\varepsilon_M(\omega)$ is given by $$\begin{aligned}
\varepsilon_M(\omega) = 1 - \lim_{\boldsymbol{q} \rightarrow 0}
v(\boldsymbol{q}) {\ensuremath{\left\langle P(\boldsymbol{q}) \left| ((\omega + i\eta) -
H)^{-1} \right| P(\boldsymbol{q}) \right\rangle}} \label{epsmacro}\end{aligned}$$ where $v(\boldsymbol{q})$ is the Fourier transform of the Coulomb interaction, $P(\boldsymbol{q})$ are the oscillator matrix elements $$\begin{aligned}
P(\boldsymbol{q})_{vc\boldsymbol{k}} =
{\ensuremath{\left\langle c\boldsymbol{k+q} \left| e^{i\boldsymbol{q}.\boldsymbol{r}} \right| v\boldsymbol{k} \right\rangle}}\end{aligned}$$ evaluated for small $\boldsymbol{q}$ and $\eta$ is a broadening factor.
The solution of the Bethe-Salpeter equation is a two-step process. First, the matrix elements of the Hamiltonian are computed from Eqs. (\[bse1\]-\[bse3b\]). Then, the macroscopic dielectric function is derived using Eq. .
In order to avoid the inversion of large matrices, Lanczos-based iterative techniques (called Lanczos algorithm in this work) can be used to obtain the macroscopic dielectric function. By using Krylov subspaces, it is possible to express Eq. in terms of a continued fraction formula.
The Lanczos algorithm can be summarized as follows. We start by setting $$\begin{aligned}
b_1 &= 0\\
{\ensuremath{\left| \psi_{1} \left\rangle \right. \right.}} &= \frac{{\ensuremath{\left| P(\boldsymbol{q}) \left\rangle \right. \right.}}}{\|{\ensuremath{\left| P(\boldsymbol{q}) \left\rangle \right. \right.}}\|}.\end{aligned}$$
Then the algorithm iterates with $i$ starting at $1$ $$\begin{aligned}
a_{i} &= {\ensuremath{\left\langle \psi_{i} \left| H \right| \psi_{i} \right\rangle}} \label{eq:ai}\\
b_{i+1} &= \| H {\ensuremath{\left| \psi_{i} \left\rangle \right. \right.}} - a_{i} {\ensuremath{\left| \psi_{i} \left\rangle \right. \right.}} - b_{i} {\ensuremath{\left| \psi_{i-1} \left\rangle \right. \right.}}
\| \label{eq:bi} \\
{\ensuremath{\left| \psi_{i+1} \left\rangle \right. \right.}} &= \frac{H {\ensuremath{\left| \psi_{i} \left\rangle \right. \right.}} - a_{i} {\ensuremath{\left| \psi_{i} \left\rangle \right. \right.}} - b_{i}
{\ensuremath{\left| \psi_{i-1} \left\rangle \right. \right.}}}{b_{i+1}} \label{eq:ci}.\end{aligned}$$
The frequency dependence of the dielectric function is computed in an efficient way in terms of the continued fraction $$\begin{aligned}
\varepsilon_M(\omega) = 1 - \lim_{\boldsymbol{q} \rightarrow 0}
v(\boldsymbol{q}) \frac{\| P(\boldsymbol{q}) \|^2}{(\omega + i\eta) - a_1 -
\frac{b_2^2}{(\omega + i \eta) - a_2 - \frac{b_3^2}{\cdots}}}
\label{eq:fraccont}\end{aligned}$$ and the iteration is stopped when $\varepsilon_M(\omega)$ is converged for each frequency.
The construction of the Krylov chain Eqs. (\[eq:ai\]-\[eq:ci\]) requires only the application of the Hamiltonian on different functions or, in linear algebra language, simple matrix-vector products. The computational cost scales as $\mathcal{O}(m N^2)$ with $m$ the number of iterations of the Lanczos algorithm and $N$ the dimension of the matrix. In our approach, the BSE Hamiltonian is expressed in the electron-hole basis thus $N = N_v N_c
N_k$ where $N_v$ is the number of valence bands, $N_c$ the number of conduction bands and $N_k$ the number of points in the BZ.
The number of iterations $m$ needed to converge $\varepsilon_M(\omega)$ is much smaller than the size of the Hamiltonian and almost independent of the size of the system. As a consequence, Lanczos methods are much more efficient than direct diagonalization techniques that scale as $\mathcal{O}(N^3)$. Unfortunately, unlike diagonalization methods, the Lanczos approach does not give direct access to the exciton levels and the corresponding wavefunctions.
The computation of the BSE matrix elements and the storage of the Hamiltonian represent the most CPU-intensive and memory demanding parts. For example, a converged computation of the dielectric function of bulk silicon requires few bands (3-4 valence bands, 4-6 conduction bands) but a large number of wavevectors in the BZ (from 14$\times$14$\times$14 to 40$\times$40$\times$40, depending on the accuracy required) which means that about $10^3$ to $10^4$ wavevectors must be sampled. This gives, for sequential computers, from days to years of computation, and in terms of memory, matrices of size ranging from 32928$\times$32928 (ca. 16 GB) to 1536000$\times$1536000 (ca. 34 TB). Such huge memory requirements and the corresponding computation time render BSE calculations challenging even on modern supercomputers. These issues are even more severe when BSE results are used to perform resonant Raman scattering calculations that, as illustrated in Ref. [@Gillet2013], require an exceedingly dense BZ sampling.
These two bottlenecks can be reduced by using the technique presented in the next section.
Presentation of the interpolation technique \[sec3\]
====================================================
The interpolation scheme we propose is based on two meshes of wavevectors in the BZ (double-grid technique). Later, we will distinguish different levels of interpolation, all based on this double-grid technique.
To facilitate the discussion, we introduce the following notation. The coarse mesh contains $\tilde{N}_k$ homogeneous wavevectors, denoted as $\tilde{\boldsymbol{k}}$. The dense mesh contains $N_k = \tilde{N}_k \times N_{div}$ homogeneous wavevectors obtained by refining the coarse mesh. The refining in each direction is done by defining equally spaced $n_i$ points in the $i$-th direction. The wavevectors of the coarse mesh are given by $$\begin{aligned}
\tilde{\boldsymbol{k}}_{(i_1,i_2,i_3)} = i_1 \hat{\boldsymbol{k}}_1 + i_2
\hat{\boldsymbol{k}}_2 + i_3 \hat{\boldsymbol{k}}_3,\end{aligned}$$ where $i_j$ are integer coordinates and $\hat{\boldsymbol{k}}_j$ are the basis vectors of the coarse mesh.
The dense wavevectors have fractional coordinates $$\begin{aligned}
\boldsymbol{k}_{(i_1,i_2,i_3),(j_1,j_2,j_3)} = (i_1 + \frac{j_1}{n_1})
\hat{\boldsymbol{k}}_1 + (i_2 + \frac{j_2}{n_2}) \hat{\boldsymbol{k}}_2 + (i_3
+
\frac{j_3}{n_3}) \hat{\boldsymbol{k}}_3 \end{aligned}$$ where $n_1$, $n_2$ and $n_3$ are the number of divisions along the basis vectors while $0 \le j_i \le (n_i - 1)$. For the sake of simplicity, we assume the same number of divisions, $n_1 = n_2 = n_3 = n_{div}$, along the three directions and therefore $N_{div} = n_{div}^3$.
The neighborhood of a dense point, $N(\boldsymbol{k})$, is defined as the set of the eight wavevectors around $\boldsymbol{k}$. The reduced coordinates of this set of points are given by $$\begin{aligned}
N(\boldsymbol{k}_{(i_1,i_2,i_3),(j_1,j_2,j_3)}) =
\left\{\tilde{\boldsymbol{k}}_{(i_1,i_2,i_3),(j_1,j_2,j_3)}^{lmn} \right\}
\text{ with }
l,m,n = 0,1 \label{eq:neighborhood}\end{aligned}$$ where $\tilde{\boldsymbol{k}}^{lmn}$ is the $lmn^\text{th}$-neighbor of $\boldsymbol{k}$ $$\begin{aligned}
\tilde{\boldsymbol{k}}_{(i_1,i_2,i_3),(j_1,j_2,j_3)}^{lmn} =
\tilde{\boldsymbol{k}}_{(i_1+l,i_2+m,i_3+n)}.\end{aligned}$$
The dense set of a coarse point, $S(\tilde{\boldsymbol{k}})$, is defined as $$\begin{aligned}
S(\tilde{\boldsymbol{k}}_{(i_1,i_2,i_3)}) = \left\{
\boldsymbol{k}_{(i_1,i_2,i_3),(j_1,j_2,j_3)} \right\} \forall (j_1,j_2,j_3).
\label{eq:subspace}\end{aligned}$$
Using these definitions, we can derive the following important relation $$\begin{aligned}
\sum_{\boldsymbol{k}} \sum_{\tilde{\boldsymbol{k}} \in N(\boldsymbol{k})}
f(\boldsymbol{k},\tilde{\boldsymbol{k}})&=
\sum_{\tilde{\boldsymbol{k}}'} \sum_{\boldsymbol{k} \in
S(\tilde{\boldsymbol{k}'})} \sum_{\tilde{\boldsymbol{k}} \in
N(\boldsymbol{k})} f(\boldsymbol{k},\tilde{\boldsymbol{k}}) \nonumber \\
&=
\sum_{\tilde{\boldsymbol{k}}'} \sum_{\boldsymbol{k} \in
S(\tilde{\boldsymbol{k}'})} \sum_{\tilde{\boldsymbol{k}} \in
N(\tilde{\boldsymbol{k}'})}f(\boldsymbol{k},\tilde{\boldsymbol{k}}) \nonumber \\
&=
\sum_{\tilde{\boldsymbol{k}}'}
\sum_{\tilde{\boldsymbol{k}} \in
N(\tilde{\boldsymbol{k}'})}
\sum_{\boldsymbol{k} \in
S(\tilde{\boldsymbol{k}'})}
f(\boldsymbol{k},\tilde{\boldsymbol{k}})\label{eq:ordersums}\end{aligned}$$ where Eq. has been used. This property will be used afterwards to perform the interpolation of data defined on the coarse mesh.
As discussed in the previous section, the Lanczos algorithm is based on matrix-vector products. In our method, we perform these operations on-the-fly to avoid the storage of the BSE Hamiltonian in memory. The matrix elements of the kernel are interpolated while performing the matrix-vector operations. As discussed in more detail in the next paragraphs, different levels of interpolation can be employed in this part of the algorithm.
Since the periodic parts of the Bloch states at a given wavevector form a complete basis set, any wavefunction on the dense mesh can be expressed as [@Rohlfing2000] $$\begin{aligned}
{\ensuremath{\left| u_{n\boldsymbol{k}} \left\rangle \right. \right.}} = \sum_{n'}
d_{n\boldsymbol{k}}^{n'\tilde{\boldsymbol{k}}}
{\ensuremath{\left| u_{n'\tilde{\boldsymbol{k}}} \left\rangle \right. \right.}}
\label{eq:completeness}\end{aligned}$$ where $$\begin{aligned}
d_{n\boldsymbol{k}}^{n'\tilde{\boldsymbol{k}}} =
{\ensuremath{\left\langle u_{n'\tilde{\boldsymbol{k}}} \left| u_{n\boldsymbol{k}} \right\rangle \right.}}.\end{aligned}$$
In the transition basis set, the electron-hole wavefunction $\Psi(\boldsymbol{r},\boldsymbol{r}')$ is given by a linear combination of products of single-particle orbitals according to $$\begin{aligned}
\Psi(\boldsymbol{r},\boldsymbol{r}') = \sum_{vc\boldsymbol{k}}
A_{vc\boldsymbol{k}} \phi_{vc\boldsymbol{k}}(\boldsymbol{r},\boldsymbol{r}')\end{aligned}$$ where $$\begin{aligned}
\phi_{vc\boldsymbol{k}}(\boldsymbol{r},\boldsymbol{r}') = e^{-i
\boldsymbol{k}.\boldsymbol{r}} u^*_{v\boldsymbol{k}}(\boldsymbol{r}) e^{i
\boldsymbol{k}.\boldsymbol{r}'} u_{c\boldsymbol{k}}(\boldsymbol{r}') = e^{i
\boldsymbol{k}.(\boldsymbol{r}' - \boldsymbol{r})}
U_{vc\boldsymbol{k}}(\boldsymbol{r},\boldsymbol{r}').\end{aligned}$$
One basis function on the dense mesh can then be expanded in term of the wavefunctions located on a coarse point by means of $$\begin{aligned}
{\ensuremath{\left| U_{vc\boldsymbol{k}} \left\rangle \right. \right.}} = \sum_{n_1 n_2}
(d_{v\boldsymbol{k}}^{n_1\tilde{\boldsymbol{k}}})^*
d_{c\boldsymbol{k}}^{n_2\tilde{\boldsymbol{k}}} {\ensuremath{\left| U_{n_1 n_2
\tilde{\boldsymbol{k}}} \left\rangle \right. \right.}}. \label{eq:expansion}\end{aligned}$$
The method developed by Rohlfing and Louie in Ref. [@Rohlfing2000] uses a single reference point $\tilde{\boldsymbol{k}}$ to expand the kernel according to $$\begin{aligned}
K^i_{vc\boldsymbol{k},v'c'\boldsymbol{k}'} = \sum_{n_1 n_2}
d_{v\boldsymbol{k}}^{n_1\tilde{\boldsymbol{k}}}
(d_{c\boldsymbol{k}}^{n_2\tilde{\boldsymbol{k}}})^*
\sum_{n_3 n_4}
(d_{v'\boldsymbol{k}'}^{n_3\tilde{\boldsymbol{k}'}})^*
d_{c'\boldsymbol{k}'}^{n_4\tilde{\boldsymbol{k}'}}
K_{n_1 n_2 \tilde{\boldsymbol{k}}, n_3 n_4
\tilde{\boldsymbol{k}'}} \label{eqRL}\end{aligned}$$ and we generalize their approach by including eight coarse points in the expansion of the wavefunctions $$\begin{aligned}
{\ensuremath{\left| U_{vc\boldsymbol{k}} \left\rangle \right. \right.}} = \sum_{\tilde{\boldsymbol{k}} \in
N(\boldsymbol{k})} f(\boldsymbol{k},\tilde{\boldsymbol{k}}) \sum_{n_1
n_2} (d_{v\boldsymbol{k}}^{n_1\tilde{\boldsymbol{k}}})^*
d_{c\boldsymbol{k}}^{n_2\tilde{\boldsymbol{k}}}
{\ensuremath{\left| U_{n_1 n_2 \tilde{\boldsymbol{k}}} \left\rangle \right. \right.}}, \label{eq:interpwfn}\end{aligned}$$ where $f(\boldsymbol{k},\tilde{\boldsymbol{k}})$ are interpolation prefactors. The RL interpolation scheme is a special case of Eq. in which $f(\boldsymbol{k},\tilde{\boldsymbol{k}}) =
1$ for a chosen neighbor and 0 for all the other ones.
In order to accelerate the convergence of the expansion, we perform a trilinear interpolation of the coefficients. In this case, the prefactors are given by $$\begin{aligned}
f(\boldsymbol{k},\tilde{\boldsymbol{k}}) &= 0 &\text{ if }
\tilde{\boldsymbol{k}} \not\in N(\boldsymbol{k}) \nonumber \\
f(\boldsymbol{k},\tilde{\boldsymbol{k}}^{lmn}) &=
f^{lmn}_{\boldsymbol{k}_{(i_1,i_2,i_3),(j_1,j_2,j_3)}} = f^l_{j_1}
f^m_{j_2} f^n_{j_3}
& \end{aligned}$$ with $$f^{l}_j = \left\{ \begin{split}
1-\frac{j}{n_{div}} ~ \text{if} ~ l = 0 \\
\frac{j}{n_{div}} ~ \text{if} ~ l = 1.
\end{split} \right.$$
Using Eq. , one obtains the following expression for the interpolated matrix elements $$\begin{aligned}
K^i_{vc\boldsymbol{k},v'c'\boldsymbol{k}'} =
\sum_{\tilde{\boldsymbol{k}} \in N(\boldsymbol{k})}
f(\boldsymbol{k},\tilde{\boldsymbol{k}})
\sum_{n_1 n_2} d_{v \boldsymbol{k}}^{n_1,\tilde{\boldsymbol{k}}}
(d_{c\boldsymbol{k}}^{n_2\tilde{\boldsymbol{k}}})^*
\sum_{\tilde{\boldsymbol{k}}' \in N(\boldsymbol{k}')}
f(\boldsymbol{k}',\tilde{\boldsymbol{k}}') \sum_{n_3 n_4}
(d_{v'\boldsymbol{k}'}^{n_3\tilde{\boldsymbol{k}}'})^*
d_{c'\boldsymbol{k}'}^{n_4\tilde{\boldsymbol{k}}'}
K_{n_1 n_2 \tilde{\boldsymbol{k}}, n_3 n_4
\tilde{\boldsymbol{k}'}} \label{interp1}.\end{aligned}$$
By using the overlaps of the periodic parts of the wavefunctions, we are thus able to include correctly the phases of the wavefunctions and these phases will cancel out with the oscillator matrix elements $P(\boldsymbol{q})$ computed with the wavefunctions on the dense mesh.
It should be stressed, however, that the matrix elements of the Coulomb interaction diverge when $\boldsymbol{q} = \boldsymbol{k} - \boldsymbol{k}' \rightarrow
0$. Following Ref. [@Rohlfing2000], we rewrite the matrix elements as $$\begin{aligned}
K_{v c \boldsymbol{k}, v' c'
\boldsymbol{k}'} =
\frac{a_{vc\boldsymbol{k},v'c'\boldsymbol{k}'}}{\boldsymbol{q}^2} +
\frac{b_{vc\boldsymbol{k},v'c'\boldsymbol{k}'}}{\boldsymbol{q}} +
c_{vc\boldsymbol{k},v'c'\boldsymbol{k}'} \label{diverg}\end{aligned}$$ and we note that an accurate interpolation technique should try to reproduce the divergent behavior as much as possible.
The different schemes we have implemented to treat the divergence are discussed in more detail in the next section.
Combining Lanczos algorithm with interpolation \[sec4\]
=======================================================
As previously discussed, the dimension of the matrix on the coarse mesh is $N_{coarse} = N_c N_v \tilde{N}_k$, while the dimension of the matrix on the dense mesh is $N_{dense} = N_c N_v N_k = N_c N_v \tilde{N}_k N_{div}$. The calculation of the matrix elements of the Hamiltonian as well as the Lanczos algorithm scale as $\mathcal{O}(N^2)$. The numerical complexity of the standard BSE solution on the coarse mesh is thus $$\begin{aligned}
\mathcal{O}(N^2_c N^2_v \tilde{N}^2_k)\end{aligned}$$ while the complete solution on the dense mesh scales as $$\begin{aligned}
\mathcal{O}(N^2_c N^2_v \tilde{N}^2_k N^2_{div}).\end{aligned}$$
To fix the ideas, supposing a halving of the coarse mesh for the three directions giving the dense mesh, $N_{div}=8$ and $N_{div}^2=64$, which points out the significant burden of using the dense meshes. If $\tilde{N}_k$ is kept constant, and $N_{div}$ is increased, the use of the dense mesh is even more unfavorable.
The most memory-demanding part is the storage of the Hamiltonian which scales quadratically with the size of the Hamiltonian. The interpolation technique given in Eq. can be implemented in two different ways. The interpolated matrix elements can be stored in memory and then used as a standard matrix for the Lanczos technique. This is the approach followed by Rohlfing in [@Rohlfing2000]. It is worth noting, however, that although the RL method allows one to avoid the explicit computation of the matrix elements on the dense mesh, the numerical complexity and the memory requirements of the approach are still the ones of a standard BSE.
Alternatively, one can reformulate the equations so that the interpolation is done on-the-fly without allocating extra memory for the dense Hamiltonian. This is the central result of this paper. As the Lanczos technique requires only matrix-vector multiplications, the full-matrix vector multiplication with the Hamiltonian can be written as $$\begin{aligned}
\phi^{(n+1)}_{vc\boldsymbol{k}} =& \sum_{v'c'\boldsymbol{k}'}
H^i_{vc\boldsymbol{k},v'c'\boldsymbol{k}'} \phi^{(n)}_{v'c'\boldsymbol{k}'} \\
=& \left( \varepsilon_{c\boldsymbol{k}}
- \varepsilon_{v\boldsymbol{k}} \right) \phi^{(n)}_{vc\boldsymbol{k}} +
\sum_{v'c'\boldsymbol{k}'}
K^i_{vc\boldsymbol{k},v'c'\boldsymbol{k}'} \phi^{(n)}_{v'c'\boldsymbol{k}'}\end{aligned}$$ that can be computed with $\mathcal{O}(N_c N_v \tilde{N}_k
N_{div})$ scaling. The matrix-vector product with the kernel can be rewritten as $$\begin{aligned}
s_{vc\boldsymbol{k}} =& \sum_{v'c'\boldsymbol{k}'}
K^i_{vc\boldsymbol{k},v'c'\boldsymbol{k}'} p_{v'c'\boldsymbol{k}'} \\
=& \sum_{\tilde{\boldsymbol{k}} \in N(\boldsymbol{k})}
f(\boldsymbol{k},\tilde{\boldsymbol{k}})
\sum_{n_1 n_2} d_{v \boldsymbol{k}}^{n_1,\tilde{\boldsymbol{k}}}
(d_{c\boldsymbol{k}}^{n_2\tilde{\boldsymbol{k}}})^* \nonumber \\
& \sum_{\tilde{\boldsymbol{k}}''} \sum_{\tilde{\boldsymbol{k}}' \in
N(\tilde{\boldsymbol{k}}'')} \sum_{n_3 n_4}
K_{n_1 n_2 \tilde{\boldsymbol{k}}, n_3 n_4
\tilde{\boldsymbol{k}'}} \nonumber \\
& \sum_{\boldsymbol{k}' \in
S(\tilde{\boldsymbol{k}}'')}
f(\boldsymbol{k}',\tilde{\boldsymbol{k}}') \sum_{v'c'}
(d_{v'\boldsymbol{k}'}^{n_3\tilde{\boldsymbol{k}}'})^*
d_{c'\boldsymbol{k}'}^{n_4\tilde{\boldsymbol{k}}'} p_{v'c'\boldsymbol{k}'}
\label{interpmatmul}.\end{aligned}$$ and can be computed in three steps using $$\begin{aligned}
q^{uvw}_{n_3 n_4 \tilde{\boldsymbol{k}''}} =&
\sum_{\boldsymbol{k}' \in
S(\tilde{\boldsymbol{k}}'')}
f(\boldsymbol{k}',\tilde{\boldsymbol{k}}') \sum_{v'c'}
(d_{v'\boldsymbol{k}'}^{n_3\tilde{\boldsymbol{k}}'})^*
d_{c'\boldsymbol{k}'}^{n_4\tilde{\boldsymbol{k}}'} p_{v'c'\boldsymbol{k}'}
\text{ with } \tilde{\boldsymbol{k}}' = {\tilde{\boldsymbol{k}''}}^{uvw}
\label{matmul1} \\
r_{n_1 n_2 \tilde{\boldsymbol{k}}} =& \sum_{\tilde{\boldsymbol{k}}''}
\sum_{uvw} \sum_{n_3 n_4}
K_{n_1 n_2 \tilde{\boldsymbol{k}}, n_3 n_4
({\tilde{\boldsymbol{k}''}}^{uvw})} q^{uvw}_{n_3 n_4 \tilde{\boldsymbol{k}''}}
\label{matmul2} \\
s_{vc\boldsymbol{k}} =& \sum_{\tilde{\boldsymbol{k}} \in N(\boldsymbol{k})}
f(\boldsymbol{k},\tilde{\boldsymbol{k}})
\sum_{n_1 n_2} d_{v \boldsymbol{k}}^{n_1,\tilde{\boldsymbol{k}}}
(d_{c\boldsymbol{k}}^{n_2\tilde{\boldsymbol{k}}})^* r_{n_1 n_2
\tilde{\boldsymbol{k}}}
\label{matmul3}.\end{aligned}$$
A schematic representation of the algorithm is given in Fig. \[fig:schema-interp\]. The application of the interpolated Hamiltonian is equivalent to averaging dense vector ($p$) on a coarse vector ($q$), then applying the coarse Hamiltonian ($r$) and finally rebuilding the full vector information on the dense mesh ($s$).
The numerical complexity of Eq. , and is $\mathcal{O}(N^2_c N^2_v \tilde{N}_k N_{div})$, $\mathcal{O}(N^2_c N^2_v \tilde{N}^2_k)$, and $\mathcal{O}(N^2_c N^2_v
\tilde{N}_k N_{div})$ respectively instead of the $\mathcal{O}(N^2_c N^2_v
\tilde{N}^2_k N^2_{div})$ scaling of a BSE run done on the same dense mesh without interpolation. This first approach is called “Method 1” (M1) in the rest of this work.
As stated at the end of Section \[sec3\], a better treatment of the divergence is expected to improve the accuracy of the interpolation. Considering Eq. , one can interpolate the coefficients and then divide the interpolated quantities by the $\boldsymbol{q}$ computed on the dense mesh. The drawback of this approach, however, is that it requires the computation of the whole matrix since the fast algorithm developed in Eq. is not applicable thus resulting in $\mathcal{O}(N_c^2
N_v^2 \tilde{N}_k^2 N_{div}^2)$ scaling. This approach is called “Method 2” (M2) in the rest of this work.
The last method (“Method 3”, M3) has been developed as a compromise between accuracy and numerical complexity. In this case, the divergent behavior is reproduced only in a small region along the diagonal with a width that can be adjusted by the user (see Fig. \[figdiv\]). This approximation allows us to employ the fast interpolation of Eq. for the full matrix. The computational cost needed to treat the divergence is negligible provided that the width is small with respect to the number of points on the coarse mesh. Under this assumption, the overall complexity of M3 is $\mathcal{O}(N_c^2 N_v^2 \tilde{N}_k N_{div}^2)$.
We define the width $w$ relatively to the smallest distance between two points in the coarse grid $d$. Then, all $(k,k')$ pairs in the dense mesh so that $||k-k'|| > w \times d$ are treated with Method 1 and for the other pairs, the coefficients of Eq. are interpolated and used together with the dense $\boldsymbol{q}$ to treat the divergence.
Comparison of the interpolation schemes \[sec5\]
================================================
In this section, the different interpolation schemes are tested and compared in detail. Our method has been implemented in the open source ABINIT code [@Gonze2005; @Gonze2009] and will be made available in the forthcoming release. First, three different prototype systems, silicon, gallium arsenide and lithium fluoride, are studied and the accuracy of the three schemes discussed in the previous section is studied in detail. Then, a non-physical test case with very low convergence parameters is used to analyze how the computational cost scales with the total number of points employed to sample the BZ.
Accuracy on test cases
----------------------
Silicon and gallium arsenide have relatively high dielectric constant (10.9 for GaAs and 12 for Si [@Yu2010]) and therefore small binding energies (4-5 meV for GaAs [@Rohlfing1998; @Yu2010] and 15 meV for Si [@Green2013]) and Mott-Wannier-like excitons. LiF, on the other hand, has a relatively small dielectric constant of 1.9 [@Rohlfing1998], yielding a weak screened interaction and therefore strong excitonic effects (binding energy on the order of 3 eV [@Benedict1998a; @Rohlfing1998; @Arnaud2001; @Marini2003]), and Frenkel-like excitons. We decided to use these prototype semi-conductors because, as discussed in [@Sander2015], their BSE matrices have very different behavior in k-space and it is important to understand how our interpolation scheme performs in two different scenarios.
Si and GaAs have been simulated using cut-off energies of 16 Ha for the wavefunctions and 4 Ha for the dielectric matrix, while for LiF, a cut-off energy of 50 Ha has been used for the wavefunctions and 4 Ha for the dielectric matrix. Three valence bands and four conduction bands were included in the electron-hole basis set. Lanczos chain iterations were stopped when the full dielectric spectrum reached a maximum relative difference of $1\%$ both on the real part and the imaginary part. The model dielectric function of Ref. [@Cappellini1993] has been used to avoid the computation of the inverse dielectric matrix, with the parameter $\epsilon^\infty$ set to 12, 10 and 2 for Si, GaAs and LiF respectively. A scissors shift is applied on top of the LDA Kohn-Sham eigenvalues to mimic the effect of the GW approximation (0.8 eV for Si and GaAs, 5.7 eV for LiF). The broadening factor (see Eq. ) is $\eta =
0.1$ eV.
Results with two coarse grids of $4\times4\times4$ and $8\times8\times8$, interpolated to $8\times8\times8$ and $16\times16\times16$, respectively, are presented in Fig. \[fig:interpsi\], \[fig:interpgaas\] and \[fig:interplif\] for silicon, gallium arsenide and lithium fluoride respectively. The three main peak positions and maximum amplitudes extracted from these results are presented in Table. \[tab:interpsi\], \[tab:interpgaas\] and \[tab:interplif\]. All the calculations are done with BZ meshes shifted along the $(0.011, 0.021, 0.031)$ direction in order to improve the accuracy of the sampling.
\
\
\
--------------------------- ----------- -------- ----------- ------- ----------- -------
Pos. (eV) Max. Pos. (eV) Max. Pos. (eV) Max.
4x4x4 3.19 126.03 4.19 74.53 5.13 17.26
4x4x4 + M1(8NB) 3.46 41.69 4.23 63.32 5.34 13.47
4x4x4 + M2(8NB) 3.35 41.44 4.10 64.31 5.22 14.28
4x4x4 + M3(8NB) 3.35 43.17 4.11 64.50 5.22 14.06
4x4x4 + M1(1NB) 3.44 44.02 4.21 60.66 5.33 14.68
4x4x4 + M2(1NB) 3.36 41.90 4.12 62.59 5.25 13.93
4x4x4 + M3(1NB) 3.36 43.50 4.13 62.93 5.25 13.81
4x4x4 + Ref.[@Gillet2013] 3.23 38.71 3.99 50.97 5.15 16.38
8x8x8 3.37 41.25 4.14 60.74 5.24 13.60
8x8x8 + M1(8NB) 3.55 38.13 4.28 50.99 5.25 14.71
8x8x8 + M2(8NB) 3.51 36.81 4.24 51.58 5.19 15.02
8x8x8 + M3(8NB) 3.51 37.71 4.23 51.56 5.19 14.83
8x8x8 + M1(1NB) 3.55 38.72 4.27 51.00 5.21 14.29
8x8x8 + M2(1NB) 3.51 36.62 4.24 51.05 5.19 14.80
8x8x8 + M3(1NB) 3.51 37.59 4.23 50.98 5.18 14.66
8x8x8 + Ref.[@Gillet2013] 3.58 36.49 4.21 48.34 5.17 14.43
16x16x16 3.50 38.37 4.23 50.80 5.19 14.57
--------------------------- ----------- -------- ----------- ------- ----------- -------
: Peak position (Pos.) and maximum amplitude (Max.) of the three main peaks of the absorption spectra of silicon represented in Figure \[fig:interpsi\]. See the caption of the figure for a complete description of the notations.[]{data-label="tab:interpsi"}
--------------------------- ----------- ------- ----------- -------- ----------- -------
Pos. (eV) Max. Pos. (eV) Max. Pos. (eV) Max.
4x4x4 1.70 31.67 2.62 104.08 4.34 72.43
4x4x4 + M1(8NB) 1.93 4.75 2.86 37.60 4.52 35.65
4x4x4 + M2(8NB) 1.82 4.62 2.73 36.39 4.37 33.79
4x4x4 + M3(8NB) 1.82 4.67 2.73 37.17 4.36 33.52
4x4x4 + M1(1NB) 1.93 4.85 2.83 37.79 4.48 31.67
4x4x4 + M2(1NB) 1.82 4.62 2.74 36.08 4.39 33.57
4x4x4 + M3(1NB) 1.82 4.67 2.74 36.85 4.39 33.27
4x4x4 + Ref.[@Gillet2013] 1.70 4.53 2.60 33.93 4.28 30.57
8x8x8 1.82 4.57 2.74 36.16 4.40 34.45
8x8x8 + M1(8NB) 2.04 2.95 3.00 25.08 4.47 38.08
8x8x8 + M2(8NB) 2.00 2.83 2.95 24.66 4.42 38.14
8x8x8 + M3(8NB) 2.00 2.87 2.94 25.00 4.41 38.12
8x8x8 + M1(1NB) 2.03 3.12 2.99 25.92 4.45 37.36
8x8x8 + M2(1NB) 1.98 2.88 2.95 25.06 4.42 38.02
8x8x8 + M3(1NB) 1.98 2.91 2.94 25.40 4.41 37.77
8x8x8 + Ref.[@Gillet2013] 1.91 2.69 2.91 23.32 4.38 35.58
16x16x16 1.98 2.95 2.93 25.84 4.41 37.74
--------------------------- ----------- ------- ----------- -------- ----------- -------
: Peak position (Pos.) and maximum amplitude (Max.) of the three main peaks of the absorption spectra of gallium arsenide represented in Figure \[fig:interpgaas\]. See the caption of the figure for a complete description of the notations.[]{data-label="tab:interpgaas"}
--------------------------- ----------- ------- ----------- ------ ----------- ------
Pos. (eV) Max. Pos. (eV) Max. Pos. (eV) Max.
4x4x4 11.77 15.52 12.96 7.15 14.22 2.02
4x4x4 + M1(8NB) 12.67 15.02 14.06 2.23 14.61 1.49
4x4x4 + M2(8NB) 12.24 13.77 13.37 2.11 13.83 3.37
4x4x4 + M3(8NB) 12.08 14.92 13.35 1.80 13.82 2.61
4x4x4 + M1(1NB) 12.12 15.88 13.27 1.47 13.72 1.81
4x4x4 + M2(1NB) 12.10 16.82 13.37 1.95 13.79 3.07
4x4x4 + M3(1NB) 11.91 17.91 13.35 1.57 13.78 2.34
4x4x4 + Ref.[@Gillet2013] 11.97 11.71 12.96 1.32 13.43 2.21
8x8x8 12.0 18.4 13.35 1.85 13.77 2.62
8x8x8 + M1(8NB) 12.20 18.00 13.88 3.04 14.21 1.75
8x8x8 + M2(8NB) 12.12 16.52 13.56 3.46 14.00 1.72
8x8x8 + M3(8NB) 11.95 17.80 13.54 3.05 14.00 1.51
8x8x8 + M1(1NB) 12.02 18.06 13.52 2.68 13.72 1.81
8x8x8 + M2(1NB) 12.07 17.16 13.54 3.53 13.99 1.65
8x8x8 + M3(1NB) 11.90 18.31 13.51 3.34 13.86 1.47
8x8x8 + Ref.[@Gillet2013] 11.98 18.39 13.47 2.23 13.77 1.62
16x16x16 11.99 18.49 13.52 3.22 13.99 1.51
--------------------------- ----------- ------- ----------- ------ ----------- ------
: Peak position (Pos.) and maximum amplitude (Max.) of the three main peaks of the absorption spectra of lithium fluoride represented in Figure \[fig:interplif\]. See the caption of the figure for a complete description of the notations.[]{data-label="tab:interplif"}
By comparing the interpolation schemes with 8 neighbors (8NB) and standard BSE computations done on the dense mesh, we observe that M1 (8NB) tends to shift the entire spectrum by a small energy and the excitonic binding energy is therefore underestimated. M2 (8NB) and M3 (8NB) give similar results for Si and GaAs that are almost on top of the computation done on the dense mesh. The case of lithium fluoride is more complicated to interpret. In this system, indeed, M1 (8NB) gives inaccurate results for the position of the first exciton (0.2 eV of error for a $8\times8\times8$ coarse mesh). M2 (8NB) performs better than M1 (8NB) although the error in the position of the first peak is still on the order of 0.12 eV. M3 (8NB) gives the best results: the excitonic binding energy is reproduced with 0.05 eV error and also the behavior at higher frequency is correctly reproduced. It should be stressed, however, that this agreement is somehow fortuitous and related to the particular value of the width $w$ used for the treatment of the divergence. Figure \[fig:m3widthlif\] shows the optical spectra of LiF computed with M3 and different values of $w$. Our results indicate that the value of the width used in M3 has a significant impact on the position of the first peak of LiF. Therefore some sort of convergence study is needed for M3 in order to find values of $w$ giving a good compromise between accuracy and efficiency.
If we compare the method using one neighbor (1NB) and the eight neighbors (8NB), we observe that the original Rohlfing and Louie interpolation (1NB) gives results for GaAs and Si whose quality is comparable to the multilinear interpolation and even better results for the special case of LiF. This is somewhat puzzling. Our current understanding is as follows. As already mentioned in the previous paragraph, the description of the divergent behavior along the diagonal of the Hamiltonian for LiF is extremely important to get a correct binding energy. In practice, the number of bands used in Eq. must be truncated and therefore the expansion is not exact. Furthermore, we neglect in the expansion possible contributions to valence (conduction) states coming from the conduction (valence) manifold in Eq. . This approximation is also used in Ref. [@Rohlfing2000]. Some terms are therefore neglected and they lead to some loss of information when building the interpolated matrix element from multiple neighbors.
Our results indicate that, although the multilinear interpolation was expected to give more accurate results, the practical implementation and the numerical approximations tend to favor a “simple” 1-neighbor interpolation. This interpolation gives sufficient accuracy at a lower computational cost as summations over 1 neighbor are cheaper than summations over 8 neighbors.
For the sake of completeness, we have also compared our methods with the multiple-shift technique introduced in Ref. [@Paier2008; @Gillet2013] and used recently in Ref. [@Sander2015]. Different coarse grids are obtained by shifting an initial homogeneous mesh so that the full set of points forms a much denser sampling. An approximate dielectric function is then obtained by averaging the results obtained on the coarse grids. As can be seen in Fig. \[fig:interpsi\], \[fig:interpgaas\] and \[fig:interplif\], this technique tends to smooth the spectrum and the amplitude of the peaks is underestimated. As stated in Ref. [@Sander2015], due to the localized character of the exciton in LiF, a small number of points in the coarse grids is enough to converge the peak position but the correct description of the fine details of the spectrum requires more accurate methods. The methods developed in the present article are more accurate than this technique and are significantly cheaper as they do not require multiple expensive calculations of BSE Hamiltonians.
As regards computational efficiency, one should notice that the time required to produce an interpolated spectrum for LiF in sequential with the above-mentioned parameters is respectively 22200 sec for M1 (8NB), 120000 sec for M2 (8NB), 3000 sec for M1 (1NB) and 80000 sec for M2 (1NB). As a reference, the time needed to compute the matrix elements of the BSE Hamiltonian on the coarse mesh is around 15000 sec and around $1\times10^6$ sec for the dense mesh. To sum up, M1 leads to a gain of two orders of magnitude in terms of CPU time while the high-accuracy M2 gives a speedup of one order of magnitude. The memory required by M1 is of the same order as the one needed for a calculation on the coarse mesh whereas M2 is much more memory demanding since the whole dense matrix must be stored.
The technique based on multiple shifts, on the other hand, requires 8 calculations of a coarse Hamiltonian. These calculations are independent and can be executed in parallel but the final results cannot reach the same frequency resolution as the ones obtained with a fast interpolation on a dense k-mesh.
Numerical scaling of the interpolation technique
------------------------------------------------
In order to assess the numerical scaling of our implementation, we have performed several benchmarks for silicon with unconverged parameters. A cut-off energy of 4 Ha has been used for the wavefunctions and 2 Ha for the dielectric function. Only one valence and one conduction band are included in the Bethe-Salpeter kernel. This allowed us to increase the number of wavevectors of the dense grid to more than 100000 wavevectors in the BZ.
For the three different methods, we have analyzed the time spent in the most important routines. Different benchmarks have been performed by changing the initial coarse grid as well as the final dense mesh of $\tilde{N}_k \times N_{div}$ wavevectors. The most CPU-critical sections are `Hinterp` for the calculation of the interpolated matrix elements and `Matmul` for the matrix-vector multiplications needed for the Lanczos method. The theoretical scaling is given in Table \[tab:scalings\] while the results of the tests are reported in Fig. \[fig:scalings\]. Several interesting observations on the major trends can be derived from the benchmarks. If we look at the interpolated matrix-vector product (`Matmul`), we observe that M1 is very efficient as it scales with the square of the size of the coarse mesh. On the other hand, both M2 and M3 are less performant. Finally, the time spent by M3 in the routine `Hinterp` (interpolation of matrix elements) is much smaller than the one spent by M2 at dense meshes.
-----------------------------------------------------------------------------------------------
`Matmul` `Hinterp`
---- ------------------------------------------------------------ -----------------------------
M1 $\mathcal{O}(\tilde{N}_k^2 + \tilde{N}_k N_{div})$ -
M2 $\mathcal{O}(\tilde{N}_k^2 N_{div}^2)$ $\mathcal{O}(\tilde{N}_k^2
N_{div}^2)$
M3 $\mathcal{O}(\tilde{N}_k N_{div}^2 + \tilde{N}_k^2)$\[\*\] $\mathcal{O}(\tilde{N}_k
N_{div}^2)$
-----------------------------------------------------------------------------------------------
: Theoretical scalings of the routines used in the three methods described in the text\[tab:scalings\]. \[\*\] Scaling of an optimal implementation that takes advantage of sparse matrices. The scaling becomes $\mathcal{O}(\tilde{N}_k^2 N_{div}^2)$ if the method is solved with dense matrices.
--------------------------------------------------- -------------------------------------------------- --
{width="7cm"} {width="7cm"}
{width="7cm"} {width="7cm"}
{width="7cm"} {width="7cm"}
--------------------------------------------------- -------------------------------------------------- --
Comparison with the Kammerlander double-grid technique
------------------------------------------------------
In this section, we compare our method with the technique proposed by Kammerlander in Ref. [@Kammerlander2012]. In that approach, the polarizability $L(\omega)$ is expressed in the transition space according to $$\begin{aligned}
L_{vc\boldsymbol{k},v'c'\boldsymbol{k}'}(\omega) =
\sum_{v''c''\boldsymbol{k}''} (1-L^0(\omega)
K)^{-1}_{vc\boldsymbol{k},v''c''\boldsymbol{k}''}
L^0_{v''c''\boldsymbol{k}'',v'c'\boldsymbol{k}'}(\omega),\end{aligned}$$ where the RPA polarizability $L^0(\omega)$ is given by $$\begin{aligned}
L^0_{vc\boldsymbol{k},v'c'\boldsymbol{k}'}(\omega) =
\frac{f_{c\boldsymbol{k}}-f_{v\boldsymbol{k}}}{\varepsilon_{c\boldsymbol{k}} -
\varepsilon_{v\boldsymbol{k}} - \omega - i \eta} \delta_{cc'} \delta_{vv'}
\delta_{\boldsymbol{k},\boldsymbol{k}'}.\end{aligned}$$
In order to avoid the direct inversion of the large matrix, an iterative scheme is used for the computation of $$\begin{aligned}
L_{vc\boldsymbol{k},v'c'\boldsymbol{k}'}(\omega) = \sum_{m}
\sum_{v''c''\boldsymbol{k}''} \left[
L^0(\omega)
K
\right]^m_{vc\boldsymbol{k},v''c''\boldsymbol{k}''}
L^0_{v''c''\boldsymbol{k}'',v'c'\boldsymbol{k}'}(\omega) \label{eqiterK}.\end{aligned}$$
The BSE is solved for every frequency in an iterative way and a double grid technique is used to reduce the number of k-points for which the kernel must be computed explicitly. The RPA polarizability is averaged on a dense mesh yielding $$\begin{aligned}
L^0_{vc\boldsymbol{k},v'c'\boldsymbol{k}'}(\omega) = \frac{1}{N_{nb}}
\sum_{\bar{\boldsymbol{k}} \in N(\boldsymbol{k})}
\frac{f_{c\bar{\boldsymbol{k}}}-f_{v\bar{\boldsymbol{k}}}}{\varepsilon_{
c\bar{\boldsymbol{k}}} -
\varepsilon_{v\bar{\boldsymbol{k}}} - \omega - i \eta}
\delta_{cc'}
\delta_{vv'}
\delta_{\boldsymbol{k},\boldsymbol{k}'},\end{aligned}$$ where the $\bar{\boldsymbol{k}}$ are taken from the set $N(\boldsymbol{k})$ of $N_{nb}$ dense points located around $\boldsymbol{k}$.
Finally the averaged values are used in the iterative BSE solver \[see Eq. \]. This approach has the advantage that the wavefunctions on the dense mesh are not needed but the divergence is not accurately reproduced. The scaling of the Kammerlander technique is linear with the number of frequencies and quadratic with the number of points in the coarse mesh. On the contrary, our technique is able to describe the frequency dependence with a computational cost that does not depend on the number of frequency points, since, as discussed in Section \[sec2\], $\varepsilon_M(\omega)$ is evaluated with Eq. whose cost is negligible.
Wavefunction interpolation
--------------------------
In this article, we assume that the entire set of wavefunctions on the dense set of points is available. For the systems investigated in this study, the calculation of wavefunctions starting from an already converged density is not the most computationally demanding part. Moreover, only wavefunctions in the transition basis set are required, that is a small fraction of the set of bands required for the screening, for example.
However, for some more complex systems, one might take advantage of interpolation techniques to obtain the wavefunctions on denser meshes. In the work of Kammerlander [@Kammerlander2012] presented in the previous section, Wannier functions were used to obtain eigenenergies on these dense meshes. Recently, Gilmore *et al.* [@Gilmore2015] have used optimized basis functions described in the work of Shirley [@Shirley1996] to compute wavefunctions on a dense mesh. These different techniques could be easily interfaced with our technique to compute the overlap matrix elements, that can afterwards be used in the interpolation of the BSE Hamiltonian.
Conclusions
===========
We have presented a fast and memory-efficient technique that combines the interpolation of the Bethe-Salpeter matrix elements with the Lanczos algorithm. The treatment of the matrix elements is similar in spirit to the Rohlfing and Louie approach but we avoid the storage and the diagonalization of large matrices. Three possible approaches for the treatment of the Coulomb singularity have been presented and discussed in detail.
The effectiveness of the method has been analyzed through calculations of optical spectral in Si, GaAs and LiF. According to our tests, the multilinear interpolation of the wavefunctions does not perform better than simple constant interpolation, already proposed by Rohlfing and Louie (although used by them only for the set up of the Hamiltonian on the dense mesh).
In conclusion, we suggest using Method 1 for a quick qualitative analysis of optical spectra e.g. for a high-throughput screening to rapidly identify possible candidates. Method 3 with the on-the-fly interpolation is the recommended approach for BSE calculations requiring dense k-meshes since it is significantly faster than M2 and the Coulomb divergence is taken into account. The downside is that one has to check the convergence with the width $w$, but we believe this is a small price to pay, especially when compared with the significant speedup that can be achieved.
The algorithmic improvements presented in this work will facilitate BSE calculations in complex systems and will also significantly ease the ab initio study of piezoreflectance, thermoreflectance and Raman intensities in systems with excitonic effects.
Acknowledgments
===============
Y.G. and M.G. wish to acknowledge the financial support of the Fonds National de la Recherche Scientifique (FNRS, Belgium). The authors would like to thank Yann Pouillon and Jean-Michel Beuken for their valuable technical support and help with the test and build system of ABINIT.
Computational resources have been provided by the supercomputing facilities of the Université catholique de Louvain (CISM/UCL) and the Consortium des Equipements de Calcul Intensif en Fédération Wallonie Bruxelles (CECI) funded by the Fonds de la Recherche Scientifique de Belgique (FRS-FNRS) under Grant No. 2.5020.11. This work was also supported by the FRS-FNRS Belgium through PDR Grant T.0238.13 - AIXPHO.
References
==========
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Elaborated calculations of the shear and the bulk viscosities in the hadron gas, using the ultrarelativistic quantum molecular dynamics (UrQMD) model cross sections, are made. These cross sections are analyzed and improved. A special treatment of the resonances is implemented additionally. All this allows for better hydrodynamical description of the experimental data. The previously considered approximation of one constant cross section for all hadrons is justified. It’s found that the bulk viscosity of the hadron gas is much larger than the bulk viscosity of the pion gas while the shear viscosity is found to be less sensitive to the hadronic mass spectrum. The maximum of the bulk viscosity of the hadron gas is expected to be approximately in the temperature range ${T=150-190~MeV}$ with zero chemical potentials. This range covers the critical temperature values found from lattice calculations. We comment on some important aspects of calculations of the bulk viscosity, which were not taken into account or were not analyzed well previously. Doing this, a generalized Chapman-Enskog procedure, taking into account deviations from the chemical equilibrium, is outlined. Some general properties, features, the physical meaning of the bulk viscosity and some other comments on the deviations from the chemical equilibrium supplement this discussion. Analytical closed-form expressions for the transport coefficients and some related quantities within a quite large class of cross sections can be obtained. Some examples are explicitly considered. Comparisons with some previous calculations of the viscosities in the hadron gas and the pion gas are done.'
author:
- Oleg Moroz
title: 'Analytical formulas, general properties and calculation of transport coefficients in the hadron gas: shear and bulk viscosities.'
---
Introduction
==============
The bulk and the shear viscosity coefficients are transport coefficients which enter in the hydrodynamic equations, and thus are important for studying of nonequilibrium evolution of any thermodynamic system.
There are two more additional reasons to study the shear viscosity. The first one is the experimentally observed minimum of the ratio of the shear viscosity to the entropy density $\eta/s$ near the liquid-gas phase transition for different substances, which may help in studying of the quantum chromodynamics phase diagram and finding of the location of the critical point [@Csernai:2006zz; @lacey][^1]. Such a minimum was observed in theoretical results in several models, see e. g. [@chakkap; @Dobado:2009ek]. For a counterexample see [@Chen:2010vf] and references therein. The second reason is the calculation of the $\eta/s$ in strongly interacting systems, preferably real ones, to compare physical inputs which provide small values of the $\eta/s$. The conjectured lowest bound[^2] $\eta/s=\frac1{4\pi}$ [@adscftbound] was violated with different counterexamples. For some reasonable ones see [@Buchel:2008vz; @Sinha:2009ev]. Also see the recent review [@Cremonini:2011iq]. The bulk viscosity, being very sensitive to violation of the equation of state and being connected with fluctuations through the fluctuation-dissipation theorem [@Callen:1951vq], can have a maximum near a phase transition [@Kharzeev:2007wb; @Karsch:2007jc; @chakkap; @Dobado:2012zf]. In [@chakkap] and [@Dobado:2012zf] sharp maxima were observed in the bulk viscosity $\xi$ and the ratio $\xi/s$ in the linear $\sigma$-model for the vacuum $\sigma$ mass $900~MeV$. Decreasing the vacuum $\sigma$ mass the maximum eventually disappears. Any maximum of the $\xi/s$ was not observed in the large-N limit of the linear $\sigma$-model in the [@Dobado:2012zf]. Any maximum of the $\xi$ was not observed in the large-N limit of the ${1+1}$-dimensional Gross-Neveu model [@FernandezFraile:2010gu] (see also Sec. \[physmeansec\] for comments).
Whether one uses the Kubo[^3] formula or the Boltzmann equation one faces nearly the same integral equation for the transport coefficients [@jeon; @jeonyaffe; @Arnold:2002zm]. The preferable way to solve it is the variational (or Ritz) method. Due to its complexity the relaxation time approximation is used often in the framework of the Boltzmann equation. Though this approximation is inaccurate, does not allow to control precision of approximation and can potentially lead to large deviations. The main difficulty in the variational method is in calculation of collision integrals. To calculate any transport coefficient in the lowest order approximation in a mixture with a very large number of components $N'$ (like in the hadron gas) one would need to calculate roughly ${N'}^2$ 12-dimensional integrals if only the elastic collisions are considered. Fortunately, it’s possible to simplify these integrals considerably and perform these calculations in a reasonable time.
This paper contains calculations of the shear and the bulk viscosity coefficients for the hadron gas using the (corrected, see Sec. \[hardcorsec\]) UrQMD cross sections. The calculations are done in the framework of the Boltzmann equation with the classical Maxwell-Boltzmann statistics, without medium effects and with the ideal gas equation of state. The Maxwell-Boltzmann statistics approximation allows one to obtain some relatively simple analytical closed-form expressions. Originally the calculations in the same approximations for the hadron gas but with one constant cross section for all hadrons were done in [@Moroz:2011vn]. The deviations in the worst cases are relatively small. In that paper some analytical formulas of the viscosities for 1-, 2- (explicitly) and N-component (up to solution of the matrix equation) gases with constant cross sections were obtained. Analogical formulas can be written down for quite a large class of non-constant cross sections, in particular, for the ones which appear in the chiral perturbation theory. The final expressions may become somewhat more cumbersome; anyway this is better than numerical integration at least in the speed of the computation. Explicit formulas for the viscosities with the elastic pion-pion isospin averaged cross section and somewhat more general one are obtained in the present paper. The results of the [@Moroz:2011vn] are partially reproduced in the present paper, improving the text and adding more detailed explanations. The presented calculations can be considered as quite precise ones at low temperatures where the elastic collisions dominate and the equation of state is close to the ideal gas equation of state. At higher temperatures the calculations with the total cross section are expected to give the qualitative description.
For comparison the calculations of the viscosities are performed for the pion gas (throughout the paper the chemical potentials are equal to zero if else is not stated). The results are relatively close to the results in [@prakash; @davesne]. There the calculations are made in the same approximations except for the [@davesne], where the Bose-Einstein statistics is used instead of the Maxwell-Boltzmann one. The discrepancies from the used classical statistics are not large at zero chemical potential and become larger as the chemical potential grows (see Sec. \[condappl\] for the errors and comments). The comparison is made with the results of the [@prakash], see fig. \[CompViscosPions\]. The discrepancies up to a factor of $3.5$ for the bulk viscosity and up to a factor of $2.5$ for the shear viscosity come most probably from somewhat different ${\pi\pi}$ elastic plus the quasielastic (through the intermediate $\rho$-resonance) cross section of the [@prakash][^4] (the averaging over the scattering angle is expected to give small errors; see also comments below the formula (\[xica\])). The minima of the shear viscosities near ${T=60-70~MeV}$ are attributed to the peaks from the $\rho$-resonances in the ${\pi\pi}$ cross sections. It’s not noticeable in the figure for the dash-dotted line. Nonzero values of the bulk viscosities and theirs maxima are solely due to the masses of the pions. The paper [@prakash] implements also the isospin averaged current algebra elastic cross sections. These cross sections can be reproduced in the lowest order in the chiral perturbation theory [@Scherer:2002tk]. They obviously have quite large deviations from the experimental data at high enough energies and wrong $\sqrt{s}$ asymptotic dependence, which can be seen from the comparison of them with the isospin averaged elastic plus the quasielastic experimental cross sections in the [@prakash]. The elastic $\pi\pi$ cross sections are rather close to the constant $5~mb$ [@Bass:1998ca; @Bleicher:1999xi].
In several papers the bulk viscosity $\xi$ was calculated for the pion gas, using the chiral perturbation theory (or the unitarized chiral perturbation theory) and some other approaches, with quite large discrepancies between the quantitative results. In [@fernnicola] the calculations were done by the Kubo formula in a rough approximation. There the number-changing ${2\leftrightarrow4}$ processes were neglected too, and the non-vanishing value of the bulk viscosity is obtained due to a trace anomaly and the pions’ masses. At small temperatures, where the effects of the trace anomaly are small, the magnitude of the bulk viscosity is large in compare to the results of this paper and the [@prakash; @davesne]. For example, at ${T=20~MeV}$ (${60~MeV}$) it’s larger approximately in 39 (8) times than the bulk viscosity in this paper. The maximal values differ in approximately 24 times. In [@lumoore] the calculations are done in the framework of the Boltzmann equation and have a divergent dependence of the $\xi$ for ${T\rightarrow0}$ because of remained weak ${2\leftrightarrow4}$ number-changing processes (at ${T=140~MeV}$ the bulk viscosity is nearly 57 times larger than the bulk viscosity calculated in this paper). This dependence should change at low enough temperatures, $T=180~MeV$ or higher ones for the pion gas, see Sec. \[physmeansec\] for explanations. Joining the results of the calculations at low and high temperatures, the function $\xi(T)$ may turn out to be not continuous at the middle temperatures (which is not a physical effect, see Sec. \[physmeansec\]), and the smooth function $\xi(T)$ is to be obtained through some interpolation. In [@dobado] the bulk viscosity was calculated in the framework of the Boltzmann equation with the ideal gas equation of state and only the elastic collisions taken into account. The Inverse Amplitude Method was used to get the scattering amplitudes of the pions. The quantitative results are close to the results in this paper (discrepancies up to a factor of $2.7$). In [@chenwang] the calculations are done in the framework of the Boltzmann equation for the massless pions. There the bulk viscosity increases rapidly so that the ratio $\xi/s$ increases with the temperature.
Calculations of the shear viscosity in the hadron gas with a large number of components were done in [@Gorenstein:2007mw], using some approximate phenomenological formula, and in [@toneev], using the relaxation time approximation. These results are in good agreement with the calculations of this paper. Hence, as long as the ratio $\xi/\eta$ calculated in the [@toneev] for the free massive pion gas is $8-58$ times larger (in the temperature range $0.02~GeV<T<0.14~GeV$ with the deviations growing as the temperature decreases) than the one calculated in this paper, one can suspect that the difference comes from the bulk viscosity because of the used relaxation time approximation[^5] and likely not conserved particle numbers at low temperatures, provided that the SHMC model’s cross sections, used in the [@toneev], don’t have large deviations from the UrQMD cross sections or the experimental data, which seems to be the case. Also note that the results in the [@toneev] for the free particles and the SHMC model don’t differ very much. These facts may explain why the $\xi/s$ of the hadron gas in the [@toneev] is $3.65-11.3$ times larger (in the temperature range $0.1~GeV<T<0.18~GeV$) than the $\xi/s$ calculated in this paper. At the low temperature ${T=100~MeV}$ and the vanishing chemical potentials it is 11.3 times more (at the same temperature the factor is 8.2 for the case of the pion gas). In [@nhngr] the calculation of the bulk viscosity is done for the hadron gas (with an excluded-volume equation of state) with the masses less than $2~GeV$ using some special formula, obtained though some ansatz [@Kharzeev:2007wb]. Its quantitative accuracy has not been clarified. The ratio $\xi/s$ in the [@nhngr] deviates from the $\xi/s$ of this paper up to a factor of 1.8 in the temperature range $0.14~GeV<T<0.18~GeV$ and is different on $4\%$ at $T=0.14~GeV$.
Also the shear viscosity has been calculated using the Kubo formula (or the Green-Kubo formula) in a gas of mesons and their resonances [@Muronga:2003tb]. There the UrQMD simulations are performed to calculate the energy-momentum tensor, used in the calculations by the Kubo formula. The $\eta$ in the [@Muronga:2003tb] is $1.14-1.77$ times smaller then the $\eta$ for the hadron gas in this paper. At $T=0.15~GeV$ it is $1.77$ times smaller. In [@Muroya:2004pu] similar calculations, using the URASiMA event generator, are done for the shear viscosity with close results.
The structure of the paper is the following. A misleading viewpoint on the bulk viscosity, connected with the inelastic processes, is commented on in Sec. \[physmeansec\] together with some properties, features and physical meaning of the bulk viscosity. In that section some questions concerning the deviations from the chemical equilibrium are addressed too. Sec. \[hardcorsec\] contains some comments on the constant cross sections, which are used in approximating calculations, and some other general comments on cross sections. Also it contains a description of the UrQMD cross sections, which are used in the main calculations, together with their analysis, corrections and the consequences of the corrections for the freeze-out temperatures. The applicability of the used through the paper approximations is discussed in Sec. \[condappl\]. The system of the Boltzmann equations, its solution and formal expressions of the transport coefficients can be found in Sec. \[CalcSec\]. The numerical calculations for the hadron gas are presented in Sec. \[numcalc\]. In Sec. \[singcomsec\] analytical results for the single-component gas are presented. In particular, an analytical expression for the first order single-component shear viscosity coefficient with constant cross section, found before in [@anderson], is corrected while the bulk viscosity coefficient remains the same. The nonequilibrium distribution function in the same approximation is written down. Also the viscosities with some non-constant cross sections are written down. Some analytical results for the binary mixture with constant cross sections are considered in Sec. \[binmixsec\]. Integrals of source terms needed for the calculation of the transport coefficients can be found in Appendix \[appA\]. The general entropy density formula can be found in Appendix \[appTherm\]. It is used in the numerical calculations for the hadron gas. Transformations of collision brackets, being the 12-dimensional integrals which enter in the viscosities, and some analytical formulas for them can be found in Appendix \[appJ\]. The closed-form expressions for collision rates, mean free paths and mean free times are included in Appendix \[appmfp\].
Some features and properties of the bulk viscosity \[physmeansec\]
===================================================================
First, it should be reminded that the transport coefficients are defined as coefficients next to their gradients in the formal expansion of the energy-momentum tensor and the charge density flows over the gradients of the thermodynamic functions or the flow velocity (see e. g. [@landau6], Section 136). The Kubo formulas are not definitions of the transport coefficients, as one might think. They may introduce some assumptions. In particular, the Kubo formulas in the form as in the [@jeon] have zero frequency and zero momentum limits, which neglect finite size and finite time effects. Zero momentum limit implies the thermodynamical limit. This limit is needed to avoid possible nonphysical contributions from inappropriate choice of a current and an ensemble [@Kadanoff]. The Kubo formulas in the form as in [@Muroya:2004pu; @Kubo] suppose thermal equilibrium in the initial moment of time $t=-\infty$. So that any infinite space-time scale cannot be connected with the transport coefficients by their *definitions*.
The Boltzmann equations will be used in what follows. In the case of the elastic collisions they can be derived from the Liouville equation[^6] in the approximations $n_k r_{kl}^3 \ll 1$ ($r_{kl}$ is the effective radius of two particle interactions between the particles of the species $k$ and $l$) that is for rarified gases with short-range interactions[^7]. Also the linear integral equations for the viscosities and other transport coefficients, derivable from the Boltzmann equation, can be obtained (with some corrections) from the perturbative calculations for quantum field theories at finite temperature (including the inelastic processes) using the Kubo formulas [@jeon; @Gagnon:2006hi; @Gagnon:2007qt], which justifies application of the Boltzmann equation when the inelastic processes are present.
The bulk viscosity can reveal itself only when there is a nonzero divergence of the flow velocity. This nonequilibrium perturbation should not be confused with another possible *independent* perturbation (as was done in several papers, some of which are mentioned below; the roots of the misleading viewpoint, perhaps, can be found in [@landau6], Section 81). Namely, this is the homogeneous perturbation. It can be both the chemical and the kinetic one[^8]. Then it can be generalized and made dependent on the coordinate. It just should not be proportional to any gradient. Then the constraints of the local conservation laws should be imposed on these perturbations. The perturbations for the particle numbers should be such that don’t violate conservation of all charges. Considering the case of homogeneous chemical perturbation in a gas with fixed volume, one concludes that the temperature should change with time, being some energy per particle. So that energy conservation should be obtained varying the temperature. Adding an infinitesimal correction to the temperature one gets a perturbation of the form $C p_k^\mu U_\mu$. Such perturbations don’t contribute to all collision integrals. To describe purely chemical perturbations they have to be chosen in the form of the momentum-independent terms (except for the $C p_k^\mu U_\mu$ terms), otherwise there will be contributions from the elastic collision integrals. Such perturbations can be considered as chemical potentials-like ones (being small, one can expand the distribution functions over them and get these momentum-independent terms) with the arguments for maximization of the entropy. To find the evolution of these terms they should be separated. Let’s write this in some formulas. Multicomponent gas distribution functions with the leading perturbations can be represented in the form (detailed definitions can be found in Sec. \[CalcSecA\]) [$$\begin{aligned}
\label{distfunc}
f_k=f_k^{(0)}(1+\tilde \varphi_k)(1+\varphi_k)\approx f_k^{(0)}(1+\tilde \varphi_k+\varphi_k),
\quad |\tilde \varphi_k|\ll 1, \quad |\varphi_k|\ll 1,
\end{aligned}$$]{} where $\varphi_k$ are the perturbation due to the gradients and $\tilde \varphi_k$ are the chemical perturbations[^9]. Following steps of Sec. \[CalcSecA\], one can get the following linearized equations from the Boltzmann equations: [$$\begin{aligned}
\label{generlinbeqn}
p_k^\mu(U_\mu D+\nabla_\mu)f^{(0)}_k +
f^{(0)}_k p_k^\mu (U_\mu D + \nabla_\mu )\tilde \varphi_k
\approx -f_k^{(0)}{\mathcal{L}}_k[\varphi_k] - f_k^{(0)}{\mathcal{L}}_k^{inel}[\tilde \varphi_k],
\end{aligned}$$]{} where ${\mathcal{L}}_k$ and ${\mathcal{L}}_k^{inel}$ is the sum of the linearized collision integrals (divided on the $-f_k^{(0)}$) of all the processes and of the inelastic processes correspondingly. The 2-nd order gradients and the squared 1-st order gradients are neglected in the l. h. s. because they are of the next order[^10] and should be cancelled in the next iteration by the next corrections to the distribution functions. Also the smallness of the $\tilde \varphi_k$ is used. If the spatial covariant gradients $\nabla_\mu \tilde \varphi_k(t=0)$ (at the initial moment of time) are of the same order as the gradients of the thermodynamic functions or the flow velocity, then the $\nabla_\mu \tilde \varphi_k$ terms in the l. h. s. should be retained[^11]. The covariant temporal derivatives $D \tilde \varphi_k$ are needed to describe the temporal evolution of the $\tilde \varphi_k$. Then the equations (\[generlinbeqn\]) can be split onto the separate equations for the $\tilde
\varphi_k$ and the $\varphi_k$ [$$\begin{aligned}
\label{generlinbeqn1}
p_k^\mu U_\mu D \tilde \varphi_k
+p_k^\mu \nabla_\mu \tilde \varphi_k
\approx - {\mathcal{L}}_k^{inel}[\tilde \varphi_k],
\end{aligned}$$]{} [$$\begin{aligned}
\label{generlinbeqn2}
p_k^\mu U_\mu D f^{(0)}_k + p_k^\mu \nabla_\mu f^{(0)}_k
\approx -f_k^{(0)}{\mathcal{L}}_k[\varphi_k].
\end{aligned}$$]{} The equations (\[generlinbeqn\]) can be split within the framework of the perturbation theory over the gradients. Let’s consider also the condition $\varphi_k \ll \tilde \varphi_k$ in the (\[distfunc\]). Then neglecting the $\varphi_k$ in the (\[distfunc\]) and repeating the steps of Sec. \[CalcSecA\], one can get the following linearized equations: [$$\begin{aligned}
\label{fadingeq}
p_k^\mu U_\mu D \tilde \varphi_k \approx - {\mathcal{L}}_k^{inel}[\tilde
\varphi_k].
\end{aligned}$$]{} The equations (\[fadingeq\]) are precise in the homogeneous case (the approximation is only from the linearization). The 1-st order gradients and the $\nabla_\mu \tilde \varphi_k$ are neglected. Then using the (\[fadingeq\]) and the (\[generlinbeqn\]), one can get [$$\begin{aligned}
\label{generlieqn}
(p_k^\mu U_\mu D+p_k^\mu \nabla_\mu)f^{(0)}_k
+f^{(0)}_k p_k^\mu \nabla_\mu \tilde \varphi_k
\approx -f_k^{(0)}{\mathcal{L}}_k[\varphi_k].
\end{aligned}$$]{} Solving the system of equations (\[fadingeq\]) in the local rest frame, one gets the leading exponential fading dependencies on time[^12] (in a covariant form this should be an explicit space-time dependence). Such dependencies were obtained in some previous studies, see e. g. [@Matsui:1985eu; @Song:1996ik]. The equations (\[generlieqn\]) are different from the ones obtained from the common Chapman-Enskog procedure (see e. g. [@groot], Chap. V) because of the $\nabla_\mu \tilde \varphi_k$ terms. The contributions from the small chemical perturbations can be neglected in the considered order in the transport coefficients because they are multiplied on the 1-st order gradients. The $\nabla_\mu \tilde \varphi_k$ terms can be cancelled, introducing terms proportional to the $\nabla_\mu \tilde \varphi_k(t=0)$ into the $\varphi_k$ terms. If the spatial distributions of the $\tilde
\varphi_k(t=0)$ are such that $\nabla_\mu \tilde \varphi_k(t=0)$ are of the 2-nd or a higher order, then the $\nabla_\mu \tilde
\varphi_k$ can be neglected. This assumption or approximation is used in the calculations of this paper. In the linear response theory one can also introduce independent small chemical perturbations with the same conclusions for the 1-st order transport coefficients and find evolution of the perturbations with time.
Note that the deviation from the chemical equilibrium itself is not necessarily a source of the bulk viscosity, as is stated in [@Paech:2006st]. If the bulk viscosity is not equal to zero only because of the particles’ masses and they are tended to zero, the bulk viscosity source term and the bulk viscosity tend to zero even if there are inelastic processes (see the end of Sec. \[CalcSecA\]). In the [@Paech:2006st] the independent chemical perturbations and the perturbations due to the gradients were just connected through the perturbations of particle numbers, and the bulk viscosity became proportional to the chemical relaxation time. Formally infinite chemical relaxation time doesn’t imply any divergencies in the chemical perturbations $\tilde \varphi_k$, but rather approximation of conserved particle numbers. Note that the dependence on the strength of the inelastic processes is different for the chemical perturbations and the perturbations due to the gradients. Increasing the strength of the inelastic processes the chemical relaxation time decreases. And the gradients’ relaxation time increases, because the transport coefficients, at least in rarified gases with short-range interactions, roughly speaking, are inversely proportional to the integrated cross sections (in an ideal liquid the gradients’ relaxation time is infinite). What happens with the bulk viscosity if the inelastic processes become weaker is discussed below.
Making the inelastic processes weaker in compare to the elastic ones, the bulk viscosity eventually gets a formal dominant contribution from them because of the approximate zero mode(s) [@jeon], connected with possible conservation of particle number(s)[^13]. As long as it’s clear that the bulk viscosity is not responsible for the chemical equilibration, it’s also clear that there may be the approximation of conserved particle numbers if the momentum spectrum, as well as the gradients, can relax by means of only the elastic collisions (which is usually the case) and the elastic processes make a dominant contribution to the collision rates. The question is only at what concrete temperature does this approximation sets in. Let’s make an illustrative example of what nonphysical contributions one can get from formally remained weak inelastic processes. Consider infinitely weak inelastic processes and the perturbation of the flow velocity such that the energy-momentum tensor gets a sizable contribution from the bulk viscosity term, not large in compare to the pressure (cf. (\[T0\]), (\[T1\])) to remain the perturbation theory applicable. Then it’s obvious that this contribution is not physical because it is created by the practically absent processes and the infinitesimal perturbation of the flow velocity. Instead, this system is practically described by the equilibrium thermodynamic functions. This also answers positively the question whether the thermodynamic chemical potential can be introduced for approximately conserved particle number in principle. As far as the author knows, the first correct comment (albeit somewhat inaccurate) on this issue can be found in the [@jeonyaffe]. However, note that in fact there is no divergent mean free paths, corresponding to the inelastic processes (IMFP) in this case. They are cut by the mean free paths, corresponding to the elastic processes (and the overall collision rate have the dominant contribution from the elastic collisions). So that it may be not necessary for the chemical relaxation time to be much larger than any relevant time scale (like the gradients relaxation time or the time of existence of the thermal part of the system) to switch off the inelastic processes. That’s why a criterion based on comparison of collision rates of elastic and inelastic processes can be considered to switch off the inelastic processes. Such a comparison is done in the UrQMD studies of the hadron gas in [@Bleicher:2002dm] (see Sec. \[condappl\] for farther discussions). According to [@Goity], the chemical relaxation time of the $2\leftrightarrow4$ processes in the pion gas is much larger than the thermal relaxation time. And e. g. at $T=180~MeV$ the chemical relaxation time is equal to $40~fm$, which is larger than the typical lifetime of the thermal part of the expanding fireball (see e. g. [@Bleicher:2002dm]). So that it’s the inelastic $2\leftrightarrow4$ processes which should be neglected in the pion gas at $T=180~MeV$ or even higher temperatures, which wasn’t done in the [@lumoore]. To show importance of the gradients relaxation time, let’s consider the following possible case. Let’s consider the only perturbation - propagating sound wave, perturbed in a point. It’s possible for the IMFP to be much larger than the gradients relaxation size (on which the wave can be considered as damped) and be much smaller than the system’s size at the same time. Then, the bulk viscosity cannot be defined by the IMFP in this case, because it enters in the sound attenuation constant. Thus, the gradients relaxation size and time are cutting parameters. Note that they exists even in infinite systems considered during infinite time interval.
The bulk viscosity source terms increases substantially if particle numbers are not conserved (cf. (\[alfrac1\]), (\[alfrac2\]); in mixtures these particle numbers should also be not small). This reflects additional fluctuations from not conserved particle numbers. Though the inelastic processes have to be effective enough to consider the approximation of not conserved particle numbers. Perhaps, the point at which the bulk viscosities in the different approximations cross can provide a criterion for switching on/off the inelastic processes. If this is not so, then one would have to make some interpolation in the intermediate region[^14]. Note that e. g. in the calculations by the Kubo formulas through the direct calculations of the energy-momentum tensor as in the [@Muronga:2003tb] it’s not needed to use the approximation of conserved or not conserved particle numbers (which defines the number of independent thermodynamic chemical potentials, through which the chemical potentials of all particles are expressed, cf. (\[mukdef\])). There the energy-momentum tensor should be a smooth function of time and the thermodynamic functions as long as the inelastic processes fade smoothly. Then the bulk viscosity should be a smooth function of the temperature and particles’ chemical potentials regardless of the number of the independent chemical potentials.
In the [@Arnold:2006fz] a bottleneck for the relaxation to equilibrium characterized by the bulk viscosity due to the weakest processes’ rates is assumed. Instead, there are rather dominant contributions from some test-functions[^15] (as is commented in the footnote \[footn2\]), which should not be specially treated though, except for the ones which are the approximate zero modes making a dominant contribution. A similar dominance[^16] is present also in other transport coefficients, in particular, when there is only one type of processes. Although in QCD at high enough temperatures the equilibrium $2\leftrightarrow 2$ elastic collisions rate is parametrically the largest one[^17], $O(\alpha_s T)$, because of cancellations the momentum transfer takes place with the rate $O(\alpha_s^2\ln(1/\alpha_s) T)$, which is parametrically smaller than the particle number change rate $O(\alpha_s^{3/2} T)$ from the effective “$1\leftrightarrow 2$” processes. This provides an example when the equilibrium collision rates may differ substantially from the relevant collision rates. The “$1\leftrightarrow 2$” processes provide small chemical relaxation time in compare to the thermal relaxation time, which justifies the approximation of not conserved particle numbers and the enhancement of the bulk viscosity from the source terms at least at small enough $\alpha_s$, whereas the contributions to the bulk viscosity from the collision integrals of the “$1\leftrightarrow 2$” processes are suppressed at small enough $\alpha_s$ (the inelastic $2\leftrightarrow 2$ processes are not suppressed, but they are of the order $O(\alpha_s^2\ln(1/\alpha_s)
T)$). To avoid misunderstanding it may be mentioned that taking the total collision rate of the “$1\leftrightarrow 2$” processes as formally infinite by taking the corresponding matrix elements as formally infinite ones, one gets zero bulk viscosity and zero mean free paths as long as both the gluons and quarks take part in these processes (see also footnote \[footn1\]).
In the case of a ${1+1}$-dimensional single-component gas the elastic collisions cannot result in the relaxation of the momentum spectra and, hence, cannot stimulate the system to evolute towards equilibrium[^18]. As a result, the exponentially divergent bulk viscosity was obtained in the paper [@FernandezFraile:2010gu]. Considering again the example about the infinitely small perturbation of the flow velocity and assuming also a finite size of the system, it’s again obvious that the weak inelastic processes may make nonphysical contributions (in this case the mean free path is formally cut by the system’s size). If this is the case, then the hydrodynamical description becomes inapplicable, and might use simulations of particles’ collisions or the Boltzmann equations in the approximation without collisions (on a time scale much smaller than the chemical relaxation time). If the ${1+1}$-dimensional description is only an approximate one (that is with small angle elastic scatterings in higher dimensions), the relaxation of the momentum spectrum by the elastic collisions should be considered. And if a $1+1$-dimensional gas has at least two components with different masses, then a nontrivial momentum exchange in the elastic collisions is possible. This results in the possibility of the relaxation of the momentum spectra by only the elastic collisions [@Cubero].
Let’s summarize this section with formulation of the physical meaning of the bulk viscosity. The bulk viscosity reflects deviation of the value of the pressure from its local equilibrium value (as can be seen from the (\[T1\])), appearing when the system expands/compresses, because of the delay in the equilibration. The bulk viscosity is not responsible for the restoration of the chemical or the kinetic equilibria - it’s responsible for the relaxation of the divergence of the flow velocity. If there are inelastic processes, then the particle numbers also get nonequilibrium contributions (cf. (\[pflow\]), (\[fpert\]), (\[varphi\])) such that the charge is conserved locally (cf. (\[cofchf\]))[^19]. Though these contributions together with the contribution to the pressure may become nonphysical because of the approximate zero modes (if such ones appear in the calculations). The magnitude of the bulk viscosity changes from theory to theory. Under some quite general assumptions a nonzero value of the bulk viscosity can be connected with violation of the scale invariance due to a nonzero value of the energy-momentum tensor [@Coleman:1970je; @Callan:1970ze]. Of course, the beta function can contribute to the energy-momentum tensor and the bulk viscosity too [@jeon].
The hard core interaction model and the UrQMD cross sections \[hardcorsec\]
============================================================================
In a non-relativistic classical theory of particle interactions there is a widespread model, used in approximate calculations, called the hard core repulsion model or the model of hard spheres with some radius $r$. For its applications to the high-energy nuclear collisions see [@Gorenstein:2007mw] and references therein. The differential scattering cross section for this model can be inferred from the problem of scattering of point particle on the spherical potential ${U(r)=\infty}$ if ${r\leq a}$ and ${U(r)=0}$ if ${r>a}$ [@landau1]. In this model the differential cross section is equal to $a^2/4$. To apply this result to the gas of hard spheres with the radius $r$ one can notice that the scattering of any two spheres can be considered as the scattering of the point particle on the sphere of the radius $2r$, so that one should take ${a=2r}$. The total cross section $\sigma_{tot}$ is obtained after integration over the angles of the $r^2 d\Omega$ which results in the $\sigma_{tot}=4\pi r^2$. For collisions of hard spheres of different radiuses one should take ${a=r_k+r_l}$ or replace the $r$ on the $\frac{r_k+r_l}2$: [$$\begin{aligned}
\label{hccs}
\sigma_{tot,kl}=\pi (r_k+r_l)^2.
\end{aligned}$$]{} The relativistic generalization of this model is the constant (not dependent on the scattering energy and angle) differential cross sections model.
The hard spheres model is classical, and connection of its cross sections to cross sections, calculated in any quantum theory, is needed. For particles, having a spin, the differential cross sections averaged over the initial spin states and summed over the final ones will be used.[^20] If colliding particles are identical and their differential cross section is integrated over the momentums (or the spatial angle to get the total cross section) then it should be multiplied on the factor $\frac12$ to cancel double counting of the momentum states. These factors are exactly the factors $\gamma_{kl}$ next to the collision integrals in the Boltzmann equations (\[boleqs\]). The differential cross sections times these factors will be called the classical differential cross sections.
The UrQMD cross sections are used in the numerical calculations of Sec. \[numcalc\][^21]. These cross sections are described in [@Bass:1998ca; @Bleicher:1999xi]. More details can be found in the UrQMD program codes. Below there is some description mainly of what is different or new.
The UrQMD cross sections are averaged over the initial spin states and summed over the final ones. As long as the UrQMD cross sections are total ones (integrated over the scattering angle), the factors $\gamma_{kl}$ are already absorbed into them (in what follows only such cross sections will be considered in this section tacitly). Dividing them on the $4\pi$, one gets the classical differential cross sections, averaged over the scattering angle.
The UrQMD codes (version 1.3) were modified to get accurately tabulated (with a step of ${25~MeV}$) cross sections. Resonances’ masses and widths (they are tuned in their uncertainty regions to describe the experimental data better), used in the UrQMD codes, have somewhat different values than the ones in the [@Bass:1998ca]. Influence of variation of these parameters was studied in [@Gerhard:2012fj]. The UrQMD codes implement somewhat different averaging of the c. m. momentums over the resonances’ masses[^22] than in the papers [@Bass:1998ca; @Bleicher:1999xi]. It was found that using the resonance dominating cross sections from the papers [@Bass:1998ca; @Bleicher:1999xi] some of these cross sections could have a large rise at small c. m. momentums if constant widths are used in the calculations of the averaged c. m. momentums in the energy dependent widths. So that one should be aware of this fact[^23]. The UrQMD codes have a low energy cut-off at ${\sqrt{s} \sim m_k+m_l+0.01~GeV}$ (and a similar one over the c. m. momentum if triggered) for the resonance dominating cross sections, and no large low energy rise was found there.
An important ingredient of the UrQMD model is the Additive Quark Model (AQM), which is used for unknown cross sections. Universality of hadrons, based on jet quenching arguments, is used to support this model. This model describes the experimentally known cross sections well at sufficiently high energies. Application of this model is better than elimination of the corresponding hadrons, which is the same as equating their all cross sections to zero and, hence, exclusion of their contributions from the thermodynamic functions (infinite mean paths, no thermalization).
At this point an interruption should be made to consider some important questions related to different types of the UrQMD cross sections. These different types are used due to several reasons and are the following: the elastic cross section(s) (ECS(s)), the elastic plus the quasielastic cross section(s) (EQCS(s)), the total cross section(s) (TCS(s)) and the previous two types with enhanced in some way resonances’ cross sections (index “2” is appended in the abbreviations).
Of course, the system of the Boltzmann equations would have a solution with any of these cross sections. Usage of the ECSs is completely self-consistent as long as only the elastic ${2\leftrightarrow 2}$ collision integrals are used in the calculations of the viscosities. However, there are reasons to consider also the EQCSs. Exactly this type of cross sections, being averaged over the isospin, is implemented in [@prakash]. The quasielastic cross sections ${2\rightarrow 1\rightarrow 2}$ can be used as rightful contributions to the ECSs in the approximation that the 4-momentum of the intermediate resonance does not change (the effects of the exclusion of the resonances as independent particles are considered in Sec. \[numcalc\]). The mean free paths of the intermediate resonances without contributions of the decays, being not equal to zero, also introduce some errors, which are neglected. The EQCSs conserve particle numbers, which is consistent with the only elastic collision integrals, implemented in the calculations. There are also some additional arguments for the usage of these cross sections. From the phenomenological considerations one can take into account shortening of the mean free paths (or enlarging of the collision rates) due to creation of the resonances. In other words, there would have to be contributions from the inelastic ${2\leftrightarrow 1}$ collision integrals next to the elastic collision integrals, and they are taken into account approximately by the contributions from the quasielastic cross sections.
Resonances are not just intermediate particles, and they can collide with other particles. They make not negligibly small contribution to the thermodynamic functions and the viscosities. So that they are also included in the calculations as independent particles with their parameters and corresponding ${2\leftrightarrow 2}$ collision integrals. They would have to have shortening of their mean free paths from their decays and contributions from the inelastic ${2\leftrightarrow 1}$ collision integrals too. These contributions may be taken into account from the following collision rate considerations. A resonance’s decay rate can be approximately replaced with just its total width. Then, given a resonance, one would have to redistribute its width (that is not changing the whole collision rate containing the contribution of the decay rate) in such a way that the cross section of the collision of this resonance with a resonance of the same species gets an addition[^24]. Using an approximate expression for the collision rates (in the nonrelativistic approximation, applicable in this case) from Appendix \[appmfp\], one easily finds the addition ${\Gamma_k/(\sqrt{2} n_k \langle {\left\vert\vec v_k\right\vert}\rangle)}$ (where $\Gamma_k$ is the width) to the ${4\pi \sigma^{cl}_{kk}}$. Such cross sections seem to be the most physically preferable ones because they take into account more realistic mean free paths than in the previous case while not violating the conservation of the particle numbers too.
The TCSs are used to take into account even larger shortening[^25] of the mean free paths than in the case of the EQCSs. However, such cross sections introduce some inconsistency, implying that the conservation of the particle numbers is violated. As long as there may be contributions from some partial cross sections to the UrQMD ECSs or the EQCSs which were not taken into account (see below), the TCSs can be used as the upper bounds for the ECSs and the EQCSs. However, it’s expected that these bounds are excessively high. If so, the TCSs (rather TCS2s) can be considered not only as the approximation taking into account real mean free paths but also as some measure of deviation from the approximation of only the elastic and the quasielastic collisions with the following arguments. If the TCSs were approximately equal to the ECSs or the EQCSs, or the numbers of particles with large inelastic cross sections were small, then one could expect small errors due to the negligibility of the inelastic collisions.
Continuing the discussion of the details of the UrQMD cross sections, it should be mentioned that the UrQMD TCSs are the most reliable ones. The sum of the partial cross sections is not always equal to the TCSs by their construction. If this is the case, then some partial cross sections are rescaled depending on their reliability[^26].
The magnitudes of the partial cross sections, implemented in the UrQMD codes, are used to determine, what a partial cross section to choose in a given collision, using a random number generator. Among these partial cross sections there are the ECSs. Exactly these ECSs are used in the present calculations. However, if a partial cross section with a string excitation is chosen in a given collision, there is a probability to end up with the elastic collision if the $\sqrt{s}$ is too small. These contributions to the ECSs are not calculated and are not added to the ECSs. Also the string excitations can, possibly, end up with creation of a resonance. Contributions to the EQCSs from the string excitations are taken into account partially (see below).
The ECSs, if not known from the experiment, are taken in the form of some extrapolations, discussed below, or the AQM is used. The normalization on the corresponding TCSs can change the ECSs notably. The meson meson (MM) ECSs are equal to $5~mb$. The meson baryon (MB) ECSs are equal to the AQM rescaled experimental ${\pi^+ p}$ cross sections. But after the normalization they become equal to zero in the resonances dominated energy range (approximately below ${\sqrt{s}=1.7~GeV}$). The anti-baryon baryon (${\bar B B}$) ECSs are equal to the AQM rescaled experimental ${\bar p p}$ cross sections. Other ECSs are equal to the AQM ECSs.
Before discussing the quasielastic cross sections first let’s write for convenience the resonance dominated cross sections formula for a reaction ${2\rightarrow 1\rightarrow any}$. Correcting a typo and rewriting it in a somewhat different form than in [@Bass:1998ca; @Bleicher:1999xi], one gets [$$\begin{aligned}
\label{resdomxsec}
\sigma^{ij}_{tot}(\sqrt s)=\sum_{R}\frac{g_R}{g_i g_j}\frac{\pi}{p_{cm}^2}
\frac{\Gamma_{R,tot}^2 b_{R\rightarrow ij}}{(M_R-\sqrt s)^2+\Gamma_{R,tot}^2/4}, \quad
b_{R\rightarrow ij}\equiv |\langle j_i,m_i,j_j,m_j || J_R, M_R \rangle|^2
\frac{\Gamma_{R\rightarrow ij}}{\Gamma_{R,tot}},
\end{aligned}$$]{} where $\Gamma_{R\rightarrow ij}$ is the partial energy-dependent width of the decay of the resonance $R$ into particles of types $i$ and $j$ without specification of their isospin projection, $\Gamma_{R,tot}$ is the total energy-dependent width of the decay of the resonance $R$, $g_i$ is the spin degeneracy factor, $b_{R\rightarrow ij}$ is the energy-dependent branching ratio. The squared Clebsch-Gordan coefficients allow to specify the branching ratio $b_{R\rightarrow ij}$ for the pair of the particles with concrete isospin projections. The squared Clebsch-Gordan coefficients should be normalized in such a way that they give unity after summation over all isospin projections in a given multiplet. This formula represents contributions from all possible resonances through which the reaction can take place. Now it’s easy to write down the cross sections for the quasielastic ${2\rightarrow 1\rightarrow 2}$ scatterings: [$$\begin{aligned}
\sigma^{ij}_{quasi}(\sqrt s)=\sum_{R}\frac{g_R}{g_i g_j}\frac{\pi}{p_{cm}^2}
\frac{\Gamma_{R,tot}^2 b_{R\rightarrow ij}^2}{(M_R-\sqrt s)^2+\Gamma_{R,tot}^2/4}.
\end{aligned}$$]{} One more multiplier $b_{R\rightarrow ij}$ takes into account the fact that a resonance $R$ decays only into the $ij$ pair and represents the probability of this decay.
The ${K^- p}$ TCS is not described by the formula (\[resdomxsec\]) completely, and a partial cross section, attributed to the s-channel strings excitations, is added in the UrQMD model to fit the TCS to the experimental data. In the UrQMD model this s-channel strings cross section is added also to other strange meson nonstrange baryon TCSs when annihilation is possible due to the quark content. From comparison with the experimental data for the ${K^- p}$ ECS [@Beringer] (actually it’s believed to be the EQCS because smaller peaks from the resonances are reproduced there) it was found that the half of the s-channel strings cross section is enough to describe well this experimental ${K^- p}$ cross section. Then the half of the s-channel strings cross section is added to other strange meson nonstrange baryon EQCSs when annihilation is possible. These contributions from the strings excitations are the most low energetic ones. They are the only contributions from the strings excitations which are added. The next in the energy scale possible contributions to the EQCSs may be in the ${\bar B B}$ cross sections. In other pairs the string excitations appear approximately from ${\sqrt{s}=3~GeV}$.
There is an important omission, found in the UrQMD codes (present also in the last version 3.3). The function fcgk returns incorrect (two times smaller) values of the squared Clebsch-Gordan coefficients for the resonances dominated cross sections in some cases. The first case is for the pairs of unflavored mesons from the same multiplet with the isospin ${I=1}$. For example, the function fcgk returns $0.5$ for the only possible isospin decomposition of the $\rho^+$ to the ${\pi^+ \pi^0}$ pair, because the states ${\pi^+ \pi^0}$ and ${\pi^0 \pi^+}$ are counted as different ones. As a result, the peak from the $\rho$-resonance becomes two times smaller than e. g. in the $\rho^0$-resonance isospin decomposition. The second less important case is for the pairs of unflavored mesons with the isospin ${I=1}$ and anti-nucleons. The third even less important case is for the pair ${\bar K K^*}$ and it’s charge conjugate.
Let’s make some comments on the errors what the above-mentioned omissions cause in some quantities at zero chemical potentials, which in turn demonstrate sensitivity to different changes in the cross sections. The errors in the viscosities with the ECSs are less than $2\%$. The errors in the shear viscosity with the EQCSs (the TCSs) reach $57-63\%$ ($29-32\%$) at $T=0.07~GeV$. Outside the temperature range $0.03~GeV \leq T \leq 0.14~GeV$ the errors reach $11.6\%$ ($5.3\%$). The errors in the bulk viscosity with the EQCSs (the TCSs) reach $14.4-15.4\%$ ($10.6-11.4\%$) at $T=0.07~GeV$. Outside the temperature range $0.03~GeV \leq T \leq
0.13~GeV$ the errors reach $4.8\%$ ($2.1\%$). The errors in the total number of collisions per unit time per unit volume (using the TCSs and including the decay rates) reach $10.2\%$ (at $T=0.07~GeV$). Outside the temperature range $0.04~GeV \leq T \leq
0.14~GeV$ the errors reach $5.1\%$. In view of the errors for the total number of collisions the kinetic freeze-out temperatures found in the UrQMD studies [@Bleicher:2002dm] should decrease, becoming closer to the experimentally extracted ones (see [@Heinz:2007in] and references therein). The chemical freeze-out temperature may change in a less extent. This is because both the inelastic and the quasielastic processes’ cross sections (like of the quasielastic collision of the $\pi^+ \pi^0$ pair and of the reaction $\pi^+ \pi^0 \rightarrow K^+ \bar K^0$) increase, so that the temperature at which the inelastic processes cease to be dominant may almost not change.
It’s observed that some of the UrQMD detailed balance cross sections (e. g. for the ${\Delta^+ \Delta^0}$ pair) are not symmetric under the particle interchange. This is because the function W3j, calculating the Wigner ${3-j}$ symbols, doesn’t return zero in some cases. Namely, the selection rule for the sum ${J_1+J_2+J_3}$ is not included. In principle, such omission could result in negative values of the essentially non-negative viscosities but, as long as only small fraction of cross sections is affected, this omission has caused only negligibly small errors in the viscosities. But e. g. the error in the ${\Delta^+
\Delta^0}$ TCS is approximately $25\%$.
Also some fixes of the UrQMD cross sections are made. It’s found that the ${K^+ p}$ UrQMD ECS has large deviations from the experimental data [@Beringer] in the range ${1.6~GeV<\sqrt{s}<5~GeV}$ (the UrQMD ${K^+ p}$ cross section reaches ${17~mb}$ in the region ${1.9~GeV<\sqrt{s}<3.1~GeV}$). To fit this cross section to the experimental data it is replaced with the AQM ECS in the range ${3~GeV<\sqrt{s}<5~GeV}$ and is interpolated smoothly with the sine function in the range ${1.6~GeV<\sqrt{s}\leq 3~GeV}$ with the cross section being equal to ${12.5~mb}$ at ${\sqrt{s}=1.6~GeV}$. This replacement is also applied to other MB ECSs, when annihilation is not possible due to the quark content.
The next fix is for the BB ECSs. It’s found that the ${\Lambda p}$ UrQMD ECS has quite large deviations form the experimental data [@Beringer] too. To fit this cross section to the experimental data it is replaced with the AQM ECS in the range ${4~GeV<\sqrt{s}<5~GeV}$ and interpolated smoothly with the sine function in the range ${2.2~GeV<\sqrt{s}\leq 4~GeV}$ with the cross section being equal to the AQM TCS at ${\sqrt{s}=2.2~GeV}$. This replacement is also applied to other BB ECSs.
Some other found lacks result in negligible errors in the viscosities. However, errors in the corresponding mean free paths and possible other quantities may be not negligible ones. Two of such lacks can be mentioned. The first one is the following. The ${\pi p}$ and ${K N}$ cross sections are fitted to the experimental data. And their charge conjugates are calculated using general formulas and so cause deviations up to $50\%$ for ${\sqrt{s}>1.7~GeV}$. The second lack is the following. In some not large energy regions with ${\sqrt{s}<1.7~GeV}$ the resonance dominated cross sections are equal to zero for some small numbers of pairs because there is no resonances which could be created by this pair. These regions are replaced by a constant continuously.
Let’s also comment on the deviations from the fixes described in the last four preceding paragraphs. The altogether deviations in the viscosities and the total number of collisions with the TCSs are less than $0.1\%$. The altogether deviations in the viscosities with the ECS or the EQCS are in the range $21-27\%$. The largest contribution is from the MB cross sections’ fixes. At $T\leq 0.12~GeV$ the deviations are less than $5\%$ (the temperatures above $T=0.27~GeV$ are not studied).
Conditions of applicability \[condappl\]
=========================================
Before proceeding forth first the applicability of the Boltzmann equation and of the calculations of the transport coefficients should be clarified.
Although the Boltzmann equations are valid for any perturbations of the distribution functions they should be slowly varying functions of the space-time coordinates to justify that they can be considered as functions of macroscopic quantities like the temperature, the chemical potentials or the flow velocity or, in other words, that one can apply thermodynamics locally. Then one can make the expansion over the independent gradients of the thermodynamic functions and the flow velocity (the Chapman-Enskog method), which vanish in equilibrium. Smallness of these perturbations of the distribution functions in compare to their leading parts ensures the validity of this expansion and that the gradients are small[^27]. Because these perturbations are inversely proportional to coupling constants one can say that they are proportional to some product of particles’ mean free paths and the gradients. So that, in other words, the mean free paths should be much smaller than characteristic lengths, on which the macroscopic quantities change considerably[^28].
As is discussed in Sec. \[physmeansec\], the inelastic processes may need addition treatment in the calculations of the bulk viscosity. There is a need to specify reasonable conditions when the inelastic processes can be neglected. One could use the following reliable criterion, which takes into account both the particle number densities and the intensity of the interactions: [$$\begin{aligned}
\int_{t_1}^{t_2}dt \int_0^{V(t)} d^3x \sum_{n\in~\text{all channels}}\widetilde R^{inel}_{k',n}< 1,
\end{aligned}$$]{} where $V(t)$ is the system’s volume, $\sum_{n\in~\text{all
channels}}\widetilde R^{inel}_{k',n}$ is the number of reactions of particles of the $k'$-th species[^29] over all channels per unit time per unit volume (analog of (\[totratekl\])), $t_1$ is chosen to satisfy the inequality, and $t_2$ is equal to the moment of time at which the divergence of the flow velocity is relaxed (if this time can be estimated reliably with remained inelastic processes) or to the moment of time at which the system becomes practically not interacting (after expansion) because of large cumulative mean free path in compare to the system’s size. Though this criterion is likely to be too strict, and at some higher temperatures the approximation of conserved particle numbers should still work well. The main alternative criterion is based on comparison of collision rates of elastic and inelastic processes (as implemented in the [@Bleicher:2002dm]). Using this criterion and some other ones, the chemical freeze-out line[^30] in the ${T-\mu_B}$ plane can be built for the hadron gas, see e. g. [@Cleymans:2005xv; @Andronic:2005yp]. At zero chemical potentials the chemical freeze-out temperature is approximately equal to ${T_{ch. f.}=160-170~MeV}$. The remaining question is how good is the approximation of only the elastic collisions at ${T
\lesssim 160~MeV}$. From the hydrodynamical description of the elliptic flow at RHIC it’s found that $\xi/s \lesssim 0.05$ near the chemical freeze-out [@Dusling:2011fd]. The constant value $\xi/s=0.04$ provides a good description of the elliptic flow both at RHIC and LHC [@Bozek:2011ph]. It seems that the approximation of conserved particle numbers is not implemented in the bulk viscosity formula used in the [@nhngr]. The bulk viscosity obtained from it is very close to the one of this paper. These results support the choice of the approximation of only the elastic collisions at ${T\lesssim 160~MeV}$ and show that the deviations are likely no more than in 2-3 times. Anyway, the numerical calculations by the Kubo formula through simulations of collisions are desirable along and around the chemical freeze-out line for more accurate calculations (though the procedure of collisions of particles introduce some errors itself [@Bass:1998ca], which should be kept in mind).
Errors due to the Maxwell-Boltzmann statistics, used instead of the Bose-Einstein or the Fermi-Dirac ones, were found to be small for the vanishing chemical potentials[^31]. According to calculations for the pion gas in [@davesne], the bulk viscosity becomes $25\%$ larger at ${T=120~MeV}$ and $33\%$ larger at ${T=200~MeV}$ for the vanishing chemical potential. Although the relative deviations of the thermodynamic quantities of the pion gas at the nonvanishing chemical potential ${\mu=100~MeV}$ are not more than $20\%$[^32] the bulk viscosity becomes up to $2.5$ times more. The shear viscosity becomes $15\%$ less at ${T=120~MeV}$ and $25\%$ less at ${T=200~MeV}$ for the vanishing chemical potential and $33\%$ less at ${T=120~MeV}$ and $67\%$ less at ${T=200~MeV}$ for the ${\mu=100~MeV}$. The corrections to the bulk viscosity of the fermion gas, according to calculations of the bulk viscosity source term, not presented in this paper, are of the opposite sign and approximately of the same magnitude. So that for the hadron gas the error due to the used classical statistics can be even smaller than for the pion gas.
The numerical calculations in Sec. \[numcalc\] of the viscosities with the total cross sections justify the choice of one constant cross section for all hadrons. It’s approximately equal to ${20~mb}$, corresponding to the effective radius ${r=0.4~fm}$ (as given by the (\[hccs\])), which is used in the estimations below.
The condition of applicability of the ideal gas equation of state is controlled by the dimensionless parameter $\upsilon n$ which appears in the first correction from the binary collisions in the virial expansion and should be small. Here ${\upsilon=16\pi
r^3/3}$ is the so called excluded volume parameter and $1/n$ is the mean volume per particle. One finds ${\upsilon n\approx 0.09}$ at ${T=120~MeV}$, ${\upsilon n\approx 0.2}$ at ${T=140~MeV}$ and ${\upsilon n\approx 1}$ at ${T=180~MeV}$ for the vanishing chemical potentials. Along the chemical freeze-out line (its parametrization can be found in [@Gorenstein:2007mw]) the $\upsilon n$ grows from $0.07$ to $0.49$ with the temperature. From comparison with lattice calculations [@Borsanyi:2010cj] one can find that the corrections to the ideal gas equation of state are small at ${T \lesssim 140~MeV}$. One could suspect that even small corrections to the thermodynamic quantities can result in large corrections for the bulk viscosity, though this seems to be not the case. The errors in the bulk viscosity from the scale-violating contributions of the hadrons’ masses are less than the errors from the contributions to the trace of the energy-momentum tensor (for more details see Sec. \[numcalc\]).
One more important requirement, which one needs to justify the Boltzmann equation approach, is that the mean free time should be much larger than $\hbar/\Omega$ (the $\Omega$ is the characteristic single-particle energy) [@danielewicz] or the de Broglie wavelength should be much smaller than the mean free path [@Arnold:2002zm] to distinguish independent acts of collisions and for particles to have well-defined on-shell energy and momentum. This condition gets badly satisfied for high temperatures or densities. The mean free path of the particle species $k'$ is given by the formula (\[mfp\]) or the formula (\[lel\]) if the inelastic processes can be neglected. The wavelength can be written as ${\lambda_{k'}\approx\frac1{\langle
{\left\vert\vec p_{k'}\right\vert}\rangle}}$, where the averaged modulus of the momentum of the $k'$-th species $\langle {\left\vert\vec p_{k'}\right\vert}\rangle$ is [$$\begin{aligned}
\label{avmom}
\langle{\left\vert\vec p_{k'}\right\vert}\rangle=\frac{\int d^3p_{k'} |\vec p_{k'}|
f^{(0)}_{k'}(p_{k'})}{\int d^3p_{k'} f^{(0)}_{k'}(p_{k'})}=\frac{2
e^{-z_{k'}} T (3 + 3z_{k'} + z_{k'}^2)}{z_{k'}^2
K_2(z_{k'})}=\sqrt{\frac{8 m_{k'} T}{\pi}}
\frac{K_{5/2}(z_{k'})}{K_2(z_{k'})},
\end{aligned}$$]{} where ${z_{k'}\equiv m_{k'}/T}$, $K_2(x)$ is the modified Bessel function of the second kind. As it follows from the (\[avmom\]) the largest wavelength is for the lightest particles, the $\pi$-mesons. The elastic collision mean free paths are close to each other for all particle species. Hence, the smallest value of the ratio $\lambda_{k'}/l^{el}_{k'}$ is for the $\pi$-mesons. Its value is close to the value of the $\upsilon n$ and is exponentially suppressed for small temperatures too. At the temperature ${T=140~MeV}$ (${180~MeV}$) and the vanishing chemical potentials this ratio is equal to 0.18 (0.7). Along the chemical freeze-out line it grows from $0.12$ to $0.37$ with the temperature.
To go beyond these conditions one can use the Kubo (or Green-Kubo) formulas, for instance. In the [@jeon] the Kubo formulas were used to perform perturbative calculations of the viscosities in the leading order. Basing on this result, an example of effective weakly coupled kinetic theory of quasiparticle excitations with thermal masses and thermal scattering amplitudes was presented in the [@jeonyaffe]. There the function $U(q)$ (appearing because of the temperature dependence of the mass) takes into account the next in the coupling constant correction to the energy-momentum tensor and the equation of state[^33]. For further developments see [@Arnold:2002zm; @Gagnon:2006hi; @Gagnon:2007qt]. For some other approaches see [@Blaizot:1992gn; @Calzetta:1986cq; @Calzetta:1999ps] and [@Arnold:1997gh] with references therein.
Details of calculations \[CalcSec\]
=====================================
The Boltzmann equation and its solution \[CalcSecA\]
------------------------------------------------------
The calculations in this paper go close to the ones in [@groot] though with some differences and generalizations. Let’s start from some definitions. Multi-indices $k,l,m,n$ will be used to denote particle species with certain spin states. Indexes $k',l',m',n'$ will be used to denote particle species without regard to their spin states (and run from 1 to the number of the particle species $N'$) and $a,b$ to denote conserved quantum numbers[^34]. Quantifiers $\forall$ with respect to the indexes are omitted in the text where they may be needed which won’t result in a confusion. Because nothing depends on spin variables one has for every sum over the multi-indexes [$$\begin{aligned}
\sum_k ... = \sum_{k'}g_{k'}...,
\end{aligned}$$]{} where $g_{k'}$ is the spin degeneracy factor. The following assignments will be used: $$\begin{aligned}
\label{assign1}
\nonumber n\equiv\sum_{k}n_k&\equiv&\sum_{k'} n_{k'}, \quad n_a\equiv \sum_k q_{ak} n_k,
\quad x_k\equiv\frac{n_k}{n}, \quad x_{k'}\equiv\frac{n_{k'}}{n}, \quad x_a\equiv\frac{n_a}{n}, \\
\hat \mu_k&\equiv&\frac{\mu_k}{T}, \quad \hat \mu_a\equiv\frac{\mu_a}{T},
\quad z_k\equiv\frac{m_k}{T}, \quad \pi_k^\mu\equiv \frac{p_k^\mu}{T},
\quad \tau_k\equiv \frac{p_k^\mu U_\mu}{T},
\end{aligned}$$ where $q_{ak}$ denotes values of conserved quantum numbers of the $a$-th kind of the $k$-th particle species. Everywhere the particle number densities are summed, the spin degeneracy factor $g_{k'}$ appears and then gets absorbed into the $n_{k'}$ or the $x_{k'}$ by the definition. All other quantities with primed and unprimed indexes don’t differ, except for rates, the mean free times and the mean free paths defined in Appendix \[appmfp\], the $\gamma_{kl}$ commented below, the coefficients $A_{k'l'}^{rs}$, $C_{k'l'}^{rs}$ and, of course, quantities, whose free indexes set the indexes of the particle number densities $n_k$. Also the assignment ${\int \frac{d^3p_k}{p_k^0}\equiv
\int_{p_k}}$ will be used for compactness somewhere.
The particle number flows are[^35] [$$\begin{aligned}
\label{pflow}
N^{\mu}_k=\int \frac{d^3p_k}{(2\pi)^3p^0_k} p^\mu_k f_k,
\end{aligned}$$]{} where the assignment ${f_k(p_k)\equiv f_k}$ is introduced. The energy-momentum tensor is [$$\begin{aligned}
\label{enmomten}
T^{\mu\nu}=\sum_k \int \frac{d^3p_k}{(2\pi)^3p_k^0}p_k^\mu p_k^\nu f_k.
\end{aligned}$$]{} The local equilibrium distribution functions are [$$\begin{aligned}
\label{loceq}
f^{(0)}_k=e^{(\mu_k-p_k^\mu U_\mu)/T},
\end{aligned}$$]{} where $\mu_k$ is the chemical potential of the $k$-th particle species, $T$ is the temperature and $U_\mu$ is the relativistic flow 4-velocity such that ${U_\mu U^\mu=1}$ (with a frequently used consequence ${U_\mu{\partial}_\nu U^\mu=0}$). The local equilibrium is considered as perturbations of independent thermodynamic variables and the flow velocity over a global equilibrium such that they can depend on the space-time coordinate $x^\mu$. Additional chemical perturbations could also be considered, but they don’t enter in the first order transport coefficients if they are small, as is discussed in Sec. \[physmeansec\]. The chemical equilibrium implies that the particle number densities are equal to their global equilibrium values. The global equilibrium is called the time-independent stationary state with the maximal entropy[^36]. The global equilibrium of an isolated system can be found by variation of the total nonequilibrium entropy functional [@landau5] over the distribution functions with condition of the total energy and the total net charges conservation: [$$\begin{aligned}
U[f]=\sum_k \int \frac{d^3p_kd^3x}{(2\pi)^3}
f_k(1-\ln f_k)-\sum_k\int \frac{d^3p_kd^3x}{(2\pi)^3}\beta
p^0_kf_k-\sum_{a,k} \lambda_a q_{ak} \int \frac{d^3p_kd^3x}{(2\pi)^3} f_k,
\end{aligned}$$]{} where $\beta, \lambda_a$ are the Lagrange coefficients. Equating the first variation to zero, one easily gets the function (\[loceq\]) with ${U^\mu=(1,0,0,0)}$, ${\beta=\frac1{T}}$ and [$$\begin{aligned}
\label{mukdef}
\mu_k=\sum_a q_{ak}\mu_a,
\end{aligned}$$]{} where ${\mu_a=\lambda_a}$ are the independent chemical potentials coupled to the conserved net charges.
With ${f_k=f_k^{(0)}}$, substituted in the (\[pflow\]) and the (\[enmomten\]), one gets the leading contribution in the gradients expansion of the particle number flow and the energy-momentum tensor: [$$\begin{aligned}
N^{(0)\mu}_k=n_kU^\mu,
\end{aligned}$$]{} [$$\begin{aligned}
\label{T0}
T^{(0)\mu\nu}=\epsilon U^\mu U^\nu - P\Delta^{\mu\nu},
\end{aligned}$$]{} where the projector [$$\begin{aligned}
\label{proj}
\Delta^{\mu\nu}\equiv g^{\mu\nu}-U^\mu U^\nu,
\end{aligned}$$]{} is introduced. The $n_k$ is the ideal gas particle number density, [$$\begin{aligned}
\label{ignk}
n_k=U_\mu N^{(0)\mu}_k=\frac{1}{2\pi^2}T^3z_k^2K_2(z_k)e^{\hat \mu_k},
\end{aligned}$$]{} the $\epsilon$ is the ideal gas energy density, [$$\begin{aligned}
\label{epsandek}
\epsilon=U_\mu U_\nu T^{(0)\mu\nu}=\sum_k
\int \frac{d^3p_k}{(2\pi)^3}p_k^0f_k^{(0)}=\sum_k n_k e_k, \quad
e_k\equiv m_k\frac{K_3(z_k)}{K_2(z_k)}-T,
\end{aligned}$$]{} and the $P$ is the ideal gas pressure, [$$\begin{aligned}
\label{pressure}
P=-\frac13T^{(0)\mu\nu}\Delta_{\mu\nu}=\sum_k \frac13\int
\frac{d^3p_k}{(2\pi)^3p_k^0}\vec{p_k}^2 f_k^{(0)}=\sum_k n_k T=nT.
\end{aligned}$$]{} Also the following assignments are used: $$\begin{aligned}
\label{assign2}
\nonumber e&\equiv&\frac{\epsilon}{n}=\sum_kx_ke_k, \quad h_k\equiv e_k+T,
\quad h\equiv\frac{\epsilon+P}{n}=\sum_k x_kh_k, \\
\hat e_k&\equiv&\frac{e_k}{T}=z_k\frac{K_3(z_k)}{K_2(z_k)}-1,
\quad \hat e\equiv\frac{e}{T}, \quad \hat h_k\equiv\frac{h_k}{T}=
z_k\frac{K_3(z_k)}{K_2(z_k)}, \quad \hat h\equiv \frac{h}{T}.
\end{aligned}$$ Above $h$ is the enthalpy per particle, $e$ is the energy per particle and $h_k$, $e_k$ are the enthalpy and the energy per particle of the $k$-th particle species correspondingly, which are well defined in the ideal gas.
In the relativistic hydrodynamics the flow velocity $U^\mu$ needs somewhat extended definition. The most convenient condition which can be applied to the $U^\mu$ is the Landau-Lifshitz condition [@landau6] (Section 136). This condition states that in the local rest frame (where the flow velocity is zero though its gradient can have a nonzero value) each imaginary infinitesimal cell of fluid should have zero momentum, and its energy density and the charge density should be related to other thermodynamic quantities through the equilibrium thermodynamic relations (without a contribution of nonequilibrium dissipations). Its covariant mathematical formulation is [$$\begin{aligned}
\label{lLcond}
(T^{\mu\nu}-T^{(0)\mu\nu})U_\mu=0, \quad (N^\mu_a-N^{(0)\mu}_a)U_\mu=0.
\end{aligned}$$]{} The next to leading correction over the gradients expansion to the $T^{\mu\nu}$ can be written as an expansion over the 1-st order Lorentz covariant gradients, which are rotationally and space inversion invariant and satisfy the Landau-Lifshitz condition[^37] (\[lLcond\]): [$$\begin{aligned}
\label{T1}
T^{(1)\mu\nu}\equiv2\eta \overset{\circ}{\overline{\nabla^\mu
U^\nu}}+\xi \Delta^{\mu\nu} \nabla_\rho U^\rho=\eta\left(\Delta^\mu_\rho \Delta^\nu_\tau
+\Delta^\nu_\rho \Delta^\mu_\tau-\frac23\Delta^{\mu\nu}\Delta_{\rho\tau}\right)\nabla^\rho
U^\tau+\xi \Delta^{\mu\nu} \nabla_\rho U^\rho,
\end{aligned}$$]{} where for any tensor $a_{\mu\nu}$ the symmetrized traceless tensor assignment is introduced: [$$\begin{aligned}
\label{tracelessten}
\overset{\circ}{\overline{a_{\mu\nu}}}\equiv \left(\frac{\Delta_{\mu\rho}
\Delta_{\nu\tau}+\Delta_{\nu\rho} \Delta_{\mu\tau}}2-\frac13\Delta_{\mu\nu}
\Delta_{\rho\tau}\right)a^{\rho\tau}\equiv \Delta_{\mu\nu\rho\tau}a^{\rho\tau},
\quad \Delta^{\mu\nu}_{~~\rho\tau}\Delta^{\rho\tau}_{~~\sigma\lambda}=
\Delta^{\mu\nu}_{~~\sigma\lambda}.
\end{aligned}$$]{} The equation (\[T1\]) is the definition of the shear $\eta$ and the bulk $\xi$ viscosity coefficients. The $\xi \Delta^{\mu\nu}
\nabla_\rho U^\rho$ term in the (\[T1\]) can be considered as a nonequilibrium contribution to the pressure which enters in the (\[T0\]).
By means of the projector (\[proj\]) one can split the space-time derivative ${\partial}_\mu$ as [$$\begin{aligned}
{\partial}_\mu=U_\mu U^\nu {\partial}_\nu + \Delta_\mu^\nu{\partial}_\nu = U_\mu
D+\nabla_\mu,
\end{aligned}$$]{} where $D \equiv U^\nu {\partial}_\nu$, $\nabla_\mu \equiv
\Delta_\mu^\nu{\partial}_\nu$. In the local rest frame (where ${U^\mu=(1,0,0,0)}$) the $D$ becomes the time derivative and the $\nabla_\mu$ becomes the spacial derivative. Then the Boltzmann equations can be written in the form [$$\begin{aligned}
\label{boleqs}
p_k^\mu{\partial}_\mu f_k=(p_k^\mu U_\mu D + p_k^\mu \nabla_\mu )f_k
=C_k^{el}[f_k]+C_k^{inel}[f_k],
\end{aligned}$$]{} where $C_k^{inel}[f_k]$ represents the inelastic or number-changing collision integrals (it is omitted in calculations in this paper if the opposite is not stated explicitly) and $C_k^{el}[f_k]$ is the elastic ${2\leftrightarrow2}$ collision integral. The collision integral $C_k^{el}[f_k]$ has the form of the sum of positive gain terms and negative loss terms. Its explicit form is[^38] (cf. [@jeon; @Arnold:2002zm]) $$\begin{aligned}
\label{ckel}
\nonumber C_k^{el}[f_k]&=&\sum_{l} \gamma_{kl}\frac12\int
\frac{d^3p_{1l}}{(2\pi)^32p_{1l}^0}\frac{d^3p'_k}{(2\pi)^32{p'}_k^0}
\frac{d^3p'_{1l}}{(2\pi)^32{p'}_{1l}^0}(f'_{k}f'_{1l}-f_{k}f_{1l})\\
&\times& |{\mathcal{M}}_{kl}|^2(2\pi)^4\delta^{4}(p'_k+p'_{1l}-p_k-p_{1l}),
\end{aligned}$$ where ${\gamma_{kl}=\frac12}$ if $k$ and $l$ denote the same particle species without regard to the spin states and ${\gamma_{kl}=1}$ otherwise, ${|{\mathcal{M}}_{kl} (p'_k,p'_{1l};
p_k,p_{1l})|^2 \equiv |{\mathcal{M}}_{kl}|^2}$ is the square of the dimensionless elastic scattering amplitude averaged over the initial spin states and summed over the final ones. Index $1$ designates that $p_k$ and $p_{1k}$ are different variables. Introducing ${W_{kl}\equiv W_{kl}(p'_k,p'_{1l};p_k,p_{1l})}$ as [$$\begin{aligned}
W_{kl}=\frac{|{\mathcal{M}}_{kl}|^2}{64\pi^2}\delta^{4}(p'_k+p'_{1l}-p_k-p_{1l}),
\end{aligned}$$]{} one can rewrite the collision integral (\[ckel\]) in the form as in [@groot] (Chap. I, Sec. 2) [$$\begin{aligned}
\label{ckelgroot}
C_k^{el}[f_k]=(2\pi)^3\sum_{l}\gamma_{kl}
\int_{p_{1l},{p'}_k,{p'}_{1l}}\left(\frac{f'_{k}}{(2\pi)^3}\frac{f'_{1l}}{(2\pi)^3}
-\frac{f_{k}}{(2\pi)^3}\frac{f_{1l}}{(2\pi)^3}\right)W_{kl}.
\end{aligned}$$]{} The $W_{kl}$ is related to the elastic differential cross section $\sigma_{kl}$ as [@groot] (Chap. I, Sec. 2) [$$\begin{aligned}
W_{kl}=s\sigma_{kl}\delta^{4}(p'_k+p'_{1l}-p_k-p_{1l}),
\end{aligned}$$]{} where ${s=(p_k+p_{1l})^2}$ is the usual Mandelstam variable. The $W_{kl}$ has properties $W_{kl}(p'_k,p'_{1l};p_k,p_{1l}) =
W_{kl}(p_k,p_{1l};p'_{k},p'_{1l}) =
W_{lk}(p'_{1l},p'_{k};p_{1l},p_k)$ (due to time reversibility and a freedom of relabelling of order numbers of particles taking part in reaction). And e.g. $W_{kl}(p'_k,p'_{1l};p_k,p_{1l}) \neq
W_{kl}(p'_{1l},p'_{k};p_{1l},p_{k})$ in the general case. The elastic collision integrals have important properties which one can easily prove [@groot] (Chap. II, Sec. 1): [$$\begin{aligned}
\label{c22prop0}
\int \frac{d^3p_k}{(2\pi)^3p^0_k} C_k^{el}[f_k]=0,
\end{aligned}$$]{} [$$\begin{aligned}
\label{c22prop}
\sum_k\int \frac{d^3p_k}{(2\pi)^3p^0_k} p_k^\mu C_k^{el}[f_k]=0.
\end{aligned}$$]{} Also the $C^{el}_k[f_k]$ vanishes if ${f_k=f^{(0)}_k}$.
The distribution functions $f_k$ solving the system of the Boltzmann equations approximately are sought in the form [$$\begin{aligned}
\label{fpert}
f_k=f^{(0)}_k+f^{(1)}_k\equiv f^{(0)}_k+f^{(0)}_k\varphi_k(x,p_k),
\end{aligned}$$]{} where it’s assumed that $f_k$ depend on the $x^\mu$ entirely through the $T$, $\mu_k$, $U^\mu$ or their space-time derivatives. Also it is assumed that ${|\varphi_k|\ll 1}$. After substitution of ${f_k=f^{(0)}_k}$ in the (\[boleqs\]) the r. h. s. becomes zero and the l. h. s. is zero only if the $T$, $\mu_k$ and $U^\mu$ don’t depend on the $x^\mu$ (provided they don’t depend on the momentum $p^\mu_k$). The 1-st order space-time derivatives of the $T$, $\mu_k$, $U^\mu$ in the l. h. s. should be cancelled by the first nonvanishing contribution in the r. h. s. This means that the $\varphi_k$ should be proportional to the 1-st order space-time derivatives of the $T$, $\mu_k$, $U^\mu$. The covariant time derivatives $D$ can be expressed through the covariant spacial derivatives by means of approximate hydrodynamic equations, valid at the same order in the gradients expansion. Let’s derive them. Integrating the (\[boleqs\]) over the $\frac{d^3p_k}{(2\pi)^3p^0_k}$ with the ${f_k=f^{(0)}_k}$ in the l. h. s. with the inelastic collision integrals retained and using the (\[c22prop0\]) and the (\[pflow\]) one would get (which can be justified using explicit form of the inelastic collision integrals) [$$\begin{aligned}
\label{conteq}
{\partial}_\mu N^{(0)\mu}_k=Dn_k+n_k\nabla_\mu U^\mu=I_k,
\end{aligned}$$]{} where $I_k$ is the sum of the inelastic collision integrals integrated over the momentum. It is responsible for the nonconservation of the total particle number of the $k$-th particle species and has the property ${\sum_k q_{ak} I_k=0}$. If ${C^{inel}_k[f_k]=0}$, then ${I_k=0}$ which results in conservation of the total particle numbers of each particle species. Multiplying the (\[conteq\]) on the $q_{ak}$ and summing over $k$ one gets the continuity equations for the net charge flows: [$$\begin{aligned}
\label{conteq2}
{\partial}_\mu N^{(0)\mu}_a=Dn_a+n_a\nabla_\mu U^\mu=0.
\end{aligned}$$]{} Then integrating the (\[boleqs\]) over the $p_k^\mu\frac{d^3p_k}{(2\pi)^3p^0_k}$ with the ${f_k=f^{(0)}_k}$ in the l. h. s. one gets [$$\begin{aligned}
\label{encons0}
{\partial}_\rho T^{(0)\rho\nu}={\partial}_\rho(\epsilon U^\rho U^\nu-P\Delta^{\rho\nu})=0.
\end{aligned}$$]{} There is zero in the r. h. s. even if the inelastic collision integrals are retained because they respect energy conservation too. Note that the Boltzmann equations (\[boleqs\]) (without any thermal corrections) permit a self-consistent description only if the energy-momentum tensor and the net charge flows of the ideal gas are used. After the convolution of the (\[encons0\]) with the $\Delta^\mu_\nu$ one gets the Euler’s equation: [$$\begin{aligned}
\label{eulereq}
DU^\mu=\frac1{\epsilon+P}\nabla^\mu P=\frac1{hn}\nabla^\mu P.
\end{aligned}$$]{} After the convolution of the (\[encons0\]) with the $U_\nu$ one gets equation for the energy density: [$$\begin{aligned}
\label{encons1}
D\epsilon=-(\epsilon+P)\nabla_\mu U^\mu = - hn\nabla_\mu U^\mu.
\end{aligned}$$]{}
To proceed farther one needs to expand the l. h. s. of the Boltzmann equations (\[boleqs\]) over the gradients of thermodynamic variables and the flow velocity. Let’s choose the $\mu_a$ and the $T$ as the independent thermodynamic variables. Then for the $Df^{(0)}_k$ one can write the expansion [$$\begin{aligned}
\label{Dfk}
Df^{(0)}_k=\sum_a \frac{{\partial}f^{(0)}_k}{{\partial}\mu_a}D\mu_a+\frac{{\partial}f^{(0)}_k}{{\partial}T}DT+\frac{{\partial}f^{(0)}_k}{{\partial}U^\mu}DU^\mu.
\end{aligned}$$]{} Writing the expansion for the $Dn_a$ and the $D\epsilon$ one gets from the (\[conteq2\]) and the (\[encons1\]): [$$\begin{aligned}
\label{Dna}
Dn_a=\sum_b \frac{{\partial}n_a}{{\partial}\mu_b}D\mu_b+\frac{{\partial}n_a}{{\partial}T}DT
=-n_a\nabla_\mu U^\mu,
\end{aligned}$$]{} [$$\begin{aligned}
\label{Depsilon}
D\epsilon=\frac{{\partial}\epsilon}{{\partial}T}DT+\sum_a\frac{{\partial}\epsilon}{{\partial}\mu_a}D\mu_a
=-hn\nabla_\mu U^\mu.
\end{aligned}$$]{} The solution to the system of equations (\[Dna\]), (\[Depsilon\]) can be found easily: [$$\begin{aligned}
\label{Teqn}
DT=-R T\nabla_\mu U^\mu,
\end{aligned}$$]{} [$$\begin{aligned}
\label{mueqn}
D\mu_a=T\sum_b \tilde{A}^{-1}_{ab}(R B_b-x_b)\nabla_\mu U^\mu,
\end{aligned}$$]{} where [$$\begin{aligned}
\label{Rdef}
R\equiv\frac{\hat h-\sum_{a,b} E_a \tilde{A}^{-1}_{ab}x_b}{C_{\{\mu\}}-\sum_{a,b}
E_a\tilde{A}^{-1}_{ab}B_b},
\end{aligned}$$]{} and [$$\begin{aligned}
\frac{{\partial}n_a}{{\partial}\mu_b}\equiv \frac{n}{T}\tilde{A}_{ab},
\quad \frac{{\partial}n_a}{{\partial}T}\equiv \frac{n}{T}B_a, \quad
\frac{{\partial}\epsilon}{{\partial}T}\equiv n C_{\{\mu\}},\quad
\frac{{\partial}\epsilon}{{\partial}\mu_a}\equiv n E_a.
\end{aligned}$$]{} Above it is assumed that the matrix $\tilde{A}_{ab}$ is not degenerate[^39], which is related to the self-consistency of the statistical description of the system. Using the ideal gas formulas (\[ignk\]) and (\[epsandek\]) one gets $$\begin{aligned}
\label{ABCE}
&~&\tilde A_{ab}=\sum_k q_{ak}q_{bk}x_k, \quad
E_a=\sum_k q_{ak} x_k \hat e_k, \quad B_a=E_a-\sum_b \tilde A_{ab}\hat\mu_b,\\
\nonumber C_{\{\mu\}}&=&\sum_k x_k(3\hat h_k+z_k^2-\hat \mu_k\hat
e_k)=\sum_kx_k(3\hat h_k+z_k^2)-\sum_aE_a\hat\mu_a \equiv
\widetilde C_{\{\mu\}}-\sum_aE_a\hat\mu_a,
\end{aligned}$$ and simplified expressions for the $R$ and the $D\hat\mu_a$ [$$\begin{aligned}
\label{Rdef2}
R=\frac{\hat h-\sum_{a,b} E_a \tilde{A}^{-1}_{ab}x_b}{\widetilde C_{\{\mu\}}-\sum_{a,b}
E_a\tilde{A}^{-1}_{ab}E_b},
\end{aligned}$$]{} [$$\begin{aligned}
D\hat\mu_a=\sum_b\tilde{A}^{-1}_{ab}(R E_b-x_b)\nabla_\mu U^\mu.
\end{aligned}$$]{} For the special case of the vanishing chemical potentials, $\mu_a\rightarrow 0$, (for a chargeless system the result is the same) the quantities $n_a$, $x_a$, $B_a$, $E_a$ tend to zero because the contributions from particles and anti-particles cancel each other and the chargeless particles don’t contribute. Then from the (\[Teqn\]) and the (\[mueqn\]) one finds [$$\begin{aligned}
\label{Teqnmu0}
DT|_{\mu_a=0}=-\frac{h}{\widetilde C_{\{\mu\}}}\nabla_\mu U^\mu,
\end{aligned}$$]{} [$$\begin{aligned}
D\mu_a|_{\mu_a=0}=0.
\end{aligned}$$]{} This means that for the vanishing chemical potentials one can simply exclude them from the distribution functions (if one does not study diffusion or thermal conductivity). In systems with only the elastic collisions each particle has its own charge so that one takes ${q_{ak}=\delta_{ak}}$ and gets $$\begin{aligned}
\label{elquant}
\nonumber \tilde A_{kl}&=&\delta_{kl}x_k, \quad B_k=x_k(\hat e_k-\hat \mu_k),
\quad E_k=\hat e_k x_k, \quad R=\frac1{c_\upsilon}, \\
C_{\{\mu\}}&-&\sum_{a,b}E_a \tilde A^{-1}_{ab} B_{b}=\sum_kx_k(-\hat h_k^2+5\hat
h_k+z_k^2-1)\equiv\sum_kx_k c_{\upsilon,k}\equiv c_\upsilon.
\end{aligned}$$ Then the equation for the $DT$ (\[Teqn\]) remains the same with a new $R$ from the (\[elquant\]), and the equations (\[mueqn\]) become [$$\begin{aligned}
\label{mukeqn}
D\mu_k=\left(\frac{T}{c_\upsilon}(\hat e_k-\hat \mu_k)-T\right)\nabla_\mu U^\mu.
\end{aligned}$$]{} Note that in systems with only the elastic collisions the $D\mu_k$ does not tend to zero for the vanishing chemical potentials so that the $\mu_k$ could not be omitted in the distribution functions in this case. Because the heat conductivity and diffusion are not considered in this paper their nonequilibrium gradients are taken equal to zero, $\nabla_\nu P=\nabla_\nu
T=\nabla_\nu\mu_a=0$. Using the (\[Teqn\]), (\[mueqn\]) and (\[eulereq\]) the l. h. s. of the (\[boleqs\]) can be transformed as [$$\begin{aligned}
\label{boleqnlhs}
(p_k^\mu U_\mu D+p_k^\mu\nabla_\mu)f_k^{(0)} = -Tf_k^{(0)}
\pi_k^\mu \pi_k^\nu \overset{\circ}{\overline{\nabla_\mu
U_\nu}}+Tf_k^{(0)}\hat Q_k\nabla_\rho U^\rho,
\end{aligned}$$]{} where [$$\begin{aligned}
\label{Qsource}
\hat Q_k\equiv\tau_k^2\left(\frac13-R\right)+\tau_k\sum_{a,b}q_{ak}
\tilde A^{-1}_{ab}(RE_b-x_b)-\frac13z_k^2.
\end{aligned}$$]{} Using the (\[tracelessten\]) one can notice that the useful equality ${ \pi_k^\mu \pi_k^\nu \overset{\circ}{
\overline{\nabla_\mu U_\nu} } = \overset{\circ}{ \overline{
\pi_k^\mu \pi_k^\nu } }\overset{\circ}{ \overline{ \nabla_\mu
U_\nu } } }$ holds. In systems with only the elastic collisions the $\hat Q_k$ simplifies in agreement with [@groot] (Chap. V, Sec. 1): [$$\begin{aligned}
\label{Qsource2}
\hat Q_k=\left(\frac43-\gamma\right)\tau_k^2+
\tau_k((\gamma-1)\hat h_k-\gamma)-\frac13z_k^2.
\end{aligned}$$]{} where the assignments $\gamma$ from [@groot] is used. It can be expressed through the $c_\upsilon$, defined in the (\[elquant\]), as ${\gamma\equiv \frac1{c_\upsilon}+1}$. Introducing symmetric round brackets [$$\begin{aligned}
(F,G)_k\equiv\frac1{4\pi z_k^2K_2(z_k)T^2}\int_{p_k} F(p_k)G(p_k)e^{-\tau_k}.
\end{aligned}$$]{} and assignments [$$\begin{aligned}
\alpha_k^r\equiv(\hat Q_k,\tau_k^r), \quad \gamma_k^r\equiv(\tau_k^r
\overset{\circ}{\overline{\pi_k^\mu\pi_k^\nu}},
\overset{\circ}{\overline{\pi_{k\mu}\pi_{k\nu}}}), \quad
a^r_k\equiv(1,\tau_k^r)_k,
\end{aligned}$$]{} and using explicit expressions of the $a_k^r$ from Appendix \[appA\] one finds for the $\alpha_k^0$ and the $\alpha_k^1$ in systems with elastic and inelastic collisions [$$\begin{aligned}
\label{alphak0}
\alpha_k^0=1+\sum_{a,b}q_{ak} \tilde A^{-1}_{ab}(R E_b-x_b)-\hat e_k R,
\end{aligned}$$]{} [$$\begin{aligned}
\label{alphak1}
\alpha_k^1=\hat h_k+\sum_{a,b}\hat e_k q_{ak}\tilde A^{-1}_{ab}(R E_b-x_b)
-(3\hat h_k+z_k^2)R.
\end{aligned}$$]{} Then using the (\[alphak0\]) and the (\[alphak1\]) one can show that [$$\begin{aligned}
\label{lhsnchcons}
\sum_kq_{ak}x_k\alpha_k^0=0,
\end{aligned}$$]{} [$$\begin{aligned}
\label{lhsencons}
\sum_kx_k\alpha_k^1=0.
\end{aligned}$$]{} Because the gradients $\nabla_\mu U^\mu$ and $\overset{\circ}{\overline{\nabla_\mu U_\nu}}$ are independent the (\[lhsnchcons\]) and the (\[lhsencons\]) are direct consequences of the local net charge (\[conteq2\]) and the energy-momentum (\[encons0\]) conservations. Quantities $(1,\overset{\circ}{\overline{\pi^\mu_{k}\pi^\nu_{k}}})$ and $(p_k^\lambda,\overset{\circ}{\overline{\pi^\mu_{k}\pi^\nu_{k}}})$ vanish automatically because of the special tensorial structure[^40] of the $\overset{\circ}{
\overline{ \pi^\mu_{k} \pi^\nu_{k}} }$.
The next step is to transform the r. h. s. of the Boltzmann equations (\[boleqs\]). After the substitution of the (\[fpert\]) in the r. h. s. of the (\[boleqs\]) the collision integral becomes linear and one gets [$$\begin{aligned}
\label{boleqnrhs}
C_k^{el}[f_k]\approx -f_k^{(0)}\sum_l {\mathcal{L}}_{kl}^{el}[\varphi_k],
\end{aligned}$$]{} where [$$\begin{aligned}
{\mathcal{L}}_{kl}^{el}[\varphi_k]\equiv\frac{\gamma_{kl}}{(2\pi)^3}\int_{p_{1l},{p'}_k,{p'}_{1l}}
f_{1l}^{(0)}(\varphi_k+\varphi_{1l}-\varphi'_k-\varphi'_{1l})W_{kl}.
\end{aligned}$$]{} The unknown functions $\varphi_k$ are sought in the form [$$\begin{aligned}
\label{varphi}
\varphi_k=\frac1{n\sigma(T)}\left(-A_k(p_k)\nabla_\mu U^\mu+C_k(p_k)
\overset{\circ}{\overline{\pi^\mu_k \pi^\nu_k}}
\overset{\circ}{\overline{\nabla_\mu U_\nu}}\right),
\end{aligned}$$]{} where $\sigma(T)$ is some formal averaged cross section, used to come to dimensionless quantities. Then using the (\[boleqnlhs\]) and the (\[boleqnrhs\]), and the fact that the gradients $\nabla_\mu U^\mu$ and $\overset{\circ}{\overline{\nabla_\mu
U_\nu}}$ are independent, the Boltzmann equations can be written as independent integral equations: [$$\begin{aligned}
\label{xieqn}
\hat Q_k=\sum_l x_l L_{kl}^{el}[A_k],
\end{aligned}$$]{} [$$\begin{aligned}
\label{etaeqn}
\overset{\circ}{\overline{\pi_{k}^\mu \pi_{k}^\nu}}=
\sum_l x_l L_{kl}^{el}[C_k \overset{\circ}{\overline{\pi_{k}^\mu \pi_{k}^\nu}}],
\end{aligned}$$]{} where the dimensionless collision integrals are introduced: [$$\begin{aligned}
L_{kl}^{el}[\chi_k]=\frac1{n_lT\sigma(T)}{\mathcal{L}}_{kl}^{el}[\chi_k].
\end{aligned}$$]{} In the case of present inelastic processes the l. h. s. of the (\[xieqn\]) is set by the source term (\[Qsource\]) and the r. h. s. contains the linear inelastic collision integrals. After introduction of inelastic processes the source terms in the (\[xieqn\]) become much larger as demonstrated in Sec. \[singcomsec\]. Using the equations (\[mueqn\]) and (\[Teqn\]) and the ideal gas formulas (\[ABCE\]) one can check that in the zero masses limit the source terms $\hat Q_k$ (\[Qsource\]) tend to zero and ${D\hat\mu_a=0}$ that is the $\hat\mu_a$ don’t scale and the distribution functions become scale invariant. The source term of the shear viscosity in the (\[etaeqn\]) doesn’t depend on the presence of inelastic processes in the system and originates from the free propagation term ${\frac{\vec p_k}{p_k^0} \frac{{\partial}f_k}{{\partial}\vec r}}$ in the Boltzmann equation.
The transport coefficients and their properties
-------------------------------------------------
After substitution of the $f_k^{(1)}$ with the $\varphi_k$ (\[varphi\]) into the (\[enmomten\]) and comparison with the (\[T1\]) one finds the formula for the bulk viscosity [$$\begin{aligned}
\label{bulkvisc}
\xi=-\frac13\frac{T}{\sigma(T)}\sum_k x_k(\Delta^{\mu\nu}\pi_{\mu k}\pi_{\nu k},A_k)_k,
\end{aligned}$$]{} and for the shear viscosity [$$\begin{aligned}
\label{shearvisc}
\eta=\frac1{10}\frac{T}{\sigma(T)}\sum_k x_k(\overset{\circ}{\overline{\pi^\mu_k \pi^\nu_k}},
C_k \overset{\circ}{\overline{\pi_{k\mu} \pi_{k\nu}}})_k,
\end{aligned}$$]{} where the relation ${\Delta^{\mu\nu}_{ ~~\sigma\tau }
\Delta_\mu^\sigma \Delta_\nu^\tau = 5}$ is used.
In kinetics the conditions that the nonequilibrium perturbations of the distribution functions does not contribute to the net charge and the energy-momentum densities are used as a convenient choice and are called matching conditions. They reproduce the Landau-Lifshitz condition (\[lLcond\]). The matching conditions for the net charge densities can be written as [$$\begin{aligned}
\label{cofchf}
\sum_k q_{ak} \int \frac{d^3p_k}{(2\pi)^3p_k^0}p_k^\mu U_\mu
f_k^{(0)}\varphi_k=0,
\end{aligned}$$]{} and for the energy-momentum density can be written as [$$\begin{aligned}
\label{cofemt}
\sum_k \int \frac{d^3p_k}{(2\pi)^3p_k^0}p_k^\mu p_k^\nu U_\nu f_k^{(0)}\varphi_k=0.
\end{aligned}$$]{} For the special tensorial functions $C_k \overset{\circ}{
\overline{\pi_{k\mu} \pi_{k\nu}}}$ in the (\[varphi\]) they are satisfied automatically and for the scalar functions $A_k$ they can be rewritten in the form (the 3-vector part of the (\[cofemt\]) is automatically satisfied) [$$\begin{aligned}
\label{condfit}
\sum_k q_{ak}x_k (\tau_k, A_k)_k=0, \quad \sum_k x_k (\tau_k^2, A_k)_k=0.
\end{aligned}$$]{} The conditions (\[cofchf\]) and (\[cofemt\]) exclude the nonphysical solutions ${A_k^{z.m.} = \sum_a C_a q_{ak} + C
\tau_k}$ (which cannot be solutions in inhomogeneous systems and are produced just due to shifts in the $T$, $\mu_a$) of the linearized Boltzmann equations for which the collision integrals vanish ($A_k^{z.m.}$ are zero modes). From the formula (\[bulkvisc\]) one can see that in the framework of the Boltzmann equation these conditions also eliminate ambiguity in the $\xi$ due to freedom of addition of the $A_k^{z.m.}$ to the solution $A_k$ of the (\[xieqn\]). With help of these matching conditions one can show explicitly essential positiveness of the $\xi$. Namely, using the matching conditions (\[condfit\]), the equation (\[xieqn\]) and the identity ${\Delta^{\mu\nu}
\pi_{\mu,k}\pi_{\nu,k} = z_k^2 - \tau_k^2}$, the bulk viscosity (\[bulkvisc\]) can be rewritten as [$$\begin{aligned}
\label{xipos}
\xi=\frac{T}{\sigma(T)}\sum_k x_k(\hat Q_k,A_k)_k=
\frac{T}{\sigma(T)}\sum_k x_k\left(\sum_l x_l L^{el}_{kl}[A_k],A_k\right)_k=
\frac{T}{\sigma(T)}[\{A\},\{A\}],
\end{aligned}$$]{} where the square brackets are introduced for sets of equal lengths ${\{F\}=(F_1,...,F_k,...)}$, ${\{G\}=(G_1,...,G_k,...)}$: [$$\begin{aligned}
\label{sqrbra}
[\{F\},\{G\}]\equiv\frac1{n^2\sigma(T)}\sum_{k,l}
\frac{\gamma_{kl}}{(2\pi)^6}\int_{p_k,p_{1l},{p'}_k,{p'}_{1l}}
f^{(0)}_kf^{(0)}_{1l}(F_k+F_{1l}-{F'}_{k}-{F'}_{1l})G_kW_{kl}.
\end{aligned}$$]{} Using the time reversibility property of the $W_{kl}$ one can show that the equality [$$\begin{aligned}
(F_k+F_{1l}-{F'}_{k}-{F'}_{1l})G_k=\frac14(F_k+F_{1l}-{F'}_{k}-{F'}_{1l})
(G_k+G_{1l}-{G'}_{k}-{G'}_{1l}),
\end{aligned}$$]{} holds under the integration and the summation in the (\[sqrbra\]). Then one gets the direct consequence [$$\begin{aligned}
[\{F\},\{G\}]=[\{G\},\{F\}], \quad [\{F\},\{F\}]\geq 0.
\end{aligned}$$]{} This proves the essential positiveness of the $\xi$. Similarly using the (\[etaeqn\]), the shear viscosity can be rewritten in essentially positive form $$\begin{aligned}
\label{etapos}
\nonumber\eta&=&\frac1{10}\frac{T}{\sigma(T)}\sum_kx_k
\left(\overset{\circ}{\overline{\pi_{k}^\mu \pi_{k}^\nu}},
C_k \overset{\circ}{\overline{\pi_{k\mu} \pi_{k\nu}}}\right)_k
=\frac1{10}\frac{T}{\sigma(T)}\sum_kx_k\left(\sum_l x_l L^{el}_{kl}
[C_k \overset{\circ}{\overline{\pi_{k}^\mu \pi_{k}^\nu}}],
C_k \overset{\circ}{\overline{\pi_{k\mu} \pi_{k\nu}}}\right)_k\\
&=&\frac1{10}\frac{T}{\sigma(T)}[\{C \overset{\circ}{\overline{\pi^\mu \pi^\nu}}\},
\{C \overset{\circ}{\overline{\pi_{\mu} \pi_{\nu}}}\}].
\end{aligned}$$
The considered variational method allows to find an approximate solution of the integral equations (\[xieqn\]) and (\[etaeqn\]) in the form of a linear combination of test-functions. The coefficients next to the test-functions are found from the condition to deliver extremum to some functional. One could take this functional in the form of some special norm, as in [@groot]. Or one can take somewhat different functional, like in [@Arnold:2003zc], which is more convenient, and get the same result. This generalized functional can be written in the form [$$\begin{aligned}
\label{genfunctional}
F[\chi]=\sum_k x_k(S_k^{\mu...\nu},\chi_{k\mu...\nu})_k-
\frac12[\{\chi^{\mu...\nu}\},\{\chi_{\mu...\nu}\}],
\end{aligned}$$]{} where ${S_k^{\mu...\nu}=\hat Q_k}$ and ${\chi_{k\mu...\nu}=A_k}$ for the bulk viscosity and ${S_k^{\mu...\nu} = \overset{\circ}{
\overline{\pi_{k}^\mu \pi_{k}^\nu} }}$, ${\chi_k^{\mu...\nu} = C_k
\overset{\circ}{ \overline{\pi_{k}^\mu \pi_{k}^\nu} }}$ for the shear viscosity. Equating to zero the first variation of the (\[genfunctional\]) over the $\chi_{k\mu...\nu}$ one gets [$$\begin{aligned}
\sum_kx_k(S_k^{\mu...\nu},\delta\chi_{k\mu...\nu})_k-
[\{\chi^{\mu...\nu}\},\{\delta\chi_{\mu...\nu}\}]=0.
\end{aligned}$$]{} Because the variations $\delta\chi_{k\mu...\nu}$ are arbitrary and independent the generalized integral equations follows then: [$$\begin{aligned}
\label{genvareqn}
S_k^{\mu...\nu}=\sum_l x_l L^{el}_{kl}[\chi_{k\mu...\nu}].
\end{aligned}$$]{} The second variation of the (\[genfunctional\]) is [$$\begin{aligned}
\delta^2 F[\chi]=-[\{\delta\chi^{\mu...\nu}\},\{\delta\chi_{\mu...\nu}\}]\leq 0,
\end{aligned}$$]{} which means that the solution of the integral equations (\[xieqn\]) and (\[etaeqn\]) is reduced to the variational problem of finding of the maximum of the functional (\[genfunctional\]). Using the (\[genvareqn\]), the maximal value of the (\[genfunctional\]) can be written as [$$\begin{aligned}
F_{max}[\chi]=\frac12[\{\chi^{\mu...\nu}\},\{\chi_{\mu...\nu}\}]|_{\chi=\chi_{\max}}.
\end{aligned}$$]{} Then using the (\[xipos\]) and the (\[etapos\]) one can write the bulk and the shear viscosities through the maximal value of the $F[\chi]$ [$$\begin{aligned}
\xi=\left.2\frac{T}{\sigma(T)}F_{max}\right|_{S_k^{\mu...\nu}=\hat Q_k, \, \chi_{k\mu...\nu}=A_k},
\end{aligned}$$]{} [$$\begin{aligned}
\eta=\left.\frac15\frac{T}{\sigma(T)}F_{max}\right|_{S_k^{\mu...\nu}=
\overset{\circ}{\overline{\pi_{k}^\mu \pi_{k}^\nu}}, \, \chi_k^{\mu...\nu}=
C_k \overset{\circ}{\overline{\pi_{k}^\mu \pi_{k}^\nu}} }.
\end{aligned}$$]{} This means that the precise solution of the (\[genvareqn\]) delivers the maximal values for the transport coefficients.
The approximate solution of the system of the integral equations (\[xieqn\]) and (\[etaeqn\]) are sought in the form [$$\begin{aligned}
\label{Atestf}
A_{k}=\sum_{r=0}^{n_1} A_k^r\tau^r_k,
\end{aligned}$$]{} [$$\begin{aligned}
\label{Ctestf}
C_{k}=\sum_{r=0}^{n_2} C_k^r\tau^r_k,
\end{aligned}$$]{} where $n_1$ and $n_2$ set the number of the used test-functions. Test-functions, used in [@Arnold:2003zc], would cause less significant digit cancellation in numerical calculations but there is a need to reduce the dimension of the 12-dimensional integrals from these test-functions as more as possible to perform the calculations in a reasonable time. The test-functions in the form of just powers of the $\tau_k$ seem to be the most convenient for this purpose. Questions concerning the uniqueness and the existence of the solution and the convergence of the approximate solution to the precise one are covered in [@groot] (Chap. IX, Sec. 1-2). As long as particles of the same particle species but with different spin states are undistinguishable their functions $\varphi_k$ (\[varphi\]) are equal, and the variational problem is reduced to the variation of the coefficients $A_{k'}^r$ and $C_{k'}^r$, and the bulk (\[xipos\]) and the shear (\[etapos\]) viscosities can be rewritten as [$$\begin{aligned}
\label{finxi}
\xi=\frac{T}{\sigma(T)}\sum_{k'=1}^{N'}\sum_{r=0}^{n_1}x_{k'}\alpha_{k'}^r A_{k'}^r,
\end{aligned}$$]{} [$$\begin{aligned}
\label{fineta}
\eta=\frac1{10}\frac{T}{\sigma(T)}\sum_{k'=1}^{N'}\sum_{r=0}^{n_2}x_{k'}\gamma_{k'}^rC_{k'}^r.
\end{aligned}$$]{} After the substitution of the approximate functions $A_{k'}$ (\[Atestf\]) and $C_{k'}$ (\[Ctestf\]) into the (\[genfunctional\]) and equating the first variation of the functional to zero one gets the following matrix equations (with the multi-indexes ${(l',s)}$ and ${(k',r)}$) for the bulk and the shear viscosities correspondingly[^41] [$$\begin{aligned}
\label{ximatreq}
x_{k'}\alpha_{k'}^r=\sum_{l'=1}^{N'} \sum_{s=0}^{n_1} A_{l'k'}^{sr}A_{l'}^s,
\end{aligned}$$]{} [$$\begin{aligned}
\label{etamatreq}
x_{k'}\gamma_{k'}^r=\sum_{l'=1}^{N'} \sum_{s=0}^{n_2} C_{l'k'}^{sr}C_{l'}^s,
\end{aligned}$$]{} where the introduced coefficients $A_{k'l'}^{rs}$ and $C_{k'l'}^{rs}$ are [$$\begin{aligned}
\label{A4ind}
A_{k'l'}^{rs}=x_{k'}x_{l'}[\tau^r,\tau_1^s]_{k'l'}+\delta_{k'l'}x_{k'}\sum_{m'=1}^{N'}
x_{m'}[\tau^r,\tau^s]_{k'm'},
\end{aligned}$$]{} [$$\begin{aligned}
\label{C4ind}
C_{k'l'}^{rs}=x_{k'}x_{l'}[\tau^r\overset{\circ}{\overline{\pi^{\mu} \pi^{\nu}}},
\tau_1^s\overset{\circ}{\overline{\pi_{1\mu}\pi_{1\nu}}}]_{k'l'}+\delta_{k'l'}x_{k'}\sum_{m'=1}^{N'}
x_{m'}[\tau^r\overset{\circ}{\overline{\pi^{\mu} \pi^{\nu}}},
\tau^s\overset{\circ}{\overline{\pi_{\mu} \pi_{\nu}}}]_{k'm'}.
\end{aligned}$$]{} They are expressed through the collision brackets [$$\begin{aligned}
\label{br1}
[F,G_1]_{kl}\equiv\frac{\gamma_{kl}}{T^6(4\pi)^2z_k^2z_l^2K_2(z_k)K_2(z_l)\sigma(T)}
\int_{p_k,p_{1l},{p'}_k,{p'}_{1l}}e^{-\tau_k-\tau_{1l}}(F_k-{F'}_k)G_{1l}W_{kl}.
\end{aligned}$$]{} The collision brackets $[F,G]_{kl}$ are obtained from the last formula by the replacement of the $G_{1l}$ on the $G_k$. Due to the time reversibility property of the $W_{kl}$ one can replace the $G_{1l}$ on the ${\frac12(G_{1l}-{G'}_{1l})}$ in the (\[br1\]). Then one can see that [$$\begin{aligned}
\label{br2pos}
[\tau^r,\tau^s]_{kl}>0.
\end{aligned}$$]{} Also it’s easy to notice the following symmetries [$$\begin{aligned}
[F,G_1]_{kl}=[G,F_1]_{lk}, \quad [F,G]_{kl}=[G,F]_{kl}.
\end{aligned}$$]{} They result in the following symmetric properties ${A_{k'l'}^{rs}
= A_{l'k'}^{sr}}$, ${C_{k'l'}^{rs} = C_{l'k'}^{sr}}$. Also the microscopical particle number and energy conservation laws imply for the $A_{l'k'}^{sr}$: [$$\begin{aligned}
\label{A4chcons}
A_{k'l'}^{0s}=0,
\end{aligned}$$]{} [$$\begin{aligned}
\label{A4encons}
\sum_{k'=1}^{N'}A_{k'l'}^{1s}=0.
\end{aligned}$$]{} The (\[A4chcons\]) together with the $\alpha_k^0=0$ (\[elal0\]) means that the equations with ${r=0}$ in the (\[ximatreq\]) are excluded. From the (\[A4encons\]) and (\[lhsencons\]) it follows that each one equation with ${r=1}$ in the (\[ximatreq\]) can be expressed through the sum of the other ones, reducing the rank of the matrix on 1. To solve the matrix equation (\[ximatreq\]) one eliminates one equation, for example with ${k'=1}$, ${r=1}$. One of coefficients of $A_{l'}^1$ is independent; for example, let it be $A_{1'}^1$. Using the (\[A4encons\]), the matrix equation (\[ximatreq\]) can be rewritten as [$$\begin{aligned}
\label{ximatreq2}
x_{k'}\alpha_{k'}^r=\sum_{l'=2}^{N'}A_{l'k'}^{1r}(A_{l'}^1-A_{1'}^1)
+\sum_{l'=1}^{N'} \sum_{s=2}^{n_1}A_{l'k'}^{sr}A_{l'}^s.
\end{aligned}$$]{} Then, using the (\[elal0\]) and the (\[lhsencons\]), the bulk viscosity (\[finxi\]) becomes [$$\begin{aligned}
\label{finxi2}
\xi=\frac{T}{\sigma(T)}\sum_{k'=2}^{N'}x_{k'}\alpha_{k'}^1 (A_{k'}^1-A_{1'}^1)
+\frac{T}{\sigma(T)}\sum_{k'=1}^{N'}\sum_{r=2}^{n_1}x_{k'}\alpha_{k'}^r A_{k'}^r.
\end{aligned}$$]{} Then the coefficient $A_{1'}^1$ can be eliminated by shift of other $A_{l'}^1$ and be implicitly used to satisfy one energy conservation matching condition. The particle number conservation matching conditions are implicitly satisfied by means of the coefficients $A_{k'}^0$. The first term in the (\[finxi2\]) is present only in mixtures. That’s why it is small in gases with close to each other masses of particles of different species (like in the pion gas). In gases with very different masses (like in the hadron gas) contribution of the first term in the (\[finxi2\]) can become dominant.
Analytical expressions for some lowest orders collision brackets, which enter in the matrix equations (\[etamatreq\]) and (\[ximatreq2\]), can be found in Appendix \[appJ\]. Higher orders are not presented because of their bulky form.
The numerical calculations \[numcalc\]
=======================================
The numerical calculations for the hadron gas involve roughly $2\frac{(N' n)^2}2$ the 12-dimensional integrals, where $N'$ is the number of particle species and $n$ is the number of the used test-functions (called the order of the calculations). The 12-dimensional integrals (they are the collision brackets ${[F,G_1]_{kl}}$ and ${[F,G]_{kl}}$) can be reduced to 1-dimensional integrals. For constant cross sections they are expressed through special functions, and for other ones numerical methods are used. The details of calculations are described in Appendix \[appJ\]. This allows to perform the calculations with a good precision in a reasonable time. Because the analytical expressions for the collision brackets are bulky the Mathematica package [@math] is used for symbolical and some numerical manipulations.
The calculations of the viscosities are quite reliable at $T \leq
120-140~MeV$ (throughout the paper the chemical potentials are equal to zero if else is not stated), as is discussed in Sec. \[condappl\]. The numerical calculations are done also for temperatures up to ${T=270~MeV}$ for the future comparisons and to show the position of the maximum of the bulk viscosity, when it is present only due to the hadrons’ masses. Introduction of the inelastic processes should increase the bulk viscosity, though in the approximation when its nonzero value is maintained only by the hadrons’ masses the maximum may shift not considerably. Taking into account the non-ideal gas equation of state, the maximum may shift to some extent too and become sharper, as can be seen from the speed of sound of the [@toneev], or a new smaller maximum can appear.
The UrQMD (version 1.3) particle list is used, which doesn’t contain charmed and bottomed particles and consists of 322 particle species including anti-particles. Some thermodynamical quantities of the ideal hadron gas with this mass spectrum are shown in fig. \[ThermQuant\] and fig. \[Rquantity\]. The $\varepsilon$ and $P$ are given by the (\[epsandek\]), (\[ignk\]), (\[pressure\]), and the $s$ is given by (\[entrden\]). The quantity $R$ (\[Rdef2\]), appearing in the bulk viscosity source term (\[Qsource\]) and tending to $1/3$ if the particles’ masses are tended to zero, is equal to the squared speed of sound $R_{ch.-n.}=\upsilon_s^2=\frac{{\partial}P}{{\partial}\varepsilon}$ in the case of the charge-neutral (or chargeless) system (implying equal to zero and not developing chemical potentials). The $R$ is given by the $R_{el. c.}$ (\[elquant\]) in the case of only the elastic collisions. The $R_{ch.-n.}$ is quite close to the squared speed of sound of the ideal gas in the [@toneev]. For the intermediate case when there are conserved and not conserved particle numbers it’s observed that the $R$ is above $1/3$.
The new particle list with charmed and bottomed particles[^42] (cut on $3~GeV$, which results in negligible errors $0.01\%$ or less) from the THERMUS package [@thermus] was used in the calculations at zero chemical potentials in the [@Moroz:2011vn]. The errors in the trace of the energy-momentum tensor because of neglected rhe charmed and bottomed particles grow with the temperature and are equal to $13\%$ ($21\%$) at ${T=140~MeV}$ ($270~MeV$). The errors in the $R$ are less than $3.2\%$. The errors in the shear viscosity (calculated with one constant cross section for all hadrons) are less than $0.6\%$ and in the bulk viscosity are less than $7.5\%$. An additional study of the mass spectrum dependence of the viscosities can be found in the [@Moroz:2011vn].
The results for the shear and the bulk viscosities are shown in fig. \[ShearAll\] and fig. \[BulkAll\] correspondingly. They are calculated with the different cross sections (with all corrections and fixes): ECQs, EQCSs, EQCS2s, TCSs, TCS2s. They are described in Sec. \[hardcorsec\]. The bulk viscosity with the TCS2s is not shown because it’s smaller only on $5\%$ or less than the one with the TCSs. Up to 5 (3) test-functions are used in the calculations of the bulk (shear) viscosity. The maximal errors are $11\%$ and are less than $4.2\%$ outside the range $40~MeV\leq T
\leq 90~MeV$. The best convergence is for the case of the ECSs (the errors are less than $2\%$). For quantitative results the recommended cross sections are the EQCS2s, as is commented in Sec. \[hardcorsec\]. For qualitative analysis the TCSs and the TCS2s are more suitable. Also it’s shown that the approximation of one constant cross section for all hadrons is a good one. For the shear viscosity with the EQCSs or the EQCS2s it’s somewhat worse. This can be explained in the following way. There are descending and growing cross sections, and these opposite dependencies compensate approximately each other, so that at some temperatures the cross sections manifest themselves approximately as a constant one. Some EQCSs and EQCS2s have quite steeply descending tails, which explains relatively fast rises in the shear viscosities, because of which one constant cross section doesn’t provides worse approximation. An explanation why the bulk viscosity is approximated well in the same case with one constant cross sections at some temperatures would be somewhat more complicated.
Also the calculations without resonances (the particles with the width larger or equal to $0.2~MeV$) are done with the EQCSs to find out the magnitude of their contributions. After the exclusion 26 particle species remain. The bulk viscosity decreases not more than in 2.8 times, and the shear viscosity decreases not more than in 1.6 times (using the TCSs these factors are somewhat smaller).
Note that at $T\approx 160~MeV$ the viscosities calculated with the TCS2s are approximately 2 times smaller than the viscosities calculated with the EQCS2s respectively. This reflects the fact that the contribution to the total number of collisions from the inelastic processes is approximately the same as from the elastic plus the quasielastic processes at the freeze-out temperature.
The maximum of the bulk viscosity is, of course, sensitive to the energy dependence of the cross sections, as can be seen from fig. \[BulkAll\]. E. g. if the BB and some MB EQCSs were not fixed, as is described in Sec. \[hardcorsec\], the maximum would be present at $T\approx 190~MeV$. After the fixes the BB and some MB EQCSs and EQCS2s have steeply descending tails, and the maximum shifts towards much higher temperatures. Though for the qualitative analysis the TCSs and the TCS2s are more suitable because at $T\gtrsim 160~MeV$ the inelastic processes make not small contributions. With these cross sections the bulk viscosity has the maximum at $T\approx 190~MeV$. With one constant cross section the maximum is present at $T\approx 200~MeV$ with [@Moroz:2011vn] or without charmed and bottomed particles.
In several papers the viscosities of the hadron gas were studied by the ones of the pion gas. This approximations turn out to be bad while calculating the bulk viscosity[^43]. In fig. \[BulkComparison\] the ratio of the bulk viscosity of the hadron gas to the one of the pion gas, using different cross sections, is shown. With all the used cross sections the deviations are quite large. Purely pion gas implies the ECSs. The bulk viscosity of the hadron gas with the EQCS2s is divided on the one of the pion gas with the EQCS. This ratio reaches 122 at $T=270~MeV$. Considering a closer to the pion gas approximation, excluding the resonances in the hadron gas, this ratio becomes $2.3-2.6$ times smaller at $T=120-140~MeV$. The same factor is equal to $2.0-2.1$ for the used TCS2s and the TCSs for the hadron gas and the pion gas correspondingly. Though the resonances should not be excluded. At the same time the corresponding ratios of the particle number densities and the energy densities at $T=140~MeV$ are approximately equal to 2 and 3 respectively. Note that the ratios of the viscosities are less sensitive to the corrections discussed in Sec. \[condappl\] than the viscosities themselves. The ratio of the shear viscosity of the hadron gas to the one of the pion gas is not larger than 1.6, as can be seen in fig. \[ShearComparison\]. It can even be somewhat smaller than 1 if the TCS2s and the TCS2 are used. This seems to be because the shear viscosity is not much sensitive to the mass spectrum as the bulk viscosity, and the contributions from the hadrons other than pions at high temperatures (where pion numbers don’t dominate) come with somewhat larger (on average) cross sections.
Some simplified explanations of the enlarged bulk viscosity in the hadron gas (to some extent) and the position of its maximum can be made. The bulk viscosity is sensitive to particle’s masses $m \sim
T$, and the hadron gas mass spectrum provides such masses at different temperatures. For a fixed $m/T$ and approximately constant cross section the bulk viscosity grows with the temperature, as can be seen from the formula (\[xi\]), using it as an estimate. Particles with very large masses have relatively small number densities because of the exponential suppression $e^{-m/T}$ so that the maximum is set by particles with not the largest masses at $T \approx 190~MeV$, and a relatively slow further descending follows at higher temperatures.
The ratio of the shear viscosity to the entropy density $\eta/s$ and the ratio of the bulk viscosity to the entropy density $\xi/s$ in the hadron gas are shown in fig. \[EtasXis\]. The EQCS2s are used. As long as the maximum of the bulk viscosity is not sharp, the ratio $\xi/s$ doesn’t have a maximum and is a descending function of the temperature. The entropy density is calculated by the formula (\[entrden\]) using the ideal gas formulas in the (\[ignk\]) and the (\[assign2\]). The ratio of the bulk viscosity to the shear viscosity with the EQCS2s is shown in fig. \[EtaXi\].
The dependencies from the temperature of the $\eta/s$ and the $\xi/s$ calculated along the chemical freeze-out line are found too and are depicted in fig. \[EtasXisFreezeout\]. The EQCS2s are used. As was discussed in Sec. \[condappl\], the calculations with large chemical potentials may contain large deviations, especially in the bulk viscosity, however, in the hadron gas the contributions from the bosons and the fermions may cancel substantially. The calculations along the chemical freeze-out line could have not small deviations for the bulk viscosity because of the inelastic processes. At the considered collision energies the strange particle numbers are not described well by the equilibrium statistical calculations. It’s expected that this is because they don’t reach the chemical equilibrium before the chemical freeze-out takes place. After the introduction of the strange saturation factors $\gamma_s$ [@Becattini:2005xt] the experimental data gets described better. As long as the considered chemical perturbations are not quite accurate and are not small they are used in a phenomenological way, being inserted into all particle number densities[^44]. Because of all this the calculations along the chemical freeze-out line are less reliable than the ones at zero chemical potentials.
All variables’ values of the chemical freeze-out line, including the strangeness saturation factor $\gamma_s$, are conveniently presented in the [@Gorenstein:2007mw]. The convergence of the calculations (with all the cross section types) is good with the errors less than $4\%$ for both the viscosities. Also it was checked how the results change if the chemical equilibrium is assumed. The entropy density increases up to $36\%$. The shear viscosity increases no more than on $13\%$. The bulk viscosity decreases no more than on $5.1\%$ (though with the ECSs the decrease would be on $44\%$).
Analytical results
====================
The single-component gas \[singcomsec\]
----------------------------------------
In the single-component gas, using one test-function, the matrix equations can be easily solved, and the shear (\[fineta\]) and the bulk (\[finxi2\]) viscosities become (indexes “1” of the particle species are omitted) [$$\begin{aligned}
\label{etasc}
\eta=\frac1{10}\frac{T}{\sigma(T)}\frac{(\gamma^0)^2}{C^{00}},
\end{aligned}$$]{} [$$\begin{aligned}
\label{xisc}
\xi=\frac{T}{\sigma(T)}\frac{(\alpha^2)^2}{A^{22}}.
\end{aligned}$$]{} In this approximation the explicit closed-form (expressed through special and elementary functions) relativistic formulas for the bulk and the shear viscosities were obtained in the [@anderson]. There the parameter ${a=2r}$. In [@groot] (Chap. XI, Sec. 1) they are written through the parameter ${\sigma=2r^2}$.[^45] The results are [$$\begin{aligned}
\label{etaincor}
\eta=\frac{15}{64\pi}\frac{T}{r^2}\frac{z^2K_2^2(z)\hat h^2}
{(5z^2+2)K_2(2z)+(3z^3+49z)K_3(2z)},
\end{aligned}$$]{} [$$\begin{aligned}
\label{xi}
\xi=\frac1{64\pi}\frac{T}{r^2}\frac{z^2K_2^2(z)[(5-3\gamma)\hat h-3\gamma]^2}
{2K_2(2z)+zK_3(2z)},
\end{aligned}$$]{} where ${\gamma=\frac1{c_\upsilon}+1=\frac{z^2+5\hat h-\hat
h^2}{z^2+5\hat h-\hat h^2-1}}$. Though the correct result for the shear viscosity is [$$\begin{aligned}
\label{eta}
\eta=\frac{15}{64\pi}\frac{T}{r^2}\frac{z^2K_2^2(z)\hat h^2}
{(15z^2+2)K_2(2z)+(3z^3+49z)K_3(2z)}.
\end{aligned}$$]{} This result is in agreement with the result in [@prakash; @leeuwen]. To get the (\[eta\]) and the (\[xi\]) the collision brackets in the $C^{00}$ (\[C4ind\]) and the $A^{22}$ (\[A4ind\]) can be taken from Appendix \[appJ\] with ${z_k=z_l=z}$ and the $\gamma^0$ and $\alpha^2$ can be taken from Appendix \[appA\]. In the nonrelativistic limit, ${z \gg 1}$, one gets[^46] [$$\begin{aligned}
\label{nonrel}
\eta=\frac{5}{64\sqrt{\pi}}\frac{T}{r^2}\sqrt{z}\left(1+\frac{25}{16}z^{-1}+...\right),
\end{aligned}$$]{} [$$\begin{aligned}
\xi=\frac{25}{256\sqrt{\pi}}\frac{T}{r^2}z^{-3/2}\left(1-\frac{183}{16}z^{-1}+...\right).
\end{aligned}$$]{} In the ultrarelativistic limit, ${z \ll 1}$, one gets[^47] [$$\begin{aligned}
\eta=\frac{3}{10\pi}\frac{T}{r^2}\left(1+\frac{1}{20}z^2+...\right),
\end{aligned}$$]{} [$$\begin{aligned}
\xi=\frac1{288\pi}\frac{T}{r^2}z^4\left(1+\left(\frac{49}{12}-6\ln2+6\gamma_E\right)z^2+6z^2\ln z
+...\right),
\end{aligned}$$]{} where $\gamma_E$ is the Euler’s constant, ${\gamma_E\approx
0.577}$.
The perturbation of the distribution function $\varphi$ (\[varphi\]) can be found too: [$$\begin{aligned}
\varphi=\frac1{n\sigma(T)}\left(-(A^0+A^1\tau+A^2\tau^2)\nabla_\mu U^\mu+C^0
\overset{\circ}{\overline{\pi^\mu \pi^\nu}}
\overset{\circ}{\overline{\nabla_\mu U_\nu}}\right),
\end{aligned}$$]{} where the $C^0$ is equal to [$$\begin{aligned}
C^0=\frac{15}{64\pi}\frac{\sigma(T)}{r^2}\frac{z^2K_2^2(z)\hat h}
{(15z^2+2)K_2(2z)+(3z^3+49z)K_3(2z)},
\end{aligned}$$]{} and the $A^2$ is equal to [$$\begin{aligned}
A^2=\frac1{64\pi}\frac{\sigma(T)}{r^2}\frac{z^2K_2^2(z)[(5-3\gamma)\hat h-3\gamma]}
{2K_2(2z)+zK_3(2z)}.
\end{aligned}$$]{} The $A^0$ and the $A^1$ are used to satisfy the matching conditions (\[condfit\]) and are equal to [$$\begin{aligned}
A^0=A^2\frac{a^2a^4-(a^3)^2}{\Delta_A}, \quad
A^1=A^2\frac{a^2a^3-a^1a^4}{\Delta_A}, \quad
\Delta_A\equiv a^1 a^3-(a^2)^2,
\end{aligned}$$]{} where the $a^s$ can be found in Appendix \[appA\]. In the nonrelativistic limit ${z\gg 1}$ one has [$$\begin{aligned}
\varphi=\frac{5\pi e^{z-\hat\mu}}{32\sqrt2 T^3 z^2 r^2}
\left(-(\tau^2+2z\tau-z^2)\nabla_\mu U^\mu+
2\overset{\circ}{\overline{\pi^\mu \pi^\nu}}
\overset{\circ}{\overline{\nabla_\mu U_\nu}}\right).
\end{aligned}$$]{} In the ultrarelativistic limit $z\ll 1$ one has [$$\begin{aligned}
\varphi=\frac{\pi e^{-\hat\mu}}{480 T^3 r^2}\left(-5z^2(\tau^2+8\tau-12)\nabla_\mu
U^\mu+36\overset{\circ}{\overline{\pi^\mu \pi^\nu}}
\overset{\circ}{\overline{\nabla_\mu U_\nu}}\right).
\end{aligned}$$]{} Note that although the shear viscosity diverges for ${T\rightarrow
\infty}$ the perturbative expansion over the gradients does not break down because the $\varphi$ does not diverge (it tends to zero, conversely).
The phenomenological formula, coming from the momentum transfer considerations in the kinetic-molecular theory, for the shear viscosity is ${\eta_{ph}\propto l n \langle {\left\vert\vec p\right\vert} \rangle}$ (with the coefficient of proportionality of order 1), where $\langle {\left\vert\vec p\right\vert} \rangle$ is the average relativistic momentum (\[avmom\]), $l$ is the mean free path. It gives the correct leading $m$ and $T$ parameter dependence of the (\[eta\]) with a quite precise coefficient[^48]. The mean free path can be estimated as ${l\approx 1/(\sigma_{tot}n)}$ (see Appendix \[appmfp\]). Choosing the coefficient of proportionality to match the nonrelativistic limit one gets [@Gorenstein:2007mw] [$$\begin{aligned}
\label{etaph}
\eta_{ph}=\frac{5}{64\sqrt{\pi}}\frac{\sqrt{mT}}{r^2}\frac{K_{5/2}(m/T)}{K_2(m/T)}.
\end{aligned}$$]{} If the bulk viscosity is expressed as ${\xi_{ph}\propto l n
\langle {\left\vert\vec p\right\vert} \rangle}$ the coefficient of proportionality is not of order 1. In the nonrelativistic limit it is $25/(512
\sqrt2 z^2)$ and in the ultrarelativistic limit it is $z^4/(864
\pi)$. To reproduce these asymptotical dependencies the bulk viscosity should be proportional to the second power of the averaged product of the source term $\hat Q$ and the $\tau$ that is to the $(\alpha^2)^2$.
If a system has no charges, then terms proportional to the $\tau_k$ in the (\[Qsource\]) are absent, and the $R$ quantity gets another form. This results in quite different values of the $\alpha_k^r$. In particular, for the single-component gas in the case ${z\gg 1}$ one gets [$$\begin{aligned}
\label{alfrac1}
\frac{(\alpha^2)^2|_{q_{11}=0}}{(\alpha^2)^2|_{q_{11}=1}}=\frac{4 z^4}{25}+...,
\end{aligned}$$]{} and in the case ${z\ll 1}$ one gets [$$\begin{aligned}
\label{alfrac2}
\frac{(\alpha^2)^2|_{q_{11}=0}}{(\alpha^2)^2|_{q_{11}=1}}=4+... .
\end{aligned}$$]{} In both cases these estimates suppose enhancement of the bulk viscosity (\[xisc\]) if the number-changing processes are not negligible.
Although constant cross sections are the most simple and the most universal ones in approximate calculations, let’s write down also formulas for some other simple energy dependencies of cross sections. Using the collision bracket from Appendix \[appJ\], one gets for the cross section $\sigma s/(2 m)^2 = \sigma
\upsilon^2/(2 z)^2$ ($\sigma$ is just a positive dimensional constant here) [$$\begin{aligned}
\label{eta1}
\eta_1=\frac{15 T}{32 \pi\sigma}\frac{\hat h^2 z^3 K_2^2(z)}
{9 z (3 z^2+34) K_2(2 z)+(3 z^4+157 z^2+920) K_3(2 z)},
\end{aligned}$$]{} [$$\begin{aligned}
\label{xi1}
\xi_1=\frac{T}{32 \pi \sigma}\frac{z^3 [(5-3 \gamma) \hat h-3 \gamma]^2 K_2^2(z)}
{(z^2+20) K_3(2 z)+6 z K_2(2 z)},
\end{aligned}$$]{} and for the cross section $\sigma (2m)^2/s = \sigma (2z)^2
/\upsilon^2$: [$$\begin{aligned}
\label{eta2}
\eta_2=\frac{15 T}{32 \pi \sigma}\frac{\hat h^2 z^2 K_2^2(z)}
{(3 z^2-2) K_2(2 z)+z (3 z^2+1) K_3(2 z)},
\end{aligned}$$]{} [$$\begin{aligned}
\label{xi2}
\xi_2=\frac{T}{32 \pi \sigma}\frac{z^2 [(5-3 \gamma) \hat h-3 \gamma]^2 K_2^2(z)}
{(z K_3(2 z)-2 K_2(2 z))}.
\end{aligned}$$]{} Using the low-energy current algebra isospin averaged $\pi\pi$ differential cross section [@prakash; @Weinberg:1966kf] [$$\begin{aligned}
\label{sigmaca}
\sigma_{CA}=\frac13\frac{s}{64\pi^2f_\pi^4}\left[3-8\frac{m_\pi^2}{s}+7\frac{m_\pi^4}{s^2}
+\left(1-8\frac{m_\pi^2}{s}+16\frac{m_\pi^4}{s^2}\right)\cos^2\Theta\right],
\end{aligned}$$]{} ($f_\pi=93~MeV$ is the pion decay constant) and treating all pions as identical particles at zero chemical potentials, one gets the formulas [$$\begin{aligned}
\label{etaca}
\eta_{CA}=\frac{360 \pi f_\pi^4}{T}\frac{\hat h^2 z K_2^2(z)}
{9 z (93 z^2+1730) K_2(2 z)+(69 z^4+6167 z^2+47104) K_3(2 z)},
\end{aligned}$$]{} [$$\begin{aligned}
\label{xica}
\xi_{CA}=\frac{24 \pi f_\pi^4}{T}\frac{z [(5-3 \gamma) \hat h-3 \gamma]^2 K_2^2(z)}
{(23 z^2+1024) K_3(2 z)+210 z K_2(2 z)},
\end{aligned}$$]{} which are exactly 2 times larger than the 1-st order calculations in the [@prakash][^49]. Taking the scattering angle averaged cross section instead of the (\[sigmaca\]), one would get approximately the same result for the viscosities (errors are not more than $4\%$). In the low temperature limit, $z\gg1$, one gets the following expansions for the shear viscosities (\[eta1\]), (\[eta2\]), (\[etaca\]): [$$\begin{aligned}
\eta_1=\frac{5}{32 \sqrt{\pi}}\frac{T}{\sigma}\sqrt{z}\left(1-\frac{39}{16}z^{-1}+...\right),
\end{aligned}$$]{} [$$\begin{aligned}
\eta_2=\frac{5}{32 \sqrt{\pi}}\frac{T}{\sigma}\sqrt{z}\left(1+\frac{89}{16}z^{-1}+...\right),
\end{aligned}$$]{} [$$\begin{aligned}
\eta_{CA}=\frac{120\pi^{3/2}}{23}\frac{f_\pi^4}{T}z^{-3/2}\left(1-\frac{2049}{368}z^{-1}+...\right),
\end{aligned}$$]{} and in the high temperature limit, $z\ll 1$, respectively: [$$\begin{aligned}
\eta_1=\frac{3}{92 \pi}\frac{T}{\sigma}z^2\left(1-\frac2{23} z^2+...\right),
\end{aligned}$$]{} [$$\begin{aligned}
\eta_2=\frac{6}{\pi}\frac{T}{\sigma} z^{-2}\left(1+\frac1{20}(9 - 4 \gamma_E)z^2
-\frac1{5}z^2\ln z+...\right),
\end{aligned}$$]{} [$$\begin{aligned}
\eta_{CA}=\frac{45 \pi}{92}\frac{f_\pi^4}{T}\left(1-\frac{17}{368}z^2+...\right).
\end{aligned}$$]{} In the low temperature limit, $z\gg1$, one gets the following expansions for the bulk viscosities (\[xi1\]), (\[xi2\]), (\[xica\]): [$$\begin{aligned}
\xi_1=\frac{25}{128 \sqrt{\pi}}\frac{T}{\sigma}z^{-3/2}\left(1-\frac{247}{16}z^{-1}+...\right),
\end{aligned}$$]{} [$$\begin{aligned}
\xi_2=\frac{25}{128 \sqrt{\pi}}\frac{T}{\sigma}z^{-3/2}\left(1-\frac{119}{16}z^{-1}+...\right),
\end{aligned}$$]{} [$$\begin{aligned}
\xi_{CA}=\frac{150 \pi^{3/2}}{23}\frac{f_\pi^4}{T}z^{-7/2}\left(1-\frac{6833}{368}z^{-1}+...\right),
\end{aligned}$$]{} and in the high temperature limit, $z\ll 1$, respectively: [$$\begin{aligned}
\xi_1=\frac{1}{1440 \pi}\frac{T}{\sigma}z^6\left(1+\frac1{30}(109 + 180 \gamma_E - 180 \ln 2)z^2
+6 z^2\ln z...\right),
\end{aligned}$$]{} [$$\begin{aligned}
\xi_2=\frac{1}{36 \pi}\frac{T}{\sigma}z^2\left(1+\frac13(13 + 12 \gamma_E - 18 \ln 2)z^2
+4 z^2\ln z+...\right),
\end{aligned}$$]{} [$$\begin{aligned}
\xi_{CA}=\frac{\pi}{96}\frac{f_\pi^4}{T}z^4\left(1+\frac1{24}(89 + 144 \gamma_E - 144 \ln 2)z^2
+6 z^2\ln z+...\right).
\end{aligned}$$]{} Let’s also write down the viscosities with the cross section $\sigma [1+c_1 \upsilon^2/(2 z)^2+c_2 (2 z)^2/\upsilon^2]$, unifying the previously considered ones: $$\begin{aligned}
\nonumber \eta_3&=&\frac{15}{32 \pi \sigma} T \hat h^2 z^3 K^2_2(z)/
[z (2 + 306 c_1 - 2 c_2 + 3 (5 + 9 c_1 + c_2) z^2) K_2(2 z)\\
&+& (920 c_1 + (49 + 157 c_1 + c_2) z^2 + 3 (1 + c_1 + c_2) z^4) K_3(2 z)],
\end{aligned}$$ [$$\begin{aligned}
\xi_3=\frac{T}{32 \pi \sigma} \frac{z^3 [(5-3 \gamma) \hat h-3 \gamma]^2 K^2_2(z)}
{2 (1 + 3 c_1 - c_2) z K_2(2 z) + (20 c_1 + (1 + c_1 + c_2) z^2) K_3(2 z)},
\end{aligned}$$]{} where $c_1$ and $c_2$ are some dimensionless coefficients such that $c_1 + c_2 + 1 \geq 0$ and $c_1\geq 0$ to make the cross section non-negative. The last formulas may have badly convergent expansions over the $z$ and the $z^{-1}$ for some values of the $c_1$ and the $c_2$ so that they ain’t expanded.
The binary mixture \[binmixsec\]
---------------------------------
The mixture of two species with masses $m_1$, $m_2$ and the different classical elastic differential constant cross sections $\sigma^{cl}_{11}$, ${\sigma^{cl}_{12} = \sigma^{cl}_{21}}$, $\sigma^{cl}_{22}$ is considered in this section. Using the (\[fineta\]) with ${n_2=0}$ and solving the matrix equation (\[etamatreq\]) one gets for the shear viscosity [$$\begin{aligned}
\eta=\frac{T}{10\sigma(T)}\frac1{\Delta_\eta}[(x_{1'}\gamma_1^0)^2C_{2'2'}^{00}-
2x_{1'}x_{2'}\gamma_1^0\gamma_2^0C_{1'2'}^{00}+(x_{2'}\gamma_2^0)^2C_{1'1'}^{00}],
\end{aligned}$$]{} where ${\Delta_\eta = C_{1'1'}^{00}C_{2'2'}^{00} -
(C_{1'2'}^{00})^2}$. The collision brackets for the $C_{k'l'}^{00}$ (\[C4ind\]) can be found in Appendix \[appJ\] and the $\gamma_k^0$ can be found in Appendix \[appA\].
In the important limiting case when one mass is large ${z_2\gg 1}$ ($g_2$ and $\hat \mu_2$ are finite so that ${x_{2'}\ll 1}$) and another mass is finite one can perform asymptotic expansion of the special functions. Then one has ${x_{1'} \propto O(1)}$, ${\gamma_1^0 \propto O(1)}$, ${x_{2'} \propto
O(e^{-z_2}z_2^{3/2})}$, ${\gamma_2^0 \propto O(z_2)}$. The collisions of light and heavy particles dominate over the collisions of heavy and heavy particles in the $C_{2'2'}^{00}$, and one has ${[\overset{\circ}{ \overline{\pi^{\mu} \pi^{\nu}} },
\overset{\circ}{ \overline{\pi_{\mu} \pi_{\nu}} }]_{21} \propto
O(z_2)}$, ${C_{2'2'}^{00} \propto O(e^{-z_2}z_2^{5/2})}$. In the $C_{1'1'}^{00}$ the collisions of light and light particles dominate, and one gets ${C_{1'1'}^{00} \propto O(1)}$. And ${[\overset{\circ}{ \overline{\pi^{\mu} \pi^{\nu}} },
\overset{\circ}{ \overline{\pi_{1\mu} \pi_{1\nu}} }]_{12} \propto
O(1)}$, ${C_{1'2'}^{00} \propto O(e^{-z_2}z_2^{3/2})}$. In the shear viscosity the first nonvanishing contribution is the single-component shear viscosity (\[eta\]), where one should take ${r^2=\sigma^{cl}_{11}}$ and ${z=z_1}$. The next correction is [$$\begin{aligned}
\Delta \eta=z_2^{5/2}e^{-z_2}\frac{3 Tg_2 e^{z_1-\hat\mu_1+\hat\mu_2}}
{64\sqrt{2\pi}(3+3z_1+z_1^2)g_1\sigma^{cl}_{12}}.
\end{aligned}$$]{} The approximate formula [@Gorenstein:2007mw] [$$\begin{aligned}
\eta=\sum_k\eta_kx_k,
\end{aligned}$$]{} where $\eta_k$ is given by the (\[eta\]) or the (\[etaph\]) with mass $m_k$ and cross section $\sigma^{cl}_{kk}$, would give somewhat different heavy mass power dependence $O(e^{-z_2}z_2^2)$.
Using the (\[finxi2\]) with ${n_1=1}$ and solving the matrix equation (\[ximatreq2\]) one gets for the bulk viscosity [$$\begin{aligned}
\xi=\frac{T}{\sigma(T)}\frac{(x_{2'}\alpha_2^1)^2}{A_{2'2'}^{11}}=
\frac{T}{\sigma(T)}\frac{x_{1'}x_{2'}\alpha_1^1\alpha_2^1}{A_{1'2'}^{11}}.
\end{aligned}$$]{} Using the definition of the $A_{2'2'}^{11}$ (\[A4ind\]) and the fact ${[\tau,\tau_1]_{kl} + [\tau,\tau]_{kl} = 0}$ (\[br211\]) one gets ${A_{2'2'}^{11} = x_{1'}x_{2'}[\tau,\tau]_{12}}$. Using the (\[br2pos\]) one gets ${[\tau,\tau]_{12} > 0}$. Then using ${x_{1'}\alpha_1^1 + x_{2'}\alpha_2^1 = 0}$, coming from the (\[lhsencons\]), the bulk viscosity can be rewritten as [$$\begin{aligned}
\xi=\frac{T}{\sigma(T)}\frac{x_{2'}(\alpha_2^1)^2}{x_{1'}[\tau,\tau]_{12}}
=\frac{T}{\sigma(T)}\frac{x_{1'}(\alpha_1^1)^2}{x_{2'}[\tau,\tau]_{12}}>0.
\end{aligned}$$]{} The collision bracket $[\tau,\tau]_{12}$ can be found in Appendix \[appJ\], and the $\alpha_k^1$ can be found in Appendix \[appA\].
In the limiting case ${z_2\gg 1}$ one has ${x_{1'} \propto O(1)}$, ${x_{2'} \propto O(e^{-z_2}z_2^{3/2})}$, ${\alpha_1^1 \propto
O(e^{-z_2}z_2^{3/2})}$, ${\alpha_2^1 \propto O(1)}$, ${A_{22}^{11}
\propto A_{12}^{11} \propto O(e^{-z_2}z_2^{1/2})}$, ${[\tau,\tau]_{12} \propto O(z_2^{-1})}$. Then for the bulk viscosity one gets [$$\begin{aligned}
\xi=e^{-z_2}z_2^{5/2}\frac{g_2 T e^{-\hat\mu_1+\hat\mu_2+z_1}[2 z_1^2-5-2\hat h_1^2
+10 \hat h_1]^2}{128 \sqrt{2 \pi } g_1 \sigma^{cl}_{12}(z_1^2+3 z_1+3)[z_1^2-1-\hat h_1^2+5 \hat h_1]^2}+....
\end{aligned}$$]{}
Concluding remarks
==================
The shear and the bulk viscosities of the hadron gas and the pion gas were calculated using the UrQMD cross sections.
The physics of the bulk viscosity is very interesting. In particular, in mixtures it can strongly depend on the mass spectrum. For instance, at the temperature ${T=120~MeV}$ (${140~MeV}$) and zero chemical potentials the bulk viscosity of the hadron gas is larger in 8.6-15.6 (14.6-40) times than the bulk viscosity of the pion gas. The used UrQMD cross sections have allowed to perform this comparison more accurately.
Also the bulk viscosity can strongly depend on the quantum statistics corrections, the equation of state and the inelastic processes, which can be explained by nontrivial form of its source term(s). It’s a future task to find the universal and optimal criterion for switching on/off the inelastic processes. Numerical calculations of the bulk viscosity along and around the chemical freeze-out line which don’t involve the approximations of conserved or not conserved particle numbers (like in the [@Muronga:2003tb], though the procedure of collisions of particles introduce some errors itself, which should be kept in mind) are desirable to get more accurate values of the bulk viscosity at these points. This is also needed for a better understanding of the chemical freeze-out itself and connected with it problems and to get a better description of the deviations from the chemical equilibrium.
The transport coefficients are connected with fluctuations through the fluctuation-dissipation theorem. Because of strong dependence of the bulk viscosity on the equation of state one might expect to find its maximum at or near a phase transition point. According to lattice calculations at zero chemical potentials [@Aoki:2006we], the QCD phase transition is an analytical crossover. Calculations of the Polyakov loop [@Hidaka:2008dr], the ’t Hooft loop [@Dumitru:2010mj] and some other calculations [@Ratti:2011au] suggest that hadronic degrees of freedom survive partially at some temperatures above the critical one. In the qualitative calculations for the hadron gas with the UrQMD total cross sections the maximum of the bulk viscosity has been found at $T\approx 190~MeV$. This value fits in the transition temperature range $T_c=185-195~MeV$ found from lattice calculations by the hotQCD collaboration, but calculations of the Wuppertal-Budapest collaboration with physical quark masses give the region $T_c=150-170~MeV$ [@Borsanyi:2010bp]. According to the Wuppertal-Budapest collaboration calculations of thermodynamic functions with physical quark masses [@Borsanyi:2010cj], the scaled trace of the energy-momentum tensor has its peak at $T\approx 190~MeV$. In some other lattice calculations with somewhat different quark masses [@Cheng:2009zi; @Petreczky:2009ey] this peak is somewhat sharper and still is present at $T\approx 190~MeV$. According to the lattice calculations in the [@Borsanyi:2010cj], the squared speed of sound in the charge-neutral hadron gas (implying equal to zero and not developing chemical potentials) has it’s minimum at $T\approx
150~MeV$, unlike the ideal gas calculations ($T\approx 190~MeV$) or the ones in the [@toneev] ($T\approx 180~MeV$). So that one could expect to find the maximum of the bulk viscosity somewhere in the temperature range $T=150-190~MeV$, presumably closer to the lowest bound. This range covers the critical temperature values found from the lattice calculations.
The shear viscosity is less dependent on the mass spectrum, the quantum statistics corrections, the equation of state and the inelastic processes. This may be explained by its more trivial source term(s).
The author would like to thank to Prof. Zoltan Fodor for providing important references. Also the author is grateful to Prof. Shin Muroya and Dr. Juan Torres-Rincon for discussions.
The values of the $\alpha^r_k$, $\gamma_k^r$ and $a_k^r$ \[appA\]
=================================================================
Their definitions are [$$\begin{aligned}
\label{alphagammaadef}
\alpha_k^r\equiv(\hat Q_k,\tau_k^r), \quad \gamma_k^r\equiv(\tau_k^r
\overset{\circ}{\overline{\pi_k^\mu\pi_k^\nu}},
\overset{\circ}{\overline{\pi_{k\mu}\pi_{k\nu}}}), \quad
a^s_k\equiv(1,\tau_k^s)_k,
\end{aligned}$$]{} where the round brackets are [$$\begin{aligned}
(F,G)_k\equiv\frac1{4\pi z_k^2K_2(z_k)T^2}\int_{p_k} F(p_k)G(p_k)e^{-\tau_k}.
\end{aligned}$$]{} Then one can rewrite the $a_k^s$ as [$$\begin{aligned}
\label{aksdef}
a_k^s=\frac1{z_k^2K_2(z_k)}\int_{z_k}^\infty d\tau (\tau^2-z_k^2)^{1/2}\tau^s e^{-\tau}.
\end{aligned}$$]{} There is a recurrence relation for the $a^s_k$: [$$\begin{aligned}
a^s_k=(s+1)a^{s-1}_k+z_k^2 a^{s-2}_k-(s-2)z_k^2 a^{s-3}_k.
\end{aligned}$$]{} It can be derived from the (\[aksdef\]) written in the form [$$\begin{aligned}
a_k^s=\frac1{z_k^2K_2(z_k)}\int_{z_k}^\infty d\tau
(\tau^2-z_k^2)^{3/2}\tau^{s-2} e^{-\tau}+z_k^2 a_k^{s-2}.
\end{aligned}$$]{} Then after integration by parts the recurrence relation follow. Some values of the $a_k^s$ are [$$\begin{aligned}
a^0_k=\frac1{z_k^2}(\hat h_k-4),
\end{aligned}$$]{} [$$\begin{aligned}
a^1_k=1,
\end{aligned}$$]{} [$$\begin{aligned}
a^2_k=\hat h_k-1,
\end{aligned}$$]{} [$$\begin{aligned}
a^3_k=3\hat h_k+z_k^2,
\end{aligned}$$]{} [$$\begin{aligned}
a^4_k=(15+z_k^2)\hat h_k+2z_k^2,
\end{aligned}$$]{} [$$\begin{aligned}
a^5_k=6(15+z_k^2)\hat h_k+z_k^2(15+z_k^2),
\end{aligned}$$]{} [$$\begin{aligned}
a^6_k=(630+45z_k^2+z_k^4)\hat h_k+5z_k^2(21+z_k^2).
\end{aligned}$$]{} The $\alpha_k^r$ can be expressed through the $a^r_k$ after the integration of the (\[Qsource\]) (or the (\[Qsource2\]) if only the elastic collisions are considered) over the momentum, using the definition (\[alphagammaadef\]). For systems with only the elastic collisions some values of the $\alpha_k^r$ are written below, in agreement with [@groot] (Chap. VI, Sec. 3): [$$\begin{aligned}
\label{elal0}
\alpha_k^0=0,
\end{aligned}$$]{} [$$\begin{aligned}
\alpha_k^1=\frac{2(c_\upsilon-9)\hat h_k+3\hat h_k^2-3z_k^2}{c_\upsilon}
=\frac{\gamma_k-\gamma}{\gamma_k-1},
\end{aligned}$$]{} [$$\begin{aligned}
\alpha_k^2=2\hat h_k-3\frac{c_{\upsilon,k}}{c_\upsilon}-3\frac{\hat
h_k+1}{c_\upsilon}=(5-3\gamma)\hat
h_k-3\gamma_k\frac{\gamma-1}{\gamma_k-1},
\end{aligned}$$]{} where the assignments $\gamma$ and $\gamma_k$ from [@groot] are used. They can be expressed through the $c_\upsilon$ and the $c_{\upsilon,k}$, defined in the (\[elquant\]), as [$$\begin{aligned}
\gamma\equiv \frac1{c_\upsilon}+1, \quad
\gamma_k\equiv\frac1{c_{\upsilon,k}}+1.
\end{aligned}$$]{} The $\gamma_k^r$ can be rewritten as [$$\begin{aligned}
\gamma_k^r=\frac23\frac1{z_k^2K_2(z_k)}\int_{z_k}^\infty d\tau
(\tau^2-z_k^2)^{5/2}\tau^r e^{-\tau}.
\end{aligned}$$]{} Then it can be rewritten through the $a_k^r$: [$$\begin{aligned}
\gamma_k^r=\frac23(a^{r+4}-2z_k^2a^{r+2}+z_k^4a^r).
\end{aligned}$$]{} Some values of the $\gamma_k^r$ are [$$\begin{aligned}
\gamma_k^0=10\hat h_k,
\end{aligned}$$]{} [$$\begin{aligned}
\gamma_k^1=10(6\hat h_k+z_k^2),
\end{aligned}$$]{} [$$\begin{aligned}
\gamma_k^2=10(7z_k^2+\hat h_k(42+z_k^2)).
\end{aligned}$$]{}
The entropy density formula \[appTherm\]
========================================
The Gibbs’s potential is defined as [$$\begin{aligned}
\label{Phidef}
\Phi(P,T)\equiv E(S,V)-ST+PV.
\end{aligned}$$]{} The differential of the energy is defined as [$$\begin{aligned}
dE=TdS-PdV+\sum_k\mu_kdN_k=TdS-PdV+\sum_a\mu_adN_a,
\end{aligned}$$]{} where it is rewritten through the independent chemical potentials and the particle net charges $N_a$. Then the differential of the $\Phi$ reads: [$$\begin{aligned}
\label{dPhi}
d\Phi=-SdT+VdP+\sum_a \mu_a dN_a.
\end{aligned}$$]{} Because the $\Phi$ is the function of the intrinsic variables $P$, $T$ and the extrinsic $N_a$ the only possible form of it in the thermodynamic limit is [$$\begin{aligned}
\label{Phi}
\Phi=\sum_a N_a \phi_a(P,T),
\end{aligned}$$]{} where $\phi_a$ are unknown functions. Then from the (\[dPhi\]) one gets ${\frac{{\partial}\Phi}{{\partial}N_a} = \mu_a}$, which means that ${\phi_a = \mu_a}$. Then substituting the (\[Phi\]) into the (\[Phidef\]) one gets the relation [$$\begin{aligned}
\sum_a N_a \mu_a(P,T)=E(S,V)-ST+PV.
\end{aligned}$$]{} Being written for local infinitesimal volume it transforms into the expression [$$\begin{aligned}
\sum_a n_a \mu_a=\epsilon-sT+P,
\end{aligned}$$]{} from where the entropy density $s$ can be found: [$$\begin{aligned}
\label{entrden}
s=\frac{\epsilon+P}{T}-\sum_a n_a \hat \mu_a.
\end{aligned}$$]{}
The calculation of the collision brackets \[appJ\]
==================================================
The momentum parametrization and the most transformations of the 12-dimensional integrals used below are taken from [@groot] (Chap. XI and XIII). Let’s start from some assignments. The full momentum is [$$\begin{aligned}
P^\mu = p^\mu_k+p^\mu_{1l}={p'}^\mu_k+{p'}^\mu_{1l}={P'}^\mu.
\end{aligned}$$]{} The “relative” momentums before collision $Q^\mu$ and after collision ${Q'}^\mu$ are defined as [$$\begin{aligned}
Q^\mu=\Delta_P^{\mu\nu}(p_{k\nu}-p_{1l\nu}), \quad
{Q'}^\mu=\Delta_P^{\mu\nu}({p'}_{k\nu}-{p'}_{1l\nu}),
\end{aligned}$$]{} with the assignment [$$\begin{aligned}
\Delta_P^{\mu\nu}=g^{\mu\nu}-\frac{P^\mu P^\nu}{P^2},
\end{aligned}$$]{} where $P^2\equiv P^\mu P_\mu$. The covariant cosine of the scattering angle can be expressed through the $Q^\mu$ and the ${Q'}^\mu$ as [$$\begin{aligned}
\cos\Theta=-\frac{Q \cdot Q'}{\sqrt{-Q^2}\sqrt{-{Q'}^2}},
\end{aligned}$$]{} where $\cdot$ denotes convolution of 4-vectors. One also has ${Q^2
= {Q'}^2}$ and [$$\begin{aligned}
\label{Q2}
Q^2=4m_k^2-(1+\alpha_{kl})^2P^2=-\left(P^2-M_{kl}^2\right)\left[1-\frac{M_{kl}^2}{P^2}
\left(1-\frac{4\mu_{kl}}{M_{kl}}\right)\right],
\end{aligned}$$]{} where [$$\begin{aligned}
\label{alphakl}
M_{kl}\equiv m_k+m_l, \quad \mu_{kl}\equiv \frac{m_k
m_l}{m_k+m_l}, \quad \alpha_{kl}\equiv\frac{m_k^2-m_l^2}{P^2}=\mathrm{sign}(m_k-m_l)
\sqrt{1-\frac{4\mu_{kl}}{M_{kl}}}\frac{M_{kl}^2}{P^2}.
\end{aligned}$$]{} The function $\mathrm{sign}(x)$ is equal to 1, if ${x>0}$ and equal to $-1$, if ${x<0}$. Note that not all $P^\mu$ and $Q^\mu$ are independent: [$$\begin{aligned}
P^\mu Q_\mu=0, \quad P^\mu {Q'}_\mu=0.
\end{aligned}$$]{} To come from the variables ${(p_k^\mu,p_{1l}^\mu)}$ to the variables ${(P^\mu,Q^\mu)}$ in the measure of integration first one has to come from the ${(p_k^\mu,p_{1l}^\mu)}$ to the ${(p_k^\mu+p_{1l}^\mu,p_k^\mu-p_{1l}^\mu)}$ (the determinant is equal to $16$) and then shift the relative momentum $p_k^\mu-p_{1l}^\mu$ on the $\alpha_{kl}P^\mu$. Analogically for the ${({p'}_k^\mu,{p'}_{1l}^\mu)}$ and the ${({P'}^\mu,{Q'}^\mu)}$. The inverse relations for the $p_k^\mu,p_{1l}^\mu,{p'}_k^\mu,{p'}_{1l}^\mu$ through the $P^\mu,Q^\mu,{Q'}^\mu$ are [$$\begin{aligned}
p_k^\mu=\frac12(1+\alpha_{kl})P^\mu+\frac12{Q}^\mu,
\end{aligned}$$]{} [$$\begin{aligned}
p_{1l}^\mu=\frac12(1-\alpha_{kl})P^\mu-\frac12{Q}^\mu,
\end{aligned}$$]{} [$$\begin{aligned}
{p'}_k^\mu=\frac12(1+\alpha_{kl})P^\mu+\frac12{Q'}^\mu,
\end{aligned}$$]{} [$$\begin{aligned}
{p'}_{1l}^\mu=\frac12(1-\alpha_{kl})P^\mu-\frac12{Q'}^\mu.
\end{aligned}$$]{}
There is a need to calculate the following integrals $$\begin{aligned}
\label{Jint0}
\nonumber &~& J_{kl}^{(a,b,d,e,f|q,r)} \equiv \frac{\gamma_{kl}}{T^6(4\pi)^2z_k^2z_l^2K_2(z_k)K_2(z_l)
\sigma(T)} \int_{p_k,p_{1l},{p'}_k,{p'}_{1l}} e^{-P\cdot U/T}(1+\alpha_{kl})^q\\
&~& \times(1-\alpha_{kl})^r \left(\frac{P^2}{T^2}\right)^a
\left(\frac{P\cdot U}{T}\right)^b\left(\frac{Q\cdot U}{T}\right)^d\left(\frac{Q'\cdot U}{T}\right)^e
\left(\frac{-Q\cdot {Q'}}{T^2}\right)^f W_{kl}.
\end{aligned}$$ Let’s start from the case of constant cross sections. After nontrivial transformations, described in more details in [@groot], one arrives at $$\begin{aligned}
\label{Jint}
\nonumber &~&J_{kl}^{(a,b,d,e,f|q,r)}=\frac{\pi(d+e+1)!!\sigma^{(d,e,f)}_{1kl}}{z_k^2z_l^2K_2(z_k)K_2(z_l)}
\sum_{q_1=0}^q\sum_{r_1=0}^r\sum_{k_2=0}^{\frac{d+e}2+f+1}\sum_{k_3=0}^{\frac{d+e}2+f+1}
\sum_{h=0}^{[b/2]}(z_k+z_l)^{2(q_1+r_1+k_2+k_3)} \\
&~&\times\left(\frac{z_k-z_l}{z_k+z_l}\right)^{q_1+r_1+2k_3}
(-1)^{r_1+k_2+k_3+h}(2h-1)!! \binom{b}{2h}\binom{q}{q_1}\binom{r}{r_1}
\binom{\frac{d+e}2+f+1}{k_2}\\
\nonumber &~& \times\binom{\frac{d+e}2+f+1}{k_3} I\left(2(a+f-q_1-r_1-k_2-k_3)+3,
b+\frac{d+e}2-h+1,z_k+z_l\right),
\end{aligned}$$ where [$$\begin{aligned}
\sigma_{1kl}^{(d,e,f)}=\frac{\sigma^{cl}_{kl}}{\sigma(T)}
\sum_{g=0}^{\min(d,e)} \sigma^{(f,g)} K(d,e,g),
\end{aligned}$$]{} where ${\sigma^{cl}_{kl} = \gamma_{kl}\sigma_{kl}}$ is the classical elastic differential constant cross section. The $\sigma^{(f,g)}$ is equal to the real, nonzero and non-diverging value (for any non-negative integer $g$) [$$\begin{aligned}
\label{sigmafg}
\sigma^{(f,g)}=\frac{2g+1}{2}\int_{-1}^1 dx x^f P_g(x)=(2g+1) \frac{f!}{(f-g)!!(f+g+1)!!},
\end{aligned}$$]{} if the difference ${f-g}$ is even and ${g \leq f}$. Above the $P_{\text{g}}(x)$ is the Legendre polynomial. The $K(d,e,g)$ is equal to the real, the nonzero and non-diverging quantity (for any non-negative integer $g$) [$$\begin{aligned}
K(d,e,g)=\frac{d! e!}{(d-g)!! (d+g+1)!! (e-g)!! (e+g+1)!!},
\end{aligned}$$]{} if ${g \leq \min(d,e)}$ and both the ${d-g}$ and the ${e-g}$ are even (which also implies that ${d+e}$ is even). The ${[...]}$ denotes the integer part. The integral $I$ is [$$\begin{aligned}
\label{IIntdef}
I(r,n,x)\equiv x^{r+n+1}\int_1^\infty du u^{r+n}K_n(xu).
\end{aligned}$$]{} Also there is the following frequently used combination of the $J$ integrals [$$\begin{aligned}
{J'}_{kl}^{(a,b,d,e,f|q,r)}\equiv\sum_{u=0}^f (-1)^u\binom{f}{u}
(2z_k)^{2(f-u)}J_{kl}^{(a+k,b,d+e,0,0|q+2u,r)}-J_{kl}^{(a,b,d,e,f|q,r)}.
\end{aligned}$$]{} The first term in the difference is obtained by the replacement of the $Q'$ on the $Q$ everywhere except for the $W_{kl}$. Using this fact, the $J'$ can be rewritten in the form $$\begin{aligned}
\label{Jpint}
\nonumber &~&{J'}_{kl}^{(a,b,d,e,f|q,r)}=\frac{\pi(d+e-1)!!
\sigma^{(d,e,f)}_{kl}}{z_k^2z_l^2K_2(z_k)K_2(z_l)}
\sum_{q_1=0}^q\sum_{r_1=0}^r\sum_{k_2=0}^{\frac{d+e}2+f+1}\sum_{k_3=0}^{\frac{d+e}2+f+1}
\sum_{h=0}^{[b/2]}(z_k+z_l)^{2(q_1+r_1+k_2+k_3)} \\
&~&\times\left(\frac{z_k-z_l}{z_k+z_l}\right)^{q_1+r_1+2k_3}
(-1)^{r_1+k_2+k_3+h}(2h-1)!! \binom{b}{2h}\binom{q}{q_1}\binom{r}{r_1}
\binom{\frac{d+e}2+f+1}{k_2}\\
\nonumber &~& \times\binom{\frac{d+e}2+f+1}{k_3}
I\left(2(a+f-q_1-r_1-k_2-k_3)+3, b+\frac{d+e}2-h+1,z_k+z_l\right),
\end{aligned}$$ where $$\begin{aligned}
\label{sigmadef}
\nonumber \sigma^{(d,e,f)}_{kl}&=&\frac{\sigma^{cl}_{kl}}{\sigma(T)}(d+e+1)\left(K(d+e,0,0)\sigma^{(0,0)}
-\sum_{g=0}^{\min(d,e)}K(d,e,g)\sigma^{(f,g)}\right)\\
&=&\frac{\sigma^{cl}_{kl}}{\sigma(T)}\left(1-(d+e+1)\sum_{g=0}^{\min(d,e)}K(d,e,g)\sigma^{(f,g)}\right).
\end{aligned}$$ There is a recurrence relation for the integral $I$ (\[IIntdef\]) [@groot] (Chap. XI, Sec. 1): [$$\begin{aligned}
\label{recrel}
I(r,n,x)=(r-1)(r+2n-1)I(r-2,n,x)+(r-1)x^{r+n-1}K_n(x)+x^{r+n}K_{n+1}(x).
\end{aligned}$$]{} For the calculations one needs only the integrals $I(r,n,x)$ with the positive values of the $n$ and the odd values of the $r$. If ${r \geq -2n+1}$, the $I$ integrals can be expressed through the Bessel functions $K_n(x)$, using the (\[recrel\]), when ${r=1}$ or ${r=-2n+1}$. Then using the recurrence relation for the $K_n(x)$ [@luke] [$$\begin{aligned}
\label{Krecrel}
K_{n+1}(x)=K_{n-1}(x)+\frac{2n}{x}K_n(x),
\end{aligned}$$]{} the final result can be expressed through a couple of Bessel functions. If ${r\leq-2n-1}$, then the recurrence relation (\[recrel\]) becomes singular if one tries to express the $I(r,n,x)$ through the $I(-2n+1,n,x)$. Using the (\[recrel\]), the $I$ integrals with ${r \leq -2n-1}$ can be expressed through the integrals $G(n,x)$ [$$\begin{aligned}
\label{Gdef}
G(n,x)\equiv I(-2n-1,n,x)=x^{-n}\int_1^\infty du u^{-n-1}K_n(xu).
\end{aligned}$$]{} There is a recurrence relation for the $G(n,x)$: [$$\begin{aligned}
\label{Grecrel}
G(n,x)=-\frac1{2n}(G(n-1,x)-x^{-n} K_n(x)).
\end{aligned}$$]{} It can be easily proved by the integration by parts of the (\[Gdef\]) and using the following relation for the $K_n(x)$ [@luke] [$$\begin{aligned}
\label{dKdx}
\frac{{\partial}}{{\partial}x}K_n(x)=-\frac{n}{x} K_n(x)-K_{n-1}(x).
\end{aligned}$$]{} It is found that collision brackets have the simplest form if they are expressed through $G(n,x)$ with ${n=3}$ or ${n=2}$ and the Bessel functions $K_3(x)$ and $K_2(x)$ or $K_2(x)$ and $K_1(x)$. It was chosen to take ${G(x) \equiv G(3,x)}$ and $K_3(x)$, $K_2(x)$. The $G(x)$ can be expressed through the Meijer function [@meijer] [$$\begin{aligned}
G(x)=\frac{1}{32}
G_{1,3}^{3,0}\left((x/2)^2\left|
\begin{array}{c}
1 \\
-3,0,0
\end{array}\right.
\right).
\end{aligned}$$]{} The needed scalar collision brackets can be expressed through the $J'$ as [$$\begin{aligned}
[\tau^r,\tau^s_1]_{kl}=\frac1{2^{r+s}}\sum_{u=1}^r\sum_{\upsilon=1}^s
(-1)^\upsilon \binom{r}{u} \binom{s}{\upsilon} {J'}_{kl}^{(0,r+s-u-\upsilon,u,
\upsilon,0|r-u,s-\upsilon)},
\end{aligned}$$]{} [$$\begin{aligned}
[\tau^r,\tau^s]_{kl}=\frac1{2^{r+s}}\sum_{u=1}^r\sum_{\upsilon=1}^s
\binom{r}{u} \binom{s}{\upsilon} {J'}_{kl}^{(0,r+s-u-\upsilon,u,
\upsilon,0|r+s-u-\upsilon,0)},
\end{aligned}$$]{} and the needed tensorial collision brackets can be expressed as $$\begin{aligned}
\nonumber &~& [\tau^r\overset{\circ}{\overline{\pi^{\mu} \pi^{\nu}}},
\tau_1^s\overset{\circ}{\overline{\pi_{1\mu} \pi_{1\nu}}}]_{kl}=
\frac1{2^{r+s+4}}\sum_{n_1=0}^r\sum_{n_2=0}^s \binom{s}{n_2}\binom{r}{n_1}(-1)^{s-n_2}
({J'}_{kl}^{(2,n_1+n_2,r-n_1,s-n_2,0|2+n_1,2+n_2)} \\
\nonumber &+&2{J'}_{kl}^{(1,n_1+n_2,r-n_1,s-n_2,1|1+n_1,1+n_2)}+{J'}_{kl}^{(0,n_1+n_2,r-n_1,s-n_2,2|n_1,n_2)}) \\
\nonumber &-&\frac1{2^{r+s+3}}\sum_{n_1=0}^{r+1}\sum_{n_2=0}^{s+1} \binom{s+1}{n_2}\binom{r+1}{n_1}(-1)^{s+1-n_2}
({J'}_{kl}^{(1,n_1+n_2,r+1-n_1,s+1-n_2,0|1+n_1,1+n_2)} \\
\nonumber &+&{J'}_{kl}^{(0,n_1+n_2,r+1-n_1,s+1-n_2,1|n_1,n_2)})+\frac23[\tau^{r+2},\tau^{s+2}_1]_{kl}
+\frac13 z_l^2 [\tau^{r+2},\tau^{s}_1]_{kl} \\
&+&\frac13 z_k^2 [\tau^{r},\tau^{s+2}_1]_{kl}- \frac13 z_k^2 z_l^2
[\tau^{r},\tau^{s}_1]_{kl},
\end{aligned}$$ $$\begin{aligned}
\nonumber &~& [\tau^r\overset{\circ}{\overline{\pi^{\mu} \pi^{\nu}}},
\tau^s\overset{\circ}{\overline{\pi_{\mu} \pi_{\nu}}}]_{kl}=
\frac1{2^{r+s+4}}\sum_{n_1=0}^r\sum_{n_2=0}^s \binom{s}{n_2}\binom{r}{n_1}
({J'}_{kl}^{(2,n_1+n_2,r-n_1,s-n_2,0|4+n_1+n_2,0)} \\
\nonumber &-&2{J'}_{kl}^{(1,n_1+n_2,r-n_1,s-n_2,1|2+n_1+n_2,0)}+{J'}_{kl}^{(0,n_1+n_2,r-n_1,s-n_2,2|n_1+n_2,0)}) \\
\nonumber &-&\frac1{2^{r+s+3}}\sum_{n_1=0}^{r+1}\sum_{n_2=0}^{s+1} \binom{s+1}{n_2}\binom{r+1}{n_1}
({J'}_{kl}^{(1,n_1+n_2,r+1-n_1,s+1-n_2,0|2+n_1+n_2,0)} \\
\nonumber &-&{J'}_{kl}^{(0,n_1+n_2,r+1-n_1,s+1-n_2,1|n_1+n_2,0)})+\frac23[\tau^{r+2},\tau^{s+2}]_{kl}
+\frac13 z_k^2 [\tau^{r+2},\tau^{s}]_{kl} \\
&+&\frac13 z_k^2 [\tau^{r},\tau^{s+2}]_{kl}-\frac13 z_k^4[\tau^{r},\tau^{s}]_{kl}.
\end{aligned}$$ Below some lowest orders collision brackets are presented with the following notations: $$\begin{aligned}
\nonumber \widetilde K_1 &\equiv& \frac{K_3(z_k+z_l)}{K_2(z_k)K_2(z_l)},
\quad \widetilde K_2\equiv \frac{K_2(z_k+z_l)}{K_2(z_k)K_2(z_l)},
\quad \widetilde K_3\equiv \frac{G(z_k+z_l)}{K_2(z_k)K_2(z_l)}, \\
Z_{kl} &\equiv& z_k+z_l, \quad z_{kl} \equiv z_k-z_l.
\end{aligned}$$ For the scalar collision brackets one has: [$$\begin{aligned}
\label{br211}
-[\tau,\tau_1]_{kl}=[\tau,\tau]_{kl}=\frac{\sigma^{cl}_{kl}}{\sigma(T)}\frac{\pi}{2z_k^2z_l^2Z_{kl}^2}
(P_{s1}^{(1,1)}\widetilde K_1+P_{s2}^{(1,1)}\widetilde K_2+P_{s3}^{(1,1)}\widetilde K_3),
\end{aligned}$$]{} where [$$\begin{aligned}
P_{s1}^{(1,1)}=-2 Z_{kl} (z_{kl}^4+4 z_{kl}^2 Z_{kl}^2-2 Z_{kl}^4),
\end{aligned}$$]{} [$$\begin{aligned}
P_{s2}^{(1,1)}=z_{kl}^4 (3 Z_{kl}^2+8)+32 z_{kl}^2 Z_{kl}^2+8 Z_{kl}^4,
\end{aligned}$$]{} [$$\begin{aligned}
P_{s3}^{(1,1)}=-3 z_{kl}^4 Z_{kl}^6,
\end{aligned}$$]{} and [$$\begin{aligned}
[\tau,\tau_1^2]_{kl}=[\tau^2,\tau_1]_{lk}=\frac{\sigma^{cl}_{kl}}{\sigma(T)}\frac{\pi}{4z_k^2z_l^2Z_{kl}^2}
(P_{s11}^{(1,2)}\widetilde K_1+P_{s12}^{(1,2)}\widetilde K_2+P_{s13}^{(1,2)}\widetilde K_3),
\end{aligned}$$]{} where [$$\begin{aligned}
P_{s11}^{(1,2)}=2 Z_{kl} (z_{kl}^5 Z_{kl}+8 z_{kl}^4+16 z_{kl}^3 Z_{kl}
+32 z_{kl}^2 Z_{kl}^2+16 z_{kl} Z_{kl}^3-40 Z_{kl}^4),
\end{aligned}$$]{} $$\begin{aligned}
\nonumber P_{s12}^{(1,2)}&=&-z_{kl}^5 Z_{kl} (Z_{kl}^2+8)-8 z_{kl}^4 (Z_{kl}^2+8)
-16 z_{kl}^3 Z_{kl} (Z_{kl}^2+8)\\ &+&16 z_{kl}^2 Z_{kl}^2 (Z_{kl}^2-16)
+8 z_{kl} Z_{kl}^3 (Z_{kl}^2-16)-8 Z_{kl}^4 (Z_{kl}^2+8),
\end{aligned}$$ [$$\begin{aligned}
P_{s13}^{(1,2)}=z_{kl}^5 Z_{kl}^7,
\end{aligned}$$]{} and [$$\begin{aligned}
[\tau,\tau^2]_{kl}=[\tau^2,\tau]_{kl}=\frac{\sigma^{cl}_{kl}}{\sigma(T)}\frac{\pi}{4z_k^2z_l^2Z_{kl}^2}
(P_{s21}^{(1,2)}\widetilde K_1+P_{s22}^{(1,2)}\widetilde K_2+P_{s23}^{(1,2)}\widetilde K_3),
\end{aligned}$$]{} where [$$\begin{aligned}
P_{s21}^{(1,2)}=2 Z_{kl} (z_{kl}^5 Z_{kl}-8 z_{kl}^4+16 z_{kl}^3 Z_{kl}
-32 z_{kl}^2 Z_{kl}^2+16 z_{kl} Z_{kl}^3+40 Z_{kl}^4),
\end{aligned}$$]{} $$\begin{aligned}
\nonumber P_{s22}^{(1,2)}&=&-z_{kl}^5 Z_{kl} (Z_{kl}^2+8)+
8 z_{kl}^4 (Z_{kl}^2+8)-16 z_{kl}^3 Z_{kl} (Z_{kl}^2+8)\\
&-&16 z_{kl}^2 Z_{kl}^2 (Z_{kl}^2-16)+8 z_{kl} Z_{kl}^3 (Z_{kl}^2-16)
+8 Z_{kl}^4 (Z_{kl}^2+8),
\end{aligned}$$ [$$\begin{aligned}
P_{s23}^{(1,2)}=z_{kl}^5 Z_{kl}^7,
\end{aligned}$$]{} and [$$\begin{aligned}
[\tau^2,\tau_1^2]_{kl}=\frac{\sigma^{cl}_{kl}}{\sigma(T)}\frac{\pi}{24z_k^2z_l^2Z_{kl}^2}
(P_{s11}^{(2,2)}\widetilde K_1+P_{s12}^{(2,2)}\widetilde K_2+P_{s13}^{(2,2)}\widetilde K_3),
\end{aligned}$$]{} where $$\begin{aligned}
\nonumber P_{s11}^{(2,2)}&=&-2 Z_{kl} [z_{kl}^6 (Z_{kl}^2+2)+6 z_{kl}^4
(11 Z_{kl}^2-32)-72 z_{kl}^2 Z_{kl}^2 (Z_{kl}^2+8)\\ &+&24 Z_{kl}^4 (Z_{kl}^2+96)],
\end{aligned}$$ $$\begin{aligned}
\nonumber P_{s12}^{(2,2)}&=&z_{kl}^6 (Z_{kl}^4+10 Z_{kl}^2+16)-6 z_{kl}^4
(Z_{kl}^4-56 Z_{kl}^2+256)\\ &+&144 z_{kl}^2 Z_{kl}^2 (5 Z_{kl}^2-32)
-48 Z_{kl}^4 (13 Z_{kl}^2+32),
\end{aligned}$$ [$$\begin{aligned}
P_{s13}^{(2,2)}=-z_{kl}^4 Z_{kl}^6 [z_{kl}^2 (Z_{kl}^2-6)-6 Z_{kl}^2],
\end{aligned}$$]{} and [$$\begin{aligned}
[\tau^2,\tau^2]_{kl}=\frac{\sigma^{cl}_{kl}}{\sigma(T)}\frac{\pi}{24z_k^2z_l^2Z_{kl}^2}
(P_{s21}^{(2,2)}\widetilde K_1+P_{s22}^{(2,2)}\widetilde K_2+P_{s23}^{(2,2)}\widetilde K_3),
\end{aligned}$$]{} where $$\begin{aligned}
\nonumber P_{s21}^{(2,2)}&=&-2 Z_{kl} [z_{kl}^6 (Z_{kl}^2+2)-36 z_{kl}^5
Z_{kl}+18 z_{kl}^4 (Z_{kl}^2+16)+96 z_{kl}^3 Z_{kl} (Z_{kl}^2-10)\\
&+&24 z_{kl}^2 Z_{kl}^2 (Z_{kl}^2+56)-48 z_{kl} Z_{kl}^3
(Z_{kl}^2+20)-24 Z_{kl}^4 (Z_{kl}^2+100)],
\end{aligned}$$ $$\begin{aligned}
\nonumber P_{s22}^{(2,2)}&=&z_{kl}^6 (Z_{kl}^4+10 Z_{kl}^2+16)+
12 z_{kl}^5 Z_{kl} (Z_{kl}^2-24)-6 z_{kl}^4 (Z_{kl}^4-72 Z_{kl}^2-384)\\
\nonumber &-&192 z_{kl}^3 Z_{kl} (Z_{kl}^2+40)-48 z_{kl}^2 Z_{kl}^2 (13 Z_{kl}^2-224)
+96 z_{kl} Z_{kl}^3 (7 Z_{kl}^2-80)\\ &+&48 Z_{kl}^4 (13 Z_{kl}^2+48),
\end{aligned}$$ [$$\begin{aligned}
P_{s23}^{(2,2)}=-z_{kl}^4 Z_{kl}^6 [z_{kl}^2 (Z_{kl}^2-6)+12 z_{kl} Z_{kl}-6 Z_{kl}^2].
\end{aligned}$$]{} And for the tensor collision brackets one has: [$$\begin{aligned}
[\overset{\circ}{\overline{\pi^{\mu} \pi^{\nu}}},
\overset{\circ}{\overline{\pi_{1\mu} \pi_{1\nu}}}]_{kl}=
\frac{\sigma^{cl}_{kl}}{\sigma(T)}\frac{\pi}{72z_k^2z_l^2Z_{kl}^2}(P_{T11}^{(0,0)}\widetilde K_1
+P_{T12}^{(0,0)}\widetilde K_2+P_{T13}^{(0,0)}\widetilde K_3),
\end{aligned}$$]{} where $$\begin{aligned}
\nonumber P_{T11}^{(0,0)}&=&-2 Z_{kl} [z_{kl}^6 (5 Z_{kl}^2-8)+
24 z_{kl}^4 (Z_{kl}^2-16)-144 z_{kl}^2 Z_{kl}^2 (Z_{kl}^2+8)\\
&+&48 Z_{kl}^4 (Z_{kl}^2+72)],
\end{aligned}$$ $$\begin{aligned}
\nonumber P_{T12}^{(0,0)}&=&z_{kl}^6 (5 Z_{kl}^4-40 Z_{kl}^2-64)-
24 z_{kl}^4 (5 Z_{kl}^4+8 Z_{kl}^2+128)\\&+&576 z_{kl}^2 Z_{kl}^2
(Z_{kl}^2-16)-192 Z_{kl}^4 (5 Z_{kl}^2+16),
\end{aligned}$$ [$$\begin{aligned}
P_{T13}^{(0,0)}=-5 z_{kl}^4 Z_{kl}^6 [z_{kl}^2 (Z_{kl}^2-24)-24 Z_{kl}^2],
\end{aligned}$$]{} and [$$\begin{aligned}
[\overset{\circ}{\overline{\pi^{\mu} \pi^{\nu}}},
\overset{\circ}{\overline{\pi_{\mu} \pi_{\nu}}}]_{kl}=
\frac{\sigma^{cl}_{kl}}{\sigma(T)}\frac{\pi}{72z_k^2z_l^2Z_{kl}^2}(P_{T21}^{(0,0)}\widetilde K_1
+P_{T22}^{(0,0)}\widetilde K_2
+P_{T23}^{(0,0)}\widetilde K_3),
\end{aligned}$$]{} where $$\begin{aligned}
\nonumber P_{T21}^{(0,0)}&=&2 Z_{kl} [z_{kl}^6 (8-5 Z_{kl}^2)+72 z_{kl}^4 (3 Z_{kl}^2-8)
-480 z_{kl}^3 Z_{kl} (Z_{kl}^2-4) \\ &-&336 z_{kl}^2 Z_{kl}^2 (Z_{kl}^2+8)+240 z_{kl} Z_{kl}^3
(Z_{kl}^2+8)+192 Z_{kl}^4 (Z_{kl}^2+67)],
\end{aligned}$$ $$\begin{aligned}
\nonumber P_{T22}^{(0,0)}&=&z_{kl}^6 (5 Z_{kl}^4-40 Z_{kl}^2-64)+240 z_{kl}^5
Z_{kl}^3-24 z_{kl}^4 (5 Z_{kl}^4+48 Z_{kl}^2-192)\\
\nonumber &+&1920 z_{kl}^3 Z_{kl}
(Z_{kl}^2-8)-192 z_{kl}^2 Z_{kl}^2 (17 Z_{kl}^2-112)+1920 z_{kl} Z_{kl}^3
(Z_{kl}^2-8)\\ &+&768 Z_{kl}^4 (5 Z_{kl}^2+6),
\end{aligned}$$ [$$\begin{aligned}
P_{T23}^{(0,0)}=-5 z_{kl}^4 Z_{kl}^6 [z_{kl}^2 (Z_{kl}^2-24)+48 z_{kl} Z_{kl}-24 Z_{kl}^2].
\end{aligned}$$]{} If ${z_k=z_l}$, then the $G(x)$ function is eliminated everywhere and the collision brackets simplify considerably.
The constant cross sections are not the only possible ones resulting in the analytical expressions for the collision brackets and, hence, for the transport coefficients. The analytical expressions through the Bessel and the Meijer functions (possibly, relatively simple ones) can be obtained also using some non-constant cross sections. For example, this is possible in the case when cross sections are proportional to integer powers of the $P^2$ and/or the $\cos\Theta$. As one can see from the (\[Jint0\]), an integer power of the $P^2$ would result just in shift of the index $a$ which can be easily taken into account in the (\[Jint\]) or the (\[Jpint\]). A power of the $\cos\Theta$ would result in shift of the first index of the $\sigma^{(f,g)}$ (\[sigmafg\]), and one would get [$$\begin{aligned}
\sigma^{(d,e,f|f')}_{kl}=\frac{\sigma^{cl}_{kl}}{\sigma(T)}\left((f'+1)^{-1}
-(d+e+1)\sum_{g=0}^{\min(d,e)}K(d,e,g)\sigma^{(f+f',g)}\right),
\end{aligned}$$]{} instead of the $\sigma^{(d,e,f)}_{kl}$ (\[sigmadef\]). So that the $J$ or the $J'$ integrals just get a different factor. Half-integer powers[^50] of the $P^2$ can also be taken into account in the same way as the integer powers of the $P^2$ however one would need to introduce one more special function through which to express the $I$ integrals with even $r$ using the recurrence relation (\[recrel\]). The convenient choice of this function is found to be [$$\begin{aligned}
G_2(n,x)\equiv I(-2n+2,n,x)=x^{-n+3}\int_1^\infty du u^{-n+2}K_n(xu).
\end{aligned}$$]{} Then one can express these functions through the function (the $n=3$ is found to be the convenient choice) [$$\begin{aligned}
G_2(x)\equiv G_2(3,x)=\frac{\pi}{6}-\frac{1}{4}
G_{1,3}^{2,1}\left((x/2)^2\left|
\begin{array}{c}
1 \\
-\frac32,\frac32,0
\end{array}\right.
\right),
\end{aligned}$$]{} using the recurrence relation [$$\begin{aligned}
G_2(n,x)=\frac1{2n-3}[K_n(x)x^{-n+3}-G_2(n-1,x)].
\end{aligned}$$]{} Its derivation is the same as the derivation of the (\[Grecrel\]). Using the above mentioned prescriptions one can calculate collision brackets for a quite large class of cross sections. Few examples of simple energy dependencies for cross sections are considered below.
Using the cross sections $\sigma_{kl}P^2/(T^2Z_{kl}^2) \equiv
\sigma_{kl}\upsilon^2/Z_{kl}^2$, growing with the energy, one finds for the scalar collision brackets ($\sigma_{kl}$ is just a positive dimensional constant here, and ${\sigma^{cl}_{kl} =
\gamma_{kl}\sigma_{kl}}$) [$$\begin{aligned}
-[\tau,\tau_1]_{kl}=[\tau,\tau]_{kl}=\frac{\sigma^{cl}_{kl}}{\sigma(T)}\frac{8 \pi }{z_k^2 z_l^2 Z_{kl}^2}
(P_{s1}^{(1,1|1)}\widetilde K_1+P_{s2}^{(1,1|1)}\widetilde K_2),
\end{aligned}$$]{} where [$$\begin{aligned}
P_{s1}^{(1,1|1)}=\frac{1}{4} Z_{kl} (z_{kl}^4-2 z_{kl}^2 Z_{kl}^2+Z_{kl}^4+48 Z_{kl}^2),
\end{aligned}$$]{} [$$\begin{aligned}
P_{s2}^{(1,1|1)}=-z_{kl}^4-z_{kl}^2 Z_{kl}^2+2 Z_{kl}^4,
\end{aligned}$$]{} and [$$\begin{aligned}
[\tau,\tau_1^2]_{kl}=[\tau^2,\tau_1]_{lk}=-\frac{\sigma^{cl}_{kl}}{\sigma(T)}\frac{4 \pi }{z_k^2 z_l^2 Z_{kl}^3}
(P_{s11}^{(1,2|1)}\widetilde K_1+P_{s12}^{(1,2|1)}\widetilde K_2),
\end{aligned}$$]{} where [$$\begin{aligned}
P_{s11}^{(1,2|1)}=Z_{kl} (z_{kl}^5+2 z_{kl}^4 Z_{kl}+4 z_{kl}^3 Z_{kl}^2-
10 z_{kl}^2 Z_{kl}^3-5 z_{kl} Z_{kl}^4+8 Z_{kl}^3 (Z_{kl}^2+30)),
\end{aligned}$$]{} $$\begin{aligned}
\nonumber P_{s12}^{(1,2|1)}&=&\frac{1}{2} [-z_{kl}^5 (Z_{kl}^2+8)+z_{kl}^4 Z_{kl} (Z_{kl}^2-16)
+2 z_{kl}^3 Z_{kl}^2 (Z_{kl}^2-16)\\
&-&2 z_{kl}^2 Z_{kl}^3 (Z_{kl}^2+8)-z_{kl} Z_{kl}^4 (Z_{kl}^2+8)+Z_{kl}^5 (Z_{kl}^2+80)],
\end{aligned}$$ and [$$\begin{aligned}
[\tau,\tau^2]_{kl}=[\tau^2,\tau]_{kl}=\frac{\sigma^{cl}_{kl}}{\sigma(T)}\frac{4 \pi }{z_k^2 z_l^2 Z_{kl}^3}
(P_{s21}^{(1,2|1)}\widetilde K_1+P_{s22}^{(1,2|1)}\widetilde K_2),
\end{aligned}$$]{} where [$$\begin{aligned}
P_{s21}^{(1,2|1)}=Z_{kl} (-z_{kl}^5+2 z_{kl}^4 Z_{kl}-4 z_{kl}^3 Z_{kl}^2-10 z_{kl}^2 Z_{kl}^3+5 z_{kl} Z_{kl}^4+8 Z_{kl}^3 (Z_{kl}^2+30)),
\end{aligned}$$]{} $$\begin{aligned}
\nonumber P_{s22}^{(1,2|1)}&=&\frac{1}{2} [z_{kl}^5 (Z_{kl}^2+8)+z_{kl}^4 Z_{kl} (Z_{kl}^2-16)
-2 z_{kl}^3 Z_{kl}^2 (Z_{kl}^2-16)\\
&-&2 z_{kl}^2 Z_{kl}^3 (Z_{kl}^2+8)+z_{kl} Z_{kl}^4 (Z_{kl}^2+8)+Z_{kl}^5 (Z_{kl}^2+80)],
\end{aligned}$$ and [$$\begin{aligned}
[\tau^2,\tau_1^2]_{kl}=-\frac{\sigma^{cl}_{kl}}{\sigma(T)}\frac{\pi }{6 z_k^2 z_l^2 Z_{kl}^4}
(P_{s11}^{(2,2|1)}\widetilde K_1+P_{s12}^{(2,2|1)}\widetilde K_2+P_{s13}^{(2,2|1)}\widetilde K_3),
\end{aligned}$$]{} where $$\begin{aligned}
\nonumber P_{s11}^{(2,2|1)}&=&2 [-z_{kl}^6 Z_{kl} (5 Z_{kl}^2+4)+18 z_{kl}^4 Z_{kl}^3
(Z_{kl}^2-2)\\
&-&18 z_{kl}^2 Z_{kl}^5 (Z_{kl}^2+70)+6 Z_{kl}^5 (Z_{kl}^4+222 Z_{kl}^2+5120)],
\end{aligned}$$ $$\begin{aligned}
\nonumber P_{s12}^{(2,2|1)}&=&z_{kl}^6 (-Z_{kl}^4+44 Z_{kl}^2+32)+36 z_{kl}^4 Z_{kl}^2
(3 Z_{kl}^2+8)\\
&-&36 z_{kl}^2 Z_{kl}^4 (11 Z_{kl}^2-8)+12 Z_{kl}^6 (19 Z_{kl}^2+856),
\end{aligned}$$ [$$\begin{aligned}
P_{s13}^{(2,2|1)}=z_{kl}^6 Z_{kl}^8,
\end{aligned}$$]{} and [$$\begin{aligned}
[\tau^2,\tau^2]_{kl}=\frac{\sigma^{cl}_{kl}}{\sigma(T)}\frac{\pi }{6 z_k^2 z_l^2 Z_{kl}^4}
(P_{s21}^{(2,2|1)}\widetilde K_1+P_{s22}^{(2,2|1)}\widetilde K_2+P_{s23}^{(2,2|1)}\widetilde K_3),
\end{aligned}$$]{} where $$\begin{aligned}
\nonumber P_{s21}^{(2,2|1)}&=&2 Z_{kl} [z_{kl}^6 (5 Z_{kl}^2+4)+12 z_{kl}^5 Z_{kl}
(Z_{kl}^2-10)-6 z_{kl}^4 Z_{kl}^2 (Z_{kl}^2-46)\\
\nonumber &-&24 z_{kl}^3 Z_{kl}^3 (Z_{kl}^2+20)-6 z_{kl}^2 Z_{kl}^4 (Z_{kl}^2+182)+12 z_{kl}
Z_{kl}^5 (Z_{kl}^2+98)\\
&+&6 Z_{kl}^4 (Z_{kl}^4+226 Z_{kl}^2+5440)],
\end{aligned}$$ $$\begin{aligned}
\nonumber P_{s22}^{(2,2|1)}&=&z_{kl}^6 (Z_{kl}^4-44 Z_{kl}^2-32)+24 z_{kl}^5 Z_{kl}
(Z_{kl}^2+40)+12 z_{kl}^4 Z_{kl}^2 (5 Z_{kl}^2-184)\\
\nonumber &-&48 z_{kl}^3 Z_{kl}^3 (7 Z_{kl}^2-80)-12 z_{kl}^2 Z_{kl}^4 (19 Z_{kl}^2+184)
+24 z_{kl} Z_{kl}^5 (13 Z_{kl}^2+40)\\
&+&12 Z_{kl}^6 (19 Z_{kl}^2+904),
\end{aligned}$$ [$$\begin{aligned}
P_{s23}^{(2,2|1)}=-z_{kl}^6 Z_{kl}^8.
\end{aligned}$$]{} And for the tensor collision brackets one has: [$$\begin{aligned}
[\overset{\circ}{\overline{\pi^{\mu} \pi^{\nu}}},
\overset{\circ}{\overline{\pi_{1\mu} \pi_{1\nu}}}]_{kl}=
-\frac{\sigma^{cl}_{kl}}{\sigma(T)}\frac{2 \pi }{9 z_k^2 z_l^2 Z_{kl}^4}(P_{T11}^{(0,0|1)}\widetilde K_1
+P_{T12}^{(0,0|1)}\widetilde K_2+P_{T13}^{(0,0|1)}\widetilde K_3),
\end{aligned}$$]{} where $$\begin{aligned}
\nonumber P_{T11}^{(0,0|1)}&=&2 [2 z_{kl}^6 Z_{kl} (Z_{kl}^2-1)+9 z_{kl}^4 Z_{kl}^3
(Z_{kl}^2-2)\\
&-&9 z_{kl}^2 Z_{kl}^5 (Z_{kl}^2+22)+3 Z_{kl}^5 (Z_{kl}^4+126 Z_{kl}^2+2240)],
\end{aligned}$$ $$\begin{aligned}
\nonumber P_{T12}^{(0,0|1)}&=&z_{kl}^6 (-5 Z_{kl}^4-14 Z_{kl}^2+16)-18 z_{kl}^4 Z_{kl}^2
(3 Z_{kl}^2-8)\\
&-&18 z_{kl}^2 Z_{kl}^4 (5 Z_{kl}^2-8)+6 Z_{kl}^6 (13 Z_{kl}^2+376),
\end{aligned}$$ [$$\begin{aligned}
P_{T13}^{(0,0|1)}=5 z_{kl}^6 Z_{kl}^8,
\end{aligned}$$]{} and [$$\begin{aligned}
[\overset{\circ}{\overline{\pi^{\mu} \pi^{\nu}}},
\overset{\circ}{\overline{\pi_{\mu} \pi_{\nu}}}]_{kl}=
\frac{\sigma^{cl}_{kl}}{\sigma(T)}\frac{2 \pi }{9 z_k^2 Z_{kl}^4 z_l^2}(P_{T21}^{(0,0|1)}\widetilde K_1
+P_{T22}^{(0,0|1)}\widetilde K_2+P_{T23}^{(0,0|1)}\widetilde K_3),
\end{aligned}$$]{} where $$\begin{aligned}
\nonumber P_{T21}^{(0,0|1)}&=&2 Z_{kl} [-2 z_{kl}^6 (Z_{kl}^2-1)+15 z_{kl}^5 Z_{kl}
(Z_{kl}^2-4)+6 z_{kl}^4 Z_{kl}^2 (Z_{kl}^2+23)\\
\nonumber &-&30 z_{kl}^3 Z_{kl}^3 (Z_{kl}^2+8)-3 z_{kl}^2 Z_{kl}^4 (7 Z_{kl}^2+614)+15 z_{kl}
Z_{kl}^5 (Z_{kl}^2+68)\\
&+&6 Z_{kl}^4 (2 Z_{kl}^4+377 Z_{kl}^2+8480)],
\end{aligned}$$ $$\begin{aligned}
\nonumber P_{T22}^{(0,0|1)}&=&z_{kl}^6 (5 Z_{kl}^4+14 Z_{kl}^2-16)-60 z_{kl}^5 Z_{kl}
(Z_{kl}^2-8)+6 z_{kl}^4 Z_{kl}^2 (29 Z_{kl}^2-184)\\
\nonumber &-&240 z_{kl}^3 Z_{kl}^3 (Z_{kl}^2-8)-6 z_{kl}^2 Z_{kl}^4 (85 Z_{kl}^2+184)
+60 z_{kl} Z_{kl}^5 (5 Z_{kl}^2+8)\\
&+&6 Z_{kl}^6 (67 Z_{kl}^2+2824)),
\end{aligned}$$ [$$\begin{aligned}
P_{T23}^{(0,0|1)}=-5 z_{kl}^6 Z_{kl}^8.
\end{aligned}$$]{} Using the descending cross sections $\sigma_{kl}
Z_{kl}^2/\upsilon^2$ one finds for the scalar collision brackets [$$\begin{aligned}
-[\tau,\tau_1]_{kl}=[\tau,\tau]_{kl}=\frac{\sigma^{cl}_{kl}}{\sigma(T)}\frac{\pi }{8 z_k^2 z_l^2}
(P_{s1}^{(1,1|-1)}\widetilde K_1+P_{s2}^{(1,1|-1)}\widetilde K_2+P_{s3}^{(1,1|-1)}\widetilde K_3),
\end{aligned}$$]{} where [$$\begin{aligned}
P_{s1}^{(1,1|-1)}=2 Z_{kl} (-z_{kl}^4+8 z_{kl}^2+8 Z_{kl}^2),
\end{aligned}$$]{} [$$\begin{aligned}
P_{s2}^{(1,1|-1)}=z_{kl}^4 (Z_{kl}^2-8)-8 z_{kl}^2 (3 Z_{kl}^2+8)-64 Z_{kl}^2,
\end{aligned}$$]{} [$$\begin{aligned}
P_{s3}^{(1,1|-1)}=z_{kl}^2 Z_{kl}^4 (-z_{kl}^2 (Z_{kl}^2-24)+24 Z_{kl}^2),
\end{aligned}$$]{} and [$$\begin{aligned}
[\tau,\tau_1^2]_{kl}=[\tau^2,\tau_1]_{lk}=-\frac{\sigma^{cl}_{kl}}{\sigma(T)}\frac{\pi }{64 z_k^2 z_l^2}
(P_{s11}^{(1,2|-1)}\widetilde K_1+P_{s12}^{(1,2|-1)}\widetilde K_2+P_{s13}^{(1,2|-1)}\widetilde K_3),
\end{aligned}$$]{} where [$$\begin{aligned}
P_{s11}^{(1,2|-1)}=Z_{kl} (-2 z_{kl}^5 Z_{kl}+32 z_{kl}^4+64 z_{kl}^3 Z_{kl}
+512 z_{kl}^2+256 z_{kl} Z_{kl}+512 Z_{kl}^2),
\end{aligned}$$]{} $$\begin{aligned}
\nonumber P_{s12}^{(1,2|-1)}&=&z_{kl}^5 Z_{kl} (Z_{kl}^2-16)-16 z_{kl}^4 (Z_{kl}^2+8)
-32 z_{kl}^3 Z_{kl} (Z_{kl}^2+8)\\
&-&256 z_{kl}^2 (Z_{kl}^2+8)-128 z_{kl} Z_{kl} (Z_{kl}^2+8)+128 Z_{kl}^2 (Z_{kl}^2-16),
\end{aligned}$$ [$$\begin{aligned}
P_{s13}^{(1,2|-1)}=z_{kl}^3 Z_{kl}^5 (-z_{kl}^2 (Z_{kl}^2-32)+16 z_{kl} Z_{kl}+32 Z_{kl}^2),
\end{aligned}$$]{} and [$$\begin{aligned}
[\tau,\tau^2]_{kl}=[\tau^2,\tau]_{kl}=\frac{\sigma^{cl}_{kl}}{\sigma(T)}\frac{\pi }{64 z_k^2 z_l^2}
(P_{s21}^{(1,2|-1)}\widetilde K_1+P_{s22}^{(1,2|-1)}\widetilde K_2+P_{s23}^{(1,2|-1)}\widetilde K_3),
\end{aligned}$$]{} where [$$\begin{aligned}
P_{s21}^{(1,2|-1)}=2 Z_{kl} (z_{kl}^5 Z_{kl}+16 z_{kl}^4-32 z_{kl}^3 Z_{kl}
+256 z_{kl}^2-128 z_{kl} Z_{kl}+256 Z_{kl}^2),
\end{aligned}$$]{} $$\begin{aligned}
\nonumber P_{s22}^{(1,2|-1)}&=&z_{kl}^5 Z_{kl} (Z_{kl}^2-16)-16 z_{kl}^4 (Z_{kl}^2+8)
+32 z_{kl}^3 Z_{kl} (Z_{kl}^2+8)\\
&-&256 z_{kl}^2 (Z_{kl}^2+8)+128 z_{kl} Z_{kl}(Z_{kl}^2+8)+128 Z_{kl}^2 (Z_{kl}^2-16),
\end{aligned}$$ [$$\begin{aligned}
P_{s23}^{(1,2|-1)}=z_{kl}^3 Z_{kl}^5 (z_{kl}^2 (Z_{kl}^2-32)+16 z_{kl} Z_{kl}-32 Z_{kl}^2),
\end{aligned}$$]{} and [$$\begin{aligned}
[\tau^2,\tau_1^2]_{kl}=-\frac{\sigma^{cl}_{kl}}{\sigma(T)}\frac{\pi }{3840 z_k^2 z_l^2}
(P_{s11}^{(2,2|-1)}\widetilde K_1+P_{s12}^{(2,2|-1)}\widetilde K_2+P_{s13}^{(2,2|-1)}\widetilde K_3),
\end{aligned}$$]{} where $$\begin{aligned}
\nonumber P_{s11}^{(2,2|-1)}&=&2 Z_{kl} [z_{kl}^6 (7 Z_{kl}^2-76)-60 z_{kl}^4 (5 Z_{kl}^2-32)\\
&-&11520 z_{kl}^2 (Z_{kl}^2-6)+3840 Z_{kl}^2 (Z_{kl}^2+18)],
\end{aligned}$$ $$\begin{aligned}
\nonumber P_{s12}^{(2,2|-1)}&=&z_{kl}^6 (-7 Z_{kl}^4+188 Z_{kl}^2+320)+60 z_{kl}^4 (5 Z_{kl}^4+80 Z_{kl}^2-256)\\
&+&23040 z_{kl}^2 (Z_{kl}^2-24)+7680 Z_{kl}^2 (7 Z_{kl}^2-72),
\end{aligned}$$ [$$\begin{aligned}
P_{s13}^{(2,2|-1)}=z_{kl}^4 Z_{kl}^6 (z_{kl}^2 (7 Z_{kl}^2-300)-300 Z_{kl}^2),
\end{aligned}$$]{} and [$$\begin{aligned}
[\tau^2,\tau^2]_{kl}=\frac{\sigma^{cl}_{kl}}{\sigma(T)}\frac{\pi }{3840 z_k^2 z_l^2}
(P_{s21}^{(2,2|-1)}\widetilde K_1+P_{s22}^{(2,2|-1)}\widetilde K_2+P_{s23}^{(2,2|-1)}\widetilde K_3),
\end{aligned}$$]{} where $$\begin{aligned}
\nonumber P_{s21}^{(2,2|-1)}&=&2 Z_{kl} [-z_{kl}^6 (7 Z_{kl}^2-76)-360 z_{kl}^5 Z_{kl}
+60 z_{kl}^4 (5 Z_{kl}^2+64)-11520 z_{kl}^3 Z_{kl}\\
&-&3840 z_{kl}^2 (Z_{kl}^2-22)+7680 z_{kl} Z_{kl} (Z_{kl}^2-10)+3840 Z_{kl}^2 (Z_{kl}^2+22)],
\end{aligned}$$ $$\begin{aligned}
\nonumber P_{s22}^{(2,2|-1)}&=&z_{kl}^6 (7 Z_{kl}^4-188 Z_{kl}^2-320)+120 z_{kl}^5 Z_{kl}(3 Z_{kl}^2+16)\\
\nonumber &-&60 z_{kl}^4 (5 Z_{kl}^4+48 Z_{kl}^2+512)-3840 z_{kl}^3 Z_{kl} (Z_{kl}^2-24)\\
&-&7680 z_{kl}^2 (7 Z_{kl}^2+88)+15360 z_{kl} Z_{kl} (Z_{kl}^2+40)+7680 Z_{kl}^2 (7 Z_{kl}^2-88),
\end{aligned}$$ $$\begin{aligned}
\nonumber P_{s23}^{(2,2|-1)}&=&z_{kl}^3 Z_{kl}^5 [z_{kl}^3 (300 Z_{kl}-7 Z_{kl}^3)
-120 z_{kl}^2 (3 Z_{kl}^2-32)\\
&+&60 z_{kl} Z_{kl} (5 Z_{kl}^2-32)+3840 Z_{kl}^2].
\end{aligned}$$ And for the tensor collision brackets one has: [$$\begin{aligned}
[\overset{\circ}{\overline{\pi^{\mu} \pi^{\nu}}},
\overset{\circ}{\overline{\pi_{1\mu} \pi_{1\nu}}}]_{kl}=
-\frac{\sigma^{cl}_{kl}}{\sigma(T)}\frac{\pi }{5760 z_k^2 z_l^2}(P_{T11}^{(0,0|-1)}\widetilde K_1
+P_{T12}^{(0,0|-1)}\widetilde K_2+P_{T13}^{(0,0|-1)}\widetilde K_3),
\end{aligned}$$]{} where $$\begin{aligned}
\nonumber P_{T11}^{(0,0|-1)}&=&2 Z_{kl} [z_{kl}^6 (7 Z_{kl}^2-76)-60 z_{kl}^4 (5 Z_{kl}^2-32)
-11520 z_{kl}^2 (Z_{kl}^2-6)\\
&+&3840 Z_{kl}^2 (Z_{kl}^2+18)],
\end{aligned}$$ $$\begin{aligned}
\nonumber P_{T12}^{(0,0|-1)}&=&z_{kl}^6 (-7 Z_{kl}^4+188 Z_{kl}^2+320)+60 z_{kl}^4 (5 Z_{kl}^4
+80 Z_{kl}^2-256)\\
&+&23040 z_{kl}^2 (Z_{kl}^2-24)+7680 Z_{kl}^2 (7 Z_{kl}^2-72),
\end{aligned}$$ [$$\begin{aligned}
P_{T13}^{(0,0|-1)}=z_{kl}^4 Z_{kl}^6 (z_{kl}^2 (7 Z_{kl}^2-300)-300 Z_{kl}^2),
\end{aligned}$$]{} and [$$\begin{aligned}
[\overset{\circ}{\overline{\pi^{\mu} \pi^{\nu}}},
\overset{\circ}{\overline{\pi_{\mu} \pi_{\nu}}}]_{kl}=
\frac{\sigma^{cl}_{kl}}{\sigma(T)}\frac{\pi }{5760 z_k^2 z_l^2}(P_{T21}^{(0,0|-1)}\widetilde K_1
+P_{T22}^{(0,0|-1)}\widetilde K_2+P_{T23}^{(0,0|-1)}\widetilde K_3),
\end{aligned}$$]{} where $$\begin{aligned}
\nonumber P_{T21}^{(0,0|-1)}&=&2 Z_{kl} [-z_{kl}^6 (7 Z_{kl}^2-76)-1800 z_{kl}^5 Z_{kl}
+60 z_{kl}^4 (5 Z_{kl}^2-32)\\
&-&3840 z_{kl}^2 (7 Z_{kl}^2-22)+19200 z_{kl} Z_{kl} (Z_{kl}^2-4)+7680 Z_{kl}^2 (2 Z_{kl}^2+11)],
\end{aligned}$$ $$\begin{aligned}
\nonumber P_{T22}^{(0,0|-1)}&=&z_{kl}^6 (7 Z_{kl}^4-188 Z_{kl}^2-320)
+600 z_{kl}^5 Z_{kl} (3 Z_{kl}^2-16)\\
\nonumber &-&60 z_{kl}^4 (5 Z_{kl}^4-240 Z_{kl}^2-256)-38400 z_{kl}^3 Z_{kl}^3
+7680 z_{kl}^2 (17 Z_{kl}^2-88)\\
&-&76800 z_{kl} Z_{kl} (Z_{kl}^2-8)+7680 Z_{kl}^2 (13 Z_{kl}^2-88),
\end{aligned}$$ $$\begin{aligned}
\nonumber P_{T23}^{(0,0|-1)}&=&z_{kl}^3 Z_{kl}^5 [z_{kl}^3 (300 Z_{kl}-7 Z_{kl}^3)
-600 z_{kl}^2 (3 Z_{kl}^2-64)\\
&+&300 z_{kl} Z_{kl} (Z_{kl}^2-64)+38400 Z_{kl}^2].
\end{aligned}$$ Also averaged cross sections with the considered energy dependencies can be written down easily. Below there are only averaged energy insertions: [$$\begin{aligned}
\langle\upsilon^2/Z_{kl}^2\rangle = Z_{kl}^{-2}
\frac{J_{kl}^{(1,0,0,0,0|0,0)}}{J_{kl}^{(0,0,0,0,0|0,0)}}
=\frac{Z_{kl} ((-z_{kl}^2+Z_{kl}^2+24) K_3(Z_{kl})+4 Z_{kl} K_2(Z_{kl}))}{Z_{kl}
(Z_{kl}^2-z_{kl}^2) K_3(Z_{kl})+4 z_{kl}^2 K_2(Z_{kl})},
\end{aligned}$$]{} $$\begin{aligned}
\nonumber \langle Z_{kl}^2/\upsilon^2 \rangle &=& Z_{kl}^2
\frac{J_{kl}^{(-1,0,0,0,0|0,0)}}{J_{kl}^{(0,0,0,0,0|0,0)}}\\
&=&\frac{12 z_{kl}^2 Z_{kl}^4 G(Z_{kl})+Z_{kl} (Z_{kl}^2-z_{kl}^2) K_3(Z_{kl})
-4 (z_{kl}^2+Z_{kl}^2) K_2(Z_{kl})}{Z_{kl} (Z_{kl}^2-z_{kl}^2) K_3(Z_{kl})+4 z_{kl}^2 K_2(Z_{kl})}.
\end{aligned}$$ Using these averaged insertions one could reproduce correct asymptotic temperature dependencies of the viscosities however with a somewhat different coefficient in the high-temperature limit.
For complicated forms of cross sections one can calculate numerically the 1- or 2-dimensional integrals (corresponding to the unknown energy and angular dependencies). As long as implemented UrQMD cross sections don’t depend on the scattering angle (being treated as averaged ones over it) one has to calculate only 1-dimensional integrals numerically. For this purpose the $J$ integrals can be represented in the form $$\begin{aligned}
\label{Jint2}
\nonumber &~&J_{kl}^{(a,b,d,e,f|q,r)}=\frac{\pi(d+e+1)!!}{z_k^2z_l^2K_2(z_k)K_2(z_l)}
\sum_{h=0}^{[b/2]}(-1)^h(2h-1)!! \binom{b}{2h} Z_{kl}^{2 (a + f) + b + (d + e)/2 - h + 5} \\
\nonumber &~&\times \int_1^\infty du \sigma^{(d,e,f)}_{1kl}(Z_{kl}u) (u^2 + z_{kl}/Z_{kl})^q
(u^2 - z_{kl}/Z_{kl})^r u^{2 (a - f - q - r) + b - h - 3(d + e)/2}\\
&~&\times (u^2 - 1)^{(d + e)/2 + f + 1} (u^2 - z_{kl}^2/Z_{kl}^2)^{(d + e)/2 + f + 1}
K_{(d + e)/2 + b - h + 1}(Z_{kl}u),
\end{aligned}$$ where the $\sigma_{1kl}^{(d,e,f)}(Z_{kl}u)$ is generalized to [$$\begin{aligned}
\sigma_{1kl}^{(d,e,f)}(Z_{kl}u)=\frac{\sigma^{cl}_{kl}(Z_{kl}u)}{\sigma(T)}
\sum_{g=0}^{\min(d,e)} \sigma^{(f,g)} K(d,e,g), \quad
\sigma^{cl}_{kl}(Z_{kl}u)\equiv \gamma_{kl}\sigma_{kl}(Z_{kl}u).
\end{aligned}$$]{} And the $J'$ integrals can be represented in the form $$\begin{aligned}
\label{Jpint2}
\nonumber &~&{J'}_{kl}^{(a,b,d,e,f|q,r)}=\frac{\pi(d+e-1)!!}{z_k^2z_l^2K_2(z_k)K_2(z_l)}
\sum_{h=0}^{[b/2]}(-1)^h(2h-1)!! \binom{b}{2h} Z_{kl}^{2 (a + f) + b + (d + e)/2 - h + 5} \\
\nonumber &~&\times \int_1^\infty du \sigma^{(d,e,f)}_{kl}(Z_{kl}u) (u^2 + z_{kl}/Z_{kl})^q
(u^2 - z_{kl}/Z_{kl})^r u^{2 (a - f - q - r) + b - h - 3(d + e)/2}\\
&~&\times (u^2 - 1)^{(d + e)/2 + f + 1} (u^2 - z_{kl}^2/Z_{kl}^2)^{(d + e)/2 + f + 1}
K_{(d + e)/2 + b - h + 1}(Z_{kl}u),
\end{aligned}$$ where the $\sigma^{(d,e,f)}_{kl}(Z_{kl}u)$ is generalized to [$$\begin{aligned}
\label{sigmadef2}
\sigma^{(d,e,f)}_{kl}(Z_{kl}u)=\frac{\sigma^{cl}_{kl}(Z_{kl}u)}{\sigma(T)}\left(1-(d+e+1)
\sum_{g=0}^{\min(d,e)}K(d,e,g)\sigma^{(f,g)}\right).
\end{aligned}$$]{} To calculate collision brackets faster one can bring all integrated expressions under one integral and simplify the integrand.
The collision rates and the mean free paths \[appmfp\]
=======================================================
The quantity $\frac{W_{k'l'}} {p_{k'}^0 p_{1l'}^0 {p'}_{k'}^0
{p'}_{1l'}^0} d^3{p'}_{k'} d^3{p'}_{1l'}$, which enters in the elastic collision integral (\[ckelgroot\]), represents the probability of scattering per unit time times unit volume for two particles which had momentums $\vec p_{k'}$ and $\vec p_{1l'}$ before scattering and momentums in the ranges ${(\vec {p'}_{k'},
\vec {p'}_{k'} + d\vec {p'}_{k'})}$ and ${(\vec {p'}_{1l'}, \vec
{p'}_{1l'} + d\vec {p'}_{1l'})}$ after the scattering. The quantity $ g_{k'} \frac{d^3p_{k'}}{ (2\pi)^3 } f_{k'}$ represents the number of particles per unit volume, which have momentums in the range ${(\vec p_{k'}, \vec p_{k'} + d\vec p_{k'})}$. The number of collisions of particles of the $k'$-th species with particles of the $l'$-th species per unit time per unit volume is then[^51] [$$\begin{aligned}
\label{totratekl}
\widetilde R^{el}_{k'l'}\equiv g_{k'}g_{l'}\frac{\gamma_{k'l'}^2}{(2\pi)^6}\int
\frac{d^3p_{k'}}{p_{k'}^0}\frac{d^3p_{1l'}}{p_{1l'}^0}\frac{d^3p'_{k'}}{{p'}_{k'}^0}
\frac{d^3p'_{1l'}}{{p'}_{1l'}^0}f_{k'}^{(0)}f_{1l'}^{(0)}W_{k'l'}.
\end{aligned}$$]{} To get the corresponding number of collisions of particles of the $k'$-th species with particles of the $l'$-th species per unit time *per particle of the $k'$-th species*, $R^{el}_{k'l'}$, one has to divide the (\[totratekl\]) on the $\gamma_{k'l'}n_{k'}$ (recall that ${n_{k'} \propto g_{k'}}$ by definition), which is the number of particles of the $k'$-th species per unit volume divided on the number of particles of the $k'$-th species taking part in the given type of reaction (2 for binary elastic collisions, if particles are identical, and 1 otherwise). This rate can be directly obtained averaging the collision rate with fixed momentum $p_k$ of the $k$-th particle species [$$\begin{aligned}
{\mathcal{R}}^{el}_{kl'}\equiv g_{l'}\gamma_{kl'}\int
\frac{d^3p_{1l'}}{(2\pi)^3}d^3p'_{k}d^3p'_{1l'}f_{1l'}^{(0)}
\frac{W_{kl'}}{p_{k}^0p_{1l'}^0{p'}_{k}^0{p'}_{1l'}^0},
\end{aligned}$$]{} over the momentum with the probability distribution $\frac{d^3p_{k}} {(2\pi)^3} \frac{f_{k}}{n_k}$ (and spin states which is trivial): [$$\begin{aligned}
R^{el}_{k'l'}\equiv g_{k'}g_{l'}\frac{\gamma_{k'l'}}{(2\pi)^6n_{k'}}\int
\frac{d^3p_{k'}}{p_{k'}^0} \frac{d^3p_{1l'}}{p_{1l'}^0}
\frac{d^3p'_{k'}}{{p'}_{k'}^0}\frac{d^3p'_{1l'}}{{p'}_{1l'}^0}f_{k'}^{(0)}
f_{1l'}^{(0)}W_{k'l'}=\frac{\widetilde R^{el}_{k'l'}}{\gamma_{k'l'}n_{k'}}.
\end{aligned}$$]{} So that to get the mean rate of the elastic collisions per particle of the $k'$-th species with all particles in the system one can just integrate the sum of the gain terms in the collision integral (\[ckelgroot\]) over $\frac{d^3p_k}{(2\pi)^3p_k^0n_k}$ and average it over spin: [$$\begin{aligned}
R_{k'}^{el}\equiv \sum_{l'} R_{k'l'}^{el}.
\end{aligned}$$]{} One can express the $\widetilde R_{k'l'}^{el}$ through the $J_{kl}^{(0, 0, 0, 0, 0| 0, 0)}$ integrals from Appendix \[appJ\] as [$$\begin{aligned}
\label{Rklel}
\widetilde R_{k'l'}^{el}=\gamma_{k'l'}\sigma(T)n_{k'} n_{l'}
J_{k'l'}^{(0, 0, 0, 0, 0| 0, 0)}.
\end{aligned}$$]{} For simplicity let’s consider constant cross sections in what follows. Then the (\[Rklel\]) becomes [$$\begin{aligned}
\widetilde R_{k'l'}^{el}=g_{k'}g_{l'}\gamma_{k'l'}\frac{2\sigma^{cl}_{k'l'} T^6}{\pi^3}
[(z_{k'}-z_{l'})^2 K_2(z_{k'}+z_{l'})+z_{k'} z_{l'}(z_{k'}+z_{l'})
K_3(z_{k'}+z_{l'})],
\end{aligned}$$]{} where $\sigma^{cl}_{k'l'}$ is the classical elastic differential constant cross section of scattering of a particle of the $k'$-th species on particles of the $l'$-th species. For the case of large temperature or when both masses are small, ${z_{k'} \ll 1}$ and ${z_{l'} \ll 1}$, one has expansion [$$\begin{aligned}
\widetilde R_{k'l'}^{el}=g_{k'}g_{l'}\gamma_{k'l'}
\frac{4 \sigma^{cl}_{k'l'} T^6}{\pi^3}
\left(1-\frac14(z_{k'}^2+z_{l'}^2)+...\right).
\end{aligned}$$]{} For the case of small temperature or when both masses are large, ${z_{k'}\gg 1}$ and ${z_{l'}\gg 1}$, one has expansion [$$\begin{aligned}
\widetilde R_{k'l'}^{el}=g_{k'}g_{l'}\gamma_{k'l'}\frac{\sqrt{2}\sigma^{cl}_{k'l'} T^6 z_{k'} z_{l'}
\sqrt{z_{k'}+z_{l'}} e^{-z_{k'}-z_{l'}}}{\pi^{5/2}}\left(1+\frac{8 z_{k'}^2+19 z_{k'} z_{l'}
+8 z_{l'}^2}{8 z_{k'} z_{l'} (z_{k'}+z_{l'})}+...\right).
\end{aligned}$$]{} For the case when only one mass is large, ${z_{l'}\gg 1}$, one has somewhat different expansion [$$\begin{aligned}
\widetilde R_{k'l'}^{el}=g_{k'}g_{l'}\gamma_{k'l'}\frac{\sqrt{2} \sigma^{cl}_{k'l'} T^6 (z_{k'}+1) z_{l'}^{3/2}
e^{-z_{k'}-z_{l'}}}{\pi^{5/2}}\left(1+\frac{4 z_{k'}^2+15 z_{k'}+15}{8 z_{k'}+8}z_{l'}^{-1}...\right).
\end{aligned}$$]{} The $\sigma(T) J_{k'l'}^{(0, 0, 0, 0, 0| 0, 0)}$ in the (\[Rklel\]) can be replaced in the limit of high temperatures with $4\pi \sigma^{cl}_{k'l'} \langle {\left\vert\vec v_{k'}\right\vert}\rangle$ and in the limit of low temperatures with $4\pi \sigma^{cl}_{k'l'}
\langle {\left\vert\vec v_{k'}\right\vert}\rangle \sqrt{1+m_{k'}/m_{l'}}= 4\pi
\sigma^{cl}_{k'l'} \langle {\left\vert\vec v_{rel,k'l'}\right\vert}\rangle$, where $\langle {\left\vert\vec v_{k'}\right\vert}\rangle$ is the mean modulus of particle’s velocity of the $k'$-th species, [$$\begin{aligned}
\label{avvel}
\langle {\left\vert\vec v_{k'}\right\vert}\rangle=\frac{\int d^3p_{k'} \frac{|\vec p_{k'}|}{p_{k'}^0}
f^{(0)}_{k'}(p_{k'})}{\int d^3p_{k'} f^{(0)}_{k'}(p_{k'})}
=\frac{2 e^{-z_{k'}}(1+z_{k'})}{z_{k'}^2K_2(z_{k'})}
=\sqrt{\frac{8}{\pi z_{k'}}}\frac{K_{3/2}(z_{k'})}{K_{2}(z_{k'})},
\end{aligned}$$]{} and $\langle {\left\vert\vec v_{rel,k'l'}\right\vert}\rangle$ is the mean modulus of the relative velocity, which coincides with the $\langle
{\left\vert\vec v_{k'}\right\vert}\rangle$ for high temperatures. Then the resultant collision rate $R^{el}_{k'}$ would reproduce simple nonrelativistic collision rates know in the kinetic-molecular theory. To get the (approximate) mean free time one has just to invert the $R^{el}_{k'}$: [$$\begin{aligned}
t^{el}_{k'}=\frac{1}{R^{el}_{k'}}.
\end{aligned}$$]{} The (approximate) mean free path $l^{el}_{k'}$ can be obtained after multiplication of it on the $\langle {\left\vert\vec
v_{k'}\right\vert}\rangle$: [$$\begin{aligned}
\label{lel}
l^{el}_{k'}=\frac{\langle {\left\vert\vec v_{k'}\right\vert}\rangle}{R^{el}_{k'}}.
\end{aligned}$$]{} For the single-component gas one gets [$$\begin{aligned}
\label{mfpsc}
l^{el}_{1'}=\frac{\langle {\left\vert\vec v_{1'}\right\vert}\rangle}{R^{el}_{1'1'}}=
\frac{\pi e^{-z_{1}} (z_{1}+1)}{g_{1}4 \sigma^{cl}_{11} T^3 z_{1}^3 K_3(2 z_{1})}.
\end{aligned}$$]{} The nonrelativistic limit of the (\[mfpsc\]) with the ${g_1=1}$ coincides with the same limit of the formula [$$\begin{aligned}
l^{el}_{1}=\frac{\langle {\left\vert\vec v_1\right\vert}\rangle}{4\pi\sigma^{cl}_{11}n_1
\langle {\left\vert\vec v_{rel}\right\vert}\rangle}=\frac{1}{4\pi\sigma^{cl}_{11}n_1\sqrt2},
\end{aligned}$$]{} which is the mean free path formula coming from the nonrelativistic kinetic-molecular theory obtained by Maxwell. The ultrarelativistic limit of the (\[mfpsc\]) with the ${g_1=1}$ coincides with the same limit of the formula [$$\begin{aligned}
l_1^{el}=\frac{1}{4\pi\sigma^{cl}_{11}n_1}.
\end{aligned}$$]{}
Analogically one can introduce inelastic rates $R^{inel}_k$ of any inelastic processes to occur for the $k'$-th particles species. Then the mean free time $t^{inel}_{k'}$ in what any inelastic process occurs for the particles of the $k'$-th species can be introduced as [$$\begin{aligned}
\label{tkinel}
t^{inel}_{k'}=\frac{1}{R^{inel}_{k'}}.
\end{aligned}$$]{} The mean free path for the particles of the $k'$-th species is obtained through the rate ${R^{el}_{k'} + R^{inel}_{k'}}$ and can be written as [$$\begin{aligned}
\label{mfp}
l_{k'}=\frac{\langle {\left\vert\vec v_{k'}\right\vert}\rangle}{R^{el}_{k'}+R^{inel}_{k'}}.
\end{aligned}$$]{}
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[^1]: Fireballs, created in heavy ion collisions, have finite sizes and finite times of existence of their thermalized part. This puts important restrictions on detection of the critical fluctuations of thermodynamic functions [@Stephanov:1999zu]. Because of this it’s also important to consider nonequilibrium dissipative corrections and nonequilibrium phenomenons like critical slow down/speed up.
[^2]: In [@Danielewicz:1984ww] the bound coming from the Heisenberg uncertainty principle was obtained for the $\eta/s$. However, it was obtained using a formula, which is justified in rarified gases with short-range interactions. It’s well known already from the nonrelativistic kinetic theory that dense gases get corrections over the particle number densities (see e. g. [@landau10], Section 18), corresponding to more than binary collisions, and in very dense gasses this bound can be quite inaccurate. In liquids and other substances the mechanism of appearance of the shear viscosity may be different (see [@Schafer:2009dj] for a review). In particular, the shear viscosity of water can be very well described by a phenomenological formula with an exponential dependence on the inverse temperature, see e. g. [@Sengers].
[^3]: The Kubo formulas are distinguished from the Greet-Kubo formulas e. g. in [@Muronga:2003tb; @Kadanoff].
[^4]: The author could not reproduce this plotted total cross section by its formula. In fact it was approximately 2.6 times larger. But the plotted total cross section is quite close to the isospin averaged (corrected) UrQMD $\pi\pi$ total cross section. A notable deviation is only at $\sqrt{s}<0.5~GeV$, when the UrQMD cross section becomes $1.5-2$ times smaller. At $1.2~GeV<\sqrt{s}<1.9~GeV$ the UrQMD cross section is a little larger instead.
[^5]: In the relaxation time approximation the bulk viscosity source term is treated somewhat differently: the $\xi$ becomes proportional to the integral of the squared source term (times some functions of momentum) and not to the square of the integrated source term (times some functions of momentum). Note that in the [@fernnicola] the used formula has this relaxation time approximation form. Also there the source term is the one of a system with the inelastic processes. These facts could help to understand the enlarged values of the bulk viscosity. Not small quantitative discrepancies can be noticed between the calculations of the [@chakkap] and the [@Dobado:2012zf].
[^6]: The Boltzmann equations can also be derived for the case of the inelastic collisions from some physical considerations, see [@groot] (Chap. I, Sec. 2).
[^7]: This is the case of interest. Coulomb interactions can be neglected in heavy ion collisions at all the considered energies in this paper.
[^8]: The inclusion of this kinetic perturbation is similar to the inclusion of the chemical one so that it is omitted for simplicity below. Usually this perturbation should fade first because also the inelastic processes are responsible for the relaxation of the momentum spectra. However, see comments for $1+1$-dimensional systems below.
[^9]: Note that if the $k$-th species have conserved particle numbers, then the nonequilibrium chemical potential is nonphysical or redefining the usual (thermodynamic) chemical potential.
[^10]: The question of validity of this expansion over the gradients (which coincides with the usual order counting in the formal expansion over the gradients in the hydrodynamics) for some profiles is not discussed in this paper.
[^11]: It’s a reasonable assumption in the case when the hydrodynamical description is applicable. For example, the chemical perturbations can be a result of a fast previous expansion (faster than the chemical equilibration). Then the inhomogeneities of the chemical perturbations should correlate with the inhomogeneities of the thermodynamic functions, the flow velocity or it’s divergence.
[^12]: If the expansion rate is much larger than the collision rates of the inelastic processes (e. g. because of a substantial decrease of the temperature), then the chemical perturbations should enlarge instead. If the r. h. s. of the (\[generlinbeqn1\]) is smaller than the second term of the l. h. s., then one can consider another approximation, when the $k$-th species particle numbers are conserved. Then the chemical perturbation becomes an addition to the thermodynamic chemical potential.
[^13]: \[footn2\]If the particles involved into the inelastic processes are massive, then the formal dominant contribution is the exponential one over the temperature and grows as the temperature decreases. If the particles are massless or approximately massless, as in high-temperature QCD [@Arnold:2006fz], then a more complicated situation can occur, and one may need to compare some differences of processes’ rates (and not just equilibrium collision rates), arising in the collision matrix ($\tilde C^{ab}_{mn}$ in assignments of the [@Arnold:2006fz]). Under the same pair of used test-functions (indexed by $m$, $n$ in the $\tilde C^{ab}_{mn}$), and for the same pair of particle species, smaller differences of processes’ rates can be neglected. Comparing among *different pairs of test-functions* the smallest nonzero contributions dominate, or rather as can be obtained directly from the inverted collision matrix.
[^14]: Perhaps, the bulk viscosity calculated without constant test-functions (except for zero modes of the inelastic collision integrals, used to conserve charges) can provide a good interpolation.
[^15]: Not a bottleneck from some perturbations, because one actually doesn’t have a choice in the form of the momentum dependence of the perturbations corresponding to the transport coefficients. The kinetic perturbation can be of different forms of the momentum dependence.
[^16]: Another similar dominance can exist from particle species interacting weakly with all particles.
[^17]: The estimate can be easily inferred from [@Arnold:2000dr].
[^18]: There are forward scatterings and momentum interchange. As long as the particles are not distinguishable the momentum interchange from the elastic collisions is equivalent to the forward scatterings or absence of the elastic collisions at all.
[^19]: One should keep in mind that while studying the chemical perturbations $\tilde \varphi_k$ through the thermodynamic functions first the contributions from the transport coefficients’ terms should be subtracted.
[^20]: It’s assumed that particle numbers of the same species but with different spin states are equal. If this were not so then in approximation, in which the spin interactions are neglected and probabilities to have certain spin states are equal, the numbers of the particles with different spin states would be approximately equal in the mean free time. With equal particle numbers their distribution functions are equal too. This allows one to use the summed over the final states cross sections in the Boltzmann equations.
[^21]: Very high energy dependence of any used UrQMD cross section is not important because of the exponential suppression $e^{-\sqrt{s}/T}$. The used cross sections were cut on the ${\sqrt{s}=5~GeV}$ and were continued by a corresponding constant continuously at higher energies. At small enough momentums there is another somewhat weaker suppression. The momentum space density of each particle provides $p^2$ suppression. This may (partially) suppress some deviations from the experimental data of some UrQMD cross sections (like for the ${\Lambda p}$ pair) at ${\sqrt{s}\sim m_k+m_l}$. To estimate at what temperatures some discrepancies in cross sections can appear one can equate the $\sqrt{s}$ to the sum of the averaged one-particle energies $e_k$ (\[epsandek\]) of the two colliding particles.
[^22]: Averaged powers of the momentums are used, not powers of the averaged momentums.
[^23]: It may be mentioned that one should be also aware of possible differences in storing of the floating point numbers in different programming languages or while using different compilers.
[^24]: \[footn1\]This enhancement leads to the shortening of the mean free paths of the resonances of only this species, as needed. In the *formal* limit of this infinitely large enhancement other collision integrals can be neglected and the Boltzmann equation for this species decouples. Then from the solution of the Boltzmann equation for a single-component gas (see Sec. \[singcomsec\]) one concludes that the nonequilibrium perturbation to the distribution function of this species vanishes in this limit. Note that infinitely strong interactions also with particles of all other particle species would result in zero transport coefficients.
[^25]: Not the largest one. The effect of the enhancements of the resonances’ TCSs is of $5\%$ for the bulk viscosity and of $50\%$ for the shear viscosity so that TCS2s are additionally considered.
[^26]: This information, including some other information about the cross sections, is stored in the array SigmaLn of the file blockres.f
[^27]: The magnitudes of thermodynamic quantities can also be restricted by this condition or, conversely, not restricted even if transport coefficients diverge. See also Sec. \[singcomsec\] of this paper. The smallness of the shear and the bulk viscosity gradients can also be checked by the condition of smallness of the $T^{(1)\mu\nu}$ (\[T1\]) in compare to the $T^{(0)\mu\nu}$ (\[T0\]). Of course, the next corrections should be small too.
[^28]: It’s clear that the mean free paths should be smaller than the system’s size too.
[^29]: Primed indexes run over the particle species without regard to their spin states. This assignment is clarified more in Sec. \[CalcSecA\].
[^30]: This is an approximation. In fact this should be a range in which particles of different particle species have their own freeze-out points.
[^31]: It should be mentioned that if the particles of the $k$-th particle species are bosons and if $\mu_k(x^\mu) \geq m_k$ then there is a (local) Bose-Einstein condensation for them, which should be treated in a special way.
[^32]: The relative deviations of the thermodynamic quantities grow with the temperature for some fixed value of the chemical potential and tend to some constant.
[^33]: In the hadron gas it’s believed that the vacuum masses are large in compare to their thermal corrections for the most of the hadrons at temperatures $T\lesssim 200~MeV$ or even higher ones. Then expanding over the thermal correction in the matrix elements, one would get even smaller corrections than the ones to the equation of state in coupling constants (because of coupling constants next to the matrix elements) in a perturbation theory, e. g. chiral perturbation theory.
[^34]: In systems with only the elastic collisions each particle species have their own “conserved quantum number”, equal to 1.
[^35]: The $+,-,-,-$ metric signature is used throughout the paper.
[^36]: The kinetic equilibrium implies that the momentum distributions are the same as in the global equilibrium. Thus, a state of a system with both the pointwise (for the whole system) kinetic and the pointwise chemical equilibria is the global equilibrium.
[^37]: Also this form of $T^{(1)\mu\nu}$ respects the second law of thermodynamics [@landau6] (Section 136).
[^38]: The factor $\gamma_{kl}$ cancels double counting in integration over momentums of identical particles. The factor $\frac12$ comes from the relativistic normalization of the scattering amplitudes.
[^39]: One can prove that the $N'' \times N''$ matrix $\tilde{A}_{ab}$ in (\[ABCE\]) is not degenerate if there are $N''$ linearly independent conserved charges. Then one can prove that the denominator in the (\[Rdef2\]) is not zero.
[^40]: Direct computation gives $(1,\overset{\circ}{
\overline{\pi^\mu_k \pi^\nu_k} })_k \propto (C_1 U^\sigma U^\rho +
C_2\Delta^{\sigma\rho}) \Delta_{ ~~\sigma\rho }^{\mu\nu} = 0$, $(p_k^\lambda, \overset{\circ}{ \overline{\pi^\mu_k \pi^\nu_k}
})_k \propto (C_1 U^\lambda U^\sigma U^\rho + C_2U^\lambda
\Delta^{\sigma\rho} + C_3U^\sigma \Delta^{\lambda\rho})
\Delta_{~~\sigma\rho}^{\mu\nu} = 0$.
[^41]: One can first derive the same equations for the $A_k$ and $C_k$, treating them as different functions for all $k$, with the coefficients $A_{kl}^{rs}$ and $C_{kl}^{rs}$ having the same form as the $A_{k'l'}^{rs}$ and $C_{k'l'}^{rs}$. Then after summation of the equations over spin states of identical particles and taking ${A_k=A_{k'}}$, ${C_k=C_{k'}}$ one reproduces the system of equations for the $A_{k'}$ and $C_{k'}$.
[^42]: These are the particles which are more or less reliably detected [@Nakamura:2010zzi].
[^43]: This discrepancy could be noticed earlier the results of the [@nhngr] and the [@toneev]. Though they required confirmations or justifications.
[^44]: The nonequilibrium chemical potential-like perturbations of the form $T n_{s,k}\ln\gamma_s$ ($n_{s,k}$ is the number of strange quarks in hadrons of the $k$-th species) [@Becattini:2005xt], obtained from statistical description and reflecting suppression of the strange particle numbers, obviously violate conservation laws to some extent. However, they are used because of the simplicity in phenomenological estimating calculations. There are also more elaborated calculations of the chemical perturbations, see e. g. [@Letessier:2005qe]. Also see [@Heinz:2007in] for some discussions.
[^45]: It is the differential cross section for identical particles. The total cross section is ${\int
\frac{d\Omega}2 2r^2=4\pi r^2}$.
[^46]: This reproduces the result of Chapman and Enskog in the nonrelativistic theory for the shear viscosity. The vanishing value of the bulk viscosity is obtained in the limit $m\rightarrow \infty$ [@landau10] (Sections 8, 10). The result of the vanishing bulk viscosity of a monoatomic classical gas in the nonrelativistic theory is attributed to James Clerk Maxwell, see [@Weinberg:1971mx].
[^47]: The vanishing value of the bulk viscosity of a monoatomic classical gas in the ultrarelativistic limit is attributed to I. M. Khalatnikov, see [@landau10].
[^48]: This formula is justified only for rarified systems where the ideal gas equation of state is applicable.
[^49]: So that it looks like the double counting factor $1/2$ is lost in the calculations of the viscosities in the [@prakash] (and presumably for the heat conductivity), and is not lost in the current algebra total cross section.
[^50]: Actually all powers of the $P^2$ can also be taken into account in the same way if all mutual differences between all powers of all the $P^2$ terms are integers.
[^51]: It represents some sum over all possible collisions. In the case of the same species one factor $\gamma_{k'l'}$ just cancels the double counting in momentum states after scattering and another factor $\gamma_{k'l'}$ also reflects the fact that scattering takes place for ${{n_{k'}\choose 2} \approx
\frac12n_{k'}^2}$ pairs of undistinguishable particles in a given unit volume.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In the present work we performed magnetoresistance measurement in a hybrid system consisting of an arc-shaped quantum point contact (QPC) and a flat, rectangular QPC, both of which together form an electronic cavity between them. The results highlight a transition between collimation-induced resistance dip to a magnetoresistance peak as the strength of coupling between the QPC and the electronic cavity was increased. The initial results show the promise of hybrid quantum system for future quantum technologies.'
author:
- Chengyu Yan
- Sanjeev Kumar
- Patrick See
- Ian Farrer
- David Ritchie
- 'J. P. Griffiths'
- 'G. A. C. Jones'
- Michael Pepper
title: 'Magnetoresistance in an electronic cavity coupled to one-dimensional systems '
---
Recent development in quantum technologies has stimulated research activities in integrating different quantum components in order to realize complex functionality[@HCR07; @MCG07]. It is therefore of fundamental interest to investigate coupling between discrete quantum devices. Coupling between electronic cavity and other quantum devices, such as quantum point contact[@KEA97; @DTW01; @YKP16; @YKP17; @SPK17] (QPC) and quantum dot[@ROZ15; @DLL17; @FOR17] (QD), has attracted considerable attention. A hybrid device consisting of a QPC and an electronic cavity, as an example, provides a unique platform to investigate electronic equivalent of optical phenomena. This may be understood from the fact that electrons in such a system transport ballistically and accumulate phase along the quasi-classical trajectories, which is a close analogue of an optical cavity. Previous studies based on QPC-cavity hybrid devices reported results based on classical trajectories of electrons[@KEA97; @DTW01; @HHH99; @HHH00] as well as quantum effects manifested as conductance fluctuations[@KEA97; @DTW01] and Ahronov-Bohm phase shift as a function of cavity size[@DTW01].
In the present work, we studied magnetoresistance in a hybrid system in a controlled manner with the assistance of two QPCs which form an electronic cavity between them. We show the strength of coupling between the QPC and cavity states can be monitored by oscillation in the magnitude of central peak/dip in magnetoresistance.
![The experiment setup and device characteristics. The blue trace shows the characteristic of arc-QPC as a function of gate voltage V$_{sg1}$; the red trace illustrates the behaviour of flat-QPC against gate voltage V$_{sg2}$. The series resistance was not removed. Inset depicts an illustration of the experiment setup, the yellow blocks represent electron-beam lithographically defined metallic gates while the red squares highlight the Ohmic contact. The length (width) of the flat-QPC is 700 nm (500 nm). The radius of the arc is 2 $\mu$m with an opening angle of 45$^\circ$. Both the length and width of the QPC formed in the center of the arc, i.e arc-QPC, are 200 nm. []{data-label="fig:1"}](Fig1){height="2.0in" width="3.2in"}
{height="2.5in" width="6.0in"}
{height="2.5in" width="6.0in"}
The devices studied in the work were fabricated from a high mobility two-dimensional electron gas (2DEG) formed at the interface of GaAs/Al$_{0.33}$Ga$_{0.67}$As heterostructure. The measured electron density (mobility) was 1.80$\times$10$^{11}$cm$^{-2}$ (2.17$\times$10$^6$cm$^2$V$^{-1}$s$^{-1}$) at 1.5 K, which ensured that both the calculated mean free path and phase coherence length [@AAK82; @PPP89] were over 10 $\mu$m which were larger than electron propagation length. The experiments were performed in a cryofree dilution refrigerator with a lattice temperature of 20 mK using the standard lockin technique.
The hybrid device consists of a pair of arc-shaped gates with a QPC (referred as arc-QPC) forming in the center of arc-gates and another pair of rectangular QPC (named as flat-QPC) as depicted in Fig. \[fig:1\]. The QPCs are assembled in such a way that the geometrical center of the arc (shaped gates) aligns with the saddle point of the flat-QPC. An electronic cavity is formed when QPCs are activated by depleting the 2D electrons underneath the gates[@YKP16; @YKP17]. Both the arc-QPC and flat-QPC showed well defined one-dimensional conductance quantization when they were characterised individually, Fig. \[fig:1\].
In the presence of a small transverse magnetic field, the magnetoresistance of flat-QPC or arc-QPC exhibited a weak-localization peak similar to reported previously[@BHD95; @KIL96]. However, the non-trivial features started appearing when the hybrid device was formed, i.e. both flat-QPC and arc-QPC were activated.
In the first experiment, the flat-QPC served as an emitter while the arc-QPC was used as a collector, see inset of Fig. \[fig:1\]. The voltage applied to the flat-QPC was incremented slowly corresponding to a conductance of G$_0$ (G$_0$=$\frac{2e^2}{h}$) up to 1D channel fully open while the arc-QPC was fixed at G$_0$. The magnetoresistance was investigated in three different regimes according to flat-QPC conductance.
In regime 1, the flat-QPC was incremented from G$_0$ to 4G$_0$, Fig. \[fig:2\](a). A dip in resistance (marked by the magenta dashed line) was observed around 0 T when the flat-QPC conductance G $\leqslant$ 2G$_0$ which is due to the fact that the injected electrons had a relatively small angular spread owing to strong collimation in low conductance regime[@LAC90; @HBG98]. The electrons tend to propagate from the flat-QPC through the arc-QPC directly without backscattering; however, the applied magnetic field guides the injected electrons to the arc-shaped boundary wall of the arc-QPC and thus results in backscattering, which in turn triggers a rise in resistance. In this respect, our hybrid system is similar to a long quantum wire where scattering at the boundary was suggested to introduce a central dip in magnetoresistance[@TRS89]. An offset in central dip in magnetoresistance of 3 mT could be due to magnetic hysteresis of the superconducting magnet. On increasing G to 4G$_0$, a central magnetoresistance peak started forming. The zero-field magnetoresistance peak in electronic billiards is a result of geometry induced closed loop[@BKM94] (in other words, an analogue to weak localization). A large angular spread at higher G makes injected electrons to be reflected at the boundary wall of the arc-shaped QPCs, thus forming a close loop even at zero magnetic field; on the hand, a relatively small angular spread at low conductance makes such reflection unlikely to happen without the assistance of a magnetic field. The backscattered electrons will be refocused to the saddle point of flat-QPC.
In regime 2, Fig. \[fig:2\](b), the flat-QPC was set from 4G$_0$ to 6G$_0$, the magnitude of the central peak fluctuated in the sense that the central peak gradually smeared out when the flat-QPC conductance was close to 5G$_0$, and then reappeared on further increasing the conductance of flat-QPC. The fluctuation will be discussed in detail in Fig. \[fig:5\]. Meanwhile it was also noticed that multiple weak-satellite peaks, marked by black arrows in Fig. \[fig:2\](b), occurred in this regime. It was suggested[@DTW01] in a previous work that the appearance of these satellite peaks was an indication of Aharonov-Bohm effect and each peak was associated with a particular classical orbit. We suggest that although the satellite peaks might be relevant with classical orbits, however, Aharonov-Bohm effect did not occur in our experiment considering the fact that the satellites peaks were almost absent in regime 1 or regime 3.
In regime 3 (6G$_0$ to fully open emitter), Fig. \[fig:2\](c), the central peak gradually splits into two peaks around the 1D-2D transition regime of the flat-QPC and eventually all the features smeared out and only a smooth background was observed with the flat-QPC entering into the 2D regime. The smooth background agrees well the weak-localization signal when the arc-QPC was characterised individually.
To be noted that Shubnikov-de Haas oscillation started appearing in all the three regimes when the magnetic field exceeded $\pm$0.13 T (data is not shown).
![Representative electron trajectories with flat-QPC and arc-QPC acting as emitter, respectively. The solid traces represent the trajectory of incident electrons whereas the dashed traces illustrate the reflected electrons. In plot (a), the solid and dashed traces are offset intentionally for clarity, which otherwise should overlap together. The thick black arrows indicate current injection direction. []{data-label="fig:4"}](Fig4){height="1.0in" width="3.6in"}
To ensure the observation did not simply arise from the superposition of the magneto-spectrum of two individual QPCs, we reversed the role of emitter and collector. In setup II the arc-QPC was utilized as an emitter and incremented while the flat-QPC functioned as a collector and was fixed at G$_0$. In addition, the ac signal is fed to the left Ohmic \[Fig. \[fig:1\](a)\] whereas the right Ohmic is grounded in setup II.The results are summarized in Fig. \[fig:3\]. Results in regime 1, Fig. \[fig:3\](a), was similar to that observed with setup I. However, the central dip dominated in regime 2 \[Fig. \[fig:3\](b)\] and 3 \[Fig. \[fig:3\](c)\] which was considerably different from its counterpart in Fig. \[fig:2\] where more features were resolved. The behaviour in setup II was similar to the magnetoresistance in two regular QPC in series[@HBG98]. It is interesting to mention that satellite peaks observed in Fig. \[fig:2\](b) did not occur in setup II. A comparison between setup I and II also suggests that the complicated evolution of magnetoresistance observed in Fig. \[fig:2\] did not directly arise from the form of wavefunction at different emitter conductance; otherwise, setup II should exhibit similar behaviour.
The difference between the results from two setup could be understood with a semi-classical picture as shown in Fig. \[fig:4\]. Electrons injected from the flat-QPC, which aligns with the geometrical centre of the arc (i.e. arc-QPC), experience an arc-shaped reflector which traps the electrons in an electronic cavity defined by these QPCs. The injected electrons after reflection at the boundary wall of the arc would be directed towards the flat-QPC. Owing to the geometry of cavity defined between the arc- and flat-QPCs, electrons would be trapped in a closed loop such as events 1$\rightarrow$4 as shown in Fig. \[fig:4\](a) until the total propagation length exceeded the mean free path; phase associated with such a close loop is unlikely to be averaged out, therefore corrections to the resistance, i.e. the central magnetoresistance peak, due to the accumulated phase was observable. On the other hand, the trajectory of electrons injected from the arc-QPC, i.e. setup II, did not necessarily form a closed loop, so that it was relatively easy for the injected electrons to get through the hybrid system via a series of scattering events, for instance events 1$\rightarrow$3 as depicted in Fig. \[fig:4\](b). Electron trajectory in the second scenario is more arbitrary and the trajectory-determined phase tends to be averaged out, which leads to no obvious corrections in the resistance.
![Fluctuation of the central feature as a function of flat-QPC conductance. The relative strength of the central feature, $\Delta$R = R$_M$-(R$_L$+R$_R$)/2, shows quasi-periodic oscillation, where R$_M$, R$_L$ and R$_R$ refer to resistance at the given magnetic field marked in Fig. \[fig:2\]. []{data-label="fig:5"}](Fig5){height="2.0in" width="3.6in"}
After addressing the difference between the two setup, we discuss a possible mechanism behind the observed fluctuation of the central features with flat-QPC serving as an emitter. To quantify the fluctuation, we defined the strength of the central feature (could be dip or peak) as such $\Delta$R = R$_M$-(R$_L$+R$_R$)/2, where R$_M$, R$_L$ and R$_R$ refer to the resistance measured at given magnetic field marked in Fig. \[fig:2\] (although in dip dominant regime there was not noticeble feature at $L$ or $R$, we still use the resistance at the same field for systematic investigation). It was seen that $\Delta$R followed a quasi-periodic oscillation[@SUO94; @YKP16; @YKP17; @KEA97; @DTW01] when the flat-QPC was tuned into the 1D regime (V$_{sg}$ $\leqslant$ -0.25 V); the fluctuation smeared out when the flat-QPC entered the 2D regime as shown in Fig. 5. The fact that the peak of oscillation does not necessarily occur at each conductance plateau suggesting that it is not simply associated with occupation of 1D subband or electron collimation, which would otherwise produce peaks corresponding to each conductance plateau. Instead, the oscillation was an indication of the coupling between the cavity and QPC sates. Each peak in Fig. 5 is a result of removing a cavity mode, therefore peaks in $\Delta$R should occur when the change in radius $r$ of cavity matched a condition [@KEA97], $\Delta r$ = N$\times$$\lambda_F$/2, where N is an integer and $\lambda_F$ is the Fermi wavelength.
In conclusion we have shown magnetoresistance in a hybrid system consisting of QPCs coupled via an electronic cavity. It was found that the central magneto-feature around 0 T underwent a transition from dip into peak when the cavity was present whereas resistance dip dominated when the cavity was effectively absent. An oscillation of the strength of the central magneto-feature was observed as a consequence of coupling between the QPC and cavity sates. The results provide insight of coupling between discrete quantum devices which is valuable for further development of integrated quantum systems.
The work is funded by the Engineering and Physical Sciences Research Council (EPSRC), United Kingdom.
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---
abstract: 'Many real-world domains can be expressed as graphs and, more generally, as multi-relational knowledge graphs. Though reasoning and learning with knowledge graphs has traditionally been addressed by symbolic approaches such as Statistical relational learning, recent methods in (deep) representation learning have shown promising results for specialised tasks such as knowledge base completion. These approaches, also known as distributional, abandon the traditional symbolic paradigm by replacing symbols with vectors in Euclidean space. With few exceptions, symbolic and distributional approaches are explored in different communities and little is known about their respective strengths and weaknesses. In this work, we compare distributional and symbolic relational learning approaches on various standard relational classification and knowledge base completion tasks. Furthermore, we analyse the complexity of the rules used implicitly by these approaches and relate them to the performance of the methods in the comparison. The results reveal possible indicators that could help in choosing one approach over the other for particular knowledge graphs.'
author:
- Sebastijan Dumančić$^1$
- |
Alberto García-Durán$^2$ Mathias Niepert$^3$ $^1$KU Leuven, Belgium\
$^2$EPFL, Switzerland\
$^3$NEC Labs, Germany [email protected], [email protected], [email protected]
bibliography:
- 'starai.bib'
title: |
A Comparative Study of Distributional and Symbolic Paradigms\
for Relational Learning\
---
Introduction
============
Recent years have created a dichotomy in the field of *Statistical relational learning* (SRL), concerning machine learning with relational data which contains instances and their mutual relationships. The *symbolic* paradigm [@Getoor:2007; @Raedt:2016:SRA:3027718], leveraging the representational and reasoning capacity of first-order logic to compactly represent relational data, has typically dominated the field. Inspired by the success of (deep) representation learning [@Goodfellow:2016:DL:3086952], an alternative *distributional* paradigm has emerged, with examples in *Knowledge graph embeddings* [@NickelReview] and *Graph neural networks* [@Hamilton2017RepresentationLO]. These distributional methods *re-represent* relational data as vectors and/or matrices in the Euclidean space so that the standard feature-based machine learning methods can be used.
The two research directions have largely been developed in isolation and little understanding is currently available on their respective advantages, even though the community has recognised the benefits of both [@DBLP:conf/uai/MinerviniDRR17; @NTPs; @Manhaeve2018; @Evans2018LearningER]. Moreover, they focus on different tasks and employ different evaluation metrics. Distributional methods focus on the knowledge base completion (KBC) task measuring the performance through the ranking of proposed entity completions. Symbolic methods, on the other hand, focus on learning from small relational data with complex forms of logical reasoning and evaluate the performance through the accuracy of predictions.
Contrasting the paradigms upfront, they exhibit a stark contrast on three dimensions: reasoning capabilities, scalability and interpretability. By relying on logic, the symbolic methods are capable of complex reasoning patterns and are flexible enough to answer any query over a domain (without a need to commit to a predefined target); this is sacrificed with distributional methods [@DBLP:journals/jair/TrouillonGDB19]. The distributional methods are scalable and can operate on knowledge graphs containing millions of facts, whereas scalability is the major challenge for symbolic methods. Symbolic methods inherit interpretability from first-order logic, whereas distributional methods are difficult to interpret. Moreover, distributional methods have difficulties handling unseen instances and, consequently, have to be re-trained every time new data arrives.
This work contributes towards a better understanding of the relative strengths and weaknesses of the aforementioned paradigms. We focus on the most prominent learning approaches within both paradigms, namely Inductive logic programming and Knowledge graph embeddings, and systematically compare them using standard benchmarks from both communities. We include both quantitative, in terms of performance, and qualitative analysis, in terms of various data properties, showing that there is no absolute winner amongst the paradigms but data properties (such as neighbour degree and diameter) can help decide which paradigm to use.
Background and Related work
===========================
*Symbolic SRL* methods use first-order logic to represent the data and reason with it. For instance, a popular symbolic SRL framework Problog [@De-Raedt:2007aa] represents a fact stating that two people (`marc` and `eve`) are friends as
`friends(marc,eve).`
Predictive models are expressed as a collection of rules. For instance, stating that *every person having a friend that smokes is also a smoker* can be expressed in Problog as a *clause*
`smokes(X) :- smokes(Y),friends(X,Y).`
To overcome the disadvantage of deterministic rules concluding only *true* and *false* statements, Problog annotates rules with probabilities to quantify the uncertainty of conclusions. For instance, one a probabilistic rule
0.4 :: `smokes(X) :- smokes(Y),friends(X,Y).`
states that every person that has a smoking friends has 40 % of being a smoker.
*Distributional SRL* methods replace symbols (i.e., an entity `eve` and a relation `friends`) with vectors and/or matrices in the Euclidean space. That way the prediction is performed through algebraic manipulations instead of more costly logical inference. Here we focus on knowledge graph embedding ([KGE]{}) approaches as the most prominent amongst the existing approaches. The underlying idea of KGE methods is to associate a score with atoms in a database (i.e., `friends(marc,eve)`). Learning then consists of finding the vector representation of instances and their relations by maximising the scores of the atoms in the database and minimising the score of atoms not in the database. Two prototypical examples are:
- **TransE** [@BordesNIPS2013] which interprets relations as translations between instances in the Euclidean space. For each atom `r(h,t)`, `r` being a relation and `h` and `t` *head* and *tail* instances respectively, the score equals $$s(r,h,t) = -||\mathbf{e}_h + \mathbf{e}_r - \mathbf{e}_t ||,$$ i.e., the vector representation $\mathbf{e}_h$ of a *head* instance translated by the relation vector $\mathbf{e}_r$ should be close to the vector representation $\mathbf{e}_t$ *tail* instance
- **DistMult** [@YangYHGD14a] which focuses on pairwise interactions of *latent* features with the following score $$s(r,h,t) = (\mathbf{e}_h \ocircle \mathbf{e}_r)\mathbf{e}_t^T$$ where $\ocircle$ is an element-wise product.
Checking the validity/truthfulness of any fact comes down to evaluating its score. The majority of KGE methods proposed so far [@EmbeddingsOverview] are a variation on the above scoring functions.
Comparing Symbolic and Distributional Methods
---------------------------------------------
Though limited, several works offer insights into the differences between the two paradigms. @NickleNIPS2014 \[[-@NickleNIPS2014]\] and @toutanova2015observed \[[-@toutanova2015observed]\] show that including both latent features from [KGEs]{} and the observable features, in form of random walks over knowledge graphs, in a joint model can greatly increase the performance and reduce the learning complexity. However, they offer no greater insight into possible reasons. @pujara:emnlp17 \[[-@pujara:emnlp17]\] show that [KGEs]{} have difficulties handling data with a high degree of sparsity and noise – which is the case with many automatically constructed knowledge graph. @GrefenstetteTFDS \[[-@GrefenstetteTFDS]\] introduces a formal framework for simulating logical reasoning through tensor calculation, which can be seen as a form of embeddings that does not require learning.
The work most related to ours is that of @toutanova2015observed \[[-@toutanova2015observed]\], @VigILP2017 \[[-@VigILP2017]\] and @DBLP:journals/jair/TrouillonGDB19 \[[-@DBLP:journals/jair/TrouillonGDB19]\]. @VigILP2017 \[[-@VigILP2017]\] compare symbolic SRL methods with embeddings obtained by the Siamese neural network, and focus on analysing the impact of the available *background knowledge* on the performance. Their results indicate that [KGEs]{} might be beneficial when the background knowledge about the task at hand is limited; if such knowledge is available, then the symbolic methods are preferable. @DBLP:journals/jair/TrouillonGDB19 \[[-@DBLP:journals/jair/TrouillonGDB19]\] study the which kinds of relational reasoning properties can be captured by distributional models. Our work presented in this paper differs in a way that it goes beyond quantitative analysis and includes substantial qualitative analysis w.r.t. the dataset properties.
Aims, Materials and Tasks
=========================
Aims
----
The main goal of this study is to put the distributional and symbolic relational learning approaches on equal grounds. Concretely, we focus on the following questions:
- *How do standard symbolic systems, which can manipulate the relational data directly, compare to the distributional systems, which approximate relational data in Euclidean spaces, on the standard benchmarks from both communities?*
- *Can we identify data properties which indicate the suitability of individual paradigms?*
To answer these questions, we focus exclusively on the classification and completion tasks as they give us a well-defined and clear performance measure, in contrast to the clustering task which is ill-defined.
We do not perform the runtime comparison as it is rather difficult to assess confidently – small implementation tricks can make a huge difference in runtimes of the distributional methods. Likewise, the performance of the symbolic methods depends on the provided language bias – syntactic instructions on how to construct logical formulas. This could lead us to wrong conclusions.
Materials
---------
### Datasets
We focus on standard benchmarks in both communities. From the symbolic community, we focus on the following datasets: UWCSE, Mutagenesis, Carcinogenesis, Yeast, WebKB, Terrorists and Hepatitis. The descriptions of datasets can be found in [@Dumancic2017]. From the distributional community, we focus on the FB15k-237 and WN18-RR which are accepted as standard. The description of the datasets can be found in [@dettmers2018conve].
Relational classification datasets often take the form of a *hypergraph*: an edge (a relationship) can connect more than two instances. [KGEs]{} cannot easily handle such datasets as they require relationships to be binary. To allow [KGEs]{} to operate on such datasets, we perform the *reification* – a decomposition of hyperedges to a set of binary edges.
### Symbolic Methods
Various machine learning methods impose various biases, which plays an important role in machine learning. To better understand how such biases influence the performance, we experiment with methods from three different families: a relational decision tree *TILDE* [@Blockeel1998285], a relational version of kernel machines *kFOIL* [@Landwehr:2006:KLS:1597538.1597601] and a kNN with the relational similarity measure of *ReCeNT* [@DumancicMLJ2017].
The above-mentioned learners belong to a subfield of SRL called Inductive logic programming (ILP) [@Raedt:2008:LRL:1202793], concerned with building machine learning models in the form of logic programs. We focus the ILP approaches as learning is more developed than with other symbolic SRL methods which mostly focus on reasoning.
### Distributional Methods
Due to the sheer number of the existing distributional methods [@EmbeddingsOverview], including a large sample would be infeasible. Moreover, it is not clear whether a big difference in performance is expected as many methods perform comparably when properly tuned [@DBLP:conf/rep4nlp/KadlecBK17]. Therefore, we focus on the prototypical and most influential approaches – TransE, DistMult and ComplEx [@trouillon2016complex]. Note that ComplEx produces embeddings in the *complex Euclidean space* and that each entity is associated with two embeddings - *real* and *imaginary* one. In order to create a single embedding out of these two, we concatenate the two embeddings. Even though TransE is one of the earliest methods, several recent works show that TransE’s performance is higher than that reported for the “updated” versions of TransE on the same datasets [@DBLP:conf/uai/Garcia-DuranN18].
Distributional methods are usually trained to assign a high score to all facts in a knowledge base, not to specific *target* facts. To make sure this training criterion does not put distributional methods at a disadvantage, we use the embeddings as the input data for a classifier learning the predictive model for the pre-specified target. To match the biases of the symbolic methods, we conjoin the embeddings with a decision tree, SVM and kNN classifiers. We focus on the *shallow* classifiers, as it would be difficult to disentangle the contribution of latent layers from the quality of embeddings.
As distributional methods focus on the knowledge base completion task, they assume that all instances are given at once and fill in the missing links in data. Therefore, handling unseen instances is challenging. We do not address this issue here, but simply learn the representation of both training and test data at the same time (with labels excluded). It is, however, worth noting that distributional methods have a certain advantage due to this.
### Dataset Properties
To better understand conditions under which each of the paradigms is preferable, we analyse the properties the relational classification datasets and contrast them to the performance of individual methods. The datasets were transformed into graphs (with reification if necessary) by treating each instance/entity as a node and the relationships as edges, and calculating various graph properties using the `networkX` package [@Schult08exploringnetwork]. The full list of the analysed properties is available in Table \[tab:informedgraph\]. Some of these properties are only defined over connected graphs (i.e., there is a path between any two nodes in a graph); to ensure this is satisfied, we calculate these properties over the *connected components* of a graph – a collection of subgraphs in which any two nodes are connected to each other via paths – and report an average value.
Moreover, the relational classification datasets often distinguish between entities and their attributes. However, [KGEs]{} do not make such a distinction but simply treat attribute values as nodes in a graph. To compensate for this, we analyse two types of graphs: *informed* one which is aware of the attributes, and *uninformed* one which treats attribute values as graph nodes.
Additionally, we include various meta-properties of the datasets. These reflect properties such as the number of attributes and relations in the datasets, as well as the properties of the components, their size and number. Given the two types of graphs we can construct from a relational dataset, we include measure aiming at quantifying what is the difference between the two graphs. To do so, we introduce a measure of *edge reduction* (the proportion of edges that were lost after the conversion to the informed graph) and *degree proportion* (a ratio between the average degree in the informed versus the uninformed graph).
{width=".85\linewidth"}
Tasks
-----
### Relational Classification
In the relational classification task, certain entities have an associated label and the task is to predict those labels. We perform standard nested cross-validation (respecting the provided splits) and report the relative performance of the methods in terms of differences in accuracy, $$acc_{distributional} - acc_{symbolic}$$ averaged over individual splits. The accuracy is reported as a proportion of correct predictions, within the range of \[0,1\]. The labels were excluded from the data when learning the embeddings and considered only during the training of the classifier.
The embeddings were obtained before learning a classifier. The dimensions of the embeddings were varied in $\{10, 20, 30, 50, 80, 100\}$; we include smaller dimension because standard relational datasets tend to have a much smaller number of entities than the KBC datasets. All embeddings were trained to 100 epochs and saved in steps of 20. We do not use the validation set and metrics such as mean reciprocal rank to select the best hyper-parameters for the embeddings, as they may not be sufficiently correlated with the predictive accuracy; instead, we treat the dimension and the number of epochs for training as additional parameters while training the classifiers as part of the inner cross-validation loop.
### Knowledge Base Completion
The KBC task corresponds to providing a ranking of missing triplets: given a head/tail entity and a relation, rank the possibilities for the remaining entity. For the distributional methods, we report the results from [@dettmers2018conve] which are considered to be state-of-the-art. These experiments include several methods: DistMult [@YangYHGD14a], ComplEx [@trouillon2016complex], R-CGN [@Schlichtkrull2017ModelingRD] and ConvE [@dettmers2018conve].
We include TILDE as the representative of the symbolic paradigm, learning a model for each relation. This was the only symbolic method scaling to the data of this size. We follow the evaluation procedure as described in [@dettmers2018conve] and report the *hits @ K* metric (*is the correct answer among the top K ranked answers*) [@BordesNIPS2013]. In order to calculate the hits@K metric, we need to associate a score with each possible completion to a query. However, TILDE typically produces binary scores as either true or false. Thus, to obtain fine-grained scores from TILDE we proceed in the following way: we check which rule triggered each possible completion and take its confidence (i.e., the proportion of correct predictions on the training set) as a score.
Discussion
==========
Relational Classification
-------------------------
When analysing the results of the relational classification, we are interested in answering the following three questions:
- How do the two paradigms perform relative to each other?
- Is there a significant difference in the performance of the embedding approaches?
- Are there any data properties that correlate with the trend in performance?
The results[^1] (Figure \[fig:classification\] reports the relative difference in performance: positive numbers (orange) indicate that [KGEs]{} perform better, while the negative numbers (blue) indicate that symbolic methods perform better) indicate that no paradigm is the *absolute* winner: different methods are suitable for different tasks. However, observing the performance in the case of decision trees and SVMs reveals a pattern dividing the datasets into three groups. The first group consists of the Hepatitis and Carcinogenesis datasets on which the distributional methods outperform the symbolic approaches. The second group consists of the Mutagenesis and UWCSE on which the methods either perform similarly or the decision is split (on the UWCSE datasets, symbolic methods perform better with decision trees while distributional methods perform better with SVMs). The third group contains the Terrorists, Yeast and WebKB datasets on which the symbolic methods perform better.
Interestingly, when contrasted with the dataset properties of the informed graphs (presented in Table \[tab:informedgraph\]), only two follow the observed trend: the *average neighbour degree* and the *diameter* of the graph. We focus the attention of the informed graph, as the statistics of the uninformed graphs did not show any trend related to the performance. Distributional approaches outperform the symbolic ones on the datasets with a higher neighbour degree (Hepatitis and Carcinogenesis, a split decision of UWCSE and Mutagenesis). This property, calculating *how many neighbours my neighbours have*, indicates the density of interactions among the nodes in a connected component: higher values mean there are more interactions between the nodes. It suggests that, for the relational classification, the distributional approaches outperform the symbolic ones on densely connected graphs. The diameter of a graph is the *longest shortest path between any pair of nodes*, i.e., it estimates whether the connected components are *compact* (nodes are close to each other) or *spread out* (many nodes are far apart). According to this measure, the symbolic methods perform better in the extremal part of the range (covering the lower and the upper part of the range), while distributional methods prefer the middle range.
Several meta-properties correlate with the performance trend. Firstly, the datasets on which the symbolic methods perform better have a substantially larger number of attributes than the ones on which the distributional methods win. Secondly, the average size of the components follows the trend of the diameter: the symbolic methods perform better at the extremal part of the range. Thirdly, according to the *edge reduction* measure, the symbolic methods perform better when the majority of edges in the uninformed graph actually belongs to the attribute-value assignment, i.e., the edge reduction values are higher. The exception to this is the Hepatitis dataset, which still satisfies the observation that the symbolic methods work better on the datasets with more attributes.
Though these measures indicate correlation and might not be a definite way to make a decision, they offer a simple test that indicates a preference for a certain paradigm. All of the indicative properties can be calculated efficiently using only the data. The reliance of the symbolic methods on search procedures explains why they underperform on the datasets with higher neighbour degree: increasing this value enlarges the search space and the symbolic methods, which rely on the local search procedures, might be stuck in the local optima. Distributional methods might be better suited for such scenarios as they leverage all available information by design.
The observed pattern, however, disappears with the kNN classifier. The distributional methods gain the advantage on the Mutagenesis datasets, but the symbolic approaches are favoured on the remaining datasets. However, a general trend is that the differences in performances are much less pronounced compared to the results with decision trees and SVMs, except on the WebKB dataset. These results indicate that the embedding methods work as well as the manually designed methods for estimating the similarity of relational objects and, therefore, form a viable alternative.
Comparing the performances of the embedding methods indicates that there is no significant difference between them. This is an interesting observation in itself as it indicates that, when it comes to relational classification, the choice of the embeddings methods matters less than the choice of the classifier.
Knowledge Base Completion
-------------------------
[0.33]{} {width=".7\linewidth"}
[0.33]{} {width=".7\linewidth"}
[0.33]{} {width=".7\linewidth"}
To understand better why is this the case, we analyse the rules learned by TILDE on the FB15k dataset. Specifically, we inspect their complexity – the number of relations the rules are composed of, and the number of rules per relation – and their effectiveness – estimated by the precision (out of all predictions by a rule, how many of them are correct) and coverage (how many triplets a rule covers). We discuss two observations.
First, it is interesting to contrast the number of *connected relations* in data – given a relation, *how many other relations share an entity with that relation* (Figure \[fig:numrelationsentity\]) – and the number of distinct relations TILDE uses to predict the existence of a triplet for a specific relation (Figure \[fig:numrelationsrule\]). The former tells us how much information we have about a relation, while the latter estimates how much information we need to predict the existence of the relation between two entities. Interestingly, we observe a stark contrast between the two distributions: whereas most relations are connected to more than 40 other relations, the majority of TILDE rules for a specific relation contain up to only four distinct relations. That is, out of 40 relations an entity participates in, only a small fraction of those is useful to predict the existence of other relationships. Therefore, useful information is sparse.
Second, the extracted rules have a high precision (Figure \[fig:precvscov\], right), i.e., how many of the predictions made by a rule are correct (meaning that they exist in the ground truth), despite their simplicity: for the vast majority of rules, more than 90% of predictions are correct. If we contrast that with the coverage of the rules, i.e., how many triplets a rule declares as true, we see that the rules with the highest precision are not only the ones with small coverage; instead, the rules with high coverage are spread through the coverage spectrum, including the rule covering hundreds and thousands of triplets (the last bin includes rules with a coverage higher than 400). Moreover, the rules with the lowest precision have a small coverage.
These two observations suggest that a big advantage of TILDE is that it can be selective about which information it uses to predict the existence of relations between pairs of entities. The knowledge graph embeddings do not have this ability and, by design, use all available information for link prediction.
A disadvantage of the symbolic methods are the longer training times on the KBC tasks. However, the rules have to be extracted only once and can be applied every time a new data arrives. Distributional methods, on contrary, have to be re-trained every time a new entity arrives, as they cannot handle unseen data introducing new entities whose embeddings cannot be calculated.
Conclusion
==========
Many problems nowadays are naturally expressed in the form of relational and graph structured data. This includes social and protein interaction networks, biological data, knowledge graphs and many more. Two main machine learning paradigms for analysing such data – the symbolic paradigm that relies on first-order logic to represent and manipulate relational data, and the distributional paradigm which re-represents relational data in vectorised Euclidean space – have mostly been studied in isolation. This work is among the first, to the best of our knowledge, that systematically compares these two paradigms on the standard tasks from both domains – relational classification and knowledge base completion.
We draw several conclusions from the experimental analysis. First, there is no absolute winner among the paradigms, but (meta-)data properties of the relational classification datasets can help to decide which paradigm to prefer. Second, estimating the similarity of relational objects is an open question in the symbolic community and the distributional SRL methods constitute a viable alternative to manually designed similarities. Third, the symbolic methods fall behind the distributional ones on the task of knowledge base completion.
This work is not meant as the criticism of any of the considered approaches, but rather a step towards better understanding and integration. We hope this work inspires new research directions focused on the combination of the paradigms which the community has started to explore [@DBLP:conf/uai/MinerviniDRR17; @Schlichtkrull2017ModelingRD], but many questions remain open. Most importantly, better methods quantifying reasoning abilities of distributional methods are needed.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors wish to thank Tim Dettmers and Jonas Schouterden for their help with the experiments. This work was partially funded by the VLAIO-SBO project HYMOP (150033) and FWO (K204818N).
Failure modes of symbolic methods on KBC tasks
==============================================
There are several factors that contributed to the initial overly-optimistic results by applying TILDE on the KBC datasets. They come from the un-anticipated interaction between the evaluation metric and certain properties of TILDE.
Evaluation problem
------------------
In the initial experiments, we have followed the procedure described in the work of @dettmers2018conve ([-@dettmers2018conve]), which calculates the rank of the $i$-th test triplet as $$\text{rank}^s_i = 1 + \sum_{\tilde{x_i} \in \mathcal{C}^s(x_i) \setminus \mathcal{G}} I [ \psi(x_i) < \psi(\tilde{x_i})],$$
where $\tilde{x_i}$ is a corrupted triplet and $\psi$ is the scoring function. Calculating the rank this way *implicitly assumes* that every triplet has a different rank (or at least that there are very few triplets with the same rank). Consequently, this mean that in the case of tie, the ground truth triplet is considered to be ranked first amongst the equals. To compensate for that, we use the *expected rank* (proposed by Manuel Fink) defined as
$$\begin{aligned}
\text{rank}^s_i = & 1 + 0.5\sum_{\tilde{x_i} \in \mathcal{C}^s(x_i) \setminus \mathcal{G}} I [ \psi(x_i) < \psi(\tilde{x_i})] \\
& + 0.5\sum_{\tilde{x_i} \in \mathcal{C}^s(x_i) \setminus \mathcal{G}} I [ \psi(x_i) \leq \psi(\tilde{x_i})],\end{aligned}$$
i.e., the average of (1) ranking the ground truth triplet as the first amongst equals and (2) ranking it as the last amongst equals.
**You can read more about this problem and its spread in [@EmbProblem]**
Issues with TILDE
-----------------
The above outlined issue is not necessarily a problem for symbolic approaches, even though they do produce a discrete set of confidence scores (in the case of TILDE, one per leaf of the tree) as long as the number of false positives is low. We aimed to verify this with the experiments related to Figure \[fig:precvscov\] by checking the coverage and precision of the rules. However, these were checked only on the samples of data (i.e., including only a sample of true negatives) and might give a wrong picture.
The problem with TILDE comes from the way it evaluates the performance. Instead of using ranking as the distributional approaches, the evaluation criterion TILDE uses is *logical entailment*: how many of the provided example a current model covers? As the coverage is the only thing that matters (and the number of false positives that would be generated that way does not matter), TILDE can construct rules such as
`relationA(X,Y) :- relationB(X,Z), relationC(Z,W).`\
`relationA(X,Y) :- relationB(X,W), relationC(Y,Z).`
Both of these rules would generate a lot of false positives because either (1) one of the arguments of the head atom does matter, or (2) the head arguments are disconnected. TILDE does not offer an effective way to prevent this from happening. The number of such rules can be minimised by providing a lot of negative samples and hoping that such cases would be among the negative examples. Besides that, we have filtered out every rule where either one of the head arguments does not matter or the head arguments are disconnected.
We believe this clearly outlines two important issues for the ILP community: (1) is entailment truly sufficient or should other objective function be considered?, and (2) what are the sensible classes of languages bias such that above-outlined cases can be avoided?
Hyper-parameters
================
This section provides the details on the hyper-parameter values we optimised during the experiments.
Propositional classifiers
-------------------------
We tune the following parameters of the `sci-kit learn` learners:
#### **k Nearest Neighbours**
- *k* - the number of neighbours: $[3, 5, 7, 9, 11, 13, 15]$
- *weights* of the neighbours: $["uniform", "distance"]$
#### **Decision tree**
- training **criterion**: $["gini", "entropy"]$
- *maximal depth* of the tree: $[4, 8, 12, 16, 24, 48]$
- *minimal number of samples in the leaf* of a tree: $[2, 4, 6, 8]$
- *minimal impurity decrease*: $[0.0, 0.05, 0.1, 0.15]$
#### **SVMs**
- *kernel* type: $["rbf", "linear", "poly"]$
- *degree* of the polynomial kernel: $[3, 5, 7, 9]$
- *C* parameter: $[0.1, 1.0, 10.0, 100.0]$
- *gamma* parameter: $[0.01, 0.1, 0.5, 0.05, 0.2, 0.3, 0.6, 0.7, 1]$
Relational classifiers
----------------------
We tune the following parameters of the relational classifiers:
#### **TILDE**
- *learning heuristic*: $["gain", "gainratio"]$
- *minimal accuracy of rules*: $[0.75, 0.8, 0.9, 1.0]$
- *minimal examples (cases) in the leaves*: $[2, 4, 6, 8, 10, 12, 15]$
#### **Relational kNN**
- *weights* of the ReCeNT similarity measure: the individual weights come from the range $\{0.0, 1.0\}$ by increments of $0.05$ and the constraint $\sum_{i} = 1.0$
- - *k* - the number of neighbours: $[3, 5, 7, 9, 11, 13, 15]$
#### **kFOIL**
- *maximal number of clauses*: $[25, 100, 1000, 10000]$
- *maximal number of literals in a clause*: $[3, 5, 7, 9]$
- *beam size*: $[1, 2, 3, 4, 5, 6, 7]$
- *kernel type* t: $ [0, 1, 2, 3]$
- *C* parameter (termed $g$ in kFOIL): $[0.1, 1.0, 10.0, 100.0]$
- *s* parameter of the polynomial kernel: $[0.01, 0.05, 0.1, 0.5, 1.0, 2.0, 10.0]$
- *r* parameter of the polynomial kernel: $[0.5, 2.0, 1.0, 0.1, 10.0, 0.01]$
- *used biased hyperplane*: $[true, false]$
- *Normalise kernel after iteration*: $[true, false]$
- *move points according to centre of mass in feature space*: $[true, false]$
- *trade-off between training error and margin*: $[0.1, 0.25, 1.0]$
Experimental details
====================
Code repository
---------------
The code developed for the experiments is available in the following repository: `https://bitbucket.org/sdumancic/embeddingsmeetilp`
Further elaboration on selected baselines
-----------------------------------------
The selection of symbolic methods included in the experiments might seem limited initial, especially as we ignore existing methods that combine logical and probabilistic reasoning, such as Markov Logic Networks (MLN), Probabilistic Soft Logic and ProPPR. We agree all these frameworks are an important part of SRL. However, we don’t include them for two reasons. First, learning logic programs from data is much better developed within ILP (Our interpretation of SRL is ILP + probabilities). We have experimented with the MLN learners but have encountered 2 issues: (1) some of the datasets were too big for MLNs, and (2) MLN learners cannot learn clauses with constants, which are essential for the majority of the tasks. Often these MLN learners would learn nothing unless a dataset is somehow reduced (for instance, selecting only the top 100 words in the WebKB dataset). ProPPR is likewise unable to learn clauses with constants, and we are unaware of any PSL structure learner. Regarding the RDNs, we are aware of the BoostSRL implementation, but it would be difficult to marginalise out the effect of boosting. Regarding other commonly used ILP techniques such as Aleph and Metagol, we had difficult time to make them work on all considered datasets. Regarding Metagol, it was difficult to provide a generally applicable set of templates beyond dyadic template discussed in the paper which were not sufficient. Regarding Aleph, we could not make it work for all datasets, resulting in many models that would simply predict majority class.
Second, we believe that keeping the probabilistic aspect separate allows us to make a clearer comparison. Distributional approaches aim at representing the graph-structured data in a vectorised format. Therefore, they try to capture relevant structure, and the uncertainty scores they provide reflect the certainty of the model that a link exists. This is related to capturing the relational information, but not a probabilistic aspect of it. In that sense, they are more related to the ILP methods. Moreover, distributional methods do not provide a probabilistic model of a domain and are closer to approximate data lookup techniques.
Language specification for symbolic methods
-------------------------------------------
The specification of the language bias for the symbolic methods can be found here: `https://bitbucket.org/sdumancic/embeddingsmeetilp/src/master/experiments/workflow_case_classification.py`
Absolute performance of individual methods
------------------------------------------
The performance of individual methods on the relational classification datasets, in terms of accuracy (= a proportion of correct predictions), is reported in Table \[tab:rawperformance\].
Data properties
---------------
The properties of the uninformed graph are reported in Table \[tab:uninformedgraph\].
[^1]: Supplementary material: <https://arxiv.org/abs/1806.11391>
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We describe a simple, black-box compression format for tensors with a multiscale structure. By representing the tensor as a sum of compressed tensors defined on increasingly coarse grids, we capture low-rank structures on each grid-scale, and we show how this leads to an increase in compression for a fixed accuracy. We devise an alternating algorithm to represent a given tensor in the multiresolution format and prove local convergence guarantees. In two dimensions, we provide examples that show that this approach can beat the Eckart-Young theorem, and for dimensions higher than two, we achieve higher compression than the tensor-train format on six real-world datasets. We also provide results on the closedness and stability of the tensor format and discuss how to perform common linear algebra operations on the level of the compressed tensors.'
address:
- 'Department of Mathematics, Massachusetts Institute of Technology, Massachusetts, USA'
- 'Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Massachusetts, USA'
author:
- Oscar Mickelin
- Sertac Karaman
bibliography:
- 'references.bib'
title: 'Multi-resolution low-rank tensor formats'
---
Introduction
============
High-dimensional data is often represented with tensors in $\mathbb{R}^{n_1 \times \ldots \times n_d}$. When the dimensions $n_k$ are large, compressed tensor formats are often used to cope with the large storage requirements and operational costs. The tensor-train format [@oseledets2011tensor] and the canonical decomposition [@kolda2009tensor] are two formats that have received significant attention over the last decade. These formats represent tensors using different black-box low-rank expansions, and achieve significant reduction in storage costs when the associated rank is low.
In this article, we will be interested in tensors that are not necessarily low-rank in either of these formats, namely tensors with multiple length-scales. Even when the contribution to the tensor from each length-scale has low rank, their combination might be of significantly higher rank. Multiscale data is essential in numerous scientific and engineering problems [@deMoraes2019; @lee2017stochastic; @lee2015multiscale; @zou2005multiscale]. In many cases, the memory limitations, particularly on emerging edge computing devices, require that large-scale multiscale data is compressed for storage and for processing in compressed formats.
There have been a few approaches to compressing multiscale tensors in the literature. In one approach [@ozdemir2017multiscale; @ozdemir2016multiscale; @ozdemir2017multi], the authors compress a given tensor into a tree structure of compressed subtensors, by recursively subdividing a tensor into local blocks on different scales and decomposing each block on each scale in the tree in the Tucker format using the HOSVD [@de2000multilinear]. A similar approach has been pursued for matrices [@ong2016beyond], where a convex nuclear-norm relaxation is used to recover the local low-rank structures. Wu et al. [@WXY:07b; @wu2008hierarchical] consider tensors representing visual data, which are partitioned into blocks on increasingly finer scales. All blocks on each scale are represented with a common basis in the Tucker format, to capture global correlations of a locally repeating structure. For fast, GPU-accelerated and interactive visualization of data, previous work [@SIMAEZGGP:11; @SMP:13; @BSP:18] has also considered subsampling a given tensor decomposition at runtime, into a desired visualization resolution. The data is again represented by dividing the tensor into local blocks, which are then compressed. Khoromskij and Khoromskaia [@khoromskij2009multigrid] present different multigrid-inspired techniques with the goal of improving the speed of decomposition into the Tucker format, rather than decreasing the storage size of the resulting decomposition. A subsampling onto coarser grids identifies indices carrying the most information, after which alternating least squares algorithms can be run on these subindices. A related, but distinct approach is the quantized tensor train-approximation; we refer to e.g., a recent monograph [@khoromskij2018tensor] for more information.
We will consider a simple approach to improving the compression ratios of tensors that exhibit a multiscale structure. This approach complements the existing literature, in that our approach captures multiscale tensors where each scale gives a non-local contribution to the tensor. Starting from a tensor given in a compressed format, we introduce a graded structure on the domain and capture low-rank information on different scales. These scales have spatial resolution with increasing coarseness, which leads to a corresponding decrease in the representation cost. This multiresolution format enables us to achieve higher compression while maintaining a given accuracy. In the case of matrices, we show how this makes it possible to beat the Eckart-Young theorem [@golub2013matrix], by achieving lower approximation error than the truncated singular value decomposition, for a given storage cost. We devise an alternating algorithm for decomposing a tensor into the multiresolution format, where the tensor can be provided in either full format or the underlying, non-multiresolution tensor format. We prove a local convergence result of a slightly restructured version of this algorithm. We also discuss the closedness of the multiresolution format and how to perform common linear algebra operations in the format in ways that respects the graded structure.
The remainder of the article is structured as follows. Section \[sec:notation\] explains our notation and introduces operators that convert tensors between differently coarse grids. Section \[sec:format\] motivates and introduces the multiresolution format, and Section \[sec:closedness\] contains results on closedness of the multiresolution format. Section \[sec:alg\] describes an alternating algorithm for computing a decomposition in the multiresolution format and Appendix \[sec:local\] proves local convergence of a slightly restructured algorithm. In Section \[sec:operations\], we discuss how to perform common tensor operations on the level of the compressed tensors, and we conclude by studying the performance of the multiresolution format for several examples of real-world tensors in Section \[sec:applications\]. Implementations of all algorithms in this paper are publicly available online. [^1]
Notation {#sec:notation}
========
Throughout the article, we will refer to a number of low-rank tensor formats, for instance the tensor-train format, the hierarchical format [@hackbusch2012tensor], the canonical decomposition [@kolda2009tensor], or orthogonally constrained canonical decompositions [@chen2009tensor; @anandkumar2014tensor]. We will denote the set of tensors represented in a general format by $\mathcal{F}$. Each of these comes with a corresponding notion of rank, which we will denote by $\text{rank}_{\mathcal{F}}$. We denote by $\mathcal{F}_r$ the set of tensors in $\mathcal{F}$ with corresponding rank no greater than $r$. For the canonical decomposition, $r$ is a positive integer, and for the tensor-train format, $r$ is a vector of positive integers and inequalities between these vectors are interpreted element-wise. We will also consider low-rank matrices, for which the tensor-train format and canonical format coincide with the ordinary low-rank matrix format. For tensor formats $\mathcal{F}$ that are weakly closed, there is an optimal approximation $T_{\text{opt}}$ in $\mathcal{F}_{\mathbf{r}}$ to any tensor $T$ [@hackbusch2012tensor Thm. 4.28], and we will denote this by $\text{round}_{\mathcal{F}_\mathbf{r}}(T)$. For the tensor-train format, the TT-SVD procedure [@oseledets2011tensor Alg. 1] efficiently produces a quasi-optimal tensor $\widetilde{T}$ in $\mathcal{F}_{\mathbf{r}}$, i.e., satisfying $\|T - \widetilde{T}\| \leq \sqrt{d-1}\|T - T_{\text{opt}}\|$. $T$ can be given either in full format or in the TT-format. For certain tensor formats, e.g., the tensor-train format, it is also possible to instead specify an error bound $\varepsilon$. A rounding procedure then produces an approximation $\widetilde{T}$ with lower rank than $T$, guaranteed to satisfy $\|T - \widetilde{T}\| \leq \varepsilon \|T\|$. We will denote this procedure by $\text{round}_\mathcal{F}(T, \varepsilon)$. We will denote the inner product between two tensors $T$ and $S$ in $\mathbb{R}^{n_1 \times \ldots \times n_d}$ by $$\langle T, S \rangle := \sum_{i_1 =1}^{n_1} \ldots \sum_{i_d=1}^{n_d} T(i_1, \ldots , i_d)S(i_1, \ldots , i_d),$$ and the Frobenius norm of $T$ is defined to be $\|T\| := \sqrt{\langle T, T\rangle}$.
Throughout this paper, we will fix a batch size $b_s \in \mathbb{N}$ with $b_s \geq 2$. We will decompose tensors into a sum of tensors defined on grids with increasingly coarse resolution. The $k$:th coarsest level will consist of tensors constant on blocks with side length $b_s^k$. The parameter $b_s$ therefore controls the resolution of the subsequent grids, and therefore also defines a grid-refinement scheme defined in the next paragraph. We will in the following consider $L$ levels, with the coarsest one having blocks of size $b_s^L$. This therefore requires that $b_s^L$ divides $n_1, n_2, \ldots , n_d$. In order to describe this construction in detail, we introduce the following two operations, which will be heavily used in what follows.
Given a tensor $T \in \mathbb{R}^{ b_s^k \times \ldots \times b_s^k}$ and a positive integer $\ell$, we define the block-extended tensor $\text{ext}_\ell(T) \in \mathbb{R}^{ b_s^{k+\ell} \times \ldots \times b_s^{k+\ell}}$ by $$\text{ext}_\ell(T)(i_1, \ldots , i_d) = T\left(\floor*{\frac{i_1-1}{b_s^\ell}}+1, \ldots, \floor*{\frac{i_d-1}{b_s^\ell}}+1\right),$$ i.e., the tensor obtained by replacing each entry of $T$ by a block with side lengths $b_s^\ell$, where each entry equals the replaced entry of $T$. As an example, if $T$ is a $2\times2$-tensor, and $S$ is a $2\times 2 \times 2$-tensor defined by $$T = \left[\begin{array}{@{}cc@{}}
a & b \\ c & d \end{array}\right], \qquad S = \left[\begin{array}{@{}cc|cc@{}}
a & b & e & f \\ c & d & g & h \end{array}\right],$$ and if $b_s = 2$, then their first extended tensors are given by $$\begin{aligned}
\arraycolsep=2.45pt \text{ext}_1(T) = \begin{bmatrix}
a & a & b & b \\
a & a & b & b \\
c & c & d & d \\
c & c & d & d
\end{bmatrix}\!\!, \,\,\,\, \arraycolsep=2.45pt \text{ext}_1(S) = \left[\begin{array}{@{}cccc|cccc|cccc|cccc@{}}
a &a & b &b & a &a & b &b & e &e & f &f & e &e & f & f \\
a &a & b &b & a &a & b &b & e &e & f &f & e &e & f & f \\
c &c & d &d &c & c & d &d & g &g & h&h& g &g & h& h \\
c &c & d &d &c & c & d &d & g &g & h&h& g &g & h& h
\end{array}\right],\end{aligned}$$ respectively. Similarly, we denote the left inverse of $\text{ext}_\ell$ by $\text{ave}_\ell$. For a tensor $S \in \mathbb{R}^{ b_s^{k+\ell} \times \ldots \times b_s^{k+\ell}}$, $\text{ave}_\ell(S) \in \mathbb{R}^{b_s^{k} \times \ldots \times b_s^{k}}$ is defined by $$\text{ave}_\ell (S)(i_1, \ldots , i_d) = \frac{1}{b_s^{\ell d}}\sum_{j_1 = 0}^{b_s^{\ell} - 1} \ldots \sum_{j_d = 0}^{b_s^{\ell} - 1} S(i_1 + j_1, \ldots , i_d + j_d).$$ Clearly $$\label{eq:extprop}
\begin{split}
\text{ave}_{\ell_1 + \ell_2}(T) = \text{ave}_{\ell_1} &\left( \text{ave}_{\ell_2}(T) \right), \quad \text{ext}_{\ell_1 + \ell_2}(T) = \text{ext}_{\ell_1} \left( \text{ext}_{\ell_2}(T) \right), \\
\text{ave}_\ell(\text{ext}_\ell(T)) &= T, \quad \text{ave}_\ell(\text{ext}_k(T)) = \text{ext}_{k-\ell}(T),
\end{split}$$ for any integers $\ell$, $\ell_1$, $\ell_2$ and $k$, with $ \ell \leq k$. These operations will allow us to convert tensors into finer or coarser grids.
Tensor format {#sec:format}
=============
Let $T$ be a tensor in $\mathbb{R}^{ b_s^L \times \ldots \times b_s^L}$, for some positive integer $L$. We will approximate $T$ by a sum of subtensors defined on grids with increasing coarseness. Each subtensor will be represented in a compressed tensor format denoted by $\mathcal{F}$. We make the following definition.
Let $\mathbf{r} = (r_0, \ldots, r_L)$ be a vector of rank bounds for each grid-scale. For any compressed tensor format $\mathcal{F}$, we define the multiresolution $\mathcal{F}_\mathbf{r}$-format by $$\text{MS}_{\mathcal{F}_\mathbf{r}}= \left\{ T: T = \sum_{k=0}^L \text{ext}_{L-k}(T_k), T_k \in \mathcal{F}_{r_k}, T_k \in \mathbb{R}^{b_s^k \times \ldots \times b_s^k}\right\}.$$ To represent a tensor in $MS_{\mathcal{F}_\mathbf{r}}$, we only need to store the $L+1$ tensors $T_k$, for $0 \leq k \leq L$. We will say that $T$ has the multiresolution representation $(T_0, \ldots , T_L)$.
The motivation behind the definition is that storing as much information as possible on coarser scales decreases the total storage cost of the tensor, since fewer grid points need to be kept in memory as compared to the finest scale. The multiresolution format uses the operator $\text{ext}_{L-k}$, instead of a more smooth interpolation operator as is common in multigrid methods, since we will see in Section \[sec:alg\] that this allows for a simple algorithm to approximate a given tensor in $\mathcal{F}_{\mathbf{r}}$.
Since the format $ \text{MS}_{\mathcal{F}_\mathbf{r}}$ contains rank-$r_L$ approximations on the finest scale, $\mathcal{F}_{r_L} \subseteq \text{MS}_{\mathcal{F}_\mathbf{r}}$, and the multiresolution format contains any tensor for $r_L$ large enough. The contribution $\text{ext}_{L-k}(T_k)$ on each scale is contained in $\mathcal{F}_{r_k}$. When $\mathcal{F}$ is either the tensor train format or the canonical decomposition, it is therefore also the case that $\text{MS}_{\mathcal{F}_\mathbf{r}} \subseteq \mathcal{F}_{r_0 + \ldots + r_L}$. However, the storage cost of a tensor $T$ in $\text{MS}_{\mathcal{F}_\mathbf{r}}$ is lower than that of a tensor in $\mathcal{F}_{r_0 + \ldots + r_L}$ since the corresponding tensors $T_k$ are compressed versions of tensors on the smaller index set $\mathbb{R}^{b_s^k \times \ldots \times b_s^k}$, instead of on the the larger index set $\mathbb{R}^{b_s^L \times \ldots \times b_s^L}$.
For a wide range of accuracies, our examples in Section \[sec:applications\] will show that approximations in the multiresolution format can often require lower storage costs than in $\mathcal{F}$. Note however that we do not expect any storage gains when representing a tensor to machine precision in the format $\text{MS}_{\mathcal{F}_\mathbf{r}}$, as compared to storing the tensor in the format $\mathcal{F}_{r_L}$. The following example explains why this is the case.
Motivating example {#sec:motivating_example}
------------------
We consider a function with multiple length-scales, for instance $$\label{eq:motivating_example}
f(x) = \prod_{k=1}^d\sin\left(x_k\right) + \prod_{k=1}^d\sin\left(2x_k\right)+ \prod_{k=1}^d\sin\left(4x_k\right).$$ We let $T$ be the grid-discretization of $f$ on the interval $[0,\pi]$ using a uniform grid with a total of $n$ grid points in each dimension. $T$ therefore has a canonical representation $$\label{eq:ONex}
T = \bigotimes_{k=1}^d u_k + \bigotimes_{k=1}^d v_k + \bigotimes_{k=1}^d w_k,$$ where $u_k$ is a discretization of $\sin(x_k)$ on the interval $[0,\pi]$, $v_k$ of $\sin(2x_k)$ and $w_k$ of $\sin(4x_k)$. We then have $\langle u_k, v_k\rangle = 0 = \langle u_k , w_k \rangle = \langle v_k , w_k \rangle$ when $n$ is odd. Eq. therefore describes an orthogonal canonical decomposition [@kolda2001orthogonal]. An optimal rank $2$-approximation of $T$ is then obtained by keeping the two terms in Eq. with the largest norms [@zhang2001rank]. In our case, the terms have equal norms so we retain any two terms, e.g., $\bigotimes_{k=1}^d v_k + \bigotimes_{k=1}^d w_k$. The square of the approximation error is $$\|\bigotimes_{k=1}^d u_k\|^2 = \left(\frac{n}{\pi} \int_0^\pi \sin^2(x) \text{d}x +\mathcal{O}(1) \right)^d= \left( \frac{n}{2} \right)^d + \mathcal{O}(n^{d-1}).$$ A possible (but not necessarily optimal) multiresolution approximation with $b_s = 2$ would be $\widetilde{T} = \widetilde{T}_1 + \text{ext}_1(\widetilde{T}_2) + \text{ext}_2(\widetilde{T}_3)$, where $\widetilde{T}_1 = \bigotimes_{k=1}^d w_k$, $\widetilde{T}_2 = \bigotimes_{k=1}^d \text{ave}_1(v_k)$, and $\widetilde{T}_3 = \bigotimes_{k=1}^d \text{ave}_2(u_k)$. Since $\text{ave}_1(v_k) \in \mathbb{R}^{\frac{n}{2}}$ and $\text{ave}_2(u_k) \in \mathbb{R}^{\frac{n}{4}}$, the cost of storing $\widetilde{T}_1$, $\widetilde{T}_2$, and $\widetilde{T}_3$ is less than storing the optimal rank-2 approximation. By the following result, this also results in far lower approximation error.
Let $\omega_1, \omega_2, \ldots , \omega_r$ be an increasing sequence of positive real numbers and $f$ a function with multiple length-scales, written in the form $$f(x) = \sum_{k=1}^r \prod_{j=1}^d g_{kj}\left( \frac{x_j}{\omega_k}\right),$$ where $\| g'_{kj}(x_j) \|_{\infty} \leq C_{kj}$ for all $k,j$. Let $T$ be the corresponding discretization on the hypercube $[a,b]^d$ with $n = b_s^L$ uniform grid points in each dimension, i.e., $T = \sum_{k=1}^r T_k$ with $T_k = \bigotimes_{j=1}^d u_{kj}$. The multiresolution canonical approximation $\widetilde{T} = \widetilde{T}_1 + \sum_{k=2}^r \text{ext}_{k-1}(\widetilde{T}_{k})$ with $\widetilde{T}_{k} = \bigotimes_{j=1}^d \text{ave}_{k-1}\left(u_{k,j}\right)$ then satisfies $$\label{eq:bound_mot}
\|T- \widetilde{T}\| \leq \sum_{k=2}^r \delta_k \|T_k\|,$$ where $\delta_k = \left[ \prod_{j=1}^d \left(1 + (b-a)\frac{C_{k,j}\sqrt{ \frac{1}{60}\left(7b_s^{2(k-1)} - 15 + 8b_s^{-2(k-1)}\right)}} {\sqrt{n}\omega_{k}\|u_{k,j}\|} \right) \right] - 1.$ For large $n$, the right hand side of Eq. is approximately equal to $$\sum_{k=2}^r\sum_{j=1}^d (b-a)\frac{C_{k,j} \sqrt{ \frac{1}{60}\left(7b_s^{2(k-1)} - 15 + 8b_s^{-2(k-1)}\right)} }{\sqrt{n}\omega_{k}\|u_{k,j}\|} \|T_k\|.$$
A sensitivity formula for the canonical decomposition [@hackbusch2012tensor Prop. 7.10] gives the error bound in Eq. with $$\delta_k = \left[ \prod_{j=1}^d \left(1 + \frac{\|u_{k,j} - \text{ext}_{k-1}\left(\text{ave}_{k-1}(u_{k,j})\right)\|}{\|u_{k,j}\|} \right) \right] - 1,$$ and we only need to bound the quantity on the right. Fix now the indices $k$ and $j$. For any batch $B$ of indices of length $b_s^{k-1}$, the average of $u_{k,j}$ over $B$ is $\frac{1}{b_s^{k-1}}\sum_{m\in B}u_{k,j}(m)$. We have $$\label{eq:error_mot}
\begin{split}
\|u_{k,j} - \text{ext}_{k-1}\left(\text{ave}_{k-1}(u_{k,j})\right)\|^2_F
= \sum_{\text{B}} \sum_{i \in B} \left(u_{k,j}(i) - \frac{1}{b_s^{k-1}}\sum_{m\in B}u_{k,j}(m) \right)^2 \\= \sum_{\text{B}} \sum_{i \in B} \frac{1}{b_s^{2(k-1)}}\left(\sum_{m\in B} \left[u_{k,j}(i) - u_{k,j}(m)\right] \right)^2.
\end{split}$$ Now $$\begin{split}
\abs*{u_{k,j}(i) - u_{k,j}(m)} = \abs*{g_{k,j}\left( \frac{i(b-a)}{n\omega_{k}}\right) - g_{k,j}\left( \frac{m(b-a)}{n\omega_{k}}\right)} \\ \leq
\frac{(b-a)\abs{i-m} C_{k,j}}{n\omega_{k}} .
\end{split}$$ Inserting this into Eq. results in $$\label{eq:almost_done_mot}
\begin{split}
\|u_{k,j} - \text{ext}_{k-1}\left(\text{ave}_{k-1}(u_{k,j})\right)\|^2_F
\leq \sum_{\text{B}} \sum_{i \in B} \frac{(b-a)^2C_{k,j}^2}{n^2\omega_{k}^2b_s^{2(k-1)}}\left(\sum_{m\in B} \abs{i-m} \right)^2 \\
= b_s^{L-k+1}\frac{(b-a)^2C_{k,j}^2}{n^2\omega_{k}^2b_s^{2(k-1)}} \sum_{i =1}^{b_s^{k-1}}\left(\sum_{m=1}^{b_s^{k-1}} \abs{i-m} \right)^2.
\end{split}$$ Using elementary closed-form expressions, the sum in the right hand side can be evaluated to be $$\sum_{i =1}^{b_s^{k-1}}\left(\sum_{m=1}^{b_s^{k-1}} \abs{i-m} \right)^2 = \frac{b_s^{k-1}}{60} \left( 7b_s^{4(k-1)} - 15b_s^{2(k-1)} + 8\right),$$ so inserting this into Eq. and using the fact that $n = b_s^L$, we obtain $$\begin{split}
\|u_{k,j} \! - \text{ext}_{k-1}\!\left(\text{ave}_{k-1}(u_{k,j})\right)\|^2_F
\leq \!\frac{(b-a)^2C_{kj}^2}{n\omega_k^2} \frac{1}{60}\left( 7b_s^{2(k-1)} \! - 15 + 8b_s^{-2(k-1)}\!\right),
\end{split}$$ which concludes the proof.
In the motivating example, we can take $C_{kj} = 1$, $\omega_1 = \frac{1}{4}$, $\omega_2 = \frac{1}{2}$ and $\omega_3 = 1$ to conclude $$\|T- \widetilde{T}\| \leq \sum_{k=2}^r \delta_k \|T_k\| \approx \frac{9.65 \pi }{n} \|\bigotimes_{k=1}^d u_k\|,$$ which shows that the multiresolution format can achieve far lower approximation error for a given storage cost, provided $n$ is large enough. Our computational examples in Section \[sec:applications\] demonstrate that gains in storage can be achieved also for moderate values of $n$.
However, this example also shows that the error when representing a tensor in the multiresolution format has an inherent lower bound from using a coarser grid. To achieve machine precision for a general tensor, we would therefore in general expect to need to use the rank vector $(0, \ldots , 0, r_L)$ with $r_L$ large enough. However, for lower accuracy, the example above shows that it is possible to obtain good storage gains.
Closedness and stability {#sec:closedness}
========================
When attempting to find an optimal approximation of a tensor $T$ in $\text{MS}_{\mathcal{F}_{\mathbf{r}}}$ for a fixed rank vector $\mathbf{r}$, it is important to know whether or not the set $\text{MS}_{\mathcal{F}_{\mathbf{r}}}$ is closed. If not, then a tensor $T \in \overline{\text{MS}_{\mathcal{F}_{\mathbf{r}}}} \setminus \text{MS}_{\mathcal{F}_{\mathbf{r}}}$ by definition has a corresponding sequence of tensors $T^{(n)}$ in $ \text{MS}_{\mathcal{F}_{\mathbf{r}}}$, converging to $T$. $T$ therefore does not have an optimal approximation in the set $\text{MS}_{\mathcal{F}_{\mathbf{r}}}$ so the problem is ill-posed. In the by now classical setting of the (non-multiresolution) canonical format, this is associated with an instability in that successive approximations $T^{(n)}$ to $T$ have terms with diverging norm and convergence to $T$ is achieved through unstable cancellation effects [@de2008tensor]. We now show that the same holds true for the multiresolution format, even when using a closed format on each scale.
The base example is the multiresolution low-rank matrix format for $d=2$ with $b_s = 2$ and rank vector $(1,1)$. The matrix $$T^{(n)} = \begin{bmatrix}
n & n \\
n & n
\end{bmatrix} -
\begin{bmatrix}
\sqrt{n+1} \\
\sqrt{n-1}
\end{bmatrix}
\begin{bmatrix}
\sqrt{n+1} &
\sqrt{n-1}
\end{bmatrix}$$ is contained in the multiresolution format with rank vector $(1,1)$ for any $n$, and $T^{(n)} \rightarrow T := \bigl[ \begin{smallmatrix} -1 & 0 \\ 0 & 1\end{smallmatrix}\bigr]$, since $n - \sqrt{n+1}\sqrt{n-1} = \frac{n^2 - (n^2-1)}{n + \sqrt{n^2-1}} \rightarrow 0$ as $n\rightarrow \infty$. Since $T + a \bigl[ \begin{smallmatrix} 1 & 1 \\ 1 & 1\end{smallmatrix}\bigr] = \bigl[ \begin{smallmatrix} a-1 & a \\ a & a+1\end{smallmatrix}\bigr]$ can be seen to have rank $2$ for any real number $a$ by row reduction, it follows that $T$ is not in the multiresolution format with rank vector $(1,1)$. In other words, the format is not closed. We next extend this example to general rank vectors, values of $b_s$ and higher dimension $d$. We will consider the tensor-train format as the base format. Even though the tensor-train format is closed, we will show that the resulting multiresolution format $MS_{\text{TT}_\mathbf{r}}$ is not closed, in general. Here, the multiresolution rank vector is $\mathbf{r} = (r_0, \ldots , r_L)$, where each $r_k$ is a vector of tensor-train ranks, i.e., $r_k = ((r_{k})_1, \ldots , (r_{k})_{d-1})$.
\[thm:closed\]
1. For $d=2$, the format $MS_{\text{TT}_\mathbf{r}}$ is closed if and only if $\mathbf{r}$ is of either the form $(r_0,\ldots, r_{k-1}, b_s^{k}, 0, \ldots , 0)$ or $(0, 0, \ldots, 0, r_k, 0, \ldots , 0)$.
2. For $d\geq 3$, the format $MS_{TT_\mathbf{r}}$ is not closed if $\mathbf{r} = (r_0,\ldots, r_{k-1}, r_k, 0, \ldots , 0)$, where the first tensor rank of $r_k$, $(r_k)_1$, is strictly less than $b_s^k$ and not all $(r_i)_1$ are zero, for $i < k$.
The proof of Thm. \[thm:closed\] is detailed in Appendix \[appendix:proof\_closed\]. Thm. \[thm:closed\] shows that there is no reason to expect $MS_{\mathcal{F}_{\mathbf{r}}}$ to be closed, even when the underlying format $\mathcal{F}$ is closed. However, we now prove a stability property that is only achieved when using a stable tensor format on each grid-scale. The following definition is similar to one made for the non-multiresolution canonical decomposition [@hackbusch2012tensor Def. 9.15].
A sequence of tensors $T^{(n)}$ in $MS_{\mathcal{F}_{\mathbf{r}}}$ with $$T^{(n)} = \sum_{k=0}^{L} \text{ext}_{L-k}(T_k^{(n)})$$ is called stable if there is a constant $C < \infty$ such that $\|T_k^{(n)}\| \leq C \|T^{(n)}\|$ for each $k = 0, \ldots, L$ and $n$.
The format $\mathcal{F}$ is closed if and only if, for all possible rank vectors $\mathbf{r}$, all stable, convergent sequences in $MS_{\mathcal{F}_{\mathbf{r}}}$ converge to a tensor in $MS_{\mathcal{F}_{\mathbf{r}}}$.
By taking $\mathbf{r} = (0, 0, \ldots, r)$, the “only if” part follows. For the converse, let $T^{(n)}$ be any sequence of stable tensors in $MS_{\mathcal{F}_{\mathbf{r}}}$ converging to some tensor $T$. We need to show that also $T$ is in $MS_{\mathcal{F}_{\mathbf{r}}}$. We proceed by showing that there exists a subsequence $T^{(n_j)}$ of the $T^{(n)}$ for which scale-wise convergence $T^{(n_j)}_k \rightarrow T_k$ holds, for some tensors $T_k$ in $\mathcal{F}_{r_k}$ and all $k=0, \ldots , L$. We then show that $T = \sum_{k=0}^L \text{ext}_{L-k}(T_k)$, which means that $T$ is indeed in $MS_{\mathcal{F}_{\mathbf{r}}}$.
For $n$ large enough, it follows that $\|T^{(n)}\| \leq \|T\| + 1$, so $\|T^{(n)}_k\| \leq C\|T\| + C$. For each fixed $k$, the sequence $\{T^{(n)}_k\}_{n=1}^\infty$ is then bounded, so has a convergent subsequence, by the Bolzano-Weierstrass theorem. By passing to subsequences of this subsequence, for each $k$ in turn, it follows that there is a subsequence such that $T^{(n_j)}_k \rightarrow T_k$ for each $k$, for some $T_k$ in $\mathcal{F}_{r_k}$, by closedness of $\mathcal{F}$. It then holds that $T^{(n_j)} = \sum_{k=0}^L \text{ext}_{L-k}(T_k^{(n_j)}) \rightarrow \sum_{k=0}^L \text{ext}_{L-k}(T_k)$. Since by assumption $T^{(n_j)}\rightarrow T$, we must have $T = \sum_{k=0}^L \text{ext}_{L-k}(T_k)$, so $T$ is in $MS_{\mathcal{F}_{\mathbf{r}}}$.
For this reason, and for reasons to do with the decomposition algorithm presented in the next section, we will mostly restrict to closed tensor formats $\mathcal{F}$ in practice. The tensor-train format is one good candidate for this purpose.
Alternating decomposition algorithm {#sec:alg}
===================================
This section describes a simple algorithm for computing an approximation of a tensor $T$ in $MS_{\mathcal{F}_{\mathbf{r}}}$. Because of Thm. \[thm:closed\], this approximation problem is ill-posed even when $\mathcal{F}$ is a closed tensor-format. It will therefore not be possible to compute an optimal approximation of $T$ in $MS_{\mathcal{F}_{\mathbf{r}}}$, since it might not even exist. We therefore describe an alternating algorithm, which improves the approximation in every iteration. The tensor $T$ can be given either in full format, or as an already compressed tensor in $\mathcal{F}$. The steps in the algorithm carry through for any weakly closed tensor format $\mathcal{F}$, and the tensor-train format is a good example. The following Lemma will be important for the approximation algorithm.
\[lemma:updownTT\]
1. If $T$ in $\mathbb{R}^{b_s^m \times \ldots \times b_s^m}$ has canonical decomposition $T = \sum_{k=1}^r \bigotimes_{j=1}^d u_{kj}$ with each $u_{kj} \in \mathbb{R}^{b_s^m}$, then $$\begin{dcases}
\text{ext}_{\ell}(T) = \sum_{k=1}^r \bigotimes_{j=1}^d \text{ext}_\ell(u_{kj}), \\
\text{ave}_{\ell}(T) = \sum_{k=1}^r \bigotimes_{j=1}^d \text{ave}_\ell(u_{kj})
\end{dcases}$$
2. If $T$ has a tensor-train representation $$T(i_1, \ldots , i_d) = \sum_{\alpha_1 =1}^{r_1} \cdots \!\!\! \sum_{\alpha_{d-1} = 1}^{r_{d-1}}G_1(i_1, \alpha_1)\cdot G_2(\alpha_1, i_2, \alpha_2) \cdot \ldots \cdot G_d(\alpha_{d-1}, i_d),$$ with $G_k \in \mathbb{R}^{r_{k-1}\times b_s^k \times r_k}$, then $\text{ext}_\ell(T)$ has a tensor train decomposition with cores $\text{ext}_\ell(G_k(\cdot, i_k, \cdot)) \in \mathbb{R}^{r_{k-1}\times b_s^{k+\ell} \times r_k} $. Similarly, $\text{ave}_\ell(T)$ has a tensor train decomposition with cores $\text{ave}_\ell(G_k(\cdot, i_k, \cdot)) \in \mathbb{R}^{r_{k-1}\times b_s^{k-\ell} \times r_k}$.
Clear from definitions of $\text{ext}_\ell$ and $\text{ave}_\ell$.
Note that the cost of computing $\text{ext}_1$ and $\text{ave}_1$ in the tensor-train format is $\mathcal{O}(dnr^2)$, if $r_k \leq r$ and $n_k \leq n$, for all $k = 1, \ldots , d$.
The alternating algorithm starts with an initial approximation $\sum_{k=0}^L \text{ext}_{L-k}(T_k^{(0)})$ to $T$. It proceeds by fixing all scales except for the $k$:th one, and improving the approximation on the $k$:th scale by the following update equation $$T_k^{(n)} = \operatorname*{argmin \,}_{S \in \mathcal{F}_{r_k}} \| T - \sum_{\ell < k} \text{ext}_{L-\ell}(T_\ell^{(n)}) - \sum_{\ell > k} \text{ext}_{L-\ell}(T_\ell^{(n-1)}) - \text{ext}_{L-k}(S) \| .$$ Sweeping over all indices $k = 0, \ldots , L$ in turn completes one step of the iteration, which is repeated subsequently. In order to compute the updates on each scale, we will use the following result.
\[lemma:updateS\] $$\begin{split}
&\operatorname*{argmin \,}_{S \in \mathcal{F}_{r_k}} \| T - \sum_{\ell < k} \text{ext}_{L-\ell}(T_\ell^{(n)}) - \sum_{\ell > k} \text{ext}_{L-\ell}(T_\ell^{(n-1)}) - \text{ext}_{L-k}(S) \| \\
&= \operatorname*{argmin \,}_{S \in \mathcal{F}_{r_k}} \| \text{ave}_{L-k}\left(T - \sum_{\ell < k} \text{ext}_{L-\ell}(T_\ell^{(n)}) - \sum_{\ell > k} \text{ext}_{L-\ell}(T_\ell^{(n-1)}) \right)- S \| \\
&= \operatorname*{argmin \,}_{S \in \mathcal{F}_{r_k}} \| \text{ave}_{L-k}( T )- \sum_{\ell < k} \text{ext}_{k-\ell }(T_\ell^{(n)}) - \sum_{\ell > k} \text{ave}_{\ell - k}(T_\ell^{(n-1)})- S \|.
\end{split}$$
We show the first equality in the statement; the second follows from the first together with Eq. . For any $S \in \mathbb{R}^{b_s^k \times \ldots \times b_s^k}$ and any fixed $A \in \mathbb{R}^{b_s^L \times \ldots \times b_s^L}$, we have $$\label{eq:minS}
\begin{split}
\| A - \text{ext}_{L-k}(S) \|^2 &= \|A\|^2 - 2\langle A, \text{ext}_{L-k}(S) \rangle + \|\text{ext}_{L-k}(S)\|^2 \\
&= \|A\|^2 - 2b_s^{d(L-k)}\langle \text{ave}_{L-k}(A), S \rangle + b_s^{d(L-k)}\|S\|^2,
\end{split}$$ so a minimizer of Eq. is also a minimizer $- 2\langle \text{ave}_{L-k}(A), S \rangle + \|S\|^2$. Since $A$ is fixed, $S$ is therefore also a minimizer of $$\|\text{ave}_{L-k}(A)\|^2 - 2\langle \text{ave}_{L-k}(A), S \rangle + \|S\|^2 = \| \text{ave}_{L-k}(A) - S \|^2,$$ from which the statement follows when taking $$A = T - \sum_{\ell < k} \text{ext}_{L-\ell}(T_\ell^{(n)}) - \sum_{\ell > k} \text{ext}_{L-\ell}(T_\ell^{(n-1)}).$$
Lemma \[lemma:updateS\] implies that the approximation $T^{(n)}_k$ on each scale can be updated by solving an optimal approximation problem in $\mathcal{F}_{r_k}$. Since $\mathcal{F}$ was assumed weakly closed, this problem is well-posed. In practice, it might be computationally easier to instead find a quasi-optimal approximation for this update. This is for instance the case when using the tensor-train approximation, where the standard TT-SVD algorithm [@oseledets2011tensor Alg. 1] indeed only guarantees a quasi-optimal result.
We can structure the update steps in a downward and upward sweep to prevent recalculating the tensors on the coarser grids. In the downward sweep, we calculate and store $$\begin{split}
T_{\text{down}, L} &= T, \\
T_{\text{down}, k-1} &= \text{ave}_1(T_{\text{down},k} - T_k^{(n-1)}), \text{ for }k = L, \ldots, 1.
\end{split}$$ In closed form, this results in $$T_{\text{down}, k} = \text{ave}_{L-k}\left(T - \sum_{\ell > k} \text{ext}_{L-\ell}(T_\ell^{(n-1)})\right),$$ so the update equation on the $0$:th level in Lemma \[lemma:updateS\] reads as $$T_0^{(n)} = \text{round}_{\mathcal{F}_{r_{0}}}(T_{\text{down},0}).$$ Next, in the upward sweep, we set $$\begin{split}
T_{\text{up}, 0} &= 0, \\
T_{\text{up},k} &= \text{ext}_1(T_{\text{up},k-1} + T_k^{(n)} ), \text{ for } k = 1, \ldots, L-1,
\end{split}$$ where $T_k^{(n)}$ is calculated from $T_k^{(n-1)}$ and $T_{\text{up},k-1}$ by the following procedure. We have $$T_{\text{down}, k} - T_{\text{up}, k-1} = \text{ave}_{L-k}( T )- \sum_{\ell < k} \text{ext}_{k-\ell }(T_\ell^{(n)}) - \sum_{\ell > k} \text{ave}_{\ell - k}(T_\ell^{(n-1)}),$$ so the update equation in Lemma \[lemma:updateS\] reads as $$T_k^{(n)} = \text{round}_{\mathcal{F}_{r_k}}(T_{\text{down}, k} - T_{\text{up}, k-1}).$$ This procedure is summarized in Alg. \[alg:AL-multi\].
$T_k^{(0)} = 0$ $T_{\text{down}, L} = T$ $T_{\text{down}, k-1} = \text{ave}_1(T_{\text{down},k} - T_k^{(n-1)})$ $T_{\text{up}} = 0$ $T_k^{(n)} = \text{round}_{\mathcal{F}_{r_k}}(T_{\text{down}, k} - T_{\text{up}})$ $T_{\text{up}} = \text{ext}_1(T_{\text{up}} + T_k^{(n)} )$ $T_L^{(n)} = \text{round}_{\mathcal{F}_{r_L}}(T - T_{\text{up}})$
In the case when $T$ is already given in the tensor-train format, Lemma \[lemma:updownTT\] shows that $\text{ext}_1(T)$ and $\text{ave}_1(T)$ can be computed in the compressed format with cost $\mathcal{O}(r^2dn)$, where $r$ is the maximum of the TT-ranks of $T$. Since the tensors $T_{\text{down}, k} = \text{ave}_{L-k}(T - \sum_{\ell > k} \text{ext}_{L-\ell}(T_\ell^{(n)}))$ appearing in the downward sweep have TT-representation with rank at most $R = r + r_0 + \ldots + r_L$, the downward sweep therefore has cost bounded by $$\mathcal{O}(R^2d(1 + b_s + \ldots + b_s^L)) = \mathcal{O}(R^2d\frac{ b_s^L - 1}{b_s - 1}) = \mathcal{O}(R^2db_s^L) = \mathcal{O}(R^2dn),$$
In the upward sweep, a quasi-optimal minimizer of $\operatorname*{argmin \,}_{S \in \mathcal{F}_{r_k}} \| T_{\text{down},k} - T_{\text{up},k} - S\|$ can be computed by a call to the TT-rounding procedure. Since $T_{\text{down},k} - T_{\text{up},k}$ has TT-ranks at most $r+\sum_{\ell \neq k}r_k$, one iteration of Alg. \[alg:AL-multi\] is then of cost at most $$\mathcal{O}(R^3d(1 + b_s + \ldots + b_s^L)) = \mathcal{O}(R^3d\frac{ b_s^L - 1}{b_s - 1}) = \mathcal{O}(R^3db_s^L) = \mathcal{O}(R^3dn).$$ The total cost of Alg. \[alg:AL-multi\] is therefore $\mathcal{O}(MR^3dn)$.
In general, alternating algorithms of the form in Alg. \[alg:AL-multi\] lead to a monotonically decreasing objective function $\|T - \sum_{k=0}^L \text{ext}_{L-k}(T_k^{(n)})\|$. However, there are in general no guarantees that the approximations $T_k^{(n)}$ on each scale converge, and even if they do, convergence might occur to only a local minimum. This is a typical situation when dealing with tensors in dimension higher than two, and occurs for instance when using the popular alternating least-squares algorithm for computing a (non-multiresolution) canonical decomposition of a tensor [@uschmajew2012local; @wang2014global], and when using iterative methods for computing canonical decompositions with orthogonality constraints [@chen2009tensor; @wang2015orthogonal]. Appendix \[sec:local\] states and proves a local convergence guarantee for Alg. \[alg:AL-multi\].
Tensor operations {#sec:operations}
=================
The different scales of the multiresolution format introduce a grading on $MS_{\mathcal{F}_\mathbf{r}}$, and we now show that all common tensor operations can be performed in such a way that they respect the graded structure and can be computed with cost independent of the number of levels $L$. The format can therefore be used in calculations without having to convert into full format.
Addition
--------
If $T$ and $S$ have the multiresolution representations $(T_0, \ldots , T_L)$ and $(S_0, \ldots , S_L)$, respectively, then $S+T$ has multiresolution representation $(T_0+S_0, \ldots , T_L+S_L)$.
Rounding
--------
Let $T$ have multiresolution representation $(T_0, \ldots , T_L)$ with each $T_k \in \mathcal{F}_{r_k}$. When the $T_k$ potentially have suboptimal ranks, for instance as a result of having performed addition or taking Hadamard products, a multiresolution representation with more beneficial ranks is given by $(\widetilde{T}_0, \ldots , \widetilde{T}_L)$, with $\widetilde{T}_k = \text{round}_\mathcal{F}(T_k, b_s^{-d(L-k)}\varepsilon)$. This results in an approximation error $$\|T - \sum_{k=0}^{L}\text{ext}_{L-k}(\widetilde{T}_k)\| \leq \sum_{k=0}^{L} \| \text{ext}_{L-k} (T_k - \widetilde{T}_k) \| \leq \varepsilon \sum_{k=0}^L \|T_k\|.$$
The cost of this procedure is given by $$\sum_{k=0}^L \mathcal{O}(r^3b_s^kd) = \mathcal{O}(r^3d \frac{b_s^{L+1}-1}{b_s - 1}) = \mathcal{O}(r^3d b_s^{L}) = \mathcal{O}(r^3d n).$$
Hadamard product
----------------
If $T$ and $S$ have the respective multiresolution representations $(T_0, \ldots , T_L)$ and $(S_0, \ldots , S_L)$, then $S\circ T$ has multiresolution representation $(R_0, \ldots , R_L)$ with $$R_k = T_k\circ (\text{ext}_k(S_0) + \ldots + S_{k}) + S_k \circ (\text{ext}_k(T_0) + \ldots + \text{ext}_1(T_{k-1})).$$ $R_k$ can be computed recursively, with rounding during intermediate steps to avoid rank-growth, i.e., $A_0 = T_0, A_k = \text{round}_{\mathcal{F}}(T_k + \text{ext}_1(A_{k-1}))$ and $B_0 = S_0, B_k = \text{round}_{\mathcal{F}}(S_k + \text{ext}_1(B_{k-1}))$. This results in $$R_k = T_k\circ A_k + S_k\circ B_k.$$ In the case when $\mathcal{F}$ is the tensor-train format, $\text{ext}_1(T_{k-1})$ can be computed with cost $\mathcal{O}(\text{rank}_{TT}(A_{k-1})^2b_s^kd)$, by Lemma \[lemma:updownTT\], and the rounding procedure has cost $\mathcal{O}(\text{rank}_{TT}(A_{k-1}+T_{k})^3b_s^kd)$. If we write $$R = \max_{1\leq k \leq L} \text{rank}_{TT}(\text{ext}_k(T_0) + \text{ext}_{k-1}(T_1) + \ldots + T_k) \leq r_0 + \ldots + r_L,$$ it follows that the total cost of computing the Hadamard product is $$\sum_{k=0}^L \mathcal{O}(R^3b_s^kd) = \mathcal{O}(R^3d \frac{b_s^{L+1}-1}{b_s - 1}) = \mathcal{O}(R^3d b_s^{L}) = \mathcal{O}(R^3d n).$$
Tensor-vector contraction
-------------------------
If the tensor $T$ has a multiresolution representation $(T_0, \ldots , T_L)$ then $T\times_j v$ has multiresolution representation $$(b_s^L T_0\times_j \text{ave}_L(v), b_s^{L-1}T_1\times_j \text{ave}_{L-1}(v), \ldots , T_L \times_j v).$$ The cost of computing $T_k \times_j v$ when $T_k$ is given in the tensor-train format, is $\mathcal{O}(r_k^2dn)$. Recursively computing $\text{ave}_{k}(v) = \text{ave}(\text{ave}_{k-1}(v))$ has cost $\mathcal{O}(n)$. The total cost then becomes $$\sum_{k=0}^L \mathcal{O}(r^2b_s^kd) = \mathcal{O}(r^2d \frac{b_s^{L+1}-1}{b_s - 1}) = \mathcal{O}(r^2d b_s^{L}) = \mathcal{O}(r^2d n).$$
Frobenius norm
--------------
The Frobenius norm can be computed as $(T\circ T)\times_1 v \ldots \times_d v$, where $v \in \mathbb{R}^n$ is a vector with all entries equal to $1$.
Applications {#sec:applications}
============
This section compares the compression ratios achieved using Alg. \[alg:AL-multi\] to those of using the tensor-train decomposition, for a variety or real-world datasets. In $2$D, we show how this can be used to achieve greater accuracy than a truncated singular value decomposition for given storage. In dimensions higher than two, we achieve greater compression than the tensor-train decomposition.
All computations were carried out on a MacBook Pro with a 3.1 GHz Intel Core i5 processor and 16 GB of memory.
Motivating example revisited
----------------------------
We consider the tensor $T \in \mathbb{R}^{n\times n \times n}$ in Eq. in the motivating example. In the canonical format, we compare the approximation error of a rank-$2$ approximation, obtained by the standard alternating least-squares algorithm [@TTB_Software; @TTB_Dense; @TTB_Sparse], to the multiresolution canonical approximation produced by Alg. \[alg:AL-multi\]. We used rank vector $(0, \ldots , 0, 1, 1, 1)$ which then has lower storage cost than the rank-$2$ approximation, with a single iteration of Alg. \[alg:AL-multi\]. For the alternating least-squares algorithm on each scale, we used the HOSVD as initial guess. The result is shown in Fig. \[fig:ex\_motivating\]. Consistent with Sec. \[sec:motivating\_example\], Alg. \[alg:AL-multi\] produces an approximation with relative error scaling as $\mathcal{O}(n^{-1})$. Alg. \[alg:AL-multi\] therefore results in far lower approximation error for the same compression ratio, provided $n$ is large enough.
![Compression of the tensor in Eq. as a rank-$2$ approximation in the canonical format and in the multiresolution canonical format. The relative approximation error is shown as a function of the tensor dimension $n$.[]{data-label="fig:ex_motivating"}](f1.pdf){width="96.00000%"}
Image data
----------
We consider two matrices with multiscale features. The matrices are greyscale versions of images with features on several scales [@nycphoto; @earthphoto], rescaled to be of size $2048\times 2048$ pixels. The resulting matrices in $\mathbb{R}^{2048\times 2048}$ were compressed in both the low-rank matrix format and in the multiresolution format. We used batch-size $b_s = 2$ and rank vector $(r, \ldots , r)$ for increasing values of $r$ and maximum number of iterations $M$. The results are shown in Fig. \[fig:ex\_matrix\_results\]. For accuracies for which the low-rank matrix format achieves a compression ratio of at least two, the multiresolution format achieves up to a factor $1.5$ higher compression ratio.
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![Top row: Images from [@nycphoto], [@earthphoto], respectively, Middle row: compression ratios of matrices in [@nycphoto; @earthphoto] as functions of approximation error. Bottom row: runtimes as functions of approximation error.[]{data-label="fig:ex_matrix_results"}](f2.png "fig:"){width="96.00000%"}
![Top row: Images from [@nycphoto], [@earthphoto], respectively, Middle row: compression ratios of matrices in [@nycphoto; @earthphoto] as functions of approximation error. Bottom row: runtimes as functions of approximation error.[]{data-label="fig:ex_matrix_results"}](f3.pdf "fig:"){width="96.00000%"}
![Top row: Images from [@nycphoto], [@earthphoto], respectively, Middle row: compression ratios of matrices in [@nycphoto; @earthphoto] as functions of approximation error. Bottom row: runtimes as functions of approximation error.[]{data-label="fig:ex_matrix_results"}](f4.pdf "fig:"){width="96.00000%"}
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Fig. \[fig:ex\_matrix\_show\] shows a side-by-side comparison of one of the images compressed in both the multiresolution low-rank matrix format, and the ordinary low-rank matrix format. For the same compression ratio, the multiresolution format has visibly significantly clearer features and correspondingly lower approximation error.
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![Compressed images with equal compression ratio in multiresolution format (left) and low-rank matrix format (right).[]{data-label="fig:ex_matrix_show"}](f5.png "fig:"){width="96.00000%"}
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Figs. \[fig:ex\_matrix\_first\] and \[fig:ex\_matrix\_second\] show the compressed version of the matrix $A$, decomposed into its different scales. The $i$:th subimage in Fig. \[fig:ex\_matrix\_first\] contains the sum of the $i$ highest scales of the compressed format, i.e., $\sum_{k=0}^{i-1} \text{ext}_{L-k} \left(A_k\right)$. The $i$:th subimage in Fig. \[fig:ex\_matrix\_second\] contains the sum of the $i$ lowest scales of the compressed format, i.e., the matrix $\sum_{k=L-i+1}^{L} \text{ext}_{L-k} \left(A_k\right)$.
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![The $i$th successive image shows the sum of the $i$ highest scales in the compressed multiresolution format.[]{data-label="fig:ex_matrix_first"}](f6.png "fig:"){width="96.00000%"}
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![The $i$th successive image shows the sum of the $i$ lowest scales in the compressed multiresolution format.[]{data-label="fig:ex_matrix_second"}](f7.png "fig:"){width="96.00000%"}
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Hyperspectral wavelength data
-----------------------------
We consider hyperspectral wavelength data, which are aerial view photographs of different environments, captured at multiple wavelengths across the electromagnetic spectrum. The data is organized into a tensor $S \in \mathbb{R}^{n_1\times n_2 \times n_3}$ where $n_1$ and $n_2$ are the physical dimensions, and $n_3$ the number of recorded wavelengths. Each slice $T( \cdot, \cdot, i)$ therefore contains a photograph at the $i$:th recorded wavelength.
We consider three different hyperspectral images of different environments [@urbandataset; @avirisdataset; @samsondataset]. One slice of each environment is shown in Fig. \[fig:ex\_hyperspectral\_results\]. For the data from [@urbandataset], slices $1-4$, $76$, $87$, $101-111$, $136-153$ and $198-210$ are removed because of contamination by atmospherical effects, and we consider the first $128$ slices of the upper left $256\times 256$ sub-image. For the data from [@avirisdataset], we consider the $1920\times 640$-pixel subimage starting at the index $(1, 1)$, and for the data from [@samsondataset], we consider the $896\times 896$-pixel subimage starting at index $(1, 1)$, each with $128$ slices. Since each image exhibits features on multiple scales, we expect the multiresolution format to achieve good compression.
This results in three tensors in $\mathbb{R}^{256 \times 256 \times 128},\mathbb{R}^{1920 \times 640 \times 128},\mathbb{R}^{896 \times 896 \times 128}$, respectively, which we compress using the multiresolution tensor-train format and the non-multiresolution tensor-train format. The results are shown in Fig. \[fig:ex\_hyperspectral\_results\], and shows higher compression ratio in the multiresolution format across practically all accuracies where the tensor-train achieves a compression ratio of at least $1$. The compression ratio is several times larger in the multiresolution tensor format for a wide range of accuracies. The simulations used batch-size $b_s = 2$, rank-vector $(r, \ldots, r)$ for increasing $r$ and maximum number of iterations $M$. The runtime per iteration is a small factor times that of the tensor-train decomposition. To lower the total computational time, one can also consider using randomized algorithms [@sun2019low; @che2019randomized; @wang2015fast] for the tensor approximation on each scale.
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![Top row: One wavelength of the hyperspectral images from [@urbandataset], [@avirisdataset], [@samsondataset], respectively. Middle row: compression ratios of the tensors in [@urbandataset], [@avirisdataset], [@samsondataset], respectively, as functions of approximation error and maximum number of iterations $M$. Bottom row: runtimes as functions of approximation error.[]{data-label="fig:ex_hyperspectral_results"}](f8.png "fig:"){width="96.00000%"}
![Top row: One wavelength of the hyperspectral images from [@urbandataset], [@avirisdataset], [@samsondataset], respectively. Middle row: compression ratios of the tensors in [@urbandataset], [@avirisdataset], [@samsondataset], respectively, as functions of approximation error and maximum number of iterations $M$. Bottom row: runtimes as functions of approximation error.[]{data-label="fig:ex_hyperspectral_results"}](f9.pdf "fig:"){width="96.00000%"}
![Top row: One wavelength of the hyperspectral images from [@urbandataset], [@avirisdataset], [@samsondataset], respectively. Middle row: compression ratios of the tensors in [@urbandataset], [@avirisdataset], [@samsondataset], respectively, as functions of approximation error and maximum number of iterations $M$. Bottom row: runtimes as functions of approximation error.[]{data-label="fig:ex_hyperspectral_results"}](f10.pdf "fig:"){width="96.00000%"}
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Video data
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We consider video recordings of three different scenes [@li2004statistical]. These correspond to tensors in $\mathbb{R}^{256\times 256 \times 1280}$, $\mathbb{R}^{128\times 128 \times 128}$, and $\mathbb{R}^{128\times 128 \times 1280}$, respectively. The scenes exhibit multiple physical and temporal scales due to e.g., objects moving through the scenes at different speeds. Sample frames are shown together with the compression results in Fig. \[fig:ex\_video\_results\]. We used batch-size $b_s = 2$ and rank vector $(r, \ldots , r)$ for increasing values of $r$ and maximum number of iterations $M$. The multiresolution approximation achieves up to more than twice as high compression ratio as the tensor-train decomposition, over a wide range of accuracies.
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![Top row: One frame of original videos. Middle row: compression ratios of the tensors as functions of approximation error and maximum number of iterations $M$. Bottom row: runtimes as functions of approximation error.[]{data-label="fig:ex_video_results"}](f11.png "fig:"){width="96.00000%"}
![Top row: One frame of original videos. Middle row: compression ratios of the tensors as functions of approximation error and maximum number of iterations $M$. Bottom row: runtimes as functions of approximation error.[]{data-label="fig:ex_video_results"}](f12.pdf "fig:"){width="96.00000%"}
![Top row: One frame of original videos. Middle row: compression ratios of the tensors as functions of approximation error and maximum number of iterations $M$. Bottom row: runtimes as functions of approximation error.[]{data-label="fig:ex_video_results"}](f13.pdf "fig:"){width="96.00000%"}
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A sample compressed scene is included in Movie 1[^2], with a side-by-side comparison of the multiresolution approximation and the tensor-train approximation. The approximations shown achieve the same compression ratio, and the multiresolution approximation exhibits more clearly defined features and noticeably fewer artifacts of the approximation procedure, compared to the tensor-train approximation. Movie 2[^3] presents the approximation on each scale. The $i$th submovie contains the sum of the $i$ highest scales of the compressed format, i.e., $\sum_{k=0}^{i-1} \text{ext}_{L-k} \left(T_k\right)$.
Multiresolution canonical decomposition {#sec:ex_video_candecomp}
---------------------------------------
We conclude this section with an example of compression into the multiresolution format, when the underlying tensor format is the canonical tensor format. The canonical format is not closed and there are no guarantees to find an optimal approximation in the format. However, Alg. \[alg:AL-multi\] can still be run with one of the standard approximation algorithms for approximation into each scale. We use the alternating least-squares algorithm [@TTB_Software; @TTB_Dense; @TTB_Sparse] with standard settings, on the first tensor in Fig. \[fig:ex\_video\_results\]. The results are shown in Fig. \[fig:ex\_video\_candecomp\], with higher compression ratios for a given accuracy, compared to the non-multiresolution format.
![Left: compression ratios of the tensor in \[sec:ex\_video\_candecomp\] as function of approximation error and maximum number of iterations $M$. Right: runtimes as functions of approximation error.[]{data-label="fig:ex_video_candecomp"}](f14.pdf){width="90.00000%"}
Conclusion
==========
We have studied a simple black-box tensor format for representing multidimensional data with multiple length-scales. An alternating algorithm for tensor approximation into this format was provided, and local convergence guarantees were proven. The closedness and stability properties of the format were also characterized. The efficiency of the format was numerically verified on six real-world datasets, achieving compression ratios several times higher than their counterparts for the tensor-train format, at the expense of higher run-times.
Proof of Thm. \[thm:closed\] {#appendix:proof_closed}
============================
We prove (1) first. Let $k$ be the highest index of the rank-vector for which $r_k \neq 0$. Note that the multiresolution format with rank vector $(r_0,\ldots, r_{k-1}, b_s^{k}, 0, \ldots , 0)$ is precisely the set of matrices $\text{ext}_{L-k}(T)$ for $T$ any matrix in $\mathbb{R}^{b_s^k \times b_s^k}$. It therefore coincides with the multiresolution format with rank vector $(0,\ldots, 0, b_s^{k}, 0, \ldots , 0)$, so we assume that $\mathbf{r} \neq (0, 0, \ldots, 0, r_k, 0, \ldots , 0)$ and $r_k < b_s^k$. There is then some $r_{i} > 0$ with $i < k$. We will produce a matrix $T$ in $\overline{MS_{\text{TT}_\mathbf{r}}} \setminus MS_{\text{TT}_\mathbf{r}}$, and first introduce some auxiliary variables for the construction.
Denote by $v^{(n)}$ the vector $v^{(n)} = \begin{bmatrix}
\sqrt{n+1} &
\sqrt{n-1} &
\ldots &
\sqrt{n-1}
\end{bmatrix}^T,
$ and by $u$ the vector $u = (\underbrace{1,1,\ldots , 1}_{b_s-1}, 1- \frac{b_s}{b_s-1}).$ Let $m = \floor*{\frac{r_k+1}{b_s}}$ and $w$ a vector of length $\abs{w}=r_k+1-mb_s$ defined by $w = (1, \ldots, 1, 1 - \frac{\abs{w}}{\abs{w}-1})$. The $b_s^k \times b_s^k$-matrix with all entries equal to $n$ is of the form $\text{ext}_{k-i}(S)$ with $S$ the $b_s^i \times b_s^i$-matrix with all entries equal to $n$. Viewing this matrix as a contribution from scale $i$, we study the matrix $\text{ext}_{L-k}(T^{(n)})$, where $$\begin{aligned}
T^{(n)} = \begin{bmatrix}
n & \cdots & n \\
\vdots & \ddots & \vdots \\
n & \cdots & n
\end{bmatrix} - v^{(n)}\otimes v^{(n)} \! \! - \text{diag}(0,0,\underbrace{1,1,\ldots , 1}_{b_s-2}, \underbrace{u, \ldots , u}_{m-1}, w, \!\!\! \! \underbrace{0, \ldots , 0}_{b_s^k - (r_k+1) \geq 0} \!\!\! \!),\end{aligned}$$ Since the second two terms have rank at most $r_k$, they are contained in the tensor format on the $k$:th scale. It follows that $\text{MS}_{\text{TT}_\mathbf{r}}$ contains $\text{ext}_{L-k}(T^{(n)})$. As $n \rightarrow \infty$, $n - \sqrt{n+1}\sqrt{n-1} = \frac{n^2 - (n^2-1)}{n + \sqrt{n^2-1}} \rightarrow 0$, so $T^{(n)}$ tends to the matrix $T$ defined by $$T = \begin{bmatrix}
-1 & 0 & \cdots & 0 \\
0 & 1 & \cdots & 1 \\
\vdots & \vdots & \ddots &\vdots \\
0 & 1 & \cdots & 1
\end{bmatrix} - \text{diag}(0,0,\underbrace{1,1,\ldots , 1}_{b_s-2}, \underbrace{u, \ldots , u}_{m-1}, w, \underbrace{0, \ldots , 0}_{b_s^k - (r_k+1) \geq 0}).$$ To conclude the proof, we show that $\text{ext}_{L-k}(T)$ is not contained in $MS_{\text{TT}_\mathbf{r}}$. Note that any sum of the form $\sum_{m = 0}^{k-1} \text{ext}_{k-m}(T_m)$ is necessarily constant on all batch-blocks of size $b_s$. By this, we mean the sub-matrices with row-indices and column-indices $i,j$ contained between two consecutive multiples of $b_s$. It therefore suffices to show that $\text{rank}(T+S) \geq r_k + 1$ for any matrix $S$ constant on all batch-blocks. We verify this by reducing $T+S$ to row-echelon form.
We first decompose $T+S$ into its batch-blocks, i.e., write $$T +S = \begin{bmatrix}
M_{1,1} & M_{1,2} & \cdots & M_{1,b_s^{k-1}} \\
M_{2,1} & M_{2,2} & \cdots & M_{2,b_s^{k-1}} \\
\vdots & \vdots & \ddots &\vdots \\
M_{b_s^{k-1},1}& M_{b_s^{k-1},2} & \cdots & M_{b_s^{k-1},b_s^{k-1}},
\end{bmatrix}$$ with each $M_{i,j}$ of size $b_s\times b_s$. Here, each lower triangular batch-block $M_{i,j}$ for $i > j$ has constant rows, and each upper triangular batch-block $M_{i,j}$ for $i < j$ has constant columns. We will show that $T+S$ has rank at least $r_k+1$ by showing that the lower triangular batch-blocks can be reduced to $0$ by row operations, and by counting the pivots of the diagonal batch-blocks $M_{i,i}$. We will treat the case $i=1$ and $i>1$ separately, and start by showing that $M_{1,1}$ has full rank, by induction on $b_s$. For some constant $a$, $M_{1,1}$ can be written as $$M_{1,1}=\begin{bmatrix}
a - 1 & a & a & \ldots & a \\
a& a+1 & a+1 & \ldots & a+1 \\
a& a +1& a & \ldots & a+1 \\
\vdots & \vdots & & \ddots & \vdots \\
a & a+1 & a +1& \ldots & a
\end{bmatrix}.$$ Subtracting the second row from the last and using the last row to eliminate the last column reduces $M_{1,1}$ to the block-form $$\begin{bmatrix}
a - 1 & a & a & \ldots & a & 0 \\
a& a+1 & a+1 & \ldots & a+1 & 0 \\
a& a +1& a & \ldots & a+1 & 0\\
\vdots & \vdots & & \ddots & \vdots \\
a & a+1 & a +1& \ldots & a &0\\
0 & 0 & 0 & \ldots & 0& -1
\end{bmatrix},$$ which has full rank, by the induction hypothesis. The base case $b_s = 2$ corresponds to $M_{1,1} = \bigl[ \begin{smallmatrix} a-1 & a \\ a & a+1\end{smallmatrix}\bigr] $. This has full rank, which concludes the proof that $M_{1,1}$ has full rank.
Moreover, since the batch-blocks $M_{1,2}, M_{1,3}, \ldots , M_{1,b_s^{k-1}}$ have constant columns, the row operations that reduce $M_{1,1}$ to row-echelon form preserve these constant columns. We can continue the row operations to reduce $M_{2,1}, M_{3,1} , \ldots M_{b_s^{k-1},1}$ to zero. Since each $M_{2,1}, M_{3,1} , \ldots M_{b_s^{k-1},1}$ has constant rows, after these operations, $M_{2,2}$ can still be written on the form
$$M_{2,2} = \begin{bmatrix}
c & c+1 & \ldots & c+1\\
c+1 & c & \ldots & \vdots \\
\vdots & \vdots & \ddots & c+1 \\
c+1& c+1 & \ldots & c + \frac{b_s}{b_s-1}
\end{bmatrix},$$
for some real number $c$. Subtracting the last row from each preceding row and then subtracting from the last row $c+1$ times each preceding row results in
$$\begin{bmatrix}
-1 & 0& \ldots & \frac{-1}{b_s-1} \\
0 & -1& \ldots & \frac{-1}{b_s-1} \\
\vdots & \vdots & \ddots & \vdots \\
0& 0& \ldots & \frac{1}{b_s-1}
\end{bmatrix},$$
with an additional $b_s$ pivots. Again, the row operations used in the reduction of $M_{2,2}$ do not change the fact that $M_{2,3}, M_{2,4}, \ldots , M_{2, b_s^{k-1}}$ all have constant columns.
We iterate this argument for $i = 2, 3, \ldots, m+1$, and count the number of pivots. It follows that $S+T$ has rank no less than $b_s + (m-1)b_s + r_k +1 - mb_s = r_k +1$, which concludes the proof of the first assertion.
For the second statement, let $T$ and $T^{(n)}$ be as above and write $S = T\otimes e_1^{\otimes d - 2}$, $S^{(n)} = T^{(n)}\otimes e_1^{\otimes d - 2}$. Here, each $T^{(n)}$ is a matrix of rank at most $(r_k)_1$. Each $S^{(n)}$ can therefore be written in the TT-format with rank vector $(1,(r_k)_1,1\ldots , 1)$, so clearly $S^{(n)} \in MS_{TT_{\mathbf{r}}}$. Moreover, $S^{(n)} \rightarrow S$. If $\text{ext}_{L-k}(S)$ were in $MS_{TT_{\mathbf{r}}}$, i.e., $S$ were expressible in the form $S = \sum_{m=0}^{k} \text{ext}_{k-m}(S_m)$ with $\text{rank}_{TT}(S_m) \leq r_m$, then it would follow that $$\begin{split}
T &= S\times_3 e_1 \times_4 \ldots \times_d e_1 = \sum_{m=0}^{k} \text{ext}_{k-m}(S_k)\times_3 e_1 \times_4 \ldots \times_d e_1 \\
&= \sum_{m=0}^{k} \text{ext}_{k-m}(S_k\times_3 e_1 \times_4 \ldots \times_d e_1),
\end{split}$$ so the matrix $T$ would be in $MS_{\mathbf{q}}$ with the rank vector $\mathbf{q} = ((r_0)_1, \ldots , (r_k)_1, 0, \ldots , 0)$. This contradicts the first statement and concludes the proof.
Local convergence of Alg. \[alg:AL-multi\] with restructured sweeping order {#sec:local}
===========================================================================
We will consider a modified version of Alg. \[alg:AL-multi\], with differently structured sweeps. This makes it slower than Alg. \[alg:AL-multi\], but both algorithms achieve similar compression ratios in our examples.
Alg. \[alg:AL-multi\] improves each scale in every iteration. We now consider a modification that improves only one scale until convergence, and then moves on to the remaining scales successively. In detail, fix a maximum iteration number $M$. For each $k= 0, 1, \ldots , L$ in turn, the procedure computes an approximation $T_k^{(n)}$, for $n=1, \ldots , M$ on the $k$:th level, by calling Alg. \[alg:AL-multi\] on the tensor $T - \sum_{\ell = 0}^{k-1} \text{ext}_{L-\ell}(T_\ell^{(M)})$ with the two-level rank vector $$(\underbrace{0, \ldots, 0}_{k-1}, r_k, \underbrace{0, \ldots , 0}_{L-k-1}, r_{k+1} + \ldots + r_L).$$ This produces a tensor $T_k^{(M)}$ on scale $k$ and a tensor $S^{(M)}_k$ on scale $L$. $T_k^{(M)}$ is stored and $S_k^{(M)}$ is discarded. Sweeping through all scales results in the multiresolution approximation $\sum_{\ell = 0}^{L} \text{ext}_{L-\ell}(T_\ell^{(M)}).$
We will consider tensors with a globally optimal approximation $\sum_{k = 0}^{L} \text{ext}_{L-k}(T_k)$ in the multiresolution format. Denote by $R$ the optimal approximation residual, i.e., $R = T - \sum_{k = 0}^{L} \text{ext}_{L-k}(T_k)$. While updating scales $k$ and $L$ during iteration $n$ of a run of the algorithm, the update equations in Lemma \[lemma:updateS\] read as $$\begin{aligned}
T_{k}^{(n+1)} &= \text{round}_{\mathcal{F}_{r_k}}\Bigl( \text{ave}_{L-k}\bigl(T - \sum_{\ell=0}^{k-1} \text{ext}_{L-\ell}(T_{\ell}^{(M)}) - S_k^{(n)}\bigr) \Bigr) \\
&= \text{round}_{\mathcal{F}_{r_k}}\Bigl( T_k + \text{ave}_{L-k}\bigl( R - S_k^{(n)}\\
& \qquad \qquad \qquad + \sum_{\ell=k+1}^{L} \!\!\! \text{ext}_{L-\ell}(T_{\ell}) + \!\! \sum_{\ell=0}^{k-1} \!\! \text{ext}_{L-\ell}(T_{\ell}- T^{(M)}_\ell) \bigr) \Bigr),\tag{\stepcounter{equation}\theequation} \\
S_k^{(n+1)} &= \text{round}_{\mathcal{F}_{r_{k+1} + \ldots + r_L}}\Bigl( T - \sum_{\ell=0}^{k-1} \text{ext}_{L-\ell}(T_{\ell}^{(M)}) - T_k^{(n+1)}\Bigr) \\
&= \text{round}_{\mathcal{F}_{r_{k+1} + \ldots + r_L}}\Bigl( \sum_{\ell=k+1}^{L} \!\!\! \text{ext}_{L-\ell}(T_{\ell}) \\
& \qquad \qquad \qquad + R + \text{ext}_{L-k}(T_k- T_k^{(n+1)}) + \sum_{\ell=0}^{k-1} \text{ext}_{L-\ell}(T_{\ell}- T^{(M)}_\ell ) \Bigr) .\end{aligned}$$
The key step of the convergence proof will be to analyze the following two amalgamated residual terms $$\label{eq:errors}
\begin{split}
E_{k}^{(n)} &:= \text{ave}_{L-k}\left( R - S^{(n)}_k + \sum_{\ell=k+1}^{L} \text{ext}_{L-\ell}(T_{\ell}) + \sum_{\ell=0}^{k-1} \text{ext}_{L-\ell}(T_{\ell}- T^{(M)}_\ell) \right), \\
D_{k}^{(n)} &:= R + \text{ext}_{L-k}(T_k- T_k^{(n+1)}) + \sum_{\ell=0}^{k-1} \text{ext}_{L-\ell}(T_{\ell}- T^{(M)}_\ell ) .
\end{split}$$ The idea of the proof is to show that our algorithm is locally a contraction, when the two residual terms $E_{k}^{(n)}$ and $D_{k}^{(n)}$ are sufficiently small. We will prove local convergence under the following three assumptions.
\[ass:smooth\] The set of tensors $\mathcal{F}_{r}$ is a weakly closed, smooth manifold embedded in $\mathbb{R}^{b_s^L\times \ldots \times b_s^L}$ for any $r$.
\[ass:rank\] The tensors $\text{ext}_{L-k-1}(T_{k+1}) + \ldots + T_L$ have rank $r_{k+1} + \ldots + r_L$, for each $k = 0, \ldots, L$.
\[ass:sep\] There is an angle $\theta > 0$ such that, for any $n$ and $k$, the amalgamated errors $E_{k}^{(n)}$ and $D_{k}^{(n)}$ subtend an angle greater than $\theta$ to the tangent space of $\mathcal{F}_{r_k}$ at the point $T_k$, and of $\mathcal{F}_{r_{k+1} + \ldots + r_L}$ at the point $\text{ext}_{L-k-1}(T_{k+1}) + \ldots + T_L$, respectively.
The first assumption is required for optimal approximations in $\mathcal{F}_r$ to exist. The second assumption excludes non-minimal examples. For instance in the matrix case, if $r_0 = 1$, and $r_1 = b_s$, then $\text{ext}_{L}(T_0) + \text{ext}_{L-1}(T_1)$ also has rank $b_s \neq r_0 + r_1$. It would therefore be desirable for a properly designed algorithm to converge to a tensor in the multiresolution format with rank vector $(0, r_1, \ldots, r_L)$ instead of $(r_0, r_1, \ldots, r_L)$, to reduce storage cost. We therefore exclude these cases from consideration by imposing Assumption \[ass:rank\]. The third assumption is technical and made so that we can use first-order perturbation expansions as part of our analysis. It can be viewed as a non-degeneracy condition. Even though it is not verifiable a priori, should it not hold at iteration $n$, our algorithm is still nearly a contraction, with an error that can grow additively at most by a term proportional to $\|T - \sum_{k = 0}^{L} \text{ext}_{L-k}(T_k)\|^2$, as will be shown in Lemma \[lemma:contract\] below. Our main result is the following.
\[thm:locconv\] Let $T$ be a matrix with an isolated, globally optimal approximation $\sum_{k=0}^L \text{ext}_{L-k} (T_k) $ in the multiresolution format with rank vector $(r_0, \ldots , r_k)$. Under assumptions \[ass:smooth\] – \[ass:rank\], there is then a constant $C > 0$ depending only on $T_0, \ldots , T_L$ such that
1. if $ T = \sum_{k=0}^L \text{ext}_{L-k} (T_k)$ and $\| T_k - T_k^{(0)} \| \leq C$, then when using the algorithm of this section, $T^{(n)}_k \rightarrow T_k$ linearly as $n\rightarrow \infty$.
2. More generally, if $$\begin{split}
\|T - \sum_{k=0}^L \text{ext}_{L-k} (T_k) \| \leq C, \\
\| T_k - T_k^{(0)} \| \leq C,
\end{split}$$ then when using the algorithm of this section, $$\label{eq:locbound}
\| T_k - T_k^{(n)}\| \leq C_k \|T - \sum_{k=0}^L \text{ext}_{L-k} (T_k)\|$$ for all $n$ large enough and some real constants $C_k$. In particular, there is a convergent subsequence $T^{(n_\ell)}_k$ such that $T^{(n_\ell)}_k \rightarrow S_k$ for all $k=0, \ldots , L$, where $$\label{eq:locboundconv}
\| T_k - S_k\| \leq C_k \|T - \sum_{k=0}^L \text{ext}_{L-k} (T_k)\|.$$
Assumption \[ass:smooth\] is satisfied e.g., for the low-rank matrix format as well as for the TT-format [@holtz2012manifolds; @steinlechner2016riemannian]. However, for the TT-format, note that the conclusion in Thm. \[thm:locconv\] applies when finding optimal low-rank approximations in the algorithm, rather than the quasi-optimal ones returned by the TT-SVD algorithm.
The result of Thm. \[thm:locconv\] is illustrated in Fig. \[fig:ex\_locconv\], for a matrix $T =\! \sum_{k=0}^L \text{ext}_{L-k} (T_k)$ and rank-vector $(0,0,0,8,0,10,10,10)$, which satisfy Assumption \[ass:rank\]. The matrices $T_k$ on each grid-scale were chosen with i.i.d. standard normal entries, and then normalized to have $\|T_k\| = 1$. The initial guess $\|T_k^{(0)}\|$ on each scale was chosen also with i.i.d. standard normal entries, with $\|T_k - T_k^{(0)}\| = 0.1$. We used $b_s = 2$ and $n = 128$.
![Example of local convergence guaranteed by Thm. \[thm:locconv\].[]{data-label="fig:ex_locconv"}](f15.pdf){width="100.00000%"}
We will need the following Lemma.
\[lemma:contract\] Let $x$ be a point on a smooth manifold $M$ embedded in $\mathbb{R}^N$, and $y$ a point in $\mathbb{R}^N$ s.t. the vector $y-x$ subtends an angle greater than $\theta$ with the tangent plane to $M$ at $x$. There is then a neighborhood of $x$ in $M$ and a constant $C<1$ which only depend on $x$ and $\theta$, such that the projection $\text{proj}_M$ onto the manifold, is uniquely defined in this neighborhood and satisfies $$\| \text{proj}_M(y) - x \| \leq C \|y-x\|,$$
for all $y$ in the neighborhood.
Throughout the proof, we will denote by $D$ arbitrary constants that only depend on $x_0$ and $\theta$. For ease of notation, we will not keep track of the exact expression for $D$, and the constants can be redefined even on the same line.
By the assumptions and possibly after permuting the coordinates, there is a smooth function $f$ and a neighborhood of radius $r$ around $x$ wherein points on $M$ can be written as $(z,f(z))$. Write $x = (x_0, f(x_0))$. Moreover, by the $\varepsilon$-neighborhood theorem [@guillemin2010differential p. 69], there is a neighborhood of radius $\eta$ around $x$ such that the projection onto $M$ is uniquely defined. Take $\varepsilon = \frac{1}{2}\text{min}(r,\eta)$.
Since $\text{proj}_M(y)$ is the closest point on $M$ to $y$ and hence closer than $x$, $\|\text{proj}_M(y) - y\| \leq \|x - y\|$. Any $y$ in the $\varepsilon$-neighborhood of $x$ then has $\text{proj}_M(y)$ of the form $(z_0, f(z_0))$ with $\|x_0 - z_0\| \leq 2\varepsilon$, since $\|x_0 - z_0\| \leq \|\text{proj}_M(y) - x\| \leq \|\text{proj}_M(y) - y\| + \|x - y\| \leq 2\|x-y\| \leq 2\varepsilon \leq r$.
Since $f$ is smooth, the shift of the tangent space $T_xM$ to $x$ consists of all points of the form $\bigl(z, f(x_0) + df(x_0)(z-x_0)\bigr)$. There is also a constant $D$ such that $$\label{eq:helper_manif}
\| f(z) - f(x_0) - df(x_0)(z-x_0) \| \leq D \|z-x_0\|^2,$$ for $z$ in the $r$-neighborhood of $x_0$. Denote now by $y_p$ the orthogonal projection of $y$ onto $T_x$. Let also $\text{proj}_M(y) = \bigl(z_0, f(z_0)\bigr)$ be the projection of $y$ onto $M$. Since $f$ is smooth, $d f$ varies smoothly so it follows that $$\label{eq:z2x}
\begin{split}
\| d f(z_0) - d f(x_0)\| &\leq D \|z_0 - x_0 \| \leq D \| \text{proj}_M(y) - x\| \\
&\leq D\left( \| \text{proj}_M(y) - y \| + \| y - x\| \right) \leq 2D\|x-y\|.
\end{split}$$ It follows that the line passing through $y$ and $\text{proj}_M(y)$ is not contained in $T_xM$. If it were, we would be able to write $y -\text{proj}_M(y) = (\xi, df(x_0)\xi$. Since $y -\text{proj}_M(y)$ is orthogonal to $T_{\text{proj}_M(y)}M$, this would imply $$\begin{split}
0 &= \langle (\xi, df(x_0)\xi), (\xi, df(z_0)\xi) \rangle \\
&= \langle (\xi, df(x_0)\xi), (\xi, df(x_0)\xi) \rangle + \langle df(x_0)\xi, (df(z_0) - df(x_0))\xi \rangle \\
& \geq \| (\xi, df(x_0)\xi) \|^2 \cdot (1 - \| (df(z_0) - df(x_0))\|) > 0,
\end{split}$$ by Eq. for $\varepsilon$ sufficiently small, which is a contradiction
The line through $y$ and $\text{proj}_M(y)$ therefore intersects the shift of $T_xM$ to $x$ at some point $y_i$. Lastly, we denote the lift of $\text{proj}_M(y)$ onto the shift of $T_xM$ to $x$ by $y_\ell = \bigl(z_{0}, f(x_0) + df(x_0)(z_{0}-x_0)\bigr)$. We have $$\begin{split}\label{eq:linearquadratic}
\| \text{proj}_M(y) - x \| &\leq \| \text{proj}_M(y) - y_\ell \| + \|y_\ell - y_i\| + \|y_i - y_p\| + \|y_p - x\|,
\end{split}$$ and we proceed by showing that the first three terms are bounded by constants times $\|y - x\|^2$.
For the first term, Eq. gives $$\| \text{proj}_M(y) - y_\ell \| = \| f(z_0) - f(x_0) - df(x_0)(z-x_0) \| \leq D \|z_0-x_0\|^2.$$ Now, we have $\|z_0 - x_0\| \leq \|\text{proj}_M(y) - x \| \leq \|\text{proj}_M(y) - y \| + \| y - x\| \leq 2 \|y - x\|$. It follows that $$\label{eq:1stterm}
\| \text{proj}_M(y) - y_\ell \| \leq D \|y - x\|^2.$$
We next study the third term. The vector $y-y_i$ is orthogonal to any vector in $T_{\text{proj}_M(y)}M$, which are of the form $(\xi, df(z_0)\xi)$. Likewise, $y-y_p$ is orthogonal to $T_xM$, i.e. vectors of the form $(\xi, df(x_0)\xi)$. If we write $y-y_i = \bigl( (y - y_i)_1, (y-y_i)_2 \bigr)$, this means that $$\begin{split}
0 &= \langle y-y_i, (\xi, df(z_0)\xi) \rangle - \langle y-y_p, (\xi, df(x_0)\xi) \rangle = \\
&= \langle y_p-y_i, (\xi, df(x_0)\xi) \rangle + \langle (y-y_i)_2, (df(z_0) - df(x_0))\xi \rangle
\end{split}$$ Since $y_p - y_i$ is in $T_xM$, it can be written on the form $y_p - y_i = \bigl((y_p - y_i)_1, df(x_0)(y_p - y_i)_2\bigr)$, so $$\begin{split}
- (y-y_i)_2^T (df(z_0) - df(x_0))\xi &= - \langle (y-y_i)_2, (df(z_0) - df(x_0))\xi \rangle \\
&=\langle y_p-y_i, (\xi, df(x_0)\xi) \rangle \\
&= (y_p-y_i)_1^T\left[ I + df(x_0)^Tdf(x_0)\right]\xi,
\end{split}$$ for any vector $\xi$. This implies $$(df(z_0) - df(x_0))^T (y-y_i)_2 = \left[ I + df(x_0)^Tdf(x_0)\right](y_p-y_i)_1.$$ Since $I$ is positive definite and $df(x_0)^Tdf(x_0)$ is positive semidefinite, the matrix in the right hand side is positive definite and hence invertible. This results in $$\label{eq:3dtermhelp}
\|(y_p-y_i)_1\| \leq \bigl\| \left[ I + df(x_0)^Tdf(x_0)\right]^{-1}\bigr\| \cdot \|(df(z_0) - df(x_0))^T\| \cdot \|(y - y_i)_2\|.$$ We observe that $$\begin{aligned}
&\|y_p - y_i\| = \|\bigl((y_p - y_i)_1, df(x_0)(y_p - y_i)_1\bigr)\| \leq (1+\|df(x_0)\|)\|(y_p - y_i)_1\| \\
&\|(y - y_i)_2\| \leq \|y - y_i\| \leq \|y - y_p\| + \|y_p - y_i\| \leq \|y-x\| + \|y_p - y_i \| \\
&\|(df(z_0) - df(x_0))^T\| \leq D\|z_0 - x_0\| \leq 2D\|y-x\|,\end{aligned}$$ where the second equation used the fact that $y_p$ is the closest point to $y$ on the shift of $T_xM$ to $x$ which also includes $x$, and the third equation used Eq. . Inserting this into Eq. results in $$\begin{split}
\|y_p - y_i\| &\leq 2D(1+\|df(x_0)\|)\bigl\| \left[ I + df(x_0)^Tdf(x_0)\right]^{-1}\bigr\| \cdot \\
&\cdot \|y-x\| (\|x-y\| + \|y_p - y_i \|) \leq D \|x-y\|^2 + \varepsilon D \|y_p - y_i\|.
\end{split}$$ Moving the last term to the left hand side shows that $$\label{eq:3dterm}
\|y_p - y_i\| \leq D\|x-y\|^2,$$ for $\varepsilon$ sufficiently small.
We lastly study the second term. Note that the vector $y_i - y_\ell$ is in $T_xM$ and can therefore be written as $y_i - y_\ell = \bigl( (y_i - y_\ell)_1, df(x_0)(y_i - y_\ell)_1\bigr)$. Next write $y_i - \text{proj}_M(y) = \bigl( (y_i - \text{proj}_M(y))_1, (y_i - \text{proj}_M(y))_2 \bigr)$. Since this vector is orthogonal to $T_{\text{proj}_M(y)}M$ $$\begin{split}
0 &= (y_i - \text{proj}_M(y))\cdot \left(\xi, df(z_0)\xi\right) \\
&= (y_i - \text{proj}_M(y))\cdot \left(\xi, df(x_0)\xi\right) + (y_i - \text{proj}_M(y))_2\cdot \left(df(z_0) - df(x_0)\right)\xi,
\end{split}$$ for any vector $\xi$. Taking $\xi = (y_{i} - y_{\ell})_1$ results in $$0 = (y_i - \text{proj}_M(y))\cdot (y_i - y_\ell) + (y_i - \text{proj}_M(y))_2\cdot \left(df(z_0) - df(x_0)\right) (y_{i} - y_{\ell})_1,$$ so $$\begin{split}
\abs{ (y_i - \text{proj}_M(y)) (y_i - y_\ell) } &\leq \| (y_i - \text{proj}_M(y))_2\| \| df(z_0) - df(x_0)\| \|(y_{i} - y_{\ell})_1\| \\
&\leq D\| y_i - \text{proj}_M(y)\| \| x - y\| \|y_{i} - y_{\ell}\|
\end{split}$$ This implies that the angle $\alpha$ between the vectors $y_i - \text{proj}_M(y)$ and $y_i - y_\ell$ satisfies $\cos \alpha \leq D\|x-y\| \leq D\varepsilon$. By choosing $\varepsilon$ sufficiently small, we can therefore guarantee $\sin \alpha \geq \frac{1}{2}$. The law of sines for the triangle with vertices $y_i$, $y_\ell$ and $\text{proj}_M(y)$ therefore gives $$\label{eq:2ndterm}
\| y_i - y_\ell\| \leq \frac{\|y_\ell - \text{proj}_M(y)}{\sin \alpha} \leq 2D \|x-y\|^2.$$
Inserting Eqs. ,, into Eq. results in $$\begin{split}
\| \text{proj}_M(y) - x \| &\leq \| y_p - x \| + D\|y-x\|^2.
\end{split}$$ By assumption, the angle between $y_p - x$ is at least $\theta$, so $$\|y_p - x \| \leq \|y-x\| \cos \theta.$$ By choosing $\varepsilon$ so that $D\varepsilon + \cos \theta < 1$, it follows that $$\begin{split}
\| \text{proj}_M(y) - x \| &\leq \| y_p - x \| + D\|y-x\|^2 \leq (\cos \theta + D\|x-y\|)\|x-y\| \\
&\leq (\cos \theta + D\varepsilon)\|x-y\|,
\end{split}$$ which finishes the proof with $C = 1 - (\cos \theta + D\varepsilon)$, since $0 < C < 1$.
We can now prove the main result of this section.
Write $R = T - \sum_{k=0}^L \text{ext}_{L-k} (T_k)$, and first assume that $\|R\| \neq 0$. We proceed by induction and assume that the algorithm has computed $T^{(M)}_0$, $\ldots$ , $T^{(M)}_{k-1}$ with $\| T_\ell- T_\ell^{(M)}\| \leq C_k \|R\|$, for $\ell = 1, \ldots , k-1$. When computing $T^{(n)}_k$, the algorithm proceeds by the updates $$\begin{split}
T_{k}^{(n+1)} &= \text{round}_{\mathcal{F}_{r_k}}\Bigl( T_k + E_k^{(n)} \Bigr), \\
S_k^{(n+1)} &= \text{round}_{\mathcal{F}_{r_{k+1} + \ldots + r_L}}\Bigl( \sum_{\ell=k+1}^{L} \!\!\! \text{ext}_{L-\ell}(T_{\ell}) + D_k^{(n)} \Bigr),
\end{split}$$ where $E_k^{(n)}$ and $D_k^{(n)}$ are defined in Eq. . This implies $$\begin{split}
&T_k^{(n+1)} - T_k = \text{round}_{\mathcal{F}_{r_k}}\left( T_k + E_k^{(n)}\right) - T_k, \\
& S_k^{(n+1)} -\!\!\!\! \sum_{\ell=k+1}^{L} \!\! \text{ext}_{L-\ell}(T_{\ell}) = \text{round}_{\mathcal{F}_{r_{k+1} + \ldots + r_L}}\bigl( \!\!\! \sum_{\ell=k+1}^{L} \!\!\! \text{ext}_{L-\ell}(T_{\ell}) + D_k^{(n)} \bigr) \!- \!\! \!\!\sum_{\ell=k+1}^{L} \!\! \text{ext}_{L-\ell}(T_{\ell}),
\end{split}$$ and we now show that the norm of these residual terms are bounded by the norms of $E_k^{(n)}$ and $D_k^{(n)}$, respectively. Provided the residual terms are sufficiently small, Lemma \[lemma:contract\] implies $$\label{eq:krate}
\begin{split}
\| &T_{k}^{(n+1)} - T_{k} \| \leq (1-C)\|E_k^{(n)}\| \\
&= (1-C)\| \text{ave}_{L-k}(\sum_{\ell=k+1}^{L} \text{ext}_{L-\ell}\bigl(T_{\ell}) - S_k^{(n)} + R + \sum_{\ell=0}^{k-1} \text{ext}_{L-\ell}(T_{\ell}- T^{(M)}_\ell) \Bigr)\| \\
&\leq \frac{(1-C)}{b_s^{\frac{d}{2}(L-k)}} \|\sum_{\ell=k+1}^{L} \text{ext}_{L-\ell}(T_{\ell}) - S_k^{(n)} + R + \sum_{\ell=0}^{k-1} \text{ext}_{L-\ell}(T_{\ell}- T^{(M)}_\ell)\|,
\end{split}$$ where the first step used Assumption \[ass:sep\] and the third step used Cauchy-Schwarz. Similarly, using Assumptions \[ass:sep\] and \[ass:rank\], we obtain $$\label{eq:convS}
\begin{split}
\| &S_k^{(n+1)} - \!\! \sum_{\ell=k+1}^{L} \!\! \text{ext}_{L-\ell}(T_{\ell}) \| \\
&\leq (1-C)\| \text{ext}_{L-k}(T_k- T_k^{(n+1)}) + R + \sum_{\ell=0}^{k-1} \text{ext}_{L-\ell}(T_{\ell}- T^{(M)}_\ell )\| \\
&\leq (1-C)b_s^{\frac{d}{2}(L-k)} \|T_k- T_k^{(n+1)}\| + (1-C_T)\|R\| + (1-C) \sum_{\ell=0}^{k-1} b^{\frac{d}{2}(L-\ell)}_s C_\ell \|R\| \\
&\leq (1-C)^2 \|S_k^{(n)} - \sum_{\ell=k+1}^{L} \text{ext}_{L-\ell}(T_{\ell}) - R + \sum_{\ell=0}^{k-1} \text{ext}_{L-\ell}(T_{\ell}- T^{(M)}_\ell)\| \\
&\qquad + (1-C)\|R\| + (1-C) \sum_{\ell=0}^{k-1} b^{\frac{d}{2}(L-\ell)}_s C_\ell \|R\| \\
& \leq (1-C)^2 \|S_k^{(n)} - \sum_{\ell=k+1}^{L} \text{ext}_{L-\ell}(T_{\ell}) \| + D\|R\|,
\end{split}$$ since $\|T_\ell - T_\ell^{(M)}\| \leq C_\ell \|R\|$, by the induction hypothesis. Here, $D$ is a constant dependent on $C$, $\{C_\ell\}_{\ell=0}^{k-1}$ and $b_s$. It follows that $$\begin{split}
\| S_k^{(n+1)} &\!\!\!- \!\! \!\! \sum_{\ell=k+1}^{L}\!\! \text{ext}_{L-\ell}(T_{\ell}) \| \leq (1-C)^{2n+2} \|S_k^{(0)} - \!\!\!\! \sum_{\ell=k+1}^{L} \!\! \text{ext}_{L-\ell}(T_{\ell}) \| + D\|R\| \!\! \sum_{m=0}^{n+1} (1-C)^{2m} \\
&\leq (1-C)^{2n+2} \|S_k^{(0)} - \sum_{\ell=k+1}^{L} \text{ext}_{L-\ell}(T_{\ell}) \| + \frac{D}{(1-C)^2}\|R\|.
\end{split}$$
Letting $n \rightarrow \infty$ shows that $\| S_k^{(n+1)} - \sum_{\ell=k+1}^{L} \text{ext}_{L-\ell}(T_{\ell}) \| \leq (1+\frac{D}{(1-C)^2})\|R\|$ for $n$ large enough. Inserting this into Eq. shows that $$\begin{split}
\| T_{k}^{(n+1)} - T_{k} \| &\leq \frac{(1-C)}{b_s^{\frac{d}{2}(L-k)}} \left[ 2+\frac{D}{(1-C)^2}+ \sum_{\ell=0}^{k-1} b^{\frac{d}{2}(L-\ell)}_s C_\ell\right] \|R\|,
\end{split}$$ which concludes the induction hypothesis and therefore the proof of the first statement. For the second statement, note that this implies that the $T_k^{(n)}$ are bounded, so there is a convergent subsequence by the Bolzano-Weierstrass theorem. Eq. then follows from taking the limit of Eq. .
Lastly, assume that $\|R\| = 0$. Eqs. and imply that $$\begin{split}
\| T_{k}^{(n+1)} - T_{k} \| \leq (1-C)^{2n+3} \|S_k^{(0)} - \sum_{\ell=k+1}^{L} \text{ext}_{L-\ell}(T_{\ell}) \|,
\end{split}$$ which concludes the proof.
[^1]: [`https://github.com/MultiResTF/multiresolution`](https://github.com/MultiResTF/multiresolution)
[^2]: [`https://github.com/MultiResTF/multiresolution/blob/master/movies/movie1.mp4`](https://github.com/MultiResTF/multiresolution/blob/master/movies/movie1.mp4)
[^3]: [`https://github.com/MultiResTF/multiresolution/blob/master/movies/movie2.mp4`](https://github.com/MultiResTF/multiresolution/blob/master/movies/movie2.mp4)
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In order to popularize the so called Schwinger’s method we reconsider the Feynman propagator of two non-relativistic systems: a charged particle in a uniform magnetic field and a charged harmonic oscillator in a uniform magnetic field. Instead of solving the Heisenberg equations for the position and the canonical momentum operators, ${\bf R}$ and ${\bf P}$, we apply this method by solving the Heisenberg equations for the gauge invariant operators ${\bf R}$ and $\mbox{\mathversion{bold}${\pi}$} = {\bf P}-e{\bf A}$, the latter being the mechanical momentum operator. In our procedure we avoid fixing the gauge from the beginning and the result thus obtained shows explicitly the gauge dependence of the Feynman propagator.'
author:
- 'A. Aragão'
- 'F. A. Barone'
- 'H. Boschi-Filho'
- 'C. Farina'
title: 'Non-Relativistic Propagators via Schwinger’s Method'
---
Introduction
============
In a recent paper published in this journal [@BaroneBoschiFarinaAJP2003], three methods were used to compute the Feynman propagators of a one-dimensional harmonic oscillator, with the purpose of allowing a student to compare the advantadges and disadvantadges of each method. The above mentioned methods were the following: the so called Schwinger’s method (SM), the algebraic method and the path integral one. Though extremely powerful and elegant, Schwinger’s method is by far the less popular among them. The main purpose of the present paper is to popularize Schwinger’s method providing the reader with two examples slightly more difficult than the harmonic oscillator case and whose solutions may serve as a preparation for attacking relativistic problems. In some sense, this paper is complementary to reference [@BaroneBoschiFarinaAJP2003].
The method we shall be concerned with was introduced by Schwinger in 1951 [@Schwinger1951] in a paper about QED entitled Gauge invariance and vacuum polarization". After introducing the proper time representation for computing effetive actions in QED, Schwinger was faced with a kind of non-relativistic propagator in one extra dimension. The way he solved this problem is what we mean by Schwinger’s method for computing quantum propagators. For relativistic Green functions of charged particles under external electromagnetic fields, the main steps of this method are summarized in Itzykson and Zuber’s textbook [@ItziksonZuberBook] (apart, of course, from Schwinger’s work [@Schwinger1951]). Since then, this method has been used mainly in relativistic quantum theory [@GitmanBook; @Dodonov75; @Dodonov76; @Lykken; @Ferrando:1994vt; @BFV96; @Gavrilov:1998hw; @McKeon:1998zx; @Chyi:1999fc; @Tsamis:2000ah; @Chaichian:2000eh; @Chung:2001mb; @BoschiFarinaVaidya1996].
However, as mentioned before, Schwinger’s method is also well suited for computing non-relativistic propagators, though it has rarely been used in this context. As far as we know, this method was used for the first time in non-relativistic quantum mechanics by Urrutia and Hernandez [@UrrutiaHernandez1984]. These authors used Schwinger’s action principle to obtain the Feynman propagator for a damped harmonic oscillator with a time-dependent frequency under a time-dependent external force. Up to our knowledge, since then only a few papers have been written with this method, namely: in 1986, Urrutia and Manterola [@UrrutiaManterola1986] used it in the problem of an anharmonic charged oscillator under a magnetic field; in the same year, Horing, Cui, and Fiorenza [@HoringCuiFiorenza1986] applied Schwinger’s method to obtain the Green function for crossed time-dependent electric and magnetic fields; the method was later applied in a rederivation of the Feynman propagator for a harmonic oscillator with a time-dependent frequency [@FarinaSegui1993]; a connection with the mid-point-rule for path integrals involving electromagnetic interactions was discussed in [@RabelloFarina1995]. Finally, pedagogical presentations of this method can be found in the recent publication [@BaroneBoschiFarinaAJP2003] as well as in Schwinger’s original lecture notes recently published [@SchwingerBookEnglert2001], which includes a discussion of the quantum action principle and a derivation of the method to calculate propagators with some examples.
It is worth mentioning that this same method was independently developed by M. Goldberger and M. GellMann in the autumn of 1951 in connection with an unpublished paper about density matrix in statistical mechanics [@Goldberger1951].
Our purpose in this paper is to provide the reader with two other examples of non-relativistic quantum propagators that can be computed in a straightforward way by Schwinger’s method, namely: the propagator for a charged particle in a uniform magnetic field and this same problem with an additional harmonic oscillator potential. Though these problems have already been treated in the context of the quantum action principle [@UrrutiaManterola1986], we decided to reconsider them for the following reasons: instead of solving the Heisenberg equations for the position and the canonical momentum operators, ${\bf R}$ and ${\bf P}$, as is done in [@UrrutiaManterola1986], we apply Schwinger’s method by solving the Heisenberg equations for the gauge invariant operators ${\bf R}$ and $\mbox{\mathversion{bold}${\pi}$} = {\bf P}-e{\bf A}$, the latter being the mechanical momentum operator. This is precisely the procedure followed by Schwinger in his seminal paper of gauge invariance and vacuum polarization [@Schwinger1951]. This procedures has some nice properties. For instance, we are not obligued to choose a particular gauge at the beginning of calculations. As a consequence, we end up with an expression for the propagator written in an arbitrary gauge. As a bonus, the transformation law for the propagator under gauge transformations can be readly obtained.
In order to prepare the students to attack more complex problems, we solve the Heisenberg equations in matrix form, which is well suited for generalizations involving Green functions of relativistic charged particles under the influence of electromagnetic fields (constant $F_{\mu\nu}$, a plane wave field or even combinations of both). For pedagogical reasons, at the end of each calculation, we show how to extract the corresponding energy spectrum from the Feynman propagator. Although the way Schwniger’s method must be applied to non-relativistic problems has already been explained in the literature [@UrrutiaManterola1986; @SchwingerBookEnglert2001; @BaroneBoschiFarinaAJP2003], it is not of common knowledge so that we start this paper by summarizing its main steps. The paper is organized as follows: in the next section we review Schwinger’s method, in section \[SectionExamples\] we present our examples and section \[SectionFinalRemarks\] is left for the final remarks.
Main steps of Schwinger’s method
================================
For simplicity, consider a one-dimensional time-independent Hamiltonian $\mathcal{H}$ and the corresponding non-relativistic Feynman propagator defined as $$\label{DefinicaoPropagador}
K(x,x^{\prime};\tau)=\theta(\tau) \langle x | \exp{ \Big[ {\frac{-i
\cal{H} \tau}{\hbar}} \Big]} | x^{\prime} \rangle,$$ where $\theta(\tau)$ is the Heaviside step function and $|x\rangle$, $|x^{\prime}\rangle$ are the eingenkets of the position operator $X$ (in the Schrödinger picture) with eingenvalues $x$ and $x^{\prime}$, respectively. The extension for 3D systems is straightforward and will be done in the next section. For $\tau>0$ we have, from equation (\[DefinicaoPropagador\]), that $$\label{Eq2}
i\hbar\frac{\partial}{\partial\tau}K(x,x^{\prime};\tau) =
\langle x |\mathcal{H} \exp{\Big[ {\frac{-i \mathcal{H} \tau}{\hbar}}\Big]} |
x^{\prime}\rangle.$$ Inserting the unity $\uma =\exp{[-(i/\hbar)\mathcal{H}\tau]}\exp{[(i/\hbar)\mathcal{H}\tau]}$ in the r.h.s. of the above expression and using the well known relation between operators in the Heisenberg and Schrödinger pictures, we get the equation for the Feynman propagator in the Heisenberg picture, $$i\hbar\frac{\partial}{\partial\tau}
K(x,x^{\prime};\tau)=\langle x,\tau |\mathcal{H} (X(0),P(0))| x^{\prime},0\rangle, \label{Eq3}$$ where $|x,\tau\rangle$ and $|x^{\prime},0\rangle$ are the eingenvectors of operators $X(\tau)$ and $X(0)$, respectively, with the corresponding eingenvalues $x$ and $x^{\prime}$: $X(\tau)|x,\tau\rangle=x|x,\tau\rangle$ and $X(0)|x^{\prime},0\rangle=x^{\prime}|x^{\prime},0\rangle$, with $K(x,x^{\prime};\tau)=\langle x,\tau|x^{\prime},0\rangle$. Besides, $X(\tau)$ and $P(\tau)$ satisfy the Heisenberg equations, $$i\hbar \frac{dX}{d\tau}(\tau)=[X(\tau),\mathcal{H}] \hspace{0.2cm} ;
\hspace{0.2cm} i\hbar \frac{dP}{d\tau}(\tau)=[P(\tau),\mathcal{H}].
\label{Eq4}$$ Schwinger’s method consists in the following steps:
(i)
: we solve the Heisenberg equations for $X(\tau)$ and $P(\tau)$, and write the solution for $P(0)$ only in terms of the operators $X(\tau)$ and $X(0)$;
(ii)
: then, we substitute the results obtained in [**(i)**]{} into the expression for $\mathcal{H} (X(0),P(0))$ in (\[Eq3\]) and using the commutator $[X(0),X(\tau)]$ we rewrite each term of $\mathcal{H}$ in a time ordered form with all operators $X(\tau)$ to the left and all operators $X(0)$ to the right;
(iii)
: with such an ordered hamiltonian, equation (\[Eq3\]) can be readly cast into the form $$\label{Eq5}
i\hbar\frac{\partial}{\partial\tau} K(x,x^{\prime};\tau) =
F(x,x^{\prime};\tau)K(x,x^{\prime};\tau),$$ with $F(x,x^{\prime};\tau)$ being an ordinary function defined as $$\label{Eq6}
F(x,x^{\prime};\tau)=\frac{\langle x,\tau |\mathcal{H}_{ord}
(X(\tau),X(0))| x^{\prime},0\rangle}
{\langle x,\tau|x^{\prime},0\rangle}.$$ Integrating in $\tau$, the Feynman propagator takes the form $$\label{Eq7}
K(x,x^{\prime};\tau)=C(x,x^{\prime})
\exp\left\{\!\!-\frac{i}{\hbar}\!\! \int^{\tau}
\!\!\!\!\! F(x,x^{\prime};\tau^\prime)d\tau^{\prime}\!\right\} ,$$ where $C(x,x^{\prime})$ is an integration constant independent of $\tau$ and $\int^{\tau}$ means an indefinite integral;
(iv)
: last step is concerned with the evaluation of $C(x,x^{\prime})$. This is done after imposing the following conditions $$\begin{aligned}
\label{CondicaoP(tau)}
-i\hbar\frac{\partial}{\partial x}\langle
x,\tau|x^{\prime},0\rangle &=&
\langle x,\tau|P(\tau)|x^{\prime},0\rangle \, , \\
i\hbar\frac{\partial}{\partial x^{\,\prime}}\langle
x,\tau|x^{\prime},0\rangle &=&
\langle x,\tau|P(0)|x^{\prime},0\rangle\, , \label{CondicaoP(0)}\end{aligned}$$ as well as the initial condition $$\label{CondicaoInicial}
\lim_{\tau\rightarrow 0^+}
K(x,x^{\prime};\tau) = \delta(x-x^{\prime})\ .$$
Imposing conditions (\[CondicaoP(tau)\]) and (\[CondicaoP(0)\]) means to substitute in their left hand sides the expression for $\langle x,\tau|x^{\prime},0\rangle$ given by (\[Eq7\]), while in their right hand sides the operators $P(\tau)$ and $P(0)$, respectively, written in terms of the operators $X(\tau)$ and $X(0)$ with the appropriate time ordering.
Examples {#SectionExamples}
========
Charged particle in an uniform magnetic field {#cpumf}
---------------------------------------------
As our first example, we consider the propagator of a non-relativistic particle with electric charge $e$ and mass $m$, submitted to a constant and uniform magnetic field ${\bf B}$. Even though this is a genuine three-dimensional problem, the extension of the results reviewed in the last section to this case is straightforward. Since there is no electric field present, the hamiltonian can be written as $$\label{HamiltonianaBUniforme}
\mathcal{H}=\frac{\left({\bf P} - e{\bf A}\right)^{2}}{2m}
= \frac{\mbox{\mathversion{bold}${\pi}$}^2}{2m}\ ,$$ where ${\bf P}$ is the canonical momentum operator, ${\bf A}$ is the vector potential and $\mbox{\mathversion{bold}${\pi}$}={\bf P}-e{\bf A}$ is the gauge invariant mechanical momentum operation. We choose the axis such that the magnetic field is given by ${\bf B}=B{\bf e_3}$. Hence, the hamiltonian can be decomposed as $$\mathcal{H} = \frac{\pi_{1}^{2}+\pi_{2}^{2}}{2m} +
\frac{P_3^2}{2m} \; =\; \mathcal{H}_{\bot} +
\frac{P_{3}^{2}}{2m}\ ,$$ with an obvious definition for $\mathcal{H}_{\bot}$. Since the motion along the ${\cal OX}_3$ direction is free, the three-dimensional propagator $K({\bf x},{\bf x}^{\prime};\tau)$ can be written as a product of a two-dimensional propagator, $K_{\bot}({\bf r},{\bf r}^{\prime};\tau)$, related to the magnetic field and a one-dimensional free propagator, $K_{3}^{(0)}(x_3,x_3^{\prime};\tau)$: $$\label{DecomposicaoPropagador}
K({\bf x},{\bf x}^{\prime};\tau)=K_{\bot}({\bf r},{\bf
r}^{\prime};\tau)
K_{3}^{(0)}(x_3,x_3^{\prime};\tau),\;\;\;\;
(\tau>0)$$ where ${\bf r} = x_1{\bf e_1} + x_2{\bf e_2} $ and $K_{3}^{(0)}(x_3,x_3^{\prime};\tau)$ is the well known propagator of the free particle [@FeynmanHibbsBook], $$K_{3}^{(0)}(x_3,x_3^{\prime};\tau)=\sqrt{\frac{m}{2\pi i\hbar \tau}}
\exp{ \Big[ \frac{im}{2\hbar}\frac{(x_3 - x_3^{\prime})^{2}}{\tau}\Big]}.
\label{Eq14}$$ In order to use Schwinger’s method to compute the two-dimensional propagator $K_{\bot}({\bf r},{\bf r}^{\prime};\tau) =
\langle{\bf r},\tau|{\bf r}^{\prime},0\rangle$, we start by writing the differential equation $$\label{EqDiferencialPropagador2D}
i\hbar\frac{\partial}{\partial\tau}
\langle{\bf r},\tau|{\bf r}^{\prime},0\rangle =
\langle {\bf r},\tau|\mathcal{H}_{\bot}({\bf R}_\bot(0),
\mbox{\mathversion{bold}${\pi}$}_\bot(0) )| {\bf r}^{\prime},0\rangle\ ,$$ where ${\bf R}_\bot(\tau) =
X_{1}(\tau){\bf e_1} + X_{2}(\tau){\bf e_2}$ and $\mbox{\mathversion{bold}${\pi}$}_\bot(\tau) =
\pi_{1}(\tau){\bf e_1} + \pi_{2}(\tau){\bf e_2}$. In (\[EqDiferencialPropagador2D\]) $|{\bf r},\tau\rangle$ and $|{\bf r}^{\prime},0\rangle$ are the eigenvectors of position operators ${\bf R}(\tau)=X_{1}(\tau){\bf
e_1} + X_{2}(\tau){\bf e_2}$ and ${\bf R}(0)=X_{1}(0){\bf e_1} +
X_{2}(0){\bf e_2}$, respectively. More especifically, operators $X_{1}(0)$, $X_{1}(\tau)$, $X_{2}(0)$ and $X_{2}(\tau)$ have the eigenvalues $x_{1}^{\prime}$, $x_{1}$, $x_{2}^{\prime}$ and $x_{2}$, respectively. In order to solve the Heisenberg equations for operators ${\bf R}_\bot(\tau)$ and $\mbox{\mathversion{bold}${\pi}$}_\bot(\tau)$, we need the commutators $$\begin{aligned}
\label{ComutadorXcomPi2}
\Big[X_{i}(\tau) , \pi_{j}^{2}(\tau) \Big] &=&
2i\hbar \pi_{i}(\tau)\, ,\cr
\Big[\pi_{i}(\tau) , \pi_{j}^{2}(\tau)\Big] &=& 2i\hbar eB\epsilon_{ij3}\pi_{j}(\tau),
\label{ComutadorPicomPi}\end{aligned}$$ where $\epsilon_{ij3}$ is the usual Levi-Civita symbol. Introducing the matrix notation $${\bf R}(\tau)=\left(
\begin{array}{c}
X_{1}(\tau) \\ X_{2}(\tau)
\end{array}
\right)
\hspace{0.25cm};\hspace{0.25cm}
\mbox{\mathversion{bold}${\Pi}$}(\tau)=\left(
\begin{array}{c}
\pi_{1}(\tau) \\ \pi_{2}(\tau)
\end{array}
\right)
\label{Eq20}\ ,$$ and using the previous commutators the Heisenberg equations of motion can be cast into the form $$\begin{aligned}
\label{EquacaoR}
\frac{d{\bf R}(\tau)}{d\tau}&=&\frac{\mbox{\mathversion{bold}${\Pi}$}(\tau)}{m}\ , \\
\frac{d\mbox{\mathversion{bold}${\Pi}$}(\tau)}{d\tau} &=& 2\omega
\mathbb{C} \mbox{\mathversion{bold}${\Pi}$}(\tau) \,,
\label{EquacaoPi}\end{aligned}$$ where $2\omega={eB}/{m}$ is the cyclotron frequency and we defined the anti-diagonal matrix $$\mathbb{C}=\left(
\begin{array}{cc}
0 & 1 \\
-1 & 0
\end{array}
\right)
\label{Eq21}\ .$$ Integrating equation (\[EquacaoPi\]) we find $$\begin{aligned}
\mbox{\mathversion{bold}${\Pi}$}(\tau) &=&
e^{2\omega\mathbb{C}\tau}\mbox{\mathversion{bold}${\Pi}$}(0)
\label{Eq23}\ .\end{aligned}$$ Substituting this solution in equation (\[EquacaoR\]) and integrating once more, we get $$\begin{aligned}
\label{Eq22}
\textbf{R}(\tau)-\textbf{R}(0)&=&\frac{\sin{(\omega\tau)}}{m\omega}
e^{\omega\mathbb{C}\tau}\mbox{\mathversion{bold}${\Pi}$}(0)
\ ,\end{aligned}$$ where we used the following properties of $\mathbb{C}$ matrix: $\mathbb{C}^{2}\!=\!-\uma$; $\mathbb{C}^{-1}\!=\!-\mathbb{C}=\mathbb{C}^{T}$, $e^{\alpha\mathbb{C}} =
\cos{(\alpha)}\uma + \sin{(\alpha)}\mathbb{C}$ with $\mathbb{C}^{T}$ being the transpose of $\mathbb{C}$. Combining equations (\[Eq22\]) and (\[Eq23\]) we can write $\mbox{\mathversion{bold}${\Pi}$}(0)$ in terms of the operators ${\bf R}(\tau)$ and ${\bf R}(0)$ as $$\label{Eq24}
\mbox{\mathversion{bold}${\Pi}$}(0)=\frac{m\omega}{\sin{(\omega\tau)}}
e^{-\omega\mathbb{C}\tau}\biggl({\bf R}(\tau)-{\bf R}(0)\biggr).$$ In order to express $\mathcal{H}_\bot = (\pi_1^2 + \pi_2^2)/2m$ in terms of $\textbf{R}(\tau)$ and $\textbf{R}(0)$, we use (\[Eq24\]). In matrix notation, we have $$\begin{aligned}
\mathcal{H}_{\bot}&=&\frac{1}{2m}\,\mbox{\mathversion{bold}${\Pi}$}^{T}(0)\mbox{\mathversion{bold}${\Pi}$}(0)\nonumber\\
&=&\frac{m\omega^2}{2\sin^2{(\omega\tau)}}\biggl({\bf R}^{T}(\tau){\bf R}(\tau)+{\bf R}^{T}(0){\bf R}(0)+\nonumber\\
&\ &\hspace{1.5cm}-{\bf R}^{T}(\tau){\bf R}(0)-{\bf R}^{T}(0){\bf R}(\tau)\biggr)\ .
\label{Hnaoordenado}\end{aligned}$$ Last term on the r.h.s. of (\[Hnaoordenado\]) is not ordered appropriately as required in the step ([**ii**]{}). The correct ordering may be obtained as follows: first, we write $$\textbf{R}(0)^{T} \textbf{R}(\tau)
= \textbf{R}(\tau)^{T}\textbf{R}(0) +
\sum_{i=1}^{2}[X_{i}(0) , X_{i}(\tau)]\, .$$ Using equation (\[Eq22\]), the usual commutator $[X_{i}(0),{\pi}_j(0)]=i\hbar\delta_{ij}\uma$ and the properties of matrix $\mathbb{C}$ it is easy to show that $$\label{Eq25}
\sum_{i=1}^{2}[X_{i}(0) , X_{i}(\tau)]
=
\frac{2i\hbar\sin(\omega\tau)\cos(\omega\tau)}{m\omega}\, ,$$ so that hamiltonian $\mathcal{H}_{\bot}$ with the appropriate time ordering takes the form $$\begin{aligned}
\mathcal{H}_\perp&=&\frac{m\omega^2}{2\sin^2{(\omega\tau)}}
\biggl\{{\bf R}^{2}(\tau)+{\bf R}^{2}(0)-2{\bf R}^{T}(\tau){\bf R}(0)\biggr\} \, \nonumber \\ &-& i
\hbar \omega\cot(\omega\tau).
\label{Eq27}\end{aligned}$$ Substituting this hamiltonian into equation (\[EqDiferencialPropagador2D\]) and integrating in $\tau$, we obtain $$\label{ProtoPropagador}
\langle{\bf r},\tau\vert{\bf r}^{\prime},0\rangle =
\frac{C({\bf r},{\bf r}^{\prime})}{\sin{(\omega\tau)}}
\exp\biggl\{{\frac{im\omega}{ 2\hbar}}\cot(\omega\tau)({\bf r}-{\bf
r}^{\prime})^2\biggr\} ,$$ where $C({\bf r},{\bf r}^{\;\prime})$ is an integration constant to be determined by conditions (\[CondicaoP(tau)\]), (\[CondicaoP(0)\]) and (\[CondicaoInicial\]), which for the case of hand read $$\begin{aligned}
\langle{\bf r},\tau\vert\pi_j(\tau)\vert{\bf r}^{\prime},0\rangle
\!\!&=&\!\!
\left(-i\hbar\frac{\partial}{\partial x_j}-eA_j({\bf r})\right)
\langle{\bf r},\tau\vert{\bf r}^{\prime},0\rangle
\label{Eq30}\\
\langle{\bf r},\tau\vert\pi_j(0)\vert{\bf r}^{\prime},0\rangle
\!\!&=&\!\!
\left(i\hbar\frac{\partial}{\partial x^\prime_j}-eA_j({\bf
r}^{\prime})\right) \langle{\bf r},\tau\vert{\bf
r}^{\prime},0\rangle\ ,
\label{Eq31}\\
\lim_{\tau\rightarrow 0^+}\langle{\bf r},
\tau\vert{\bf r}^{\prime},0\rangle
\!\!&=&\!\! \delta^{(2)}({\bf r}-{\bf
r}^{\prime}).\hspace{3.45cm} \label{Eq32}\end{aligned}$$ In order to compute the matrix element on the l.h.s. of (\[Eq30\]), we need to express $\mbox{{\mathversion{bold}${\Pi}$}}(\tau)$ in terms of ${\bf R}(\tau)$ and ${\bf R}(0)$. From equaitons (\[Eq23\]) and (\[Eq24\]), we have $$\mbox{{\mathversion{bold}${\Pi}$}}(\tau)=
\frac{m\omega}{\sin{(\omega\tau)}}\mbox{\large
$e^{\omega\tau\mathbb{C}}$} \biggl({\bf R}(\tau)-{\bf R}(0)\biggr),
\label{Eq33}$$ which leads to the matrix element $$\begin{aligned}
\langle{\bf r},\tau\vert\pi_j(\tau)\vert{\bf r}^{\prime},0\rangle
&=&
m\omega[\cot(\omega\tau) \left(x_j-x^\prime_j \right) \nonumber\\
&+& \mbox{\large $\epsilon_{jk3}$}\left(x_k-x^\prime_k\right)]
\langle{\bf r},\tau\vert{\bf r}^{\prime},0\rangle\ , \label{Eq34a}\end{aligned}$$ where we used the properties of matrix $\mathbb{C}$ and Einstein convention for repeated indices is summed. Analogously, the l.h.s. of equation (\[Eq31\]) can be computed from (\[Eq24\]), $$\begin{aligned}
\label{Eq34b}
\langle{\bf r},\tau\vert\pi_j(0)\vert{\bf r}^{\prime},0\rangle &=& m\omega[\cot(\omega\tau) \left(x_j-x^\prime_j \right) \nonumber\\
&-&\mbox{\large $\epsilon_{jk3}$}\left(x_k-x^\prime_k\right)] \langle{\bf r},\tau\vert{\bf r}^{\prime},0\rangle\ .\end{aligned}$$ Substituting equations (\[Eq34a\]) and (\[Eq34b\]) into (\[Eq30\]) and (\[Eq31\]), respectively, and using (\[ProtoPropagador\]), we have $$\begin{aligned}
\label{Equacao1ParaC}
\Big[i\hbar\frac{\partial }{\partial x_j} +
eA_j({\bf r})\! + {1\over 2} e
F_{jk}(x_k \!-\! x^\prime_k)\Big] C({\bf r},{\bf r}^{\;\prime})\! \!\!&=&\!\! 0,\\
\Big[i\hbar\frac{\partial }{\partial x^\prime_j} -eA_j({\bf
r}^{\;\prime}) \! + {1\over 2} e F_{jk}(x_k \! -\!x^\prime_k)\Big]\,
\!C({\bf r},{\bf r}^{\;\prime}) \!\!\!&=& \!\!0,\label{Equacao2ParaC}\end{aligned}$$ where we defined $F_{jk}={\mbox{\large $\epsilon$}}_{jk3}\, B$.
Our strategy to solve the above system of differential equations is the following: we first equation (\[Equacao1ParaC\]) assuming in this equation variables ${{\bf r}}^{\;\prime}$ as constants. Then, we impose that the result thus obtained is a solution of equation (\[Equacao2ParaC\]). With this goal, we multiply both sides of (\[Equacao1ParaC\]) by $dx_j$ and sum over $j$, to obtain $$\label{dlnC}
{1\over C}\left({\partial C\over\partial x_j}\; dx_j \right) =
{ie\over\hbar} \biggl[ A_j({\bf r})+{1\over 2}F_{jk}\left(
x_k-x^\prime_k\right)\biggr] \; dx_j\; .$$ Integration of the previous equation leads to $$\label{integracaoCrr'1}
C({\bf r},{\bf r}^{\;\prime}) = C({{\bf r}}^{\;\prime},{{\bf
r}}^{\;\prime})\;
\mbox{\Large$ e^{\{ {ie\over \hbar}_{\;\;\Gamma}
\int_{\;\;\atop{{{\bf r}}^{\;\prime}}}^{\;{\bf r}}
[ A_j (\mbox{\footnotesize{\mathversion{bold}${\xi}$}}) +
{1\over 2}\, F_{jk}\left(\xi_k-x^\prime_k\right)] \; d\xi_j\}}$}\; ,$$ where the line integral is assumed to be along curve $\Gamma$, to be specified in a moment. As we shall see, this line integral does not depend on the curve $\Gamma$ joining ${{\bf r}}^{\;\prime}$ and ${\bf r}$, as expected, since the l.h.s. of (\[dlnC\]) is an exact differencial.
In order to determine the differential equation for $C({{\bf
r}}^{\;\prime},{{\bf r}}^{\;\prime})$ we must substitue expression (\[integracaoCrr’1\]) into equation (\[Equacao2ParaC\]). Doing that and using carefully the fundamental theorem of differential calculus, it is straightforward to show that $$\label{integracaoCrr'2}
{\partial C\over\partial x^\prime_j}({{\bf r}}^{\;\prime},{{\bf
r}}^{\;\prime})=0\; ,$$ which means that $C({\bf r}^{\,\prime},{{\bf r}}^{\;\prime})$ is a constant, $C_0$, independent of ${{\bf r}}^{\;\prime}$. Noting that $$[{\bf B}\times\left(\mbox{{\mathversion{bold}${\xi}$}}- {{\bf
r}}^{\;\prime}\right)]_j =
-F_{jk}\left(\xi_k-x^\prime_k\right)\, ,$$ equation (\[integracaoCrr’1\]) can be written as $$\label{Cintegrado1}
C({\bf r},{{\bf r}}^{\;\prime}) \!= C_0\; \exp\left\{ {ie\over
\hbar}_{\;\;\Gamma}\!\! \int_{\;\;\atop{{{\bf
r}}^{\;\prime}}}^{\;{\bf r}}\!\! \bigl[{\bf A}
(\mbox{{\mathversion{bold}${\xi}$}})-{1\over 2}\, {\bf B}\times\left
(\mbox{{\mathversion{bold}${\xi}$}}-{{\bf
r}}^{\;\prime}\right)\bigr]\! \cdot\!
d\mbox{{\mathversion{bold}${\xi}$}}\right\} .$$ Observe, now, that the integrand in the previous equation has a vanishing curl, $$\mbox{{\mathversion{bold}${\nabla}$}}_{\mbox{{\mathversion{bold}${\xi}$}}}
\times\biggl[{\bf A} (\mbox{{\mathversion{bold}${\xi}$}})-{1\over
2}\, {\bf B} \times\left(\mbox{{\mathversion{bold}${\xi}$}}-{{\bf
r}}^{\;\prime}\right)\biggr]= {\bf B}-{\bf B}={\bf 0}\; ,$$ which means that the line integral in (\[Cintegrado1\]) is path independent. Choosing, for convenience, the straightline from ${{\bf r}}^{\;\prime}$ to ${\bf r}$, it can be readly shown that $${\;}_{\;\;\;\atop{\mbox{$\Gamma_{sl}$}}}\!\int_{\;\;\atop{{{\bf
r}}^{\;\prime}}}^{\;{\bf r}} [{\bf B}\times\left(
\mbox{{\mathversion{bold}${\xi}$}}-{{\bf r}}^{\;\prime} \right)]
\cdot d\mbox{{\mathversion{bold}${\xi}$}}=0\; ,$$ where $\Gamma_{sl}$ means a straightline from ${{\bf r}}^{\;\prime}$ to ${\bf r}$. With this simplification, the $C({{\bf r}}^{\;\prime},{\bf r})$ takes the form $$\label{Cintegrado1}
C({\bf r},{{\bf r}}^{\;\prime}) \!= C_0\; \exp\left\{ {ie\over
\hbar}_{\;\;\Gamma_{sl}}\!\! \int_{\;\;\atop{{{\bf
r}}^{\;\prime}}}^{\;{\bf r}}\!\! {\bf A}
(\mbox{{\mathversion{bold}${\xi}$}}) \cdot
d\mbox{{\mathversion{bold}${\xi}$}}\right\} .$$ Substituting last equation into (\[ProtoPropagador\]) and using the initial condition (\[CondicaoInicial\]), we readly obtain $C_{0}=\frac{m\omega}{2\pi i\hbar}$. Therefore the complete Feynman propagator for a charged particle under the influence of a constant and uniform magnetic field takes the form $$\begin{aligned}
\label{PropagadorFinalBUniforme}
K({\bf x},{{\bf x}}^{\prime};\tau)\hspace{6.0cm}\nonumber\\
={m\, \omega\over 2\pi i\hbar\,\sin{(\omega\tau)}} \sqrt{{m\over 2\pi i\hbar\tau}}
\exp\left\{ {ie\over \hbar}
\int_{{\bf r}^{\prime}}^{{\bf r}}\!\!\!\!\!{\bf A}
(\mbox {{\mathversion {bold}${\xi}$}}) \cdot d\mbox{{\mathversion{bold}${\xi}$}}\right\}\nonumber\\
\exp\biggl\{{im\omega\over 2\hbar}\cot(\omega\tau)({\bf r}-{{\bf r}}^{\;\prime})^2\biggr\}
\exp\biggl\{{im\over 2\hbar}{\left( x_3 - x_3^\prime\right)^2\over
\tau}\biggr\}\, ,\end{aligned}$$ where in the above equation we omitted the symbol $\Gamma_{sl}$ but, of course, it is implicit that the line integral must be done along a straightline, and we brought back the free propagation along the ${\cal OX}_3$ direction. A few comments about the above result are in order.
1. Firstly, we should emphasize that the line integral which appears in the first exponencial on the r.h.s. of (\[PropagadorFinalBUniforme\]) must be evaluated along a straight line between ${\bf r}^{\prime}$ and ${\bf r}$. If for some reason we want to choose another path, instead of integral $\int_{{\bf r}^{\prime}}^{\bf r}
{\bf A}(\mbox{{\mathversion{bold}${\xi}$}})\cdot
d\mbox{{\mathversion{bold}${\xi}$}}$, we must evaluate $\int_{{\bf r}^{\prime}}^{\bf r}[
{\bf A}(\mbox{{\mathversion{bold}${\xi}$}})-(1/2){\bf B}\times
(\mbox{{\mathversion{bold}${\xi}$}}-{{\bf r}}^{\;\prime})]\cdot
d\mbox{{\mathversion{bold}${\xi}$}}$.
2. Since we solved the Heisenberg equations for the gauge invariant operators ${\bf R}_\bot$ and $\mbox{{\mathversion{bold}${\pi}$}}_\bot$, our final result is written for a generic gauge. Note that the gauge-independent and gauge-dependent parts of the propagator are clearly separated. The gauge fixing corresponds to choose a particular expression for $\bf A(\mbox{{\mathversion{bold}${\xi}$}})$. Besides, from (\[PropagadorFinalBUniforme\]) we imediately obtain the transformation law for the propagator under a gauge transformation ${\bf A} \rightarrow {\bf A} +
\mbox{{\mathversion{bold}${\nabla}$}}\Lambda$, namely, $$\label{GaugeTransformation}
K({\bf r},{{\bf r}}^{\;\prime};\tau)\longmapsto
\mbox{\large $ e^{\frac{ie}{\hbar}\,\Lambda({\bf r})}$}\, K({\bf
r},{{\bf r}}^{\;\prime};\tau)\, \mbox{\large $
e^{-\frac{ie}{\hbar}\,\Lambda({{\bf r}}^{\;\prime})}$}\; .\nonumber$$ Although this transformation law was obtained in a particular case, it can be shown that it is quite general.
3. It is interesting to show how the energy spectrum (Landau levels), with the corresponding degeneracy per unit area, can be extracted from propagator (\[PropagadorFinalBUniforme\]). With this purpose, we recall that the partition function can be obtained from the Feynman propagator by taking $\tau=-i\hbar\beta$, with $\beta=1/(K_BT)$, and taking the spatial trace, $$Z(\beta) = \int_{-\infty}^\infty dx_1\int_{-\infty}^\infty dx_2\;
K({\bf r},{\bf r};-i\hbar\beta)\; .$$ Substituting (\[PropagadorFinalBUniforme\]) into last expression, we get $$Z(\beta) = \int_{-\infty}^\infty dx_1\int_{-\infty}^\infty dx_2\;
{m\omega\over 2\pi\hbar\,\mbox{senh}(\hbar\beta\omega)}\; ,$$ where we used the fact that $\sin(-i\theta)=-i\,\sinh\,\theta$. Observe that the above result is divergent, since the area of the ${\cal OX}_1{\cal X}_2$ plane is infinite. This is a consequence of the fact that each Landau level is infinitely degenerated, though the degeneracy per unit area is finite. In order to proceed, let us assume an area as big as we want, but finite. Adopting this kind or regularization, we write $$\begin{aligned}
\int_{-L/2}^{L/2} \!\! dx_1\!\!\int_{-L/2}^{L/2}\!\! dx_2\;
\!\!\!&K&\!\!\! ({\bf r},{\bf r};-i\hbar\beta)\approx
{L^2\, m\omega\over 2\pi\hbar\,\mbox{senh}(\hbar\beta\omega)}\nonumber\\
\nonumber\\
&=& {L^2\, eB\over 2\pi\hbar\left(\mbox{\large $
e^{\hbar\beta\omega}$}-
\mbox{\large $ e^{-\hbar\beta\omega}$}\right)}\nonumber\\
\nonumber\\
&=& {L^2\, eB\over 2\pi\hbar} {\mbox{\large $ e^{-{1\over
2}\hbar\beta\omega_c}$}\over \left(1-\mbox{\large $
e^{-\hbar\beta\omega_c}$}\right)}
\nonumber\\
\nonumber\\
&=& \sum_{n=0}^\infty {L^2\, eB\over 2\pi\hbar} \mbox{\large $
e^{-\beta(n+{1\over2})\hbar\omega_c}$}\; ,\nonumber\end{aligned}$$ where we denoted by $\omega_c = eB/2m$ the ciclotron frequency. Comparing this result with that of a partition function whose energy level $E_n$ has degeneracy $g_n$, given by $$Z(\beta) = \sum_n g_n\; \mbox{\large $ e^{-\beta E_n}$}\; ,$$ we imediately identify the so called Landau leves and the corresponding degeneracy per unit area, $$\label{niveisdeLandau}
E_n = \left( n+{1\over 2}\right)\hbar\omega_c\;\; ;\;\;
{g_n\over A} = {eB\over 2\pi\hbar}\;\;\;\; (n=0,1,...)\;
.\nonumber$$
Charged harmonic oscillator in a uniform magnetic field {#choumf}
-------------------------------------------------------
In this section we consider a particle with mass $m$ and charge $e$ in the presence of a constant and uniform magnetic field $\textbf{B}
= B{\bf e_3}$ and submitted to a 2-dimensional isotropic harmonic oscillator potential in the ${\cal OX}_1{\cal X}_2$ plane, with natural frequency $\omega_{0}$. Using the same notation as before, we can write the hamiltonian of the system in the form $$\mathcal{H} = \mathcal{H}_{\bot} + \frac{P_3^2}{2m}, \label{Eq48}$$ where $$\mathcal{H}_{\bot} = \frac{{\pi}_{1}^{2}+{\pi}_{2}^{2}}{2m} +
\frac{1}{2}m\omega_0^2\left(X_1^2 + X_2^2\right). \label{Eq49}$$ As before, the Feynman propagator for this problem takes the form $K(\textbf{x},\textbf{x}^{\prime};\tau)=K_{\bot}
(\textbf{r},\textbf{r}^{\prime};\tau)K_{3}^{(0)}(x_3,x_3^{\prime};\tau)$, with $K_{3}^{(0)}(x_3,x_3^{\prime};\tau)$ given by equation (\[Eq14\]). The propagator in the ${\cal OX}_1{\cal X}_2$-plane satisfies the differential equation (\[EqDiferencialPropagador2D\]) and will be determined by the same used in the previous example.
Using hamiltonian (\[Eq49\]) and the usual commutation relations the Heisenberg equations are given by $$\begin{aligned}
\frac{d\textbf{R}(\tau)}{d\tau}&=&\frac{\mbox{\mathversion{bold}${\Pi}$}(\tau)}{m} \,, \label{Eq51} \\
\frac{d\mbox{\mathversion{bold}${\Pi}$}(\tau)}{d\tau} &=& 2\omega
\mathbb{C} \mbox{\mathversion{bold}${\Pi}$}(\tau) - m
\omega_{0}^{2}\textbf{R}(\tau)\ , \label{Eq52}\end{aligned}$$ where we have used the matrix notation introduced in (\[Eq20\]) and (\[Eq21\]). Equation (\[Eq51\]) is the same as (\[EquacaoR\]), but equation (\[Eq52\]) contains an extra term when compared to (\[EquacaoPi\]). In order to decouple equations (\[Eq51\]) and (\[Eq52\]), we differentiate (\[Eq51\]) with respect to $\tau$ and then use (\[Eq52\]). This procedure leads to the following uncoupled equation $$\begin{aligned}
\frac{d^{2}\textbf{R}(\tau)}{d\tau^{2}}
&-&
2\omega\mathbb{C}\frac{d\textbf{R}(\tau)}{d\tau}+\omega_{0}^{2}\textbf{R}(\tau)=0
\label{Eq53}\end{aligned}$$ After solving this equation, $\textbf{R}(\tau)$ and $\mbox{\mathversion{bold}${\Pi}$}(\tau)$ are constrained to satisfy equations (\[Eq51\]) and (\[Eq52\]), respectively. A straightforward algebra yields the solution $$\begin{aligned}
\textbf{R}(\tau)&=&\mathbb{M}^{-}\textbf{R}(0)+\mathbb{N}\mbox{\mathversion{bold}${\Pi}$}(0)
\label{Eq56} \\
\mbox{\mathversion{bold}${\Pi}$}(\tau)
&=&
\mathbb{M}^{+}\mbox{\mathversion{bold}${\Pi}$}(0)-m^{2}\omega_{0}^{2}\mathbb{N}\textbf{R}(0)\
, \label{Eq57}\end{aligned}$$ where we defined the matrices $$\begin{aligned}
\mathbb{N}&=&\frac{\sin{(\Omega\tau)}}{m\Omega}e^{\omega\tau\mathbb{C}} \label{Eq58}\\
\mathbb{M}^{\pm}&=&e^{\omega\tau\mathbb{C}}\Big[\cos{(\Omega\tau)}\uma\pm
\frac{\omega}{\Omega}\sin{(\Omega\tau)}\mathbb{C}\Big]\ ,
\label{Eq59}\end{aligned}$$ and frequency $\Omega = \sqrt{\omega^{2}+\omega_{0}^{2}}$. Using (\[Eq56\]) and (\[Eq57\]), we write $\mbox{\mathversion{bold}${\Pi}$}(0)$ and $\mbox{\mathversion{bold}${\Pi}$}(\tau)$ in terms of $\textbf{R}(\tau)$ and $\textbf{R}(0)$, $$\begin{aligned}
\!\!\!\!\!\!\!\mbox{\mathversion{bold}${\Pi}$}(0)
\!\!&=&\!\!
\mathbb{N}^{-1}\textbf{R}(\tau)-\mathbb{N}^{-1}\mathbb{M}^{-}\textbf{R}(0)\,
,
\label{Pi(0)OH+B} \\
\!\!\!\!\!\!\!\mbox{\mathversion{bold}${\Pi}$}(\tau)
\!\!\!&=&\!\!
\mathbb{M}^{+}\mathbb{N}^{-1}\textbf{R}(\tau) \! -\!\!
\Big[ \mathbb{M}^{+}\mathbb{N}^{-1}\mathbb{M}^{-} \!\!\!+\!
m^{2}\omega_{0}^{2}\mathbb{N}\Big]\! \textbf{R}(0).
\label{Pi(tau)OH+B}\end{aligned}$$ Now, we must order appropriately the hamiltonian operator $\mathcal{H}_{\bot}=\mbox{\mathversion{bold}${\Pi}$}^{T}(0)
\mbox{\mathversion{bold}${\Pi}$}(0)/(2m)+m\omega_{0}^{2}\textbf{R}^{T}(0)\textbf{R}(0)/2$, which, with the aid of equation (\[Pi(0)OH+B\]), can be written as $$\begin{aligned}
\mathcal{H}_{\bot}&=& \Big[\textbf{R}^{T}(\tau)(\mathbb{N}^{-1})^T -
\textbf{R}^{T}(0)(\mathbb{M}^{-})^T (\mathbb{N}^{-1})^T \Big] \cr
&&\times \Big[\mathbb{N}^{-1} \textbf{R}(\tau) -
\mathbb{N}^{-1}\mathbb{M}^{-} \textbf{R}(0)\Big] +m\omega_0^2
\textbf{R}^{T}(0)\textbf{R}(0) \cr\cr
&=&\frac{m\Omega^{2}}{2\sin^{2}{(\Omega\tau)}}
\Big[\textbf{R}^{T}(\tau) - \textbf{R}^{T}(0)(\mathbb{M}^{-})^T
\Big] \cr &&\times \Big[\textbf{R}(\tau) - \mathbb{M}^{-}
\textbf{R}(0)\Big] +m\omega_0^2 \textbf{R}^{T}(0)\textbf{R}(0)
\cr\cr &=&\frac{m\Omega^{2}}{2\sin^{2}{(\Omega\tau)}}
\Big[\textbf{R}^{T}(\tau) \textbf{R}(\tau) - \textbf{R}^{T}(\tau)
\mathbb{M}^{-}\textbf{R}(0) \cr &&-
\textbf{R}^{T}(0)(\mathbb{M}^{-})^T \textbf{R}^{T}(\tau) +
\textbf{R}^{T}(0)(\mathbb{M}^{-})^T \mathbb{M}^{-} \textbf{R}(0)
\Big] \cr &&\;\;+m\omega_0^2 \textbf{R}^{T}(0)\textbf{R}(0) \cr\cr
&=&\frac{m\Omega^{2}}{2\sin^{2}{(\Omega\tau)}}\Big[\textbf{R}^{2}(\tau)
-\textbf{R}^{T}(\tau)\mathbb{M}^{-}\textbf{R}(0)
\cr
&& \qquad\qquad -\textbf{R}^{T}(0)(\mathbb{M}^{-})^{T}\textbf{R}(\tau) +\textbf{R}^{2}(0)\Big]\ ,
\label{Eq63}\end{aligned}$$ where superscript $T$ means transpose and we have used the properties of the matrices $\mathbb{N}$ and $\mathbb{M}^{-}$ given by (\[Eq58\]) and (\[Eq59\]). In order to get the right time ordering, observe first that $$\textbf{R}^T(0)(\mathbb{M}^{-})^T \textbf{R}(\tau) =
\textbf{R}^T(\tau) \mathbb{M}^{-}\textbf{R}(0) + \Big[\left(
\mathbb{M}^{-} \textbf{R}(0)\right)_{i},\textbf{X}_{i}(\tau)\Big]
\,,$$ where $$\Big[\left( \mathbb{M}^{-}
\textbf{R}(0)\right)_{i},\textbf{X}_{i}(\tau)\Big]
= i\hbar \mbox{Tr}\Big[\mathbb{N}(\mathbb{M}^{-})^{T} \Big]
= \frac{i\hbar}{m\Omega} \sin{(2\Omega\tau)}\, .$$ Using the last two equations into (\[Eq63\]) we rewrite the hamiltonian in the desired ordered form, namely, $$\begin{aligned}
\label{Hordenada1}
\mathcal{H}_{\bot}&=&\frac{m\Omega^{2}}{2\sin^{2}{(\Omega\tau)}}\Big[\textbf{R}^{2}(\tau)+\textbf{R}^{2}(0) - 2\textbf{R}^{T}(\tau)\mathbb{M}^{-}\textbf{R}(0) \nonumber \\
&& \qquad \qquad - \frac{i\hbar}{m\Omega} \sin{(2\Omega\tau)}
\Big]\ .\end{aligned}$$ For future convenience, let us define $$\begin{aligned}
U(\tau)&=&\cos{(\omega\tau)}\cos{(\Omega\tau)}
+ \frac{\omega}{\Omega}\sin{(\omega\tau)}\sin{(\Omega\tau)}\, , \label{Eq67} \\
V(\tau)&=&\sin{(\omega\tau)}\cos{(\Omega\tau)}
- \frac{\omega}{\Omega}\cos{(\omega\tau)}\sin{(\Omega\tau)}
\label{Eq68}\end{aligned}$$ and write matrix $\mathbb{M}^{-}$, defined in (\[Eq59\]), in the form $$\mathbb{M}^{-}=U(\tau)\uma + V(\tau)\mathbb{C}. \label{Eq66}$$ Substituting (\[Eq66\]) in (\[Hordenada1\]) we have $$\begin{aligned}
\mathcal{H}_{\bot}&=&\frac{m\Omega^{2}}{2\sin^{2}{(\Omega\tau)}}\Big[\textbf{R}^{2}(\tau)+\textbf{R}^{2}(0)
- 2U(\tau)\textbf{R}^{T}(\tau)\textbf{R}(0) \nonumber \\
&-& 2V(\tau)\textbf{R}^{T}(\tau)\mathbb{C}\textbf{R}(0)-\frac{i\hbar}{m\Omega}
\sin{(2\Omega\tau)}\Big]\,.
\label{Eq69}\end{aligned}$$ The next step is to compute the classical function $F({\bf r},{\bf r^{\prime}};\tau)$. Using the following identities $$\begin{aligned}
\frac{\Omega U(\tau)}{\sin^{2}{(\Omega\tau)}}
&=&
- \frac{d}{d\tau} \Big[\frac{\cos{(\omega\tau)}} {\sin{(\Omega\tau)}}\Big]\, , \label{Eq70} \\
\frac{\Omega V(\tau)}{\sin^{2}{(\Omega\tau)}}
&=&
- \frac{d}{d\tau}\Big[\frac{\sin{(\omega\tau)}} {\sin{(\Omega\tau)}}\Big],
\label{Eq71}\end{aligned}$$ into (\[Eq69\]), we write $F(\textbf{r},\textbf{r}^{\prime};\tau)$ in the convenient form $$\begin{aligned}
\!\!\!\! F(\textbf{r},\textbf{r}^{\prime};\tau)
\!\!&=&\!\!
\frac{m\Omega^{2}}{2}(\textbf{r}^{2}+\textbf{r}^{\prime^{2}})\mbox{csc}(\Omega\tau)^{2}
\!\!+ m\Omega\textbf{r}\cdot\textbf{r}^{\prime}\! \frac{d}{d\tau}
\! \Big[ \frac{\cos{(\omega\tau)}} {\sin{(\Omega\tau)}}\Big] \nonumber \\
&+& \; m\Omega\textbf{r}\cdot \mathbb{C}\textbf{r}^{\prime}
\frac{d}{d\tau} \Big[ \frac{\sin{(\omega\tau)}}
{\sin{(\Omega\tau)}}\Big]-i \hbar \Omega
\frac{\cos{(\Omega\tau)}}{\sin{(\Omega\tau)}}\, .
\label{Eq72}\end{aligned}$$ Inserting this result into the differential equation $$i\hbar\frac{\partial}{\partial\tau}\langle{\bf r},\tau
\vert{\bf r}^{\prime},0\rangle =
F(\textbf{r},\textbf{r}^{\prime};\tau)
\langle{\bf r},\tau
\vert{\bf r}^{\prime},0\rangle\, ,$$ and integrating in $\tau$, we obtain $$\begin{aligned}
\langle\textbf{r},\tau \vert \textbf{r}^{\prime}\!\!\!\!\!
&,&\!\!\!\!\! 0\rangle = \frac{C(\textbf{r},\textbf{r}^{\prime})}
{\sin{(\Omega\tau)}} \mbox{exp} \left\lbrace
\frac{im\Omega}{2\hbar}\Big[ (\textbf{r}^{2}
+ \textbf{r}^{\prime^{2}})\cot{(\Omega\tau)} \right. \nonumber \\
&-&
\left. 2 \left( \textbf{r}\cdot\textbf{r}^{\prime}
\frac{\cos{(\omega\tau)}}{\sin{(\Omega\tau)}}
+ \textbf{r}\cdot \mathbb{C}\textbf{r}^{\prime}
\frac{\sin{(\omega\tau)}}{\sin{(\Omega\tau)}} \right) \Big] \right\rbrace.
\label{ProtoPropagadorOH+B}\end{aligned}$$ where $C({\bf r},{\bf r}^{\;\prime})$ is an arbitrary integration constantto be determined by conditions (\[Eq30\]), (\[Eq31\]) and (\[Eq32\]). Using (\[Pi(tau)OH+B\]) we can calculate the l.h.s. of condition (\[Eq30\]), $$\begin{aligned}
\langle{\bf r}, \!\!\!\!&\tau&\!\!\!\! \vert
\pi_j(\tau)\vert{\bf r}^{\prime},0\rangle
=
\frac{m\Omega}{\sin{(\Omega\tau)}}\Big\{ \cos{(\Omega\tau)}x_j-\cos{(\omega\tau)}x^{\prime}_j \nonumber \\
&+&
\Big[\frac{\omega}{\Omega}\sin{(\Omega\tau)} x_k-\sin{(\omega\tau)}x^{\prime}_{k}\Big]
\epsilon_{jk3} \Big\}\langle{\bf r},\tau\vert{\bf r}^{\prime},0\rangle, \label{Eq74}\end{aligned}$$ and using (\[Pi(0)OH+B\]) we get the l.h.s. of condition (\[Eq31\]), $$\begin{aligned}
\langle{\bf r}, \!\!\!\!&\tau&\!\!\!\! \vert\pi_j(0)\vert{\bf
r}^{\prime},0\rangle = \frac{m\Omega}{\sin{(\Omega\tau)}}
\Big\{\cos{(\omega\tau)}x_j-\cos{(\Omega\tau)}x^{\prime}_j \nonumber \\
&+& \Big[\frac{\omega}{\Omega}\sin{(\Omega\tau)} x^{\prime}_k-\sin{(\omega\tau)}x_{k}\Big]\epsilon_{jk3} \Big\}
\langle{\bf r},\tau\vert{\bf r}^{\prime},0\rangle.
\label{Eq75}\end{aligned}$$ With the help of the simple identities $$\begin{aligned}
\frac{\partial}{\partial x_j}(\textbf{r}^{2}+\textbf{r}^{\prime^{2}})=2x_j \hspace{0.198cm}
&;&
\hspace{0.198cm} \frac{\partial}{\partial x^{\prime}_j}(\textbf{r}^{2}
+ \textbf{r}^{\prime^{2}})=2x^{\prime}_j \nonumber \\
\frac{\partial}{\partial x_j}\textbf{r}\cdot\textbf{r}^{\prime}=x^{\prime}_j \hspace{0.658cm}
&;&
\hspace{0.658cm} \frac{\partial}{\partial x^{\prime}_j}\textbf{r}\cdot\textbf{r}^{\prime}= x_j \nonumber \\
\frac{\partial}{\partial x_j}\textbf{r}\cdot
\mathbb{C}\textbf{r}^{\prime}=\epsilon_{jk3}x^{\prime}_k \hspace{0.2295cm}
&;&
\hspace{0.2295cm} \frac{\partial}{\partial x^{\prime}_j}\textbf{r}\cdot
\mathbb{C}\textbf{r}^{\prime}= -\epsilon_{jk3}x_k. \nonumber\end{aligned}$$ and also using equation (\[ProtoPropagadorOH+B\]), we are able to compute the right hand sides of conditions (\[Eq30\]) and (\[Eq31\]), which are given, respectively, by $$\begin{aligned}
\Big\{
\!\!\!\!&-&\!\!\!\!
\frac{i\hbar}{C(\textbf{r},\textbf{r}^{\prime})}\frac{\partial
C(\textbf{r},\textbf{r}^{\prime})} {\partial x_j}
+ m\Omega\frac{\cos{(\Omega\tau)}}{\sin{(\Omega\tau)}}x_j
- m\Omega\frac{\cos{(\omega\tau)}}{\sin{(\Omega\tau)}}x^{\prime}_j
\nonumber \\
&-&
m\Omega\frac{\sin{(\omega\tau)}}{\sin{(\Omega\tau)}}\epsilon_{jk3}x^{\prime}_k
- e A_j(\textbf{r}) \Big\} \langle\textbf{r},\tau|\textbf{r}^{\prime},0\rangle
\label{Eq77}\end{aligned}$$ and $$\begin{aligned}
\Big\{
\!\!\!\!\!&{\,}&\!\!\!\!\!
\frac{i\hbar}{C(\textbf{r},\textbf{r}^{\prime})}
\frac{\partial C(\textbf{r},\textbf{r}^{\prime})} {\partial x^{\prime}_j}
m\Omega\frac{\cos{(\Omega\tau)}}{\sin{(\Omega\tau)}}x^{\prime}_j
%\nonumber \\
+ m\Omega\frac{\cos{(\omega\tau)}}{\sin{(\Omega\tau)}}x_j
\nonumber\\
&-&
m\Omega\frac{\sin{(\omega\tau)}}{\sin{(\Omega\tau)}}\epsilon_{jk3}x_k
-e A_j(\textbf{r}^{\prime})\Big\} \langle\textbf{r},\tau|\textbf{r}^{\prime},0\rangle.
\label{Eq78}\end{aligned}$$ Equating (\[Eq74\]) and (\[Eq77\]), and also (\[Eq75\]) and (\[Eq78\])), we get the system of differential equations for $C(\textbf{r},\textbf{r}^{\prime})$ $$\begin{aligned}
i\hbar \frac{\partial C(\textbf{r},\textbf{r}^{\prime})}{\partial
x_j}
&+&
e\Big[A_j(\textbf{r})+\frac{F_{jk}}{2}x_k \Big] C(\textbf{r},\textbf{r}^{\prime})=0\, , \label{Eq79} \\
i\hbar \frac{\partial C(\textbf{r},\textbf{r}^{\prime})}{\partial x^{\prime}_j}
&-&
e\Big[A_j(\textbf{r}^{\prime})+\frac{F_{jk}}{2}x^{\prime}_k \Big] C(\textbf{r},\textbf{r}^{\prime})=0.
\label{Eq80}\end{aligned}$$ Proceeding as in the previous example, we first integrate (\[Eq79\]). With this goal, we multiply it by $dx_j$, sum in $j$ and integrate it to obtain $$C({\bf r},{\bf r}^{\;\prime})= C({{\bf r}}^{\;\prime},{\bf
r}^{\;\prime})
\exp\left\lbrace {ie\over \hbar}_{\;\;\Gamma}\!\!
\int_{{\bf r}^{\prime}}^{{\bf r}}
\Big[ A_j(\mbox{\mathversion{bold}${\xi}$})+{F_{jk}\over 2}\xi_k\Big] d\xi_j \right\rbrace\, , \\
\label{CrrLinha1}$$ where the path of integration $\Gamma$ will be specified in a moment. Inserting expression (\[CrrLinha1\]) into the second differential equation (\[Eq80\]), we get $$\frac{\partial}{\partial x_j^{\prime}}
C({{\bf r}}^{\;\prime},{{\bf r}}^{\;\prime}) = 0
\;\;\Longrightarrow\;\;
C({{\bf r}}^{\;\prime},{{\bf r}}^{\;\prime}) =C_0\, ,$$ where $C_0$ is a constant independent of ${{\bf r}}^{\;\prime}$, so that equation (\[CrrLinha1\]) can be cast, after some convenient rearrangements, into the form $$C({\bf r},{\bf r}^{\prime})=C_{0}\exp{\left\lbrace{ie\over\hbar}
_{\;\;\Gamma}\!\!
\int_{{\bf r}^{\prime}}^{{\bf r}}
\Big[\textbf{A}(\mbox{\mathversion{bold}${\xi}$})
- \frac{1}{2}\textbf{B}\times\mbox{\mathversion{bold}${\xi}$}\Big]\cdot
d\mbox{\mathversion{bold}${\xi}$}\right\rbrace }.
\label{CrrLinha2}$$ Note that the integrand has a vanishing curl so that we can choose the path of integration $\Gamma$ at our will. Choosing, as before, the straight line between ${\bf r}^{\prime}$ and ${\bf r}$, it can be shown that $$\int_{\textbf{r}^{\prime}}^{\textbf{r}}
\Big[ \textbf{A}(\mbox{\mathversion{bold}${\xi}$})-\frac{\textbf{B}}{2}
\times\mbox{\mathversion{bold}${\xi}$}\Big]
\cdot d\mbox{\mathversion{bold}${\xi}$}=\int_{\textbf{r}^{\prime}} ^{\textbf{r}}
\textbf{A}(\mbox{\mathversion{bold}${\xi}$})\cdot
d\mbox{\mathversion{bold}${\xi}$}
+ \frac{1}{2}B\textbf{r}\cdot\mathbb{C}\textbf{r}^{\prime}\, ,
\label{IntegralLinha1}$$ where, for simplicity of notation, we omitted the symbol $\Gamma_{sl}$ indicating that the line integral must be done along a straight line. From equations (\[CrrLinha1\]), (\[CrrLinha2\]) e (\[IntegralLinha1\]), we get $$C({\bf r},{\bf r}^{\prime}) =
C_{0}\exp{\left\lbrace{ie\over\hbar}\int_{\atop{{{\bf
r}}^{\prime}}}^{{\bf r}}
\textbf{A}(\mbox{\mathversion{bold}${\xi}$})\cdot
d\mbox{\mathversion{bold}${\xi}$}\right\rbrace }
\exp{\left\lbrace{im\omega\over\hbar}\textbf{r}
\cdot\mathbb{C}\textbf{r}^{\prime}\right\rbrace} ,
\label{Eq86}$$ which substituted back into equation (\[ProtoPropagadorOH+B\]) yields $$\begin{aligned}
\langle\textbf{r},\tau|\textbf{r}^{\prime},0\rangle
= \frac{C_{0}}{\sin{(\Omega\tau)}}\exp{\left\lbrace{ie\over\hbar}
\int_{\atop{{{\bf r}}^{\prime}}}^{{\bf r}} \textbf{A}(\mbox{\mathversion{bold}${\xi}$})
\cdot d\mbox{\mathversion{bold}${\xi}$}\right\rbrace } \nonumber \\
\mbox{exp} \Big\{ \frac{im\Omega}{2\hbar\sin{(\Omega\tau)}}
\Big\{(\textbf{r}^{2} +\textbf{r} ^{\prime^{2}}) \cos{(\Omega\tau)}
\nonumber \\
- 2\textbf{r} \cdot\textbf{r}^{\prime}\cos{(\omega\tau)}
%\nonumber \\
- 2\Big[\sin{(\omega\tau)}-\frac{\omega}{\Omega}\sin{\Omega\tau} \Big]
\textbf{r}\mathbb{C}\textbf{r}^{\prime} \Big\} \Big\}
\label{Eq87}\end{aligned}$$ The initial condition implies $C_0=m\Omega/(2\pi i\hbar)$. Hence, the desired Feynman propagator is finally given by
$$\begin{aligned}
K({\bf x},{{\bf x}}^{\prime};\tau)
&=&
K_{\bot}
(\textbf{r},\textbf{r}^{\prime};\tau)K_{3}^{(0)}(x_3,x_3^{\prime};\tau)\nonumber\\
&=&
\frac{m\Omega}{2\,\pi\, i\, \hbar\,\sin{(\Omega\tau)}}
\sqrt{{m\over 2\pi i\hbar\tau}} \exp{\left\lbrace{ie\over\hbar}
\int_{\atop{{{\bf r}}^{\prime}}}^{{\bf r}} \textbf{A}(\mbox{\mathversion{bold}${\xi}$})
\cdot d\mbox{\mathversion{bold}${\xi}$}\right\rbrace } \mbox{exp} \left\lbrace
\frac{im\Omega}{2\hbar\sin{(\Omega\tau)}} \left\lbrace \cos{(\Omega\tau)}
(\textbf{r}^{2} +\textbf{r} ^{\prime^{2}}) \right. \right. \nonumber \\
&{\;}& \, -\;\;
\left. \left. 2 \cos{(\omega\tau)} \textbf{r} \cdot\textbf{r}^{\prime} -
2 \Big[\sin{(\omega\tau)}-\frac{\omega}{\Omega}\sin{(\Omega\tau)} \Big]
\textbf{r}\cdot\mathbb{C}\textbf{r}^{\prime} \right\rbrace \right\rbrace
\exp\biggl\{{im\over 2\hbar}{\left(x_3 - x_3^\prime\right)^2\over \tau}\biggr\}\, ,
\label{Eq90}\end{aligned}$$
where we brought back the free part of the propagator corresponding to the movement along the ${\cal OX}_3$ direction. Of course, for $\omega_0=0$ we reobtain the propagator found in our first example and for ${\bf B}={\bf 0}$ we reobtain the propagator for a bidimensional oscillator in the ${\cal OX}_1{\cal X}_2$ plane multiplied by a free propagator in the ${\cal OX}_3$ direction, as can be easily checked.
Regarding the gauge dependence of the propagator, the same comments done before are still valid here, namely, the above expression is written for a generic gauge, the transformation law for the propagator under a gauge transformation is the same as before, etc. We finish this section, extracting from the previous propagator, the corresponding energy spectrum. With this purpose, we first compute the trace of the propagator,
$$\begin{aligned}
\int_{-\infty}^\infty\!\!\! dx_1\!\!
\int_{-\infty}^\infty\!\!\! dx_2 \,
K_\perp^{\,\prime}(x_1,x_1,x_2,x_2;\tau) &=& {m\Omega\over 2\pi
i\hbar\,\sin(\Omega\tau) }
\int_{-\infty}^\infty\!\!\! dx_1
\!\!\int_{-\infty}^\infty\!\!\! dx_2
\exp\biggl\{ {im\Omega\over
2\hbar\,\sin(\Omega\tau)}\left[
2\Bigl(\mbox{cos}(\Omega\tau)-\mbox{cos}(\omega\tau)\Bigr)(x_1^2+x_2^2)\right]
\biggr\} \nonumber\\
&=& {1\over 2[\mbox{cos}(\Omega\tau)-\mbox{cos}(\omega\tau)]}\; ,\end{aligned}$$
where we used the well known result for the Fresnel integral. Using now the identity $$\cos(\Omega\tau)-\cos(\omega\tau) = -2\,\sin[(\Omega+\omega)\tau/2]
\,\sin[(\Omega-\omega)\tau/2)]\, ,$$ we get for the corresponding energy Green function $$\begin{aligned}
\label{FGreenG(E)}
\!\!&{\cal G}&\!\!\!(E) =\! -i\!\!\int_0^\infty\!\!\! d\tau\,
e^{{i\over\hbar}E\tau} \!\!\int_{-\infty}^\infty\, \!\!\!\!
dx_1\!\!\int_{-\infty}^\infty\!\!\!\! dx_2\,
K_\perp^{\,\prime}(x_1,x_1,x_2,x_2;\tau)\cr\cr
&=&{i\over4}\int_0^\infty d\tau {e^{{i\over\hbar}E\tau}\over
\,\mbox{sen}({\Omega\tau\over2}\tau)\,\mbox{sen}({\Omega-\omega\over2}\tau)}\cr\cr
&=&-i\!\!\int_0^\infty\!\!\!\! d\tau\,
e^{{i\over\hbar}E\tau}\left(\sum_{l=0}^\infty
e^{-(l+{1\over2})(\Omega+\omega)\tau}\!\!\right)\!\!
\left(\sum_{n=0}^\infty
e^{-i(n+{1\over2}(\Omega-\omega)\tau}\!\!\right) ,\nonumber\end{aligned}$$ where is tacitly assumed that $E\rightarrow E-i\varepsilon$ and we also used that (with the assumption $\nu\rightarrow \nu-i\epsilon$) $${1\over\,\mbox{sen}({\nu\over2}\tau)}=2i\sum_{n-0}^\infty
e^{-i(n+{1\over2}) \nu\tau}\; .$$ Changing the order of integration and summations, and integrating in $\tau$, we finally obtain $${\cal G}(E)=\sum_{l,n=0}^\infty{1\over E-E_{nl}}\; ,$$ where the poles of ${\cal G}(E)$, which give the desired energy levels, are identified as $$E_{nl}=(l+n+1)\hbar\Omega+(l-n)\hbar\omega\, ,\;\;\; (l,n = 0,1,...)\; .$$ The Landau levels can be reobtained from the previous result by simply taking the limit $\omega_0\rightarrow 0$: $$E_{nl}\longrightarrow(2l+1)\hbar\omega
= (l+{1\over2})\hbar\omega_c\; ,$$ with $l=0,1,...$ and $\omega_c = eB/m$, in agreement to the result we had already obtained before.
${\;}$
Final Remarks {#SectionFinalRemarks}
=============
In this paper we reconsidered, in the context of Schwinger’s method, the Feynman propagators of two well known problems, namely, a charged particle under the influence of a constant and uniform magnetic field (Landau problem) and the same problem in which we added a bidimensional harmonic oscillator potential. Although these problems had already been treated from the point of view of Schwinger’s action principle, the novelty of our work relies on the fact that we solved the Heisenberg equations for gauge invariant operators. This procedure has some nice properties, as for instance: the Feynman propagator is obtained in a generic gauge; [*(ii)*]{} the gauge-dependent and gauge-independent parts of the propagator appear clearly separated and [*(iii)*]{} the transformation law for the propagator under gauge transformation can be readly obtained. Besides, we adopted a matrix notation which can be straightforwardly generalized to cases of relativistic charged particles in the presence of constant electromagnetic fields and a plane wave electromagnetic field, treated by Schwinger [@Schwinger1951]. For completeness, we showed explicitly how one can obtain the energy spectrum directly from que Feynman propagator. In the Landau problem, we obtained the (infinitely degenerated) Landau levels with the corresponding degeneracy per unit area. For the case where we included the bidimensional harmonic potential, we obtained the energy spectrum after identifying the poles of the corresponding energy Green function. We hope that this pedagogical paper may be useful for undergraduate as well as graduate students and that these two simple examples may enlarge the (up to now) small list of non-relativistic problems that have been treated by such a powerful and elegant method.
Acknowledgments {#acknowledgments .unnumbered}
===============
F.A. Barone, H. Boschi-Filho and C. Farina would like to thank Professor Marvin Goldberger for a private communication and for kindly sending his lecture notes on quantum mechanics where this method was explicitly used. We would like to thank CNPq and Fapesp (brazilian agencies) for partial financial support.
[99]{}
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Given a smooth foliation by complex curves (locally around a point $x\in\mathbb{C}^2\setminus\{0\}$) which is “compatible” with the foliation by spheres centered at the origin, we construct a smooth real-valued function $g$ in a neighborhood of said point, which is positive, homogeneous and constant along the leaves. A corollary we obtain from this is relevant to the problem of “bumping out” certain pseudoconvex domains in $\mathbb{C}^3$.'
address: 'Lars Simon, Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway'
author:
- Lars Simon
bibliography:
- 'refspaper3.bib'
title: A Homogeneous Function Constant Along The Leaves Of A Foliation
---
[^1]
Introduction
============
The technique of “bumping out” bounded, smoothly bounded pseudoconvex domains of finite D’Angelo $1$-type in $\mathbb{C}^{n+1}$, $n\geq{1}$, has proven to be useful both in the construction of peak functions (see e.g. [@MR0492400], [@MR1016439]) and in the construction of integral kernels for solving the $\overline{\partial}$-equation (see e.g. [@MR835766], [@MR1070924]).
As in [@MR2452636], a local bumping of a smoothly bounded pseudoconvex domain $\Omega\subseteq\mathbb{C}^{n+1}$, $n\geq{1}$, at a boundary point $\zeta\in\partial\Omega$ is defined to be a triple $(\partial\Omega{},U_{\zeta},\rho_{\zeta})$, such that:
- [$U_{\zeta}\subseteq\mathbb{C}^{n+1}$ is an open neighborhood or $\zeta$,]{}
- [$\rho_{\zeta}\colon{}U_{\zeta}\to\mathbb{R}$ is smooth and plurisubharmonic,]{}
- [$\rho_{\zeta}^{-1}(\{0\})$ is a smooth hypersurface in $U_{\zeta}$ that is pseudoconvex from the side $U_{\zeta}^{-}:=\{z\colon{}\rho_{\zeta}(z)<0\}$,]{}
- [$\rho_{\zeta}(\zeta)=0$, but $\rho_{\zeta}<0$ on $U_{\zeta}\cap\left(\overline{\Omega}\setminus{\{\zeta\}}\right)$.]{}
A priori, such a local bumping needs to have additional properties for the upper mentioned constructions to work; specifically, when assuming $\Omega$ to be of finite type (“type” refers to the D’Angelo $1$-type), one desires the order of contact between $\partial\Omega$ and $\rho_{\zeta}^{-1}(\{0\})$ at $\zeta$ to not exceed the type of $\zeta$ in any direction.
As seen in, e.g., [@MR2452636], attempts to construct such a local bumping with the desired additional properties naturally lead to the problem of bumping homogeneous plurisubharmonic polynomials on $\mathbb{C}^n$. While it is not obvious how bumping results for homogeneous plurisubharmonic polynomials can be used to obtain [*useful*]{} bumping results for the domains that motivate their study, Noell [@MR1207878] and Bharali [@MR2993440] have been successful in doing so.\
Hence, bumping results for homogeneous plurisubharmonic polynomials are an important first step towards obtaining useful bumping results for domains.
Specifically, assume we are given a real-valued polynomial $P\not\equiv{0}$ with complex coefficients in $n$ complex variables $z_1,\dots{},z_n$ and their conjugates $\overline{z_1},\dots{},\overline{z_n}$. Furthermore assume that
- [$P$ is $\mathbb{R}$-homogeneous of degree $2k$, for some positive integer $k\geq{2}$,]{}
- [$P$ is plurisubharmonic,]{}
- [$P$ does not have any pluriharmonic terms.]{}
In this setting the question becomes, roughly speaking, how much one can subtract from $P$ without destroying plurisubharmonicity and while preserving homogeneity.\
If $P$ is additionally assumed to not be harmonic along any complex line through $0\in\mathbb{C}^n$, then there exists a smooth function $F\colon{\mathbb{C}^n\setminus\{0\}}\to\mathbb{R}$, such that $F$ is positive, $\mathbb{R}$-homogeneous of degree $2k$ and such that $P-F$ is strictly plurisubharmonic on $\mathbb{C}^n\setminus\{0\}$ (the assumption that $P$ does not have any pluriharmonic terms is clearly not necessary for this). In the case $n=1$ this follows from a stronger result by Forn[æ]{}ss and Sibony [@MR1016439 Lemma 2.4]. In the case $n\geq{2}$ this was shown by Noell [@MR1207878]. Since the case $n=1$ is completely solved by this, we will assume $n\geq{2}$ from now on.
If, however, $P$ is allowed to be harmonic along complex lines through $0$, then one cannot expect to obtain such a strong result. The next best bumping result one could hope for is the existence of a function $H\colon\mathbb{C}^n\to\mathbb{R}$ having the following properties:
- [$H$ is $\mathbb{R}$-homogeneous of degree $2k$ and smooth away from 0,]{}
- [$H\geq{0}$ everywhere with equality precisely in $0$ and along all complex lines through $0$ along which $P$ is harmonic (i.e. vanishes, since $P$ does not have any pluriharmonic terms),]{}
- [$P-H$ is plurisubharmonic,]{}
- [$\left|{\frac{P}{H}}\right|$ is bounded on ${\mathbb{C}}^n\setminus{}H^{-1}(\{0\})$.]{}
In dimension $n=2$, Bharali and Stens[ø]{}nes [@MR2452636] have obtained such bumping results in two cases, which, in some sense, can be interpreted as the two “extremal behaviors” the Levi-degeneracy set of $P$ can exhibit, when $P$ is allowed to be harmonic along complex lines through $0$.
In one of the upper mentioned cases studied by Bharali and Stens[ø]{}nes [@MR2452636], the polynomial $P\colon\mathbb{C}^2\to\mathbb{R}$ is assumed to be harmonic along the smooth part of every level set of a non-constant entire function. They proceed by showing that $P$ can be written as the composition of a subharmonic homogeneous polynomial on $\mathbb{C}$ with a holomorphic homogeneous polynomial on $\mathbb{C}^2$. The bumping for $P$ is then constructed by applying the result by Forn[æ]{}ss and Sibony [@MR1016439 Lemma 2.4]. So, roughly speaking, they identify a foliation by complex curves along which $P$ is harmonic and then bump with something that is homogeneous of degree $2k$, constant along the leaves of the foliation and positive away from a small singular set.
If this assumption, that such an entire function exists, is replaced by the [*weaker*]{} assumption that the determinant of the Complex Hessian Matrix of $P$ vanishes identically on $\mathbb{C}^2$, then no bumping results for $P$ are known thus far. Applying the Frobenius theorem, however, one [*does*]{} obtain a foliation as above, albeit not necessarily a holomorphic one (see also the paper by Bedford and Kalka [@MR0481107]).\
Therefore, it seems natural to replicate the previously explained bumping method of Bharali and Stens[ø]{}nes. In this setting, however, it is not clear whether there even exist [*locally defined*]{} smooth functions, which are positive, homogeneous of degree $2k$ and constant along the (local) leaves of the foliation. This is the content of a question asked by Stens[ø]{}nes.
The purpose of this paper is to give an [*affirmative answer*]{} to this question in a slightly more general setting: given a smooth $\mathbb{R}$-homogeneous vector field on an open subset of $\mathbb{C}^2$, such that the collection of complex vector spaces spanned by said vector field is an involutive distribution of real dimension $2$, we construct, assuming that a certain compatibility condition is satisfied, a positive smooth function that is homogeneous of any desired degree and constant along the (local) leaves of the foliation induced by the Frobenius theorem. As a corollary we obtain a positive answer to Stens[ø]{}nes’ question. A precise statement of these results can be found in the following section.
Preliminaries and Statement of Results
======================================
From now on, we fix a point $x\in\mathbb{C}^2\setminus{\{0\}}$, an open neighborhood $N$ of $x$ in $\mathbb{C}^2\setminus\{0\}$ and a vector field $$\begin{aligned}
\mathcal{V}=
\begin{pmatrix} V_1+i\cdot{V_2} \\ V_3+i\cdot{V_4}\end{pmatrix}
\colon{N}\to\mathbb{C}^2\text{,}\end{aligned}$$ such that $\mathcal{V}$ vanishes nowhere on $N$.
\[propo\][Assumption]{}
\[topmanfolipaper\] We make the following additional assumptions:
1. \[condi1\][$\mathcal{V}$ is of class $\mathcal{C}^{\infty}$,]{}
2. \[condi3\][The collection of $\mathbb{C}$-vector spaces spanned by $\mathcal{V}$ at the points in $N$ is involutive (as a $\mathcal{C}^{\infty}$ distribution of real dimension $2$ on $N$),]{}
3. \[condi4\][$\mathcal{V}$ is $\mathbb{R}$-homogeneous of degree $m$ for some positive integer $m$,]{}
4. \[condi5\][$p$ and $\mathcal{V}(p)$ are $\mathbb{C}$-linearly independent for all $p\in{N}$.]{}
[We denote the foliation obtained from applying the Frobenius theorem to the distribution in Property \[condi3\] as $\mathcal{F}$. ]{}
\[propo\][Remark]{}
\[compacondi\] Properties \[condi4\] and \[condi5\] can be interpreted as a compatibility condition between $\mathcal{F}$ and the foliation by spheres centered at the origin.
Under these assumptions, the main result of this paper can be stated as follows:
\[propo\][Theorem]{}
\[maintheoremfoliation\] There exist an open neighborhood $W\subseteq{N}$ of $x$ in $\mathbb{C}^2$ and, given an arbitrary positive integer $n$, a function $g\colon{W}\to\mathbb{R}$ with the following properties:
- [$g$ is constant along the leaves of the restriction of $\mathcal{F}$ to $W$,]{}
- [$g$ is $\mathbb{R}$-homogeneous of degree $n$,]{}
- [$g>0$ on $W$,]{}
- [$g$ is of class $\mathcal{C}^{\infty}$.]{}
\[propo\][Remark]{}
\[moregeneralgogogo\] In order to show the existence of the function $g$ in Theorem \[maintheoremfoliation\], one needs to find a function with prescribed behavior with respect to [*both*]{} the foliation $\mathcal{F}$ and the foliation by spheres centered at the origin. This will be possible because, as mentioned in Remark \[compacondi\], the two foliation satisfy a certain “compatibility condition”.\
Both the setting in $\mathbb{C}^2$ and the foliation by spherical shells are quite specific. It is likely that one can adjust the method of proof in this paper to derive compatibility conditions that two (or more) foliations need to satisfy in order to admit non-trivial functions with prescribed behavior with respect to each of the foliations, which might be of independent interest.\
Furthermore, it [*might*]{} be possible to use an appropriate global version of the implicit function theorem in order to get a global result in the spirit of Theorem \[maintheoremfoliation\].
We end this section by stating the following corollary, which is relevant for the bumping problem:
\[propo\][Corollary]{}
\[bumppingfoliacoro\] Let $x$ be as above and let $P$ be a real-valued polynomial with complex coefficients in two complex variables $(z,w)$ and their conjugates $(\overline{z},\overline{w})$. Assume that
- [$P$ is $\mathbb{R}$-homogeneous of degree $2k$ for some integer $k\geq{2}$,]{}
- [the Complex Hessian matrix of $P$ does not vanish at $x$,]{}
- [$x$ does not lie on a complex line through $0\in\mathbb{C}^2$ along which $P$ is harmonic,]{}
- [the Levi determinant of $P$ vanishes identically on a neighborhood of $x$ and hence, by real-analyticity, on all of $\mathbb{C}^2$.]{}
Then there exist an open neighborhood $W$ of $x$ in $\mathbb{C}^2$ and a function $g\colon{W}\to\mathbb{R}$ with the following properties:
- [There exists a smooth foliation of $W$ by complex curves along which $P$ is harmonic,]{}
- [$g$ is constant along the leaves of said foliation,]{}
- [$g$ is $\mathbb{R}$-homogeneous of degree $2k$,]{}
- [$g>0$ on $W$,]{}
- [$g$ is of class $\mathcal{C}^{\infty}$.]{}
Proof of Theorem \[maintheoremfoliation\]
=========================================
This section is devoted to proving Theorem \[maintheoremfoliation\]. We have to find an open neighborhood $W$ of $x$ in $N$ having certain properties. By a slight [*abuse of notation*]{}, we will (instead of defining the set $W$) shrink the open neighborhood $N$ of $x$ a finite amount of times and establish the existence of a function $g$ with the desired properties on $N$. Each time we shrink $N$, we also restrict the foliation $\mathcal{F}$ accordingly, which we will not always comment on.\
We begin with the following lemma (as usual, $\mathbb{D}$ denotes the open unit disc centered at $0$ in $\mathbb{C}$):
\[propo\][Lemma]{}
\[localfolichartsubm\] After shrinking $N$ if necessary and restricting $\mathcal{F}$ accordingly, there exist $0<{\delta}<1$, smooth functions ${u_1},{u_2}\colon{N}\to\mathbb{R}$ and a smooth function $\phi\colon{N}\to\mathbb{C}$, such that:
1. \[folipaperannoying1\][the real gradients $\nabla{u_1}$ and $\nabla{u_2}$ are $\mathbb{R}$-linearly independent at every point in $N$ (in particular they vanish nowhere on $N$),]{}
2. \[folipaperannoying2\][for $j\in\{1,2\}$, the real gradient $\nabla{u_j}$ is orthogonal to both $\mathcal{V}$ and $i\cdot\mathcal{V}$ at every point in $N$ with respect to the standard inner product on the $\mathbb{R}$-vector space $\mathbb{R}^4$,]{}
3. \[folipaperannoying3\][the leaves of $\mathcal{F}$ are precisely the level sets of $u:={u_1}+i{u_2}$, which are complex submanifolds of $\mathbb{C}^2$ of complex dimension $1$,]{}
4. \[folipaperannoying4\][the map $\Phi{:=}({\phi},u)$ is a $\mathcal{C}^{\infty}$ diffeomorphism from $N$ onto $\mathbb{D}\times\mathbb{D}$ and ${\Phi}(x)=0$, i.e. ${\phi}(x)=0$ and $u(x)=0$,]{}
5. \[folipaperannoying5\]
For all $t\in{(1-{\delta},1+{\delta})}$, $a,b\in{N}$ we have the following:
If $u(a)=u(b)$ and if $ta,tb\in{N}$, then $u(ta)=u(tb)$.
By assumption, the collection of $\mathbb{C}$-vector spaces spanned by $\mathcal{V}$ at the points in $N$ is an involutive $\mathcal{C}^{\infty}$ distribution of real dimension $2$ on $N$.
Hence Properties \[folipaperannoying1\], \[folipaperannoying2\], \[folipaperannoying3\] and \[folipaperannoying4\] will follow by applying the Frobenius theorem and the submersion theorem, while shrinking $N$ appropriately several (finitely many) times and restricting the foliation $\mathcal{F}$ accordingly.
Regarding the level sets of $u$ being complex submanifolds of $\mathbb{C}^2$, we note that they are (embedded) smooth submanifolds of $\mathbb{C}^2$ of real dimension $2$, whose tangent spaces at every point can easily be seen to be complex linear subspaces of $\mathbb{C}^2$.
It remains to address Property \[folipaperannoying5\]. Let $u$ and $\phi$ satisfy Properties \[folipaperannoying1\], \[folipaperannoying2\], \[folipaperannoying3\] and \[folipaperannoying4\]. We make the following claim within the proof ($\mathbb{D}_{1/2}$ denotes the open disc of radius $1/2$ in $\mathbb{C}$ centered at $0$):
There exists $0<{\delta}<1$, such that for all $t\in{(1-{\delta},1+{\delta})}$, $a,b\in{\Phi}^{-1}({\mathbb{D}_{1/2}}\times{\mathbb{D}_{1/2}})$ we have the following:
- [$ta,tb\in{N}$,]{}
- [if $u(a)=u(b)$, then $u(ta)=u(tb)$.]{}
After having shown the claim, we can finish the proof by replacing $N$ by ${\Phi}^{-1}({\mathbb{D}_{1/2}}\times{\mathbb{D}_{1/2}})$ and rescaling $\phi$ and $u$. Hence it suffices to prove the claim.
To this end, we note that ${\Phi}^{-1}({\mathbb{D}_{1/2}}\times{\mathbb{D}_{1/2}})\Subset{N}$, i.e. there exists $0<{\delta}<1$, such that $t\cdot{p}\in{N}$, whenever $t\in{(1-{\delta},1+{\delta})}$ and $p\in{{\Phi}^{-1}({\mathbb{D}_{1/2}}\times{\mathbb{D}_{1/2}})}$.
Now let $a,b\in{\Phi}^{-1}({\mathbb{D}_{1/2}}\times{\mathbb{D}_{1/2}})$ with $u(a)=u(b)$. By choice of $\delta$ we have $ta,tb\in{N}$, whenever $t\in{(1-{\delta},1+{\delta})}$. Set $c:=u(a)=u(b)\in\mathbb{D}_{1/2}$ and, for $t\in{(1-{\delta},1+{\delta})}$, define a map $$\begin{aligned}
\Gamma_t\colon\mathbb{D}_{1/2}\to{N}\text{, }\gamma\mapsto{t\cdot}{\Phi}^{-1}({\gamma},c)\text{,}\end{aligned}$$ which is welldefined by choice of $\delta$. For $j\in\{1,2\}$, we compute the real gradient of $u_j\circ\Gamma_t\colon\mathbb{D}_{1/2}\to\mathbb{R}$ in real coordinates. We get for $\gamma\in{\mathbb{D}_{1/2}}$: $$\begin{aligned}
\mathbb{R}^{1\times{2}}\ni\nabla{({u_j\circ\Gamma_t})}({\gamma}) & =\nabla{(u_j)({\Gamma_t}({\gamma}))}\cdot{\operatorname{J}_{\Gamma_t}}({\gamma})\\
& =\nabla{(u_j)(t\cdot{\Gamma_1}({\gamma}))}\cdot{t}\cdot{\operatorname{J}_{\Gamma_1}}({\gamma})\text{,}\end{aligned}$$ where ${\operatorname{J}_{\Gamma_t}}({\gamma})\in\mathbb{R}^{4\times{2}}$ denotes the Jacobian matrix of $\Gamma_t$ evaluated at $\gamma$. But $u\circ\Gamma_1\equiv{c}$, so $\nabla{(u_j\circ\Gamma_1)}\equiv{0}$ for $j\in\{1,2\}$, i.e. we have $$\begin{aligned}
\nabla{(u_j)({\Gamma_1}({\gamma}))}\cdot{\operatorname{J}_{\Gamma_1}}({\gamma})=0\end{aligned}$$ for all $\gamma\in\mathbb{D}_{1/2}$, $j\in\{1,2\}$. Hence both columns of ${\operatorname{J}_{\Gamma_1}}({\gamma})$ are orthogonal to $\nabla{(u_j)({\Gamma_1}({\gamma}))}$ with respect to the standard inner product on $\mathbb{R}^4$. Since $\mathcal{V}$ vanishes nowhere on $N$ and since we are in real dimension $4$, we can use Properties \[folipaperannoying1\] and \[folipaperannoying2\] to deduce that the columns of ${\operatorname{J}_{\Gamma_1}}({\gamma})$ are contained in the [*complex*]{} vector space spanned by $\mathcal{V}({\Gamma_1}({\gamma}))$.\
But, by \[condi4\] in Assumption \[topmanfolipaper\], we immediately get that the columns of ${\operatorname{J}_{\Gamma_t}}({\gamma})=t\cdot{}{\operatorname{J}_{\Gamma_1}}({\gamma})$ are contained in the [*complex*]{} vector space spanned by $\mathcal{V}(t\cdot{\Gamma_1}({\gamma}))$ and hence orthogonal to $\nabla{(u_j)(t\cdot{\Gamma_1}({\gamma}))}$, $j\in\{1,2\}$, with respect to the standard inner product on $\mathbb{R}^4$. Since $\mathbb{D}_{1/2}$ is connected, this shows together with the above calculation, that $u\circ\Gamma_t$ is constant. Noting that ${\phi}(a),{\phi}(b)\in\mathbb{D}_{1/2}$, we compute: $$\begin{aligned}
u(ta)=u({{t\cdot}{\Phi}^{-1}({{\phi}(a)},u(a))})=u({{t\cdot}{\Phi}^{-1}({{\phi}(a)},c)})=({u\circ\Gamma_t})({\phi}(a))\text{.}\end{aligned}$$ Analogously we get that $u(tb)=({u\circ\Gamma_t})({\phi}(b))$. Since $u\circ\Gamma_t$ is constant, we obtain $u(ta)=u(tb)$, as desired.
\[propo\][Remark]{}
\[localglobalmiregalleaf\] It should be noted that the distinction between local and global leaves disappears, whenever $N$ is shrunk in a way that it coincides with the open set associated to a foliation chart containing $x$, since we always restrict the foliation appropriately. Because of this, we will not distinguish between local and global leaves of the foliation $\mathcal{F}$ for the remainder of this section, unless stated otherwise.
\[propo\][Remark]{}
\[leavesscalehomog\] Property \[folipaperannoying5\] in Lemma \[localfolichartsubm\] says that, roughly speaking, the leaves of the foliation scale homogeneously. While this may appear trivial at first glance, it should be noted that this is a property which a priori could easily be destroyed by restricting the foliation to the “wrong” open set.
Armed with Lemma \[localfolichartsubm\], we now set for $\tau\in\mathbb{D}$: $$\begin{aligned}
p_{\tau}:={\Phi}^{-1}(0,{\tau})\in{N}\text{,}\end{aligned}$$ i.e. for each leaf $\{u={\tau}\}$ of $\mathcal{F}$ we pick one point on it, such that this choice depends smoothly on the leaf. Since $N$ is open, we find a ${\delta}_{p_{\tau}}>0$, such that $t\cdot{p_{\tau}}\in{N}$, whenever $1-2{{\delta}_{p_{\tau}}}<{t}<{1}+2{\delta}_{p_{\tau}}$. Hence, for all $\tau\in\mathbb{D}$, we can define a smooth map $$\begin{aligned}
S_{\tau}\colon{}({{1}-2{\delta}_{p_{\tau}}},{{1}+2{\delta}_{p_{\tau}}})\to\mathbb{D}\text{, }t\mapsto{u(t\cdot{p_{\tau}})}\text{.}\end{aligned}$$ Intuitively speaking, we go along the real ray through $0\in\mathbb{C}^2$ and $p_{\tau}\in{N}$ and apply $u$, which amounts to checking which leaf we are on. So the derivative $S_{\tau}'$ of $S_{\tau}$ measures “how the leaf changes along the ray”.
Computing the derivative at $t=1$ in real coordinates, the map $$\begin{aligned}
\colon\mathbb{D}\to\mathbb{R}^2\text{, }\tau\mapsto{S_{\tau}'}(1)=\begin{pmatrix}
\nabla{u_1}({p_{\tau}}) \\
\nabla{u_2}({p_{\tau}})
\end{pmatrix}\cdot{p_{\tau}}\end{aligned}$$ (where $p_{\tau}$ is considered as an element of $\mathbb{R}^{4\times{1}}$) defines a smooth vector field on $\mathbb{D}$. If ${S_{\tau}'}(1)$ was to vanish for some $\tau\in\mathbb{D}$, then (analogously to the proof of Property \[folipaperannoying5\] in Lemma \[localfolichartsubm\]) that would imply that $p_{\tau}$ was contained in the [*complex*]{} vector space spanned by $\mathcal{V}({p_{\tau}})$, in contradiction to \[condi5\] in Assumption \[topmanfolipaper\]. Hence we have ${S_{\tau}'}(1)\neq{0}$ for all $\tau\in\mathbb{D}$.
Consequently, the collection of [*real*]{} vector spaces spanned by ${S_{\tau}'}(1)\in\mathbb{R}^2\setminus\{0\}$ at the points $\tau\in\mathbb{D}$ yields a $\mathcal{C}^{\infty}$ distribution of real dimension $1$ on $\mathbb{D}$, which is trivially involutive. The Frobenius theorem implies the following:
\[propo\][Lemma]{}
\[frobbyondiscfolipaper\] There exist an open subset $\Omega$ of $\mathbb{D}$ containing $u(x)=0$ and a $\mathcal{C}^{\infty}$ diffeomorphism $$\begin{aligned}
\omega{=}({\omega_1},{\omega_2})\colon\Omega\to{(-1,1)\times{(-1,1)}}\text{,}\end{aligned}$$ such that
- [the real gradient $\nabla\omega_2$ vanishes nowhere on $\Omega$,]{}
- [$\nabla\omega_2{(\tau)}$ and ${S_{\tau}'}(1)$ are orthogonal with respect to the standard inner product on $\mathbb{R}^2$ for all $\tau\in\Omega$.]{}
This follows from the above considerations.
If $p_1,p_2\in{\Phi}^{-1}(\mathbb{D}\times{\Omega})$ are points on the same $\mathbb{R}_{\geq{0}}$-ray originating at $0\in\mathbb{C}^2$, then $u({p_1})$ and $u({p_2})$ are not necessarily contained in the same level set of $\omega_2$, since, roughly speaking, one might temporarily leave the set ${\Phi}^{-1}(\mathbb{D}\times{\Omega})$ when going from $p_1$ to $p_2$ along the ray. If, however, we restrict our attention to a suitable smaller open neighborhood of $x$, where this problem does not arise, then the level sets of $\omega_2$ exhibit the desired behavior. That is the content of the following lemma:
\[propo\][Lemma]{}
\[lemmaafolipaper\] There exists an open neighborhood $W_x$ of $x$ in $N$ with the following properties:
1. \[lemmaafolipaperprop1\][$W_x\subseteq{{\Phi}^{-1}(\mathbb{D}\times{\Omega})}$ and $W_x\cap{(-W_x)}=\emptyset$,]{}
2. \[lemmaafolipaperprop2\][$u(W_x)\Subset\Omega$,]{}
3. \[lemmaafolipaperprop3\]
there exist an open subset $B_x$ of $\{p\in\mathbb{C}^2\colon{\Vert{p}\Vert{=\Vert{x}\Vert}}\}$ (which is equipped with the subspace topology it inherits from $\mathbb{C}^2$) and a real number ${0<}d_x{<1}$, such that:
- [$x\in{B_x}$,]{}
- [${W_x}=\{t\cdot{p}\in\mathbb{C}^2\colon{{1-d_x}<t<{1+d_x}\text{ and }p\in{B_x}}\}$,]{}
- [if $q\in{W_x}$ and if $(1-{d_x})/(1+{d_x})<t<(1+{d_x})/(1-{d_x})$, then $t\cdot{q}\in{N}$ and $u(t\cdot{q})\in\Omega$,]{}
4. \[lemmaafolipaperprop4\][if ${p_1},{p_2}\in{W_x}$ lie on the same $\mathbb{R}_{\geq{0}}$-ray originating at $0\in\mathbb{C}^2$, then ${\omega_2}(u({p_1}))={\omega_2}(u({p_2}))$.]{}
In fact, whenever $W_x$ is an open neighborhood of $x$ in $N$ having Properties \[lemmaafolipaperprop1\], \[lemmaafolipaperprop2\] and \[lemmaafolipaperprop3\], then it will necessarily have Property \[lemmaafolipaperprop4\].
It is clear that there exists an open neighborhood $W_x$ of $x$ in $N$ having Properties \[lemmaafolipaperprop1\], \[lemmaafolipaperprop2\] and \[lemmaafolipaperprop3\]. We have to show that such a neighborhood necessarily has Property \[lemmaafolipaperprop4\].
To this end, let ${p_1},{p_2}\in{W_x}$ lie on the same $\mathbb{R}_{\geq{0}}$-ray originating at $0\in\mathbb{C}^2$. By Property \[lemmaafolipaperprop3\], there exist $p\in{B_x}$ and ${t_1},{t_2}\in{({1-d_x},{1+d_x})}$, such that ${p_1}=t_1{p}$ and ${p_2}=t_2{p}$. Hence it suffices to show that the derivative of the (clearly welldefined) smooth map $$\begin{aligned}
\chi\colon{({1-d_x},{1+d_x})}\to\mathbb{R}\text{, }t\mapsto{\omega_2}(u(t\cdot{p}))\end{aligned}$$ vanishes identically. So, given $t_0\in{({1-d_x},{1+d_x})}$, we need to show that ${\chi}'({t_0})=0$.
Let $\tau{:=}u(t_0\cdot{p})\in\Omega$ and recall that $p_{\tau}={\Phi}^{-1}(0,{\tau})\in{N}$. We trivially have $u(t_0\cdot{p})=u({p_\tau})$, so, using Property \[folipaperannoying5\] in Lemma \[localfolichartsubm\], we find a $0<\widetilde{\delta}\ll\delta$, such that we have the following for all $t\in{({1-\widetilde{\delta}},{1+\widetilde{\delta}})}$:
- [$t\cdot{t_0}\cdot{p}$ and $t\cdot{p_{\tau}}$ are contained in $N$,]{}
- [$u({t\cdot{t_0}\cdot{p}})=u({t\cdot{p_{\tau}}})$,]{}
- [$t\cdot{t_0}\in{({1-d_x},{1+d_x})}$, i.e. $t\cdot{t_0}\cdot{p}\in{W_x}$.]{}
Using this, we can define a map $$\begin{aligned}
\widetilde{\chi}\colon{({1-\widetilde{\delta}},{1+\widetilde{\delta}})}\to\mathbb{R}\text{, }t\mapsto\chi{(t\cdot{t_0})}\text{.}\end{aligned}$$ Since $t_0\neq{0}$ and $\widetilde{\chi}'(t)={\chi}'(t\cdot{t_0})\cdot{t_0}$ for all $t\in{({1-\widetilde{\delta}},{1+\widetilde{\delta}})}$, it suffices to show that $\widetilde{\chi}'(1)=0$. But, using that $u({t\cdot{t_0}\cdot{p}})=u({t\cdot{p_{\tau}}})$ for all $t\in{({1-\widetilde{\delta}},{1+\widetilde{\delta}})}$ and that $u(p_{\tau})=\tau$, one readily computes $$\begin{aligned}
\widetilde{\chi}'(1) & =\left(({\nabla\omega_2})(u(t\cdot{p_\tau}))\cdot\begin{pmatrix}
\nabla{u_1}(t\cdot{p_{\tau}}) \\
\nabla{u_2}(t\cdot{p_{\tau}})
\end{pmatrix}\cdot{p_{\tau}}\right)\Bigg\rvert_{t=1}\\
& ={\nabla\omega_2}({\tau})\cdot{{S_{\tau}'}(1)}\\
& =0\text{,}\end{aligned}$$ where the last equality follows from Lemma \[frobbyondiscfolipaper\].
From now on, we fix an open neighborhood $W_x$ of $x$ as in Lemma \[lemmaafolipaper\]. Furthermore, we choose an open subset $\widetilde{W_x}$ of $\mathbb{C}^2$ and a $0<\widetilde{\delta_x}\ll{1}$ with the following properties:
- [$\widetilde{\delta_x}<\delta$ (see Lemma \[localfolichartsubm\]),]{}
- [$x\in\widetilde{W_x}\Subset{W_x}$,]{}
- [$t\cdot{q}\in{W_x}$, whenever $q\in\widetilde{W_x}$ and $t\in{(1-{\widetilde{\delta_x}},1+{\widetilde{\delta_x}})}$.]{}
Owing to these properties and Lemma \[lemmaafolipaper\], the following maps are welldefined and smooth: $$\begin{aligned}
\mathcal{M}\colon\widetilde{W_x}\times{(1-{\widetilde{\delta_x}},1+{\widetilde{\delta_x}})} & \to\mathbb{R}\text{,}\\
(q,t) & \mapsto{\omega_1}(u({t\cdot{q}}))-{\omega_1}(u(x))\text{,}\\
\mathcal{N}\colon\widetilde{W_x}\times{(1-{\widetilde{\delta_x}},1+{\widetilde{\delta_x}})} & \to\mathbb{R}\text{,}\\
(q,t) & \mapsto{\omega_2}(u({t\cdot{q}}))-{\omega_2}(u(x))\text{.}\end{aligned}$$ Noting that $p_{0}={\Phi}^{-1}(0,0)=x$ and using Property \[lemmaafolipaperprop4\] in Lemma \[lemmaafolipaper\] we compute: $$\begin{aligned}
\begin{pmatrix}
\frac{\partial\mathcal{M}}{\partial{t}}(x,1)\\
0
\end{pmatrix} & =\begin{pmatrix}
\frac{\partial\mathcal{M}}{\partial{t}}(x,1)\\[1ex]
\frac{\partial\mathcal{N}}{\partial{t}}(x,1)
\end{pmatrix}\\
& =\left(\operatorname{J}_{\omega}(u(tx))\cdot\begin{pmatrix}
\nabla{u_1}(tx) \\
\nabla{u_2}(tx)
\end{pmatrix}\cdot{x}\right)\Bigg\rvert_{t=1}\\
& =\operatorname{J}_{\omega}(u(x))\cdot\begin{pmatrix}
\nabla{u_1}({p_{0}}) \\
\nabla{u_2}({p_{0}})
\end{pmatrix}\cdot{p_{0}}\\
& =\operatorname{J}_{\omega}(u(x))\cdot{{S_{0}'}(1)}\text{.}\end{aligned}$$ But we have ${S_{\tau}'}(1)\neq{0}$ for all $\tau\in\mathbb{D}$ and $\omega$ is a $\mathcal{C}^{\infty}$ diffeomorphism, so we can conclude that $$\begin{aligned}
\frac{\partial\mathcal{M}}{\partial{t}}(x,1)\neq{0}\text{.}\end{aligned}$$ So, since $\mathcal{M}$ is smooth and we clearly have $\mathcal{M}(x,1)=0$, the implicit function theorem implies the following lemma:
\[propo\][Lemma]{}
\[defofcalitfolipaper\] There exist an open neighborhood $V_x$ of $x$ in $\widetilde{W_x}$, an open neighborhood $\mathcal{I}$ of $1$ in ${(1-{\widetilde{\delta_x}},1+{\widetilde{\delta_x}})}$ and a smooth map $\mathcal{T}\colon{V_x}\to\mathcal{I}$ with $\mathcal{T}(x)=1$, such that for all $(q,t)\in{V_x}\times\mathcal{I}$ we have: $$\begin{aligned}
\mathcal{M}(q,t)=0\text{ if and only if }t=\mathcal{T}(q)\text{.}\end{aligned}$$
This follows from the above considerations.
Pick an open subset $\widetilde{V_x}$ of $\mathbb{C}^2$, an open subset $\widetilde{B_x}$ of $\{p\in\mathbb{C}^2\colon{\Vert{p}\Vert{=\Vert{x}\Vert}}\}$ and a $0<{\lambda_x}\ll{1}$, such that:
- [$x\in\widetilde{V_x}\Subset{V_x}$ and $\widetilde{V_x}\cap{(-\widetilde{V_x})}=\emptyset$ and $x\in\widetilde{B_x}$,]{}
- [$\widetilde{V_x}=\{t\cdot{p}\in\mathbb{C}^2\colon{{1-\lambda_x}<t<{1+\lambda_x}\text{ and }p\in\widetilde{B_x}}\}$.]{}
Let $n$ be a positive integer, as in the statement of Theorem \[maintheoremfoliation\]. We now define: $$\begin{aligned}
g\colon\widetilde{V_x}\to\mathbb{R}\text{, }q\mapsto{\left({\frac{1}{\mathcal{T}(q)}}\right)}^n\text{,}\end{aligned}$$ which is clearly welldefined. Since we can shrink $N$, it suffices to show that $g$ has the desired properties on $\widetilde{V_x}$.
It is obvious that $g$ is of class $\mathcal{C}^{\infty}$ and everywhere $>0$. Now assume that $q_1,q_2\in\widetilde{V_x}$ lie on the same leaf of the restriction of $\mathcal{F}$ to $\widetilde{V_x}$. In particular we have $u({q_1})=u({q_2})$. We have to show that $g({q_1})=g({q_2})$; so it suffices to prove that $\mathcal{T}({q_1})=\mathcal{T}({q_2})$.\
Owing to the choices we made, we have $\mathcal{T}({q_2})\in{(1-{\delta},1+{\delta})}$ and the points $q_1$, $q_2$, $\mathcal{T}({q_2})\cdot{q_1}$ and $\mathcal{T}({q_2})\cdot{q_2}$ are contained in $N$. Since $u({q_1})=u({q_2})$, we can hence apply Lemma \[localfolichartsubm\] to obtain $u({\mathcal{T}({q_2})\cdot{q_1}})=u({\mathcal{T}({q_2})\cdot{q_2}})$. Since $({q_1},\mathcal{T}({q_2}))$ and $({q_2},\mathcal{T}({q_2}))$ are contained in $V_x\times\mathcal{I}$, we get $$\begin{aligned}
\mathcal{M}({q_1},\mathcal{T}({q_2}))=\mathcal{M}({q_2},\mathcal{T}({q_2}))=0\text{;}\end{aligned}$$ Lemma \[defofcalitfolipaper\] then implies that $\mathcal{T}({q_1})=\mathcal{T}({q_2})$, as desired.
It remains to show that $g$ is $\mathbb{R}$-homogeneous of degree $n$. To this end, let $q\in\widetilde{V_x}$, $t\in\mathbb{R}$ and assume $t\cdot{q}\in\widetilde{V_x}$. We have to show that $g(tq)=t^n\cdot{g(q)}$. By choice of $\widetilde{V_x}$ one readily reduces to the case that $q\in\widetilde{B_x}$ and $t\in{(1-{\lambda_x},1+{\lambda_x})}$.
The map $$\begin{aligned}
\colon{(1-{\lambda_x},1+{\lambda_x})} & \to\mathbb{R}^2\text{,}\\
s & \mapsto{(\mathcal{M}{(sq,\mathcal{T}(sq))},\mathcal{N}{(sq,\mathcal{T}(sq))})}\end{aligned}$$ is welldefined and constant by Lemmas \[defofcalitfolipaper\] and \[lemmaafolipaper\] and by the choices we made. Since $\omega$ is a $\mathcal{C}^{\infty}$ diffeomorphism, that implies that the following map is welldefined and constant: $$\begin{aligned}
\colon{(1-{\lambda_x},1+{\lambda_x})}\to\mathbb{C}\text{, }
s\mapsto{u(\mathcal{T}(sq)\cdot{sq})}\text{.}\end{aligned}$$ By differentiating and thereupon using Property \[condi5\] in Assumption \[topmanfolipaper\] analogously to above, we obtain: $$\begin{aligned}
\mathcal{T}(sq)+s\cdot\nabla\mathcal{T}(sq)\cdot{q}=0\text{ for all }s\in{(1-{\lambda_x},1+{\lambda_x})}\text{,}\end{aligned}$$ which directly implies that the map $$\begin{aligned}
\colon{(1-{\lambda_x},1+{\lambda_x})}\to\mathbb{R}\text{, }
s\mapsto{\mathcal{T}(sq)\cdot{s}}\end{aligned}$$ is constant. Since $t\in{(1-{\lambda_x},1+{\lambda_x})}$, we get $\mathcal{T}(q)=\mathcal{T}(tq)\cdot{t}$. A straightforward calculation then gives $g(tq)=t^n\cdot{g(q)}$, as desired.
Proof of Corollary \[bumppingfoliacoro\]
========================================
This section is devoted to proving Corollary \[bumppingfoliacoro\]. To this end, let $P$ and $x$ be as in the statement of Corollary \[bumppingfoliacoro\].
\[propo\][Notation]{}
\[xxcomplehessiannotationx2\] We denote the Complex Hessian Matrix or the Levi Matrix of $P$ as $H_P$, i.e.$$\begin{aligned}
\arraycolsep=0.2pt\def\arraystretch{1.4}
H_P=\left( \begin{array}{ccc}
\frac{\partial^2 P}{\partial{z}\partial{\overline{z}}} & \frac{\partial^2 P}{\partial{w}\partial{\overline{z}}} \\
\frac{\partial^2 P}{\partial{z}\partial{\overline{w}}} & \frac{\partial^2 P}{\partial{w}\partial{\overline{w}}} \end{array} \right)\text{.}\end{aligned}$$
Firstly we note that, due to the formulation of Corollary \[bumppingfoliacoro\] and the proof of Theorem \[maintheoremfoliation\], we neither have to concern ourselves with the difference between local leaves and global leaves nor with the smoothness of the foliation.
Furthermore, after having checked the assumptions for applying Theorem \[maintheoremfoliation\], it will be immediate that, after shrinking $W$ if necessary, the leaves are submanifolds of $\mathbb{C}^2$ of real dimension $2$ and hence complex curves, since their tangent spaces at every point are [*complex*]{} linear subspaces of $\mathbb{C}^2$. We also remark that harmonicity of $P$ along the leaves of the foliation will be a welldefined notion due to the leaves being complex curves.
Because of these remarks we will simply check that the assumptions for applying Theorem \[maintheoremfoliation\] are satisfied.
By assumption we have $H_P{(x)}\neq{0}$, so that at least one of the two vector fields $$\begin{aligned}
(z,w)\mapsto\begin{pmatrix} -\frac{\partial^2 P}{\partial{w}\partial{\overline{z}}} \\[1ex] \phantom{-}\frac{\partial^2 P}{\partial{z}\partial{\overline{z}}}\end{pmatrix}(z,w)\text{ and }(z,w)\mapsto\begin{pmatrix} -\frac{\partial^2 P}{\partial{w}\partial{\overline{w}}} \\[1ex] \phantom{-}\frac{\partial^2 P}{\partial{z}\partial{\overline{w}}}\end{pmatrix}(z,w)\end{aligned}$$ does not vanish in $x$. Let $\mathcal{V}$ be one of these two vector fields, such that $\mathcal{V}(x)\neq{0}$. Since the Levi determinant of $P$ vanishes on $\mathbb{C}^2$, we get $H_P\cdot\mathcal{V}\equiv{0}$ by choice of $\mathcal{V}$, which shows that $P$ is indeed harmonic along the leaves of the foliation whose existence we are about to establish.\
Now we pick an open neighborhood $N\subseteq\mathbb{C}^2\setminus\{0\}$ of $x$, such that $\mathcal{V}$ vanishes nowhere on $N$ and such that $N$ does not meet a complex line through $0$ along which $P$ is harmonic. The latter is possible by assumption on $P$ and $x$.
It remains to verify the properties in Assumption \[topmanfolipaper\]. Properties \[condi1\] and \[condi4\] are clear. Noting that $\mathcal{V}$ does not vanish on $N$ and hence defines a $\mathcal{C}^{\infty}$ distribution of real dimension $2$ on $N$, Property \[condi3\] follows from directly computing the Lie bracket $[\mathcal{V},i\cdot\mathcal{V}]$ in real Cartesian coordinates and making use of the fact that $\det{H_P}\equiv{0}$ on $\mathbb{C}^2$.\
By restricting the foliation on $N$ obtained from Frobenius theorem to a foliation chart containing $x$ and subsequently replacing $N$ by the open set associated to said chart, we can assume that the leaves of the foliation are (embedded) complex submanifolds of $N$. The leaves of the restricted foliation are precisely the plaques of the original foliation in the foliation chart of consideration.
Using the assumptions on $P$ and $x$ and the properties established thus far, we can (by an argument similar to the one appearing in the previous section) find smooth functions $u={u_1}+i{u_2}\colon{N}\to\mathbb{D}$ and $\phi\colon{N}\to\mathbb{D}$ having Properties \[folipaperannoying1\], \[folipaperannoying2\], \[folipaperannoying3\] and \[folipaperannoying4\] from Lemma \[localfolichartsubm\] (it should be noted that this is potentially accompanied by shrinking $N$ and restricting the foliation yet again).
In order to verify Property \[condi5\] in Assumption \[topmanfolipaper\], we assume for the sake of a contradiction that there exists a point $p\in{N}$, such that $p$ and $\mathcal{V}(p)$ are $\mathbb{C}$-linearly dependent. Since $\mathcal{V}(p)\neq{0}$, we have that $p$ in contained in the complex vector subspace of $\mathbb{C}^2$ spanned by $\mathcal{V}(p)$. Since $P$ is homogeneous, we find a small $0<{\delta}\ll{1}$ (not to be confused with the $\delta$ appearing in the previous section), such that for all $t\in\left({1-{\delta},1+{\delta}}\right)$ we have: $$\begin{aligned}
t\cdot{p}\in{N}\text{ and }{p}\in\operatorname{Span}_{\mathbb{C}}\left(\left\{\mathcal{V}(t\cdot{p})\right\}\right)\text{.}\end{aligned}$$ We consider the map $$\begin{aligned}
S\colon{\left({1-{\delta},1+{\delta}}\right)}\to\mathbb{D}\text{, }t\mapsto{u(t\cdot{p})}\text{.}\end{aligned}$$ In real coordinates, the derivative at $t\in\left({1-{\delta},1+{\delta}}\right)$ computes to $$\begin{aligned}
S'(t)=\begin{pmatrix}
\nabla{u_1}(t\cdot{p}) \\
\nabla{u_2}(t\cdot{p})
\end{pmatrix}\cdot{p}\text{,}\end{aligned}$$ where $p$ is considered as an element of $\mathbb{R}^{4\times{1}}$ and the gradients are considered as elements of $\mathbb{R}^{1\times{4}}$. As seen above, $p$ is contained in the $\mathbb{R}$-vector space spanned by $\mathcal{V}(t\cdot{p})$ and $i\cdot\mathcal{V}(t\cdot{p})$, which implies $S'(t)=0$ by the defining properties of $u$. It follows that $S$ is constant, so the points $t\cdot{p}$, $t\in\left({1-{\delta},1+{\delta}}\right)$, are all contained in $L:=\{q\in{N}\colon{u(q)=u(p)}\}$. Recalling the behavior of the level sets of $u$, we find an open neighborhood $\mathcal{Z}$ of $p$ in $N$ and a holomorphic coordinate system $({\zeta}_1,{\zeta}_2)\colon\mathcal{Z}\to\mathbb{C}^2$, such that $$\begin{aligned}
\{q\in\mathcal{Z}\colon{\zeta_2}(q)=0\}=\mathcal{Z}\cap{L}\text{.}\end{aligned}$$ Let $X\subseteq\mathbb{C}$ be a small open disc centered at $1$, such that $s\cdot{p}\in\mathcal{Z}$ for all $s\in{X}$. The map $$\begin{aligned}
\colon{X}\to\mathbb{C}\text{, }s\mapsto{\zeta_2}(s\cdot{p})\end{aligned}$$ is holomorphic and vanishes on $X\cap{\left({1-{\delta},1+{\delta}}\right)}$; hence said map vanishes on all of $X$. But this immediately gives that $$\begin{aligned}
(\colon{X}\to\mathbb{C}\text{, }s\mapsto{u(s\cdot{p})})\equiv{u(p)}\text{.}\end{aligned}$$ Writing $s=a+ib$, applying $\partial{}/\partial{a}$ and considering $p$ as an element of $\mathbb{R}^{4\times{1}}$ again, we get the following in real coordinates: $$\begin{aligned}
\begin{pmatrix}
\nabla{u_1}(s\cdot{p}) \\
\nabla{u_2}(s\cdot{p})
\end{pmatrix}\cdot{p}=0\text{ for all }s\in{X}\text{.}\end{aligned}$$ So, with respect to the standard inner product on $\mathbb{R}^4$, we get that $p$ is contained in the orthogonal complement of the $\mathbb{R}$-span of $\nabla{u_1}(s\cdot{p})$ and $\nabla{u_2}(s\cdot{p})$, for all $s\in{X}$. But $\nabla{u_1}(s\cdot{p})$ and $\nabla{u_2}(s\cdot{p})$ are linearly independent over $\mathbb{R}$, i.e. said orthogonal complement has real dimension $2$ and hence equals $\operatorname{Span}_{\mathbb{C}}\left(\left\{\mathcal{V}(s\cdot{p})\right\}\right)$. We get $$\begin{aligned}
{p}\in\operatorname{Span}_{\mathbb{C}}\left(\left\{\mathcal{V}(s\cdot{p})\right\}\right)\text{ for all }s\in{X}\text{.}\end{aligned}$$ Since $H_P\cdot\mathcal{V}\equiv{0}$, this implies that $H_P{(s\cdot{p})}\cdot{p}=0$ in complex coordinates for all $s\in{X}$. Noting that this expression is real-analytic in $s\in\mathbb{C}$, we deduce that $H_P{(s\cdot{p})}\cdot{p}=0$ for all $s\in\mathbb{C}$. But that implies that $P$ is harmonic along the complex line through $0$ and $p$. Since, however, $N$ was chosen to not meet a complex line through $0$ along which $P$ is harmonic, we get $p\not\in{N}$. We have arrived at the desired contradiction.
[^1]: Part of this work was done during the international research program “Several Complex Variables and Complex Dynamics” at the Centre for Advanced Study at the Academy of Science and Letters in Oslo during the academic year 2016/2017.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study the emergency of mutual cooperation in evolutionary prisoner’s dilemma games when the players are located on a square lattice. The players can choose one of the three strategies: cooperation ($C$), defection ($D$) or “tit for tat” ($T$), and their total payoffs come from games with the nearest neighbors. During the random sequential updates the players adopt one of their neighboring strategies if the chosen neighbor has higher payoff. We compare the effect of two types of external constraints added to the Darwinian evolutionary processes. In both cases the strategy of a randomly chosen player is replaced with probability $P$ by another strategy. In the first case, the strategy is replaced by a randomly chosen one among the two others, while in the second case the new strategy is always $C$. Using generalized mean-field approximations and Monte Carlo simulations the strategy concentrations are evaluated in the stationary state for different strength of external constraints characterized by the probability $P$.'
address: |
$^*$Research Institute for Technical Physics and Materials Science\
P.O.Box 49, H-1525 Budapest, Hungary\
$^{\dagger}$Department of Theoretical Physics, University of Geneva, 1211 Geneva 4, Switzerland\
$^{\ddagger}$Department of Ecology, József Attila University, H-6721 Szeged, Egyetem u. 2, Hungary
author:
- 'György Szabó$^*$, Tibor Antal$^{\dagger}$, Péter Szabó$^{\ddagger}$, and Michel Droz$^{\dagger}$'
title: 'On the Role of External Constraints in a Spatially Extended Evolutionary Prisoner’s Dilemma Game'
---
Introduction {#introduction .unnumbered}
============
The successful applications of game theory in the area of economics and political decisions initiated its increasing development after the second world war [@vNM]. Originally, the game theory is devoted to find the optimal strategy for a given game between two intelligent players. The straightforward developments involve the generalization toward the iterated games of $n$ players with assuming local interactions among the spatially distributed players. The spatial evolutionary prisoner’s dilemma games (SEPDG) has attracted a particular attention because of its applicability in the human and behavior sciences as well as in biology [@HS; @sigmund; @msmith; @axelrod; @weibull]. Nowadays the prisoner’s dilemma game is considered to be the metaphor for studying the emergence of cooperation among selfish individuals. The emerging cooperation appears to be crucial at many transitions in evolution [@MSSz]. The first numerical investigations have shown that the cooperation can be maintained by very simple strategies in the iterated games [@axelrod]. Very recently it is demonstrated that the players can be as simple as bacteriophages (viruses that infect bacteria)[@TC; @NS99].
In these systems the players wish to maximize their individual income coming from games with other players. The prisoner’s dilemma game is a simple version of the two-player matrix games where the players’ income depend on their simultaneous choice between two options. Following the widely accepted expressions each player can choose defection or cooperation with the other player. The defector reaches the highest payoff $t$ (called temptation to defect) against the cooperator, which receives then the lowest reward $s$ (called sucker’s payoff). For mutual cooperation \[defection\] each player receives the same payoff $r$ (reward for mutual cooperation) \[$p$ (punishment)\]. The game is symmetric in the sense that player’s income is independent of the player itself, it depends only on their choice. The mentioned payoff values satisfy the inequalities $t > r > p > s$ and $2r > t+s$. These assumptions provide the largest total payoff for the mutual cooperators. Comparing to this situation the defector reaches extra income against the cooperator whose loss exceeds the defector’s benefit. Consequently, the choice of defection can be interpreted as an exploiting behavior. These are the main features for which the prisoner’s dilemma games are used to study the emergence of mutual cooperation, altruism and ethic norms among selfish individuals [@axelrod; @NMS].
The rational players should defect as this choice provides the larger income, independently of the partner’s decision. However, this situation creates a dilemma for intelligent players as mutual cooperation would result in higher income for each of them than mutual defection does.
In the iterated round-robin prisoner’s dilemma games we can introduce some simple evolutionary processes without assuming intelligent players (who are capable to find the best strategy if it exists). These games are started from an initial set of strategies, which defines the player’s decision in the knowledge of their previous choices. The evolutionary process is devoted to model the Darwinian selection principle among $n$ (selfish) players whose total income comes from $n-1$ games within a given round. In the simplest evolutionary models the worst player adopts the winner’s strategy round by round.
The numerical simulations have demonstrated the crucial role of the so-called “tit for tat” strategy in the emergence of mutual cooperation. Despite of its simplicity the “tit for tat” strategy won the computer tournaments conducted by Axelrod [@axelrod]. The “tit for tat” strategy cooperates in the first step and then always repeats his co-player’s previous decision. This strategy cooperates forever with all the other so-called nice strategies which never defect first. Furthermore, its defection and cooperation can be interpreted as a punishment and forgiveness when reacting to the previous decision for other strategies. The most remarkable feature of this strategy is that it is capable to sustain the mutual cooperation among themselves in the presence of defectors.
Early numerical investigations have also indicated the importance of local interactions because it favors the formation of cooperating colonies. In the simplest models the players are distributed on a lattice and the interaction (the games between two player as well as the strategy adoption) is limited to a given neighborhood. Evidently, the short range interactions enhance the role of fluctuations at the same time. These models can be well investigated by sophisticated methods of non-equilibrium statistical physics.
For the numerical investigation of the spatial effects Nowak and May [@NM] have introduced an SEPDG model, which is equivalent to a two-state cellular automaton. Each lattice site can be in one of the two states $C$ and $D$, representing the two simple strategy “always cooperate” and “always defect” respectively. The income for a given player (site) comes from games with its neighbors (and also with itself in some version of the model). According to the cellular automaton rule the players modify their strategy simultaneously in discrete time steps. Namely, each player adopts the best strategy found in its neighborhood. The step by step visualization of the strategy distribution on a two-dimensional lattice exhibits different spatio-temporal patterns (homogeneous and coexisting strategies, transitions between these states, competing interfacial invasions, etc.) depending on the payoff matrix, which is characterized by a single parameter. In these models the randomness is restricted to the initial states. In a subsequent work Nowak [*et al.*]{} [@NBM] have extended the former models by allowing irrational strategy adoptions with some probability. The simulations indicated that the randomness favors the spreading of $D$ strategies. These results have initiated systematic numerical investigations of many stochastic cellular automata [@KD96; @KD98; @KDK; @CO].
The study of spatio-temporal patterns observed in nature, however, requires continuous time description [@HG; @NM]. Moreover, it is difficult to analyze the above mentioned stochastic cellular automata in the framework of generalized mean-field approximation, which is often used in non-equilibrium physics. To reduce the technical difficulties Szabó and Tőke have suggested a simplified dynamics [@epdg2s]. The systematic investigations of this model have justified that when tuning the model parameters the stationary state undergoes two consecutive phase transitions which belong to the directed percolation (DP) universality class[@epdg2s; @CO]. Very recently this SEPDG model has been extended by allowing three strategies for the players [@epdg3s]. In the present work this three-strategy model will be compared with its simplified version. During the model descriptions and discussion, our attention will be focused on the motivations, the elementary processes and their consequences as well as on the universal features relating the SEPDGs to the area of complex systems.
Spatial evolutionary model with three strategies {#spatial-evolutionary-model-with-three-strategies .unnumbered}
================================================
In the present spatial evolutionary prisoner’s dilemma game the players are located on the sites ${\bf x}=(i,j)$ of a square lattice, where $i,j=1, \ldots , L$. To avoid the undesired boundary effects we assume periodic boundary conditions. Each player follows one of the three strategies: $D$ defects always; $C$ cooperates unconditionally; $T$ accommodating to the partner’s strategy chooses defection against $D$ and cooperation with $C$ and $T$. In fact the name $T$ refers to the strategy “tit for tat” which first cooperates and later repeats the partner’s previous decision. Consequently, after the first step the decisions of these two strategies are equivalent against $C$, $D$ and themselves. The consequences of the different first decisions become irrelevant if the strategy changes (defined below) are rare comparing to the frequency of games. At the site ${\bf x}$ the player’s strategy is denoted by a three-component unit vector whose possible values are $${\bf s}({\bf x})=\left( \matrix{1 \cr 0 \cr 0 \cr}\right)~,~~~
\left( \matrix{0 \cr 1 \cr 0 \cr}\right)~,~~~
\left( \matrix{0 \cr 0 \cr 1 \cr}\right)~$$ corresponding to the $D$, $C$, and $T$ strategies respectively. At a given time the state of the whole system is described by the variables ${\bf s}({\bf x})$.
For each player the total payoff comes from the games with its four nearest neighbors. Using the above formalism the total payoff $m({\bf x})$ for the player at site ${\bf x}$ is given as $$m({\bf x})=\sum_{\delta {\bf x}} {\bf s}^{\star}({\bf x}){\bf M}{\bf s}
({\bf x}+\delta{\bf x})$$ where ${\bf s}^{\star}({\bf x})$ is the transpose matrix of ${\bf s}({\bf x})$ and the summation runs over the four nearest neighbors ($\delta
{\bf x}$). Accepting the simplified payoff matrix suggested by Nowak and May [@NM] ${\bf M}$ is given by the following expression: $${\bf M}=\left( \matrix{0 & b & 0 \cr
0 & 1 & 1 \cr
0 & 1 & 1 \cr}\right)$$ where the only free parameter $b$ ($1<b<2$) measures the temptation to defect. In the above mentioned notation the present payoff matrix corresponds to the choices: $r=1$, $p=0$, $t=b$, and $s=-\varepsilon$ in the limit $\varepsilon \to 0$.
To model the Darwinian selection rule the players are allowed to modify their strategy. In the simplest case the system evolution is governed by random sequential updates. It means that a randomly chosen player (e.g. at site ${\bf x}$) adopts one of its neighboring strategy, ${\bf s}({\bf x}+\delta {\bf x})$, if $m({\bf x}+\delta {\bf x})
> m({\bf x})$ and this elementary process is iterated many times.
Here it is worth mentioning that a state consisting only of $C$ and $T$ strategies leads to a uniform payoff distribution \[$m({\bf x})=4$\] and the above dynamics leaves this state unchanged. An example of a more complicated situation is given in Figure \[fig:podistrn\]. The payoffs associated with the three different strategies, $D$, $C$ and $T$, are explicitly given.
The reader can easily check that inside a $D$ region the defectors receive zero payoffs. The same is true for a solitary $T$ surrounded by defectors. In the absence of $C$ strategies, however, two (or more) neighboring $T$ strategies will invade the $D$ territories because their mutual cooperation gives them some incomes, while the defectors’ payoff remain zero.
In the presence of $C$ strategies, however, the above situation becomes quite different as the exploitation provides large incomes for the defectors. As a result, the defectors can invade the neighboring $C$ or $T$ sites for some configurations. This process dominates the time evolution for small $T$ and large $C$ concentrations as illustrated in a ternary diagram (see Figure \[fig:flowp\]). Note that the trajectories are two dimensional projections of a many dimensional space. Accordingly, there can be crossing of trajectories. As the average defector’s payoff decreases with the $C$ concentration, sooner or later the $T \to D$ invasion processes will govern the system evolution and, finally, all the $D$ strategies extinct. Figure \[fig:flowp\] shows clearly that the ratio of $C$ and $T$ strategies in the final (frozen) state depends on the initial state.
It is emphasized that in the absence of $T$ strategies the defectors will dominate the present system in the final state. It is not evident as in Figure \[fig:podistrn\] one can find many $D$-$C$ pairs where $C$ beats $D$. In general, these pairs are located along the horizontal and vertical straight fronts separating the $D$ and $C$ domains. The random sequential invasions, however, makes the smooth fronts irregular and this situation generally prefers the $D \to C$ invasion to the opposite one. As a result, the “sharp” $D$ fronts cut the $C$’s domains into small pieces and finally all the $C$s will be eliminated.
The reader can easily recognize that in most of the $C$-$D$ (or $T$-$D$) confrontations the direction of dominance is not affected by the value of $b$ within the prescribed region ($1<b<2$). The systematic analysis shows that there is only one situation when the value of $b$ becomes important. Namely, if a defector has a payoff of $2b$ while its $C$ (or $T$) counterpart has 3. In this case, $D$ wins if $b>3/2$, otherwise $D$ will be invaded. These types of elementary processes, however, do not modify the system behavior drastically [@epdg3s], therefore the subsequent investigations will be focused on the case $b>3/2$.
The above dynamical rules introduce some noises (irrational choices) in the system evolution. Now an additional (superimposing) noisy term is introduced by allowing the appearance of mutants with probability $P$. In fact, the effect of two different external constraints (mutation mechanisms) will be studied in models A and B as a function of $P$.
Model A {#model-a .unnumbered}
-------
In the first model, the above evolutionary rule is modified as follows. Each randomly chosen player adopts with probability $P$ a randomly chosen strategy among the two other strategies. With probability $1-P$ it follows the old rule.
This model can describe the behavior of those biological and economical problems where the appearance of mutants cannot be neglected [@msmith; @HS]. The main feature of this model is that this mutation mechanism does not allow the extinction of any strategy.
Model B {#model-b .unnumbered}
-------
In the second model the mutation mechanism is restricted to the adoption of $C$ strategies [@epdg3s]. In other words, the randomly chosen player adopts the $C$ strategy with probability $P$, otherwise it adopts one of its neighboring strategy if this neighbor has higher income. Note that in this case the extinction of the $D$ and/or $T$ strategies is permitted.
Model B is devoted to describe the effect of an external constraint which enforces the cooperative behavior naively by supporting an unconditional cooperation. Such a phenomenon can be observed in human societies in which any kind of social pressure enforces the $D$ and $T$ players to choose the $C$ strategy. Furthermore, a $T$ player surrounded by only cooperating strategies ($C$ or $T$) is motivated to adopt the $C$ strategy also because of its convenience. In fact, playing $C$ is simpler than playing $T$, which requires the knowledge of the previous decision of your neighbors.
Mean-field approximation {#mean-field-approximation .unnumbered}
========================
In the classical mean-field approximation the system is described by the strategy concentrations which satisfy the normalization condition $c_D(t)+c_C(t)+c_T(t)=1$. In this approach the average payoffs are given as: $$\begin{aligned}
m_D&=&b c_C \ ,\nonumber \\
m_C&=&c_C+c_T \
\label{eq:mfpayoff} ,\\
m_T&=&c_C+c_T \ . \nonumber\end{aligned}$$ For model A, the time dependent concentrations satisfy the following equations of motion: $$\begin{aligned}
\dot{c}_D&=&{P \over 2}(c_C+c_T-2c_D) \mp (1-P) c_D(c_C+c_T) \ , \nonumber \\
\dot{c}_C&=&{P \over 2}(c_T+c_D-2c_C) \pm (1-P) c_D c_C \ ,
\label{eq:mfa} \\
\dot{c}_T&=&{P \over 2}(c_D+c_C-2c_T) \pm (1-P) c_D c_T \ , \nonumber\end{aligned}$$ where the upper (lower) signs are valid if $m_D < m_C=m_T$ ($m_D > m_C=m_T$). In these expressions the first terms describe the effect of external constraint, the second terms come from the Darwinian selection mechanism.
After some algebraic manipulations one can easily get the following stationary solution (for $P<1$): $$\begin{aligned}
c_D&=&{1+P/2-\sqrt{1-P+9P^2/4} \over 2(1-P)} \ ,\nonumber \\
c_C&=&c_T={1-c_D \over 2} \ \ .
\label{eq:MFsolA}\end{aligned}$$ Here all the three strategies are present for arbitrary values of $P$. Notice that the concentrations of $C$ and $T$ strategies are the same due to the symmetries of Eqs. (\[eq:mfa\]). In the limit $P \to 0$, however, the concentration of $D$ strategy vanishes. Evidently, the concentration of the three strategies becomes equal when the evolution is governed exclusively by the mutation ($P=1$).
For model B the corresponding equations of motion are similar to those given by Eqs. (\[eq:mfa\]), the differences appear in the first terms proportional to $P$. Namely, $$\begin{aligned}
\dot{c}_D&=&-Pc_D \mp (1-P) c_D(c_C+c_T) \ , \nonumber \\
\dot{c}_C&=&+P(c_T+c_D) \pm (1-P) c_D c_C \ ,
\label{eq:mfb} \\
\dot{c}_T&=&-P2c_T \pm (1-P) c_D c_T \ , \nonumber\end{aligned}$$ where the average payoff values are given by Eqs. (\[eq:mfpayoff\]) and the conditions of validity of the upper and lower signs are defined as above. The analytical solution of these equations predicts strikingly different behavior in the stationary state [@epdg3s], that is, for $0<P<1/2$ $$\begin{aligned}
c_D&=&{1-2P \over 1-P} \ ,\nonumber \\
c_C&=&{P \over 1-P}
\label{eq:MFsolB}\ , \\
c_T&=&0 \ ,\nonumber\end{aligned}$$ while the system goes to the absorbing state ($c_C=1$ and $c_D=c_T=0$) for $P>1/2$. The most surprising result is the extinction of $T$ strategy if $P > 0$.
We have to emphasize the non-analytical behavior in the limit $P \to 0$. As illustrated in the upper plot of Figure \[fig:flowzab\], without the mutation ($P=0$) the system evolves toward either a homogeneous $D$ state ($c_D=1$) or a mixed state composed of $C$ and $T$ strategies with a ratio depending on the initial conditions. However, the homogeneous $D$ state is unstable against $T$ invasions, therefore in the close vicinity of this state some small perturbations can drive the system toward the state of $C$+$T$. Conversely, this mixed state becomes unstable at a given concentration (where $m_D=m_C=m_T$) against small perturbations increasing $c_C$ and $c_D$ simultaneously. In other words, the system evolves toward the $D$ dominance when the state is positioned on the right hand side of dashed line (see the upper plot in Figure \[fig:flowzab\] as a results of fluctuations. This feature explains why the system is so sensitive to applied external constraints.
Figure \[fig:flowzab\] illustrates that for model A the mutation drives the (concentration) trajectories away from the boundaries. On the right hand side of the dashed line ($m_D>m_C=m_T$), $c_C$ and $m_D$ decrease while $c_D$ and $c_T$ increase until one crosses the dashed line. On the left hand side all the initial states tend toward the only fixed point given by Eq. (\[eq:MFsolA\]). For model B, however, there is no fixed point on the left hand side. In this region the external insertion of $C$ strategies increases the value of $c_C$ until $m_D$ becomes larger than $m_C=m_T$ and then the $D$ invasion drives the system toward the fixed point defined by Eq. (\[eq:MFsolB\]). During the $D$ invasion the external constraint can compensate only the loss of $C$ strategies. Consequently, the $T$ strategies die out exponentially fast.
Notice that the variation of $b$ leaves the fixed points unchanged, but modifies only the slope of the dashed line separating the two regions mentioned above.
Within the framework of mean-field theory, the extinction of $T$ strategies is a consequence of the fact that here $m_T=m_C$ \[see Eq. (\[eq:mfpayoff\])\] in contrary to the spatially extended case, as illustrated in Figure \[fig:podistrn\].
Monte Carlo simulations {#monte-carlo-simulations .unnumbered}
=======================
Systematic Monte Carlo simulations have been performed on a square lattice consisting of $L \times L$ sites with periodic boundary conditions, $L$ varying from 200 to 1500. The larger sizes were used in the vicinity of the critical points. Each run started from a random initial state. During the simulations we have monitored the number of players playing a given strategy ($N_{\alpha}$; $\alpha = D$, $C$ or $T$) and the payoffs related to a given strategy. After some relaxation time we have determined the average concentrations $$c_{\alpha} = \langle N_{\alpha}\rangle / L^2
\label{eq:c}$$ and the fluctuations $$\chi_{\alpha}=L^2 \langle (N_{\alpha}/L^2 - c_{\alpha})^2 \rangle
\label{eq:chi}$$ by averaging over a sampling time varying between $10^4$ and $10^6$ Monte Carlo steps (MCS) per sites. The results obtained respectively for model A and B are the following.
Results for Model A {#results-for-model-a .unnumbered}
-------------------
Figure \[fig:assp02\] shows a typical strategy distribution for the stationary state at a small value of $P$. In contrary to the mean-field prediction \[see Eqs. (\[eq:MFsolA\])\] the system is dominated by the $T$ strategies. The randomly inserted $D$ and $C$ strategies form small islands. Occasionally the larger $C$ islands are occupied by $D$s, however, a consecutive $T$ invasion will eliminate the larger $D$ territories and maintains the $T$ dominance. At the same time this process prevents the formation of large $C$ islands inside a $T$ domain.
One can observe in Figure \[fig:s3mmc\] that when increasing the value of $P$, the concentration of $D$ and $C$ strategies increases monotonously. In the limit $P \to 1$, the strategy distribution on the lattice tends toward a random (uncorrelated) one $c_D=c_C=c_T=1/3$ in agreement with the classical mean-field theory \[see Eq. (\[eq:MFsolA\])\]. In this case, instead of the neighbor invasions, the system evolution is ruled by the stochastic mutation mechanism.
As shown in Figure \[fig:s3mmc\], the Monte Carlo data agree remarkably well with the results of the pair approximation. This pair approximation is considered as a generalized mean-field theory taking the nearest-neighbor correlations explicitly into account. The details of this calculation are available in many previous works [@epdg2s; @epdg3s; @ST]. The good agreement refers to the absence of long-range correlations which is observable in the “homogeneous” strategy distribution (see Figure \[fig:assp02\]). It is worth mentioning that the pair approximation is capable to describe the dominance of $T$ strategies in the limit $P \to 0$.
Results for Model B {#results-for-model-b .unnumbered}
-------------------
In order to visualize the relevant differences between the two models at small $P$ values the strategy distribution for model B is displayed in Figure \[fig:bssp02\]. When comparing the corresponding snapshots (Figures \[fig:assp02\] and \[fig:bssp02\]) the reader can easily recognize the most striking differences. Namely, the appearance of a strongly correlated spatial structure for model B. In this case the formation of large $C$ domains inside the sea of $T$ strategies is not prevented by the random appearance of $D$ mutants as happened in the previous case. The large $C$ domains (white areas in Figure \[fig:bssp02\]), however, are unprotected against the $D$ invasion. Figure \[fig:bssp02\] shows some $D$ domains (black areas) invading the $C$’s territories. These $D$ domains are “strip-like” because their territories are invaded simultaneously by the $T$ strategies. This invasion process is similar also for larger values of $P$ (but $0<P<P_{c1}$, see later), and only the average invasion velocity changes. On the other hand, the randomly inserted $C$ strategies survive and accumulate in the $T$ domains. Consequently, far behind the $T$-$D$ invasion front the $T$’s territory will be occupied by the externally inserted $C$s and then this area becomes unprotected against the $D$ invasion. Sooner or later this area will be invaded by $D$s and the above process repeats itself. This means that the cyclic invasion maintains a self-organizing domain structure. Here we have to emphasize that this cyclic (rock-scissors-paper game like) dominance is provided by this external constraint.
Similar processes are observed in the forest-fire models [@GK; @DS] introduced by Bak et al. [@BCT] to model the phenomenon of self-organized criticality. In these models each cell can be in one of the following three states: non-burning tree, burning tree and ash. The dynamics are governed by cyclic dominance, similarly to our model B. Note that the consequence of cyclic invasion with three (or more) states are studied in Lotka-Volterra models [@Lotka; @Volterra; @AD; @ST] and in cyclically dominated voter models [@TI; @Tain93; @T94; @FKB; @SSM].
In model B the transition from the $T$ to $C$ state introduces a characteristic length and time unit, both proportional to $1/P$. In other words, this length unit is characteristic to the typical (linear) size of the $T$+$C$ domain, and the time unit corresponds to periodic time of cyclic invasion processes at a given site.
When increasing the value of $P$, the typical size of $T$+$C$ domains decreases and the concentration of $D$’s increases. It is found that the $T$ strategies die out if $P>P_{c1}=0.1329(1)$. Figure \[fig:bssp13\] shows a typical snapshot in the vicinity of this critical value. In this case the external support is sufficiently strong to maintain small $C$ clusters inside the $D$ domains. The most remarkable feature of this snapshot is that the $T$’s form non-uniformly distributed small (isolated) colonies. The observation of time evolution of configuration shows that these $T$ colonies walk randomly, they can extinct spontaneously, a single colony can split into two, or two colonies can merge. This phenomenon is analogous to the branching annihilating random walks (BARW) exhibiting a critical transition when varying the control parameters [@CT96; @CT98]. The corresponding critical transitions, both for our model B and for BARW, belongs to the so-called directed percolation universality class [@Janssen; @Grassb].
For $P > P_{c1}$, the concentration of $D$ decreases monotonously if $P$ is increased and vanishes at $P=P_{c2}=0.3678(1)$. This extinction process is similar to the previous one, i.e. it also belongs to the DP universality class. The similarity in the correlations is recognizable in the spatial distribution of the extincting strategies when comparing the snapshots displayed in Figures \[fig:bssp13\] and \[fig:bssp366\].
For $P > P_{c2}$, any initial state evolves toward the absorbing state where all the players follow the $C$ strategy.
The results of our systematic investigations are summarized in Figure \[fig:s3mc\]. Systematic numerical investigations in the close vicinity of the critical points show that the vanishing concentrations follow the same power law behavior. Namely, $$\begin{aligned}
c_T&=&(P_{c1}-P)^{\beta} \ , \nonumber \\
c_D&=&(P_{c2}-P)^{\beta} \ ,
\label{eq:beta}\end{aligned}$$ in the limits $P_{c1}-P \to 0$ and $P_{c2}-P \to 0$ respectively and $\beta = 0.57(3)$ in both cases [@epdg3s]. Within the statistical error this value of the exponent $\beta$ agrees with the one of the 2+1 dimensional directed percolation [@BFM; @JFD].
As expected, these critical transitions are accompanied with the divergence of concentration fluctuations, i.e. $$\begin{aligned}
\chi_T&=&(P_{c1}-P)^{-\gamma} \ , \nonumber \\
\chi_D&=&(P_{c2}-P)^{-\gamma} \ ,
\label{eq:gamma}\end{aligned}$$ in the vicinity of the corresponding critical points. The numerical fitting yields $\gamma =0.37(9)$ in agreement with the DP values [@BFM; @JFD; @Janssen; @Grassb].
Despite the same universal behavior there is a remarkable difference between the two extinction processes. The second extinction process (at $P=P_{c2}$) results in a frozen (time independent) absorbing state. Conversely, the transition at $P=P_{c1}$ is an example where the extinction of $T$ strategies happens on a fluctuating background. In other words, the properties of the absorbing state (frozen or fluctuating) do not affect the critical behavior of our model.
As demonstrated in Figure \[fig:s3mc\] the results of Monte Carlo simulations are reproduced qualitatively well by the pair approximation [@epdg3s]. The striking differences are related to the long-range correlations accompanying the critical transitions at $P=P_{c1}$ and $P_{c2}$. Due to the strongly correlated domain structure, illustrated in Figure \[fig:bssp02\], the largest deviation can be observed for small $P$ values. We note that the concentration fluctuations, defined by Eq. (\[eq:chi\]), also diverge in the limit $P \to 0$. Unfortunately, in this particular case, we could not deduce a reliable value for the exponent $\gamma$ because of the significant size effects. Further systematic analyzes are required to clarify what happens in this limit.
Conclusions {#conclusions .unnumbered}
===========
We have studied quantitatively the effect of external constraints on the emergence of cooperation in an evolutionary prisoner’s dilemma game with three possible strategies (cooperation, defection and tit for tat). In the present spatial model the players are distributed on a square lattice and their interactions are restricted to nearest neighbors. The Darwinian selection rule is modeled by the adoption of the neighboring successful strategies. This evolutionary process is superimposed by two types of mutation mechanisms (external constraints) whose strength is characterized by a control parameter $P$.
The choice of these three possible strategies yields non-analytical behavior in the limit $P \to 0$ for both the mean-field approximation and Monte Carlo simulation. The time-dependent predictions of mean-field theory are sensitive to the small perturbations.
According to the Monte Carlo simulations, in the absence of external constraint the system tends toward a frozen state composed from $C$ and $T$ strategies whose ratio depends on the initial concentrations. For both types of external constraints (models A and B) the system evolves toward a stationary state independently of the initial condition, and the defector concentration vanishes linearly as $P\to 0$. In the limit $P \to 0$, however, model A and B will exhibit different ratio of $C$ and $T$ strategies. This difference is related to the appearance of self-organizing patterns for model B. The present investigation indicates that such a society of strategies (or species) are very sensitive to the type of external supports (or the ration of mutation rates).
The measure of mutual cooperation can be well characterized by the average payoff whose maximum (4) can be reached only in the absence of defectors. Figure \[fig:apayoff\] compares the Monte Carlo results for the models A and B. Surprisingly, for weak external support (small $P$) the average payoff is larger for model A than for model B. In contrary to the naive expectation, the weak support of defenseless cooperators results in opposite consequence. Namely, this mechanism feeds the defectors and simultaneously prevents their elimination by the retaliatory ($T$) strategies.
Examples from the political and economical world justifies the above conclusions. In general, the exploiters are preferred by the governmental support for the defenseless layer of a society. The most dangerous effect is the reduction in the $T$ type population which can maintain the mutual cooperation against the exploiters. From the view point of cooperation, it is better to help those individuals who are able to prevent themselves against the exploitation.
Evidently, for sufficiently large $P$ values the random insertion of $C$ strategies can provide their dominance. In this case the A type external support is preferred to the B one if we wish to improve cooperation. Above a threshold value this type of external constraints yields a homogeneous $C$ state which is defenseless against any defector appearing occasionally in a real system. Further systematic research is required to clarify what happens in those models where the mutation mechanism is characterized by three independent control parameters.
The present study confirms that the $T$ strategy is able to prevent the spreading of defection in the spatial models. We have to emphasize, however, that according to the simplest mean-field theory, $T$ dies out if the external support is of B type. Consequently, the defectors will dominate those systems where the mean-field theory is exact (e.g. infinite range of interaction, or randomly chosen partnership). In these mean-field like systems, the games between the “parent“ and “its offspring” is not emphasized (they are not neighbors), which is an advantage for the defectors comparing to spatially extended models. In the light of this feature our investigations imply many interesting questions related to the transition from the “short range” spatially extended systems to the “long range” of mean-field like ones.
#### Acknowledgments. {#acknowledgments. .unnumbered}
This work was supported by the Hungarian National Research Fund under Grant No. T-23552 and by the Swiss National Foundation.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We construct a suite of discrete chemo-dynamical models of the giant elliptical galaxy NGC 5846. These models are a powerful tool to constrain both the mass distribution and internal dynamics of multiple tracer populations. We use Jeans models to simultaneously fit stellar kinematics within the effective radius $R_{\rm e}$, planetary nebula (PN) radial velocities out to $3\, R_{\rm e}$, and globular cluster (GC) radial velocities and colours out to $6\,R_{\rm e}$. The best-fitting model is a cored DM halo which contributes $\sim 10\%$ of the total mass within $1\,R_{\rm e}$, and $67\% \pm 10\%$ within $6\,R_{\rm e}$, although a cusped DM halo is also acceptable. The red GCs exhibit mild rotation with $v_{\rm max}/\sigma_0 \sim 0.3$ in the region $R > \,R_{\rm e}$, aligned with but counter-rotating to the stars in the inner parts, while the blue GCs and PNe kinematics are consistent with no rotation. The red GCs are tangentially anisotropic, the blue GCs are mildly radially anisotropic, and the PNe vary from radially to tangentially anisotropic from the inner to the outer region. This is confirmed by general made-to-measure models. The tangential anisotropy of the red GCs in the inner regions could stem from the preferential destruction of red GCs on more radial orbits, while their outer tangential anisotropy – similar to the PNe in this region – has no good explanation. The mild radial anisotropy of the blue GCs is consistent with an accretion scenario.'
author:
- 'Ling Zhu$^1$[^1], Aaron J. Romanowsky$^{2,3}$, Glenn van de Ven$^1$, R. J. Long$^{4,5}$,'
- 'Laura L. Watkins$^6$, Vincenzo Pota$^7$, Nicola R. Napolitano$^7$, Duncan A. Forbes$^8$,'
- |
Jean Brodie$^9$, Caroline Foster$^{10}$,\
$^1$Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany\
$^2$Department of Physics & Astronomy, San José State University, One Washington Square, San Jose, CA 95192, USA\
$^3$University of California Observatories, 1156 High Street, Santa Cruz, CA 95064, USA\
$^4$National Astronomical Observatories, Chinese Academy of Sciences, A 20 Datun Rd, Chaoyang District, Beijing 100012, China\
$^5$Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester M13 9PL, UK\
$^6$Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA\
$^7$INAF - Osservatorio Astronomico di Capodimonte, Salita Moiariello, 16, I-80131 Napoli, Italy\
$^8$Centre for Astrophysics & Supercomputing, Swinburne University, Hawthorn, VIC 3122, Australia\
$^9$University of California Observatories, 1156 High Street, Santa Cruz, CA 95064, USA\
$^{10}$Australian Astronomical Observatory, PO Box 915, North Ryde, NSW 1670, Australia\
bibliography:
- 'ngc5846.bib'
date: 'Accepted 0000 Month 00. Received 0000 Month 00; in original 0000 Month 00'
title: 'A discrete chemo-dynamical model of the giant elliptical galaxy NGC 5846: dark matter fraction, internal rotation and velocity anisotropy out to six effective radii'
---
\[firstpage\]
Galaxy: individual: NGC 5846 – Globular Cluster: kinematics– model: chemo-dynamical
Introduction {#S:intro}
============
Globular clusters (GCs) and planetary nebulae (PNe) are two powerful tracers of the dark matter distributions in giant elliptical galaxies. At the same time, they are important tracers of the outer halo as a fossil record of a galaxy’s formation history.
How the outer haloes of giant ellipticals are assembled is still under debate. Numerical simulations of these galaxies have been carried out in a cosmological context [@Cooper2013; @Wu2014; @Rottgers2014; @Pillepich2014], and it has been found that the stellar mass surface density profiles show an upward break at the radius where stars accreted from previously distinct galaxies ([*ex situ*]{}) start to dominate over the stars formed in the galaxy itself ([*in situ*]{}). The break disappears in the most massive galaxies, where [*ex situ*]{} stars dominate at all radii. The motions of [*ex situ*]{} stars are typically radially biased, while [*in situ*]{} stars can become tangentially biased if dissipation was significant during the later stages of assembly of the galaxy. Thus, in simulations, more massive galaxies with a large fraction of [*ex situ*]{} stars have a radially anisotropic velocity distribution outside of one effective radius. Tangential anisotropy is seen only for galaxies with high fractions of [*in situ*]{} stars, and is particularly rare in “slow rotator” ellipticals [@Wu2014].
When comparing galaxies, or comparing observations and simulations, different tracers are often used and their kinematics are not necessarily the same. Elliptical galaxies usually have more than one population of GCs–metal rich (red) and metal poor (blue) (e.g. @Brodie2006; @Brodie2012)–that exhibit different properties. The red GCs usually have a density distribution and velocity dispersion profile that follows the galaxy’s field stars, while the blue GCs extend further spatially and have a higher velocity dispersion. The velocity anisotropy of the blue GCs is also different from the red GCs and stellar tracers [@Zhang2015; @Pota2015b]. These different properties of the red and blue GCs indicate different formation scenarios. The expectation is that the red GCs are predominantly born [*in situ*]{}. The blue GCs are thought to be predominantly accreted from low mass galaxies. This accretion is evidenced by the known correlations between the average metallicity of a GC population and its host galaxy’s mass [@Peng2006]. However, after the dissolution of some GCs by tidal forces, even the red GCs may obtain kinematic properties different from the galaxy’s main stellar population. With increasing data volumes, recent analyses have examined the kinematics of each population of GCs separately (e.g. @Schuberth2010 [@Pota2013; @Pota2015a; @Zhang2015]).
Modelling the kinematics of all tracers simultaneously, with distinct dynamical properties for each population, is a powerful technique for addressing degeneracies: tracers with different velocity anisotropies may help to alleviate the mass–anisotropy degeneracy, while kinematic constraints on different spatial scales can alleviate the luminous and dark matter mass degeneracy [@Napolitano2014; @Agnello2014; @Pota2015b]. However, several key aspects of this multi-population approach could be improved. Models are typically restricted to spherical symmetry, and their fitting processes do not function with unbinned velocities and, so, do not maximize the constraining power of the data (see @Romanowsky2001 [@Bergond2006; @Wu2006] for single-population examples). GC subpopulations are separated through their distribution of colour or metallicity before dynamical modelling occurs. In this context, the discrete chemo-dynamical modelling technique that we introduced in Zhu et al., 2016 (submitted) is a powerful tool to separate multiple populations that overlap chemically, to investigate their dynamical properties and to constrain the underlying gravitational potential simultaneously. In this paper, we extend our axisymmetric Jeans modelling technique to include three dynamical populations (the stars/PNe and two GC subpopulations) and, in addition to the discrete data, also fit the integrated-light stellar kinematic data in the inner region. The only comparable work from the literature is the @Oldham2016 spherical Jeans models of M87, using stars, two GC subpopulations and satellite galaxies.
The paper is organised as follows: in Section \[S:obs\] we present the observational data; in Section \[method\], we describe our model algorithms; in Section \[S:modelfits\], we present the results of modelling the kinematics; in Section \[S:kin\], we present the internal rotation and velocity anisotropy profiles obtained by the chemo-dynamical Jeans models as well as those from the Made-to-Measure (M2M) models for each population. In Section \[S:mass\], we show the mass profiles. We discuss the implications of the results in Section \[S:discussion\] and summarise in Section \[S:conclusions\]. In the Appendix, we show kinematic figures for different models and describe in more detail the M2M models we employed.
Data {#S:obs}
====
NGC 5846 is a giant elliptical that is the brightest galaxy at the centre of a group at a distance of 24.2 Mpc away (@Tonry2001, corrected by subtracting 0.06 mag from the distance modulus as suggested by @Mei2007), so that $1'' = 117\,\mathrm{pc}$. It appears nearly spherical in shape with Hubble type E0 and is kinematically classified as a centrally-slow rotator [@Emsellem2011]. The effective radius of the galaxy is $R_{\rm e} = 81'' \simeq 9.5 \, \mathrm{kpc}$. We adopt a major-axis position angle of $60^\circ$, based on the stellar isophotes at large radii [@Kronawitter2000].
Throughout, $(x, y, z)$ indicate the coordinates of the de-projected three-dimensional (3D) system; $r$ indicates the spherical shell radius. We place the projected major and minor axes of the galaxy along $(x', y')$, while $z'$ is along the line of sight; $R$ indicates the projected radius $R = \sqrt{x'^2 + y'^2}$. We also use the projected semi-major elliptical annular radius defined as $$\label{eqn:RR}
R' \equiv {\rm sign}(x') \times \sqrt{x'^2 + (y'/q)^2},$$ with an average flattening $q = 0.85$ (axis ratio between minor and major axis $b/a$) adopted for all populations.
Projected density profiles {#SS:sd}
--------------------------
![Density profiles for NGC 5846. [**Top panel:**]{} The stellar surface brightness and the GC surface number density profiles along the major axis. The black diamonds and solid curve are the observed data and MGE fit to the stellar surface brightness, respectively. The red plus-symbols with error bars and dash-dotted curves are those for the red GCs, the blue asterisks with error bars and dashed curves are those for the blue GCs. The stellar data have been arbitrarily normalized for comparison with the GCs. [**Bottom panel:**]{} The axis ratio $q$ profiles. The black solid curve represents the stars. For the GCs, we adopt constant values of $q=0.83$ for the red GCs and 0.88 for the blue GCs, as indicated by the horizontal red dash-dotted and blue dashed lines. The vertical dashed line indicates the position of $3\,R_{\rm e}$ for the galaxy.[]{data-label="fig:sdsb"}](figure/sbsd_2d){width="\hsize"}
Photometric observations provide the stellar surface brightness and the projected number density profiles for GCs, as shown in Fig. \[fig:sdsb\]. We take the observed $V$-band stellar surface brightness profile along the projected major axis of the galaxy from @Kronawitter2000 (black diamonds), and we fit the surface brightness profile with a multi-Gaussian expansion (MGE; @Cappellari2002; @Emsellem1994). Eight Gaussian components are used for the best-fitting MGE to the surface brightness (black solid lines). This surface brightness profile has been normalized for comparison purposes. The surface number density profile of the PNe (not shown in the figure) is almost identical with the stellar surface brightness profile [@Coccato2009].
The surface number density profiles of red GCs (red pluses) and blue GCs (blue asterisks) are calculated from the GC photometric data. The photometric data were derived from [*Hubble Space Telescope*]{} ([*HST*]{})/Wide-Field Planetary Camera 2 and Subaru/Suprime-Cam images, as described in @Napolitano2014, but updated slightly here for ellipticity effects. Briefly, the GC density profiles were computed for the [*HST*]{} and Subaru datasets separately and combined post-facto. The two datasets have different magnitude normalizations. Therefore, we first selected GCs 0.5 mag brighter than the turn-over-magnitude to eliminate this normalization difference. Then we divided the GCs into blue GCs and red GCs using $(g-i) < 0.90$ and $(g-i) > 0.95$ for the Subaru data sets, and using $(V - I ) < 1.05$ and $(V-I) > 1.09$ for the [*HST*]{} datasets. Note that the separation on $(g-i)$ is only used to determine the projected number density profiles to be used as model inputs.
The surface number density profiles of red and blue GCs are described well by an MGE plus a constant background component, shown as red dash-dotted and blue dashed lines in Fig. \[fig:sdsb\], respectively. The constant background term is 0.25 for the blue GCs and 0.167 for the red GCs, which will be subtracted from the profiles before the modeling.
The projected density profiles are not spherical but spheroidal. In the bottom panel of Fig. \[fig:sdsb\], we show the projected flattening profiles along the major axis for the three populations. The black solid line represents that of the stellar surface brightness, with values varying from $q \sim 0.9$ in the center to $\sim 0.7$ in the outer regions. We adopt a constant flattening for the red and blue GCs: $q=0.83$ for red GCs (red dash-dotted line) and 0.88 for the blue GCs (blue dashed line), which are obtained from all the GC photometric data. The spectroscopic data we will use for dynamical constraints are far from complete and have complicated selection functions, leaving little ability to constrain the density or brightness profiles of the tracer populations. These profiles–determined beforehand from photometric data (not the spectroscopic data), and converted into MGE form–will provide input to our modelling process.
Kinematic data {#SS:kin}
--------------
{width="\hsize"}
For NGC 5846, we have 123 PNe with LOS velocity measurements extending to $\sim 37 \, \mathrm{kpc} \simeq 4\,R_{\rm e}$ from the Planetary Nebulae Spectrograph Survey [@Coccato2009], and 214 GCs with line-of-sight (LOS) velocity measurements extending to $\sim 62\, \mathrm{kpc} \simeq 6.5\, R_{\rm e}$ from the SAGES Legacy Unifying Globulars and GalaxieS (SLUGGS) Survey [@Brodie2014][^2]. The GCs are mostly the same data as described in @Napolitano2014, with the addition of $\sim$ 20 new, preliminary velocities from an observing run with Keck/DEIMOS on 2012-04-17 (to be discussed in full in D. Forbes et al., in preparation).
We also have integrated-light stellar kinematic measurements at 80 discrete positions extending to $\sim 1\, R_{\rm e}$, obtained by the SLUGGS survey. These measurements provide at each point the mean velocity and velocity dispersion which we will use for modelling, as well as the Gauss-Hermite coefficients $h_3$ and $h_4$. In our modelling, we discard the measurements that are around the satellite galaxy NGC 5846A toward the South.
The SAURON integral-field unit (IFU) data for NGC 5846 [@Emsellem2011], extending to $\sim 0.5\,R_{\rm e}$, are also included in our model. The stellar velocity dispersions from the SAURON data are systematically higher than in SLUGGS by $\sim$ 10 [kms$^{-1}$]{} – a known issue found in comparing data from these two surveys (@Foster2016; @Boardman2016). Here we add 10 [kms$^{-1}$]{} to the SLUGGS stellar velocity dispersions. We check the effects on our models by alternatively subtracting 10 [kms$^{-1}$]{} from the SAURON data (see Section \[S:mass\]).
The kinematic data in the projected plane are shown in Fig. \[fig:xyplot\]. The black pluses are the positions of the GCs, the green diamonds are the PNe, the purple asterisks are the positions of stellar kinematics measurements from SLUGGS and the area filled with orange asterisks is the SAURON data. The three dashed circles indicate scale radii of $1\,R_{\rm e}$, $3\,R_{\rm e}$ and $6\,R_{\rm e}$ for the galaxy.
Discrete chemo-dynamical models {#method}
===============================
GCs themselves contribute very little to the total stellar luminosity of the galaxy, which comes predominately from the stars in the central galactic structure. PNe simply mark dead low-mass stars, are thus assumed to follow the distribution of the main stellar population. The stellar tracers (PNe + central galaxy stars), the red GCs and the blue GCs are three independent populations that trace the underlying galactic potential.
We use the discrete chemo-dynamical modelling technique developed in Zhu et al., 2016 (submitted), and refer to this paper for the details of the technique. The dynamical properties of each population are described by an axisymmetric Jeans model with free velocity anisotropy and rotation parameters [@Cappellari2008]. The discrete data are modelled directly without spatial binning [@Watkins2013], and the red and blue GCs are not separated via the colour distribution before the modelling. In this paper, we extend the method to also include integrated-light stellar kinematics.
Gravitational potential {#SS:potential}
-----------------------
The gravitational potential is modelled by a combination of the stellar and dark matter mass distributions. We de-project the 2D MGE fit to the galaxy’s surface brightness in $V$ band to get the 3D luminosity density of the galaxy. After assuming a constant stellar mass-to-light ratio $\Upsilon_V$, we obtain the 3D stellar mass density that generates the stellar contribution to the potential. When de-projecting the 2D MGE of the image to a 3D MGE for the luminosity, we assume that the line-of-sight inclination angle of the galaxy is $90^\circ$ (edge-on). The inclination angle is fixed in our model because, especially for a slowly rotating galaxy, it is poorly constrained by kinematic data [@vdB2009].
For the dark matter contribution, we adopt a generalized @NFW1996 (NFW) density distribution as in @Zhao1996 $$\rho(r) = \frac{\rho_{s} }{ (r/r_s)^{\gamma}(1 + (r/r_s)^{\eta})^{(3-\gamma)/\eta} },
\label{eq:densgNFW}$$ with $r^2 = x^2 + y^2 + z^2/p_z^2$ in the case of an oblate axisymmetric distribution. Since the flattening of the dark matter halo is, to a large degree, degenerate with its radial profile, line-of-sight data alone are expected to provide weak constraints if both the flattening and the radial profile of DM are left free. Numerical simulations show that dark matter halos are usually rounder than luminous matter [@Wu2014], which, in the case of a massive elliptical galaxy, is not far from spherical, and hence we adopt a spherical DM halo with $p_z = 1$. There are four free halo parameters: the scale radius $r_s$, the scale density $\rho_s$, and the inner and outer density slopes $\gamma$ and $\eta$. When ($\eta= 1,\, \gamma =1$), the halo reduces to a classic cusped profile [@NFW1996], while, for ($\eta =2, \,\gamma =0$), there is a core in the centre. We expand the density $\rho$ as an MGE to simplify various calculations such as the computation of the gravitational potential [@Emsellem1994] and the solution of the axisymmetric Jeans equation [@Cappellari2008]. The total density we use to generate the potential is just the combination of the 3D MGE of stellar and DM density. The central black hole is ignored, a $\sim 10^8 \,M_{\odot}$ central black hole has an influence radius less than $1''$, which won’t affect our results.
Tracer probabilities
--------------------
Consider a dataset of $N$ measured points such that the $i^{\rm th}$ point has sky coordinates $(x'_i, y'_i)$ and line-of-sight velocity $v_{z',i} \pm \delta{v_{z',i}}$ with measured metallicity $Z_i\pm\delta{Z_i}$.
Multiple populations are permitted in the model, each with its own chemical, spatial and dynamical distributions. For each population $k$, we adopt a Gaussian metallicity distribution (see @Forte2007 for an alternative distribution) with mean metallicity $Z_0^k$ and metallicity dispersion $\sigma_Z^{k} $ with both the mean and dispersion being taken as free parameters. Here we do not use direct observations of metallicities (cf. @Usher2012) but instead adopt $g-i$ colour as a proxy for metallicity. Note that the exact equivalence between colour and metallicity does not matter in this context; we are effectively modelling a two-component population of GC colours, which could in principle be re-interpreted according to some other stellar population parameters.
The metallicity probability of point $i$ in population $k$ is $$\label{eq:pchm}
P_{\mathrm{chm},i}^k = \frac{1}{\sqrt{2\pi[(\sigma_{Z}^k)^2+\delta{Z_i}^2]}}
\exp\left[ -\frac12\frac{(Z_i-Z_0^k)^2}{[(\sigma_Z^k)^2 + \delta{Z_i}^2]}\right].$$
Each population has its own spatial distribution through its observed projected number density $\Sigma^k(x',y')$, and so the spatial probability of point $i$ in population $k$ is $$\label{eq:pspa}
P_{\mathrm{spa},i}^k = \frac{\Sigma^k(x'_i,y'_i)}{\Sigma_\mathrm{obj}(x'_i,y'_i)},$$ where $\Sigma_\mathrm{obj} = \Sigma^1 + \Sigma^2 + \dots$ is the combined density of all populations that belong to the object under consideration (for example, a galaxy).
The dynamical properties of each population are described by an axisymmetric Jeans model with a specific density profile, velocity anisotropy and rotation parameter. This model assumes that the velocity ellipsoid is aligned with the cylindrical coordinates and the velocity anisotropy in the meridional plane is constant. The dynamical probability of point $i$ in population $k$ for an assumed Gaussian velocity distribution is $$\label{eq:pdyn}
P_{\mathrm{dyn},i}^k = \frac{1}{\sqrt{(\sigma^k_i)^2 + (\delta v_{z',i})^2}}
\exp\left[ -\frac12\frac{(v_{z', i} - \mu^k_{i})^2 }{(\sigma^k_i)^2 + (\delta v_{z',i})^2}\right],$$ where $\mu_i^k$ and $\sigma_i^k$ are the line-of-sight mean velocity and velocity dispersion as predicted by a dynamical model at the sky position $(x'_i,y'_i)$. To investigate the velocity anisotropy and the rotation properties, we include two free parameters for each population $k$ in an axisymmetric Jeans model (in a fixed potential): the constant velocity anisotropy in the meridional plane $\beta_z^{k}$ ($= 1 - \overline{v_z^2} / \overline{v_R^2}$) and the rotation parameter $\kappa^{k}$ ($= [\overline{v_{\phi}}] / ([\overline{v_{\phi}^2}] - [\overline{v_{R}^2}] )^{1/2}$) [@Cappellari2008].
In principle, we could let the density profiles of all populations be free within a model, i.e., the density profiles could be constrained by the actual spatial distribution of the tracers if we have sufficient tracers to sample adequately the true distributions. However, the kinematic data are usually far from complete, with complicated selection functions, and thus provide almost no constraints on the population density profiles. In practice, the projected density profiles are taken from photometric data, with different populations being separated by metallicity (as shown in Fig. \[fig:sdsb\]) and used as model inputs.
### Discrete GCs and PNe
With discrete GC and PN data, the model can be considered as a generalization of the two-component discrete chemo-dynamical model (Zhu et al., 2016, submitted) to include three populations. However, the stellar tracers (PNe) and GCs are independent tracers from each other, hence the likelihood is $$L_{i \in {\rm GC}} = \sum_{k={\rm GC}} P_{\mathrm{chm},i}^k \,P_{\mathrm{spa},i}^k \, P_{\mathrm{dyn},i}^k$$ where point $i$ represents a GC, and the other subscripts indicate chemical, spatial or dynamical probability. Once the best-fitting parameters in a model have been obtained, the likelihood of a GC $i$ belonging to each GC population $k$ is $$P_i^k = P_{{\rm spa},i}^k P_{{\rm chm},i}^k P_{{\rm dyn},i}^k,$$ where $k$ can be either a red GC or a blue GC. The relative probability $P_i^{'k}$, defined as $$\label{eqn:Pik}
P_i^{'k} = P_i^k / \sum^{k={\rm GC}} P_i^k,$$ can be used to identify a GC as being red or blue by utilising the end of run model.
When the point $i$ is a PN, the likelihood is $$L_{i \in {\rm PN}} =P_{\mathrm{dyn},i}^{\mathrm{star}},$$ which is just the probability of an independent single component Jeans model. Both $L_{i \in {\rm GC}}$ and $L_{i \in {\rm PN}}$ are evaluated in the same gravitational potential.
The total log likelihood of the discrete data is $$\begin{gathered}
\label{eqn:LogL}
\mathcal{L} = \sum_{i=1}^{N_{\rm GC}} \log{L_{i \in {\rm GC}}} + \sum_{i=1}^{N_{\rm PN}} \log{L_{i \in {\rm PN}}}
\equiv \mathcal{L}_{\rm GC} + \mathcal{L}_{\rm PN}.\end{gathered}$$
### Integrated-light stellar kinematics {#SS:combine}
The SAURON IFU data and the SLUGGS stellar kinematics provide the mean velocity and velocity dispersion $ (\mu_j^{\rm s}\pm \delta \mu_{j}^{\rm s} , \sigma_j^{\rm s}\pm \delta \sigma_{j}^{\rm s} )$ of the stars across the projected plane of the galaxy in bins with centroids ($x'_j, y'_j$).
We directly compare the mean velocity and velocity dispersion of the data with that predicted by the stellar tracer Jeans model, resulting in $$\chi^2_{\rm star} = \sum_{j=1}^{N_{\rm s}} { (\frac{\mu_j^{\rm s}-\mu_{j}^{\rm star}}{\delta \mu_{j}^{\rm s} })^2} + \sum_{j=1}^{N_{\rm s}} { (\frac{\sigma_j^{\rm s}-\sigma_{j}^{\rm star}}{\delta \sigma_{j}^{\rm s} })^2} ,$$ where $N_{\rm s}$ is the number of bins in the integrated-light stellar kinematic data, and $(\mu_{j}^{\rm star}, \sigma_{j}^{\rm star})$ are the model predicted mean velocity and velocity dispersion for the stellar tracer – which we also used for the PNe.
The minimum of $\chi^2_{\rm star}$ and the maximum of the log likelihood for the discrete stars can be determined at the same time by maximising the combined likelihood defined as $$\label{eqn:Ltot}
\mathcal{L}_{\rm tot} = \mathcal{L} - \frac{1}{2} \alpha_{\rm s} \chi^2_{\rm star}
\\
= \mathcal{L}_{\rm GC} + \mathcal{L}_{\rm PN} - \frac{1}{2} \alpha_s \chi^2_{\rm star},$$ where $\alpha_{\rm s}$ is a weight parameter which will be adapted manually to balance the relative influence of the integrated stellar kinematic data and the GC and PN discrete data (see Section \[SS:modelparam\]).
Model parameters and optimization {#SS:modelparam}
---------------------------------
Using the prescriptions laid out in Section \[SS:potential\], there are three free parameters in the gravitational potential:
- $\Upsilon_V$, the $V$-band stellar mass-to-light ratio;
- $\rho_s$, DM scale density;
- $d_s \equiv \log (\rho_s^2 r_s^3)$ is a proxy for the scale radius $r_s$, reducing the strong degeneracy between $\rho_s$ and $r_s$.
We have two GC populations, red and blue, with different chemical, spatial and dynamical distributions. The projected number density of each population is fixed, as shown in Fig. \[fig:sdsb\], leaving two chemical and two dynamical free parameters for each population, and thus there are four free parameters for the red population:
- $Z_0^\mathrm{red}$, mean of the Gaussian colour distribution;
- $\sigma_Z^\mathrm{red}$, dispersion of the Gaussian colour distribution;
- $\lambda^\mathrm{red}$, $\equiv - \ln \left( 1 - \beta^\mathrm{red}_z \right)$, a symmetric recasting of the constant velocity anisotropy in the meridional plane $\beta^\mathrm{red}_z$;
- $\kappa^\mathrm{red}$, rotation parameter.
There are correspondingly four free parameters for the blue population:
- $Z_0^\mathrm{blue}$;
- $\sigma_Z^\mathrm{blue}$;
- $\lambda^\mathrm{blue}$;
- $\kappa^\mathrm{blue} $.
The PNe and stars are treated together as stellar tracers with chemical properties decoupled from the GCs. The projected stellar density profile is set by the surface brightness profile in Fig. \[fig:sdsb\], and there are two free dynamical parameters left to be defined:
- $\lambda^\mathrm{star}$, $\equiv - \ln \left( 1 - \beta^\mathrm{star}_z \right)$, the velocity anisotropy parameter of the stars/PNe;
- $\kappa^\mathrm{star} $, the rotation parameter of the stars/PNe.
In order to understand the ability of our model to distinguish between different DM haloes, we run two sets of models: one set using cored DM haloes and one set using cusped DM haloes. For each model, we have 13 free parameters as described above.
The <span style="font-variant:small-caps;">emcee</span> package [@Foreman-Mackey2013], a Python implementation of the an affine-invariant Markov Chain Monte Carlo (MCMC) ensemble sampler, is used to run the models. For each set of models, 200 walkers with 300 steps are employed, where walkers are the members of the ensemble. The walkers are similar to separate Metropolis–Hasting chains but the proposal distribution for a given walker depends on the positions of all the other walkers in the ensemble. We burn in the chain at the time step 250, only the last 50 steps are used for the post-burn distributions as shown in Fig. \[fig:mcmc\_postburn\].
{width="\hsize"}
Parameter $\alpha_{\rm s}$ in equation (\[eqn:Ltot\]) is used to numerically balance the equation and set the relative influence of the discrete data extending to large scales and the integrated stellar kinematic data at small scales. We manually vary $\alpha_{\rm s}$ and find that, at small values, the integrated-light stellar kinematic data provides a weak constraint on the model, with large variations in $\Upsilon_V$, and hence in the DM mass. When $\alpha_{\rm s}$ is large, the integrated stellar kinematic data dominate the evolution of the log likelihood, and cause a poor fit to the discrete data. For our NGC 5846 model, the SAURON IFU data have about $1000$ good quality data points and so we find that a small value, $\alpha_{\rm s} = 0.1$, leads to good fits to both the integrated stellar kinematic data and the discrete PNe velocities.
Model fits {#S:modelfits}
==========
The best-fitting parameters obtained by the MCMC process for the models with cored and cusped DM potentials are presented in Table \[tab:para\] (with $d_s$ converted to the scale radius $r_s$, and $\lambda^\mathrm{red}$, $\lambda^\mathrm{blue}$ and $\lambda^\mathrm{star}$ converted to the velocity anisotropy parameters $\beta_z^\mathrm{red}$, $\beta_z^\mathrm{blue}$ and $\beta_z^\mathrm{star}$ for convenience).
Fig. \[fig:mcmc\_postburn\] shows the resulting distributions for 9 of the 13 free parameters for the models with a cored DM halo; the 4 parameters related to the colour distribution of the GCs are omitted as they are not directly related to the dynamics. The scatter plots show the projected 2D distributions, with the points coloured by their likelihoods from blue (low) to red (high). The ellipses represent the $1\sigma$, $2\sigma$ and $3\sigma$ regions of the projected covariance matrix. The histograms show the projected 1D distributions, with bell-like curves representing the $1\sigma$ projected covariance matrix. In most cases there are no degeneracies between the parameters, except for some covariance between the stellar mass-to-light ratio $\Upsilon_V$, stellar velocity anisotropy $\lambda^{\mathrm{star}}$, and DM scale density $\rho_s$ (representating classic mass–anisotropy and mass decomposition degeneracies). Unlike the spherical Jeans model, the mass–anisotropy degeneracy is not strong in our model. The MCMC process works similarly for the cusped model and the corresponding parameters show similar degeneracies.
We calculate the virial mass $M_{200}$ and the concentration $c$ of the DM halo. The virial mass $M_{200}$ is defined as the enclosed mass within the virial radius $r_v$, the virial radius $r_v$ taken as $r_{200}$, where the average density inside $r_{200}$ is 200 times the critical density ($\rho_{\mathrm{crit}} = 1.37 \times 10^{-7} \,M_{\odot}\,\mathrm{pc}^{-3}$). The concentration $c$ is defined as the ratio between the DM virial radius $r_v$ and the DM scale radius $r_s$. $M_{200}$ and $c$ are shown in Table \[tab:para\] as well as the scale density $\rho_s$ and the scale radius $r_s$. The cusped DM halo has a remarkably low concentration and large virial mass, which is not expected from cosmological simulations (and discussed in Section \[S:mass\]).
We run a constrained cusped (CC) model with $\rho_s > 1\times10^{-3} M_{\odot}$ pc$^{-3}$ (this roughly corresponds to a standard NFW halo as a prior) to see statistically how strongly we can constrain the concentration of the halo. The best-fitting parameters of this constrained cusped model are also listed in Table \[tab:para\].
The inclination angle is fixed at $90^\circ$. We have also investigated the effect of leaving the inclination angle as a free parameter. We find that it is poorly constrained in the range from $50^\circ$ to $90^\circ$. The rotation parameter of the red GCs does increase slightly, but otherwise a free inclination angle does not have a significant effect on our results.
[llllllllll]{} DM & $\rho_s $ & $c$ & $r_s $ & $\log{M_{\rm vir}}$ & $\Upsilon_V$ & $Z_0^\mathrm{red}$ & $Z_0^\mathrm{blue}$ & $\sigma_Z^\mathrm{red}$ & $\sigma_Z^\mathrm{blue}$\
\
& $\beta_z^\mathrm{star}$ & $\beta_z^\mathrm{red}$ & $\beta_z^\mathrm{blue}$ & $\kappa ^{\rm star}$& $\kappa ^{\rm red}$ & $\kappa ^{\rm blue}$ & & $\mathcal{L}_{\rm max}$ & $\delta L (1\sigma)$\
Cored & $4.0^{+2.3}_{-1.5}$ & - & $50^{+30}_{-20} $ & $13.1^{+0.7}_{-0.9}$ & $9.0\pm0.3$ & $1.09\pm0.02$ & $0.87\pm0.02$ & $0.15\pm0.02$ & $0.12\pm0.02$\
& $ 0.05^{+0.05}_{-0.05}$ & $-0.34^{+0.32}_{-0.50}$ & $0.32^{+0.13}_{-0.16}$ & $-0.07\pm0.04$ & $0.3\pm0.2$ & $-0.1\pm0.4$ & & $-7415$ & 6\
Cusped & $0.025^{+0.17}_{-0.022}$ & $1.2^{+1.2}_{-0.6}$ & $1149^{+6000}_{-600} $ & $15.4^{+2.4}_{-2.6}$ & $8.7\pm0.3$ & $1.10\pm0.02$ & $0.86\pm0.02$ & $0.15\pm0.02$ & $0.12\pm0.02$\
& $ 0.05^{+0.04}_{-0.05}$ & $-0.34^{+0.36}_{-0.50}$ & $0.35^{+0.13}_{-0.17}$ & $0.08\pm0.01$ & $0.3\pm0.2$ & $-0.1\pm0.3$ & & $-7415$ & 7\
CC & $1.6^{+0.9}_{-0.6}$ & $5.7_{-0.7}^{+1.2}$ & $74^{+37}_{-23} $ & $12.9^{+0.8}_{-0.4}$& $8.3\pm0.3$ & $1.10\pm0.02$ & $0.86\pm0.02$ & $0.15\pm0.02$ & $0.12\pm0.02$\
& $ 0.08^{+0.01}_{-0.01}$ & $-0.49^{+0.36}_{-0.50}$ & $0.33^{+0.13}_{-0.17}$ & $0.08\pm0.01$ & $0.3\pm0.2$ & $-0.1\pm0.3$ & & $-7417$ & 6\
The chemo-dynamical GCs {#SS:chemo}
-----------------------
![Chemo-dynamical modelling results for the spatial, colour and kinematic distributions of GCs in NGC 5846. [**Top scatter panel**]{}: The projected semi-major elliptical shell radius $R'$ vs. the colour $g-i$. [**Bottom scatter panel**]{}: $R'$ vs. line-of-sight velocity $v_{z'}$. GCs are plotted with points coloured by $P_i^\mathrm{'red}$– the probability of belonging to the red population–from blue (low) to red (high). The red and blue histograms show the distributions in radius, color and velocity for the GCs classified as red and blue. The solid curves over-plotted are the model distributions for each population. The radial distributions, inferred from the surface number density profiles, do not match the histograms in the inner regions owing to spectroscopic incompleteness. Note also that the velocity dispersions actually vary with radius, but here we plot the overall velocity distribution for each population. The two vertical dashed lines indicate the positions of $-3\,R_{\rm e}$ and $3\,R_{\rm e}$. []{data-label="fig:chem-dyn"}](figure/chemo-dyn){width="\hsize"}
The key approach in our chemo-dynamical modelling is that the 213 GCs with velocity measurements are not separated into red and blue groups before performing the dynamical modelling. Instead, from the best model obtained with the MCMC process, we calculate the probabilities of an individual GC being red ($P_i^\mathrm{'red}$) or blue ($P_i^\mathrm{'blue}$ ) following equation (\[eqn:Pik\]). We show the chemo-dynamical separation of the GCs in our best-fitting cored model in Fig. \[fig:chem-dyn\]. The results for the cusped DM halo and the constrained DM halo models are very similar, so we do not include them here. The upper and lower scatter plots show the distribution of GCs in colour $g-i$, and the line-of-sight velocity $v_{z'}$ versus the projected semi-major elliptical shell radius $R'$ defined in equation (\[eqn:RR\]). The GCs are plotted with points coloured by $P_i^\mathrm{'red}$: the red points representing GCs with high probability of being in the red population, and the blue points representing GCs with high probability of being in the blue population. The red and blue histograms are constructed directly from the GCs identified by $P_i^\mathrm{'red} > 0.5$ for red and $P_i^\mathrm{'blue} > 0.5$ for blue. This criterion for separation is used in all the following sections of the paper. The solid curves in the histograms are the best-fit model-predicted distributions for each population. The model-predicted radial distributions $f(R')$ are inferred from the photometric surface number density profiles $\Sigma(R)$, with $f(R') = f(R)/2 = \pi qR\Sigma(R)dR$, where $R'$ is the semi-major elliptical annular radius defined in Equation \[eqn:RR\].
The model input surface number density profiles of the red and blue GCs were determined by photometrically-detected GCs with separation at $g-i < 0.90$ for blue and $g-i> 0.95$ for red (Fig. \[fig:sdsb\]). In the final colour distribution obtained by the chemo-dynamical model in Fig. \[fig:chem-dyn\], the colour distributions of the two GC populations with velocity measurements are clearly separated with a small overlap around $g-i \sim 0.9-1.0$. These distributions are reasonably consistent with the results from a standard, non-dynamical bimodal colour modelling approach [@Napolitano2014]. The surface number density profiles used for the modelling were created in a way that turns out to be nicely consistent with the separation inferred from the dynamical tracers.
Fit to the kinematic data {#SS:fit}
-------------------------
{width="\hsize"}
The kinematic maps from the best-fitting cored model are shown in Fig. \[fig:kin\_map\], the left panels are the kinematic maps for the red GCs and the blue GCs; while the right panels are for the stellar tracers corresponding to the PNe and integrated-light stellar kinematic measurements. The kinematic maps are the predicted mean velocity (left columns) and velocity dispersion (right columns) in the projected plane, with each point representing a measurement position colour-coded with the predicted value (as indicated by the corresponding colour bar). The velocity dispersion anisotropy is encoded in the opening angle of the kinematic maps (@Li2016, @Cappellari2008). Red GCs have higher dispersions along the major axis while the blue GCs have higher dispersions along the minor axis, indicating their different velocity anisotropies. We will discuss this further in Section \[SS:anisotropy\].
We bin the data and the model predictions to show a direct comparison in Fig. \[fig:kin\]. We divide the projected plane into two cones, one spanning a $\pm45$ deg from the major axis, and the other a $\pm45$ deg from the minor axis as indicated on the top-left panel. The data and model predictions are binned along the major and minor axes using points in the corresponding cones. Both the data and model are axis-symmetrized and point-symmetrized to increase the number of points by a factor of 4 before binning.
Fig \[fig:kin\] shows the binned mean velocity and velocity dispersion profiles. In each panel, the coloured points are constructed from the observational data along the major axis, while the black stars are along the minor axis. The coloured solid curves are the mean and $1\sigma$ uncertainties of the model predictions along the major axis, and the black dashed curves are along the minor axis.
Equally-populated radial bins are used for the GC and PN discrete data with 30 (40 for PN) points per bin counting from left to right. Adjacent bins are independent. The mid-way position from the minimum to maximum $R'$ of the 30 points is taken as the $R'$ value of a bin. The horizontal error bar covers the $R'$ region that the 30 points span. For the stellar kinematic data, we omit the horizontal error bars, while the SAURON data are binned 40 points per bin and the SLUGGS data are binned with 20 points per bin, with adjacent bins independent from each other.
The data points fluctuate substantially, with the typical uncertainty shown in each panel. The observed rotation in the red GCs and in the integrated starlight in the inner regions is significant and matched well by the model prediction. The stellar kinematics exhibit a weak kinematically decoupled core (KDC) inside $\sim$ 30 arcsec which is not matched well by our model. The observed PNe mean velocity profile has large fluctuations and shows no obvious rotation, which is consistent with our model predicts, agrees with the result from @Coccato2009. The rotation of these different tracers will be discussed further in Section \[SS:rot\].
The model-predicted velocity dispersion profiles for all the tracers match the observational data reasonably well. The dispersion profile is matched radially, and is the velocity dispersion difference between the major and minor axes.
The cusped model fits the data as well as the cored model. The constrained cusped model provides a poorer fit to the stellar kinematics, with a too large velocity dispersion around $20-100$ arcsec predicted by the model. The kinematics of the cusped model and the constrained cusped model are shown in Appendix \[S:kin\_c5\].
Kinematic properties of the three tracers {#S:kin}
=========================================
Internal rotation {#SS:rot}
-----------------
{width="\hsize"}
![ The variation of line-of-sight velocity $v_{z'}$ along the azimuthal angle ($\phi$) measured from the major axis of the galaxy. [**Top panel:**]{} The points are the SAURON data, and the black curve is the sinusoidal fit to the data. [**Bottom panel:**]{} The thin black pluses represent the red GCs at $R< R_{\rm e}$, and the red thick pluses are the red GCs at $R>R_{\rm e}$, with the red curve as the corresponding sinusoidal fit. []{data-label="fig:rot_fit"}](figure/rot_fit){width="\hsize"}
We identify the red GCs based on their likelihoods in the best-fitting model. As shown in Figure \[fig:kin\], the field stars and red GCs show significant rotation. In our axisymmetric model, only the rotation about the minor axis is fitted by our modelling: see the complicated rotation pattern shown in Figure \[fig:kin\]. As we discuss below, the PNe and the blue GCs are consistent with no rotation, so we only show the red GCs and the field stars in the following two figures.
Figure \[fig:xy\_rot\] shows the positions of the tracers coloured by their LOS velocities, illustrating the KDC in the inner region. The velocity gradient of the field stars traced by the SLUGGS data (asterisks) at $\sim 0.5$–1.0 $R_{\rm e}$ starts differ from that in the inner $0.5\,R_{\rm e}$ as traced by the SAURON data (the velocity map showing the Voronoi binning). Further out, the red GCs (plus symbols) show rotation in the opposite direction from the SAURON data, and thus the region traced by the SLUGGS stellar kinematics is likely to be at the transition of these counter-rotating inner and outer regions.
The rotation directions of the SAURON data and the red GCs are roughly around the photometric minor axis of the galaxy. The variation of LOS velocities along the azimuthal angle $\phi$ measured from the major axis of the galaxy is shown in Fig. \[fig:rot\_fit\]. A simple sinusoidal fit to the SAURON data, $v = v_{\rm max} \sin(\phi + \phi_0)$, yields $\phi_0 = -60^\circ \pm 20^\circ$ and $v_{\rm max} = 7.0\pm 2.2$ [kms$^{-1}$]{}, as indicated by the solid black curve in the top panel. The sinusoidal fit to the red GCs at $R > \,R_{\rm e}$ yields the red solid curves with $\phi_0 = 93^\circ \pm 6^\circ$ and $v_{\rm max} = 61 \pm 21$ [kms$^{-1}$]{}, with hints of the rotation increasing with radius. For the red GCs with $\sigma^\mathrm{red} \approx 200$ [kms$^{-1}$]{}, we get $v_{\rm max}/\sigma \sim 0.3$ at $R>R_{\rm e}$. At $R < R_{\rm e}$, there are only a few red GCs, and no obvious rotation pattern. These fits confirm that the central galaxy light and the red GCs both have rotation around the minor axis (along the major axis) but in opposite directions.
We also tried the sinusoidal fit to the PNe and blue GCs, both of which are consistent with zero rotation, as well as that suggested by the dynamical model fitting as shown in the last section. The PNe are, if anything, counter-rotating to the red GCs at the same radii. This raises the possibility that the red GCs do not surprisingly trace the field stars (PNe) spatial distribution - as noted earlier. The blue GCs are also consistent with no rotation at all radii; they may share the similar rotate properties with the PNe rather than the red GCs.
Velocity dispersion anisotropy {#SS:anisotropy}
------------------------------
![The velocity anisotropy profiles in the cored models. The orange, green, red and blue symbols represent field stars, PNe, red GCs and blue GCs. The diamonds with error bars are the profiles obtained from the discrete chemo-dynamical Jeans model. The solid lines are the results from our M2M models. The large orange plus symbol is the stellar velocity anisotropy obtained by a M2M model from @Long2012. The three vertical lines represent the positions of $R_{\rm e}$, $3\,R_{\rm e}$ and $6\,R_{\rm e}$. []{data-label="fig:beta"}](figure/betar){width="\hsize"}
We calculate the velocity anisotropy profiles from our dynamical models. We use the usual definition for the velocity anisotropy with $\beta_r = 1 - (\sigma_{\phi}^2 + \sigma_{\theta}^2)/2\sigma_{r}^2$ where $\sigma_{\phi}$, $\sigma_{\theta}$ and $\sigma_{r}$ being the velocity dispersion in spherical coordinates, are obtained from the $\sigma_{R}$, $\sigma_{\phi}$ and $\sigma_{z}$ cylindrical values from our axisymmetric models. The radial profiles for the three populations are shown in Fig \[fig:beta\]. The profiles are for the models with a cored DM halo, but the results are similar for the cusped halo models. The orange, black, red and blue symbols and lines represent the field stars, PNe, red GCs and blue GCs, respectively.
The diamonds with error bars are the velocity anisotropy profiles obtained from the discrete chemo-dynamical Jeans model. In the models described in Section \[SS:fit\], the velocity anisotropy of the stellar tracers including PNe is dominated by the stellar kinematic data in the inner region. This is plotted as the orange diamond. The velocity anisotropy in the region covered by the PNe only is plotted with green diamonds.
We find very mild radial velocity anisotropy for the stars in the inner regions. The PNe match up with the stars, and in the outer parts become mildly tangentially anisotropic. The blue GCs are slightly radially anisotropic, and the red GCs are tangentially anisotropic. Note that these anisotropy profiles are fairly flat by construction, since in our chemo-dynamical model we assumed a constant $\beta_z$ for each population.
In an axisymmetric system, radial velocity anisotropy causes the velocity dispersions along the minor axis to be higher, while tangential velocity anisotropy increase the dispersions along the major axis. The anisotropy results we obtained for the different populations illustrate this scenario. The blue GCs have higher dispersions along the minor axis (Fig. \[fig:kin\]) and they are found to be radially anisotropic. The red GCs at all radii and the outermost PNe have higher dispersions along the major axis and are they are found to be tangentially anisotropic.
Having obtained the best-fitting model, the red and blue GCs can be separated via likelihood as described in Section \[SS:chemo\] to give three independent discrete tracers: PNe, red GCs and blue GCs. To investigate the velocity anisotropies further, we create independent single component particle-based made-to-measure (M2M) models for each population of tracers (@deLorenzi2007; @Long2010; @Zhu2014). A cored DM halo is adopted with the parameters of the potential fixed by the values obtained by the best-fitting chemo-dynamical Jeans models. The details of the M2M models are described in the Appendix \[S:m2m\].
The M2M models are not used to constrain the potential but to assist with the velocity anisotropy profile analysis, which should provide more robust result by using the information about the opening angle of the kinematic map as well as the whole velocity distribution. The solid lines in Fig. \[fig:beta\] are the velocity anisotropy profiles obtained from the M2M models. Again, the red, blue and green lines represent the red GCs, blue GCs and PNe anisotropies. The large orange plus symbol shows the velocity anisotropy of the stars obtained by M2M models using SAURON data alone [@Long2012]. The PNe are found to be radially anisotropic within $3\,R_{\rm e}$, after which the velocity anisotropy starts to decrease, reaching isotropy by the projected radial limit of the data. The further continuation at larger radii to tangential anisotropy is uncertain. Red GCs are found to be tangentially anisotropic around $1\, R_{\rm e}$, becoming isotropic around $3\,R_{\rm e}$, after which the velocity anisotropy decreases to become tangentially anisotropic again. Blue GCs are found to have radial anisotropy which increases gradually with radius, in agreement with the results from the axisymmetric Jeans model, and in contrast to the profiles for the other tracers.
Overall, the results from Jeans and M2M are reasonably consistent, keeping in mind both the limitations of Jeans models, and the possible uncertainties in the M2M models. Considering the differences in more detail: (1) the Jeans models have limited ability to produce radial variations in the velocity anisotropy profiles as seen in the M2M models; (2) the velocity anisotropy profiles obtained using the Jeans models are, on average, more tangential than those obtained using M2M. This effect could be caused by deviations of the galaxy’s velocity ellipsoid from the zero tilt angles assumed by the axisymmetric Jeans model [@Cappellari2008].
Mass profile constraints {#S:mass}
========================
![[**Top panel**]{}: The enclosed mass profiles of NGC 5846. The black solid, black dashed and red dot-dashed curves represent the total mass profiles of the models with cored, cusped and constrained cusped DM halos, while the orange solid, dashed and dot-dashed curves represent the corresponding stellar mass profiles. The pluses with error bars indicate the typical uncertainty in the total mass at those positions. The blue solid curve represents the standard NFW modelling results from @Napolitano2014, based on stars and GCs. The green solid curve is the PN-based mass model from @Deason2012, and the purple solid curve is the mass profile obtained from modelling the X-ray emission [@Das2010]. [**Middle panel**]{}: The enclosed DM fraction, where the black solid, black dashed and red dot-dashed curves represent the cored DM model, the cusped model and the constrained cusped model. The pluses with error bars indicate the typical uncertainty at those positions. [**Bottom panel**]{}: The relative mass difference of the constrained cusped model compared to the cored model (solid) and the cusped model (dashed). The vertical dashed lines represent the positions of $R_{\rm e}$, $3\,R_{\rm e}$ and $6\,R_{\rm e}$. []{data-label="fig:mass"}](figure/mass){width="\hsize"}
The enclosed mass profiles obtained by our best-fitting models are shown in the upper panel of Fig. \[fig:mass\]. The middle panel displays the DM fraction profile and the bottom panel, the mass profile difference between the constrained cusped model and the other two models.
The black solid and dashed curves represent the total mass profile with respectively a cored and a cusped DM halo, the orange solid and dashed curves represent the corresponding stellar mass profiles. These two mass profiles are consistent with each other to within $1\sigma$ uncertainty, and they fit all the data with almost equal quality (see Table \[tab:para\]). A stellar mass-to-light ratio of $\Upsilon_V = 9.0 \pm 0.3$ is obtained with the cored DM halo, while we estimate $\Upsilon_V = 8.7 \pm 0.1$ with the cusped DM halo (Table \[tab:para\]). The DM fraction within $1\,R_{\rm e}$ of these two models is only $\sim 10\%$ and $\sim 15\%$ respectively (see lower panel). By comparison @Cappellari2013b found a DM fraction of 14% through modelling the SAURON data with an axisymmetric Jeans model.
The cored model yields reasonable values for the DM density around the scale radius and for the virial mass for a giant elliptical galaxy (Table \[tab:para\]; @Dutton2014). In contrast, the cusped halo has a remarkably low density and large scale radius (low concentration and an implausibly large virial mass of $> 10^{15} M_\odot$; see Table \[tab:para\]). This may seem to be evidence against a cusped halo. To investigate this further, we have tried a constrained cusped halo, using a density prior of $\rho_s > 1\times 10^{-3} \,M_{\odot}\,\mathrm{pc}^{-3}$ which concentrates more DM in the inner regions (red dash-dotted). As a consequence of the higher DM concentration, we obtain a lower stellar mass-to-light ratio of $\Upsilon_V = 8.3 \pm 0.3$ (orange dash-dotted lines), and the DM fraction within $1\,R_{\rm e}$ increases to $\sim 20\%$. As shown in Table \[tab:para\], although this model is not favoured having a smaller maximum likelihood than the previous two models, the statistical significance of the difference is less than $1\,\sigma$.
The mass profile differences between the constrained cusped model and the other two models are shown in the bottom panel of Fig \[fig:mass\]. The solid curve represents the comparison with the cored model and the dashed curve represents the comparison with the cusped model. The differences are most significant in the regions $\sim~2\,R_{\rm e}$ and $\gtrsim 4\,R_{\rm e}$. The velocity dispersion of the PNe and GCs has a typical uncertainty of $\sim 10\%$, so variations in the mass profiles at the $\sim$ 10–20% level are to be expected. The stellar kinematics around $1\,R_{\rm e}$ ($\sim$ 80 arcsec) are most sensitive to this mass profile difference (see Figure \[fig:kin\] and Figure \[fig:kin\_c5\]). More data points around 1–$3\,R_{\rm e}$ or in the outer most regions would improve our ability to distinguish between the cored model and the constrained cusped model.
As discussed earlier, there is a $\sim$ 10 [kms$^{-1}$]{} offset between the stellar velocity dispersions from SAURON and from SLUGGS. In our default models, we have shifted the SLUGGS data up to match SAURON, but if we instead move the SAURON dispersions down by 10 [kms$^{-1}$]{}, then the stellar mass-to-light ratios in the three models decrease by $\Delta \Upsilon_V \sim 0.2$, The DM parameters do not change significantly and a cored model is still preferred with similar significance. If we do not try to match the SLUGGS and SAURON data, then the model fits become slightly worse, as the decreasing stellar velocity dispersions from $0-20$ arcsec to $20-100$ arcsec becomes harder to reproduce. This is particularly true for the constrained cusped model which is then excluded with $1\sigma$ significance.
Mass profile modelling of NGC 5846 has been performed previously using stellar kinematics within $2\,R_{\rm e}$ (e.g. @Kronawitter2000 [@Cappellari2013a]). However, tracers extending to larger radii are needed in order to probe the DM distribution adequately. @Das2010 derived a mass profile for NGC 5846 using X-ray gas properties from *Chandra* and *XMM-Newton* observations (shown as the purple solid line in Fig. \[fig:mass\]). This mass is higher than we have found here, probably because of disturbances in the gas that violate the assumption of hydrostatic equilibrium required in the X-ray analysis. @Deason2012 modelled the PNe dynamics using a power-law distribution function model (shown as the green solid line), and obtained results which are consistent with our results over the region covered by most of the PNe.
@Napolitano2014 utilized a spherical Jeans model of field stars and two GC populations and included velocity distribution kurtosis constraints. The blue solid curve represents these results obtained with a standard NFW DM model, which is in general agreement with our mass profiles. The density of their DM halo is higher than our preferred cored model but is and consistent with our constrained cusped model, They found a larger scale radius with the DM mass at all radii larger than our constrained cusped model. With a stellar mass-to-light ratio of $\Upsilon_V = 8.2$, they obtained a DM fraction within $1\,R_{\rm e}$ of $\sim 30\%$.
As we discussed in Section \[SS:fit\], the concentration of the DM halo is sensitive to the velocity dispersions in the region of $20-100$ arcsec. @Napolitano2014 only fit the upper limit of the velocity dispersion of the stellar kinematic data, which is $210-220$ [kms$^{-1}$]{}in that region. This is $\sim 10$ [kms$^{-1}$]{}higher that our data and our model prediction. This could explain the higher concentration of DM @Napolitano2014 obtained. @Napolitano2014 used similar GC data in the outer regions. The higher mass they obtained at larger radius could be a result of the mass-anisotropy degeneracy. As they obtained a higher mass as well as a more radially velocity dispersion anisotropy.
Our estimated stellar mass-to-light ratio of $\Upsilon_V \sim$ 8–9 may be compared with the ATLAS$^{\rm 3D}$ result [@Cappellari2013b]. They found $\Upsilon_r = 7.0$, which we convert to $V$-band based on the galaxy’s $g-r = 0.9$ colour from SDSS, and arrive at $\Upsilon_V \simeq 8.5$, which is nicely consistent with our results. Spectroscopy-based stellar population synthesis modelling for NGC 5846 gives a stellar $\Upsilon_V =8.5$ assuming a Salpeter initial mass function (IMF; @Cappellari2013b). A Kroupa IMF would give $\Upsilon_V \simeq 5.3$, which we tentatively rule out with dynamic modelling, with the caveat that we have not allowed for adiabatically contracted halo models which would decrease the inferred stellar mass. With total stellar mass of $\sim 0.7\times 10^{12}\, M_{\odot}$, our galaxy is consistent with the trend of more massive galaxies displaying more Salpeter-like IMFs [@Cappellari2012]. Although things are actually more complicated, higher stellar mass-to-light ratio could also be accomplished by allowing the high-mass end slope of a Chabrier-like IMF to be steeper. As recently shown in @Lyubenova2016, the latter seems to be so far the preferred option for elliptical galaxies whereas a Salpeter-like single power-law is ruled out for $75\%$ of the galaxies in their sample. Moreover, there are also indications that the IMF might radially vary within galaxies [@Martn2015]. An interesting avenue for future work will be to combine our dynamical constraints with stellar population modelling that include IMF variations. Our modelling result also agrees with the full-spectral fitting of the galaxy’s central light with a free IMF [@Conroy2012], which yielded $\Upsilon_V = 8.8$.
The situation is thus different from the multi-population dynamical modelling of NGC 1407, where the difference between cored and cusped DM halo models was degenerate with the IMF assumption [@Pota2015b]. For our NGC 5846 modelling, the primary limitation comes instead from the total dynamical mass estimate, rather than from the stellar mass-to-light ratio.
Overall, the dark and luminous mass distributions are individually well constrained in our model. Marginalizing over the different DM models, we find a stellar mass-to-light ratio of $\Upsilon_V = 8.8\pm0.5$. We prefer a cored DM halo, which leads to only $\sim 10\%$ DM fraction within $1\,R_{\rm e}$, increasing to $67\pm10 \%$ at $6\,R_{\rm e}$. A standard NFW model is not favoured, although only with low statistical significance.
Discussion: connections to other galaxies and formation clues {#S:discussion}
=============================================================
![The extended velocity anisotropy profiles of four giant ellipticals. The solid lines represent the profiles for PNe, red GCs and blue GCs in NGC 5846 based on our M2M modelling, the dash-dotted lines represent red GCs and blue GCs in M87 from @Zhang2015, the dashed black line shows PNe in M84 from @Zhu2014, and the dotted horizontal lines are for red and blue GCs in NGC 1407 from @Pota2015b. []{data-label="fig:betar_compare"}](figure/betar_compare){width="\hsize"}
We now consider some broader implications of our modelling results for NGC 5846. In early-type galaxies, it is thought that the kinematics of red GCs are generally coupled to the field stars of the host galaxy, with rotation around the photometric minor axis, while blue GCs are more dominated by random motions [@Pota2013]. This is consistent with what we have found for NGC 5846, although we did not find the same red GC rotation as @Pota2013 for this galaxy. Considering the large fluctuations in rotation that we found (Fig. \[fig:xy\_rot\] and Fig. \[fig:rot\_fit\]), the differences in results from @Pota2013 could be due to their hard cut on the colour distribution in separating the two GC populations. In any case, both studies, as well as the PN study by @Coccato2009, agree on very low outer rotation in NGC 5846. This result is consistent with cosmological simulations of massive galaxy formation [@Wu2014], where slow rotation is a consequence of multiple minor mergers from random directions (e.g. @Moody2014). A similar formation scenario for NGC 5846 is also inferred from a detailed analysis of its extended stellar kinematics by @Forbes16.
Additional clues to galaxy formation come from velocity dispersion anisotropy profiles, which we derived for NGC 5846 using three tracers: stars+PNe, red GCs, and blue GCs (Section \[SS:anisotropy\]). These profiles can be compared to the generic expectations for elliptical galaxies that have formed through mergers. Such galaxies have mild anisotropy in their centres, yielding to strong radial anisotropy in their outer regions. This applies to both stars and GCs, whether they were formed in mergers or were passively accreted (e.g. @Dekel2005 [@Kruijssen2012]). The blue GCs in NGC 5846 show this pattern, but the stars and red GCs, less so. For the red GCs, the tangential anisotropy interior to $\sim$ 20 kpc may simply relate to the effects of tidal forces, which preferentially dissolve the GCs that are on more radial orbits. The isotropic to tangential trends for the stars and red GCs at $\gtrsim 30$ kpc are more difficult to understand[^3], but might connect to a component of [*in situ*]{} formation [@Wu2014; @Rottgers2014].
In Figure \[fig:betar\_compare\], we compared results for the velocity anisotropy at large radii from NGC 5846 with other massive slow rotating ellipticals: NGC 1407, M84, and M87. For each galaxy, PNe, red GCs and blue GCs were treated as independent tracers. Among these four galaxies, there is a wide scatter of inferred anisotropy at both small and large radii, with no consistent pattern within each galaxy or each tracer type. The overall impression is of complex and inhomogeneous formation histories for this type of galaxy. On the other hand, similar results for the field stars (including PNe) of fast-rotator ellipticals (NGC 3379, NGC 4494, NGC 4697, M60) seem to have a more consistent pattern of isotropy in the centre, transitioning to radial anisotropy in the outer regions as expected (based on field stars and PNe; @deLorenzi2008 [@deLorenzi2009; @Das2011; @Morganti2013]). Tangential anisotropy for blue GCs from inner to outer regions is inferred from the negative kurtosis of the velocity distribution for the SLUGGS galaxies [@Pota2013].
Given the uncertainties in our DM profile results for NGC 5846, considering their implications would take careful analysis that is beyond the scope of this paper. However, we can make an interesting comparison of the [*total*]{} dynamical mass profile with the recent results of @Cappellari2015 and @Serra2016, who studied a large sample of fast-rotator early-type galaxies using both extended stellar kinematics (from ATLAS$^{\rm 3D}$+SLUGGS) and HI kinematics. They found a remarkable homogeneity in the total mass density profiles of these galaxies, with power-law slopes clustered tightly around $-2.2$ dex per dex. The slope that we find for NGC 5846 over the radial range of 1–4 $R_{\rm e}$ is much shallower at $-1.7$ dex per dex. This difference presumably arises from the location of NGC 5846 within a massive group-central halo, which is the dominant structure, but may also be a reflection of a difference in assembly histories between fast and slow rotators.
Summary {#S:conclusions}
=======
We have applied our discrete chemo-dynamical modelling techniques to the giant elliptical galaxy NGC 5846, incorporating kinematic constraints from its red GCs, blue GCs and PNe – along with field stars. We used 214 GCs extending to $\sim 6 \,R_{\rm e}$ with LOS velocities from the SLUGGS survey, 123 PNe extending to $\sim 4\, R_{\rm e}$ with LOS velocities, SAURON IFU data extending to $\sim 0.5 \, R_{\rm e}$, and integrated stellar velocity and velocity dispersion measurements at 80 discrete positions extending to $\sim 1\,R_{\rm e}$ from SLUGGS. In a departure from usual practice, the GCs were [*not*]{} separated into red and blue GCs via a hard metallicity cut before modelling. Instead, a free colour distribution for each GC population was included in the chemo-dynamical model. The three populations orbit in the same potential but with their own unique surface density profiles, internal rotation and velocity anisotropy properties. By applying an MCMC process, the model was able to find, simultaneously, the region in parameter space that matches the observed kinematics. We have found:
- The mass profiles at all scales covered by the data points are constrained well. We constrain the total mass within $6\, R_{\rm e}$ ($=480'' = 57 \, \mathrm{kpc}$) to be $(1.7\pm0.3) \times 10^{12}\, M_{\odot}$, and obtain a $V$-band stellar mass-to-light ratio of $\Upsilon_V = 8.8\pm0.5$ (corresponding to a Salpeter IMF). A cored DM halo is weakly preferred, and implies a DM fraction of $\sim 10\%$ within $1\,R_{\rm e}$, increasing to $67\pm 10 \%$ within $6\,R_{\rm e}$. A standard NFW halo, with $\sim 20\%$ DM within $1\,R_{\rm e}$, is not favoured although with only low statistical significance.
- The red and blue GCs are naturally separated out by the likelihood analysis. We find weak rotation for the red GCs, with $v_{\rm max}/\sigma_0 \sim 0.3$ at $R> 1\, R_{\rm e}$. The rotation is about the photometric minor axis of the galaxy, and is opposite to the rotation of the inner field stars – indicating a KDC. We find no significant rotation for the PNe or blue GCs.
- We have created particle-based M2M models for the three populations, and used them to confirm that the velocity anisotropy profiles for the three populations as obtained by the axisymmetric Jeans models are acceptable, and are consistent with those obtained by the M2M models.
The more general velocity anisotropy profiles obtained by the M2M models show that neither the red nor the blue GCs exactly follow the stars. In the inner regions, the red GCs are more tangentially anisotropic than the stellar tracers, which probably reflects tidal forces destroying the GCs on radial orbits. In the outer regions, the blue GCs are radially anisotropic, as expected in accretion scenarios. The outer PNe and red GCs show similar isotropic to tangential orbits. However, the red GCs have significant rotation which is not detected in the PNe or blue GCs. From their anisotropy and rotation, it raises the possibility that neither the red GCs nor the blue GCs have the same formation history as the stellar tracers. The outer PNe and red GCs show isotropic to tangential orbits, which is a feature seen in the PNe or blue GCs of other slow rotator ellipticals
Acknowledements
===============
We thank XiangXiang Xue for useful discussions, and Alis Deason for sending her mass-profile results. Computer runs were mainly performed on the MPIA computer clusters *queenbee* and *theo*. This work was supported by Sonderforschungsbereich SFB 881 “The Milky Way System” (subprojects A7 & A8) of the German Research Foundation (DFG) and the National Science Foundation grant AST-1211995, NRN is supported by Legge 5/2002 Regione Campania: “Dark Matter in Elliptical Galaxies”, and DAF thanks the ARC for financial support via DP130100388.
Cusped dark matter models {#S:kin_c5}
=========================
Figure \[fig:kin\_cusp\] and Figure \[fig:kin\_c5\] show the fit to the kinematic data by the cusped model and the constrained cusped model with $\rho_s > 1\times 10^{-3}\, M_{\odot}\,\mathrm{pc}^{-3}$. The data are binned in the same way as Figure \[fig:kin\]. The red and blue GCs are categorised slightly differently in different models, so that the kinematics of each population differ slightly between models. Compared to the cored model in Figure \[fig:kin\], the cusped model predicts a higher dispersion between $20-100$ arcsec, but still fits the data with almost equal quality. The constrained cusped model predicts even higher dispersions between $20-100$ arcsec, and thus overfits the stellar kinematics from the SLUGGS data in this region. At the same time, it predicts lower dispersions in the outermost regions, and thus the kinematics of the blue GCs are not fitted as well but the outer most data points of the red GCs are fitted somewhat better. With the large error bars in the data, we can see that, even for constrained cusped models, the model predictions are still consistent with the data within $1\sigma$ uncertainty. The differences in the velocity dispersions predicted by these three models are consistent with their differences in mass profiles.
Note that the velocity dispersions from of SLUGGS data have already been adjusted by adding 10 [kms$^{-1}$]{} to match the SAURON data in the inner region. If we leave the SLUGGS data unchanged, then the constrained cusped model provides a poorer fit to the kinematics in this region.
{width="\hsize"}
{width="\hsize"}
The M2M models {#S:m2m}
==============
We create single-population M2M models for the red GCs, blue GCs and PNe respectively. The models are created following @Zhu2014, to which we refer for further details. The same potential is used in all three models, and is the cored potential matching the black solid line in Fig \[fig:mass\]. The M2M models are not created to determine any potential parameters, but to investigate the dynamical properties of each tracer population. The surface number density and discrete LOS velocities of the corresponding tracers are used as constraints in each model.
The initial conditions
----------------------
We initially give the particles equal luminosity weights. The particle number density profile follows the 3D number density profile deprojected from the observed surface number density for each population. The velocity distribution at each radius is a Gaussian distribution, with a mean velocity of zero and velocity dispersion following the solution of an isotropic, spherical Jeans model using the M2M potential and the population tracer number density profile. No net rotation of the particle system is used. In total, we create 400000 particles for each model.
Fit to the data
---------------
Discrete LOS velocity data from our GCs and PNe are included in the model. Overall we have 100 red GCs, 96 blue GCs, and 123 PNe. To reduce noise from the data, axis-symmetrization and point-symmetrization (e.g., @deLorenzi2008; @vdB2008) are applied with the data sizes being enlarged by a factor of 4 as a result.
Model data values are formed by binning the particle data onto the projected plane using $4\time4$ bins in projected radius and azimuthal angle as shown in Fig \[fig:bin\_scheme\]. The binning schemes are constructed in the same way for each model, differing only in bin radius because the three tracers cover slightly different spatial regions.
![The binning schemes for the M2M models of the red GCs, blue GCs and PNe from top to bottom. The black points in each panel represent the positions of the discrete data, the circles indicate the binning in radius and the solid lines represent the binning in azimuthal angle for each model. The binning schemes about radially are different from each other in the three models.[]{data-label="fig:bin_scheme"}](figure/bin_scheme){width="6cm"}
Given the small number of data points available, regularisation is used to ensure indirectly a smooth final velocity distribution [@Long2010]. Models with different regularisation were tried, and we show here three models with high regularisation ($\mu = 1.5$), low regularisation ($\mu = 0.5$) and with no regularisation ($\mu = 0.0$). The absolute value of the parameter $\mu$ has no physical meaning: it merely sets the relative influence of regularisation with reference to the other modelling constraints. Parameter $\mu$’s role is not dissimilar to parameter $\alpha_s$’s role in Section \[SS:combine\].
For each model, the particle weights converged well with more than $95\%$ of particles having weight variations less than $5\%$ in the last 20 half mass dynamical time units of modelling. Fig \[fig:M2Mfit\] shows how well the M2M models fit the data for the models of the red GCs, blue GCs and PNe reading from top to bottom. We only show the velocity distributions for the bins with a central azimuthal angle of $0^{\circ}$ or $90^{\circ}$. The other bins are not shown, as the velocity distributions are similar but with oppositely signed velocities in the bins with an azimuthal angle difference of $180^{\circ}$. The models with no or little regularisation are quite noisy and have some local peaks in the velocity distributions. The model with $\mu = 1.5$ is generally smooth. The velocity anisotropy profiles we use in Fig \[fig:beta\] are directly calculated from the highly regularised models.
![The velocity distributions for the models of red GCs, blue GCs and PNe from top to bottom. The histograms are the velocity distributions of the data in each bin. The black solid lines, blue dashed lines and red dashed lines are those of the models with regularisation $\mu = 1.5, \, 0.5, \, 0.0$ respectively. []{data-label="fig:M2Mfit"}](figure/rGC "fig:"){width="10cm" height="6cm"} ![The velocity distributions for the models of red GCs, blue GCs and PNe from top to bottom. The histograms are the velocity distributions of the data in each bin. The black solid lines, blue dashed lines and red dashed lines are those of the models with regularisation $\mu = 1.5, \, 0.5, \, 0.0$ respectively. []{data-label="fig:M2Mfit"}](figure/bGC "fig:"){width="10cm" height="5.5cm"} ![The velocity distributions for the models of red GCs, blue GCs and PNe from top to bottom. The histograms are the velocity distributions of the data in each bin. The black solid lines, blue dashed lines and red dashed lines are those of the models with regularisation $\mu = 1.5, \, 0.5, \, 0.0$ respectively. []{data-label="fig:M2Mfit"}](figure/PN "fig:"){width="10cm" height="5.5cm"}
\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: <http://sluggs.ucolick.org>
[^3]: @Napolitano2014 inferred radial anisotropy for both red and blue GCs in NGC 5846 by spherical Jeans modelling using kurtosis constraints. However, by taking a spherical assumption, they lost the information of velocity dispersion anisotropy encoding the opening angle of the kinematic map (as we shown in Fig \[fig:kin\_map\] and Fig \[fig:kin\]) and the uncertainties in the kurtosis were quite high. In our discrete axisymmetric models, we have used the discrete data which encodes information about the opening angle of the kinematic map as well as the whole velocity distribution (for M2M; including the kurtosis). Therefore our anisotropy results should be more robust albeit still with substantial uncertainties. Considering the uncertainties of both sides, the @Napolitano2014 results are still consistent with our results from M2M.
|
{
"pile_set_name": "ArXiv"
}
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---
author:
- |
Ahsan Habib\
School of Information Technology\
Deakin University\
Geelong, Australia 3225\
`[email protected]`\
Chandan Karmakar\
School of Information Technology\
Deakin University\
Geelong, Australia 3225\
`[email protected]`\
John Yearwood\
School of Information Technology\
Deakin University\
Geelong, Australia 3225\
`[email protected]`\
bibliography:
- 'main.bib'
title: Choosing a sampling frequency for ECG QRS detection using convolutional networks
---
Introduction
============
Methodology {#method}
===========
ECG Data {#ecgdata}
========
Results
=======
Discussion
==========
Conclusion and Future Work {#conclusion}
==========================
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We reinvestigate the pressure dependence of the crystal structure and antiferromagnetic phase transition in MnTe$_2$ by the rigorous and reliable tool of high pressure neutron powder diffraction. First-principles density functional theory calculations are carried out in order to gain microscopic insight. The measured Néel temperature of MnTe$_2$ is found to show unusually large pressure dependence of $12$ K GPa$^{-1}$. This gives rise to large violation of Bloch’s rule given by $\alpha=\frac{d\log T_N}{d\log V}=-\frac{10}{3} \approx -3.3$, to a $\alpha$ value of -6.0 $\pm$ 0.1 for MnTe$_2$. The ab-initio calculation of the electronic structure and the magnetic exchange interactions in MnTe$_2$, for the measured crystal structures at different pressures, gives the pressure dependence of the Neél temperature, $\alpha$ to be -5.61, in close agreement with experimental finding. The microscopic origin of this behavior turns to be dictated by the distance dependence of the cation-anion hopping interaction strength.'
author:
- 'Tapan Chatterji$^1$, Antonio M. dos Santos$^2$, Jamie J. Molaison$^2$, Thomas C. Hansen$^1$, Stefan Klotz$^3$ and Mathew Tucker$^4$'
- 'Kartik Samanta$^{5}$ and Tanusri Saha-Dasgupta$^{5}$'
title: 'Origin of anomalous breakdown of Bloch’s rule in the Mott-Hubbard insulator MnTe$_2$'
---
Introduction
============
Long time ago in 1966, Bloch [@bloch66] studied the pressure variation of the Néel temperature, $T_N$ and that of volume ($V$) of several transition-metal (TM) based antiferromagnetic insulators (AFI) and came up with the general relationship $$\alpha=\frac{d\log T_N}{d\log V}=-\frac{10}{3} \approx -3.3.$$ In the localized-electron limit where perturbative superexchange theory is applicable, the Néel temperature can be related to the effective TM-TM hopping interaction ($b$), charge transfer energy ($\Delta$) and Coulomb interaction ($U$), as $$T_N \sim b^2 \left[\frac{1}{U}+\frac{1}{2\Delta}\right].$$ The first term in the above equation is the Anderson superexchange term and the second term involves the two electron transfer from the anion. A theoretical rationalization of the Bloch’s rule comes from the calculations of the variation of the cation-anion transfer integral $b^{ca}$ with the cation-anion bond length $r$, which varies as $r^{-n}$. The calculated values[@smith69; @shrivastava76] of $n$ using molecular orbital theory or configuration interaction method on transition-metal oxides and flourides, turn out to be in the range 2.5-3. This leads to $T_N \sim r^{-10} \sim V^{-3.3}$, assuming $b = (b^{ca})^{2}/\Delta$. Experimentally Bloch’s rule is obeyed by a variety of Mott insulators. However, there are exceptions too. For example, while Bloch rule is found to be obeyed in YCrO$_3$ and CaMnO$_3$, it was found to fail in LaMnO$_3$.[@goodenough] The failure has been explained in terms of breakdown of localized approach used in Bloch’s formulation. We note that the cases discussed so far on the pressure dependence of T$_N$, all involve oxygen or flourine, [*i.e.*]{} anions with 2$p$ electrons. As is well known, the nature of anionic wavefunction changes as one moves down the column of the periodic table, from 2$p$ to 3$p$ series and even more to 4$p$ and 5$p$ series, effecting the TM-anion bonding. It would therefore be of interest to consider the validity of Bloch’s criterion in case of TM compounds containing anions like Te.
A compound of interest in this context is MnTe$_2$. MnTe$_2$ belongs to a large class of pyrite type and related marcasite and arsenopyrite type compounds MX$_2$ (M = transition element, X = Chalcogen or pnictogen element) with diverse magnetic and electrical properties. They range from insulator to metal or even superconductor. They can be diamagnetic, weakly paramagnetic, ferromagnetic or antiferromagnetic etc. The magnetic semiconductor MnTe$_2$ having a pyrite type crystal structure, as shown in Fig. \[structure\], orders below $T_N \approx 88$ K in type-I antiferromagnetic structure[@hastings59; @chattopadhyay87a; @burlet97] with the propagation vector ${\bf k} = (1,0,0)$. The magnetic phase transition at $T_N$ was found to be of second order within experimental resolution [@chattopadhyay87a; @burlet97] although the related other manganese dichalcogenides MnS$_2$ and MnSe$_2$ undergo first-order phase transitions[@hastings76; @chattopadhyay84; @chattopadhyay91; @chattopadhyay87b] at $T_N$. The magnetic structure of MnTe$_2$ had been subject to controversy,[@hastings59; @pasternak69; @hastings70] regarding whether the magnetic structure of MnTe$_2$ is of collinear single-[**k**]{} or non-collinear triple-[**k**]{} type, or whether there is any spin reorientation transition. Burlet [*et al.*]{}[@burlet97] resolved this controversy and determined the magnetic structure to be of non-collinear triple-[**k**]{} type. The structure was found to be stable below $T_N$ down to 4.2 K, the lowest temperature at which the magnetic structure was investigated.
The high pressure X-ray diffraction was carried out previously,[@fjellvag85; @fjellvag95] to study the pressure induced volume changes in MnTe$_2$, though no detailed structural analysis was carried out in terms of determination of atomic positions. Also, in a separate study the pressure dependence of Neél temperature was obtained from resistivity and Mössbauer measurements.[@vulliet01] The results of these two studies put together show a large violation of Bloch’s rule, which however has not been stressed before. More importantly, a microscopic understanding of this phenomena was lacking.
In the present study, we take up this issue by experimentally revisiting the pressure dependence of the structure and magnetic ordering temperature of MnTe$_2$ in terms of high-pressure powder neutron diffraction measurment, together with first-principles density functional theory (DFT) calculation to provide the microscopic understanding. The neutron diffraction study carried out in the present work, is undoubtedly a more reliable tool to measure the magnetic transition temperatures, compared to resistivity or Mössbauer. In addition, the present neutron diffraction study provide the detailed structural information, which was not available before, based on which our first-principles calculations have been carried out. Our rigorous study confirms and rationalizes the breakdown of Bloch’s rule in MnTe$_2$.
Methodology
===========
High pressure neutron diffraction investigations were done on three neutron powder diffractometers, viz. PEARL at the ISIS Facility in UK, D20 of Institute Laue-Langevin, Grenoble and also SNAP at SNS, Oak Ridge. Pressure was generated by Paris-Edinburgh pressure cells [@besson92; @klotz05] and a mixture of 4:1 deuterated methanol:ethanol was used as pressure transmitting medium. The PEARL measurements used anvils made of tungsten carbide and a scattering geometry which restricted the available d-spacing range to below 4.2 $\AA$, i.e. the magnetic (100) reflection was not recorded. Rietveld refinements of the patterns to the crystal structure were carried out by the GSAS program [@vondreele86]. The experiments on D20 and SNAP used anvils made of cubic boron nitride [@klotz05] and a scattering geometry which gives access to reflections with larger d-spacings. The sample temperature was controlled using closed-cycle cryostats; fast cooling to 77 K was achieved by flooding the cell assembly with liquid N$_2$. The pressure was determined from the known pressure variation [@fjellvag85; @fjellvag95] of the lattice parameter of MnTe$_2$.
DFT calculations on the experimentally measured structures were carried out in the plane-wave basis, within the generalized gradient approximation (GGA) for the exchange-correlation functional, as implemented in the Vienna Ab-initio Simulation Package.[@vasp] We used Perdew-Burke-Ernzerhof implementation of GGA.[@pbe] The projector augmented wave potential was used. For the total energy calculation of different spin configurations, we considered a 2 $\times$ 2 $\times$ 1 super-cell containing a total of 48 atoms in the cell. For the self-consistent field calculation, an energy cut-off of 600 eV and 4 $\times$ 4 $\times$ 8 Monkhorst-pack K-point mesh were found to provide good convergence of the total energy. The missing correlation at the Mn sites beyond GGA, was taken into account through supplemented Hubbard $U$ (GGA+$U$) calculation[@gga+u] following the Dudarev implementation, with choice of $U$ = 5.0 eV and Hund’s coupling, J$_H$ of 0.8 eV. Variation in $U$ value has been studied and found to have no significant effect on the trend.
Evolution of the Structural parameters under Pressure
=====================================================
The pyrite type crystal structure of MnTe$_2$ in the $Pa\bar{3}$ space group has Mn atom at $4(a)~(000)$ and Te atom at $8(c) ~(xxx)$ position. The cubic lattice parameter $a$ and the Te positional parameter $x$ were refined along with the isotropic atomic displacement parameters of Mn and Te atoms. Figure \[param\] shows the pressure dependence of the structural parameters of MnTe$_2$, viz. lattice parameter, positional parameter $x$ of Te atom, Mn-Te and Te-Te bond lengths and the Mn-Te-Mn and Mn-Te-Te bond angles. The results are very remarkable and contrary to our naive expectation that the Te-Te bond distance would continuously decrease with pressure. Instead the bond distance seems to increase slightly at lower pressure but after reaching a maximum at P = 2 GPa it decreases and becomes somewhat flat at about P = 10 GPa. The two bond angles also show anomalous pressure dependence. This is expected since all the relevant bond distances and angles are derived from the single Te positional parameter $x$ and the cubic lattice parameter $a$ that decreases with pressure in the usual way. In contrast, the Mn-Te bond length is highly pressure sensitive and almost entirely responsible for the pressure-induced volume reduction, suggesting relatively weak Mn -Te bonds which are susceptible to changes upon application of pressure. The remarkable pressure response of the Te-Te bond distance and Mn-Te-Mn and Mn-Te-Te angles should be reflected in the pressure dependence of the superexchange interaction that decides the pressure variation of the Néel temperature, as obtained in our first-principles calculations. Fig. \[murnaghan\] shows the pressure variation of the unit cell volume of MnTe$_2$ and its fit with Murnaghan equation of state. The fit gave $B_0 = 34.6 \pm 1.0$ GPa and $Bo^{\prime} = 8.8 \pm 0.4$ where $B_0$ is the bulk modulus and $Bo^\prime$ is its pressure derivative. The values agree well with the values determined previously from high pressure X-ray diffraction [@fjellvag85; @fjellvag95].
Measured Pressure Variation of T$_N$
====================================
The antiferromagnetic phase transition of MnTe$_2$ was first investigated at ambient pressure with the sample (outside the pressure cell) fixed to the cold tip of the standard orange cryostat. Fig. \[ambient\] shows neutron powder diffraction intensities of the $100$ and $110$ magnetic peaks along with the $111$ nuclear peak at several temperatures below and close to the antiferromagnetic Néel temperature $T_N = 88$ K. Fig. \[critical\](a) shows the temperature variation of the integrated intensity of the $100$ magnetic Bragg peak. The intensity of this reflection decreases continuously with increasing temperature and becomes zero at about $T_N \approx 88$ K. The data just below $T_N$ could be fitted by a power-law exponent $$I=I_0\left(\frac{T_N-T}{T_N}\right)^{2\beta}$$ where $I$ is the integrated intensity, $I_0$ is the saturation value of the intensity at $T = 0$, $T_N$ is the critical temperature and $\beta$ is the power-law exponent. Least-squares fit of the data in the temperature range from $T = 60 $ to $T = 88$ K gave $ \beta = 0.29 \pm 0.04$ and a Néel temperature $T_N = 88 \pm 2$ K. The fitted value of the Néel temperature was used to determine the reduced temperature $t = (T_N-T)/T_N$. We then produced a standard log-log plot shown in Fig. \[critical\] (b) to extract the critical exponent $ \beta= 0.29$ from the slope that agreed well with that determined by the least-squares fit.
Fig. \[5GPa\] shows neutron powder diffraction intensities of the $100$ and $110$ magnetic peaks along with the $111$ nuclear peak of MnTe$_2$ under $P = 4.75$ GPa at several temperatures below and above the antiferromagnetic Néel temperature. It is immediately noticed that the application of hydrostatic pressure P = 4.75 GPa increases the Néel temperature $T_N = 88$ K of MnTe$_2$ substantially. By fitting the temperature dependence of the intensity of the $100$ magnetic peak and fitting the data by a power law we determine $T_N = 145 \pm 7$ K. Fig. 7(a) shows this fit. Similarly we determined the Néel temperatures of MnTe$_2$ at several pressures. The result is shown in Fig. 7(b). The obtained trend agrees well with that obtained from resistivity and Mössbauer spectroscopy[@vulliet01], as also shown in Fig. 7(b). The neutron diffraction results show that $T_N$ of MnTe$_2$ increases linearly in the pressure range $0-8$ GPa at a rate of about $12$ K GPa$^{-1}$, determined from the slope of the linear plot. From this linear relationship we calculated the T$_N$ values for the pressures at which we determined the lattice and positional parameters of MnTe$_2$ from the high pressure neutron diffraction experiment on the PEARL diffractometer. Fig. 7(c) shows the log-log plot of Néel temperature T$_N$ vs. unit cell volume of MnTe$_2$. The slope of this plot gives $\alpha = -6.0 \pm
0.1$ which is much larger than the Bloch rule value of $\alpha = -3.3$. Our result therefore point towards a spectacular breakdown of Bloch’s rule in MnTe$_2$. We note that transition to a non-magnetic state of the Mn$^{2+}$ ions in MnTe$_2$ was reported[@vulliet01] from the resistivity and Mössbauer study, and also evidenced by the pressure variation of infrared reflectivity investigated by Mita et al. [@mita08]. Our experiments however did not show the volume collapse observed in high pressure X-ray diffraction experiments [@fjellvag85; @fjellvag95]. It is therefore plausible that we did not reach the transition pressure during the present high pressure neutron diffraction experiments. The exact pressure at which the transition to a non-magnetic state is expected to happen depends sensitively on the experimental conditions.
The present neutron diffraction data contain in principle the magnetic moment information because neutron diffraction probes both crystal and the magnetic structures and the intensities of the magnetic reflections when put to the absolute scale by using the intensities of the nuclear reflections can give the ordered moment values. However this is not an easy task especially in a high pressure experiment using a large Paris-Edinburg pressure cell. The high absorption of the pressure cell and also a very high background hinders accurate determination of the nuclear and magnetic intensities. In the present case despite our efforts the determination of the pressure dependence of ordered moment from the neutron diffraction data was not successful. We know however from our calculations (using pressure dependence of the structural parameters) that the ordered moment of Mn ions does not change at all (or very little) in the range 0 - 9 GPa investigated. The present high pressure neutron diffraction data seem to support this result. The intensity ratio of the magnetic and nuclear Bragg peaks do not change very much and is within the accuracy in the pressure range investigated.
First-Principles Study
======================
Calculated Pressure Variation of Neél Temperature
-------------------------------------------------
To gain understanding on the significantly large pressure dependence of the Neél temperature in MnTe$_2$ we carried out theoretical investigation in terms of first-principles DFT calculations. Fig. \[Fig2\] shows the comparison of the spin-polarized density of states of MnTe$_2$ at ambient pressure and at a pressure of 9.16 GPa, the highest pressure studied in the present calculations. The spin-polarized calculations within GGA+$U$ gave rise to a magnetic moment of 4.6 $\mu_B$ (4.5 $\mu_B$) at Mn site together with a moment of 0.03 $\mu_B$ (0.05 $\mu_B$) at Te site for the ambient (P = 9.16 GPa) pressure condition, suggesting the high-spin state of Mn at both ambient and high pressure conditions, in agreement with experimental findings. Both ambient pressure and high pressure phases were found to be insulating, with a gap at Fermi energy, marked as zero in the figure. The Mn-$d$ states are fully occupied in the majority spin channel and completely empty in minority spin channel, in correspondence with high spin state of Mn in its nominal 2+ valence state. The comparison of the density of states between ambient pressure and at high pressure though, shows enhancement of the Mn-$d$ band width by $\approx$ 1 eV, indicating the hopping interaction between Mn-$d$ and Te$-p$ to increase substantially in moving from ambient to high pressure phase.
\[bloch\]
To extract the various magnetic interactions ($J$’s) between the Mn spins, we calculated the GGA+$U$ total energies for various configurations of Mn spins and mapped the total energy onto an underlying S = 5/2 Heisenberg model. Calculations were carried out for six different pressures, 0.0 GPa, 0.4 GPa, 2.16 GPa, 5.32 GPa, 8.43 GPa and 9.16 GPa. The dominant magnetic interactions considered in our calculation of $J$’s, were $J_1$, between the first nearest neighbor (1NN) Mn atoms, connected to each other by the corner-shared Te atoms, and $J_2$, between the second nearest neighbor (2NN) Mn atoms, connected to each other through Te-Mn-Te bridges.
Apart from the ferromagnetic (FM) configuration, with all Mn spins in the supercell pointing in the same direction, two different antiferromagnetic (AFM) configurations, AFM1 and AMF2 were considered, with antiferromagnetic arrangement of 1NN Mn and 2NN Mn spins. The GGA+$U$ total energies corresponding to AFM configurations, measured with respect to the energy of FM configuration, turned out to be negative, for all the studied pressures, in accordance with dominance of anti-ferromagnetic interactions. Extracting J$_{1}$ and J$_{2}$ by mapping the total energy onto the spin Hamiltonian, given by $H= -J{_1}\sum_{nn}S^{i}_{Mn}.S^{j}_{Mn} - J{_2}\sum_{2nn}S^{i}_{Mn}.S^{j}_{Mn}$, gave $J_{2}$ a small fraction of $J_1$, with $J_{1}$/$J_{2}$ = 0.09 at ambient pressure and 0.12 at 9.16 GPa, suggesting the magnetism being primarily governed by $J_1$. The pressure dependence of exchange interactions is shown in Figure \[Fig4\].
\[bloch\]
With the knowledge of J’s, we calculated the Ne[é]{}l temperature T$_{N}$ using Mean-field theory, given by $T^{mf}_{N}=\frac{S(S+1)J_{0}}{3K_{B}}$, where $J_{0}$ is the net effective interaction 12$J_{1}$ + 4$J_{2}$, S = 5/2, and $K_{B}$ is the Boltzmann constant. Mean field is expected to overestimate the transition temperature, though the trend is expected to captured well, which is governed by $J$’s. The computed log\[$T_{N}$(P)/$T_{N}$(0)\] plotted as a function of log($V$) is shown in Fig. \[Fig5\]. The straight line fit to the calculated data points gives rise to a slope of -5.61, close to the experimental estimate of -6.0 $\pm$ 0.1. Both our experimental results and ab-initio calculations, thus establish that Bloch’s rule is largely violated in MnTe$_2$. In the following we theoretically investigate the microscopic origin of this behavior.
\[bloch\]
Microscopic Origin of breakdown of Bloch rule
---------------------------------------------
Bloch’s rule has been found to be very successful with many magnetic insulators, especially the transition metal oxides and fluorides. The question then arises: what makes the Bloch’s rule fail for MnTe$_2$? In order to explore this, we extracted the hopping interactions $b^{ca}$, where $c$ and $a$ signifies Mn and Te, respectively, by carrying out N-th order muffin-tin orbital (NMTO) based NMTO-downfolding calculations.[@nmto] This involved construction of real-space representation of the Mn $d$- O $p$ Hamiltonian in Wannier function basis out of the full DFT calculations, by integrating out all the degrees of freedom other than Mn $d$ and Te $p$. Our NMTO-downfolding calculations to extract the dependence of $b^{ca}$ on cation-anion distance $r$ gave, $b^{ca} \sim \frac{1}{r^{4.2}}$, instead of $b^{ca} \sim \frac{1}{r^{2.5}}$ assumed for derivation of the Bloch’s rule. This gives $T_N ~\sim \frac{1}{r^{17}} \sim \frac{1}{V^{5.67}}$, very close to the estimate obtained from total energy calculations, as well as from the experiment. This points to the fact that violation of Bloch’s rule in case of MnTe$_2$ is caused due to the deviation in the distance dependence of $b^{ca}$, from the $\frac{1}{r^{2.5}}$ behavior rather than that by $U$ or $\Delta$. We find that the distance dependence of $b^{ca}$ found for MnTe$_2$, is more like the canonical behaviour,[@oka] in which the interatomic matrix elements are supposed to scale with distance as $\frac{1}{r^{l+l'+1}}$ where $l$ and $l'$ are the angular momenta of the orbitals involved. In case of several TM oxides, on the other hand, anaylysis of DFT band structure,[@dd] gave rise to a $\frac{1}{r^{l+l'}}$ behavior, similar to that obtained from molecular orbital theory or configuration interaction method on KNiF$_4$ or MnO or MnF$_2$[@smith69; @shrivastava76]. This presumably originates from differential nature of Te 4$p$ wavefunctions compared with that of 2$p$ or 3$p$, which together with non monotonic pressure dependence of position parameter, x, influences the super-exchange interaction in a differential manner.
Summary
=======
In conclusion, the Néel temperature of MnTe$_2$ was found to show unusually large pressure dependence of about $12$ K GPa$^{-1}$, which has been confirmed in the present study through more rigorous and reliable high pressure neutron diffraction experiments compared to that in literature, as well as through first-principles density functional theory calculations. Our measured pressure dependence of the Néel temperature and unit cell volume gave $\alpha = -6.0 \pm 0.1$ which is much larger than that expected from the Bloch’s rule $\alpha=\frac{d\log T_N}{d\log V}=-\frac{10}{3} \approx -3.3$. The calculated pressure dependence of Néel temperature gave rise to $\alpha= -5.61$ in good agreement with the experimental estimate. We provided a microscopic understanding of this behavior in terms of the distance dependence of Mn-Te hopping interaction upon application of pressure, which showed significant deviation from that for NiF$_4$ or MnO or MnF$_2$[@smith69; @shrivastava76].
Finally, the large pressure dependence of magnetic interactions and magnetic ordering temperature provide us with a handle to tune the properties of magnetic materials, which can lead to important technological applications. The present study, should have important bearing on this topic.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider the transmission of classical information through a degraded broadcast channel, whose outputs are two quantum systems, with the state of one being a degraded version of the other. Yard *et al. *\[[*IEEE Trans. Inf. Theory*, **57**(10):7147–7162, 2011](https://ieeexplore.ieee.org/document/6034754)\] proved that the capacity region of such a channel is contained in a region characterized by certain entropic quantities. We prove that this region satisfies the strong converse property, that is, the maximal probability of error incurred in transmitting information at rates lying outside this region converges to one exponentially in the number of uses of the channel. In establishing this result, we prove a second-order Fano-type inequality, which might be of independent interest. A powerful analytical tool which we employ in our proofs is the tensorization property of the quantum reverse hypercontractivity for the quantum depolarizing semigroup.'
address: |
$^{1}$Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences\
University of Cambridge, Cambridge CB3 0WA, United Kingdom\
$^{2}$Technische Universit[ä]{}t M[ü]{}nchen, 80333 M[ü]{}nchen, Germany
author:
- '[Hao-Chung Cheng$^{1}$, Nilanjana Datta$^{1}$, Cambyse Rouzé$^{1,2}$ ]{}'
title: '****'
---
Introduction
============
A broadcast channel models noisy one-to-many communication, examples of which abound in our daily lives. It can be used to transmit information to two[^1] receivers (say, Bob and Charlie) from a single sender (say, Alice). It was introduced by Cover in 1972 [@Cov72]. In the most general case, part of the information (the [*common part*]{}) is intended for both the receivers, while part of the information ([*the private part*]{}) consists of information intended for Bob and Charlie separately. Classically, the so-called discrete memoryless broadcast channel is modelled by a conditional probability distribution $\{p_{YZ|X}(y,z|x)\}$, where the random variables $X,Y,Z$ take values in ${\mathcal{X}}$ (the [*input alphabet*]{}), and alphabets ${\mathcal{Y}}$ and ${\mathcal{Z}}$ (the [*output alphabets*]{}), respectively. Hence, $X$ models Alice’s input to the channel, while $Y$ and $Z$ correspond to the outputs received by Bob and Charlie, respectively. Suppose Alice sends her messages (or information) through multiple (say $n$) successive uses of such a channel, with $R_B$ and $R_C$ being the rates at which she transmits private information to Bob and Charlie, respectively, and $R$ being the rate at which she transmits common information to both of them. A triple $(R_B, R_C, R)$ is said to be an [*achievable rate triple*]{} if the probability that an error is incurred in the transmission of the messages vanishes in the limit $n \to \infty$. In other words, these rates correspond to reliable transmission of information. Obviously, there is a tradeoff between these three rates: if one of them is high, the others are lowered in order to ensure that the common- as well as private information are transmitted reliably. The set of all achievable rate triples defines the [*achievable rate region*]{}, and its closure defines the [*capacity region*]{} of the broadcast channel.
Determining the capacity region for a general broadcast channel remains a challenging open problem. However, certain special cases have been solved (see e.g. [@Ber73; @Wyn73; @Ber74; @Ber77; @AGK76; @Gal74; @vdM75; @Cov75; @KM77; @vdM77; @Sat78; @Mar79; @Gam79; @GvdM81; @Cov98; @Nai10; @Ooh15a; @Ooh15b; @Ooh16; @GK11]), the first of these being the case of the so-called [*degraded broadcast channel*]{} (DBC). This is a broadcast channel for which the message that Charlie receives is a degraded version of the message that Bob receives. In other words, there exists a stochastic map which when acting on the message that Bob receives, yields the message that Charlie receives. Hence $p_{Y,Z|X}(y,z|x) = p_{Z|Y}(z|y)p_{Y|X}(y|x)$, and the three random variables $X,Y$ and $Z$ form a Markov Chain $X-Y-Z$. Let us focus on the case in which there is no common information[^2] and hence the capacity region is specified by achievable rate pairs $(R_B, R_C)$. In this case, the capacity region has been shown to be given by [@Ber73; @Gal74; @AK75] $$\begin{aligned}
& \bigcup \{(R_B, R_C) \,: R_B \leq I(X;Y|U), R_C \leq I(U;Z)\},\end{aligned}$$ where the union is over all joint probability distributions $\{p_{UX}(u,x)\}_{u \in {{\mathcal{U}}}, x \in {{\mathcal{X}}}}$, with $U$ being an auxiliary random variable taking values in an alphabet ${\mathcal{U}}$ with cardinality $|{\mathcal{U}}| \leq \min \{|{\mathcal{X}}|,|{\mathcal{Y}}|, |{\mathcal{Z}}|\} + 1.$ Here $I(X;Y|U)$ and $I(U;Z)$ denote the conditional mutual information (between $X$ and $Y$ conditioned on $U$) and the mutual information between $U$ and $Z$, respectively, and are the entropic quantities characterizing the achievable rate region.
In this paper, we consider a [*classical-quantum degraded broadcast channel*]{} (c-q DBC), which we denote by $\mathscr{W}^{X \to BC}$. Here too, the input to the channel is classical and denoted by a random variable $X$ but the outputs are states of quantum systems $B$ and $C$. The channel is degraded in the sense that there exists some other quantum channel (say ${{\mathcal{N}}}$), which when acting on the state of the system $B$ yields the state of the system $C$. Bob and Charlie receive the systems $B$ and $C$ respectively, and perform measurements on them in order to infer the classical messages that Alice sent to each of them. The channel is assumed to be memoryless and the achievable rates are computed in the asymptotic limit ($n \to \infty$, where $n$ denotes the number of successive uses of the channel). The achievable rate region for this channel was studied by Yard [*et al.*]{} [@YHD11] and later by Savov and Wilde [@SW12]. Let $R_B$ and $R_C$ denote the rates at which Alice sends private information to Bob and Charlie respectively, and let $R$ be her rate of transmission of common information to both of them. See Figure \[fig:protocol\] for the illustration.
It was shown in [@YHD11] (see also [@SW12]) that any rate triple $(R, R_B, R_C) $ satisfying $$\begin{aligned}
\begin{split}
&R_B \leq I(X;B|U)_\sigma,\\
&R + R_C \leq I(U;C)_\sigma,
\end{split}
\label{rf}\end{aligned}$$ lies in the achievable rate region[^3]. Here the entropic quantities, appearing in the above inequalities are, taken with respect to a state $\sigma_{UXBC}$ of the following form $$\begin{aligned}
\sigma_{UXBC} &= \sum_{(u,x)\in \mathcal{U}\times \mathcal{X} } p_U(u)\, p_{X|U}(x|u)\, |u \rangle \langle u|_U \otimes |x \rangle \langle x|_X \otimes \sigma^x_{BC} .
$$ Here we use $X$ and $U$ to denote both random variables (taking values in finite sets ${{\mathcal{X}}}$ and ${{\mathcal{U}}}$, respectively), as well as quantum systems whose associated Hilbert spaces, ${{\mathcal{H}}}_X$ and ${{\mathcal{H}}}_U$, have complete orthonormal bases $\{|x\rangle\}$ and $\{|u\rangle\}$ labelled by the values taken by these random variables[^4]. Hence, $|\mathcal{U}| := \dim \mathcal{H_U}$ and $|\mathcal{X}| := \dim \mathcal{H_X}$.
Moreover, Yard [*et al.*]{} [@YHD11 Theorem 2] established that the capacity region for such a c-q DBC is contained in a region specified by the following inequalities: $$\begin{aligned}
\begin{split} \label{rfc}
&R_B \leq I(X;B|U)_\omega,\\
&R + R_C \leq I(U;C)_\omega,
\end{split}\end{aligned}$$ for some state $\omega_{UXBC}$ of the following (more general) form, in which the system $U$ is a quantum system: $$\omega_{UXBC} = \sum_{x\in\mathcal{X} } p_X(x) \rho^x_{U}\otimes |x \rangle \langle x| \otimes \rho^x_{BC},$$ where $\forall$ $x \in {{\mathcal{X}}}$, $\rho^x_{U}$ is a state of the quantum system $U$.
The above result establishes that for any rate triple $(R_B, R_C, R)$ which does not satisfy the inequalities for $\rho_{UXBC}$ of the above form, the [[maximum]{}]{} probability of incurring an error in the transmission of information is bounded away from zero, even in the asymptotic limit. In this paper, we show that the region spanned by such rate triples satisfies the so-called [*strong converse property*]{}, that is, for any rate triple which lies outside this region, the [maximal]{} probability of error in the transmission of information is not only bounded away from zero but goes to one in the asymptotic limit. Moreover, the convergence to one is exponential in $n$. A precise statement of this result is given by Corollary \[coro:exponential\] of Section \[main\] below. We first establish this strong converse property in the case in which no common information is sent (i.e. $R=0$) and then discuss how this result can be extended to the general case in which both private and common information is sent by Alice.
**Organization of the paper:** In Section \[sec:notation\], we introduce necessary notation and the information-theoretic protocol of c-q DBC coding. In Section \[main\], we state our main results. In Section \[sec:sc\], we prove a second-order Fano-type inequality for c-q channel coding, which is a main ingredient for establishing the second-order strong converse bound, which we prove in Section \[sec:sc\_DBC\].
Notations and Definitions {#sec:notation}
=========================
Throughout this paper, we consider finite-dimensional Hilbert spaces, and discrete random variables which take values in finite sets. The subscript of a Hilbert space (say $B$), denotes the quantum system (say ${{\mathcal{H}}}_B$) to which it is associated. We denote its dimension as $d_B := {\rm{dim}}\,{\mathcal{H}}_B$. Let $\mathbb{N}$, $\mathbb{R}$, and $\mathbb{R}_{\geq 0}$ be the set of natural numbers, real numbers, and non-negative real numbers, respectively. Let ${\mathcal{B}}({\mathcal{H}})$ denote the algebra of linear operators acting on a Hilbert space ${\mathcal{H}}$, ${\mathcal{P}}({\mathcal{H}}) \subset {\mathcal{B}}({\mathcal{H}})$ denote the set of positive semi-definite operators, ${\mathcal{D}}({\mathcal{H}}) \subset {\mathcal{P}}({\mathcal{H}})$ the set of quantum states (or density matrices): ${\mathcal{D}}({\mathcal{H}}) :\{ \rho \in {\mathcal{P}}({\mathcal{H}})\,:\, \operatorname{Tr}[\rho] = 1\}$. A quantum operation (or quantum channel) is a superoperator given by a linear completely positive trace-preserving (CPTP) map. A quantum operation ${\mathcal{N}}^{A \to B}$ maps operators in ${\mathcal{B}}({\mathcal{H}}_A)$ to operators in ${\mathcal{B}}({\mathcal{H}}_B)$. A superoperator $\Phi: {\mathcal{B}}({\mathcal{H}}) \to {\mathcal{B}}({\mathcal{H}})$ is said to be unital if $\Phi(\mathbb{I}) = \mathbb{I}$, where $\mathbb{I}$ denotes the identity operator in ${\mathcal{B}}({\mathcal{H}})$ . We denote the identity superoperator as ${\rm{id}}$. For any finite set $\mathcal{M}$, a positive-operator valued measure (POVM) on $\mathcal{H}$ is a set of positive semi-definite operators $\{\Pi^m\}_{m \in \mathcal{M}}$ satisfying $\Pi^m \in \mathcal{P(H)}$, for every $m\in\mathcal{M}$, and $\sum_{m\in\mathcal{M}} \Pi^m = \mathbb{I}$.
The von Neumann entropy of a state $\rho$ is defined as $S(\rho):= - \operatorname{Tr}[\rho \log \rho]$, with the logarithm being taken to base $2$. The quantum relative entropy between a state $\rho \in {\mathcal{D}}({\mathcal{H}})$ and a positive semi-definite operator $\sigma$ is defined as $$\begin{aligned}
D(\rho||\sigma) &: = \operatorname{Tr}\left[\rho (\log \rho - \log \sigma)\right].\end{aligned}$$ It is well-defined if ${\rm{supp}}\,\rho \subseteq {\rm{supp}}\,\sigma$, and is equal to $+\infty$ otherwise. Here ${\rm{supp}}\,A$ denotes the support of the operator $A$. The quantum relative Rényi entropy of order $\alpha$, for $\alpha \in (0,1)$, is defined as follows [@Pet86]: $$\begin{aligned}
D_\alpha(\rho||\sigma) &:= \frac{1}{\alpha-1} \log \operatorname{Tr}\left[ \rho^\alpha \sigma^{1-\alpha} \right].\end{aligned}$$ It is known that $D_\alpha(\rho||\sigma) \to D(\rho||\sigma)$ as $\alpha \to 1$ (see e.g. [@Tom16 Corollary 4.3], [@Ume62]). An important property satisfied by these relative entropies is the so-called [*[data-processing inequality]{}*]{}, which is given by $ D_\alpha(\Lambda(\rho)\|\Lambda(\sigma)) \leq D_\alpha(\rho\|\sigma)$ for all $\alpha\in(0,1)$ and quantum operations $\Lambda$. This induces corresponding data-processing inequalities for the quantities derived from these relative entropies, such as the quantum mutual information information and the conditional entropy .
For a bipartite state $\rho_{AB} \in \mathcal{D}(\mathcal{H}_A \otimes \mathcal{H}_B)$, the quantum mutual information and the conditional entropy are given in terms of the quantum relative entropy as follows: $$\begin{aligned}
I(A;B)_\rho &= D\left( \rho_{AB} \| \rho_A \otimes \rho_B \right); \label{eq:mutual} \\
H(A|B)_\rho &= -D\left( \rho_{AB} \| \mathbb{I}_A \otimes \rho_B \right). \label{eq:conditional}\end{aligned}$$ The following inequality plays a fundamental role in our proofs.
\[Araki-Lieb-Thirring inequality [[@Ari76; @LT76]]{}\] \[lemm:ALT\] For any $A,B\in \mathcal{P}(\mathcal{H})$, and $r\in[0,1]$, $$\begin{aligned}
\operatorname{Tr}\left[ B^{\frac{r}{2}} A^r B^{\frac{r}{2}} \right] \leq \operatorname{Tr}\left[ \left( B^\frac12 A B^\frac12 \right)^r\, \right].
\end{aligned}$$
The proof of one of our main results (Theorem \[theo:Fano\]) employs a powerful analytical tool, namely, the so-called [*[quantum reverse hypercontractivity]{}*]{} of a certain quantum Markov semigroup (QMS) and its tensorization property. Let us introduce these concepts and the relevant results in brief. For more details see e.g. [@BDR18] and references therein. The QMS that we consider is the so-called [*[generalized quantum depolarizing semigroup (GQDS)]{}*]{}. In the Heisenberg picture, for any state $\sigma >0$ on a Hilbert space ${{\mathcal{H}}}$, the GQDS with invariant state $\sigma$ is defined by a one-parameter family of linear completely positive (CP) unital maps $\left(\Phi_t\right)_{t \geq 0}$, such that for any $X \in {{\mathcal{B}}}({{\mathcal{H}}})$, $$\begin{aligned}
\label{eq-gqds}
\Phi_t(X) = \operatorname{\mathrm{e}}^{-t}X + (1-\operatorname{\mathrm{e}}^{-t}) \operatorname{Tr}[\sigma X] \,\mathbb{I}.\end{aligned}$$ In the Schrödinger picture, the corresponding QMS is given by the family of CPTP maps $\left(\Phi^\star_t\right)_{t \geq 0}$, such that $$\operatorname{Tr}[Y \Phi_t (X)] = \operatorname{Tr}[\Phi^\star_t(Y) X], \quad \forall, \,\, X,Y \in {{\mathcal{B}}}({{\mathcal{H}}}).$$ The action of $\Phi^\star_t$ on any state $\rho \in {{\mathcal{D}}}({{\mathcal{H}}})$ is that of a generalized depolarizing channel, which keeps the state unchanged with probability $e^{-t}$, and replaces it by the state $\sigma$ with probability $(1 - \operatorname{\mathrm{e}}^{-t})$: $$\Phi^\star_t(\rho) = \operatorname{\mathrm{e}}^{-t}\rho + (1-\operatorname{\mathrm{e}}^{-t}) \sigma\,.$$ Note that $ \Phi^\star_t(\sigma) = \sigma$ for all $t\geq 0$, and that $\sigma$ is the unique invariant state of the evolution.
To state the property of quantum reverse hypercontractivity, we define, for any $X\in\mathcal{B(H)}$, the non-commutative weighted $L_p$ norm [with respect to]{} the state $\sigma \in \mathcal{D}({{\mathcal{H}}})$, for any $p \in \mathbb{R}\backslash\{0\}$[^5]: $$\begin{aligned}
\label{eq:wLp}
\left\| X \right\|_{p, \sigma} := \left( \operatorname{Tr}\left[ \left| \sigma^{\frac{1}{2{p}}} X \sigma^{\frac{1}{2 {p}}} \right|^{{p}} \right] \right)^{\frac{1}{{p}}}.\end{aligned}$$ A QMS $\left(\Phi_t\right)_{t \geq 0}$ is said to be reverse $p$-contractive for $p <1$, if $$\begin{aligned}
\label{contract}
|| \Phi_t(X)||_{p,\sigma} \geq ||X||_{p,\sigma}, \quad \forall \, X >0.\end{aligned}$$ The GQDS can be shown to satisfy a stronger inequality: $\forall$ $ p<q <1$, $$\begin{aligned}
\label{QRHC}
|| \Phi_t(X)||_{p,\sigma} &\geq ||X||_{q,\sigma}, \quad \forall \, X >0,\end{aligned}$$ for $$\begin{aligned}
\label{tcond}
t &\geq \frac{1}{4 \alpha_1 ({{\mathcal{L}}})} \log \left(\frac{p-1}{q-1}\right),\end{aligned}$$ where $\alpha_1 ({{\mathcal{L}}})>0$ is the so called [*[modified logarithmic Sobolev constant]{}*]{}, and ${{\mathcal{L}}}$ denotes the generator of the GQDS, which is defined through the relation $\Phi_t(X) = \operatorname{\mathrm{e}}^{-t{{\mathcal{L}}}}(X)$ and is given by $${{\mathcal{L}}}(X) = X - \operatorname{Tr}[\sigma X] \mathbb{I}.$$ The inequality is indeed stronger than since the map $p \mapsto ||X||_{p,\sigma}$ is non-decreasing.
In the context of this paper, instead of the GQDS defined through , we need to consider the QMS $\left(\Phi_{t, x^n}\right)_{t \geq 0}$, with $\Phi_{t, x^n}$ being a CP unital map acting on ${{\mathcal{B}}}({{\mathcal{H}}}^{\otimes n})$, and being labelled by sequences $x^n \equiv (x_1,x_2, \ldots, x_n) \in {{\mathcal{X}}}^n$, where ${{\mathcal{X}}}$ is a finite set. For any $x \in {{\mathcal{X}}}$, let $\rho^x \in {{\mathcal{D}}}({{\mathcal{H}}})$. Further, let $$\begin{aligned}
\rho^{x^n} &:= \rho^{x_1} \otimes \cdots \otimes \rho^{x_n} \,\in {{\mathcal{D}}}({{\mathcal{H}}}^{\otimes n}).
\end{aligned}$$ Then, $$\begin{aligned}
\Phi_{t,x^n} &:= \Phi_{t,x_1} \otimes \cdots \otimes \Phi_{t,x_n},
\end{aligned}$$ where $(\Phi_{t,x})_{t\ge 0}$ is a GQDS with invariant state $\rho^x$. We denote by ${{\mathcal{K}}}_{x^n}=\sum_{i=1}^n\widehat{{{\mathcal{L}}}}_{x_i}$ the generator of $(\Phi_{t,x^n})_{t\ge 0}$ where $\widehat{{{\mathcal{L}}}}_{x_i}={\rm{id}}^{\otimes i-1} \otimes {{{\mathcal{L}}}}_{x_i} \otimes {\rm{id}}^{\otimes n-i}$, with ${{{\mathcal{L}}}}_{x_i}$ being the generator of the GQDS $(\Phi_{t,x_i})_{t\geq 0}$. If the modified logarithmic Sobolev constant $\alpha_1(\mathcal{K}_{x^n})$ is independent of $n$, or satisfies an $n$-independent lower bound, then it is called the *tensorization property* of the GQDS. The following tensorization property of the quantum reverse hypercontractivity of the above tensor product of GQDS was established in [@BDR18] ( [See [@mossel2013reverse] for its classical counterpart, as well as [@CKMT15] for its extension to doubly stochastic QMS]{}):
\[lemm:RHC\] \
For the QMS $\left(\Phi_{t, x^n}\right)_{t \geq 0}$ introduced above, for any $\mathsf{p}\leq \mathsf{q}< 1$ and for any $t$ satisfying $t\geq \log \frac{\mathsf{p}-1}{\mathsf{q}-1}$, the following inequality holds: $$\begin{aligned}
\left\| \Phi_{t,x^n}(G_n) \right\|_{\mathsf{p},\rho^{x^n}}
\geq \left\| G_n \right\|_{\mathsf{q},\rho^{x^n}}, \quad \forall\, G_n>0.
\end{aligned}$$ In other words, $\alpha_1({{\mathcal{K}}}_{x^n})\ge \frac{1}{4}$.
Classical-quantum (c-q) broadcast channel {#sec:c-q_DBC}
-----------------------------------------
We define the *classical-quantum (c-q) degraded broadcast channel* as follows.
\[defn:channel\] A *classical-quantum broadcast channel* ${\mathscr{W}}^{X\to BC}$ is a quantum operation defined as follows: $$\begin{aligned}
\begin{split}
\mathscr{W} \equiv {\mathscr{W}}^{X\to BC}: \mathcal{X} &\to \mathcal{D}(\mathcal{H}_B\otimes \mathcal{H}_C); \\
x &\mapsto \rho_{BC}^x.
\end{split}
\end{aligned}$$ Here $X$ is a random variable which takes values in a finite set ${\mathcal{X}}$. A classical input $x \in {\mathcal{X}}$ to this channel, yields a quantum state $\rho_{BC}^x$ as output. Moreover, such a channel is said to be a c-q degraded broadcast channel (c-q DBC), if there exists a quantum channel ${\mathcal{N}}^{B \to C}$ such that $ \forall \, x \in {\mathcal{X}}$ the reduced state of the system $C$, $\rho^x_{C} = \operatorname{Tr}_B (\rho^x_{BC}),$ satisfies $$\begin{aligned}
\rho^x_{C} &= {\mathcal{N}}^{B \to C}(\rho^x_B), \quad {\hbox{with}}\,\, \rho^x_B= \operatorname{Tr}_C (\rho^x_{BC}).
\end{aligned}$$ Here $\operatorname{Tr}_B$ and $\operatorname{Tr}_C$ denote the partial traces over ${\mathcal{H}}_B$ and ${\mathcal{H}}_C$, respectively. As in the classical case, we consider Alice to be the sender (she hence holds $X$) while the quantum systems $B$ and $C$ are received by Bob and Charlie, respectively.
As in the classical case, we assume the channel to be *memoryless* and consider multiple (say $n\in\mathbb{N}$) successive uses of it. In this scenario, one hence considers a sequence of channels $\{\mathscr{W}^{\otimes n} \}_{n\in\mathbb{N}}$ such that for all $x^n \equiv (x_1,x_2,\ldots,x_n) \in \mathcal{X}^n$, $$\begin{aligned}
\mathscr{W}^{\otimes n}(x^n) := \mathscr{W}(x_1) \otimes \mathscr{W}(x_2) \otimes\cdots \otimes \mathscr{W}(x_n).\end{aligned}$$
As mentioned above, we first focus on the case in which there is no common information, and hence $R=0$. In this case, the inequalities reduce to $$\begin{aligned}
\begin{split} \label{rfcnc}
R_B & \leq I(X;B|U)_\omega,\\
R_C & \leq I(U;C)_\omega,
\end{split}\end{aligned}$$ for some state $$\begin{aligned}
\omega_{UXBC} &= \sum_{x \in \mathcal{X} } p_X(x) \rho^x_{U}\otimes |x \rangle \langle x| \otimes \rho^x_{BC}.
\label{r2}\end{aligned}$$
Let Alice’s private messages to Bob and Charlie be labelled by the elements of the index sets ${{\mathcal{M}}}:= \{1,2,\ldots , |{{\mathcal{M}}}|\}$ and ${{\mathcal{K}}}:= \{1,2,\ldots , |{{\mathcal{K}}}|\}$, respectively. For any $(R_B, R_C) \in {\mathbb{R}}^2_{\geq 0}$, and any $n \in {\mathbb{N}}$, an $(n, R_B, R_C)$ code is given by the pair $\left(\mathcal{E}_n, \mathcal{D}_n\right)$, where ${{\mathcal{E}}}_n$ is the encoding map $$\begin{aligned}
\begin{split}
{{\mathcal{E}}}_{n} : {{\mathcal{M}}}\times {{\mathcal{K}}}&\to {{\mathcal{X}}}^n;\\
(m,k) &\mapsto x^n(m,k) = (x_1(m,k),x_2(m,k) \ldots,x_n(m,k)),
\end{split}\end{aligned}$$ with $|{{\mathcal{M}}}| = \lfloor 2^{nR_B}\rfloor$ and $|{{\mathcal{K}}}| = \lfloor 2^{nR_C}\rfloor$. Henceforth, for simplicity we assume that $2^{nR_B}$ and $2^{nR_C}$ are integers. The decoding map ${{\mathcal{D}}}_n$ consists of two POVMs: $\Pi_{B^n}:=\{\Pi_{B^n}^m\}_{m \in {{\mathcal{M}}}}$, and $\Pi_{C^n}:=\{\Pi_{C^n}^k\}_{k \in {{\mathcal{K}}}}$ where $\Pi_{B^n}^m \in {{{\mathcal{P}}}({{\mathcal{H}}}_B^{\otimes n})}$ and $\Pi_{C^n}^k \in {{{\mathcal{P}}}({{\mathcal{H}}}_C^{\otimes n})}$ for any $(m,k) \in {{\mathcal{M}}}\times {{\mathcal{K}}}$ and $\sum_{ m \in {{\mathcal{M}}}} \Pi_{B^n}^m = \mathds{1}_{B^n}$ and $\sum_{ k \in {{\mathcal{K}}}} \Pi_{C^n}^k = \mathds{1}_{C^n}$.
If the classical sequence $x^n(m,k)$ (which is the codeword corresponding to the message $(m,k)$) is sent through $n$ successive uses of the memoryless c-q DBC $\mathscr{W}^{X\to BC}$, the output is the product state $$\begin{aligned}
\rho_{B^n C^n}^{x^n(m,k)} &= \rho_{BC}^{x_1(m,k)}\otimes \rho_{BC}^{x_2(m,k)}\ldots \otimes \rho_{BC}^{x_n(m,k)} \in \mathcal{D}\left( \mathcal{H}_{BC}^{\otimes n} \right),\end{aligned}$$ where $\mathcal{H}_{BC} \equiv \mathcal{H}_B \otimes \mathcal{H}_C$. The probability that an error is incurred in sending the message $(m,k)$ is then given by $$1-\operatorname{Tr}\left[ \rho_{B^n C^n}^{x^n(m,k)} \left( \Pi_{B^n}^{m} \otimes \Pi_{C^n}^{k} \right) \right] .$$ The [*maximal probability of error*]{} for the code $\left(\mathcal{E}_n, \mathcal{D}_n\right)$ is then defined as follows: $$\begin{aligned}
p_{\max}\left(\mathcal{E}_n, \mathcal{D}_n\right) &:= \max_{(m,k) \in {{\mathcal{M}}}\times {{\mathcal{K}}}}
\left( 1- \operatorname{Tr}\left[ \rho_{B^n C^n}^{x^n(m,k)} (\Pi_{B^n}^{m} \otimes \Pi_{C^n}^{k} )\right]\right).\end{aligned}$$ and the [*average probability of error*]{} for the code $\left(\mathcal{E}_n, \mathcal{D}_n\right)$ is defined as $$\begin{aligned}
\label{eq:avg}
p_{\text{avg}}\left(\mathcal{E}_n, \mathcal{D}_n\right) &:= \frac{1}{|\mathcal{M}||\mathcal{K}| } \sum_{(m,k) \in {{\mathcal{M}}}\times {{\mathcal{K}}}}
\left( 1- \operatorname{Tr}\left[ \rho_{B^n C^n}^{x^n(m,k)} (\Pi_{B^n}^{m} \otimes \Pi_{C^n}^{k} )\right]\right).\end{aligned}$$ For any $\eps \in [0,1]$, an $(n, R_B, R_C)$ code $\left(\mathcal{E}_n, \mathcal{D}_n\right)$ is said be an $(n, R_B, R_C, \eps)$ code if $p_{\max}\left(\mathcal{E}_n, \mathcal{D}_n\right) \leq \eps$. For a fixed $\eps \in [0,1)$, a rate pair $(R_B, R_C)$ is said to be *$\eps$-achievable* (under the maximal error criterion) if there exists a sequence of $(n, R_B, R_C, \eps_n)$ codes such that $\eps_n \to \eps$ as $n \to \infty$.
A rate pair $(R_B, R_C)$ is [*achievable*]{} if $\eps=0$. It is clear that any rate pair which is achievable is also $\eps$-achievable for all $\eps \in (0,1)$. For any $\eps \in [0, 1)$, let us define the *$\eps$-achievable rate region* and the *$\eps$-capacity region* of $\mathscr{W}$ as follows: $$\begin{aligned}
\begin{split} \label{eq:C_W_eps}
{{\mathcal{R}}}_\mathscr{W}(\eps) &:= \left\{ (R_B, R_C) \in \mathbb{R}_{\geq 0}^2 \,:\text{ $(R_B,R_C)$ is $\eps$-achievable} \right\}; \\
{{\mathcal{C}}}_\mathscr{W}(\eps) &:= \overline{{{\mathcal{R}}}_\mathscr{W}(\eps)},
\end{split}\end{aligned}$$ where $\overline{ \mathcal{R}_\mathscr{W} } (\eps)$ denotes the closure of the set ${ \mathcal{R}_\mathscr{W} }(\eps)$. The *capacity region* of $\mathscr{W}$ is then ${{\mathcal{C}}}_\mathscr{W}(0)$. It is clear that $$\begin{aligned}
{{\mathcal{R}}}_\mathscr{W}(0) &= \bigcap_{\eps \in (0,1)} {{\mathcal{R}}}_\mathscr{W}(\eps); \quad {{\mathcal{C}}}_\mathscr{W}(0) = \bigcap_{\eps \in (0,1)} {{\mathcal{C}}}_\mathscr{W}(\eps).\end{aligned}$$ Similarly, one can introduce the $\ep$-capacity region under the average error criterion, which we denote as ${{\mathcal{C}}}_{\mathscr{W},\operatorname{avg}}(\eps)$. Since the average probability of error of a code is always less than or equal to the associated maximal probability of error, the inclusion ${{\mathcal{C}}}_{\mathscr{W}}(\eps) \subseteq {{\mathcal{C}}}_{\mathscr{W}, \text{avg}}(\eps)$ holds for all $\eps \in[0,1)$. Furthermore, a standard *codebook expurgation method* [@Wil90], [@GK11 Problem 8.11] shows that it is possible to construct a sequence of $(n, R_B - \frac2n \log n, R_C - \frac2n \log n)$ code with maximal probability of error less than $\sqrt{\eps_n}$ if a sequence $(n, R_B, R_C)$ code with average probability of error $\eps_n$ exists such that $\eps_n \to 0$ as $n\to \infty$. Hence, $${{\mathcal{C}}}_{\mathscr{W},\text{ave}}(0) = {{\mathcal{C}}}_{\mathscr{W}}(0).$$ For convenience, we will only focus on the $\eps$-capacity region ${{\mathcal{C}}}_{\mathscr{W}}(\eps)$ under the maximal error criterion throughout this paper.
Let us define the following entropic regions $$\begin{aligned}
\begin{split}
\mathcal{R}_\mathscr{W}^{\text{ent}} &:= \bigcup \left\{ (R_B, R_C) \in \mathbb{R}_{\geq 0}^2 \, : R_B \leq I(X;B|U)_\omega, R_C \leq I(U;C)_\omega \right\}; \\
\mathcal{C}_\mathscr{W}^{\text{ent}} &:= \overline{ \mathcal{R}_\mathscr{W} },
\end{split} \label{eq:C_W}\end{aligned}$$ where the union is taken over all states $\omega_{UXBC}$ of the form . Yard [*et al.*]{} showed that [@YHD11 Theorem 2] $$\begin{aligned}
\label{eq:Yard}
{{\mathcal{C}}}_\mathscr{W}(0) \subseteq {{\mathcal{C}}}_\mathscr{W}^{\text{ent}}.\end{aligned}$$ We now have all the definitions needed to state our main results.
Main Results {#main}
============
For the memoryless c-q DBC $\mathscr{W}\equiv \mathscr{W}^{X\to BC}$ defined above, the results that we obtain can be briefly summarized as follows. For more detailed and precise statements of these results, see the relevant corollaries and theorems given in Section \[sec:sc\].
***[Result 1 \[Strong converse property, Corollary \[coro:epsilon\]\]]{}*** For any $\eps\in (0,1)$ $$\begin{aligned}
{{\mathcal{C}}}_\mathscr{W} (\eps) &\subseteq {{\mathcal{C}}}_\mathscr{W}^{\text{ent}},\end{aligned}$$ where ${{\mathcal{C}}}_\mathscr{W} (\eps)$ denotes its $\eps$-capacity region (defined in ), whereas $ {{\mathcal{C}}}_\mathscr{W}^{\text{ent}}$ is the region characterized by entropic quantities given in .
This result implies that for any sequence of $(n, R_B, R_C)$ codes $\left(\mathcal{E}_n, \mathcal{D}_n\right)$, for which the rate pair $(R_B, R_C)$ lies outside the region $ {{\mathcal{C}}}_\mathscr{W}^{\text{ent}}$ of the c-q DBC $\mathscr{W}$, $$\begin{aligned}
p_{\max} \left(\mathcal{E}_n, \mathcal{D}_n\right) &\to 1 \quad {\hbox{as}} \; n \to \infty.\label{conv}\end{aligned}$$ This establishes the strong converse property of ${{\mathcal{C}}}_\mathscr{W}^{\operatorname{ent}}$.
***[Result 2 \[Exponential convergence, Corollary \[coro:exponential\]\]]{}*** The convergence in is exponential in $n$: $$\begin{aligned}
p_{\max} \left(\mathcal{E}_n, \mathcal{D}_n\right) & \geq 1 - \operatorname{\mathrm{e}}^{-nf}, \quad \forall n \in \mathbb{N},\end{aligned}$$ where $f = ( \sqrt{ (\sqrt{d_B} + \sqrt{d_C})^2 + \eta } - \sqrt{d_B} - \sqrt{d_C} )^2 > 0$ for some $\eta >0$, which depends only on how far the rate pair $(R_B, R_C)$ is from the region $\mathcal{C}_\mathscr{W}^{\operatorname{ent}}$.
[**[Proof Ingredients:]{}**]{} We prove the above results by first strengthening [@YHD11 Theorem 2] by establishing second order (in $n$) upper bounds on $\eps$-achievable rate pairs $(R_B,R_C)$; see Theorem \[theo:SC\_DBC\] of Section \[sec:sc\]. The key ingredient of the proof of this result is a *second-order Fano-type inequality for c-q channel coding* (Theorem \[theo:Fano\]), which we consider to be a result of independent interest. The latter in turn employs the powerful analytical tool described in Theorem \[lemm:RHC\], namely the tensorization property of the [*quantum reverse hypercontractivity*]{} for the quantum depolarizing semigroup [@BDR18; @LHV18; @CDR19a].
Second-Order Fano-type inequality {#sec:sc}
=================================
In this section we give precise statements of our results (which were summarized in Section \[main\]) and their proofs. In Theorem \[theo:Fano\] below, we establish a second-order Fano-type inequality for standard classical-quantum (c-q) channel[^6] coding. This theorem is a key ingredient in the proof of the second-order strong converse bound for the c-q degraded broadcast channel $\mathscr{W}^{X \to BC}$ (Theorem \[theo:SC\_DBC\]), which leads to our main results (Corollaries \[coro:epsilon\] and \[coro:exponential\]).
\[Second-order Fano-type inequality for c-q channel coding\] \[theo:Fano\] Let $\mathcal{M},\mathcal{K}$, and $\mathcal{X}$ denote arbitrary finite sets, and let the map $x\mapsto \rho_B^x \in \mathcal{D}(\mathcal{H}_B)$ denote a c-q channel for all $x\in\mathcal{X}$. Consider the following encoding map: $\forall$ $m\in\mathcal{M}$, $${{\mathcal{E}}}_n: m \mapsto x^n(m,k) \in \mathcal{X}^n\quad {\hbox{with probability}}\,\, q(k),$$ where $\{q(k)\}_{k \in {{\mathcal{K}}}}$ denotes an arbitrary probability distribution on $\mathcal{K}$. Further, let ${{\mathcal{D}}}_n$ denote a decoding map given by a POVM $\{\Pi^m_{B_n}\}_{m \in \mathcal{M}}$. If $({{\mathcal{E}}}_n,{{\mathcal{D}}}_n)$ are such that for some $\eps \in (0,1)$, $$\begin{aligned}
\label{eq:Fano_condition}
\prod_{(m,k) \in \mathcal{M}\times \mathcal{K}} \left( \operatorname{Tr}\left[ \rho_{B^n}^{x^n(m,k)} \Pi_{B^n}^m \right] \right)^{\frac{1}{|\mathcal{M}|}q(k)} \geq 1-\eps,
\end{aligned}$$ then $$\begin{aligned}
\label{ineq1}
\log |\mathcal{M}| \leq I(M;B^n)_\rho + 2 \sqrt{ n d_B \log \frac{1}{1-\eps} } + \log\frac{1}{1-\eps}.
\end{aligned}$$ In the above, the mutual information is taken with respect to a state $\rho_{MB^n}$ which is the reduced state of $$\rho_{MKB^n} := \frac{1}{|\mathcal{M}|} \sum_{(m,k)\in\mathcal{M}\times\mathcal{K}} q(k) |m\rangle \langle m| \otimes |k\rangle \langle k| \otimes \rho_{B^n}^{x^n(m,k)},$$ with $\rho_{B^n}^{x^n(m,k)}= \bigotimes_{i=1}^n\rho_B^{x_i(m,k)}$ being an $n$-fold product state on $\mathcal{H}_B^{\otimes n}$.
We refer to it as a Fano-type inequality because of the following. The usual (classical) Fano inequality [@Fan61] can be cast in the following form: Let $M, \widehat{M}$ denote two random variables taking values in the same finite set $\mathcal{M}$. If $\Pr(M\neq \widehat{M}) = \eps \in [0,1)$. Then, the (classical) Fano inequality [@Fan61] states that $$\begin{aligned}
\label{Fanoclass}
H(M) \leq I(M;\widehat{M}) + h(\eps) + \eps\log \left(|\mathcal{M}| - 1 \right),
\end{aligned}$$ where $h(\eps) := -\eps \log \eps - (1-\eps) \log (1-\eps)$ is the binary entropy function.
In Theorem \[theo:Fano\], the random variable $M$ is equiprobable and hence $H(M) = \log |\mathcal{M}|$. Considering $\widehat{M}$ to be the random variable denoting the outcome of the POVM $\{\Pi^m_{B_n}\}_{m \in \mathcal{M}}$ on the state $\rho_{B^n}^{x^n(m,k)}$, and using the data-processing inequality for the mutual information, one can upper bound the right-hand side of by $$\begin{aligned}
\log |\mathcal{M}| &\leq I(M;B^n) + h(\eps) + \eps\log \left(|\mathcal{M}| - 1 \right)\\
&\le I(M;B^n) + h(\eps) + \eps \log |\mathcal{M}|
\end{aligned}$$ which can be rewritten as $$\begin{aligned}
\label{eq:usual_Fano}
\log |\mathcal{M}| \leq \frac{1}{1-\eps} I(M;B^n)_\rho + f(\eps),
\end{aligned}$$ where $f(\eps) = \frac{h(\eps) }{1-\eps}$. The similarity between and lead Liu *et al.* [@LHV18] to refer to the latter as a Fano-type inequality in the classical case. The phrase ‘second-order’ is used because the right-hand side of explicitly gives a term of order $\sqrt{n}$.
The above theorem is a generalization of Theorem 32 of [@BDR18], in which an inequality similar to was obtained[^7]. The main difference between the two is that in [@BDR18], the mutual information, arising in the inequality, was evaluated with respect to a state which is a direct sum of tensor product states. In contrast, in Theorem \[theo:Fano\], the mutual information is with respect to states which have a more general form, namely, they are direct sums of mixtures of tensor product states (i.e. separable states): $$\begin{aligned}
\rho_{MB^n} =\frac{1}{|\mathcal{M}|} \sum_{m\in\mathcal{M}} |m\rangle \langle m| \otimes \rho^m_{B^n}, \end{aligned}$$ where $$\begin{aligned}
\label{eq:Fano17}
\rho^m_{B^n} := \sum_{k\in\mathcal{K}} q(k) \rho_{B^n}^{x^n(m,k)}, \quad \forall\, m\in\mathcal{M},
\end{aligned}$$ where $ \rho_{B^n}^{x^n(m,k)}= \bigotimes_{i=1}^n\rho_B^{x_i(m,k)}$. This generalization is crucial for our proof of the strong converse property of a c-q DBC.
\[remark:error\] The condition given by the inequality is called the *geometric average error criterion*. It is stronger than the *average error criterion*, $$\frac{1}{|\mathcal{M}|} \sum_{(m, k) \in \mathcal{M}\times \mathcal{K} } q(k) \operatorname{Tr}[ \rho_{B^n}^{x^n(m,k)} \Pi_{B^n}^m ] \geq 1-\eps,$$ in the classical Fano inequality [@Fan61], but is weaker than the *maximal error criterion*, $$\min_{(m, k) \in \mathcal{M}\times \mathcal{K} } \operatorname{Tr}[ \rho_{B^n}^{x^n(m,k)} \Pi_{B^n}^m ] \geq 1-\eps.$$ Since the Fano-type inequality is a tool to prove converse results in network information theory, one might wonder if it still holds under a weaker error criterion. In the classical case, Liu *et al.* showed that an analogous second-order Fano-type inequality does not hold if the geometric average error criterion is replaced by the average error criterion [@LCV17], [@LHV18 Remark 3.3]. However, by a standard technique known as the *codebook expurgation* (see e.g. [@GK11 Problem 8.11], [@Wil90], [@CK11], [@Ahl78]), which consists of discarding codewords corresponding to large error probabilities, one might still be able to show a second-order converse bound under the average error criterion in certain network information-theoretic tasks.
\[Proof of Theorem \[theo:Fano\]\] Before starting the proof, we introduce necessary definitions that will be used later. Consider the QMS $\left(\Phi_{t,x^n(m,k)}\right)_{t\geq 0}$, where for all $(m, k) \in {{\mathcal{M}}}\times {{\mathcal{K}}}$ $$\begin{aligned}
&\Phi_{t,x^n(m,k)} := \Phi_{t,x_1(m,k)} \otimes \cdots \otimes \Phi_{t,x_n(m,k)}, \label{eq:semigroup}\end{aligned}$$ and $\forall$ $x(m,k) \in {{\mathcal{X}}}$, $\Phi_{t,x(m,k)}$ denotes the superoperator defining the GQDS : $$\begin{aligned}
\Phi_{t,x(m,k)}( T) &:= \operatorname{\mathrm{e}}^{-t} T + (1-\operatorname{\mathrm{e}}^{-t}) \operatorname{Tr}\left[ \rho_B^{x(m,k)} T \right] \mathbb{I}_B, \quad \forall T \in\mathcal{B}(\mathcal{H}_B),\, t>0\end{aligned}$$ Further, we define the following superoperator $$\begin{aligned}
&\Psi_{t}(T) := \operatorname{\mathrm{e}}^{-t} T + (1-\operatorname{\mathrm{e}}^{-t}) \operatorname{Tr}\left[ T \right] \mathbb{I}_B, \quad \forall\, T \in\mathcal{B}(\mathcal{H}_B).
\end{aligned}$$ For any $\rho, \sigma \in\mathcal{D}(\mathcal{H}_B)$, the *projectively measured Rényi relative entropy* is defined as [@Don86; @HP91; @Pet86b]: $$\begin{aligned}
{D^{\mathbb{P}}}_\alpha(\rho\|\sigma) &:=
\frac{1}{\alpha-1} \log {Q^{\mathbb{P}}}_\alpha(\rho\|\sigma), \quad \forall \alpha \in (0,1), \\
{\hbox{with}}\quad {Q^{\mathbb{P}}}_\alpha(\rho\|\sigma) &:= \inf_{ \{P_i\}_{i=1}^{d_B }} \left\{ \sum_{i=1}^{d_B } \left(\operatorname{Tr}[ P_i \rho ] \right)^\alpha \left(\operatorname{Tr}[ P_i \sigma ] \right)^{1-\alpha} \right\}, \quad \forall \alpha \in (0,1), \label{eq:QP}
\end{aligned}$$ where the optimization is over all sets of mutually orthogonal projectors $ \{P_i\}_{i=1}^{d_B } $ on $\mathcal{H}_B$. Let $m \in \mathcal{M}$, $t>0$, $p\in(0,\sfrac12)$ and let $\hat{p} = (1-\sfrac1p)^{-1} \in (-1,0)$ be its Hölder conjugate. We commence the proof by invoking a variational formula for ${Q^{\mathbb{P}}}_p$ [@BFT17 Lemma 3]: $$\begin{aligned}
{Q^{\mathbb{P}}}_p( \rho_{B^n} \| \rho_{B^n}^m ) &= \inf_{G_n>0} \left\{ \left( \operatorname{Tr}\left[ \rho_{B^n} G_n \right] \right)^p \left( \operatorname{Tr}\left[ \rho_{B^n}^m G_n^{ \hat p} \right]\right)^{1-p} \right\} \\
&\leq \left( \operatorname{Tr}\left[ \rho_{B^n} \Psi_t^{\otimes n}(\Pi_{B^n}^m) \right] \right)^p \left( \operatorname{Tr}\left[ \rho_{B^n}^m (\Psi_t^{\otimes n}(\Pi_{B^n}^m))^{ \hat p} \right]\right)^{1-p} \\
&= \left( \operatorname{Tr}\left[ \rho_{B^n} \Psi_t^{\otimes n}(\Pi_{B^n}^m) \right] \right)^p \left( \sum_{k\in\mathcal{K}} q(k) \operatorname{Tr}\left[ \rho_{B^n}^{x^n(m,k)} (\Psi_t^{\otimes n}(\Pi_{B^n}^m))^{ \hat p} \right]\right)^{1-p}. \label{eq:Fano1}
\end{aligned}$$ In the above, we remark that $\Psi_t^{\otimes n}(\Pi_{B^n}^m) >0$ for all $t>0$ due to the definition of $\Psi_t$ and the condition .
Applying the Araki-Lieb-Thirring inequality, Lemma \[lemm:ALT\], with $r= -\hat{p} \in(0,1)$, $A = (\Psi_t^{\otimes n}(\Pi_{B^n}^m))^{-1} > 0$, and $B^{r} = \rho_{B^n}^{x^n(m,k)} $ yields $$\begin{aligned}
\operatorname{Tr}\left[ \rho_{B^n}^{x^n(m,k)} (\Psi_t^{\otimes n}(\Pi_{B^n}^m))^{\hat{p}} \right] &\leq \operatorname{Tr}\left[ \left( \left(\rho_{B^n}^{x^n(m,k)} \right)^{- \frac{1}{2\hat{p}} } \left( \Psi_t^{\otimes n}(\Pi_{B^n}^m) \right)^{-1} \left(\rho_{B^n}^{x^n(m,k)} \right)^{- \frac{1}{2\hat{p}} } \right)^{-\hat{p}} \right] \\
&= \left\| \Psi_t^{\otimes n}(\Pi_{B^n}^m) \right\|_{\hat{p}, \rho_{B^n}^{x^n(m,k)} }^{\hat{p}}. \label{eq:Fano2}
\end{aligned}$$ On the other hand, it is clear from the the definition that ${Q^{\mathbb{P}}}_p( \rho_{B^n} \| \rho_{B^n}^m ) = {Q^{\mathbb{P}}}_{1-p} (\rho_{B^n}^m \| \rho_{B^n})$. Combining and , taking logarithms of both sides of the resulting inequality, and dividing by $p$, yields $$\begin{aligned}
{D^{\mathbb{P}}}_{1-p} \left(\rho_{B^n}^m \| \rho_{B^n}\right) &\geq \frac{1}{\hat{p}} \log \left( \sum_{k\in\mathcal{K}} q(k) \left\| \Psi_{t}^{\otimes n}( \Pi_{B^n}^m ) \right\|_{\hat{p}, \rho_{B^n}^{x^n(m,k)} }^{\hat{p}} \right) - \log \operatorname{Tr}\left[ \rho_{B^n} \Psi_{t}^{\otimes n}( \Pi_{B^n}^m ) \right]. \label{eq:Fano16}
\end{aligned}$$ Further, the left-hand side of can be upper bounded using the data processing inequality for the *relative Rényi entropy* with respect to projective measurements, i.e. for $p\in(0,\sfrac12)$, $$\begin{aligned}
{D^{\mathbb{P}}}_{1-p} \left(\rho_{B^n}^m \| \rho_{B^n}\right) \leq D_{1-p} \left(\rho_{B^n}^m \| \rho_{B^n}\right) := \frac{1}{-p} \log \operatorname{Tr}\left[ (\rho_{B^n}^m)^{1-p} (\rho_{B^n})^p \right].
\end{aligned}$$ Averaging over all $m \in \mathcal{M}$, we have $$\begin{aligned}
\begin{split}
\frac{1}{|\mathcal{M}|} \sum_{m\in\mathcal{M}} D_{1-p} \left(\rho_{B^n}^m \| \rho_{B^n}\right) &\geq \frac{1}{ |\mathcal{M}|} \sum_{m\in\mathcal{M}} \frac{1}{\hat{p}}\log \left( \sum_{k\in\mathcal{K}} q(k)\left\| \Psi_{t}^{\otimes n}( \Pi_{B^n}^m ) \right\|_{\hat{p}, \rho_{B^n}^{x^n(m,k)} }^{\hat{p}} \right) \\
&\quad - \frac{1}{|\mathcal{M}|} \sum_{m\in\mathcal{M}}\log \operatorname{Tr}\left[ \rho_{B^n} \Psi_{t}^{\otimes n}( \Pi_{B^n}^m ) \right].
\end{split} \label{eq:Fano3}
\end{aligned}$$
In the following, we lower bound the right-hand side of . The superoperator $(\Psi_{t}^{\otimes n} - \Phi_{t,x^n(m,k)})$, where $\Phi_{t,x^n(m,k)}$ is the superoperator defined through , is positivity-preserving for every $(m,k) \in \mathcal{M}\times \mathcal{K}$, since $\rho_B^x \leq \mathbb{I}_B$ for all $x\in\mathcal{X}$. (This can be proved by induction in $n$, as in the proof of [@BDR18 Theorem 29]). Further, the non-commutative weighted $L_\mathsf{p}$-norm $\|\cdot \|_{\mathsf{p}, \rho}$ is monotone non-decreasing in its argument for every $\mathsf{p} \in \mathbb{R}\backslash \{0\}$ (which can be immediately verified from the definition by using Weyl’s Monotonicity Theorem [@Bha97 Corollary III.2.3]). Hence, for every $m\in\mathcal{M}$, $$\begin{aligned}
\sum_{k\in\mathcal{K}} q(k) \left\| \Psi_{t}^{\otimes n}( \Pi_{B^n}^m ) \right\|_{\hat{p}, \rho_{B^n}^{x^n(m,k)} }^{\hat{p}} \leq
\sum_{k\in\mathcal{K}} q(k)\left\| \Phi_{t,x^n(m,k)}^{\otimes n}( \Pi_{B^n}^m ) \right\|_{\hat{p}, \rho_{B^n}^{x^n(m,k)} }^{\hat{p}}. \label{eq:Fano4}
\end{aligned}$$ Then, we employ the Reverse Hypercontractivity, Lemma \[lemm:RHC\], on the right-hand side of with $\mathsf{p} = \hat{p} \in (-1,0)$ and any $\mathsf{q} = q \in (0,1)$ satisfying $t = \log \frac{\hat{p}-1}{q-1}$ to obtain $$\begin{aligned}
\left\| \Phi_{t,x^n(m,k)}(\Pi_{B^n}^m) \right\|_{\hat{p}, \rho_{B^n}^{x^n(m,k)} }
&\geq \left\| \Pi_{B^n}^m \right\|_{q, \rho_{B^n}^{x^n(m,k)} } \\
&= \left( \operatorname{Tr}\left[ \left( (\rho_{B^n}^{x^n(m,k)})^{\frac{1}{2q}} \Pi_{B^n}^m (\rho_{B^n}^{x^n(m,k)})^{\frac{1}{2q}} \right)^q \right] \right)^{\frac{1}{q}} \\
&\geq \left( \operatorname{Tr}\left[ \rho_{B^n}^{x^n(m,k)} (\Pi_{B^n}^m)^q \right] \right)^{\frac{1}{q}} \label{eq:Fano5}\\
&\geq \left( \operatorname{Tr}\left[ \rho_{B^n}^{x^n(m,k)} \Pi_{B^n}^m \right] \right)^{\frac{1}{q}}. \label{eq:Fano6}
\end{aligned}$$ Here, we used the Araki-Lieb-Thirring inequality, Lemma \[lemm:ALT\], with $A = \Pi_{B^n}^m$, $B = (\rho_{B^n}^{x^n(m,k)})^{\frac1q}$, and $r = q \in(0,1)$ to obtain the inequality . The inequality holds because $0\leq \Pi_{B^n}^m \leq \mathbb{I}_{B^n}$, so that $(\Pi_{B^n}^m)^q \geq \Pi_{B^n}^m$ for $q\in(0,1)$. From and , the first term on the right-hand side of is hence lower bounded by $$\begin{aligned}
\frac{1}{ |\mathcal{M}|} \sum_{m\in\mathcal{M}} \frac{1}{\hat{p}}\log \left( \sum_{k\in\mathcal{K}} q(k) \left( \operatorname{Tr}\left[ \rho_{B^n}^{x^n(m,k)} \Pi_{B^n}^m \right] \right)^{\frac{\hat{p}}{q}} \right). \label{eq:Fano9}
\end{aligned}$$
Next, we lower bound the second term on the right-hand side of . The concavity of the logarithm function implies that $$\begin{aligned}
- \frac{1}{|\mathcal{M}|} \sum_{m\in\mathcal{M}} \log \operatorname{Tr}\left[ \rho_{B^n} \Psi_{t}^{\otimes n}( \Pi_{B^n}^m ) \right] &\geq - \log \left( \frac{1}{|\mathcal{M}|} \sum_{m\in\mathcal{M}} \operatorname{Tr}\left[ \rho_{B^n} \Psi_{t}^{\otimes n}( \Pi_{B^n}^m ) \right] \right) \\
&= - \log \left( \frac{1}{|\mathcal{M}|} \operatorname{Tr}\left[ \rho_{B^n} \Psi_{t}^{\otimes n} (\mathbb{I}_B^{\otimes n} ) \right] \right), \\
&\geq \log |\mathcal{M}| - dnt, \label{eq:Fano10}
\end{aligned}$$ where the last inequality follows from the fact that[^8] $$\begin{aligned}
\Psi_t^{\otimes n}(\mathbb{I}_B^{\otimes n}) = (\operatorname{\mathrm{e}}^{-t} + d (1-\operatorname{\mathrm{e}}^{-t}))^n \mathbb{I}_B^{\otimes n} \leq \operatorname{\mathrm{e}}^{(d-1)nt} \mathbb{I}_B^{\otimes n} \leq \operatorname{\mathrm{e}}^{dnt} \mathbb{I}_B^{\otimes n}.
\end{aligned}$$
Combining , , and yields $$\begin{aligned}
&\frac{1}{|\mathcal{M}|} \sum_{m\in\mathcal{M}} D_{1-p} \left(\rho_{B^n}^m \| \rho_{B^n}\right) \notag \\ &\geq \frac{1}{ |\mathcal{M}|} \sum_{m\in\mathcal{M}} \frac{1}{\hat{p}}\log \left( \sum_{k\in\mathcal{K}} q(k)\left( \operatorname{Tr}\left[ \rho_{B^n}^{x^n(m,k)} \Pi_{B^n}^m \right] \right)^{\frac{\hat{p}}{q}} \right)
+ \log |\mathcal{M}| - dnt. \label{eq:Fano11}
\end{aligned}$$ Next we take the limits $p\to 0$ and $\hat{p} \to 0$ (which in turn ensures that $q \to 1-\operatorname{\mathrm{e}}^{-t}$) on both sides of the above inequality. Then the left-hand side of becomes $$\begin{aligned}
\lim_{p\to 0}
\frac{1}{|\mathcal{M}|} \sum_{m\in\mathcal{M}} D_{1-p} \left(\rho_{B^n}^m \| \rho_{B^n}\right) &= \frac{1}{|\mathcal{M}|} \sum_{m\in\mathcal{M}} D \left(\rho_{B^n}^m \| \rho_{B^n}\right) \label{eq:Fano15} \\
&= D( \rho_{MB^n}|| \rho_M \otimes \rho_{B^n}) = I(M; B^n)_\rho, \label{eq:Fano12}
\end{aligned}$$ where the equality follows from the fact that the the quantum relative Rényi entropy $D_{1-p}$ converges to the quantum relative entropy $D$ as $p\to 0 $.
On the other hand, the first term on the right-hand side of becomes $$\begin{aligned}
&\lim_{ \hat{p} \to 0} \frac{1}{ |\mathcal{M}|} \sum_{m\in\mathcal{M}} \frac{1}{\hat{p}}\log \left( \sum_{k\in\mathcal{K}} q(k) \left( \operatorname{Tr}\left[ \rho_{B^n}^{x^n(m,k)} \Pi_{B^n}^m \right] \right)^{\frac{\hat{p}}{q}} \right) \notag \\
&= \frac{1}{ |\mathcal{M}|} \sum_{(m,k) \in \mathcal{M}\times \mathcal{K}} q(k)\, \frac{1}{1-\operatorname{\mathrm{e}}^{-t}} \log \operatorname{Tr}\left[ \rho_{B^n}^{x^n(m,k)} \Pi_{B^n}^m \right] \label{eq:Fano18}\\
&\geq \frac{1}{1-\operatorname{\mathrm{e}}^{-t}} \log (1-\eps ) \label{eq:Fano19}\\
&\geq -\left(1+\frac1t\right) \log \frac{1}{1-\eps }. \label{eq:Fano13}
\end{aligned}$$ In the above, equality is due to L’Hôspital’s rule; inequalities and hold because of the assumption given in and the fact that $\frac{1}{1-\operatorname{\mathrm{e}}^{-t}} \leq 1+ \frac1t$. Finally, , , and , together imply that $$\begin{aligned}
\log |\mathcal{M}| \leq I(M;B^n)_\rho + dnt + \left(1+\frac1t\right) \log \frac{1}{1-\eps }. \label{eq:Fano14}
\end{aligned}$$ The above bound can be shown to be optimized when $$\begin{aligned}
\label{tee}
t = \sqrt{ \frac{ - \log (1-\eps) }{ d n } },
\end{aligned}$$ which satisfies the requirement $t>0$ since $\eps \in (0,1)$. Substituting in yields the desired result.
Second-Order Strong Converse Bound for a Classical-Quantum Degraded Broadcast Channel {#sec:sc_DBC}
=====================================================================================
Let us now revert to the c-q degraded broadcast channel $\mathscr{W}^{X \to BC}$ which was introduced in Section \[sec:c-q\_DBC\], and is the focus of this paper. In the following theorem we establish second-order (in $n$) upper bounds to rate pairs $(R_B, R_C)$ of any $(n, R_B, R_C, \eps)$ code for such a channel $\mathscr{W}^{X \to BC}$.
\[Second-order strong converse bound for a c-q DBC\] \[theo:SC\_DBC\] For a c-q DBC $\mathscr{W}^{X\to BC}$ as given in Definition \[defn:channel\], any $(n, R_B, R_C, \eps)$ code satisfies $$\begin{aligned}
& R_B \leq I(X;B|U)_\omega + 2 \sqrt{ \frac{d_B }{n} \log \frac{1}{1-\eps} } + \frac1n\log\frac{1}{1-\eps}; \\
& R_C \leq I(U;C)_\omega + 2 \sqrt{ \frac{ d_C }{n} \log \frac{1}{1-\eps} } + \frac1n\log\frac{1}{1-\eps}, \label{eq:SC_DBC}
\end{aligned}$$ for some $\omega_{UXBC}$ of the form $$\begin{aligned}
\label{eq:DBC_condition}
\omega_{UXBC} &= \sum_{x \in {{\mathcal{X}}}} p_X(x) |x \rangle \langle x| \otimes \rho_U^x \otimes \rho_{BC}^x
\end{aligned}$$ for some probability distribution $p$ on $\mathcal{X}$, and some collection of density matrices $\{\rho_U^x\}_{x\in\mathcal{X}}$.
Theorem \[theo:SC\_DBC\] can be extended to the case in which Alice transmits common information (at a rate $R$, say), in addition to private information. It can be verified that if $(R, R_B, R_C)$ is an $\eps$-achievable rate triple, then $(0, R_B, R+ R_C)$ is also $\eps$-achievable. An intuitive way to see this is as follows: Bob can disregard the common information that he receives, while Charlie can consider the common information that he receives as part of his private information, without affecting the error probability of the protocol. This was stated for the case $\eps=0$ for a c-q DBC in [@YHD11] and is well-known for the case of classical broadcast channels (see e.g. [@GK11]). Due to this reason it suffices to incorporate the common information rate into the rate of private information transmission to Charlie. If the common information rate $R$ is non-zero, then the left-hand side of should be read as $R+R_C$.
\[Proof of Theorem \[theo:SC\_DBC\]\]
Let $$\begin{aligned}
\rho_{MKX^nB^nC^n} = \frac{1}{|\mathcal{M}||\mathcal{K}|} \sum_{m\in \mathcal{M}} \sum_{k \in \mathcal{K}} |m\rangle\langle m| \otimes |k\rangle\langle k| \otimes |x^n(m,k)\rangle\langle x^n(m,k) |\otimes \rho_{B^n C^n}^{x^n(m,k)}.
\end{aligned}$$ Observe that $$\begin{aligned}
\min\left\{ \operatorname{Tr}\left[ \rho_{B^n}^{x^n(m,k)} \Pi_{B^n}^{m} \right] , \operatorname{Tr}\left[ \rho_{ C^n}^{x^n(m,k)} \Pi_{C^n}^{k} \right] \right\} \geq
\operatorname{Tr}\left[ \rho_{B^n C^n}^{x^n(m,k)} \Pi_{B^n}^{m} \otimes \Pi_{C^n}^{k} \right] \geq 1-\eps
\end{aligned}$$ by definition of an $(n,R_B, R_C, \eps)$ code. Hence, the $(n,R_B, R_C, \eps)$-code satisfies the geometric average error criterion given by (cf. Remark \[remark:error\]) We then apply the second-order Fano-type inequality, Theorem \[theo:Fano\], with the choice $q(k) = \frac{1}{|\mathcal{K}|}$ for every $k\in\mathcal{K}$ and $n R_B = \log |\mathcal{M}|$, $n R_C = \log |\mathcal{K}|$ to obtain the following upper bounds for the rate pair: $$\begin{aligned}
\begin{split} \label{eq:DBC0}
n R_B &\leq I(M;B^n)_\rho + 2 \sqrt{ n d_B \log \frac{1}{1-\eps} } + \log\frac{1}{1-\eps}; \\
n R_C &\leq I(K;C^n)_\rho + 2 \sqrt{ n d_C \log \frac{1}{1-\eps} } + \log\frac{1}{1-\eps}.
\end{split}
\end{aligned}$$
To complete the proof, we need to find upper bounds on $I(M;B^n)_\rho$ and $I(K;C^n)_\rho$ in terms of single-letter entropic quantities. This was done by Yard *et al.* [@YHD11 Theorem 2] following the same idea that was used by Gallager [@Gal74] in the classical case, and which is often referred to as *identification of the auxiliary random variable* (see also [@GK11 Chapter 5.4]). The upper bounds obtained are given by $$\begin{aligned}
\begin{split} \label{eq:upper}
I(M;B^n)_\rho &\leq n I(X;B|U)_{\omega}, \\
I(K; C^n)_\rho &\leq n I(U;C)_{\omega},
\end{split}\end{aligned}$$ for some quantum state $\omega_{UXBC}$ of the form given in of the statement of Theorem \[theo:SC\_DBC\]. For the sake of completeness, we include the proof in Appendix \[proof:upper\]. This concludes the proof of Theorem \[theo:SC\_DBC\].
Taking the limit $n\to \infty$, on both sides of the inequalities in Theorem \[theo:SC\_DBC\] directly shows that the $\eps$-capacity region $\mathcal{C}_{\mathscr{W}}(\eps)$ is contained in $\mathcal{C}^\text{ent}_{\mathscr{W}}$ for all $\eps \in(0,1)$. This in turn demonstrates the strong converse property for the c-q DBC, stated in Corollary \[coro:epsilon\]. In other words, for any sequence of codes with rate pair $(R_B, R_C) \not\in \mathcal{C}^{\operatorname{ent}}_{\mathscr{W}}$, transmission of private information from Alice to Bob and Charlie fails with certainty, no matter how many times the channel is used.
\[Strong Converse Property\] \[coro:epsilon\] For a c-q DBC $\mathscr{W}^{X\to BC}$ as given in Definition \[defn:channel\], the following holds: $$\begin{aligned}
\mathcal{C}_{\mathscr{W}}(\eps) \subseteq \mathcal{C}_{\mathscr{W}}^{\textnormal{ent}}, \quad \forall \eps\in(0,1).
\end{aligned}$$
In fact, Theorem \[theo:SC\_DBC\] yields a finite blocklength strong converse, namely, that the maximal error of any $(n, R_B, R_C)$-code converges to $1$ exponentially fast (in $n$) whenever $(R_B, R_C) \not\in \mathcal{C}^\text{ent}_{\mathscr{W}}$. This is stated in the following corollary.
\[Exponential Strong Converse\] \[coro:exponential\] For a c-q DBC $\mathscr{W}^{X\to BC}$ as given in Definition \[defn:channel\] and any non-negative rate pair $(R_B, R_C) \not\in \mathcal{C}_{\mathscr{W}}^{\operatorname{ent}}$, the maximal error of any $(n, R_B, R_C)$ code $\left( \mathcal{E}_n, \mathcal{D}_n \right)$ satisfies $$\begin{aligned}
p_{\max}\left( \mathcal{E}_n, \mathcal{D}_n \right) \geq 1 - \operatorname{\mathrm{e}}^{-nf},
\end{aligned}$$ where $$\begin{aligned}
f = \left( \sqrt{ (\sqrt{ d_B } + \sqrt{ d_C })^2 + \eta } - \sqrt{d_B} - \sqrt{d_C} \right)^2 > 0
\end{aligned}$$ for some $\eta >0$ depending only on how far the rate pair $(R_B, R_C)$ is from the region $\mathcal{C}_\mathscr{W}^{\operatorname{ent}}$.
\[Proof of Corollary \[coro:exponential\]\] Let us first define the function $$\begin{aligned}
F(t) &:= \sup_{ { \rho } } \left\{ I(X;B|U)_\rho: I(U;C)_\rho \geq t \right\}, \quad \forall t\geq 0,
\end{aligned}$$ where the supremum is taken over all states $\rho\equiv \rho_{UXBC}$ of the form of . By the definition of $\mathcal{C}_\mathscr{W}^{\operatorname{ent}}$ given in , $(R_B, R_C) \not\in \mathcal{C}_{\mathscr{W}}^{\operatorname{ent}}$ implies that $$\begin{aligned}
R_B > F(R_C). \label{eq:exp2}
\end{aligned}$$ In Appendix \[app:concavity\], we prove that $F(t)$ is a concave function in $t\geq 0$. Therefore, by the method of Lagrange multipliers, inequality can be further written as $$\begin{aligned}
R_B > \inf_{\mu\geq 0} \sup_{ { \rho } } \left\{ I(X;B|U)_\rho + \mu I(U;C)_\rho - \mu R_C \right\}.
\end{aligned}$$ Hence, there must exist some $\mu^\star \in \mathbb{R}_{\geq 0}$ and $\gamma >0$ such that $$\begin{aligned}
R_B + \mu^\star R_C \geq \sup_{ \rho } \left\{I(X;B|U)_\rho + \mu^\star I(U;C)_\rho \right\} + \gamma, \label{eq:exponential1}
\end{aligned}$$
On the other hand, Theorem \[theo:SC\_DBC\] guarantees that any $(n, R_B, R_C)$ code $\left( \mathcal{E}_n, \mathcal{D}_n \right)$ with $p_{\max} \left( \mathcal{E}_n, \mathcal{D}_n \right)\leq \eps \in (0,1)$ satisfies $$\begin{aligned}
R_B \leq I(X;B|U)_\omega + 2 \sqrt{ \frac{d_B }{n} \log \frac{1}{1-\eps} } + \frac1n\log\frac{1}{1-\eps}, \\
R_C \leq I(U;C)_\omega + 2 \sqrt{ \frac{d_C }{n} \log \frac{1}{1-\eps} } + \frac1n\log\frac{1}{1-\eps}
\end{aligned}$$ for some $\omega_{UXBC}$ of the form . Defining $x_n^2 := \log \frac{1}{1-\eps}$, then we have $$\begin{aligned}
R_B + \mu^\star R_C &\leq I(X;B|U)_\omega + \mu^\star I(U;C)_\omega + 2\frac{(1+\mu^\star)}{\sqrt{n}} (\sqrt{d_B} + \sqrt{d_B}) x_n + \frac{(1+\mu^\star)}{n} x_n^2 \\
&\leq \sup_{ \rho } \left\{I(X;B|U)_\rho + \mu^\star I(U;C)_\rho \right\} + 2\frac{(1+\mu^\star)}{\sqrt{n}} (\sqrt{d_B} + \sqrt{d_C}) x_n + \frac{(1+\mu^\star)}{n} x_n^2. \label{eq:exponential2}
\end{aligned}$$ Combining and gives $$\begin{aligned}
(1+\mu^\star) x_n^2 + 2(1+\mu^\star)(\sqrt{n d_B} + \sqrt{n d_C}) x_n - n \gamma \geq 0.
\end{aligned}$$ Solving this and choosing $\eta = \frac{\gamma}{1+\mu^\star} > 0$ concludes the proof of the corollary.
Acknowledgements {#acknowledgements .unnumbered}
================
HC was supported by the Cambridge University Fellowship and the Ministry of Science and Technology Overseas Project for Post Graduate Research (Taiwan) under Grant 108-2917-I-564-042. CR is supported by the TUM University Foundation Fellowship. We thank Jingbo Liu for helpful discussions.
Proof of (\[eq:upper\]) {#proof:upper}
========================
For each $i \in \{1,\ldots, n\}$, we introduce an auxiliary composite quantum system $U_i = (K,B^{i-1})$. We upper bound the first term in as follows: $$\begin{aligned}
I(M;B^n)_\rho &\leq I(M; KB^n)_\rho \label{eq:DBC8} \\
&= I(M; KB^n)_\rho - I(M;K)_\rho \label{eq:DBC1}\\
&= I(M; B^n | K)_\rho \label{eq:DBC2}\\
&= \sum_{i=1}^n I(M; B_i | K, B^{i-1})_\rho \label{eq:DBC3}\\
&= \sum_{i=1}^n I(M; B_i | U_i )_\rho \\
&\leq \sum_{i=1}^n I(M; B_i | U_i )_\rho + I(X_i; Y_i| M U_i)_\rho \label{eq:DBC4}\\
&= \sum_{i=1}^n I(X_i, M; B_i | U_i )_\rho \label{eq:DBC5} \\
&= \sum_{i=1}^n I(X_i ; B_i| U_i )_\rho + I(M; B_i | X_i U_i )_\rho \label{eq:DBC6}\\
&= \sum_{i=1}^n I(X_i ; B_i | U_i )_\rho. \label{eq:DBC7}\end{aligned}$$ Here, inequality is due to monotonicity of the mutual information with respect to the partial trace. Identity is because $M$ and $K$ are uncorrelated. Equalities , , , and follow from the chain rule of quantum mutual information: $I(A^n:C |B)_\rho = \sum_{i=1}^n I(A_i;C|B, A^{i-1})_\rho$ and $I(A; C^n| B)_\rho = \sum_{i=1}^n I(A;C_i|B, C^{i-1})_\rho$. Inequality is due to the non-negativity of the conditional quantum mutual information. The last line holds because of the Markov chain: $M - (K,X_i, B^{i-1}) - B_i$. To see this, the right quantum system $B_i$ can be produced by knowing the value of $X_i$.
Next, we consider the second term in : $$\begin{aligned}
I(K; C^n)_\rho &= \sum_{i=1}^n I(K; C_i| C^{i-1} )_\rho \label{eq:DBC10} \\
&= \sum_{i=1}^n H(C_i| C^{i-1})_\rho - H(C_i| K C^{i-1})_\rho \\
&\leq \sum_{i=1}^n H(C_i)_\rho - H(C_i| K C^{i-1})_\rho \label{eq:DBC11} \\
&\leq \sum_{i=1}^n H(C_i)_\rho - H(C_i| K B^{i-1})_\rho \label{eq:DBC12} \\
&= \sum_{i=1}^n I(K, B^{i-1}; C_i)_\rho. \label{eq:DBC9}\end{aligned}$$ Here, equalities and are again by the chain rule. Inequality is because conditioning reduces entropies. Inequality follows from the data processing with respect to the tensor product of the degrading quantum operation $\mathcal{N}^{B\to C}$.
Now, we introduce a *time-sharing* random variable $T$ that is uniform on $\{1,\ldots, n\}$ and independent of other systems. Identify $U = (T,K,B^{T-1})$, which clearly satisfies . We have the following bounds of and , respectively: $$\begin{aligned}
\sum_{i=1}^n I(X_i, ; B_i | U_i )_\rho &= n I(X_T; B_T| TK B^{T-1} )_{T\otimes \rho} \\
&= n I(X;B|U)_{\omega},\end{aligned}$$ and $$\begin{aligned}
\sum_{i=1}^n I(K B^{i-1}; C_i)_\rho &= n I(K, B^{T-1}; C_T | T)_{T\otimes \rho} \\
&\leq n \left[ I(K B^{T-1}; C_T | T)_{T\otimes \rho} + I(T; C_T)_{T\otimes \rho} \right] \\
&= n I(I, K B^{T-1}; C_T)_{T\otimes \rho} \\
&= n I(U;C)_{\omega}.\end{aligned}$$
A Concavity Property {#app:concavity}
====================
We define the following function: $$\begin{aligned}
F(t) &:= \sup_{ { \rho \in \Sigma(\mathscr{W}) } } \left\{ I(X;B|U)_\rho: I(U;C)_\rho \geq t \right\}, \quad \forall t\geq 0; \label{eq:rate_function} \\
\Sigma(\mathscr{W}) &:= \left\{
\rho_{UXBC} = \bigoplus_{ x \in\mathcal{X} }\, p(x) \rho_U^x\otimes \rho_{BC}^x: p \text{ is a probability distribution on } \mathcal{X}, \text{and } \left\{\rho_U^x \right\}_{x\in\mathcal{X}} \subset \mathcal{D}(\mathcal{H}_U)
\right\}. \label{eq:set}\end{aligned}$$
The following concavity of the function $F(t)$ can be proved by following similar idea of Ahlswede and Körner [@AK75]. For completeness, we provide a proof here.
\[theo:concavity\] The function $F(t)$ defined in is concave for all $t\geq 0$. Moreover, $$\begin{aligned}
\label{eq:infimum}
F(t) = \inf_{ \mu \geq 0 } \sup_{\rho\in\Sigma(\mathscr{W}) } \left\{
I(X;B|U)_\rho + \mu I(U;C)_\rho - \mu t
\right\}.
\end{aligned}$$
We aim to prove $$\begin{aligned}
F(\lambda t_0 + (1-\lambda) t_1) \geq \lambda F(t_0) + (1-\lambda) F(t_1),
\end{aligned}$$ for all $\lambda \in [0,1]$ and $t_0, t_1 \geq 0$. For every $\gamma>0$, let $\rho_0, \rho_1 \in \Sigma(\mathscr{W})$ such that $I(X;B|U)_{\rho_i} \geq F(t_i) - \gamma $ and $ I(U;C)_{\rho_i} \geq t_i$ for $i \in \{0,1\}$.
Now, we introduce a new Bernoulli random variable $V$ with $\Pr\{V=0\} = \lambda$ and $ \Pr\{V = 1\} = (1-\lambda)$ such that $$\begin{aligned}
\rho_{VUBC} := \lambda |0\rangle\langle 0| \otimes \rho_0 + (1-\lambda) |1\rangle\langle 1| \otimes \rho_1 \in \Sigma(\mathscr{W}).
\end{aligned}$$ From the choice of $\rho_0, \rho_1$, and $\rho$, we have $$\begin{aligned}
\lambda F(t_0) + (1-\lambda) F(t_1) - \gamma
&\leq \lambda I(X;B|U)_{\rho_0} + (1-\lambda) I(X;B|U)_{\rho_1} \\
&= I(X;B|UV)_\rho.
\end{aligned}$$ On the other hand, using the chain rule and non-negativity of quantum mutual information, we have $$\begin{aligned}
\lambda t_0 + (1-\lambda) t_1 &\leq
\lambda I(U;C)_{\rho_0} + (1-\lambda) I(U;C)_{\rho_1} \\
&= I(U;C|V)_{\rho} \\
&= I(VU; C)_\rho - I(V;C)_\rho \\
&\leq I(VU;C)_\rho.
\end{aligned}$$ This means that $\rho$ satisfies the constraint of in the definition of $F(\lambda t_0 + (1-\lambda) t_1)$. Therefore, $$\begin{aligned}
F(\lambda t_0 + (1-\lambda) t_1) &\geq I(X;B|VU)_\rho \\
&\geq \lambda F(t_0) + (1-\lambda) F(t_1) - \gamma.
\end{aligned}$$ Since this holds for every $\gamma>0$, we conclude the proof by letting $\gamma\to 0$. The second assertion in follows from the method of Lagrange multipliers and the concavity of $F(t)$.
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R. Bhatia, *Matrix Analysis*.1em plus 0.5em minus 0.4em Springer New York, 1997.
[^1]: More generally, one can consider even more than two receivers.
[^2]: The capacity region with common information can be obtained from the one without common information (see e.g. [@GK11 Chapter 5.7]).
[^3]: For a precise definition of the achievable rate region and the capacity region, see Section \[sec:c-q\_DBC\].
[^4]: Yard [*et al.*]{} [@YHD11] showed that it suffices to consider a random variable $U$ for which $|{\mathcal{U}}| \leq \min\{|{\mathcal{X}}|, d_B^2 + d_C^2 -1 \}$.
[^5]: For $p<1$, these are pseudo-norms, since they do not satisfy the triangle inequality. For $p<0$, they are only defined for $X>0$ and for a non-full rank state by taking them equal to $\big( \operatorname{Tr}\big[ \big| \sigma^{-\frac{1}{2{p}}} X^{-1} \sigma^{-\frac{1}{2 {p}}} \big|^{{-p}} \big]\big)^{1/p}$.
[^6]: That is, for a point-to-point channel with a single user and a single receiver, as opposed to a broadcast c-q channel.
[^7]: A classical analogue of Theorem 32 of [@BDR18] was earlier proved in [@LHV18].
[^8]: Note that the convexity of $h(u):= u^d$ for $d\geq 2$ implies that $(h(u)-h(1))/(u-1) \geq h'(1)$ for ever $u\geq1$. Hence, $\operatorname{\mathrm{e}}^{dt} - 1 \geq d(\operatorname{\mathrm{e}}^t-1)$ for every $t\geq 0$, and $\operatorname{\mathrm{e}}^{-t} + d(1-\operatorname{\mathrm{e}}^{-t}) \leq \operatorname{\mathrm{e}}^{(d-1)t}$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
In extant quantum secret sharing protocols, once the secret is shared in a quantum network (<span style="font-variant:small-caps;">qnet</span>) it can not be retrieved back, even if the dealer wishes that her secret no longer be available in the network. For instance, if the dealer is part of two <span style="font-variant:small-caps;">qnet</span>s, say $\mathcal{Q}_1$ and $\mathcal{Q}_2$ and she subsequently finds that $\mathcal{Q}_2$ is more reliable than $\mathcal{Q}_1$, the dealer may wish to transfer all her secrets from $\mathcal{Q}_1$ to $\mathcal{Q}_2$. Known protocols are inadequate to address such a revocation. In this work we address this problem by designing a protocol that enables the source/dealer to bring back the information shared in the network, if desired. Unlike classical revocation, no-cloning-theorem automatically ensures that the secret is no longer shared in the network.
The implications of our results are multi-fold. One interesting implication of our technique is the possibility of [*routing*]{} qubits in [*asynchronous*]{} <span style="font-variant:small-caps;">qnets</span>. By asynchrony we mean that the requisite data/resources are intermittently available (but not necessarily simultaneously) in the <span style="font-variant:small-caps;">qnet</span>. For example, we show that a source $S$ can send quantum information to a destination $R$ even though (a) $S$ and $R$ share no quantum resource, (b) $R$’s identity is [*unknown*]{} to $S$ at the time of sending the message, but is subsequently decided, (c) $S$ herself can be $R$ at a later date and/or in a different location to bequeath her information (‘backed-up’ in the <span style="font-variant:small-caps;">qnet</span>) and (d) importantly, the path chosen for routing the secret may hit a dead-end due to resource constraints, congestion etc. (therefore the information needs to be [*back-tracked*]{} and sent along an alternate path). Another implication of our technique is the possibility of using [*insecure*]{} resources. For instance, if the quantum memory within an organization is insufficient, it may safely store (using our protocol) its private information with a neighboring organization without (a) revealing critical data to the host and (b) losing control over retrieving the data.
Putting the two implications together, namely routing and secure storage, it is possible to envision applications like quantum mail (qmail) as an outsourced service.
author:
- 'S. Sazim'
- 'V. Chiranjeevi'
- 'I. Chakrabarty'
- 'K. Srinathan'
title: Retrieving and Routing Quantum Information in a Quantum Network
---
Quantum entanglement [@einstein] not only gives us insight in understanding the deepest nature of reality but also acts as a very useful resource in carrying out various information processing protocols like quantum teleportation [@bennett2], quantum cryptography [@gisin] and quantum secret sharing [@hillery], to name a few.0.1cm
In a secret sharing protocol the sender/dealer of the secret message, who is unaware of the individual honesty of the receivers, shares the secret in such a way that none of the receivers get any information about the secret. Quantum secret sharing (QSS) [@hillery; @cleve99] deals with the problem of sharing of both classical as well as quantum secrets. A typical protocol for quantum secret sharing, like many other tasks in quantum cryptography, uses entanglement as a cardinal resource, mostly pure entangled states. Karlsson et al.[@karlsson] studied quantum secret sharing protocols using bipartite pure entangled states as resources. Many authors investigated the concept of quantum secret sharing using tripartite pure entangled states and multi partite states like graph states [@bandyopadhyay; @bagherinezhad; @lance; @gordon; @zheng; @markham; @markham08]. Q. Li et al. [@li] proposed semi-quantum secret sharing protocols taking maximally entangled GHZ state as resource.
In a realistic situation, the secret sharing of classical or quantum information involves transmission of qubits through noisy channels that entails mixed states. Recently in [@satya], it is shown that Quantum secret sharing is possible with bipartite two qubit mixed states (formed due to noisy environment or otherwise). Subsequently in [@indranil] authors propose a protocol for secret sharing of classical information with three qubit mixed state. Quantum secret sharing has also been realized in experiments [@tittel; @schmid; @schmid1; @bogdanski].
In quantum secret sharing(QSS), it is typically assumed that the system consists of solely the dealer and the receivers. However, in practical settings the dealer/receivers are part of a quantum network. One important question of how information can be transferred through a quantum network is addressed in [@ind]. In this work we focus on two different situations in a given quantum network (<span style="font-variant:small-caps;">qnet</span>). In the first situation, we consider the problem of revoking the secret in QSS. For instance, if the dealer finds the receivers to be dishonest, she can stop them from accessing it. Moreover, she may choose to retrieve back the secret completely. In our model we consider the receivers to be semi-honest – that is the receivers, though dishonest to eavesdrop on their share and process it, diligently participate in the protocol. On the other hand, note that Byzantinely malicious receivers can easily destroy the secret, making revocation impossible. In the second situation we have extended the above idea to design routing mechanism for multi-hop transmission of [*secret*]{} qubits in the shared domain itself.
Although the above two situations appear to solve unrelated problems namely, revocable secret sharing and quantum routing, the following is an interesting symbiosis of the two to solve problems posed by resource constraints and asynchrony in the network. Consider a situation where quantum storage is constrained and therefore Alice needs to store her private data in some untrusted memory available in the network. This she can do using [*revocable*]{} quantum secret sharing. Further, if she wants to send this data to Bob, (for security reasons) she should be able to do it without reconstructing the quantum secret anywhere in the network. This she can achieve using the quantum routing in shared domain. Incidentally, our solution also takes care of scenarios where Bob too is in short supply of trusted quantum memory and uses network storage.\
***Sharing of a Message :***\
First of all, we consider a simple situation where we have three parties Alice, Bob and Charlie. They share a three qubit maximally entangled GHZ state, i.e., ${| GHZ \rangle}_{ABC}=\frac{1}{\sqrt{2}}({| 000 \rangle}+{| 111 \rangle})$. Here the first qubit is with Alice, second is with Bob and the third one is with Charlie. Here Alice is the dealer and she wishes to secret-share a qubit ${| S \rangle}=\alpha{| 0 \rangle}+\beta{| 1 \rangle}$ (where $\lvert\alpha\rvert^2+\lvert\beta\rvert^2=1$; $\alpha,\beta$ are amplitudes) with both the parties Bob and Charlie. In order to do so Alice has to do two-qubit measurements in Bell basis $\{ |\phi_{\pm}\rangle,|\psi_{\pm}\rangle\}$ jointly on her resource qubit and the message qubit she wants to share (see Appendix 1). In correspondence to various measurement outcomes obtained by Alice, Bob and Charlie’s qubits collapse into the states given in TABLE I.
---------------------- --------------------------------------------
${| \phi^+ \rangle}$ $\alpha{| 00 \rangle}+\beta{| 11 \rangle}$
${| \phi^- \rangle}$ $\alpha{| 00 \rangle}-\beta{| 11 \rangle}$
${| \psi^+ \rangle}$ $\alpha{| 11 \rangle}+\beta{| 00 \rangle}$
${| \psi^- \rangle}$ $\alpha{| 11 \rangle}-\beta{| 00 \rangle}$
---------------------- --------------------------------------------
: **Sharing of Quantum Information**
At this point if Alice finds both Bob and Charlie to be dishonest, she can stop them from accessing the message. She does this by not communicating about her measurement results to any one of them. So there is no transfer of classical bits at this stage. At this point there lies the question of security from Bob and Charlie sides. If we have malicious (parties who are not going to follow the protocol and do whatever they wish to do ) Bob and Charlie can destroy the message by doing local operations in their respective qubits and by communicating classically between them. However, they will never be successful in obtaining the message without Alice’s help.\
***Revocation of Quantum Information :***
If Bob and Charlie are semi-honest (i.e., they are faithful executors of the protocol but curious to learn Alice’s secret), we ask [*can Alice revoke her shared secret ${| S \rangle}$*]{}? The ability to revoke the shared secret is important for several reasons, some of which are (a) Alice decides to change her secret (for instance, ${| S \rangle}$ might have been inadvertently shared) (b) Alice conjectures that the recipients are no longer trustworthy (c) there is an update of data/secrets in the higher-level application using secret sharing as a subroutine and (d) Alice has found a more economical alternative <span style="font-variant:small-caps;">qnet</span> to safeguard ${| S \rangle}$\
To make the revocation possible Alice needs an additional resource (a Bell state) shared with Bob. Consider a very simple case when Alice and Bob are sharing the Bell state $|Bell\rangle_{AB}=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$ in addition to the GHZ state shared by Alice, Bob and Charlie. Let us also assume the first case in the above TABLE I, when Bob and Charlie share the entangled state $\alpha|00\rangle+\beta|11\rangle$ as a result of Alice’s measurement. Now Alice asks Bob to do Bell measurement on his two qubits (one from the shared resource and one from shared secret) and Charlie to do measurement (on his qubit of shared secret) in Hadamard basis (see Appendix 2, see FIG. 1). In TABLE II we show how Alice can retrieve back her message by enlisting down the respective local operations corresponding to Bob’s and Charlie’s measurement outcomes.
$$\begin{array}{cc}
\includegraphics[width=0.4\textwidth]{SS4.png}
\end{array}$$
------------------------ ----------------- ------------
${| \phi^{+} \rangle}$ ${| + \rangle}$ $I$
${| \phi^{+} \rangle}$ ${| - \rangle}$ $\sigma_z$
${| \phi^{-} \rangle}$ ${| + \rangle}$ $\sigma_z$
${| \phi^{-} \rangle}$ ${| - \rangle}$ $I$
${| \psi^{+} \rangle}$ ${| + \rangle}$ $I$
${| \psi^{+} \rangle}$ ${| - \rangle}$ $\sigma_z$
${| \psi^{-} \rangle}$ ${| + \rangle}$ $\sigma_z$
${| \psi^{-} \rangle}$ ${| - \rangle}$ $I$
------------------------ ----------------- ------------
: **Retrieving Quantum Information**
***Quantum Routing in shared domain :***
If Alice has shared her secret qubit ${| S \rangle}$ in some part of a (huge) <span style="font-variant:small-caps;">qnet</span>, we ask [*can she/anyone else retrieve ${| S \rangle}$ at some other part of the network* ]{}? A naive way-out is to reconstruct ${| S \rangle}$ and teleport it, possibly via successive entanglement swapping. However, this severely compromises the security of ${| S \rangle}$. A superior approach is to retain ${| S \rangle}$ in the shared domain while the shares are being routed across the <span style="font-variant:small-caps;">qnet</span>. However, since the shares are themselves entangled and distributed across multiple parties, it is non-trivial to teleport them over the <span style="font-variant:small-caps;">qnet</span>. We address the problem in two parts. First, we show its possible for Alice to dynamically choose the receiver (of her secret), [*after*]{} the sharing phase. Second, we show that quantum information can be transmitted in the shared domain; that is, the information secret shared among a set of nodes is transferred to another set of nodes. Putting the two together, Alice can now move her shared secret close to the desired receiver in the <span style="font-variant:small-caps;">qnet</span> and also remotely control the reconstruction of the secret at the receiver.
Consider a situation where we have $(3+n)$ parties. Here Alice is the sender, both Charlie and Bob act as agents, the remaining $n$ parties {$R_1, R_2, R_3,...,R_n$} are the potential receivers. Alice desires to send the message in form of a qubit to any one of them. Here the role of Bob and Charlie are changed as they are no longer receivers of information but they now act as agents for holding the information in the network. In broader sense they together act like a router and play a vital role in sending the information to the desired receiver.\
Once again we start with Alice, Bob and Charlie sharing a three qubit maximally entangled GHZ state, i.e., ${| GHZ \rangle}_{ABC}=\frac{1}{\sqrt{2}}({| 000 \rangle}
+{| 111 \rangle})$ and Charlie shares Bell’s states, i.e, ${| Bell \rangle}_{CR_i}=\frac{1}{\sqrt{2}}({| 00 \rangle}+{| 11 \rangle})$ with each of the receivers ($R_i$). (In principle, receivers can share resource with any one of the agents Bob and Charlie. Without any loss of generality we assume the receivers share resources with Charlie only.) Suppose Alice wishes to send a qubit ${| S \rangle}=\alpha{| 0 \rangle}+\beta{| 1 \rangle}$ to $R_i$ through the parties Bob and Charlie. First Alice shares her secret with Bob and Charlie in the same way as it is shown in the (TABLE 1). At this point, Alice sends her measurement outcomes encoded in the form of two classical bits to $R_i$. Once the two bits of classical information are obtained, the receiver can easily get back the Alice’s secret ${| S \rangle}$, provided Bob and Charlie perform the actions as described next. We assume that the identity of the receiver is authentically known to Alice, Bob and Charlie, perhaps through a classically secure authentication/identification protocol.\
$$\begin{array}{cc}
\includegraphics[width=0.3\textwidth]{RR3.png}
\end{array}$$
The agents Bob and Charlie do the following. Bob measures his qubit (part of the GHZ state) in the Hadamard basis. Charlie measures two qubits ( one from GHZ state and one from Bell state shared with $R_i$ ) in the Bell basis. After performing these measurements both the agents will send their outcomes through classical channels to the receiver $R_i$. With these measurement outcomes the receiver can retrieve the message which Alice intended to send (see Appendix 3, see Fig(\[router\])). Let us consider the case, when Alice and Bob share the entangled state $\alpha|00\rangle+\beta|11\rangle$, obtained as a result of Alice’s measurement. TABLE III gives an elaborate view of the unitary operations the receiver $R_i$ has to do upon getting various measurement outcomes from Bob and Charlie.
------------------------ ----------------- --------------------
${| \phi^{+} \rangle}$ ${| + \rangle}$ $I$
${| \phi^{-} \rangle}$ ${| + \rangle}$ $\sigma_z$
${| \phi^{+} \rangle}$ ${| - \rangle}$ $\sigma_z$
${| \phi^{-} \rangle}$ ${| - \rangle}$ $I$
${| \psi^{+} \rangle}$ ${| + \rangle}$ $\sigma_x$
${| \psi^{-} \rangle}$ ${| + \rangle}$ $\sigma_z\sigma_x$
${| \psi^{+} \rangle}$ ${| - \rangle}$ $\sigma_z\sigma_x$
${| \psi^{-} \rangle}$ ${| - \rangle}$ $\sigma_x$
------------------------ ----------------- --------------------
: **Sending Quantum Information**
$$\begin{array}{cc}
\includegraphics[width=0.35\textwidth]{QQ3.png}
\end{array}$$
Finally, we address the problem of transferring secret qubits in the shared domain till it comes close to the desired receiver. If we have a source $(S)$ and receivers $R_1, R_2,...., R_n$ and we want to send the information to the receiver $R_i$ through a huge network with pair of agents $(A_1,B_1),(A_2,B_2),.........,(A_n,B_n)$ at each blocks. So every pair shares Bell state with consecutive pair say $A_i$ with $A_{i+1}$ and $B_i$ with $B_{i+1}$. The above setting is depicted in FIG. 3. Once the source shares the information with $i^{th}$ pair the information can be transferred to $(i+1)^{th}$ pair by the process of entanglement swapping in the following way. $A_i$ performs the Bell measurement on two qubits one from the shared secret and other from the Bell state shared with $A_{i+1}$, similarly $B_i$ performs the Bell measurement on two qubits one from the shared secret and other from the Bell state shared with $B_{i+1}$. This sequence of measurements goes on till the closest pair gets the Shared secret. The classical outcomes of each measurement are sent to Alice immediately after the measurement to keep track of the state of the shared secret. The receivers can stay in the network in between each pairs. The source is not going to send the classical information until the quantum information (shared secret) reaches the pair $(A_i,B_i)$ close to the desired receiver. Thus, in a <span style="font-variant:small-caps;">qnet</span> we can share, retrieve, hold and as well as transfer the quantum information.\
***Concluding remarks and Outlook:***\
This paper addresses the problem of [*revocable*]{} quantum secret sharing. The ability to revoke a quantum shared secret has implications on the possibility of quantum routing ( backtracking etc.) in shared domain. An interesting consequence of the above is that critical/private information ${| S \rangle}$ can be [*q-mailed*]{} across public <span style="font-variant:small-caps;">qnet</span>s, first by secret sharing ${| S \rangle}$ and then routing ${| S \rangle}$ (in the shared domain) to the desired receiver. We have assumed the resources to be pure entangled states, however working out with resources being mixed entangled states still remains an open question. ***Acknowledgment:*** This work is done at Center for Security, Theory and Algorithmic Research (CSTAR), IIIT, Hyderabad. S Sazim gratefully acknowledge their hospitality. We acknowledge Prof. P. Agrawal for having useful discussions.
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***Appendix 1:***\
Consider a 3-qubit GHZ state $\frac{1}{\sqrt{2}}\{|000\rangle$+$|111\rangle\}$ among Alice, Bob and Charlie and let $|\psi\rangle = \alpha|0\rangle+\beta|1\rangle$ be the message with Alice. $$\begin{aligned}
&&|\psi\rangle\otimes\frac{1}{\sqrt{2}}\{|000\rangle+|111\rangle\}{}\nonumber\\&&
= \{\alpha|0\rangle+\beta|1\rangle\}\otimes\frac{1}{\sqrt{2}}\{|000\rangle+|111\rangle\} {}\nonumber\\&&
= \frac{1}{\sqrt{2}}\{\alpha|0000\rangle+\alpha|0111\rangle+\beta|1000\rangle+\beta|1111\rangle\} {}\nonumber\\&&
= \frac{1}{\sqrt{2}}\{|00\rangle\alpha|00\rangle+|01\rangle\alpha|11\rangle+|10\rangle\beta|00\rangle+|11\rangle\beta|11\rangle\} {}\nonumber\\&&
= \frac{1}{2}\{[|\phi^+\rangle + |\phi^-\rangle]\alpha|00\rangle+[|\psi^+\rangle + |\psi^-\rangle]\alpha|11\rangle+[|\psi^+\rangle - |\psi^-\rangle]\beta|00\rangle+[|\phi^+\rangle - |\phi^-\rangle]\beta|11\rangle\}{}\nonumber\\&&
= \frac{1}{2}\{|\phi^+\rangle[\alpha|00\rangle +\beta|11\rangle] + |\phi^-\rangle[\alpha|00\rangle-\beta|11\rangle]+|\psi^+\rangle[\alpha|11\rangle+\beta|00\rangle]+|\psi^-\rangle[\alpha|11\rangle-\beta|00\rangle]\}\end{aligned}$$
***Appendix 2:***\
Suppose Alice and Bob share a bell state $\frac{1}{\sqrt{2}}\{|00\rangle+|11\rangle\}_{AB}$ and the secret is already being shared between Bob and Charlie is $\{\alpha|00\rangle+\beta|11\rangle\}_{BC}$. $$\begin{aligned}
&&\frac{1}{\sqrt{2}}\{|00\rangle+|11\rangle\}_{AB}\otimes\{\alpha|00\rangle+\beta|11\rangle\}_{BC}{}\nonumber\\&&
= \frac{1}{\sqrt{2}}\{\alpha|0\rangle|00\rangle|0\rangle+\beta|0\rangle|01\rangle|1\rangle+\alpha|1\rangle|10\rangle|0\rangle+\beta|1\rangle|11\rangle|1\rangle\}_{ABBC} {}\nonumber\\&&
= \frac{1}{2\sqrt{2}}\{
[|+\rangle+|-\rangle][|\phi^+\rangle+|\phi^-\rangle]\alpha|0\rangle
+ [|+\rangle+|-\rangle][|\psi^+\rangle+|\psi^-\rangle]\beta|1\rangle {}\nonumber\\&&
+ [|+\rangle-|-\rangle][|\psi^+\rangle-|\psi^-\rangle]\alpha|0\rangle
+ [|+\rangle-|-\rangle][|\phi^+\rangle-|\phi^-\rangle]\beta|1\rangle
\} {}\nonumber\\&&
=\frac{1}{2\sqrt{2}}\{
|+\rangle|\phi^+\rangle[\alpha|0\rangle+\beta|1\rangle]
+ |+\rangle|\phi^-\rangle[\alpha|0\rangle-\beta|1\rangle]
+ |-\rangle|\phi^+\rangle[\alpha|0\rangle-\beta|1\rangle]{}\nonumber\\&&
+ |-\rangle|\phi^-\rangle[\alpha|0\rangle+\beta|1\rangle]
+ |+\rangle|\psi^+\rangle[\alpha|0\rangle+\beta|1\rangle]
+ |+\rangle|\psi^-\rangle[\beta|1\rangle-\alpha|0\rangle]{}\nonumber\\&&
+ |-\rangle|\psi^+\rangle[\beta|1\rangle-\alpha|0\rangle]
+ |-\rangle|\psi^-\rangle[\alpha|0\rangle+\beta|1\rangle]
\}\end{aligned}$$
***Appendix 3:***\
Suppose $R_i$ is the authorized receiver sending request to Charlie and sharing a bell state $\frac{1}{\sqrt{2}}\{|00\rangle+|11\rangle\}_{CR_i}$ with charlie. Suppose $\{\alpha|00\rangle+\beta|11\rangle\}_{BC}$ is shared among Bob and charlie. $$\begin{aligned}
&&\{\alpha|00\rangle+\beta|11\rangle\}_{BC}\otimes\frac{1}{\sqrt{2}}\{|00\rangle+|11\rangle\}_{CR} {}\nonumber\\&&
=\frac{1}{\sqrt{2}}\{\alpha|0\rangle|00\rangle|0\rangle+
\alpha|0\rangle|01\rangle|1\rangle+\beta|1\rangle|10\rangle|0\rangle+\beta|1\rangle|11\rangle|1\rangle\}_{BCCR} {}\nonumber\\&&
= \frac{1}{2\sqrt{2}}\{
[|+\rangle+|-\rangle][|\phi^+\rangle+|\phi^-\rangle]\alpha|0\rangle
+ [|+\rangle+|-\rangle][|\psi^+\rangle+|\psi^-\rangle]\alpha|1\rangle {}\nonumber\\&&
+ [|+\rangle-|-\rangle][|\psi^+\rangle-|\psi^-\rangle]\beta|0\rangle
+ [|+\rangle-|-\rangle][|\phi^+\rangle-|\phi^-\rangle]\beta|1\rangle
\} {}\nonumber\\&&
=\frac{1}{2\sqrt{2}}\{
|+\rangle|\phi^+\rangle[\alpha|0\rangle+\beta|1\rangle]
+ |+\rangle|\phi^-\rangle[\alpha|0\rangle-\beta|1\rangle] {}\nonumber\\&&
+ |-\rangle|\phi^+\rangle[\alpha|0\rangle-\beta|1\rangle]
+ |-\rangle|\phi^-\rangle[\alpha|0\rangle+\beta|1\rangle]
+ |+\rangle|\psi^+\rangle[\alpha|1\rangle+\beta|0\rangle]{}\nonumber\\&&
+ |+\rangle|\psi^-\rangle[\alpha|1\rangle-\beta|0\rangle]
+ |-\rangle|\psi^+\rangle[\alpha|1\rangle-\beta|0\rangle]
+ |-\rangle|\psi^-\rangle[\alpha|1\rangle+\beta|0\rangle]
\}\end{aligned}$$
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We prove that the cardinality of the torsion subgroups in homology of a closed hyperbolic manifold of any dimension can be bounded by a doubly exponential function of its diameter. It would follow from a conjecture by Bergeron and Venkatesh that the order of growth in our bound is sharp.
We also determine how the number of non-commensurable closed hyperbolic manifolds of dimension at least $3$ and bounded diameter grows. The lower bound implies that the fraction of arithmetic manifolds tends to zero as the diameter goes up.
address: 'Mathematisches Institut der Universität Bonn, Germany'
author:
- Bram Petri
bibliography:
- 'TorsDiam.bib'
title: 'Counting non-commensurable hyperbolic manifolds and a bound on homological torsion'
---
Introduction
============
Recently there has been a lot of progress on understanding the relation between the volume of a negatively curved manifold and its toplogical complexity. In this note, we will instead consider the relation between complexity and diameter. We will restrict to closed hyperbolic (constant sectional curvature $-1$) manifolds. The main upshot of considering the diameter instead of the volume is that we obtain bounds in dimension $3$.
New results
-----------
Recall that two manifolds are called commensurable if they have a common finite cover. The diameter of a commensurability class of manifolds is the minimal diameter realized by a manifold in that class. Given $d\in\RR_+$ and $n\geq 3$, let $NC^{\diam}_n(d)$ denote the number of commensurability classes of closed hyperbolic $n$-manifolds of diameter $\leq d$. We will prove
\[thm\_diam\] For all $n \geq 3$ there exist $0<a<b$ so that $$a\cdot d \leq \log(\log(NC^{\diam}_n(d))) \leq b \cdot d$$ for all $d\in\RR_+$ large enough.
Note that the analogous statement in dimension $2$ is false. Since surfaces have large deformation spaces of hyperbolic metrics (see eg. [@Bus] for details), it is not hard to produce an uncountable number of non-commensurable hyperbolic surfaces that have both bounded diameter and bounded volume.
Our upper bound follows directly from a result by Young [@You] (see also Equation ) that estimates the number of manifolds up to a given diameter. However, for his lower bound, Young uses finite covers of a fixed manifold, which are all commensurable. We will instead consider a collection of non-commensurable manifolds constructed by Gelander and Levit [@GelLev] and will use results on random graphs due to Bollobás and Fernandez de la Vega [@BolFer] to argue that most of these manifolds have small diameters.
Because $NC^{\diam}_n(d)$ is finite for all $n\geq 3$ and $d\in \RR_+$, we can turn the set of commensurability classes of closed hyperbolic $n$-manifolds of diameter $\leq d$ into a probability space by equipping it with the uniform probability measure. That is, given $n\geq 3$ and $d\in \RR_+$ and a set $A$ of commensurability classes of closed hyperbolic manifolds of diameter $\leq d$, we set $$\PP_{n,d}[A] = \frac{\card{A}}{NC^{\diam}_n(d)},$$ where $\card{A}$ denotes the cardinality of $A$.
It follows from Theorem \[thm\_diam\] together with results by Belolipetsky [@Bel] for $n\geq 4$ and Belolipetsky, Gelander, Lubotzky and Shalev [@BelGelLubSha] for $n=3$ that most maximal lattices are not arithmetic:
\[cor\_arithm\] Let $n\geq 3$. We have $$\lim_{d\to\infty}\PP_{n,d}[\text{The manifold is arithmetic}] = 0.$$
Similar results have been proved in dimension $\geq 4$ by Gelander and Levit [@GelLev] with diameter replaced by volume and by Masai [@Mas] for a different model of random $3$-manifolds: random $3$-dimensional mapping tori built out of punctured surfaces.
We show that for closed hyperbolic manifolds, the size of homological torsion can also be bounded in terms of the diameter of the manifold.
\[thm\_torsion\] For every $n\geq 2$ there exists a constant $C >0$ so that $$\log\log\left(\card{H_i(M,\ZZ)_{\mathrm{tors}}}\right) \leq C\cdot \diam(M)$$ for all $i=0,\ldots,n$ for any closed hyperbolic $n$-manifold $M$.
Let us first note that for $n=2$ our theorem is automatic, since all torsion subgroups in the homology of a closed hyperbolic surface are trivial. Moreover, in dimension at least $4$ results by Bader, Gelander and Sauer [@BadGelSau] (see also Equation ) together with a comparison between volume and diameter (Lemma \[lem\_diamvol\]) also prove Theorem \[thm\_torsion\]. However, it is known that in dimension $3$, this method cannot work.
We also note that to prove that our theorem is sharp, the most likely examples would be sequences hyperbolic manifolds with exponentially growing torsion and logarithmically growing diameter. Examples of such sequences are abelian covers [@SilWil; @Rai1; @Fellows], Liu’s construction [@Liu] and random Heegaard splittings [@Kow; @Fellows]. However, these manifolds are not known to have small diameters and in some cases are even known not to have small diameters. On the other hand, certain sequences of covers of arithmetic manifolds are conjectured to have exponential torsion growth by Bergeron and Venkatesh [@BerVen]. It is known that the corresponding lattice has Property $(\tau)$ with respect to this sequence of covers [@SarXue], from which it follows that their diameter is small by a result of Brooks [@Bro] (see Sections \[sec\_diambds\] and \[sec\_arithm\]). So assuming the conjecture by Bergeron and Venkatesh, these manifolds would saturate the bound in Theorem \[thm\_torsion\] up to a multiplicative constant.
The proof of Theorem \[thm\_torsion\] consists of two steps. First we use Young’s method from [@You] to build a simplicial complex that models our manifold and has a bounded number of cells of any dimension. We then use a lemma due to Bader, Gelander and Sauer [@BadGelSau] (based on a lemma of Gabber) that bounds the homological torsion in terms of the number of cells (Lemma \[lem\_tors\]) to derive our bound.
Volume and complexity
---------------------
The main motivation for our work is formed by results that bound the homological complexity of a complete negatively curved $n$-manifold $M$ in terms of its volume.
A classical result due to Gromov and worked out by Ballmann, Gromov and Schröder [@Gro2; @BalGroSch], states that there exists a constant $C >0$, depending only on $n$, so that the Betti numbers $b_i(M)$ satisfy $$\label{eq_volbetti}
b_i(M) \leq C \cdot \vol(M),$$ for $i=0,\ldots,n$, where $\vol(M)$ denotes the volume of $M$.
More recently, Bader, Gelander and Sauer [@BadGelSau] have shown that, when the dimension $n$ is at least $4$, the cardinality of the torsion subgroups $H_i(M,\ZZ)_{\mathrm{tors}}$ in homology can also be bounded in terms of the volume. They show that for all $n\geq 4$, there exists a constant $C>0$, depending only on $n$, so that $$\label{eq_voltors}
\log\left(\card{H_i(M,\ZZ)_{\mathrm{tors}}}\right) \leq C\cdot \vol(M).$$
Counting manifolds by volume
----------------------------
We will again restrict ourselves to closed hyperbolic manifolds. A classical result due to Wang [@Wan] states that in dimension $n\geq 4$ the number of closed hyperbolic manifolds of volume $\leq v$ is finite for any $v\in \RR_+$. Let $N^{\vol}_n(v)$ denote the number such manifolds. The first bounds on the growth of $N^{\vol}_n(v)$ are due to Gromov [@Gro1]. Burger, Gelander, Lubotzky and Mozes [@BurGelLubMoz] showed that for all $n\geq 4$ there exist $0<a<b \in \RR$ such that $$\label{eq_volcount}
a \cdot v\log(v)\leq \log\left(N^{\vol}_n(v)\right)\leq b\cdot v\log(v),$$ for all $v\in \RR_+$ large enough.
Analogously to the case of the diameter, the volume of a commensurability class of manifolds is the minimal volume realized in that class. Let $NC^{vol}_n(v)$ denote the number of commensurability classes of hyperbolic $n$-manifolds of volume $\leq v$. The first lower bounds on this number are due to Raimbault [@Rai2]. Gelander and Levit [@GelLev] showed that for all $n\geq 4$ there exist $0<a<b \in \RR$ such that $$\label{eq_volcommcount}
a\cdot v\log(v)\leq \log\left(NC^{\vol}_n(v)\right)\leq b\cdot v\log(v),$$ for all $v\in \RR_+$ large enough.
$3$-dimensional manifolds
-------------------------
As opposed to in dimension $4$ and above, the number of closed hyperbolic $3$-manifolds of bounded volume is not finite. This for instance follows from work of Thurston (see for example [@BenPet Chapter E]). This means that there is also no reason to suppose that a statement like Equation holds in dimension $3$. In fact, in [@BadGelSau], the authors prove that no such bound can exist, even for sequences of hyperbolic $3$-manifolds manifolds that Benjamini-Schramm converge to $\HH^3$. We note that Frczyk [@Fra] has however proved that for arithmetic manifolds, a similar bound to that of Bader, Gelander and Sauer does hold.
Diameters
---------
The number of closed hyperbolic $n$-manifolds of diameter at most $d$ is finite for any $n\geq 3$ and $d\in \RR_+$. The best known estimates on the number $N^{\diam}_n(d)$ of closed hyperbolic $n$-manifolds of diameter $\leq d$ are due to Young [@You]. He proved that for every $n\geq 3$ there exist constants $0<a<b\in\RR^+$ so that $$\label{eq_diamcount}
a\cdot d \leq \log(\log(N^{\diam}_n(d)))\leq b\cdot d$$ for all $d\in \RR_+$ large enough.
Equation implies that the Betti numbers of a closed hyperbolic manifold $M$ can also be bounded by its diameter $\diam(M)$ as follows $$b_i(M) \leq C \cdot e^{(n-1)\cdot \diam(M)},$$ for all $i=0,\ldots n$, where $C>0$ is a constant depending only on $n$ (see Lemma \[lem\_diamvol\]). Moreover, because there are hyperbolic surfaces with linearly growing genus and logarithmically growing diameter (for instance random surfaces [@BroMak; @Mir]), a bound of this generality is necessarily exponential in diameter.
Acknowledgement {#acknowledgement .unnumbered}
---------------
The author’s research was supported by the ERC Advanced Grant “Moduli”. The author also thanks the organizers of the Borel Seminar 2017, during which the research for this article was done. Finally, the author thanks Filippo Cerocchi, Jean Raimbault and Roman Sauer for useful conversations.
Background material
===================
In what follows, $n$ will be a natural number, $M$ a closed oriented hyperbolic $n$-manifold. We will use $\vol(M)$, $\diam(M)$, $\inj(M)$ and $\lambda_1(M)$ to denote the volume, the diameter, the injectivity radius and the first non-zero eigenvalue of the Laplace-Beltrami operator of $M$ respectively. Moreover, $\dist:M\times M\to \RR_+$ will denote the distance function on $M$. Finally, $\HH^n$ will denote hyperbolic $n$-space and $\Isom^+(\HH^n)$ will denote its group of orientation preserving isometries.
Bounds involving the diameter {#sec_diambds}
-----------------------------
Many of our bounds are based on the following well known fact.
\[lem\_diamvol\] Let $n\geq 2$. There exists a constant $C>0$, depending only on $n$, so that for every closed hyperbolic $n$-manifold $M$ we have $$C\cdot \log(\vol(M)) \leq \diam(M).$$
The crucial observation is that the ball $B_M(p,\diam(M))$ of radius $\diam(M)$ around any point $p\in M$, by definition of the diameter, covers $M$. This implies that $$\vol(M)\leq \vol(B_{M}(p,\diam(M))).$$ On the other hand, the volume of $B_{M}(p,\diam(M))$ is at most the volume of a ball of the same radius in $\HH^n$. The volume of a ball $B_{\HH^n}(p,R)$ of radius $R$ around $p\in\HH^n$ is equal to $$\vol(B_{\HH^n}(p,R))=\vol(\Sphere^{n-1})\int_0^R \sinh^{n-1}(t)dt,$$ where $\Sphere^{n-1}$ denotes the $(n-1)$-sphere equipped with the round metric (see for instance [@Rat §3.4]). Putting this together with the inequality above gives the lemma.
Moreover, we shall need a bound on the injectivity radius in terms of the diameter. The following was proved by Young [@You], based on work by Reznikov [@Rez]:
\[lem\_diaminj\] Let $n\geq 2$. There exists a constant $C>0$, depending only on $n$, so that $$\inj(M) \geq \exp(-\diam(M)/C)$$ for all closed hyperbolic $n$-manifolds $M$.
In order to control the diameter of a sequence of congruence covers later on, we will use the eigenvalue of their Laplacian in combination with the following theorem due to Brooks [@Bro Theorem 1]:
\[thm\_diamlaplace\] Let $M$ be a closed hyperbolic manifold and Let $\{M_i\}_{i\in\mathcal{I}}$ be a family of finite covers of $M$. If there exists a constant $C>0$ so that $\lambda_1(M_i)> C$ for all $i\in\mathcal{I}$, then there exist constants $a,b,c>0$ such that $$a < \frac{\log(\vol(M_i))+c}{\diam(M_i)} < b$$ for all $i\in\mathcal{I}$.
Gelander and Levit’s construction
---------------------------------
The lower bound in Theorem \[thm\_diam\] will come from a construction due to Gelander and Levit, which is inspired by a classical construction due to Gromov and Piatetski-Shapiro [@GroPia] (see also [@Rai2]). We will briefly describe some, but not all, of the details of their construction. For more information we refer to [@GelLev].
Assume we are given six compact hyperbolic $n$-manifolds with boundary $V_0$, $V_1$, $A_+$, $A_-$, $B_+$ and $B_-$ so that
- $V_0$ and $V_1$ both have four boundary components and $A_+$, $A_-$, $B_+$ and $B_-$ all have two boundary components.
- All the boundary components of these manifolds are isometric to a fixed closed hyperbolic $(n-1)$-manifold.
- Each of these six manifolds is embedded in an arithmetic manifolds without boundary that are pairwise non-commensurable (see Section \[sec\_arithm\] for a definition of an arithmetic group).
In [@GelLev Section 4], Gelander and Levit explain how to construct these manifolds.
We will glue these manifolds according to Schreier graphs for finite index subgroups of the free group $\FF_2=\langle a,b\rangle$. Let $\Cay(\FF_2,\{a,b\})$ denote the Cayley graph of $\FF_2$ with respect to the generating set $\{a,b\}$. Given $H < \FF_2$, the Schreier graph $\Gamma_H$ is the graph $$\Gamma_H = \Cay(\FF_2,\{a,b\}) / H.$$ Since the edges in $\Cay(\FF_2,\{a,b\})$ come with a natural labeling with the symbols $\{a^\pm,b^\pm\}$, the edges $\Gamma_H$ come with such a labeling as well. Furthermore, note that the number of vertices of $\Gamma_H$ is equal to the index $[\FF_2:H]$. Let us denote the vertex and edge set of $\Gamma_H$ by $V(\Gamma_H)$ and $E(\Gamma_H)$ respectively.
Given a finite index subgroup $H < \FF_2$ and a map $\tau:V(\Gamma_H)\to\{0,1\}$, we construct the closed hyperbolic $n$-manifold $M(H,\tau)$ as follows:
- To each vertex $v\in V(\Gamma_H)$, associate a copy of $V_{\tau(v)}$
- and to each edge $e\in E(\Gamma_H)$, associate a copy of the pair $A^+,A^-$ or $B^+,B^-$, according to whether it is labeled with an $a^\pm$ or a $b^\pm$.
- Glue the manifolds together according to the incidence relations in $\Gamma_H$. In particular, the order in which to glue the two blocks associated to an edge depends on whether or not the edge is labeled with an inverse.
Note that there is some ambiguity in the construction above: there is for instance a choice which boundary component to glue to which. Since we are using the construction for a lower bound, this won’t make a difference to us. We will from now on assume some choice of gluing is given for every pair $(H,\tau)$. Figure \[pic\_construction\] shows a cartoon of what the local picture of $M(H,\tau)$ might look like:
[Pic\_Construction.pdf]{} (11,31) [$V_0$]{} (23,36.5) [$A_+$]{} (33,36.5) [$A_-$]{} (47,30) [$V_1$]{} (61,36.5) [$B_+$]{} (71,36.5) [$B_-$]{} (83,31) [$V_0$]{} (29.5,16) [$a$]{} (65,16) [$b^{-1}$]{} (47,3) [$\Gamma_H$]{} (42,40) [$M(H,\tau)$]{} (17,11) [$v_1$]{} (51,11) [$v_2$]{} (80,11) [$v_3$]{}
In [@GelLev Proposition 3.3], Gelander and Levit show:
\[prp\_noncomm\] Let $H,H'<\FF_2$ be distinct finite index subgroups and let $\tau:V(\Gamma_H)\to \{0,1\}$ and $\tau':V(\Gamma_{H'})\to \{0,1\}$ be so that $$\card{\tau^{-1}(1)} = \card{(\tau')^{-1}(1)} = 1.$$ Then $M(H,\tau)$ and $M(H',\tau')$ are not commensurable.
The upshot of this proposition is that the construction of Gelander and Levit gives rise to at least $a_N(\FF_2)$ non-commensurable manifolds on built out of graphs with $n$ vertices, where $a_N(\FF_2)$ denotes the number of index $N$ subgroups of $\FF_2$.
Graphs and groups
-----------------
To work with Gelander and Levit’s construction, we will need two bounds. We need a lower bound on $a_N(\FF_2)$ and we need to know what the typical diameter of a Schreier graph of an index $N$ subgroup of $\FF_2$ is.
For details on the subgroup growth of $\FF_2$, we refer to Chapter 2 in the monograph by Lubotzky and Segal [@LubSeg]. We will use the following bound, that can be found as a special case of [@LubSeg Theorem 2.1]:
\[thm\_subgrps\] We have $$a_N(\FF_2)\sim N \cdot N!$$ as $N\to\infty$.
In the theorem above, we write $f(N)\sim g(N)$ as $N\to\infty$ to mean that $$\lim_{N\to\infty} f(N)/g(N) =1.$$
The distance between two vertices in a connected graph is the minimal number of edges in a path between these two vertices. The diameter $\diam(\Gamma)$ of a finite graph $\Gamma$ is the maximal distance realized by two vertices in $\Gamma$.
To control the diameter of a typical Schreier graph we will use results from random graphs. The fact that the diameter of a random regular graph is bounded by a logarithmic function of the number of vertices can for instance be derived from the fact that a random regular graph has a large spectral gap with probability tending to $1$ as the number of vertices tends to infinity (see [@Fri; @Pud; @LinHooWig; @BroSha]). The sharpest result however does not use this method and is due to Bollobás and Fernandez de la Vega [@BolFer Theorem 3]. We will state their result only in the case of $4$-regular graphs.
\[thm\_diamgraph\] Let $\PP_N$ denote the uniform probability measure on the set of isomorphism classes of $4$-regular graphs on $N$. There exists a function $E:\NN\to\RR$ so that $$E(N) = o(\log(N))$$ as $N\to\infty$ and $$\PP_N[\text{The graph has diameter }\leq \log_3(N) + E(N)]\to 1$$ as $N\to\infty$.
We note that the bound is basically as low as one could possibly expect. Indeed, with an argument very similar to that in Lemma \[lem\_diamvol\] it can be shown that the diameter of a $4$-regular graph is at least of the order $\log_3(N)$.
We won’t go into random regular graphs in this note and refer the reader to [@BolFer; @Bol; @Wor] for the details. We do however note that uniformly picking an index $N$ Schreier graph is a slightly different model for random $4$-valent graphs than the uniform probability measure on isomorphism classes of $4$-valent graphs. It turns out that the two models are what is called contiguous: they have the same asymptotic $0$-sets, which is enough for our purposes. More details on this can be found in [@Wor Section 4] and [@GreJanKimWor].
Arithmetic manifolds {#sec_arithm}
--------------------
Let $G$ be a semisimple Lie group of noncompact type that is defined over $\QQ$ (in our case $G$ will always be $\Isom^+(\HH^n)$). A discrete subgroup $\Gamma < G(\QQ)$ will be called arithmetic if there is a $\QQ$-embedding $\rho:G\to \GL_m(\RR)$ such that $\rho(\Gamma)$ is commensurable with $G(\ZZ) = \GL_m(\ZZ)\cap \rho(G)$. Arithmetic groups come with a sequence of finite index subgroups called congruence groups. For general background on arithmetic groups, we refer to [@MacRei; @Mor].
Recall that a lattice $\Gamma < \Isom^+(\HH^n)$ is called uniform if $\Gamma\backslash\Isom^+(\HH^n)$ is compact. We will call $\Gamma$ maximal if it is not properly contained in another lattice. The result we will need is a bound on the number of maximal uniform arithmetic lattices up to a given covolume in $\Isom^+(\HH^n)$ for $n\geq 3$. To this end, let $\mathrm{MAL}^u_n(v)$ denote the number of maximal uniform arithmetic lattices of covolume $\leq v$ in $\Isom^+(\HH^n)$. The following theorem is due to Belolipetsky [@Bel] in dimension $n\geq 4$ and Belolipetsky, Gelander, Lubotzky and Shalev [@BelGelLubSha] in dimensions $2$ and $3$.
\[thm\_mal\] Let $n\geq 2$ and $\varepsilon >0$. There exist constants $\alpha=\alpha(n) \in \RR_+$ and $\beta=\beta(n,\varepsilon)\in\RR_+$ so that $$v^\alpha \leq \mathrm{MAL}^u_n(v) \leq v^{\beta\cdot (\log v)^\varepsilon}$$ for all $v\in\RR^+$ large enough.
Let us now restrict to hyperbolic $3$-manifolds. To get bounds on the diameters of congruence covers of arithmetic manifolds, we will use the following theorem due to Sarnak and Xue [@SarXue]:
\[thm\_tau\] Let $\Gamma<\Isom^+(\HH^3)$ be a uniform arithmetic lattice. Then there exists a constant $C=C(\Gamma)>0$ so that for all congruence subgroups $\Gamma'<\Gamma$ we have $$\lambda_1(\Gamma'\backslash \HH^3) \geq C.$$
If a lattice $\Gamma<\SL_2(\CC)$ has a sequence $\{\Gamma_N\}_{N}$ of finite index subgroups so that $\lambda_1(\Gamma_N\backslash \HH^3)$ is uniformly bounded from below for all $N\in\NN$, then $\Gamma$ is said to have Property ($\tau$) with respect to this sequence. In the special case where $\Gamma$ is arithmetic and the sequence consists of congruence subgroups, Property ($\tau$) is sometimes also called the Selberg property.
Congruence covers are also believed to have large torsion subgroups in their homology. Specifically, there is the following conjecture, due to Bergeron and Venkatesh [@BerVen], which we state in the special case of hyperbolic $3$-manifolds:
\[con\_berven\] Let $\Gamma<\SL_2(\CC)$ be a uniform arithmetic lattice and $\ldots<\Gamma_N<\Gamma_{N-1}<\ldots<\Gamma_1<\Gamma$ a sequence of congruence subgroups of $\Gamma$ so that $\cap_N \Gamma_N = \{1\}$. Then $$\lim_{N\to\infty} \frac{\log\left(\card{H_1(\Gamma_N \backslash \HH^3,\ZZ)_\mathrm{tors}}\right)}{\vol(\Gamma_N \backslash \HH^3)} = \frac{1}{6\pi}.$$
The reason for the constant $1/6\pi$ above is that it is the $\ell^2$-torsion of $\HH^3$.
Torsion and the nerve lemma
---------------------------
To bound torsion in our manifolds we will use a lemma by Bader, Gelander and Sauer [@BadGelSau Lemma 5.2], that they derived from a lemma due to Gabber (which can for instance be found in [@Sou Lemma 1]). In this lemma, the degree of a vertex ($0$-cell) in a simplicial complex is the degree of that vertex in the $1$-skeleton of the given complex.
\[lem\_tors\] For all $D,p \in \NN$ there exists a constant $C=C(D,p)>0$ so that for any simplicial complex $X$ with $\leq V$ vertices that all have degree $\leq D$ we have: $$\log\left(\card{H_p(X,\ZZ)_{\mathrm{tors}}}\right) \leq C\cdot V.$$
The simplicial complex we will use will be the nerve of an open cover of our manifold. Recall that an open cover of a space $X$ is a collection $\mathcal{U}=\{U_i\}_{i\in\mathcal{I}}$ of open subsets of $X$ so that $$X = \bigcup_{i\in\mathcal{I}}U_i.$$ The nerve $\mathcal{N}(\mathcal{U})$ of this cover is the simplicial complex that has the sets $U_i$ as vertices and contains a $k$-simplex for every $k$-tuple of elements in $\mathcal{U}$ that have a non-trivial intersection. See [@Hat Section 4G] for more details.
The following statement is known as the nerve lemma and can for instance be found as [@Hat Corollary 4G.3]:
\[lem\_nerve\] If $\mathcal{U}$ is an open cover of a paracompact space $X$ such that every nonempty intersection of finitely many sets in $\mathcal{U}$ is contractible, then $X$ is homotopy equivalent to the nerve $\mathcal{N}(\mathcal{U})$.
Counting
========
We are now ready to prove Theorem \[thm\_diam\]:
[\[thm\_diam\]]{} For all $n \geq 3$ there exist $0<a<b$ so that $$a\cdot d \leq \log(\log(NC^{\diam}_n(d))) \leq b \cdot d$$ for all $d\in\RR_+$ large enough.
The upper bound is direct from Young’s result (Equation ), so we focus on the lower bound.
Consider the building blocks defined by Gelander and Levit and set $$D=\max\{\diam(V_0),\diam(V_1),\diam(A_+),\diam(A_-),\diam(B_+),\diam(B_-)\}.$$ Because all the building blocks are compact, this is a finite number. Given a finite index subgroup $H<\FF_2$ and a map $\tau:V(\Gamma_H)\to\{0,1\}$, we have $$\diam(M(H,\tau)) \leq 2D\cdot\diam(\Gamma_H)+2D.$$ Indeed, suppose $x,y\in M(H,\tau)$. Then the number of building blocks that need to be crossed to get from $x$ to $y$ is at most $2\diam(\Gamma_H)+2$, including the building blocks containing $x$ and $y$.
Because of Theorem \[thm\_subgrps\] combined with Theorem \[thm\_diamgraph\] and Stirling’s approximation, there exists a $C>0$ so that the index $N$ subgroups of $\FF_2$ for $N$ large enough produce at least $$C\cdot N^{CN}$$ non-isomorphic graphs of diameter $\leq \log_3(N) + o(\log(N))$. If we now define maps $\tau:V(\Gamma)\to\{0,1\}$ that assign the value $1$ to only one vertex per graph $\Gamma$, we obtain $C\cdot N^{CN}$ manifolds of diameter $d \leq 2D\log_3(N) + o(\log(N))$. Working this out, we see that this number of manifolds is at least $$\exp(C'\cdot d \cdot \exp(C'\cdot d)),$$ for some $C'>0$. Proposition \[prp\_noncomm\] tells us that none of the resulting manifolds will be commensurable.
Corollary \[cor\_arithm\] now also easily follows.
[\[cor\_arithm\]]{} Let $n\geq 3$. We have $$\lim_{d\to\infty}\PP_{n,d}[\text{The manifold is arithmetic}] = 0$$
The only thing we need to control is the number of maximal uniform arithmetic lattices of diameter $\leq d$ in $\Isom^+(\HH^n)$. Let us call this number $\mathrm{MALD}^u_{n}(d)$. By Lemma \[lem\_diamvol\] we have $$\mathrm{MALD}^u_{n}(d) \leq \mathrm{MAL}^u_n(e^{d/C_n})$$ for some $C_n>0$ independent of $d$. As such, Theorem \[thm\_mal\] implies that for every $\varepsilon>0$ there exists a $\beta'>0$ so that $$\mathrm{MALD}^u_{n}(d) \leq \exp(\beta'\cdot d^{1+\varepsilon}).$$ Comparing this to Theorem \[thm\_diam\] gives the result.
Torsion
=======
In this section we prove Theorem \[thm\_torsion\] and explain how a positive answer to Conjecture \[con\_berven\] would lead to a sequence of manifolds that saturates the bound in that theorem up to a multiplicative constant.
An upper bound for torsion in homology
--------------------------------------
We will start with:
[\[thm\_torsion\]]{} For every $n\geq 2$ there exists a constant $C >0$ so that $$\log\log\left(\card{H_i(M,\ZZ)_{\mathrm{tors}}}\right) \leq C\cdot \diam(M)$$ for all $i=0,\ldots,n$ for any closed hyperbolic $n$-manifold $M$.
Our final goal is to employ Lemma \[lem\_tors\]. In the language of [@BadGelSau]: we need to show that a closed hyperbolic manifold $M$ is homotopy equivalent to a $(D,C\cdot\diam(M))$-simplicial complex, where $C,D>0$ are constants depending only its dimension. The simplicial complex we build is the same as that used for the upper bound in Young’s result (Equation ).
Set $r=\inj(M)$ and let $S\subset M$ be a maximal set of points so that $$\dist(s,s') \geq r/4$$ for all $s,s'\in S$. Now consider the collection $$\mathcal{U}=\{B_M(s,r/2)\}_{s\in S},$$ where $B_M(s,r/2)$ denotes the open ball in $M$ of radius $r/2$ around $s$. It follows from maximality of $S$ that these balls form an open cover. Moreover, because their radius is half the injectivity radius they are isometrically embedded $n$-dimensional hyperbolic balls. As such they are convex, which means that their intersections are convex and thus contractible. Hence, the nerve lemma (Lemma \[lem\_nerve\]) applies. This means that the homology groups we are after are those of $\mathcal{N}(\mathcal{U})$.
So we need to find bounds on the number of vertices and their degrees in $\mathcal{N}(\mathcal{U})$ in order to apply Lemma \[lem\_tors\].
The number of vertices, or equivalently the number of points in $S$, can be bounded by $$\card{S} \leq \frac{\vol(M)}{\vol(B_{\HH^n}(p,r/4))} \leq D\cdot \frac{\vol(M)}{(r/4)^n},$$ where $B_{\HH^n}(p,r/4)$ is the ball of radius $r/4$ around some point $p\in\HH^n$ and $D>0$ is some constant depending only on the dimension. The second of these bounds again follows from the closed formula for the volume of a ball in $\HH^n$. Now we use Lemma \[lem\_diaminj\] and Lemma \[lem\_diamvol\] to tell us that $$(r/4)^n \geq \exp(-C\diam(M)) \;\;\text{and}\;\; \vol(M) \leq \exp(C\diam(M))$$ for some $C>0$ depending only on $n$. So we obtain $$\card{S} \leq A\cdot \exp(B\diam(M)).$$ for some $A,B>0$ depending only on $n$.
All that remains is to show that each vertex has a bounded number of neighbors. First of all note that all the neighbors of a point $s\in S$ lie in $B_M(s,r)\subset M$. By definition of $S$, the balls of radius $r/8$ around the neighbors of $s$ are all disjoint and all lie in $B_{M}(s,9r/8)$. This means that the number of neighbors is at most $$\frac{\vol(B_M(s,9r/8))}{\vol(B_M(s,r/8))} \leq \frac{\vol(B_{\HH^n}(s,9r/8))}{\vol(B_{\HH^n}(s,r/8))},$$ which, for $r$ small enough, is uniformly bounded in each fixed dimension.
Sharpness of the bound
----------------------
Like we said in the introduction, if Conjecture \[con\_berven\] holds then we would get a sequence of closed hyperbolic $3$-manifolds $\{M_N\}_{N\in\NN}$ such that $\diam(M_N)\to \infty$ as $N\to\infty$ and $$\log\log\left(\card{ H_1(M_N,\ZZ)_{\mathrm{tors}}}\right) \geq C \diam(M_N).$$ for some $C>0$ independent of $N$. Indeed, it follows from Theorem \[thm\_diamlaplace\] together with Theorem \[thm\_tau\] that for a uniform arithmetic group $\Gamma < \SL_2(\CC)$ and a sequence of congruence subgroups $\{\Gamma_N\}_N$, the manifolds $M_N = \Gamma_N \backslash \HH^3$ satisfy $$\diam(M_N) \leq A \cdot \log(\vol(M_N)).$$ It would follow from Conjecture \[con\_berven\] that $$\vol(M_N)\leq B\cdot \log\left(\card{ H_1(M_N,\ZZ)_{\mathrm{tors}}}\right)$$ for all $N$ and some $B>0$ independent of $N$. Putting these two together would give the desired result.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In the first part of this paper we give a short review of the hierarchy of stochastic models, related to physical chemistry. In the basement of this hierarchy there are two models — stochastic chemical kinetics and the Kac model for Boltzman equation. Classical chemical kinetics and chemical thermodynamics are obtained as some scaling limits in the models, introduced below. In the second part of this paper we specify some simple class of open chemical reaction systems, where one can still prove the existence of attracting fixed points. For example, MichaelisMenten kinetics belongs to this class. At the end we present a simplest possible model of the biological network. It is a network of networks (of closed chemical reaction systems, called compartments), so that the only source of nonreversibility is the matter exchange (transport) with the environment and between the compartments.'
author:
- 'V.A. Malyshev'
title: Fixed Points for Stochastic Open Chemical Systems
---
Keywords: chemical kinetics, chemical thermodynamics, Kac model, mathematical biology
Introduction
============
Relation between the existing (mathematical) physical theory and future mathematical biology seems to be very intimate. For example, equilibrium is a common state in physics but in biology equilibrium means death. Biology should be deeply dynamical but this goal seems unreachable in full extent: even in simplest physical situations the time consuming complexity of any study of local dynamics is out of the present state of art. Thus the only possibility would be to consider simpler dynamical models (mean field etc.) but to go farther in their structure. The obvious first step should have been related to chemical kinetics and chemical thermodynamics. Here we present a review of these first results and discuss what should be the second step.
In the first part of this paper we give a short review (in more general terms than in [@Mal3]) of the hierarchy of stochastic models, related to physical chemistry. In the basement of this hierarchy there are two models — stochastic chemical kinetics and the Kac model for Boltzman equation. Classical chemical kinetics and chemical thermodynamics are obtained as some scaling limits in the models, introduced below.
If some physical conditions, as reversibility, are assumed for a closed (without matter exchange) system, then we have sufficiently simple behaviour: one can prove convergence to a fixed point. However, in many models of physical chemistry and biology, no reversibility condition is assumed, and the behaviour can be as complicated as one can imagine. Here we have already some gap between physics and biology, and it is necessary to fill in this gap. In the second part of this paper we specify some simple class of open chemical reaction systems, where one can still prove the existence of attracting fixed points. For example, MichaelisMenten kinetics belongs to this class. At the end we present a simplest possible model of the biological network. It is a network of networks (of closed chemical reaction systems, called compartments), so that the only source of nonreversibility is the matter exchange (transport) with the environment and between the compartments.
Microdynamics
=============
Any molecule of mass $m$ can be characterized by translational degrees of freedom (velocity $v\in\R^{3}$, coordinate $x\in\R^{3}$) and internal, or chemical (for example, rotational and vibrational) degrees of freedom. Internal degrees of freedom include the type $j=1,\ldots,J$ of the molecule and internal energy functionals $K_{j}(z_{j}),z_{j}\in\mathbf{K}_{j}$, in the space $\mathbf{K}_{j}$ of internal degrees of freedom. It is often assumed, see [@LanLif], that the total energy of the molecule $i$ is$$E_{i}=T_{i}+K_{j}(z_{j,i}).$$ We consider here the simplest choice when $K_{j}$ is the fixed nonnegative number, depending only on $j$. It can be interpreted as the energy of some chemical bonds.
We consider the set $\mathbf{X}$ of countable locally finite configurations $X\!\!=\!\{x_{i},v_{i},j_{i}\}$ of particles (molecules) in $\R^{3}$, where each particle $i$ has a coordinate $x_{i}$, velocity $v_{i}$ and type $j_{i}$. Denote by $\mathfrak{M}$ the system of all probability measures on $\mathbf{X}$ with the following properties:
- Coordinates of these particles are distributed as the homogeneous Poisson point field of particles on $\R^{3}$ with some density $c$.
- The vectors $(v_{i},j_{i})$ are independent of the space coordinates and of the other particles. The velocity $v$ of a particle is assumed to be uniformly distributed on the sphere with the radius defined by the kinetic energy $T={m}(v_{1}^{2}+v_{2}^{2}+v_{3}^{2})/2$ of the particle, and the pairs $(j_{i},T_{i})$ are distributed via some common density $p(j,T),$ $$\sum_{j}\int p(j,T)\, dT=1.$$
Our first goal will be to define random dynamics on $\mathbf{X}$ (or deterministic dynamics on $\mathfrak{M}$). It is defined by a probability space $(\mathbf{X}^{0,\infty},\mu)$, where $\mu=\mu^{0,\infty}$ is a probability measure on the set $\mathbf{X}^{0,\infty}$ of countable arrays $X^{0,\infty}(t)=\{x_{i}(t),v_{i}(t),j_{i}(t)\}$ of trajectories $x_{i}(t),v_{i}(t),j_{i}(t)$ on intervals $I_{i}=(\tau_{i},\eta_{i})$, where $0\leq\tau_{i}<\eta_{i}\leq\infty$. The measure $\mu$ belongs to the set of measures $\mathfrak{M}^{0,\infty}$ on $X^{0,\infty}(t)$, defined by the following properties:
- If for any fixed $0\leq t<\infty$ we denote by $\mu(t)$ the measure induced by $\mu$ on $\mathbf{X}$, then $\mu(t)\in\mathbf{\mathfrak{M}}$.
- The trajectories $x_{i}(t),v_{i}(t),j_{i}(t)$ are independent, each of them is a Markov process (not necessary time homogeneous). This process is defined by initial measure $\mu(0)$ on $\mathbf{X}$, by birth and death rates, defining time moments $\tau_{i},\eta_{i}$, and by transition probabilities at time $t$, independent of the motion of individual particles but depending on the concentration densities $c_{t}(j,T)$ at time $t$.
- The evolution of the pair $(j,T)$ for the individual particle in-between the birth and death moments is defined by the following Kolmogorov equations, which control the one-particle process $$\frac{\partial p_{t}(j_{1},T_{1})}{\partial t}=\sum_{j}\!\int\!(P(t;j_{1},T_{1}|j,T)\, p_{t}(j,T)-P(t;j,T|j_{1},T_{1})p_{t}(j_{1},T_{1}))\, dT\label{kol_1}$$ defining Markov process with distributions $p_{t}(j,T)$. The probability kernel $P$ depends however on $p_{t}(j,T)$ itself, we shall make it precise below.
The dynamics we will describe here is based on some earlier mathematical models and central dogmas of physical chemistry. The simplest way to rigorously introduce the measure $\mu$ is by the limit of finite volume random dynamics. Initial conditions for this dynamics are as follows: at time $0$ some number $n^{(\Lambda)}(0)$ of molecules are thrown uniformly in the cube $\Lambda$, their parameters $(j,T)$ are independent and have some common density $p_{0}(j,T)$, not depending on $\Lambda$. Let $n_{j}(t)=n_{j}^{(\Lambda)}(t)$ be the number of type $j$ molecules at time $t$.
Heuristically, our time scale is such that for the unit of time each molecule does $O(1)$ transitions. Then for any macroquantity $q\textrm{ }$of substance its $O(q)$ part may change. One should choose time scales for input-output processes correspondingly.
The (output) rate of the jumps $n_{j}\rightarrow n_{j}-1$ is denoted by $\lambda_{j}^{(0)}$, that is with this rate a molecule of type $j$ is chosen randomly and deleted from $\Lambda$. Similarly, the (input) rate of the jumps $n_{j}\rightarrow n_{j}+1$ is denoted by $\lambda_{j}^{(i)}$, that is a molecule of type $j$ is put uniformly in $\Lambda$ with this rate. Dependence of both rates on the concentrations can be quite different. To get limiting I/O process after (canonical) scaling one can assume that $$\lambda_{j}^{(0)}=f_{j}^{(0)}\Lambda,\qquad\lambda_{j}^{(i)}=f_{j}^{(i)}\Lambda,\label{IOfunc}$$ where $f_{j},g_{j}>0$ are some functions of all $c_{1}^{(\Lambda)},\ldots,c_{J}^{(\Lambda)}$, $c_{j}^{(\Lambda)}={n_{j}}/{\Lambda}$ are the concentrations. However, mostly we restrict ourselves to the case when $f_{j}$ are functions of $c_{j}$ only. In other words, an individual type $j$ molecule leaves the volume with rate ${f_{j}^{(0)}(c_{j}^{(\Lambda)})}/{c_{j}^{(\Lambda)}}$. Denote $$f_{j}=f_{j}^{(i)}-f_{j}^{(0)}.\label{IOfunc1}$$
The hierarchy presented here depends on what parameters of a molecule are taken into account. In stochastic chemical kinetics only type is taken into account. The state of the system is given by the vector $(n_{1},\ldots,n_{J})$. There are also $R$ reaction types and the reaction of the type $r=1,\ldots,R$, can be written as$$\sum_{j}\nu_{jr}M_{j}=0$$ where we denote by $M_{j}$ a $j$ type molecule, and the stoihiometric coefficients $\nu_{jr}>0$ for the products and $\nu_{jr}<0$ for the substrates. One event of type $r$ reaction corresponds to the jump $n_{j}\rightarrow n_{j}+\nu_{jr},\; j=1,\ldots,J.$ Classical polynomial expressions (most commonly used)$$\lambda_{r}=A_{r}\prod\limits _{j:\nu_{jr}<0}n_{j}^{-\nu_{jr}}$$ for the rates of these jumps define a continuous time Markov process, a kind of random walk in $\Z_{+}^{J}$. This dependence can be heuristically deduced from local microdynamics. However, polynomial dependence is not the only possibility, see [@Sava]. Moreover, there can be various scalings for these rates. The scaling$$A_{r}=a_{r}\Lambda^{\gamma_{r}+1},\qquad\gamma_{r}=\sum_{j:\nu_{jr}<0}\nu_{jr}$$ where $a_{r}$ are some constants and $\Lambda$ is some large parameter, is called canonical because the classical chemical kinetics equations $$\frac{dc_{j}(t)}{dt}=\sum_{r}R_{j,r}(\vec{c}(t))\label{cck}$$ for the densities$$c_{j}(t)=\lim_{\Lambda\rightarrow\infty}\Lambda^{-1}n_{j}^{(\Lambda)}(t)$$ follow in the large $\Lambda$ limit, with some polynomials $R_{j,r}$ (see below and [@MaPiRy]).
It is important to give (at least heuristic) local probabilistic models to explain other than polynomial dependence and scalings for the rates. For example for arbitrary homogeneous functions as in [@Sava].
It is assumed that at time $t=0$ as $\Lambda\rightarrow\infty,$ $\Lambda^{-1}n^{(\Lambda)}(0)\longrightarrow c(0).$
Chronologically, the first paper in stochastic chemical kinetics was by Leontovich [@Leo], which appeared from discussions with A.N. Kolmogorov. Other references see in [@GadLeeOth]. In 70s stochastic chemical kinetics for small $R,J$ was studied intensively, see reviews [@McQua; @Kal]. At the same time the general techniques to get limiting equations (\[cck\]) appears in probability theory [@VenFre; @EthKur]. Now there are many experimental arguments in favor of introducing stochasticity in chemical kinetics [@AdAr; @ArRoAd; @GadLeeOth].
In the classical Kac model [@Kac] the molecules $i=1,\ldots,N$ have the same type, but each molecule $i$ has a velocity $v_{i}$ or kinetic energy $T_{i}$. In collisions the velocities (or the kinetic energies) change somehow. There is still continuing activity with deeper results concerning the Kac model, in particular convergence rate, see for example [@CaCaLo].
One should merge Kac type models with stochastic chemical kinetics. Then each molecule $i$ acquires a pair $(j_{i},T_{i})$ of parameters: type $j$ and kinetic energy $T$. However this is not sufficient to get energy redistribution. One should introduce also “chemical” energy. As it is commonly accepted, the general idea is that the energy of chemical bonds of a substrate molecule can be redistributed between product molecules, part of the energy transforming into heat. To describe this phenomena in well-defined terms we introduce fast and slow reactions. Fast reactions do not touch chemical energy, that is types, but slow reactions may change both kinetic and chemical energies, thus providing energy redistribution between heat and chemical energy.
Examples of reactions:
1\. All chemical reactions are assumed slow — unary (unimolecular) $A\rightarrow B$, binary $A+B\rightarrow C+D$, synthesis $A+B\rightarrow C$, decay $C\rightarrow A+B$ etc. In any considered reaction the total energy conservation is assumed, that is the sum of total energies in the left side is equal to the sum of total energies in the right side of the reaction equation.
2\. Fast binary reactions of the type $A+B\rightarrow A+B$, which correspond to elastic collisions and draw the system towards equilibrium.
3\. Fast process of heat exchange with the environment, with reactions of the type $A+B\rightarrow A+B$, but where one of the molecules is an outside molecule.
If there is no input and output, then the Markov jump process is the following. Consider any subset $i_{1}<\ldots<i_{m(r)}$ of $m(r)=-\sum_{j:\nu_{jr}<0}\nu_{jr}$ substrate molecules for reaction of type $r$.
On the time interval $(t,t+dt)$ these molecules have a “collision” with probability ${\Lambda^{-(m(r)-1)}}b_{r}\, dt$, where $b_{r}$ is some constant. Let the parameters of these molecules be $j_{k}=j(i_{k}),T_{k}=T(i_{k})$. Denote $$T=\sum_{i=1}^{m}T_{i},\qquad K=\sum_{i=1}^{m}K_{j_{i}}$$ and $T^{\prime},K^{\prime}$ are defined similarly for the parameters $j_{1}^{\prime},\ldots,j_{m}^{\prime},T_{1}^{\prime},\ldots,T_{m^{\prime}}^{\prime}$ of $m^{\prime}$ product molecules. The reaction occurs only if $$T+K-K^{\prime}\geq0\label{energycon}$$ and then the energy parameters of the product particles at time $t+0$ have the distribution defined by some conditional density $P_{r}(T_{1}^{\prime},\ldots,T_{m^{\prime}-1}^{\prime}|T_{1},\ldots,T_{m})$ on the set $0\leq T_{1}^{\prime}+\ldots+T_{m^{\prime}-1}^{\prime}\leq T+K-K^{\prime}$. By energy conservation then$$T_{m^{\prime}}^{\prime}=T+K-K^{\prime}-\sum_{i=1}^{m^{\prime}-1}T_{i}^{\prime}.$$ This defines a Markov process $M_{\bar{A}}(t)$ on the finite-dimensional space (note that $T\in R_{+}$) $$Q_{\bar{A}}=\bigcup_{(n_{1},\ldots,n_{J})}R_{+}^{n_{1}}\times\ldots\times R_{+}^{n_{J}}$$ where the union is over all vectors $(n_{1},\ldots,n_{J})$ such that for the array $\bar{A}=(A_{1},\ldots,A_{Q})$ of positive integers and for any atom type $q=1,\ldots,Q,$ $$\sum_{j}n_{j}a_{jq}=A_{q}$$ where $a_{jq}$ is the number of atoms of type $q$ in the $j$ type molecule. In other words, each atom type defines the conservation law $A_{q}=\mathrm{const}$.
Now, using conditional densities $P_{r}$, we define the “one-particle” transition kernel $$P(t;j_{1},T_{1}|j,T)=\sum_{r}P^{(r)}(t;j_{1},T_{1}|j,T),\label{kernel}$$ that is the sum of terms $P^{(r)}$ corresponding to reactions $r$, which we define for some reaction types. For unimolecular reactions $j\rightarrow j_{1}$ the product kinetic energy $T_{1}$ is uniquely defined, thus $P_{j\rightarrow j_{1}}$ is trivial and for some constants $u_{jj_{1}},$ $$P^{(j\rightarrow j_{1})}=u_{jj_{1}}\delta(T+K-K_{1}-T_{1}).$$ For binary reactions $j,j'\rightarrow j_{1},j'_{1},$ $$\begin{aligned}
P^{(j,j'\rightarrow j_{1},\, j'_{1})} & =\sum_{j^{\prime},\, j_{1}^{\prime}}\int dT^{\prime}dT_{1}^{\prime}b_{j,j'\rightarrow j_{1},\, j'_{1}}P_{j,j'\rightarrow j_{1},\, j'_{1}}(T_{1}|T,T^{\prime})\\
& \quad\times c_{t}(j^{\prime},T^{\prime})\,\delta(T+K+T^{\prime}+K^{\prime}-K_{1}-T_{1}-K_{1}^{\prime}-T_{1}^{\prime}).\end{aligned}$$ In particular, for “fast” collisions (which do not change type) we have the same transition kernel but with $j=j_{1},j'=j'_{1}$. We see that $P^{(j,j'\rightarrow j_{1},j'_{1})}$ depend on the concentrations $c_{t}(j,T)=p_{t}(j,T)\, c(t).$ They are defined via the Boltzman type equation $$\begin{aligned}
\frac{\partial c_{t}(j_{1},T_{1})}{\partial t} & =f_{j}(c_{j})+\sum_{j}\int\big(P(t;j_{1},T_{1}|j,T)\, c_{t}(j,T)\notag\\
& \quad-P(t;j,T|j_{1},T_{1})\, c_{t}(j_{1},T_{1})\big)\, dT\label{bol_1}\end{aligned}$$ which is similar to the Kolmogorov equation but includes also birth and death terms.
All technicalities about the derivation of the limiting processes see in Appendix of [@Mal3].
To get thermodynamics we need also volume, pressure etc. Thus it is necessary to define space dynamics and also scaling limit.
In the jump process, defined above, each particle $i$ independently of the others, in random time moments $$\tau_{i}(\omega)<t_{1i}(\omega)<\ldots<t_{in}(\omega)<\ldots<\sigma_{i}(\omega)$$ changes its type and kinetic energy (thus velocity). For each trajectory $\omega$ of the jump process we define the local space dynamics as follows. It does not change types, energies, velocities, but only coordinates. If at jump moment $t$ of the trajectory $\omega$ the particle acquires velocity $\vec{v}(\omega)=\vec{v}(t+0,\omega)$ and has coordinate $\vec{x}(t,\omega)$, then at time $t+s$ $$\vec{x}(t+s,\omega)=\vec{x}(t,\omega)+\vec{v}(\omega)s\label{trans}$$ unless the next event (jump), concerning this particle, of the trajectory $\omega$ occurs on the time interval $\left[t,t+s\right]$. We assume periodic boundary conditions or elastic reflection from the boundary. We denote this process by $X_{\Lambda}(t)$, the state space of this process is the sequence of finite arrays $X_{i}=\left\{ j_{i},\vec{x}_{i},\vec{v}_{i}\right\} $. Thus each particle $i$ has a piecewise linear trajectory in the time interval $(\tau_{i},\sigma_{i})$.
The thermodynamic limit $X^{0,\infty}(t)=\mathfrak{X}_{c}(t)$ of the processes $X_{\Lambda}(t)$ exists and its distribution belongs to $\mathfrak{M}^{0,\infty}$.
**Proof** See [@Mal3].
Scaling limit
=============
Now we define more restricted (than $\mathfrak{M}$) manifolds of probability measures on $\mathbf{X}$: the grand canonical ensemble for a mixture of ideal gases with one important difference — fast degrees of freedom are gaussian and slow degrees of freedom are constants $K_{j}$, depending only on $j$.
We consider a finite number $n_{j}$ of particles of types $j=1,\ldots,J$ in a finite volume $\Lambda$. Remind that for the ideal gas of the $j$ type particles the grand partition function of the Gibbs distribution is $$\begin{aligned}
\Theta(j,\beta) & =\sum_{n_{j}=0}^{\infty}\frac{1}{n_{j}!}\bigg(\prod\limits _{i=1}^{n_{j}}\int_{\Lambda}\int_{\R^{3}}\int_{\mathbf{I}_{j}}d\vec{x}_{j,i}\, d\vec{v}_{j,i}\bigg)\exp\beta\bigg(n_{j}(\mu_{j}-K_{j})-\sum_{i=1}^{n_{j}}\frac{m_{j}v_{j,i}^{2}}{2}\bigg)\\
& =\sum_{n_{j}=0}^{\infty}\frac{1}{n_{j}!}(\Lambda\lambda)^{n_{j}}\exp\beta(\mu_{j}-K_{j})n_{j}=\exp(\Lambda\lambda_{j}\exp\beta\hat{\mu}_{j})\end{aligned}$$ where $$\lambda_{j}=\beta^{-{3}/{2}}\Bigl(\frac{2\pi}{m_{j}}\Bigr)^{{3}/{2}},\qquad\hat{\mu}_{j}=\mu_{j}-K_{j}.$$ General mixture distribution of $J$ types is defined by the partition function $\Theta=\prod_{j=1}^{J}\Theta(j,\beta)$. The limiting space distribution of type $j$ particles is the Poisson distribution with concentration $c_{j}$. We will need the formulas relating $c_{j}$ and $\mu_{j}$: $$\begin{aligned}
c_{j} & =\frac{\langle n_{j}\rangle_{\Lambda}}{\Lambda}=\beta^{-1}\frac{\partial\ln\Theta}{\partial\mu_{j}}=\lambda_{j}\exp\beta\hat{\mu}_{j},\notag\\
\mu_{j} & =\beta^{-1}\ln\Bigl(\frac{\langle n_{j}\rangle}{\Lambda}\lambda_{j}^{-1}\Bigr)=\mu_{j,0}+\beta^{-1}\ln c_{j}+K_{j},\end{aligned}$$ where $\mu_{j,0}=-\beta^{-1}\ln\lambda_{j}$ is the so called standard chemical potential, it corresponds to the unit concentration $c_{j}=1$. We put $c=c_{1}+\ldots+c_{J}$.
We will need Gibbs free energy $G$ and the limiting Gibbs free energy per unit volume$$g=\lim_{\Lambda\rightarrow\infty}\frac{G}{\Lambda}=\sum\mu_{j}c_{j}.$$
Define by $\mathfrak{M}_{0}\subset\mathfrak{M}$ the set of all such measures for any $\beta,\mu_{1},\ldots,\mu_{J}$, and by $\mathfrak{M}_{0,\beta}$ its subset with fixed $\beta$.
In the process defined above the kinetic energies are independent but may have not $\chi^{2}$ distributions, that is the velocities may not have Maxwell distribution. We force them to have it by specifying some trend to equilibrium process (elastic collisions) and heat transfer (elastic collisions with outside molecules) processes.
Assume that there is a family $M(a),0\leq a<\infty$, of distributions $\mu_{a}$ on $R_{+}$ with the following property. Take two i.i.d.random variables $\xi_{1},\xi_{2}$ with the distribution $M(a)$. Then their sum $\xi=\xi_{1}+\xi_{2}$ has distribution $M(2a)$. We assume also that $a$ is the expectation of the distribution $M(a)$. Denote $p(\xi_{1}|\xi)$ the conditional density of $\xi_{1}$ given $\xi$, defined on the interval $[0,\xi]$. We put$$P^{(f)}(T_{1}|T,T^{\prime})=p(T_{1}|T+T^{\prime})$$ and of course $T_{1}^{\prime}=T+T^{\prime}-T_{1}$. Denote the corresponding generator by $H_{N}^{(f)}$.
We model heat transfer similarly to the fast binary reactions, as random “collision” with outside molecules in an infinite bath, which is kept at constant inverse temperature $\beta$. The energy of each outside molecule is assumed to have $\chi^{2}$ distribution with $3$ degrees of freedom and with parameter $\beta$. More exactly, for each molecule $i$ there is a Poisson process with some rate $h$. Denote by $t_{ik},k=1,2,\ldots,$ its jump moments, when it undergoes collisions with outside molecules. At this moments the kinetic energy $T$ of the molecule $i$ is transformed as follows. The new kinetic energy $T_{1}$ after transformation is chosen correspondingly to conditional density $p$ on the interval $[0,T+\xi_{ik}]$, where $\xi_{ik}$ are i.i.d. random variables having $\chi^{2}$ distribution with density $cx^{{1}/{2}}\exp(-\beta x)$. Denote the corresponding conditional density by $P^{(\beta)}(T_{1}|T)$. In fact, this process amounts to $N$ independent one-particle processes, denote the corresponding generator $H_{N}^{(\beta)}$.
Thus we can write the generator as
$$H=H(s_{f},s_{\beta})=H^{(r)}+s_{f}H^{(f)}+s_{\beta}H^{(\beta)}$$ where $H^{(r)}$ corresponds to slow reactions and $s_{f},s_{\beta}$ are some large scaling factors, which eventually will tend to infinity.
We will force the kinetic energies to become $\chi^{2}$ using the limit $s_{f}\rightarrow\infty$.
The limits in distribution $$\mathfrak{C}_{c}(t)=\lim_{s_{f}\rightarrow\infty}\mathfrak{X}_{c}(t),\qquad\mathfrak{O}_{c,\beta}(t)=\lim_{s_{\beta}\rightarrow\infty}\mathfrak{C}_{c}(t)$$ exist for any fixed $t$. Moreover, the manifold $\mathfrak{M}_{0}$ is invariant with respect to the process $\mathfrak{C}_{c}(t)$ for any fixed rates $u,b,h$. The manifolds $\mathfrak{M}_{0,\beta}$ are invariant with respect to $\mathfrak{O}_{c,\beta}(t)$.
Thus, in the process $\mathfrak{C}_{c}(t)$ the velocities have Maxwell distribution at any time moment. For the process $\mathfrak{O}_{c,\beta}(t)$ moreover, at any time $t$ the inverse temperature is equal to $\beta$, that is there is heat exchange with the environment. Our individual molecules still undergo Markov process, but simplified. At the same time, the macrovariables undergo deterministic evolution on $\mathfrak{M}_{0,\beta}$.
Note that initially the jump rates depend on the energies. We show that, after the scaling limit, the process restricted on the types will also be Markov. We assume that there are only unary and binary reactions but we do not need reversibility assumption here.
The process, projected on types, that is the process $(n_{1}(t),\ldots,$ $n_{J}(t))$ is Markov. It is time homogeneous for unary reaction system and time inhomogeneous in general.
**Proof** Recall that the jump rates were assumed to have simplest energy dependence, that is collisions occur independently of the energies, but reactions occur only if energy condition (\[energycon\]) is satisfied. Write $g_{\beta}(r)=\P(\left|\xi\right|>r)$ for the $\chi^{2}$ random variable $\xi$ with inverse temperature $\beta$.
Assume $K_{1}\leq\ldots\leq K_{J}$ and consider first the case of unary reactions. It is easy to see that the process $\mathfrak{O}_{c,\beta}(t)$ can be reduced to the Markov chain on $\left\{ 1,\ldots,J\right\} $ with rates $v_{jj^{\prime}}=u_{jj^{\prime}}$ if $j\geq j^{\prime}$, and $v_{jj'}=g_{\beta}(K_{j^{\prime}}-K_{j})u_{jj^{\prime}}$ if $j<j^{\prime}$. We used here that the kinetic energy distribution is $\chi^{2}$ at any time moment.
Similarly for the binary reaction $j,j^{\prime}\rightarrow j_{1},\, j_{1}^{\prime}$ we define the renormalized Markov transition rates as $c(j,j^{\prime}\rightarrow j_{1},\, j_{1}^{\prime})=b_{j,j^{\prime}\rightarrow j_{1},j_{1}^{\prime}}$ if $K_{j}+K_{j^{\prime}}\geq K_{j_{1}}+K_{j_{1}^{\prime}}$ and $$c(j,j^{\prime}\rightarrow j_{1},j_{1}^{\prime})=b_{j,j^{\prime}\rightarrow j_{1},j_{1}^{\prime}}\P\{\left|\xi_{1}+\xi_{2}\right|>K_{j_{1}}+K_{j_{1}^{\prime}}-(K_{j}+K_{j^{\prime}})\}$$ if $K_{j}+K_{j^{\prime}}<K_{j_{1}}+K_{j_{1}^{\prime}}$. Here $\xi_{i}$ are independent and $\chi^{2}$ with inverse temperature $\beta$. It is crucial here the use of the scaling limit for fast reactions.
Thus, in the thermodynamic limit we get the equations without the energies, that is the classical chemical kinetics $$\frac{dc_{j}(t)}{dt}=\sum_{r}R_{j,r}(\vec{c}(t))+f(c_{j}).\label{occk}$$
Assume now that the continuous time Markov chain on $\{1,\ldots,J\}$ with rates $u_{jj'}$ is irreducible. We say that this Markov chain is compatible with the equilibrium conditions $$\mu_{1}=\ldots=\mu_{J}\label{equi}$$ if its stationary probabilities $\pi_{j}$, or stationary concentrations $c_{j,e}=\pi_{j}c$, satisfy the following conditions $$\ln c_{1,e}+(\mu_{1,0}+K_{1})=\ldots=\ln c_{J,e}+(\mu_{J,0}+K_{J}).$$
**Remark** This compatibility condition should appear naturally in local dynamics, but it is not clear how to deduce it in the mean field dynamics. Note that reversibility is not a sufficient condition for the compatibility condition.
To exhibit monotonicity for dynamics one needs special Lyapounov functions in the space of distributions. For Markov chains this is the Markov entropy with respect to stationary measure $\pi_{j},$ $$S_{M}=\sum p_{j}\ln\frac{p_{j}}{\pi_{j}},$$ see for example [@Ligg].
Recall that the equilibrium function — Gibbs free energy $g(t)$ — undergoes deterministic evolution together with the parameters $\mu_{j}$ or $c_{j}$. We will show that at any time moment it coincides with the Markov entropy up to multiplicative and additive constants.
If the compatibility condition (\[equi\]) holds, then $$g(t)=\mu c+\frac{1}{\beta C}S_{M}(t)\label{GFE1}$$ and monotone behaviour of the Gibbs free energy density follows.
**Proof** We have $$\begin{aligned}
g & =\lim_{\Lambda}\frac{G}{\Lambda}=\sum_{j}c_{j}\mu_{j}=\beta^{-1}\sum_{j}c_{j}\ln c_{j}+\sum_{j}c_{j}(\mu_{j,0}+K_{j})\label{free_1}\\
& =\beta^{-1}\sum_{j}c_{j}\ln c_{j}+\sum_{j}c_{j}(\mu-\beta^{-1}\ln c_{j,e})\notag\\
& =\mu c+\beta^{-1}\sum_{j}c_{j}\ln\frac{c_{j}}{c_{j,e}}\notag\end{aligned}$$ where the first and the second equalities are the definitions, in the third and the fourth equalities we used the formula $$\begin{aligned}
\mu_{j} & =\beta^{-1}\ln\Big(\frac{\langle n_{j}\rangle}{\Lambda}\lambda_{j}^{-1}\Big)=\mu_{j,0}+\beta^{-1}\ln c_{j}+K_{j},\intertext{where}\mu_{j,0} & =-\beta^{-1}\ln\lambda_{j}=-\beta^{-1}\Big(-\frac{d_{j}}{2}\ln\beta+\ln B_{j}\Big)\label{standard}\end{aligned}$$ is the so called standard chemical potential, it corresponds to the unit concentration $c_{j}=1$ for the equilibrium density, see for example [@Mal3].
At the same time$$S_{M}=\sum p_{j}\ln\frac{p_{j}}{\pi_{j}}=C\sum c_{j}\ln\frac{c_{j}}{c_{j,e}}.$$
We see that for unary reactions one does not need reversibility assumption.
For binary reactions a similar result holds (we will not formulate it formally). However, we do not have Markov evolution for the concentrations anymore. Instead, we have the Boltzman equation for the concentrations, that is the so called nonlinear Markov chain on $\{1,\ldots,J\}$. Then, instead of the Markov entropy one should take the Boltzman entropy with respect to some one-point distribution $p_{j}^{(0)}$ (see definitions in [@MaPiRy])$$S_{H}(t)=-\sum p_{j}(t)\ln\frac{p_{j}(t)}{p_{j}^{(0)}}$$ which coincides with the Markov entropy for ordinary Markov chains. For the monotonic behaviour of the Boltzman entropy, one should assume reversibility or a more general condition — unitarity, called local equilibrium in [@MaPiRy]. Under this condition the monotonicity of the Boltzman entropy was proved in [@MaPiRy]. We get the same formula as (\[GFE1\]) if we replace $S_{M}$ by $-S_{H}$.
Note that under these conditions $p_{j}(t)$ is a time inhomogeneous Markov chain. In fact, in the long run, that is as $t\rightarrow\infty$, the transition rates for one-particle inhomogeneous Markov chain, in the vicinity of the fixed point, is asymptotically homogeneous. This shows that binary case is asymptotically close to the unary case.
Open thermodynamic compartments
===============================
Our systems in finite volume evolve via Markov dynamics. It is not known when and how this dynamics could rigorously be deduced from the local physical laws. However, there are many arguments that reversibility is a necessary condition for this. Reversibility is a particular case of the unitarity property of the scattering matrix of a collision process. It was called local equilibrium condition in [@MaPiRy; @FaMaPi]).
The reversibility gives strong corollaries for the scaling limits — 1) Boltzman monotonicity and 2) attractive fixed points. We call chemical networks with properties 1) and 2) [*thermodynamic compartments*]{}. Denote the class of such systems $\mathbf{T}$. These systems are a little bit more general than the systems, corresponding to the systems with local physical laws (in particular, having convergence to equilibrium property). For example, any unimolecular reaction system belongs to $\mathbf{T}$, because, as we saw above, the Markov entropy is the Boltzman entropy here. However, biological systems obviously are not of class $\mathbf{T}$. There are different ways to generalize class $\mathbf{T}$ systems.
The first one is quite common: in chemical and biological systems stochastic processes usually are not assumed to be reversible. However, without the reversibility assumption the time evolution could be as complicated as possible (periodic orbits, strange attractors etc.). That has advantages — one can adjust to real biological situations, and disadvantages — too many parameters, even arbitrary functions. Normally, the rate functions $R_{j,r}$ can be rather arbitrarily chosen, typical example where this methodology is distinctly pronounced is [@CCCCNT], connections with physics lost etc. In other words, theory becomes meaningless when one can adjust it to any situation.
Another way could be a hierarchy of procedures to introduce nonreversibility in a more cautious way. Each further step to introduce nonreversibility is as simple as possible and each is related to time scaling, for example, reversible dynamics is time scaled and projected on a subsystem. We start to study here the simplest type of such procedures. In our case the Markov generator will be the sum of two terms, $$H=H_{\mathit{rev}}+H_{\mathit{nonrev}},\label{rev-nonrev}$$ where the first one is reversible and the other one is not, but the latter corresponds only to input and output processes. One of technical reasons to choose such nonreversible hamiltonian is to keep invariance of the manifolds $\mathfrak{M},\mathfrak{M}_{0},\mathfrak{M}_{0,\beta}$.
In principle, another philosophy is possible — large deviation or other rare event conditioning, this we do not discuss here.
We consider the case with $J=2$ and unary reactions only, however the following assertions help to understand how more general open systems can behave. Consider first the thermodynamic limit, and then the stochastic finite volume problem.
In the thermodynamic limit the following equations for the concentrations $c_{j}(t),j=1,2$, hold: $$\frac{dc_{1}}{dt}=-\nu_{1}c_{1}+\nu_{2}c_{2}+f_{1},\qquad\frac{dc_{2}}{dt}=\nu_{1}c_{1}-\nu_{2}c_{2}+f_{2},$$ where $\nu_{1}=u_{12},\;\nu_{2}=u_{21}$ and $f_{j}$ are defined by (\[IOfunc1\]). Possible positive (i.e., $c_{1},c_{2}>0$) fixed points satisfy the following system: $$f_{1}(c_{1})+f_{2}(c_{2})=0,\qquad-\nu_{1}c_{1}+\nu_{2}c_{2}+f_{1}(c_{1})=0.$$
For example, for constant $f_{j}$ a positive fixed point exists for any $c$ sufficiently large and equals$$c_{1}=\frac{\nu_{2}c-f_{2}}{\nu_{1}+\nu_{2}},\qquad c_{2}=\frac{\nu_{1}c-f_{1}}{\nu_{1}+\nu_{2}}.$$ In the linear case, that is for $f_{j}=a_{j}c_{j}$, for the existence of a positive fixed point it is necessary and sufficient that $a_{j}$ have different signs and $|a_{j}|<\nu_{1}+\nu_{2}$. Then the positive fixed point is unique and is defined by$$c_{1}=\frac{\nu_{2}c}{\nu_{1}+\nu_{2}-a_{1}}.$$ For faster than linear growth of $f_{j}$ fixed points cannot exist for large $c$.
We see from these formulas that the equilibrium fixed point$$c_{1}=\frac{\nu_{2}c}{\nu_{1}+\nu_{2}},\qquad c_{2}=\frac{\nu_{1}c}{\nu_{1}+\nu_{2}}$$ (for the corresponding closed system) is slightly perturbed if $f_{j}$ (or $a_{j}$) are small. Moreover, the perturbed fixed point is still attractive. This is true in more general situations as well.
Now consider the stochastic (finite volume) case.
Assume that $f_{j}$ are constants. In a finite volume the process is ergodic if $\sum f_{j}<0$, transient if $\sum f_{j}>0$ and null recurrent if $\sum f_{j}=0$.
**Proof** Note that the number of particles is conserved and the number of states is finite if there is no I/O, otherwise the Markov chain is countable: a random walk on $Z_{+}^{2}=\{(n_{1},n_{2}):n_{1}n_{2}\geq0\}$. There are jumps $(n_{1},n_{2})\rightarrow(n_{1}-1,n_{2}+1)$ or $(n_{1},n_{2})\rightarrow(n_{1}+1,n_{2}-1)$ due to reactions, denote their rates $\nu_{1}n_{1},\nu_{2}n_{2}$ correspondingly. There are also jumps $(n_{1},n_{2})\rightarrow(n_{1}\pm1,n_{2}),(n_{1},n_{2})\rightarrow(n_{1},n_{2}\pm1)$ due to input-output with the parameters $a_{j}\Lambda$ and $b_{j}\Lambda$ correspondingly.
Transience and ergodicity can be obtained using Lyapounov function $n_{1}+n_{2}$ and the results from [@FaMaMe]. To prove null recurrence note that for sufficiently large $c$ the system should be in the neighbourhood of the fixed point, which exists for $c$ sufficiently large. Thus one can also use the same Lyapounov function.
General conclusion is that only null recurrent case is interesting. However, models with constant rates are too naive. It is reasonable that there are regulation mechanisms which give more complex dependence of $f_{j}$ on the rates. Unfortunately, there is no firm theoretical basis to get exact dependence of reaction and I/O rates on the densities.
The generator for MichaelisMenten kinetics is of type (\[rev-nonrev\]) only in some approximation. This model has 4 types of molecules: $E$ (enzyme), $S$ (substrate), $P$ (product) and $ES$ (substrate-enzyme complex). There are 3 reactions $$E+S\rightarrow ES,\quad ES\rightarrow E+S,\quad ES\rightarrow E+P$$ with the rates $k_{1}\Lambda^{-1}n_{E}n_{S},k_{-1}n_{ES},k_{2}n_{ES}$ correspondingly. We can also fix somehow the output rate for $P$ and input rate for $S$.
If $k_{2}=0$ then, as a zero’th approximation, we have a reversible Markov chain. In fact, there are conservation laws $$n_{E}+n_{ES}=m(E),\qquad n_{S}+n_{ES}=m(S)$$ for some constants $m(E),m(S)$. Thus we will have random walk for one variable, say $n_{ES}$, on the interval $[0,\mathrm{min}(m(E),m(S))]$, with jumps $n_{ES}\rightarrow n_{ES}\pm1$. Such random walks are always reversible. The stationary probabilities for this random walk are concentrated around the fixed point of the limiting equations of the classical kinetics $$\frac{dc_{ES}}{dt}=k_{1}c_{S}c_{E}-(k_{-1}+k_{2})c_{ES}\label{MM1}$$ defined by$$c_{ES}=\frac{c_{S}}{a+bc_{S}}$$ for some constants $a,b$, defined by $m(E),m(S)$. If $k_{2}>0$ but small compared to $k_{1},k_{-1},$ then up to the first order in $k_{2}$ we have the $P$ production speed$$\frac{dc_{P}}{dt}=k_{2}c_{ES}=k_{2}\frac{c_{S}}{a+bc_{S}}.$$ We could also look on this kinetics as on the simple random walk. We have to introduce (arbitrarily) output rate for the product $P$ and adjust the input rate of $S$ so that the system becomes null-recurrent. In fact, due to the conservation law $n_{E}+n_{ES}=m(E)$ we have random walk on the half strip $\left\{ (n_{S},n_{ES})\right\} =Z_{+}\times(0,m(E))$. The null-recurrence condition can be obtained using methods of [@FaMaMe], we will not discuss this here.
Network of thermodynamic compartments
=====================================
We call thermodynamic compartments, introduced above, [*networks of rank*]{} 1. We saw that they have fixed points, and thermodynamics plays the central role there. It can be some tightly dependent and/or space localized system of chemical reactions.
Network of rank 2 consists of vertices $\alpha$ — networks of rank 1, and directed edges, that is compartments are organized in a directed graph. Directed edge from compartment $\alpha$ to compartment $\alpha^{\prime}$ means that there is a matter flow from $\alpha$ to $\alpha^{\prime}$. Matter exchange between two compartments suggests some transport mechanism. It is natural that there is a time delay between the moments of departure from $\alpha$ and arrival to $\alpha^{\prime}$. The simplest probabilistic model could be the following. Each $j$ type molecule leaves $\alpha$ for the destination $\alpha'$ with rates $f_{j,\alpha,\alpha'}$, similar to defined in (\[IOfunc\]), and after some random time $\tau(j,\alpha,\alpha')$ arrives to $\alpha^{\prime}$. Times $\tau(j,\alpha,\alpha')$ are independent and their distribution depends only on $j,\alpha,\alpha^{\prime}$. One can imagine that there is an effective distance $L(\alpha,\alpha^{\prime})$ between $\alpha$ and $\alpha^{\prime}$ and some transportation mechanism, which defines effective speed to go through this distance. For example, it can be transport through membrane, which can be represented as a layer $\left[0,L\right]\times R^{2}$ of thickness $L$. During time $\tau(j,\alpha,\alpha')$ the particle is absent from the network, it has left $\alpha$ but has not yet arrived to $\alpha'$.
Denote by $c_{\alpha,j}$ the concentration of type $j$ molecules in the compartment $\alpha$. Limiting equations are $$\begin{aligned}
\frac{dc_{\alpha',j}(t)}{dt} & =f_{\alpha',j}^{(i)}(c_{\alpha'}(t))-f_{\alpha',j}^{(0)}(c_{\alpha'}(t))+\sum_{\alpha}f_{j,\alpha,\alpha'}(c_{\alpha}(t-\tau(j,\alpha,\alpha'))\\
& \quad-\sum_{\alpha}f_{j,\alpha',\alpha}(c_{\alpha'}(t))+\sum_{r}\nu_{\alpha',jr}R_{\alpha',r}(c_{\alpha'}(t))\end{aligned}$$ where $c_{\alpha}=(c_{\alpha,1},\dots c_{\alpha,J})$, $f_{\alpha,j}^{(i)}$ is the input rate to $\alpha$ from external environment, $f_{\alpha,j}^{(0)}$ is the output rate from $\alpha$ to the external environment. Note that these equations are random due to random delay times $\tau$. In the first approximation one can consider $\tau$ constant, however random time delays seem very essential to restore randomness on the time scale, higher than microscopic, in the otherwise deterministic classical chemical kinetics.
Note that the above written equations follow from a similar microscopic model — we will not formally formulate it, because it is obvious from our previous constructions: the corresponding manifold is $\times_{\alpha\in A}\mathfrak{M}_{\alpha}$, where $A$ is the set of compartments, $\mathfrak{M}_{\alpha}$ is the manifold for the compartment $\alpha$.
The following problems and phase transitions can be discussed in the defined model on the rigorous basis (in progress):
1\. The method of thermodynamic bounds in the thermodynamic networks, defined in [@Mavr].
2\. (Phase transitions due to transport rates.) Normal functioning of the network can be close to the system $\left\{ c_{\alpha,j,e}\right\} $ of equilibrium fixed points in each compartment $\alpha$. Such situation can be called homeostasis. Homeostatic regulation — keeping the system close to some system $\left\{ c_{\alpha,j,e}\right\} $. If there is no transport, then the compartments are independent and the fixed points inside them are pure thermodynamic. Under some transport rates the fixed points change in a stable way, they smoothly depend on the transport parameters. However, under some change of the transport rates, the fixed points may change drastically: the system goes to other basin of attraction.
3\. (Phase transition due to time desynchronization.) It is known now that even a decease can be a consequence of timing errors. For a network of rank 2, having for instance a cyclic topology (this is called circuit in [@Tho]), assume that the input rates change periodically or randomly in time. The question is: to what process the concentrations converge and with what speed ? This time behaviour could be the next step in the analysis of the structure of logical networks in the sense of [@Tho].
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A bi-layer quantum frustrated antiferromagnet is studied using an effective action approach. The action derived from the microscopical Hamiltonian has the form of the $O(3)$ non-linear sigma model. It is solved in the mean field approximation with the ultraviolet cut off chosen to fit numerical results. The obtained phase diagram displays a decrease in the critical value of interlayer coupling with increase of in-plane frustration. The critical point for a single-layer frustrated antiferromagnet (the $J_1$-$J_2$ model) is estimated to be .'
address: ' School of Physics, The University of New South Wales, Sydney 2052, Australia'
author:
- 'A. V. Dotsenko'
date: 'Internet: [[email protected]]{} '
title: 'Effective action approach to a bi-layer frustrated antiferromagnet'
---
In the past few years, a lot of attention has been given to order-disorder transitions in two-dimensional antiferromagnets. Much of the interest stems from relevance of the problem to high-temperature superconductors where such transitions occur. Although in superconductors the transition is driven by mobile holes, it can also be studied in a frustrated antiferromagnet. The antiferromagnet with frustrating interaction between next nearest neighbours (the $J_1$-$J_2$ model) is the simplest case and has often been studied.[@ds]
In this paper, I study a [*bi-layer*]{} frustrated antiferromagnet. The bi-layer problem has emerged from the studies of YBa$_2$Cu$_3$O$_{6+x}$ which consists of pairs of close CuO$_2$ planes. Experimentally, there are a number of differences between bi-layer and single layer compounds.[@experim] Recent infrared measurements[@IR] suggest that the antiferromagnet interlayer coupling within bi-layers may be as strong as 60 meV (about half of the in-plane coupling) in which case it must be taken into account in any theoretical study. Several authors have discussed pairing in bi-layer models.[@pairing] This work is restricted to studying magnetic order in the static antiferromagnet.
The Hamiltonian is taken in the form $$\begin{aligned}
\label{H}
H = J_1 \sum_{\rm NN}
( {\bf S}_{1{\bf r}} \cdot {\bf S}_{1{\bf r} '}
+ {\bf S}_{2{\bf r}} \cdot {\bf S}_{2{\bf r} '} )
+ J_2 \sum_{\rm 2N}
( {\bf S}_{1{\bf r}} \cdot {\bf S}_{1{\bf r} '}
+ {\bf S}_{2{\bf r}} \cdot {\bf S}_{2{\bf r} '} ) && \\
+ J_3 \sum_{\rm 3N}
( {\bf S}_{1{\bf r}} \cdot {\bf S}_{1{\bf r} '}
+ {\bf S}_{2{\bf r}} \cdot {\bf S}_{2{\bf r} '} )
+ J_\perp \sum_{{\bf r}} {\bf S}_{1{\bf r}} \cdot {\bf S}_{2{\bf r}},
&& \nonumber\end{aligned}$$ where the summation is performed over a square lattice, NN denotes nearest neighbours, 2N (3N) stands for second (third) neighbours, and the two planes within the bi-layer are labelled as 1 and 2 (it is assumed that all interactions $J$ are positive).
In case $J_\perp = 0$ the Hamiltonian (\[H\]) reduces to a frustrated Heisenberg antiferromagnet. Increasing $J_2$ and $J_3$ leads to a quantum phase transition into a disordered phase. The numerical value of the critical coupling $J_{2c}$ (at $J_3 = 0$) is, however, known very poorly.
The case of $ J_2 = J_3 = 0 $ (a bi-layer antiferromagnet) has also received a lot of attention.[@mfswt; @sw; @cui; @series; @mm; @ss; @scs; @cm] Availability of reliable numerical results[@series; @ss] for this case makes it a convenient model[@scs] for testing theoretical results. The bi-layer antiferromagnet has also been used[@mm] to explain some experimental results in bi-layer superconductors. However, the value of interlayer coupling required for this ($J_\perp \gtrsim 2.5 J_1$) is too large to be realistic. Apparently, both in-plane and inter-plane effects are important. In the presence of in-plane frustration the critical value of interlayer coupling may be drastically reduced.
In both the bi-layer antiferromagnet and single layer frustrated antiferromagnet, the mean field spin wave and mean field Schwinger boson theories significantly overestimate the region of stability of the Néel ordered phase. The main reason for this appears to be that these theories assume long range order from the start and do not include some types of fluctuations (see discussion of longitudinal spin fluctuations in bi-layer antiferromagnet in Ref. ).
In this study, I use an effective action approach which is based on mapping the microscopic Hamiltonian onto the quantum $O(3)$ non-linear sigma model and which has been extensively applied to two-dimensional antiferromagnets. Using a $1/N$ expansion of the non-linear sigma model, Chubukov, Sachdev, and Jinwu Ye[@csj] have recently presented a comprehensive analysis of the general properties of clean two-dimensional antiferromagnets in the vicinity of the order-disorder transition. Good agreement with numerical and experimental data has been obtained.
In this work, I first derive the effective action for the bi-layer frustrated antiferromagnet. It is observed then that for the purposes of the study, the action can be reduced to that of the standard one-band non-linear sigma model. The phase diagram is then found using the mean field (saddle point) solution with the ultraviolet cut off (or, equivalently, the critical coupling) adjusted so as to fit numerical results for the bi-layer antiferromagnet.
There are several ways to obtain the action. I follow the derivation based on coherent state representation.[@trieste] Each spin is represented by a single unit vector ${\bf N}$. Then all spin configurations are expressed as a combination of four fields (two fields per layer) $$\begin{aligned}
\label{decomp}
{\bf N}_{1{\bf r}} = \eta_{\bf r}
{\bf n}_{1{\bf r}} (1- a^2 {\bf L}^2_{1{\bf r}} )^{1/2}
+ a {\bf L}_{1{\bf r}},\nonumber\\
{\bf N}_{2{\bf r}} =- \eta_{\bf r}
{\bf n}_{2{\bf r}} (1- a^2 {\bf L}^2_{2{\bf r}} )^{1/2}
+ a {\bf L}_{2{\bf r}},\end{aligned}$$ where $${\bf n}_1^2 = {\bf n}_2^2 = 1,~~~
{\bf n}_1\cdot{\bf L}_1 = {\bf n}_2\cdot{\bf L}_2 = 0,~~~
a^2 {\bf L}_{1,2}^2 \ll 1,$$ $\eta_{\bf r} = \pm 1$ for the two sublattices, and $a$ is the lattice constant. The unit length fields $ {\bf n}_{i{\bf r}} $ in the classical limit describe orientation of the local Néel ordering and the fields ${\bf L}_{i{\bf r}}$ represent small local fluctuations.
Now the Hamiltonian is expressed using Eq. (\[decomp\]) and the continuum limit is taken by making an expansion in gradients of ${\bf n}_1$ and ${\bf n}_2$ and powers of ${\bf L}_1$ and ${\bf L}_2$ $$\begin{aligned}
\label{H-cont}
H = \int d^2 {\bf r}
\biggl[
\case{1}{2} \rho_{s1} [ (\nabla {\bf n}_1)^2 + (\nabla {\bf n}_2)^2 ]
+ S^2 J_\perp ({\bf n}_1 - {\bf n}_2)^2
& & \nonumber \\
+ 4 S^2 (J_1 + \case{1}{8} J_\perp ) ({\bf L}_1^2 + {\bf L}_2^2)
+ S^2 J_\perp {\bf L}_1 \cdot {\bf L}_2
\biggr], & &\end{aligned}$$ where $\rho_{s1} = S^2 (J_1 - 2J_2 - 4J_3)$ is the bare spin stiffness.
Decomposition of the partition function is a straightforward extension of the procedure for the single layer antiferromagnet[@trieste] and leads to $$\label{Z1}
Z = \int {\cal D} {\bf n}_1 {\cal D} {\bf n}_2
{\cal D} {\bf L}_1 {\cal D} {\bf L}_2
\delta ({\bf n}_1^2 - 1) \delta ({\bf n}_2^2 - 1) \exp (S_n)$$ with the action $$S_n = S_B -
\int d^2 {\bf r}
\int_0^\beta d\tau
\bigl[ H (\tau) + i S {\bf L}_1
\cdot ({\bf n}_1 \times \partial_\tau {\bf n}_1)
+ i S {\bf L}_2
\cdot ({\bf n}_2 \times \partial_\tau {\bf n}_2)
\bigr],$$ where $S_B$ is the residual Berry phase which is zero for all smooth spin configurations and which is ignored hereafter. Finally, the fields ${\bf L}_1$ and ${\bf L}_2$ are integrated out of Eq. (\[Z1\]) $$\begin{aligned}
\label{Z1-5}
Z = \int&& {\cal D} {\bf n}_1 {\cal D} {\bf n}_2
\delta ({\bf n}_1^2 - 1) \delta ({\bf n}_2^2 - 1) \nonumber\\
&\times& \exp
\Biggl\{
- \int d^2 {\bf r} \int_0^{c\beta} d \tau
\biggl[
\case{1}{2} \rho_{s1} [(\nabla {\bf n}_1)^2 + (\nabla {\bf n}_2)^2]
+ S^2 J_\perp ({\bf n}_1 - {\bf n}_2)^2
\nonumber \\
&+& \case{1}{8} (J_1 + \case{1}{4} J_\perp)^{-1}
({\bf n}_1 \times \partial {\bf n}_1
+ {\bf n}_2 \times \partial {\bf n}_2)^2
+ \case{1}{8} J_1^{-1}
({\bf n}_1 \times \partial {\bf n}_1
- {\bf n}_2 \times \partial {\bf n}_2)^2
\biggr]
\Biggr\}.\end{aligned}$$ At $J_\perp = 0$, the action (\[Z1-5\]) represents two independent sigma models each possessing a Goldstone mode at $T=0$ due to spontaneous symmetry breaking. At finite $J_\perp$, one mode acquires a gap $\Delta \propto \sqrt{J_\perp J_1}$. There is no general solution in this case because of the presence of the term $ ({\bf n}_1 \times \partial {\bf n}_1 )
\cdot ({\bf n}_2 \times \partial {\bf n}_2)^2) $ in the action. However, at larger $J_\perp$ and $T\ll\sqrt{J_\perp J_1}$ the action can be simplified due to the fact that it is dominated by configurations with ${\bf n}_1 \approx {\bf n}_2$. In this regime of coupled planes, I restrict myself to the Goldstone mode and set ${\bf n} = {\bf n}_1 = {\bf n}_2$ in Eq. (\[Z1-5\]). The action takes the form $$\label{Z2}
Z = \int {\cal D} {\bf n} \delta ({\bf n}^2 - 1) \exp
\left\{
- {1\over 2g} \int d^2 {\bf r} \int_0^{c\beta} d \tau
\left[
(\nabla {\bf n})^2 + (\partial_\tau {\bf n})^2
\right]
\right\} ,$$ where time has been rescaled ($c \tau \rightarrow \tau$ ) and the coupling constant is $ g = c/\rho_s $ with the spin wave velocity $$\label{velocity}
c = 2 S \bigl[ 2 (J_1-2J_2-4J_3) (J_1 + \case{1}{4} J_\perp) \bigr]^{1/2}$$ and the bare spin stiffness $$\label{rho}
\rho_s = 2 S^2 (J_1 - 2J_2 - 4J_3).$$ From Eq. (\[velocity\]) for the bi-layer antiferromagnet the spin wave velocity at the critical point $J_\perp = 2.5 J_1$ is $c_c = 1.8J_1$. It is in good agreement with numerical results[@ss] which suggests that $1/S$ corrections are negligible at this point. The spin stiffness Eq. (\[rho\]) is twice as large as in the regime of independent planes. Thus when $J_\perp$ is introduced the system first becomes more “classical” (in other words, the effective spin $S^\ast$ increases) and then evolves towards the quantum disordered phase. This effect was observed in earlier studies[@cui; @mm; @cm] at a fairly small value of $J_\perp \approx 0.1 J_1$. With in-plane frustration present the crossover will occur at even smaller values of $J_\perp$. (In a recent exact diagonalisation study of the bi-layer $t$-$J$ model,[@eder] it was noticed that the two layers become essentially correlated at $J_\perp \approx 0.2 J$.)
Equation (\[Z2\]) represents the quantum $O(3)$ non-linear sigma model. Not being interested in details of the critical behaviour, I use the simplest mean field solution which is exact for the $O(\infty)$ non-linear sigma model. An ultraviolet cut off must be introduced in momentum integration and a Pauli-Villars cut off is a convenient choice.[@trieste] In the mean field approximation, there are spin-1 excitations with a gapped spectrum and no damping. Straightforward calculations show that the gap $m$ is determined form $$\label{m}
\log \left( \sinh {m \over 2T} \right) = - { 2\pi \over T}
\left(
{1\over g} - {1\over g_c}
\right)$$ where $g _c = 4\pi/ \Lambda$ is the critical coupling ($\Lambda$ is the cut off). The system is ordered at $T=0$ when $ g<g_c$. Knowing the critical point at $J_2 = J_3 = 0$ allows to eliminate uncertainty from the cut off. Using $J_{\perp c} = 2.5 J$, the cut off is found to be $\Lambda = 1.1 \pi$. It is easy to see that the critical line $g = g_c$ has the form $$\label{crit-line}
J - 2J_2 - 4J_3 =
{
J + \case{1}{4} J_\perp
\over
J + \case{1}{4} J_{\perp c}
}.$$ While Eq. (\[m\]) is only the mean field solution and is not quite accurate, Eq. (\[crit-line\]) is independent of $1/N$ corrections. The phase diagram is presented in Figs. 2 and 3. Figure 2 is the phase diagram for $J_3 = 0$. In the absence of interlayer coupling, the critical value of $J_2$ is found to be $J_{2c}=0.19 J_1$. It is compatible with the results obtained by other methods except for numerical results (see Table I). However, numerical studies[@numericalJ1J2] face severe size restrictions and are not very reliable quantitatively. Figure 3 is the phase diagram for $J_2 = 0$. Both Figs. 2 and 3 show that in the presence of an interlayer coupling smaller values of in-plane frustration are sufficient for the disordering transition.
In conclusion, I have studied the bi-layer frustrated antiferromagnet using the effective action approach. It was demonstrated that the essential physics can be described by a one-band non-linear sigma model. The region of stability of the Néel ordered phase has been identified The critical value of next nearest neighbour interaction for the $J_1$-$J_2$ model is estimated to be $J_2 = 0.19 J_1$.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
I am grateful to O. P. Sushkov for discussions and to A. V. Chubukov for his comments on the paper. This work forms part of a project supported by a grant of the Australian Research Council.
--------------------------- -----------------
method $J_{2c}$
Green functions 0.12
renormalization group 0.15
mean field bond operators 0.19
series expansion $0.33 \pm 0.08$
present work 0.19
exact diagonalisations $ > 0.34 $
--------------------------- -----------------
: The critical coupling in the $J_1$-$J_2$ model obtained by different methods.
A. V. Dotsenko and O. P. Sushkov, , 13$\ $821 (1994) and references therein. T. Siegrist et al, , 7137 (1989). J. M. Tranquada, G. Shirane, B. Keimer, S. Shamoto, and M. Sato, , 4503 (1989) M. Grüninger, J. Münzel, A. Gaymann, A. Zibold, H. P. Geserich, and T. Kopp, preprint MG-94-1, [cond-mat/9501065]{}. K. Kuboki and P. A. Lee, [cond-mat/9501030]{}. A. I. Liechtenstein, I. I. Mazin, and O. K. Andersen, (in press); [cond-mat/9501118]{}. R. R. dos Santos, [cond-mat/9502043]{}. K. Hida, J. Phys. Soc. Jpn, [**59**]{}, 2230 (1990). T. Matsuda and K. Hida, J. Phys. Soc. Jpn [**59**]{}, 2223 (1990). Shi-Min Cui, Commun. in Theor. Phys. [**16**]{}, 401 (1991). K. Hida, J. Phys. Soc. Jpn [**61**]{}, 1013 (1992). A. J. Millis and H. Monien, , 2810 (1993); , 16 606 (1994). A. W. Sandvik and D. J. Scalapino, , 2777 (1994). A. W. Sandvik, A. V. Chubukov, and S. Sachdev, [cond-mat/9502012]{}. A. V. Chubukov and D. K. Morr, [cond-mat/9503029]{}. A. V. Chubukov, S. Sachdev, and Jinwu Ye, , 11 919, (1994). S. Sachdev, [*Low Dimensional Quantum Field Theories for Condensed Matter Physicists*]{}, Proceedings of the Trieste Summer School 1992 (World Scientific, Singapore, 1993). [cond-mat/9303014]{}. R. Eder, Y. Ohta, and S. Maekawa, [cond-mat/9504013]{}. A. F. Barabanov and V. M. Berezovsky, Phys. Lett. A, [**186**]{}, 175 (1994); Zhur. Eksp. Teor. Fiz. [**106**]{}, 1156 (1994) \[JETP [**79**]{}, 627 (1994)\]; J. Phys. Soc. Jpn [**63**]{}, 3974 (1994). T. Einarsson, P. Fröjdh, and H. Johannesson, , 13 121 (1992). S. Sachdev and R. N. Bhatt, , 9323 (1990). M. P. Gelfand, R. R. P. Singh, and D. A. Huse, , 10 801 (1989); J. Stat. Phys. [**59**]{}, 1093 (1990); M. P. Gelfand, , 8206 (1990). H. J. Schulz, T. A. L. Ziman, and D. Poilblanc, [cond-mat/9402061]{}.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A self-consistent particle-phonon coupling model is used to investigate the properties of the isobaric analog resonance in $^{208}$Bi. It is shown that quantitative agreement with experimental data for the energy and the width can be obtained if the effects of isospin-breaking nuclear forces are included, in addition to the Coulomb force effects. A connection between microscopic model predictions and doorway state approaches which make use of the isovector monopole resonance, is established via a phenomenological ansatz for the optical potential.'
address:
- '$^1$ Dipartimento di Fisica, Università degli Studi and INFN, Sezione di Milano, Via Celoria 16, 20133 Milano, Italy'
- '$^2$ Center for Mathematical Sciences, The University of Aizu, Aizu-Wakamatsu, Fukushima 965, Japan'
- '$^3$ Division de Physique Théorique, Institut de Physique Nucléaire, 91406 Orsay Cedex, France'
- '$^4$ Department of Physics, College of Humanities and Sciences, Nihon University, Sakurajosui 3-25-40, Setagaya-ku, Tokyo 156, Japan'
author:
- 'G. Colò$^1$, H. Sagawa$^2$, N. Van Giai$^3$, P.F. Bortignon$^1$, and T. Suzuki$^4$'
title: 'Widths of Isobaric Analog Resonances: a microscopic approach'
---
Introduction
============
Since its discovery some 35 years ago, the isobaric analog resonance (IAR) has always attracted a considerable interest because it is one of the clearest manifestations of approximate isospin symmetry in atomic nuclei. The main causes of isospin symmetry breaking are essentially the long-range Coulomb force, and to a lesser extent the short-range nuclear forces of the charge-symmetry breaking (CSB) type and charge-independence breaking (CIB) type [@hen]. Quoting a sentence from a review paper on this subject [@Aue83], the Coulomb interaction “is strong enough so that its effects are quite easily detectable and weak enough not to destroy or alter considerably what the nuclear force has produced.” Despite this favourable situation, we are still missing a microscopic model which is able to reproduce one of the physical observables related to the breaking of isospin symmetry in nuclei, namely the total width of the IAR, taking into account all the isospin-breaking effects.
The general features of the IAR are qualitatively well understood. In the absence of isospin-breaking effects, the analog state would be degenerate with the ground state of the ($N,Z$) parent nucleus and it would have zero width. The effect of the Coulomb and other isospin-breaking forces is to shift up the analog state by several MeV (the so-called Coulomb displacement energy). However, these isospin-breaking forces do not induce strong isospin mixing of the states. The analog state, just as its parent state, has practically pure $T=T_0$ isospin ($T_0 \equiv (N-Z)/2$) whereas the neighbouring states in the daughter nucleus have mostly $T=T_0 - 1$. Thus, a simple argument based on the Fermi golden rule shows that a small spreading width for the IAR must be expected. In addition, the escape width must also be small because the main escape channels are those of low energy protons.
When one attempts to build a microscopic model of IAR, the requirement that isospin symmetry must be restored if isospin-breaking forces are switched off should be taken into account. It is known that in $N \ne Z$ systems the Hartree-Fock (HF) approximation introduces a spurious isospin symmetry breaking and that a self-consistent charge-exchange random phase approximation (RPA) restores this broken symmetry [@Lan_find]. By self-consistent RPA we mean that the HF single-particle spectrum and the residual particle-hole interaction are derived from the same effective two-body force. Therefore, we shall consider here only this self-consistent framework. Charge-exchange RPA was applied for the first time to IAR studies in Ref. [@Aue80] (see also [@Aue83] and references therein). Calculations of other types of excitations using charge-exchange RPA can be found in Ref. [@Aue83b]. However, the RPA description can at best give information on escape widths if continuum effects are included but it cannot shed any light on spreading widths because this spreading is due to states beyond the one particle-one hole (1p-1h) space. An extension of the model space to include 2p-2h configurations and leading to second RPA would be a more appropriate scheme. In heavy nuclei where the neutron excess is large one can safely replace the RPA by the Tamm-Dancoff approximation (TDA). In Ref. [@Ada_find] such a second TDA calculation was performed for the IAR in $^{208}$Bi and it gave a reasonable estimate of the escape width $\Gamma^{\uparrow}$ and spreading width $\Gamma^{\downarrow}$. More recently, calculations based on a particle-phonon coupling model [@Col94] also led to satisfactory values of $\Gamma^{\uparrow}$ and $\Gamma^{\downarrow}$ but some discrepancies with experimental data still remain. In the above mentioned works, only isospin breaking due to the inclusion of the Coulomb force in the HF mean field was considered.
The purpose of the present paper is twofold. Firstly, we wish to examine the effects of the isospin-breaking nuclear forces on the properties of the IAR. It is known that the CSB and CIB interactions affect the mass number dependence of the Coulomb displacement energy anomalies and bring them in better agreement with experiment [@Sag_1]. Moreover, these isospin-breaking nuclear forces lead to an increase of about 50% of the isospin mixing in nuclear ground states [@Sag_2]. Thus, one should expect also some sizable effects on the values of $\Gamma^{\downarrow}_{IAR}$. We study these effects within the particle-phonon coupling model [@Col94] together with the short range parametrizations of the CSB and CIB forces of Ref. [@Sag_2]. The calculations are performed for the typical case of $^{208}$Bi. Secondly, it is instructive to study the connection between a microscopic model such as the present one or the second TDA model of Ref. [@Ada_find], and IVMR doorway state approaches [@Mek70; @Suz96] where the shift and width of the IAR result from the coupling via the Coulomb interaction of an ideal analog state with the isovector monopole resonance (IVMR) playing the role of a doorway state. Here, we show that this connection can be established if one makes a phenomenological ansatz for the isospin dependence of the nucleon optical potential.
In Sect. II the microscopic model is presented and its results are discussed in Sect. III. The connection between the microscopic model and the approach of Ref. [@Suz96] is shown in Sect. IV. Conclusions are drawn in Sect. V.
The microscopic model
=====================
The RPA extended so as to include the continuum coupling and the particle-phonon coupling has been described in detail in Ref. [@Col94] and therefore, we shall first recall only the main features of the model and then proceed to the specific points of the present calculation.
Starting from an effective Hamiltonian $H$ with two-body Skyrme interaction [@Bei75; @SGII], Coulomb interaction and CSB-CIB interactions which will be described below, the HF equations determine the self-consistent mean field of the parent nucleus. This mean field is diagonalized on a basis of 15 shells of harmonic oscillator wave functions ($\hbar
\omega_{osc}$ = 6.2 MeV for $^{208}$Pb). This procedure provides a discrete set of levels (occupied and unoccupied). We select all occupied levels, and 6 unoccupied levels for each value of $(l,j)$ with increasing values of the radial quantum number $n$. This determines the subspace $Q_1$ of discrete 1p-1h (proton particle-neutron hole) configurations.
To account for the escape width $\Gamma^\uparrow$ and spreading width $\Gamma^\downarrow$ of the IAR, we build two other subspaces $P$ and $Q_2$. The space $P$ is made of particle-hole configurations where the particle is in an unbound state orthogonal to all the above discrete single-particle levels. The method to calculate these unbound states is described in Ref.[@Col94]. On the other hand, the space $Q_2$ is built with the main configurations which are known to play a major role in the damping process of nuclear giant resonances: these configurations are 1p-1h states coupled to a collective vibration. We have included in our model space all the isoscalar vibrations of multipolarity 2$^+$, 3$^-$ and 4$^+$ up to the energy of 20 MeV and which exhaust more than 1% of the energy-weighted sum rule (EWSR) of the corresponding multipole operator. These collective vibrations are calculated consistently in HF-RPA within the $Q_1$ space and they constitute the phonons of our particle-phonon coupling model.
Using the projection operator formalism one can easily find that the effects of coupling the subspaces $P$ and $Q_2$ to $Q_1$ are described by the following effective Hamiltonian acting in the $Q_1$ space: $$\begin{aligned}
\ & {\cal H} & (E) \equiv Q_1 H Q_1 + W^\uparrow(E)
+ W^\downarrow(E) \nonumber \\
= & Q_1 & H Q_1 + Q_1 H P {\textstyle 1 \over \textstyle
E - PHP + i\epsilon} P H Q_1
+ Q_1 H Q_2 {\textstyle 1 \over \textstyle
E - Q_2 H Q_2 + i\epsilon}
Q_2 H Q_1, \nonumber \\
\ & \ &
\label{H_eff}\end{aligned}$$ where $E$ is the excitation energy. For each value of $E$ the RPA equations corresponding to this effective, complex Hamiltonian ${\cal H} (E)$ are solved. The resulting sets of eigenstates enable one to calculate all relevant quantities such as giant resonances energies and widths (see Ref. [@Col94]). In practice, we use the HF-RPA states (corresponding to positive and negative eigenvalues) as a basis for the $Q_1$ space because we can truncate this basis and neglect states which have negligible $T_{-}$ strength.
A simplifying approximation is made when calculating the matrix elements of $W^\downarrow$ by neglecting the interactions among the states within $Q_2$. Each matrix element $W^\downarrow_{ph,p^\prime h^\prime}$ is a sum of four terms whose diagrammatic representation is shown in Fig. 1. To evaluate these diagrams we use the following expression for the particle-vibration vertices: $$V = \sum_{\alpha\beta} \ \sum_{LnM} \langle \alpha |
\varrho^{(L)}_n (r) v(r) Y_{LM}(\hat r) | \beta \rangle
\ a^\dagger_\alpha a_\beta.
\label{pvc}$$ This form comes from the particle-vibration coupling model where the vibration (phonon) $|n\rangle$ is characterized by its angular momentum $L$ and its radial transition density $\varrho^{(L)}_n(r)$. The form factor $v(r)$ appearing in (\[pvc\]) is related to the particle-hole interaction derived from the Skyrme force by $V_{ph}(\vec r_1,\vec r_2)=v(r_1)\delta(\vec r_1 -\vec r_2)$. When deriving this form factor, the velocity-dependent terms of the residual Skyrme interaction are dropped.
It was pointed out in Ref.[@Col94] that the particle-phonon coupling model does not automatically insure the isospin symmetry properties of the nuclear forces (contrarily to a fully microscopic 2p-2h model like in Ref.[@Ada_find]). One must therefore enforce isospin symmetry in the evaluation of $W^\downarrow_{ph,p^\prime h^\prime}$ by an appropriate isospin projection procedure[@Col94].
In the present calculations the Skyrme interactions SIII [@Bei75] and SGII [@SGII] have been employed for the isospin symmetric part of the Hamiltonian. In addition, CSB and CIB effective nucleon-nucleon forces determined in Ref.[@Sag_2] are also included. These forces were obtained from a short-range expansion of Yukawa-type interactions and they have a form similar to that of Skyrme forces: $$\begin{aligned}
V_{CSB} = & {1\over 4} & \{\tau_z(1) + \tau_z(2)\} \{s_0 (1+y_0 P_{\sigma})
+ {1\over 2}s_1 (1+y_1 P_{\sigma}) \hfill\nonumber \\
\ & \times & ({\vec k}^2 + {\vec k}^{\prime 2})
+ s_2 (1+y_2 P_{\sigma})
{\vec k}^\prime \cdot {\vec k} \},
\label{V_CSB}\end{aligned}$$ and $$\begin{aligned}
V_{CIB} = & {1\over 2} & \tau_z(1)\tau_z(2) \{u_0 (1+z_0 P_{\sigma})
+ {1\over 2}u_1 (1+z_1 P_{\sigma}) \hfill\nonumber \\
\ & \times & ({\vec k}^2 + {\vec k}^{\prime 2})
+ u_2 (1+z_2 P_{\sigma})
{\vec k}^\prime \cdot {\vec k} \}.
\label{V_CIB}\end{aligned}$$ The parameters $s_i$ and $u_i$ are given in Ref. [@Sag_2], and all exchange parameters $y_i$ and $z_i$ are -1 because of the singlet-even character of $V_{CSB}$ and $V_{CIB}$. Therefore, they do not contribute as residual particle-hole interactions in the isovector channel and their only influence is through their contributions to the HF mean field.
Discussion of results
=====================
Three types of calculations have been done which are labeled by I, II, III respectively. In calculation I the starting Hamiltonian contains only the Skyrme interaction without the Coulomb force between protons and without CSB-CIB nuclear forces. In calculation II the Coulomb interaction is also included. In calculation III the CSB-CIB forces are added. Thus, the three calculations have an increasing degree of isospin breaking and they are expected to lead to increasing values of the IAR width according to the arguments recalled before. In calculations II and III the Coulomb exchange contributions to the mean field are treated in the Slater approximation whereas the Coulomb p-h residual interaction is dropped. The results obtained by using respectively the interactions SIII and SGII are shown in Tables 1 and 2. Calculation II is in principle equivalent to what was done in Ref. [@Col94] except for two changes in the model spaces $P$ and $Q_2$, and in the averaging parameter $\Delta$. Because of these changes, the values of the mean energies and widths of the IAR we quote as calculation II are slightly different from the correspnding values reported in [@Col94]. The spaces $P$ and $Q_2$ have been enlarged with respect to the calculation of Ref. [@Col94] by including a larger (96 instead of 64) number of unoccupied proton single-particle states. This affects mainly the escape and spreading widths calculated with the interaction SIII which both increase by about 10 keV whereas the widths calculated with the interaction SGII are less affected. Also, the averaging parameter $\Delta$ (see Ref. [@Col94]) employed in the calculation has been changed in the present work from 100 keV, as it was in [@Col94], to 200 keV. In fact, we have carefully studied the dependence of the spreading width on the parameter $\Delta$, and we have found that $\Gamma^\downarrow$ increases by about 20 keV when $\Delta$ is changed from 100 keV to 200 keV (as it was already discussed in Ref. [@Col94]) and also increases by another 20 keV when $\Delta$ is changed from 200 keV to 400 keV. On the other hand, $\Gamma^\downarrow$ remains constant if $\Delta$ is set above 400 keV. Therefore, we have adopted the value of 200 keV for $\Delta$ since this value is intermediate between 0 keV and 400 keV (the value at which $\Gamma^\downarrow$ saturates). This study allows us to say that the uncertainity on the values of $\Gamma^\downarrow$ due to the freedom of choice of the parameter $\Delta$ is about $\pm$ 20 keV.
The calculation without any isospin-breaking force (calculation I) has been performed to show that our procedure can recover the isospin symmetry reasonably well at each step. The first step (a) is discrete TDA since discrete RPA is not possible in this case because of the negative energy configurations. Indeed, without the Coulomb interaction the neutron states lie higher in energy than their proton partners and the energies $\varepsilon_p - \varepsilon_h$ of the unperturbed proton particle-neutron hole 0$^+$ configurations are negative. For the six excess neutron hole levels and the corresponding proton particle levels, these values are between -7.3 MeV and -8.3 MeV in the case of the force SGII and between -8.6 MeV and -10 MeV in the case of the force SIII. In an ideally accurate numerical calculation these negative energy configurations would be coherently pushed up to form a collective state at zero energy. Small numerical inaccuracies can weaken the residual interaction, and as a result complex RPA eigenvalues may appear. This can be easily understood in the simple case of a schematic model, or in a case containing a single state, where the RPA and TDA energies are related by $E_{RPA}^2 = E_{TDA}^2 - V^2$ with $V$ representing the interaction term. In the limit $V_C=0$ the r.h.s. may become slightly negative and consequently $E_{RPA}$ is imaginary, if $E_{TDA}^2$ and $V^2$ do not exactly cancel numerically. This difficulty of complex RPA solutions in the case $V_C=0$ is well known and, for example, in Ref. [@Aue83] the way out was to slightly renormalize the nuclear residual interaction. Therefore, only TDA is possible for the study of the IAR without the Coulomb interaction. One should also note that, contrary to a naive view, adding the Coulomb interaction does not result in an overall shift of the RPA eigenvalues. This can again be seen, e.g., in the framework of a schematic model. The second step (b) is RPA with the coupling to the continuum, which gives no width since the IAR is below the proton emission threshold. The aim of the third step (c) (inclusion of the spreading width) is to know whether the particle-phonon coupling model we have adopted can introduce some spurious width because of the approximations made: finite size of the set of 1p-1h plus phonon states, overcompleteness of the 1p-1h plus phonon basis, violation of the Pauli principle, simple form of the particle-phonon vertex function. The fact that we obtain only 4 keV for the spurious $\Gamma^\downarrow$ with the force SGII is very satisfactory since it means that the approximations we have mentioned are safe in this case.
In the case of the force SIII the spurious width (24 keV) is larger than that obtained with the force SGII. We have reached the conclusion that this spuriosity is due by half to the particle-phonon coupling and by half to the fact that HF-TDA is not completely able to recover the isospin symmetry. To estimate the spuriosity due to the particle-phonon coupling, we have considered an ideal analog state at zero energy with the following schematic wave function: $$|A\rangle = {1\over \sqrt{2T_0}}\ \hat T_- |HF\rangle
= \sum_{\pi,\nu^{-1}} X_{\pi,\nu^{-1}} |\pi,\nu^{-1}\rangle,$$ with $\nu^{-1}$ restricted to the excess neutrons and $$X_{\pi,\nu^{-1}} = {1\over \sqrt{2T_0}}\ \delta(l_\pi,l_\nu)
\delta(j_\pi,j_\nu)\ \sqrt{2j_\pi+1}\ \int dr\ u_\pi(r) u_\nu(r).$$ We have calculated the coupling of this schematic IAR with the 1p-1h plus phonon states adopted for all the calculations of Table I, and we have obtained a state at 27.6 keV whose width is 13 keV. Therefore, 13 keV is the broadening of the IAR introduced spuriously by the coupling with particle-hole-phonon configurations. The remaining 11 keV is still larger than the value obtained with the interaction SGII. This is on one hand related to the different single-particle levels obtained for the two interactions. As mentioned above, proton particle and neutron hole levels are more separated in the case of the force SIII. Consequently, the isospin breaking in the HF field is restored less efficiently by TDA and the energy of the IAR is less close to zero than in the case of the force SGII. Therefore, in the case of SIII we diagonalize the effective Hamiltonian (\[H\_eff\]) at a higher value of the energy $E$ and the spreading width of the IAR at higher excitation energy is larger. Moreover, the imaginary parts of the self-energy terms in our model would cancel exactly to give a zero spreading width if the single-particle radial wave functions of neutrons and protons with the same quantum numbers $(n,l,j)$ were identical. This cancellation can be seen by looking at the expressions of the four diagrams of Fig. 1 [@Col94]. However, the cancellation is not complete if the single-particle radial wave functions are different. The difference between radial wave functions is larger in the case of the force SIII, so again the spurious width is expected to be larger than that of SGII.
If we now include the Coulomb interaction between protons in the HF mean field, the proton levels are pushed up and become higher than the corresponding neutron levels. The energies of the unperturbed proton particle-neutron hole 0$^+$ configurations have positive values between 11 and 11.8 MeV for SGII and between 8.8 and 10.3 MeV for SIII (for the six main configurations already considered above). This difference between the two forces is essentially related to the fact that in the case of the force SGII the neutron holes are more bound. Therefore, the proton particles which enter the IAR wave function have less energy available ($\varepsilon_p = E_{IAR} + \varepsilon_h$) and a smaller probability to escape. This explains why the $\Gamma^\uparrow$ is considerably smaller than in the case of the force SIII.
Finally, the most important result of our calculation is that the total width obtained by employing the force SIII (in the case of the complete calculation III, last column of Table 1) nicely agrees with the experimental finding $\Gamma_{TOT}^{(exp)}$ = 230 keV. The improvement with respect to calculation II (without CSB-CIB forces) is about 15%. This shows that CSB-CIB forces can contribute significantly to the total width of the IAR.\
Comparison with the IVMR doorway state approach
===============================================
We have seen that microscopic approaches, like the particle-phonon coupling model described above or the 2p-2h TDA model of Ref.[@Ada_find] can give a reasonable description of IAR widths in spite of some sensitivity to the effective interactions. The spreading widths come from the coupling terms $Q_1HQ_2$ appearing in $W^{\downarrow}$ of Eq. (\[H\_eff\]), and this coupling between the simple $Q_1$ configurations and more complex $Q_2$ configurations is produced mostly by the isospin-conserving Skyrme interaction (in fact, this is the only residual interaction we keep in the calculation of $W^{\downarrow}$ in the previous sections). Thus, the fact that the resulting $\Gamma^{\downarrow}$ is non-zero is due entirely to the effects of the Coulomb and other isospin-breaking forces in the mean field as they produce a finite density of states with the isospin of the parent nucleus at the IAR energy. On the other hand, in the approaches of Refs. [@Mek70; @Aue83; @Suz96] $\Gamma^{\downarrow}$ originates from the coupling of an ideal analog state $\vert A \rangle$ with a specific doorway state, namely the IVMR in the daughter nucleus, via the isospin-breaking part of the Hamiltonian (usually the isovector component of the Coulomb force). Here, we show that these apparently different points of view can be connected.
Let us first recall the expression for $\Gamma^{\downarrow}$ obtained in Ref.[@Suz96]. The Hamiltonian is assumed to be a sum of an isospin-conserving part plus the Coulomb interaction, $H = H_0 +V_C $, and the parent ground state which is eigenstate of $H_0$ is denoted by $\vert 0 \rangle$. The three isospin components of the IVMR in the daughter nucleus are schematically written as $$\begin{aligned}
\vert M; T_0 +i, T_0 -1 \rangle & = & \vert \{ \vert 0 \rangle^{T=T_0}
\otimes \ |ph^{-1} \rangle^{T=1} \}^{T+i}_{T_0 - 1} \rangle~,
\label{eq1}\end{aligned}$$ where $i=-1,0,1$ and $\vert ph^{-1} \rangle$ stands for a combination of monopole p-h excitations. In Ref.[@Suz96] the IAR spreading width $\tilde \Gamma^{\downarrow}$ (we use this notation for the spreading width calculated by following the doorway state approach) was expressed in terms of the analog state energy $E_A$, IVMR energies $E_M^{T_0 +i}$, the width $\Gamma_M (E_A)$ of the IVMR evaluated at energy $E=E_A$, and the reduced Coulomb matrix element: $$\tilde v_{C} = \frac{1}{\sqrt{3}}
\langle (ph^{-1})^{T=1} \parallel V_C^{(1)} \parallel 0 \rangle~,
\label{not_suz}$$ where $V_C^{(1)}$ is the isovector part of the Coulomb potential. If one neglects the isospin splittings of the IVMR and adopts a common value $E_M^{T_0 +i} \simeq E_M$ the expression of $\tilde
\Gamma^{\downarrow}$ takes the simple form $$\tilde \Gamma^\downarrow_A = \Gamma_M(E_A) {\vert {\tilde v_C}\vert^2 \over
(E_A - E_M)^2 + ({\Gamma_M\over 2})^2 }~.
\label{eq2}$$ Furthermore, it was shown that the isospin mixing probability of the $T_0 + 1$ component of IVMR in the parent ground state $|\pi\rangle$ is given in second order perturbation theory by $$\begin{aligned}
\vert c_{T_0+1}\vert ^2 & \equiv & {1\over 2(T_0+1)} \langle \pi | T_- T_+
| \pi \rangle \nonumber \\
& = & { 1 \over T_0+1 }{ \vert{\tilde v_C}\vert^2 \over \vert \Delta
E_M \vert ^2 }~,
\label{eq3}\end{aligned}$$ where $\Delta E_M$ is the excitation energy of the IVMR in the parent nucleus which is approximately equal to $E_A - E_M$ of Eq. (\[eq2\]). We can safely neglect $\Gamma_M$ in Eq. (\[eq2\]) and thus obtain $$\tilde \Gamma^\downarrow_A
\sim {1\over 2} \Gamma_M(E_A) \langle \pi |
T_- T_+ | \pi \rangle~.
\label{gamma1}$$
In the microscopic models [@Ada_find; @Col94] the spreading width of the IAR results from the couplings mediated by the isospin-conserving operator $W^\downarrow$ defined in section II. Denoting the RPA eigenstate corresponding to the IAR by $|A\rangle$, we can write its width as $$\Gamma^\downarrow_A = -2\ Im\ \langle A | W^\downarrow | A
\rangle.
\label{gamma2_1}$$ The IAR wave function can be well approximated by $$|A\rangle = {1\over \sqrt{2T_0}}\ T_- | \pi \rangle~.
\label{schem_wf}$$ For the isospin-conserving $W^\downarrow$ interaction we make the ansatz: $$\begin{aligned}
W^\downarrow & = & a({{\vec T}}\cdot{{\vec T}} - b) \nonumber \\
& = & (a_R + i a_I)({{\vec T}}\cdot{{\vec T}} - b)~,
\label{schem_Wdown}\end{aligned}$$ and obtain $$\begin{aligned}
\langle A | W^\downarrow | A \rangle & = & {a\over {2T_0}} \langle
\pi | T_+ ({ {\vec T}}\cdot{ {\vec T}} - b) T_-
| \pi \rangle
\hfill\nonumber \\
\ & = & {a\over {2T_0}} \langle \pi | T_+ T_- ({ {\vec
T}}\cdot{ {\vec T}} - b)
| \pi \rangle
\hfill\nonumber \\
\ & \approx &
{a\over {2T_0}} \langle \pi | T_+ T_- | \pi
\rangle \langle \pi\vert ({ {\vec T}}\cdot{ {\vec T}}
- b) | \pi \rangle~.
\label{15_of_fax}\end{aligned}$$ In the last step of the above equation the contributions of the excited states to the closure relation have been dropped because they are of order $\vert c_{T_0 +1}\vert ^2$. Let us introduce $\tilde T$ by: $$\langle \pi | { {\vec T}}\cdot{ {\vec T}}
| \pi \rangle \equiv
\tilde T (\tilde T + 1)~,
\label{ttilde}$$ ($\tilde T$ differs slightly from $T_0$ because $\vert \pi \rangle$ has isospin mixing) and choose $b={\tilde T}^2$. Then: $$\langle A | W^\downarrow | A \rangle = a [ \tilde T (\tilde T + 1) - b ]
= a\ \tilde T.
\label{end2}$$ Thus, the two expressions (\[gamma1\]) and (\[gamma2\_1\]) are equal if the following condition is satisfied: $$a_I = -{\Gamma_M(E_A)\over 4\tilde T} \langle \pi | T_- T_+ |
\pi \rangle.
\label{end3}$$
Next, we diagonalize the TDA schematic model with the interaction $$v_{eff} = (a_R + i a_I) ({ {\vec T}}\cdot{ {\vec T}} - b)~.
\label{veff}$$ The complex eigenvalue is $$\begin{aligned}
E_A - i {\Gamma_A \over 2} & = & \varepsilon_{ph} + {v_{eff}\over 2}
\sum_{ph} | \langle
(ph^{-1})^{T=1,T_z=-1} | T_- | 0 \rangle |^2 \hfill\nonumber \\
\ & = & \varepsilon_{ph} + {a_R\over 2}\cdot 2T_0 + i {a_I\over 2}\cdot
2T_0~,
\label{schem_TDA}\end{aligned}$$ where $\varepsilon_{ph}$ is the degenerate unperturbed energy of the 0$^+$ proton particle-neutron hole configurations. Replacing $a_I$ by its value (\[end3\]) we see that $\Gamma_A$ becomes $$\Gamma_A = {1 \over 2}{\Gamma_M(E_A)}{T_0 \over \tilde T}
\langle \pi | \hat T_- \hat T_+ | \pi \rangle~,
\label{fine3}$$ which is consistent with Eq. (\[gamma1\]). Thus, the adopted interaction $W^\downarrow$ results in a value of the spreading width coming from the coupling of the IAR with states of 2p-2h type through the nuclear interaction, which is comparable with the result of the IVMR doorway state approach in which the spreading width is obtained through the coupling of the IAR to the IVMR due to the Coulomb interaction. It is left as a future problem the justification of the choice of (\[schem\_Wdown\]) from a microscopic study of optical potentials.
Conclusion
==========
Within the framework of a microscopic model based on self-consistent HF-RPA plus coupling with continuum configurations as well as with 1p-1h plus phonon configurations, we have calculated the total width of the IAR in $^{208}$Bi. We have shown that if the nuclear isospin-breaking forces of CSB and CIB type are included in the Hamiltonian in addition to the Coulomb interaction, the width of the IAR is increased by 15-20%. Thus, the nuclear isospin-breaking interactions which were already known to increase the isospin mixing of ground states have also significant contributions to the total width of the IAR. As far as comparison with experiment is concerned, the values of $\Gamma^{\downarrow}$ and $\Gamma_{TOT}$ calculated with SIII are in satisfactory agreement with the data whereas those obtained with SGII are not so good due to the peculiarities of the single-particle spectra of SGII.
Our microscopic model introduces some spuriosity in the evaluation of the total width of the IAR. This spuriosity turns out to be quite small and we have shown that it is due partly to the incomplete restoration of symmetry by TDA and partly to the 1p-1h plus phonon model. Indeed, this model does not have the full self-consistency of a second RPA calculation which would be free in principle of spurious isospin violations. However, a second RPA calculation including also continuum effects would be extremely difficult and it has never been done so far.
Finally, we have been able to propose for the first time a connection between the microscopic model and IVMR doorway state approaches for the spreading width of the IAR. This connection is possible by making a phenomenological ansatz for the isospin dependence of the nucleon optical potential.
Acknowledgments {#acknowledgments .unnumbered}
===============
G.C. likes to acknowledge the nice hospitality of the Division de Physique Théorique (IPN, Orsay) where part of the work has been done. H.S. and P.F.B. acknowledge the warm hospitality of the Institute for Nuclear Theory of Seattle, where this project started during the 1995 workshop on nuclear structure.
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N. Auerbach and A. Klein, Nucl. Phys. [**A395**]{}, 77 (1983). S. Adachi and S. Yoshida, Nucl. Phys. [**A462**]{}, 61 (1987).
G. Colò, N. Van Giai, P.F. Bortignon and R.A. Broglia, Phys. Rev. C [**50**]{}, 1496 (1994).
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----- ----------- ------------ ----------- ------------------- ------------ ----------- ---------------- --------------------- ------------
E$_{IAR}$ % of m$_0$ E$_{IAR}$ $\Gamma^\uparrow$ % of m$_0$ E$_{IAR}$ $\Gamma_{TOT}$ $\Gamma^\downarrow$ % of m$_0$
I 0.268 99.9 - - - 0.267 24 24 99.7
II 18.50 85 18.50 124 97 18.36 194 70 97
18.28 16
III 18.64 80 18.65 128 96 18.54 228 100 96
18.39 11
----- ----------- ------------ ----------- ------------------- ------------ ----------- ---------------- --------------------- ------------
: IAR results with the interaction SIII. Three different types of interactions based on SIII are used in the calculations: I) Skyrme interaction without Coulomb force; II) Skyrme interaction with Coulomb force; III) Skyrme interaction with Coulomb force and CSB-CIB forces. Three different microscopic models are also adopted: a) TDA without the coupling to the continuum; b) RPA with the coupling to the continuum; c) RPA with the couplings to both the continuum and the phonons. Energies are given in MeV and widths are in keV. The percentage of total strength m$_0$=(N-Z)/2 exhausted by the IAR is also shown.
----- ----------- ------------ ----------- ------------------- ------------ ----------- ---------------- --------------------- ------------
E$_{IAR}$ % of m$_0$ E$_{IAR}$ $\Gamma^\uparrow$ % of m$_0$ E$_{IAR}$ $\Gamma_{TOT}$ $\Gamma^\downarrow$ % of m$_0$
I 0.185 99.8 - - - 0.185 4 4 99.8
II 18.50 87 18.61 40 96 18.52 138 98 95
III 18.65 87 18.77 42 96 18.69 164 112 96
----- ----------- ------------ ----------- ------------------- ------------ ----------- ---------------- --------------------- ------------
: IAR results obtained with the interaction SGII. For details, see the caption to the previous table.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We have measured X-ray magnetic circular dichroism (XMCD) spectra at the Pu $M_{4,5}$ absorption edges from a newly-prepared high-quality single crystal of the heavy fermion superconductor $^{242}$PuCoGa$_{5}$, exhibiting a critical temperature $T_{c} = 18.7~{\rm K}$. The experiment probes the vortex phase below $T_{c}$ and shows that an external magnetic field induces a Pu 5$f$ magnetic moment at 2 K equal to the temperature-independent moment measured in the normal phase up to 300 K by a SQUID device. This observation is in agreement with theoretical models claiming that the Pu atoms in PuCoGa$_{5}$ have a nonmagnetic singlet ground state resulting from the hybridization of the conduction electrons with the intermediate-valence 5$f$ electronic shell. Unexpectedly, XMCD spectra show that the orbital component of the $5f$ magnetic moment increases significantly between 30 and 2 K; the antiparallel spin component increases as well, leaving the total moment practically constant. We suggest that this indicates a low-temperature breakdown of the complete Kondo-like screening of the local 5$f$ moment.'
author:
- 'N. Magnani'
- 'R. Eloirdi'
- 'F. Wilhelm'
- 'E. Colineau'
- 'J.-C. Griveau'
- 'A. B. Shick'
- 'G. H. Lander'
- 'A. Rogalev'
- 'R. Caciuffo'
bibliography:
- 'PuCoGa5.bib'
date: 'September 14, 2017'
title: 'Probing magnetism in the vortex phase of PuCoGa$_5$ by X-ray magnetic circular dichroism'
---
PuCoGa$_{5}$ is a prototypical heavy-fermion compound that becomes a superconductor below $T_{c} \simeq$ 18.5 K [@sarrao02], the highest critical temperature of any heavy-fermion material. Fifteen years on its discovery, the nature of the pairing boson in PuCoGa$_{5}$ remains an open question. Superconductivity (SC) mediated by spin fluctuations (SFs) associated with the proximity to an antiferromagnetic (AFM) quantum critical point (QCP) was initially proposed. The hypothesis was supported by the observation in the normal phase of a Curie-Weiss (CW) behavior of the magnetic susceptibility $\chi_m$, suggesting the presence of Pu atoms carrying a local magnetic moment. Further arguments in favor of SFs-controlled SC were provided by NMR studies [@curro05], revealing a nodal-gap function separating the condensate from the unpaired states. Subsequent point-contact spectroscopy measurements confirmed that the wavefunction of the paired electrons has an unconventional $d$-wave symmetry [@daghero12]. However, the SF conjecture was questioned [@jutier08; @ummarino09] after polarized neutron diffraction failed to observe a local magnetic moment in the normal state of PuCoGa$_{5}$ [@hiess08], pointing to an extrinsic origin of the reported temperature dependent $\chi_m$. This observation is confirmed in the present article by showing that the magnetic susceptibility of an almost defect-free PuCoGa$_{5}$ single crystal is weak and temperature-independent from $T_{c}$ up to room temperature.
Other members of the Pu$MX_{5}$ family ($M$ = Co, Rh, and $X$ = Ga, In) also become superconductors, with $T_c$ ranging from $\sim$1.7 K in the case of PuRhIn$_{5}$ to $\sim$8.7 K for PuRhGa$_{5}$ [@bauer15]. The much larger $T_{c}$ of PuCoGa$_{5}$ could indicate that a different pairing mechanism is acting in the various compounds of the family. Indeed, Bauer *et al.* [@bauer12] proposed that SC in PuCoIn$_{5}$ ($T_{c}$ = 2.5 K) is related to an AFM QCP, whereas PuCoGa$_{5}$ would reside on a larger SC dome in the temperature-pressure (hybridization strength) phase diagram, around a valence fluctuation (VF) QCP. The hypothesis is supported by dynamical mean field theory (DMFT) calculations resulting in a quasiparticle peak at the Fermi level that is sharper in PuCoIn$_{5}$ than in PuCoGa$_{5}$, which suggests a more localized 5$f$-electron character in the less dense $X$ = In compound [@zhu12]. For $X$ = Ga, the Pu atom would be in an intermediate-valence ground state between $5f^{5}$ and $5f^{6}$, with a fractional occupation number $n_{f} \sim 5.2$ [@pezzoli11]. Similar conclusions are reached by electronic structure calculations combining the density-functional theory (DFT) with an exact diagonalization (ED) of the Anderson impurity model [@shick13]. For $X$ = Ga, these calculations provide an electron density of states in reasonable agreement with photoemission measurements [@joyce03; @eloirdi09] and a non-magnetic singlet for the plutonium ground state. In this framework, the 5$f$ local magnetic moment is quenched by the combination of intermediate valence and hybridization with the surrounding cloud of conduction electrons. On the other hand, for $X$ = In, the predicted Pu ground state is magnetic as a result of a weaker hybridization strength [@shick13].
The occurrence of valence fluctuations in PuCoGa$_{5}$ has been recently suggested by resonant ultrasound spectroscopy measurements, showing that the three compressional elastic moduli exhibit anomalous softening upon cooling, which is truncated at the SC transition [@ramshaw15]. These results have been interpreted as evidence for a valence transition at a $T_{V} < T_{c}$ that is avoided by the superconducting state [@ramshaw15]. On the other hand, the relaxation rate isotope ratio T$_{1}^{-1}$($^{71}$Ga)/T$_{1}^{-1}$($^{69}$Ga) provided by Nuclear Quadrupole Resonance is not compatible with the presence of charge fluctuations in the normal state, but rather indicates the presence of anisotropic SFs [@koutroulakis16] that could, nevertheless, be associated with charge (valence) fluctuations with a higher energy scale. High-resolution powder x-ray-diffraction recently showed that the volume expansion of PuCoGa$_5$ deviates from the curve expected for a simple Grüneisen-Einstein model, but the observed variations are too small to be taken as an indication for the proximity of the system to a valence instability [@eloirdi17]. The origin of SC in PuCoGa$_{5}$ remains therefore unclear. Alternative models have also been proposed, for instance assuming a *composite* pairing in a lattice of Kondo ions screened by two distinct channels [@flint08; @flint10], or interband pairing with a sign-changing gap driven by SFs arising from spin-orbit split 5$f$ states and 5$f$-5$f$ and 3$d$-5$f$ particle-hole transitions [@graf15].
To shed further light on the extraordinary properties of PuCoGa$_{5}$ we have measured X-ray magnetic circular dichroism (XMCD) spectra at the Pu $M_{4,5}$ absorption edges from a newly-prepared high-quality single crystal of this material. The XMCD experiment was performed at the ID12 beamline [@rogalev01] of the ESRF in Grenoble. Data have been collected between 2 and 30 K on a single crystal sample (with approximate size 2.5 $\times$ 1.0 $\times$ 0.05 mm) grown in a Ga flux at the Karlsruhe establishment of the JRC. The sample was prepared using $^{242}$Pu metal obtained by amalgamation process to avoid effects from radiation damage and self-heating. The isotopic composition of the PuCoGa$_5$ sample used for the experiment (99.99 wt% $^{242}$Pu, 0.0009 wt% $^{241}$Pu, 0.0063 wt% $^{240}$Pu, 0.0021 wt% $^{239}$Pu, 0.00057 wt% $^{238}$Pu on October 2014) was checked by ICP-MS. The sample mass was 1.00 mg, corresponding to a plutonium mass of 0.37 mg and an activity of $\sim$54 kBq. The crystal was glued with Stycast$^{\circledR}$ 1266 transparent epoxy resin on an aluminum holder, with the crystallographic $c$-axis parallel to the incident X-ray beam and to the applied magnetic field. The sample holder was then introduced into a hermetic Al capsule with two Kapton windows of 62 $\mu$m thickness in total, following a protocol developed for XMCD measurements on other transuranium elements [@halevy12; @magnani15]. In addition, magnetic susceptibility measurements were also carried out, in the temperature range 2$-$300 K, with an external magnetic field up to 7 T on a 697 mg sample using the MPMS-7 superconducting quantum interference device (SQUID) from Quantum Design available at JRC-Karlsruhe.
The SQUID susceptibility curves for the investigated sample are shown in Fig. \[suscept\]. From these data one obtains a critical temperature $T_{c}$ = 18.7 K (confirmed by heat capacity measurements not shown here). Contrary to magnetization measurements reported in earlier papers, but in agreement with neutron scattering results [@hiess08], the magnetic susceptibility in the normal phase, $\chi_{m}$, is practically temperature independent between $T_{c}$ and room temperature. This is the typical behavior of intermediate-valence systems well below the characteristic charge fluctuation temperature $T_{fc}$ [@khomskii79].
![(Color online) Magnetic susceptibility in the normal state of PuCoGa$_{5}$ measured with an applied field of 1 mT on warming the sample after zero-field cooling (filled black dots). The blue open circles represent values deduced from polarized neutron diffraction measurements [@hiess08]. Inset: Temperature dependence of the magnetic susceptibility measured under zero-field cooling (filled black dots) and field cooling conditions (open red dots) in an applied field of 1 mT, providing $T_{c} = 18.7~{\rm K}$. \[suscept\]](suscept.eps){width="7.5cm"}
X-ray absorption spectroscopy (XAS) and XMCD data have been collected at several temperatures in the photon energy range between 3720 and 4040 eV, across the $M_{4,5}$ edges of Pu. The XAS spectra were recorded in backscattering geometry using the total-fluorescence-yield detection mode. The beam intensity was measured for parallel $\mu^{+}(E)$ and antiparallel $\mu^{-}(E)$ photon helicity, in a magnetic field $B_{\parallel} = 17~{\rm T}$. The XAS, ($\mu^{+}(E)+\mu^{-}(E))/2$, and the XMCD spectra, $\mu^{+}(E)-\mu^{-}(E)$, were obtained after applying self-absorption and incomplete polarization corrections using standard procedures discussed in [@wilhelm13]. Any variation of the irradiated volume (for example due to sample motion) is corrected by normalizing the spectra to the edge jump. In the superconducting phase the magnetic field penetrates into the sample forming vortices. The XMCD signal is different from zero only if the atomic shells are polarized by the applied field, therefore only the vortex cores of the superconducting state contribute to it. On the other hand, the XANES signal is not affected by the superconducting transition. This means that XMCD provides atomic quantities averaged over all plutonium atoms in the irradiated volume both above and below $T_c$.
The penetration depth for PuCoGa$_{5}$ at 2 K and $B = 60~{\rm mT}$ is $\lambda = 265~{\rm nm}$ [@ohishi07] and is bound to increase for larger fields [@sonier97], whereas the Ginzburg-Landau coherence length is $\xi \sim 2.1~{\rm nm}$ [@sarrao02]. These values must be compared with the penetration of the X-ray beam at the $M_{4}$ edge, which is $\sim 200$ nm. Assuming a critical field $B_{c2} = 63(1-T^{2}/T_{c}^{2})$ [@ummarino09b] and $\kappa = \lambda/\xi = 126$, the temperature variation of the volume average of the magnetic field is less than 0.1%. We can therefore be confident that any temperature dependence of quantities probed by XMCD is not related to changes in the flux line lattice.
After cooling the sample to 2.1 K in zero magnetic field, $B_{\parallel}$ was applied and data were collected at several temperatures up to 30 K (according to [@ummarino09b], $T_{c} \sim 15.4~{\rm K}$ for $B_{\parallel} = 17~{\rm T}$). The spectra at 2.1 K are shown in Fig. \[XASXMCD\]. The XAS branching ratio $B = I_{M_{5}}/(I_{M_{5}}+I_{M_{4}})$ is proportional to the expectation value of the angular part of the valence states spin-orbit operator $2 \langle {\bf l} \cdot {\bf s} \rangle = 3 n_{7/2} - 4 n_{5/2}$ [@thole88], $$\label{so}
\frac{2\langle {\bf l} \cdot {\bf s} \rangle}{3n_{h}}-\Delta = -\frac{5}{2}(B-\frac{3}{5})$$ where $n_h = 14 - n_f$ is the number of holes in the 5$f$ shell, $I_{M_{4,5}}$ is the integrated intensity of the isotropic X-ray absorption spectra at the $M_{4,5}$ edge, and $\Delta$ is a quantity dependent from the electronic configuration, which we will neglect here since it is equal to zero for Pu$^{3+}$ [@vanderlaan04]. No appreciable temperature variation is observed for the branching ratio. Inserting in Eq. \[so\] the experimental values at 2 K, $I_{M_{5}}$ = 51.04(8) and $I_{M_{5}}+I_{M_{4}}$ = 63.5(1), we find $B = 0.804(3)$, which within experimental errors coincides with the value measured by XAS for PuFe$_{2}$ [@wilhelm13] and by electron energy-loss spectroscopy for $\alpha$-plutonium [@vanderlaan04]. It is also close to the value expected for a $5f^5$ configuration assuming intermediate coupling (IC) ($B = 0.83 $) [@vanderlaan04] and slightly smaller than the value measured for PuSb ($B = 0.848(8)$) [@janoschek15].
![(Color online) The X-ray absorption near-edge structure (XANES, solid black lines) and X-ray magnetic circular dichroism (XMCD) spectra as a function of photon energy through the Pu $M_5$ (red line) and $M_4$ (green line) edges in PuCoGa$_5$. \[XASXMCD\]](XASXMCD.eps){width="8.0cm"}
The orbital contribution to the magnetic moment carried by the Pu atoms can be determined as [@thole92] $$\label{om}
\langle L_{z}\rangle = \frac{n_h}{I_{M_{5}}+I_{M_{4}}} (\Delta I_{M_{5}}+\Delta I_{M_{4}})$$ where $\Delta I_{M_{4,5}}$ is the partial integrated dichroic signal at the Pu $M_{4,5}$ edge. Applying this sum rule to the spectra recorded at 30 K in a 17-tesla field ($\Delta I_{M_{5}}$ = -0.16(1); $\Delta I_{M_{4}}$ = -0.21(1)), we obtain the orbital moment on Pu as $\mu_L=-\langle L_{z}\rangle$ = + 0.052(2) $\mu_B$. Interestingly, from the spectra measured at 2 K ($\Delta I_{M_{5}}$ = -0.20(1); $\Delta I_{M_{4}}$ = -0.28(1)) we obtain a slightly larger induced orbital moment of + 0.068(2) $\mu_B$.
[cddddd]{} Quantity & & &\
(units) & & & & &\
$\mu_L = -\langle L_z \rangle$ ($\mu_B$) & +0.068(2) & +0.052(2) & +0.068(2) & +0.052(2) & +0.048\
$\langle S_{\rm eff} \rangle = \langle S_z \rangle + 3 \langle T_z \rangle$ ($\mu_B$) & +0.016(1) & +0.011(1) & +0.016(1) & +0.011(1) & +0.010\
$\mu_S = -2\langle S_z \rangle$ ($\mu_B$) & -0.040(3) & -0.028(3) & -0.053(8) & -0.037(2) & -0.067\
$\mu = \mu_L + \mu_S$ ($\mu_B$) & +0.028(4) & +0.024(4) & +0.015(8) & +0.015(1) & -0.019\
$R = \mu_L / \mu_S$ & -1.7(1) & -1.9(2) & -1.3(2) & -1.4(1) & -0.72\
$r = 3 \langle T_z \rangle / \langle S_z \rangle$ & -0.218 & -0.218 & -0.41(8) & -0.41(8) & -0.72\
A second sum rule correlates the measured dichroic signal and the spin polarization $\langle S_{z} \rangle$, stating that [@carra93] $$\label{sm}
\langle S_{\rm eff} \rangle \equiv \langle S_{z}\rangle + 3\langle T_{z}\rangle = \frac{n_{h}}{2(I_{M_{5}}+I_{M_{4}})} (\Delta I_{M_{5}}-\frac{3}{2}\Delta I_{M_{4}}).$$ Therefore, in order to determine the spin component of the magnetic moment ($\mu_S = -2 \langle S_z \rangle$) from XMCD measurements it is necessary to extract the value of $\langle T_z \rangle$, the $z$ component of the expectation value of the magnetic dipole operator $\bf{T} = \sum_i[\bf{s}_i-3\bf{r}_i(\bf{r}_i \cdot \bf{s}_i)/r_i^2]$. This can be done either by using a theoretical estimate for $\langle T_z \rangle$ or by combining XMCD with another experimental technique which provides the value of the total magnetic moment $\mu = \mu_L+\mu_S$. Each of these approaches has its advantages and disadvantages, some of which will be clarified below. However, an inspection of Table \[summary\] shows that in this case one obtains the same qualitative result with both methods: the total $5f$ magnetic moment is temperature-independent even below $T_c$.
For the first approach, we assume that the ratio $r = 3 \langle T_z \rangle / \langle S_z \rangle$ has the temperature-independent value calculated in IC for a $5f^5$ configuration, $r_{\rm IC} = -0.218$, and use it to calculate $\langle T_z \rangle$ from the experimentally measured $\langle S_{\rm eff} \rangle$. At first one might argue that this simple choice is not suitable to describe the multiconfigurational ground state of PuCoGa$_5$, but considering that the weight of the $5f^4$ wavefunction is expected to be small and that the $5f^6$ states only contribute with very weak induced moments it actually appears to be a good approximation (a similar situation is found for example in the well-known intermediate-valence compound CePd$_3$, whose XMCD signal at the Ce-$L_{2,3}$ edges arises only from the $4f^1$ final state [@kappler04]). Nevertheless, analyzing our XMCD data with this method we obtain a value $\mu = 0.024(4)~\mu_B$ at 30 K, which agrees only qualitatively with the induced magnetic moment obtained by SQUID measurements at the same temperature, $\mu_{\rm SQUID} = \chi_m(T) \times B_{\parallel}$ (Fig. \[suscept\]); in fact, taking into account the diamagnetic contributions for the argon (on the Co and Ga sites) and radon (on the Pu site) core electrons, $\chi_d \approx 0.5 \times 10^{-4}$ emu/mol, we estimate that at 30 K $\mu_{\rm SQUID} = 0.015(1) ~ \mu_B$. Whereas in principle $\mu$ and $\mu_{\rm SQUID}$ cannot be compared directly, since the latter represents the total magnetic moment whereas the former only accounts for the $5f$-electron contribution, it must be remarked that the magnetic susceptibility at the Pu sites determined by neutron scattering is very close to the SQUID result. On the other hand, the most interesting result is that treating the data at 2 K leads to $\mu = 0.028(4)~\mu_B$, which coincides to the value determined at 30 K within the experimental uncertainties.
For the second approach, we assume that the value of $\mu$ measured by XMCD at 30 K equals $\mu_{\rm SQUID}$; since $\mu_L = 0.052(2)~\mu_B$ is known from XMCD, we obtain $\mu_S = -0.037(2)~\mu_B$ and $r = -0.41(8)$. Although it does not exactly coincide with $r_{\rm IC}$, the value is quite reasonable for an electronic configuration close to 5$f^{5}$ [@magnani15]. SQUID measurements cannot determine the magnetic moment below the SC transition temperature; however, $r$ is not expected to change with temperature so we can use its value at 30 K to extract the Pu spin moment at 2 K from the XMCD measurements, obtaining $\mu_S = -0.053(8)~\mu_B$. The estimated total moment at 2 K is therefore $\mu = 0.015(8)~\mu_B$; despite the larger uncertainty, once again it turns out to be equal to that measured at 30 K.
The moments experimentally obtained with both approaches are listed in Table \[summary\]. As explained above, $\mu_S$ and $\mu_L$ are averages over all irradiated plutonium atoms, and therefore their values represent lower limits for those inside the vortex phase. The Table also shows a comparison with selected results of DFT+ED calculations; although some discrepancies between theory and experiment remain (for example, the calculations wrongly predict that the spin component would be larger than the orbital moment), the DFT+ED magnetic moments are in qualitative agreement with XMCD data, as opposed to the much larger values ($\mu_S=4.08~\mu_B$ and $\mu_L=-2.32~\mu_B$) calculated with DFT [@opahle03]. It is also worth noticing that the ratio $R = \mu_L / \mu_S$ obtained with the $r = r_{\rm IC}$ assumption is in very good agreement with the theory of Pezzoli [*et al.*]{} [@pezzoli11]. On the other hand, the large $\mu_L / \mu$ ratio obtained assuming $\mu = \mu_{\rm SQUID}$ would cause a “hump” in the neutron form factor which was not seen in the experiments [@hiess08], although these neutron measurements were performed with the magnetic field applied in a different direction with respect to XMCD.
![(Color online) Temperature dependence of the XMCD signal measured at the $M_{4}$ absorption edge of Pu for PuCoGa$_5$, with a 17-T magnetic field applied along $c$. The red circles correspond to the integral of the XMCD spectra over the whole $M_4$-edge energy range, whereas the black triangles are the XMCD value measured at the $M_4$ peak energy (3968 eV). The line is a guide to the eye. Inset: XMCD spectra at the $M_{4}$ absorption edge of Pu measured for PuCoGa$_5$ (present work) and PuFe$_{2}$ [@wilhelm13]. The spectra are normalized to their respective maxima. Note that the signal measured on PuFe$_{2}$ was about 30 times larger. \[MS\]](MSvsT.eps){width="8.0cm"}
Figure \[MS\] shows the experimentally measured temperature dependence of the XMCD signal at the Pu $M_4$ edge. According to the sum rules $-\Delta I_{M_4}$ is proportional to $\mu_L - \mu_S$. This quantity increases monotonically from 30 to 2 K; we therefore expect that the low-temperature increase observed for $\mu_L$ and $\left| \mu_S \right|$ (Table \[summary\]) is monotonous as well. In particular, we note that no clear anomaly is visible around $T_c$. Since the transition is second order, and therefore the volume fraction occupied by the vortex phase has no discontinuity, this means that also the magnetic moments change continuously. The apparent discrepancy with the local spin susceptibility measured by NMR for PuCoGa$_5$ [@curro05], which *decreases* upon lowering the temperature below $T_c$, is due to the fact that these measurements probed the Co and Ga centers, whereas XMCD is sensitive only to the moments of the 5$f$ shell. On the other hand, the increase of the Pu spin moment suggests that the Kondo-like screening required to reproduce the flat magnetic susceptibility in the normal phase partially breaks down at low temperature, possibly because of a change in the 5$f$ and conduction electron hybridization. De Luca [*et al.*]{} [@deluca10] have observed a similar effect in the spin susceptibility of high-$T_c$ superconductors, which they attributed to a field-induced reorientation of the fluctuating spins perpendicular to the CuO$_2$ planes.
The overall shape of the XMCD signal is also interesting. One single peak is observed at the $M_{4}$ edge, whith a full width at half maximum (FWHM) of about 5 eV. A narrow peak of similar width was also seen in PuSb [@janoschek15] but not in PuFe$_{2}$ where the $M_{4}$ peak is significantly broader (FWHM $\simeq$ 7.5 eV) and shifted by about 1 eV towards higher energy (Fig. \[MS\]). The $M_{5}$ spectra show two peaks, separated by $\simeq$ 5 eV. Again, the $M_{5}$ spectral shape for PuFe$_{2}$ is different, being characterized by a sharp negative peak followed by a small positive upturn. The close similarity between the spectral shape of PuCoGa$_{5}$ and PuSb (a well-known localized system), as well as the difference with PuFe$_{2}$ (a well-known itinerant system), means that we are probing the localized 5$f$ electron states in PuCoGa$_{5}$; as expected, treating its electronic states as completely itinerant is an incorrect approximation.
We thank P. Colomp (ESRF radioprotection services) for his cooperation during the execution of the experiment, P. Amador-Celdran (JRC) for technical support, C. Brossard and M. Schulz (JRC) for their assistance in organizing the sample transport, and E. Zuleger (JRC) for the isotope analysis. A. B. S. acknowledges financial support provided by the Czech Science Foundation (GACR) grant No. 15-07172S.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- Emmanuel Rousseau
- Didier Felbacq
bibliography:
- 'biblio\_Quantum\_Hamiltonian\_Reply.bib'
title: 'Reply to “The equivalence of the Power-Zineau-Woolley picture and the Poincaré gauge from the very first principles” by G. Kónya *et al.*[@2018arXiv180105590K]'
---
The canonical momentum conjugated to $\vec{A}_p/\vec{A}_p^\bot$ is $-\varepsilon_0 \vec{E}$
===========================================================================================
In order to recover the Power-Zienau-Woolley Hamiltonian, it is mandatory to find that the canonical momentum $\vec{\pi}$ conjugated to the vector potential is the displacement vector $\vec{D}$, *i.e.* $\vec{\pi} = -\vec{D}$. We show in the following that cannot be correct. We do the calculations in two ways. The first one follows the comment by G. Kónya *al.*[@2018arXiv180105590K] in writing the explicit dependence between the dynamical variables (see subsection \[expl\] ). The second way (subsection \[impl\] ) follows the Dirac procedure for constraint hamiltonians. This method highlights unambiguously the contribution of the constraint $\partial_t \pi_\phi=0$. In both ways, we show that the canonical momentum$\vec{\pi}$ equal the electric field $\vec{\pi} = -\varepsilon_0 \vec{E}$ because of the constraints $\pi_\phi=0$ and $\partial_t \pi_\phi=0$ coming from the fact that the Lagrangian is free from $\partial_t \phi(\vec{x},t)$-term.
For completeness let us first recall our main assumptions and the starting points share by G. Kónya *al.*[@2018arXiv180105590K] and our work[@rousseau]. We consider one single electron with position $\vec{r}$ and electric charge $q$ evolving in a binding potential $V(\vec{r})$. The electron interacts with the electromagnetic field. The electromagnetic-field dynamical-variables are the vector potential $\vec{A}(\vec{x},t)$ and the scalar potential $\phi(\vec{x},t)$. The gauge is fixed to the Poincaré gauge defined by $\vec{x}.\vec{A}_p(\vec{x},t)=0$ for all points $\vec{x}$ in space.
In the Poincaré gauge the Lagrangian reads as
$$\begin{aligned}
L_p(\vec{r}, \vec{A}_p,\phi_p) &=& \frac{1}{2}m\dot{\vec{r}}^{~2} - V(\vec{r}) \label{eq:Lpoin1} \\
&+& \int d\vec{x} \frac{1}{2}\varepsilon_0[(\partial_t \vec{A}_p(\vec{x},t)+\nabla \phi_p(\vec{x},t))^2 - \frac{1}{2\mu_0}(\vec{\nabla} \times \vec{A}_p(\vec{x},t))^2] \\
&+& q\dot{\vec{r}}~.~\vec{A}_p(\vec{r},t)-q\phi_p(\vec{r})
\label{eq:Lpoin3}\end{aligned}$$
This is the eq.(38) in ref.[@2018arXiv180105590K] except that these authors have expressed the potentials with their values in the Poincaré gauge:
$$\begin{aligned}
\vec{A}_p(\vec{r},t) &=& - \vec{r} \times \int_0^1 udu \vec{B}(u\vec{r}) = - \vec{r} \times \int d\vec{x} \int_0^1 udu \vec{B}(\vec{x}) \delta(\vec{x} - u\vec{r}) \label{eq:Ap} \\
\phi(\vec{r},t)_p &=& - \vec{r} . \int_0^1 du \vec{E}(u\vec{r}) = - \vec{r} . \int d\vec{x} \int_0^1 du \vec{E}(\vec{x}) \delta(\vec{x} - u\vec{r})\label{eq:phip}\end{aligned}$$
To understand the discrepancies between G.Kónya *et al.* results [@2018arXiv180105590K] and our results[@rousseau] one should have in mind the similarities and the differences between our two works. Our approach[@rousseau] is based on the Dirac procedure for constrained hamiltonians[@Dirac; @henneaux] whereas the approach by Kónya *et al.* [@2018arXiv180105590K] writes explicitly the constraints between the dynamical variables. Similarities and discrepancies are summarized in the table (\[table:comp\]). This table shows that the two approaches share similar inputs except one constraint that is missing in Kónya *et al.* [@2018arXiv180105590K]. This constraint imposes that the canonical momentum $\pi_\phi$ associated to the scalar potential $\phi$ has to remain null at any time. *This is the origin of the discrepancy between our results*.
Physical quantities Rousseau and Felbacq[@rousseau] Kónya *et al.* [@2018arXiv180105590K]
--------------------------------------------------------------------------------------- ------------------------------------- -----------------------------------------
Lagrangian
Constraint; $\chi_1 = \pi_\phi=0$ Yes Yes
Constraint; $\chi_2 = \partial_t \pi_\phi $ Yes
$\chi_2 = -[q \delta(\vec{x}-\vec{r}) + \vec{\nabla}.\vec{\pi}(\vec{x},t)]=0$
Constraint; Yes Yes, through vector potential equation;
$\chi_3 =\vec{x}.\vec{A}(\vec{x},t)=0$ eq.(\[eq:Ap\])
Constraint; $\chi_4 = \vec{x}.\partial_t \vec{A}(\vec{x},t) $ Yes Yes, through scalar potential equation
$\chi_4 = \vec{x}.[\frac{\vec{\pi}}{\varepsilon_0} - \vec{\nabla} \phi(\vec{x},t)]=0$ eq.(\[eq:phip\])
Dirac procedure Explicit writing of the constraints
methods for constrained hamiltonian and at the Lagrangian level
Dirac Brackets
: $-\varepsilon_0 \vec{E}(\vec{x},t)$ $-\vec{D}(\vec{x},t)$
Canonical momentum $\vec{\pi}$
electric field displacement field
: Similarities and differences in Rousseau and Felbacq results[@rousseau] and in Kónya *et al.* [@2018arXiv180105590K]. The main difference lyies in the method used to get the hamiltonian. Rousseau and Felbacq[@rousseau] used the Dirac theory for constrained hamiltonien whereas Kónya *et al.* [@2018arXiv180105590K] wrote explicitly the constraints at the level of the Lagrangian.[]{data-label="table:comp"}
Although, both papers share lots of similarities, the final results are different: this comes from a different canonical momentum $\vec{\pi}$. In order to understand the origin of the discrepancy let us emphasize the main advantages of Dirac theory. In this theory, although the dynamical variables may be dependent upon each other, they are considered as being independent variables, while their interdependences are taken into account through constraints used to compute the Dirac Brackets. The Dirac brackets add corrective terms to the Poisson brackets, these corrective terms arising from the constraints. Among several advantages, the Dirac theory is algorithmic in the sense that the procedure is based on theorems[@henneaux]. If the procedure is done in a correct way, a unique solution exists for the transformation from the Lagrangian formalism to the Hamiltonian formalism[@Dirac; @henneaux]. On another hand, from a more a concrete point of view, an important consequence of the theory is that the dynamical variables are considered as being independent from each other. As a consequence, one has not “to use explicit expressions for the dependent variables in terms of the independent ones”[@WeinbergField p.347]. In such a way, taking the functional derivative of any quantity can be done safely without having to take into account for the constraints between the variables.
When computing the canonical momentum conjugated to the vector potential, using Dirac theory as we have done, we can compute the functional derivative safely since all dynamical variables are assumed to be independent from each other. In other words, in Dirac theory, the functional derivative has to be understood as $ \frac{\delta L_p}{\delta \partial_t A_p} = \frac{\delta L_p}{\delta \partial_t A_p}\big |_{\phi_p=cst}$. The previous notation means that the scalar potential $\phi_p$ is considered as a constant with respect to the functional derivative. In such a case, we found[@rousseau]:
$$\begin{aligned}
\vec{\pi} = \frac{\delta L_p}{\delta \partial_t A_p}\bigg |_{\phi_p=cst} = -\varepsilon_0 \vec{E}
\label{eq:piR}\end{aligned}$$
In G. Kónya *et al.*’s approach the difficulty is to perform variations in phase-space only on the manifold allowed by the constraints. Following their approach, the transverse part of the vector potential $\vec{A}^\bot_p$ is the dynamical variable. The longitudinal part of the vector potential $\vec{A}^\parallel_p$ and the scalar potential $\phi_p$ are considered as functions of the transverse part. Concerning the canonical momentum, they found:
$$\begin{aligned}
\vec{\pi}' = -\varepsilon_0 \vec{E} -q\vec{r}\int_0^1 du \delta(\vec{x}-u\vec{r})
\label{eq:piK}\end{aligned}$$
As in our result, the first term originates from variations of the term $\frac{\varepsilon_0}{2}\vec{E}^2$ in the Lagrangian. The second term arises from variation of the term $- \vec{r} . \int_0^1 du \vec{E}(u\vec{r})$ that is nothing else but the scalar potential in the Poincaré gauge \[see eq.(\[eq:phip\])\]. Indeed $q \frac{\delta}{\delta \partial_t A_p^\bot} [- \vec{r} . \int_0^1 du \vec{E}(u\vec{r})] = q\int_0^1 du \delta(\vec{x}-u\vec{r}) $. To summarize, the canonical momentum $\vec{\pi}'$ computed by Kónya *et al.* [@2018arXiv180105590K] includes also variations along the scalar potential variable.
$$\begin{aligned}
\vec{\pi}' = \frac{\delta L_p}{\delta \partial_t A_p^\bot}\Bigg |_{\phi_p=cst} + \frac{\delta L_p}{\delta \phi_p} \Bigg |_{A_p^\bot=cst}\frac{\delta \phi_p[A_p^\bot]}{\delta \partial_t A_p^\bot}
\nonumber\end{aligned}$$
In the next subsection, we compute the canonical momentum $\vec{\pi}'$ one the manifold allowed by the constraints.
Calculation of the canonical momentum conjugated to $\vec{A}$ with an explicit writing of the constraints \[expl\]
------------------------------------------------------------------------------------------------------------------
To do so, we consider the Lagrangian given by the set of equations (\[eq:Lpoin1\]-\[eq:Lpoin3\]). The scalar potential $\phi_p(\vec{x},t)$ is expressed with the help of the equation (\[eq:phip\]). The longitudinal part of the vector potential $\vec{A^\parallel_p}$ is given in the Poincaré gauge by the gauge-generating function. It reads $\vec{A^\parallel_p} = - \vec{\nabla} \int_0^1 du~ \vec{x}.\vec{A^\bot_p}(u\vec{x},t)$. We are actually assuming a transition from the Coulomb gauge to the Poincaré gauge. As a consequence, we consider the transverse part of the vector potential $\vec{A^\bot_p}(\vec{x},t)$ as the only dynamical variable for the electromagnetic field. Here we are mimicking Kónya *et al.* [@2018arXiv180105590K]. We are assuming an explicit dependence of the longitudinal part of the vector potential $\vec{A}^\parallel[\vec{A}^\bot]$ and of the scalar potential $\phi[\vec{A}^\bot]$ with the transverse part of the vector potential.
$$\begin{aligned}
\vec{\pi}' = \frac{\delta L_p}{\delta \partial_t A_p^\bot}\Bigg |_{\phi_p=cst} + \frac{\delta L_p}{\delta \phi_p} \Bigg |_{A_p^\bot=cst}\frac{\delta \phi_p[A_p^\bot]}{\delta \partial_t A_p^\bot}
\nonumber\end{aligned}$$
The functional derivative occurs along a path for which the action is extremal. Then the Euler-Lagrange equation holds $$\frac{\delta L_p}{\delta \phi}\Bigg |_{A_p^\bot=cst} =\partial_t \frac{\delta L_p}{\partial_t \delta \phi}\Bigg |_{A_p^\bot=cst}$$
One finds $\frac{\delta L_p}{\delta \phi}\big |_{A_p^\bot=cst} = -\varepsilon_0\vec{\nabla}.(\partial_t \vec{A}_p + \vec{\nabla} \phi_p) - q \delta(\vec{x}-\vec{r}) = 0$. This is the Maxwell-Gauss equation. But, most importantly for our purpose, this equation is a consequence of the first constraint $\chi_1= \pi_\phi=0$. Indeed since this constraint has to hold at any time $\partial_t \pi_\phi = \partial_t \frac{\delta L_p}{\partial_t \delta \phi}|_{A_p^\bot=cst} = 0$. If one wishes that the system remains on the surface defined by $\pi_\phi = 0$ at any time, one must have $\frac{\delta L_p}{\delta \phi}|_{A_p^\bot=cst} =0$. The canonical momentum conjugated to $A_p^\bot$ is then:
$$\begin{aligned}
\vec{\pi}' = \frac{\delta L_p}{\delta \partial_t A_p^\bot}\Bigg |_{\phi_p=cst} = \vec{\pi}
\nonumber\end{aligned}$$
It reduces to the same equation as ours \[eq:(\[eq:piR\])\]. As a consequence of the constraint $\partial_t \pi_\phi = 0$, the functional derivative must be evaluated as if the scalar potential is a constant. The result $\vec{\pi}' = \vec{\pi} = -\varepsilon_0 \vec{E}$ is recovered as in our paper[@rousseau] but following Kónya *et al.* [@2018arXiv180105590K] method. Because the scalar potential and the transverse part of the vector potential are coupled through the equation (\[eq:phip\]) a small variation $ \delta \partial_t \vec{A_p^\bot}$ induces a variation of the time derivative of the scalar potential $ \delta \partial_t \phi_p$. In the hamiltonian formalism this last variation $\partial_t \delta \phi_p$ implies a variation of the canonical momentum $\delta \pi_\phi$. But $\pi_\phi$ and its variations are constrained. So $\delta \pi_\phi$ must be null, *i.e.* $\delta \pi_\phi= 0$. This is the error done by Kónya *et al.* [@2018arXiv180105590K]. They did not realize that the term $-q\int_0^1 du \delta(\vec{x}-u\vec{r})$ arises from variations of the scalar potential. They have differentiated the term arising from the scalar potential in the Pioncaré gauge as if it were an independent term, which is not.
To be exhaustif, as noted by G. Kónya *al.*[@2018arXiv180105590K], there is an ambiguity in the definition of the canonical momentum $\vec{\pi}(\vec{x},t)$ \[see also ref.[@WeinbergField p.348] for more details\]. As a matter of fact, changing $ \partial_t \vec{A_p^\bot}(\vec{x},t)$ by the amount $\delta \partial_t \vec{A_p^\bot}(\vec{x},t)$ changes the Lagrangian by the quantity $\delta L_p[\partial_t \vec{A}^\bot \rightarrow \partial_t \vec{A}^\bot + \delta \partial_t \vec{A}^\bot] = \int d\vec{x} \vec{\pi}(\vec{x},t) . \delta \partial_t \vec{A^\bot_p}(\vec{x},t)$. But since we change only the transverse part of the vector potential, we must have $\vec{\nabla} . \delta \partial_t \vec{A^\bot_p}(\vec{x},t) = 0$. So we can add to $\vec{\pi}(\vec{x},t)$ the gradient of a scalar function $f(\vec{x},t)$ without changing the variations $\delta L_p$. Indeed,
$$\begin{aligned}
\delta L_p[\partial_t \vec{A}^\bot \rightarrow \partial_t \vec{A}^\bot + \delta \partial_t \vec{A}^\bot] &=& \int d\vec{x} (\vec{\pi}(\vec{x},t) + \vec{\nabla} f(\vec{x},t)) . \delta \partial_t \vec{A^\bot_p}(\vec{x},t) \nonumber \\
&=& \int d\vec{x} [\vec{\pi}(\vec{x},t). \delta \partial_t \vec{A^\bot_p}(\vec{x},t) - f(\vec{x},t)) \vec{\nabla}. \delta \partial_t \vec{A^\bot_p}(\vec{x},t)] \nonumber \\
&=& \int d\vec{x} \vec{\pi}(\vec{x},t). \delta \partial_t \vec{A^\bot_p}(\vec{x},t) \nonumber\end{aligned}$$
The most general solution is $ \vec{\pi}(\vec{x},t) = -\varepsilon_0 \vec{E}(\vec{x},t) + \vec{\psi}(\vec{x},t)$ with $\vec{\psi}(\vec{x},t) = \vec{\nabla} f(\vec{x},t)$. As shown in the following, the vector field $\vec{\psi}(\vec{x},t)$ is not a dynamical variable since it does not modify the equations of motion. The vector field $\vec{\psi}(\vec{x},t)$ contributes to a shift of the total energy. Fixing the reference of the energy to zero when all fields are null leads to the condition $\vec{\psi}(\vec{x},t) = \vec{0}$. Then $\vec{\pi}(\vec{x},t) = -\varepsilon_0 \vec{E}(\vec{x},t)$ as it is found in our paper[@rousseau] or in many books[@Weinberg; @WeinbergField; @Dirac; @henneaux].
Calculation of the canonical momentum conjugated to $\vec{A}$ following Dirac procedure for constrained Hamiltonian \[impl\]
----------------------------------------------------------------------------------------------------------------------------
As explained above, the Dirac procedure for constrained hamiltonian considers dynamical variables as being independent from each other[@Dirac][@henneaux p.29][@WeinbergField p.347]. Relationships are taken into account by a set a constraints denoted $\chi_i$ with $i=1,...,4$ in our paper[@rousseau]. For completeness, they are recalled in the table (\[table:comp\]). This table shows that except for the constraint $\chi_2$ forgotten by G. Kónya *al.* [@2018arXiv180105590K] both results include the same list of constraints. Particularly, in our result also, the scalar and the vector potential are respectively given by the eq.(\[eq:Ap\]) and the eq.(\[eq:phip\]). Consequently, the independent degrees of freedom are exactly the same in both papers. As demonstrated above, the difference lies in the constraint $\chi_2$ which has not been taken into account in G. Kónya *al.* [@2018arXiv180105590K] but not from the consideration of different dynamical variables.
Using the Dirac formalism for constrained hamiltonian, we show in the following that the proposition $\vec{\pi}' = -\varepsilon_0 \vec{E} -q\vec{r}\int_0^1 du \delta(\vec{x}-u\vec{r})$ as a momentum is excluded. From the set of equations (\[eq:Lpoin1\]-\[eq:Lpoin3\]) the canonical momentum conjugated to the vector potential is given by :
$$\begin{aligned}
\pi_i(\vec{x},t) = \frac{\delta L_p(\vec{r}, \vec{A}_p,\phi_p)}{\delta \partial_t {A}^i_p} = \varepsilon_0 [\partial_t A_i(\vec{x},t)+ \partial_i \phi(\vec{x},t)] = -\varepsilon_0 E_i(\vec{x},t)
\label{eq:pi}\end{aligned}$$
As noted by G. Kónya *al.* [@2018arXiv180105590K] there is an ambiguity in the definition of the canonical momemtum $\vec{\pi}(\vec{x},t)$ see also ref.[@WeinbergField p.348] for more details.
Changing $\partial_t \vec{A}(\vec{x},t)$ by the amount $\delta \partial_t \vec{A}(\vec{x},t)$ changes the Lagrangian by the quantity $\delta L = \int d\vec{x} \vec{\pi}(\vec{x},t) . \delta \partial_t \vec{A}(\vec{x},t)$. But since variations must also satisfy the gauge constraints, for example in the Coulomb gauge, we can add the gradient of a scalar function $f(\vec{x},t)$ without changing the variations $\delta L_c$. Indeed, $$\delta L_c = \int d\vec{x} [\vec{\pi}(\vec{x},t) + \vec{\nabla} f(\vec{x},t)] . \delta \partial_t \vec{A}_c(\vec{x},t)= \int d\vec{x} \vec{\pi}(\vec{x},t). \delta \partial_t \vec{A}_c(\vec{x},t)$$
In a similar fashion, in the Poincaré gauge, the vector-potential satifies $\vec{x}.\delta \vec{A}(\vec{x},t) = x\vec{e}_r. \delta \vec{A}(\vec{x},t)= \delta A_r(\vec{x},t) = 0 $ where $A_r(\vec{x},t)$ is the component along the basis vector $\vec{e}_r$. So we can add any radial vector-field $\vec{\psi}(\vec{r}) = f(\vec{r}) \vec{e}_r$ without changing the Lagrangian variations $\delta L_p$:
$$\delta L_p = \int d\vec{x} [\vec{\pi}(\vec{x},t) + f(\vec{r}) \vec{e}_r ] . \delta \partial_t \vec{A}_p(\vec{x},t)= \int d\vec{x} \vec{\pi}(\vec{x},t) . \delta \partial_t \vec{A}_p(\vec{x},t)$$
Since the previous conditions have to hold at any time, we must have $\partial_t \vec{\psi}(\vec{r},t) = 0$ in both case. The field $\vec{\psi}(\vec{r})$ depends only on space variables.
At this stage, neither in the Coulomb gauge nor in the Poincaré gauge, the canonical momentum $\pi$ is uniquely defined by the eq.(\[eq:pi\]). In the Poincaré gauge, in a generic way, it reads $\vec{\pi}(\vec{x},t) = \varepsilon_0 [\partial_t \vec{A}_p(\vec{x},t)+ \vec{\nabla} \phi_p(\vec{x},t) ] + \vec{\psi}(\vec{r}) $ where $\vec{\psi}(\vec{r})$ is a radial vector-field.
We can specify the vector-field $\vec{\psi}(\vec{x},t)$ with the help of the constraints and the Maxwell-Gauss equation. First, we need to find the Hamiltonian:
$$\begin{aligned}
H_p(\vec{r}, \vec{A}_p,\phi_p) &=& \vec{\mathcal{P}} . \dot{\vec{r}} + \int d\vec{x} \vec{\pi}(\vec{x},t).\partial_t \vec{A}_p(\vec{x},t) - L_p(\vec{r}, \vec{A}_p,\phi_p) \nonumber \\
&=& \frac{1}{2m} [\vec{\mathcal{P}} - q \vec{A}_p(\vec{x},t)]^2 +V(\vec{r}) + q \phi_p(\vec{r},t) \nonumber \\
&+& \int d\vec{x} \{ \frac{1}{2\varepsilon_0}[ 2 \vec{\pi}^2(\vec{x},t) - (\vec{\pi}(\vec{x},t)-\vec{\psi}(\vec{x},t))^2 ] + \frac{1}{2\mu_0}\vec{B}^2(\vec{x},t) - \vec{\pi}(\vec{x},t).[\vec{\nabla} \phi_p(\vec{x},t) +\frac{ \vec{\psi}(\vec{x},t)}{\varepsilon_0}]\} \label{eq:Hpsi} \end{aligned}$$
where $ \vec{\mathcal{P}}$ is the canonical momentum associated to the particle position $\vec{r}$.
To obtain this expression one writes $\partial_t \vec{A}_p(\vec{x},t) = [\frac{1}{\varepsilon_0} \vec{\pi}(\vec{x},t) - \vec{\nabla} \phi_p(\vec{x},t) - \frac{1}{\varepsilon_0} \vec{\psi}(\vec{x})]$
At this step, all dynamical variables $\vec{A}_p(\vec{x},t), \vec{\pi}(\vec{x},t)$, $\phi(\vec{x},t)$, $\pi_\phi(\vec{x},t)$ are assumed to be independent. Since $\frac{\delta L}{\delta \partial_t \phi} = \pi_\phi = 0$, we have found one constraint that applies to the dynamics. This constraint has to hold at any time. So, the following should hold:
$$\begin{aligned}
\partial_t \vec{\pi}_\phi(\vec{x},t) = \{ \pi_\phi, H_p\} = -\frac{\delta \pi_\phi}{\delta \pi_\phi}\frac{\delta H_p}{\delta \phi} = -[q \delta(\vec{x}-\vec{r}) + \vec{\nabla}.\vec{\pi}(\vec{x},t)] = 0
\label{eq:cons}\end{aligned}$$
The Hamiltonian $H_p$ is given by the equation (\[eq:Hpsi\]).
On the other hand, the Maxwell-Gauss equation has to hold too: $q \delta(\vec{x}-\vec{r}) - \varepsilon_0 \vec{\nabla}.\vec{E}(\vec{x},t) = 0$ leading to the constraint $\vec{\nabla}.\vec{\psi}(\vec{x}) = 0$ for the field $\vec{\psi}(\vec{x})$.
To conclude, there is a freedom in the choice of the canonical momentum associated to the vector potential but with some constraints as summarized in table (\[table:psi\]).\
---------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------
$$\begin{aligned} $$\begin{aligned}
&& \text{Poincar\'e~ gauge} \nonumber \\ && \text{Coulomb~gauge} \nonumber \\
&&\vec{\pi}(\vec{x},t) = -\varepsilon_0 \vec{E}_p(\vec{x},t) + \vec{\psi}(\vec{x}) \nonumber \\ && \vec{\pi}(\vec{x},t) = -\varepsilon_0 \vec{E}_p(\vec{x},t) + \vec{\psi}(\vec{x}) \nonumber \\
&&\text{with~} \vec{\psi}(\vec{x}) = f(\vec{x}) \vec{e}_r ~,~\partial_t \vec{\psi}(\vec{x}) = 0 \nonumber \\ &&\text{with~} \vec{\psi}(\vec{r}) = \vec{\nabla}f(\vec{x}) ~,~\partial_t \vec{\psi}(\vec{x}) = 0 \nonumber \\
&&\text{~and~} \vec{\nabla}.\vec{\psi}(\vec{x}) = 0 \nonumber \end{aligned}$$ &&\text{~and~} \vec{\nabla}.\vec{\psi}(\vec{x}) = 0 \nonumber \end{aligned}$$
---------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------
: Conditions satisfied by the vector field $\vec{\psi}$ in the Poincaré and in the Coulomb gauge.[]{data-label="table:psi"}
The condition $\vec{\nabla}.\vec{\psi}(\vec{x}) = 0$ does not depend on the chosen gauge. It results from $\partial_t \pi_\phi =0$. As a consequence there is here an additional argument against G. Kónya *al.* [@2018arXiv180105590K] proposition \[eq:(\[eq:piK\])\] as a canonical momentum $\vec{\pi}$. The vector field $\vec{\psi}(\vec{x}) = -q\vec{r}\int_0^1 du \delta(\vec{x}-u\vec{r}) $ that they proposed is not divergence-free. So the constraint $\chi_2$ given by the eq.(\[eq:cons\]) excludes this solution. Again, the constraint $\partial_t \pi_\phi =0$ excludes G. Kónya *al.* [@2018arXiv180105590K] result as a valid solution for the canonical momentum $\vec{\pi}$.
For completeness we specify a bit more the radial field $\vec{\psi}$ and show that it does only change the reference of the energy. For this we need the Hamiltonian. The Dirac procedure for constrained hamiltonian can be continued as in our paper[@rousseau]. The Dirac brackets can be computed[@rousseau]. After these computations the constraints act effectively. The hamiltonian can be simplify by taking all the constraints into account. After, integration by part, it reads:
$$\begin{aligned}
H_p(\vec{r}, \vec{A}_p,\phi_p) &=& \frac{1}{2m} [\vec{\mathcal{P}} - q \vec{A}_p(\vec{x},t)]^2 + V(\vec{r}) \nonumber \\
&+& \int d\vec{x} \{ \frac{\vec{\pi}^2(\vec{x},t)}{2\varepsilon_0} + \frac{\vec{\pi}(\vec{x},t).\vec{\psi}(\vec{x},t)}{\varepsilon_0} -\frac{\vec{\psi}^2(\vec{x},t)}{2\varepsilon_0} ] - \frac{\vec{\pi}(\vec{x},t). \vec{\psi}(\vec{x},t)}{\varepsilon_0} + \frac{1}{2\mu_0}\vec{B}^2(\vec{x},t) ]\} \nonumber \\
&+& \int d\vec{x} \phi_p(\vec{x},t)[\vec{\nabla}.\vec{\pi}(\vec{x},t) + q\delta(\vec{x}-\vec{r})] \nonumber\end{aligned}$$
It can be simplified with the help of the constraint $\chi_2$ \[eq:(\[eq:cons\])\].
$$\begin{aligned}
H_p(\vec{r}, \vec{A}_p,\phi_p) &=& \frac{1}{2m} [\vec{\mathcal{P}} - q \vec{A}_p(\vec{x},t)]^2 + V(\vec{r}) \nonumber \\
&+& \int d\vec{x} \{ \frac{\vec{\pi}^2(\vec{x},t)}{2\varepsilon_0} + \frac{1}{2\mu_0}\vec{B}^2(\vec{x},t) \} \nonumber \\
&-& \int d\vec{x} \frac{\vec{\psi}^2(\vec{x},t)}{2\varepsilon_0} \end{aligned}$$
This last expression shows that the field $\vec{\psi}(\vec{x},t)$ is not a dynamical variable. It does not contribute to any equations of motion. Actually, it just adds up a constant contribution to the total energy. But the energy can only be defined up to a constant. If we make the usual choice $H=0$ as the origin of the energy when there is no electromagnetic field then $\vec{\psi}(\vec{x},t)=0$. we recover then the usual results:
$$\begin{aligned}
H_p(\vec{r}, \vec{A}_p,\phi_p) &=& \frac{1}{2m} [\vec{\mathcal{P}} - q \vec{A}_p(\vec{x},t)]^2 + V(\vec{r}) + \int d\vec{x} \{ \frac{\vec{\pi}^2(\vec{x},t)}{2\varepsilon_0} + \frac{1}{2\mu_0}\vec{B}^2(\vec{x},t) \}
\label{eq:Hpoin} \\
\vec{\pi}(\vec{x},t) &=& -\varepsilon_0 [\partial_t \vec{A}_p(\vec{x},t)+ \vec{\nabla} \phi_p(\vec{x},t) ] = -\varepsilon_0 \vec{E}(\vec{x},t)\end{aligned}$$
As a conclusion, the Power-Zienau-Woolley hamiltonian is not the minimal-coupling hamiltonian written in the Poincaré gauge. As previously shown[@Kobe1988], it remains form-invariant through a gauge transformation.
Inserting $A_p^\perp = A_C$ and $A_p^\parallel$ in our result does not lead to the Power-Zienau-Woolley Hamiltonian
===================================================================================================================
On the contrary to Kónya *et al.*[@2018arXiv180105590K] we do not consider that our choice of dynamical variables is mandatory. It just considers the vector potential as a whole quantity. Nevertheless as in Kónya *et al.*[@2018arXiv180105590K] paper the transverse part of the vector potential is the only independent variable. Nonetheless once our result is known one is always free to exhibit the only independent variable and can write $\vec{A}_p = \vec{A}_p^\bot + \vec{A}_p^\parallel[ \vec{A}_p^\bot]$. So we do not share the conclusion that our choice of variable is an “awkward choice”. It has been described as an “awkward choice” by Kónya *et al.* based on a quote from Weinberg’s book[@Weinberg]. In fact, Kónya *et al.* quotation of Weinberg writings is approximative and changes its very meaning. Page 15 of their manuscript[@2018arXiv180105590K], they wrote: *“As explained by Weinberg in Section 11.3 of his book \[13\], $\Pi'_C=-\varepsilon_0 E$ is ”an awkward choice“ (quote: Weinberg) for the canonical field momenta because when quantized, it does not commute with the particle momenta”*
Here is the exact citation[@Weinberg p.315-316]: .
The “awkward feature” is actually not related to the choice of the dynamical variables but to a transition to the interaction picture. Weinberg made another statement in Ref.[@WeinbergField p.348-349] confirming our understanding of his previous citation:
Weinberg then expresses the dynamical variables in terms of the longitudinal and transverse part. He then writes $\vec{A} = \vec{A}^\perp + \vec{A}^\parallel$ and $\vec{\pi} = \vec{\pi}^\perp + \vec{\pi}^\parallel$ and inserts these expressions into the commutators and the hamiltonian previously derived.
So can the Power-Zienau-Woolley hamiltonian be obtained by writing $\vec{A}_p = \vec{A}_p^\perp + \vec{A}_p^\parallel$ and $\vec{\pi}_p = \vec{\pi}_p^\perp + \vec{\pi}_p^\parallel$ in our Hamiltonian \[eq.(8) of the main manuscript or equation (\[eq:Hpoin\]) in these paper\]?
Of course not. If we do the same procedure as Weinberg’s and write $\vec{A}_p = \vec{A}_p^\perp + \vec{A}_p^\parallel$ and $\vec{\pi}_p = \vec{\pi}_p^\perp + \vec{\pi}_p^\parallel$, we do not recover the Power-Zienau-Woolley hamiltonian since $\vec{\pi}_p^\perp = -\varepsilon_0 \vec{E}^\perp \neq \vec{D}$ where $\vec{D}$ is the displacement vector as shown previously.
As a conclusion, the Power-Zienau-Woolley hamiltonian be derived from the minimal-coupling hamiltonian through a gauge transformation.
Some weaknesses of the Power-Zienau-Woolley hamiltonian
=======================================================
The Power-Zienau-Woolley hamiltonian reads:
$$\begin{aligned}
H_{PZW} &=& \frac{1}{2m}[\mathcal{P}+q \vec{r} \times \int_0^1 du u \vec{B}(u\vec{r},t)]^2 \\
&+& \int d^3 x \frac{1}{2 \varepsilon_0} \vec{D}^2(\vec{x},t) + \frac{1}{2 \mu_0} \vec{B}^2(\vec{x},t) \\
&-& - \frac{1}{\varepsilon_0} \int d^3 x ~\vec{D}(\vec{x},t).\vec{P}(\vec{x},t) \\
&+& + \frac{1}{2 \varepsilon_0} \int d^3 x ~ \vec{P}^2(\vec{x},t)\end{aligned}$$
This is the equation eq:(46) in Kónya *et al.* comment[@2018arXiv180105590K p.14] and according to them this is also the minimal coupling hamiltonian in the Poincaré gauge. They made the following comments quote in italic:
1. *The PZW Hamiltonian is free from the A-square term*,
Maybe it is here a question of semantic. But it can not be said that the PZW hamiltonian is free from the A-square term since precisely $\vec{r} \times \int_0^1 du u \vec{B}(u\vec{r},t) = -\vec{A}_p(\vec{r},t)$ is the vector potential in the Poincaré gauge. This term is usually neglected in the so-called electric-dipole approximation but it does contribute in the complete theory.
2. *accounts for the light-matter interaction in the form of the $\vec{D}(\vec{x},t).\vec{P}(\vec{x},t)$ term*,
3. *contains a P-square term.*
By definition $\vec{D}(\vec{x},t) = \varepsilon_0\vec{E}(\vec{x},t) + \vec{P}(\vec{x},t)$. Replacing this definition into the set of equations (1-4), one can remark that the so-called light-matter interaction term and the P-square term cancel out. It is dramatic in the electric-dipole approximation since there is then no interaction term. Indeed, the Power-Zienau-Woolley Hamiltonian reduces to:
$$\begin{aligned}
H_{PZW} & \simeq & \frac{1}{2m}\mathcal{P}^2 + \int d^3 x ~ \frac{1}{2 \varepsilon_0} \vec{E}^2(\vec{x},t) + \frac{1}{2 \mu_0} \vec{B}^2(\vec{x},t)
\nonumber \end{aligned}$$
Those criticisms previously raised in our paper weaken strongly the validity of the Power-Zienau-Woolley Hamiltonian. Nevertheless Kónya *et al.* did not comment on them.
Conclusion {#conclusion .unnumbered}
==========
To conclude, we have shown that if all the constraints are taken into account correctly then the canonical momentum $\vec{\pi}(\vec{x},t)$ conjugated to the vector potential is $-\varepsilon_0 \vec{E}(\vec{x},t)$ provided that the reference of the energy is taken to be null. This result has been derived following Kónya *et al.* methodology[@2018arXiv180105590K p.14] where all quantities are written with the help of the independent dynamical variable $\vec{A}^\bot_p(\vec{x},t)$. We have shown that the constraint $\partial_t \pi_\phi = 0$ leads to this result. We have also recalled our derivation based on the Dirac theory for constrained hamiltonian. We have obtained the same result $\vec{\pi}(\vec{x},t) = -\varepsilon_0 \vec{E}(\vec{x},t)$ based on the same argument $\partial_t \pi_\phi = 0$. Moreover, this derivation allowed us to conclude that the following proposition for the momentum $\vec{\pi}'(\vec{x},t) = -\varepsilon_0 \vec{E}(\vec{x},t) -q\vec{r}\int_0^1 du \delta(\vec{x}-u\vec{r})$ cannot be considered as correct since $-q\vec{r}\int_0^1 du \delta(\vec{x}-u\vec{r})$ is not divergence-free. We have also explained that the differences between our both results cannot be attributed to the consideration of different dynamical variables since they are similar in both papers. The independent dynamical variables are taken into account explicitly in Kónya *et al.* [@2018arXiv180105590K] work and implicitly in our work through the constraints $\chi_i$. Nevertheless they are exactly the same.
In order to obtain the Power-Zienau-Woolley hamiltonian one needs $\vec{\pi}'(\vec{x},t) = - \vec{D}(\vec{x},t)$, which is impossible as we have demonstrated. As a consequence, we conclude that this hamiltonian cannot be derived from the minimal-coupling hamiltonian. We ended these notes in highlighting some weaknesses of the Power-Zienau-Woolley hamiltonian. More weaknesses can be found in our paper. Particularly the physical meaning of the term $\frac{1}{2 \varepsilon_0} \vec{D}^2(\vec{x},t) + \frac{1}{2 \mu_0} \vec{B}^2(\vec{x},t)$ is questionable since this is neither the electromagnetic-field energy-density in vacuum nor in matter. As far as we know, the weaknesses of the Power-Zienau-Woolley hamiltonian have never been commented (and answered) into the literature.
Author contributions statement {#author-contributions-statement .unnumbered}
==============================
DF initiated this work. ER did the calculations. ER and DF have discussed the results and reviewed the manuscript.
Additional information {#additional-information .unnumbered}
======================
The authors declare no competing financial interests.
The corresponding author is responsible for submitting a [competing financial interests statement](http://www.nature.com/srep/policies/index.html#competing) on behalf of all authors of the paper. This statement must be included in the submitted article file.
|
{
"pile_set_name": "ArXiv"
}
|
---
bibliography:
- 'ws-book-Schomaker.bib'
title: 'Handwritten Historical Document Analysis, Recognition, and Retrieval – State of the Art and Future Trends'
---
[Lifelong learning for text retrieval and recognition in historical handwritten document collections]{}\
Lambert Schomaker\
To appear as chapter in book:\
Handwritten Historical Document Analysis, Recognition, and Retrieval – State of the Art and Future Trends\
in the book series:\
Series in Machine Perception and Artificial Intelligence\
World Scientific\
ISSN (print): 1793-0839
chapter-Schomaker.tex
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The Gell-Mann grading, one of the four gradings of ${\operatorname{sl}(3,{\mathbb{C}})}$ that cannot be further refined, is considered as the initial grading for the graded contraction procedure. Using the symmetries of the Gell-Mann grading, the system of contraction equations is reduced and solved. Each non–trivial solution of this system determines a Lie algebra which is not isomorphic to the original algebra ${\operatorname{sl}(3,{\mathbb{C}})}$. The resulting $53$ contracted algebras are divided into two classes — the first is represented by the algebras which are also continuous Inönü–Wigner contractions, the second is formed by the discrete graded contractions.'
author:
- 'Jiří Hrivnák$^{1}$'
- Petr Novotný$^1$
date:
title: 'Graded contractions of the Gell-Mann graded ${\operatorname{sl}(3,{\mathbb{C}})}$'
---
Introduction
============
This paper extends the work undertaken in [@PC4; @HN1] where the graded contractions of the Cartan and Pauli graded ${\operatorname{sl}(3,{\mathbb{C}})}$ are classified. In this article, the Gell-Mann matrices and their corresponding grading [@HPP3; @PZ2] of ${\operatorname{sl}(3,{\mathbb{C}})}$ are considered as a starting point for the graded contraction procedure. The Gell-Mann matrices represent a hermitian generalization of the two–dimensional Pauli matrices [@Georgi]. The results of this procedure are Gell-Mann graded complex Lie algebras, with characteristics specifically linked to the original algebra ${\operatorname{sl}(3,{\mathbb{C}})}$.
The graded contraction procedure is introduced in [@PC1] as an alternative to continuous Inönü–Wigner (IW) contractions. In contrast to continuous contractions, graded contractions represent an algebraical perspective — instead of a limit of sequences of mutually isomorphic Lie algebras, the commutation relations among graded subspaces are multiplied by contraction parameters. These parameters then have to satisfy the system of equations resulting from the Jacobi identities. The complexity of this contraction system depends on the number of grading subspaces and the structure of a given grading.
The graded contraction procedure of Lie algebras was applied to a number of cases and was also generalized to other graded algebras — graded contractions of inhomogeneous algebras [@dAzca], central extensions [@Guise], affine algebras [@Hus; @dM2], Jordan algebras [@KP], Virasoro algebras [@Kostyak], Lie algebra $\operatorname{e}(2,1)$ [@PC5] were studied. Physically motivated cases related to the kinematical and conformal group of space–time were presented in [@dM1; @T2]. Partial general results, for so called generic cases of gradings of Lie algebras, were constructed in [@WW4; @WW5]. This approach solved the contraction systems simultaneously for all Lie algebras which allow a given grading. However, it appears that in the generality of this approach, a significant part of the solutions would be omitted for non-generic cases. As the Gell-Mann grading of ${\operatorname{sl}(3,{\mathbb{C}})}$ cannot be considered a generic case, the detailed analysis of possible outcome of graded contraction procedure is thus at present necessary.
The results of the graded contraction procedure fall into two distinct categories — continuous graded contractions, which correspond to IW contractions, and discrete graded contractions. In contrast to the IW contraction, a discrete graded contraction may yield a continuous parametric family of non–isomorphic Lie algebras — such continuous parametric families can be used in the deformation theory of Lie algebras. Additional advantages of continuous graded contractions are consequences of them also being IW contractions. Besides the physical application of IW contractions — studying the possibility of existence of a correspondence principle — they can be used for obtaining invariants and representations of the contracted algebras [@IW].
The concept of the graded contractions can be also used for contractions of representations [@PC2; @PN2]. The process of obtaining representations of the contracted algebras is not completely straightforward, the grading and the representation have to satisfy the so called compatibility condition. Even though the compatibility condition is always satisfied for root space decompositions, i.e. Cartan gradings, in other cases it may be a more challenging task [@Leng; @PT; @T1]. The general representation theory which exists for simple Lie algebras, can be then used for the construction of representations of other types of Lie algebras, especially solvable ones. The method is mostly valuable for discrete graded contractions — then the representations cannot be obtained by the standard way using IW contractions.
Thus, the goals of this work are:
(i) to obtain all IW contractions which start from the Gell-Mann grading of ${\operatorname{sl}(3,{\mathbb{C}})}$ and preserve the Cartan subalgebra; classifying these contractions is motivated by their direct applicability for obtaining Casimir operators and representations,
(ii) to obtain discrete graded contractions – these may yield new representations of the contracted algebras in cases where the direct IW method is not possible,
(iii) to contribute to the classification problem of Lie algebras — solvable Lie algebras are classified only for dimensions not greater than six.
The paper is organized as follows. In Section 2, the Gell-Mann grading and its symmetries are recalled. In Section 3, the graded contraction procedure is formulated and a suitable equivalence of the solutions introduced. The contraction system, its reduction by the symmetries and the solution, together with the set of higher-order identities is located in Section 4. The procedure, which is necessary for identification of the results, is applied to the contracted Lie algebras in Section 5. Concluding remarks and follow-up questions are contained in Section 6. In Appendix A.1, the contraction matrices are located, Appendix A.2 contains the final classification of graded contractions of the Gell-Mann graded ${\operatorname{sl}(3,{\mathbb{C}})}$ in a tabulated form. Appendix A.3 contains the invariant functions of the one–parametric graded contractions.
The Gell–Mann grading
======================
The grading ${\Gamma}$
----------------------
Throughout this article, we use the following basis of the Lie algebra of three-dimensional complex traceless matrices ${\operatorname{sl}}(3,{\mathbb{C}})$: $$\begin{array}{llll}
e_1=\begin{pmatrix}
1&0&0\\
0&-1&0\\
0&0&0
\end{pmatrix}, &
e_2=\begin{pmatrix}
0&0&0\\
0&1&0\\
0&0&-1
\end{pmatrix}, &
e_3=\begin{pmatrix}
0&1&0\\
1&0&0\\
0&0&0
\end{pmatrix}, &
e_4=\begin{pmatrix}
0&0&1\\
0&0&0\\
1&0&0\\
\end{pmatrix},
\\[24pt]
e_5=\begin{pmatrix}
0&0&0\\
0&0&1\\
0&1&0
\end{pmatrix}, &
e_6=\begin{pmatrix}
0&-1&0\\
1&0&0\\
0&0&0
\end{pmatrix}, &
e_7=\begin{pmatrix}
0&0&0\\
0&0&-1\\
0&1&0
\end{pmatrix}, &
e_8=\begin{pmatrix}
0&0&-1\\
0&0&0\\
1&0&0
\end{pmatrix}.
\end{array}$$ The corresponding non–zero commutation relations $[e_i,e_j]=e_ie_j-e_je_i$, $i,j\in \{1,2,\dots,8\}$ of ${\operatorname{sl}}(3,{\mathbb{C}})$ written in this basis are $$\label{comrel}
\arraycolsep=2pt
\begin{array}{llllll}
[e_1,e_3]=-2e_6, & [e_1,e_4]=-e_8,& [e_1,e_5]=e_7,& [e_1,e_6]=-2e_3,& [e_1,e_7]=e_5,& [e_1,e_8]=-e_4, \\[2pt] [e_2,e_3]=e_6, & [e_2,e_4]=-e_8,& [e_2,e_5]=-2e_7,& [e_2,e_6]=e_3,& [e_2,e_7]=-2e_5,& [e_1,e_8]=-e_4,\\[2pt]
[e_3,e_4]=-e_7,& [e_3,e_5]=-e_8,& [e_3,e_6]=2e_1,& [e_3,e_7]=-e_4,& [e_3,e_8]=-e_5,\\[2pt]
[e_4,e_5]=-e_6,& [e_4,e_6]=-e_5,& [e_4,e_7]=e_3,& \multicolumn{2}{l}{[e_4,e_8]=2(e_1+e_2),}\\[2pt]
[e_5,e_6]=e_4,& [e_5,e_7]=2e_2,& [e_5,e_8]=e_3,& [e_6,e_7]=-e_8,& [e_6,e_8]=e_7,& [e_7,e_8]=-e_6.
\end{array}$$ The two-dimensional subspace ${\operatorname{span}}_{\mathbb{C}}\{e_1,e_2\} $ forms a Cartan subalgebra of ${\operatorname{sl}}(3,{\mathbb{C}})$. In the following, it is convenient to take into account ordered triplets $(i_1,i_2,i_3)$ with $i_1,i_2,i_3\in\{0,1\}$ which form the additive abelian group ${\mathbb{Z}}_2^3$ with addition mod $2$ and introduce a set of seven indices $I$ such that $$I={\mathbb{Z}}_2^3\setminus \{(0,0,0)\}.$$ Let us denote the Cartan subalgebra by $$L_{001}= {\operatorname{span}}_{\mathbb{C}}\{e_1,e_2\}$$ and the one-dimensional subspaces corresponding to the remaining basis vectors by $$\begin{alignedat}{3}
L_{111}&= {\mathbb{C}}e_3,\quad & L_{101}&= {\mathbb{C}}e_4,\quad & L_{011}&={\mathbb{C}}e_5, \\
L_{110}&= {\mathbb{C}}e_6,\quad & L_{010}&= {\mathbb{C}}e_7,\quad & L_{100}&={\mathbb{C}}e_8.
\end{alignedat}$$ If we assign $L_{000}=\{0\}$, we obtain a decomposition ${\Gamma}$ of the vector space ${\operatorname{sl}}(3,{\mathbb{C}})$ into the direct sum of the subspaces $$\begin{aligned}
\nonumber\label{ord_gell}
{\Gamma}:\quad {\operatorname{sl}}(3,{\mathbb{C}}) &= L_{001}\oplus L_{111}\oplus L_{101} \oplus
L_{011}\oplus L_{110} \oplus L_{010}\oplus L_{100}.\end{aligned}$$ with the grading property $$[L_j,L_k]\subseteq L_{j+k}, \quad j,k\in I.$$
The grading ${\Gamma}$ is commonly known as the orthogonal grading of ${\operatorname{sl}}(3,{\mathbb{C}})$ and is described in [@Kostrikin] as a special case of orthogonal gradings of ${\operatorname{sl}}(n,{\mathbb{C}})$. Since the grading subspaces of ${\Gamma}$ are minimal and its index set is embedded in ${\mathbb{Z}}_2^3$, the grading ${\Gamma}$ forms so called fine group grading. Bases of the grading subspaces of ${\Gamma}$ are formed by the Gell-Mann matrices [@GellMann] and therefore, we also refer to ${\Gamma}$ as to the Gell-Mann grading.
The symmetry group of ${\Gamma}$
--------------------------------
The symmetry group of the grading ${\Gamma}$ consists of such automorphisms of ${\operatorname{sl}}(3, {\mathbb{C}})$ which permute the grading subspaces. For the case of the Gell-Mann grading, the induced permutation group can be realized as a certain finite matrix group which acts on the index set $I$. Such a matrix group, realizing permutations of ${\Gamma}$ which correspond to automorphisms, is described in [@HPPT1; @HPPT3]. It can be realized as the stability subgroup $G$ of the point $(0,0,1)$ in the finite matrix group $SL(3,{\mathbb{Z}}_2)$, i.e. $$G= \left\{\left(
\begin{array}{ccc}
a&b&e\\
c&d&f\\
0&0&1\\
\end{array}
\right)\ \bigg|\ a,b,c,d,e,f\in\mathbb{Z}_2,\ ad-bc= 1 \pmod 2
\right\}.$$ The matrix group $G$ has 24 elements and its action on the index set $I$ is given as the right matrix multiplication, i.e. for $A=\left(\begin{smallmatrix} a & b & e \\ c & d & f
\\ 0 & 0 & 1 \\
\end{smallmatrix}\right) \in G$ and $i=(i_1,i_2,i_3)\in I$ it holds $$i \mapsto iA = ((i_1a+i_2c)_{\hspace{-6pt}\mod2}, (i_1b+i_2d)_{\hspace{-6pt}\mod2},
(i_1e+i_2f+i_3)_{\hspace{-6pt}\mod2}).$$ In the following, the symmetry matrix group $G$ is crucial for construction and solving of the contraction system.
The Gell–Mann graded contractions
==================================
The graded contractions of ${\Gamma}$
-------------------------------------
The graded contraction procedure for the grading ${\Gamma}$ constructs new Lie algebras by introducing complex parameters ${\varepsilon}_{ij}\in {\mathbb{C}}$ and defining contracted commutation relations of grading subspaces $L_i$ via $$[x_i,x_j]_{\varepsilon}= {\varepsilon}_{ij}[x_i,x_j],$$ for all $x_i\in L_i$, $x_j\in L_j$ and $i,j\in I$. If for a pair of the subspaces $[L_i,L_j]=0$ holds then the corresponding contraction parameter ${\varepsilon}_{ij}$ is irrelevant and we put ${\varepsilon}_{ij}=0$. Relevant are only such contraction parameters ${\varepsilon}_{ij}$ for which $[L_i,L_j]\neq 0$ — these have to fulfill the following conditions in order to $[.\,,.]_{\varepsilon}$ become a Lie bracket.
Firstly, the relevant contraction parameters have to satisfy the symmetry condition which corresponds to the required antisymmetry of $[.\,,.]_{\varepsilon}$: $${\varepsilon}_{ij}={\varepsilon}_{ji}.$$ Secondly, the equation $e_{(i\,j\,k)}$ corresponding to Jacobi identity has to be satisfied for all $i,j,k\in I$: $$\label{rovnice}
\begin{array}{ll}
e_{(i\,j\,k)}: & {\varepsilon}_{jk}{\varepsilon}_{i,j+ k}[x_i,[x_j,x_k]] +
{\varepsilon}_{ki}{\varepsilon}_{j,k+i}[x_j,[x_k,x_i]] + {\varepsilon}_{ij}{\varepsilon}_{k,i+
j}[x_k,[x_i,x_j]] = 0 \\[4pt]
& \forall x_i\in L_i, \forall x_j\in L_j, \forall x_k\in L_k. \\
\end{array}$$ A contraction equation $e_{(i\,j\,k)}$ as well as a contraction parameter ${\varepsilon}_{ij}$ do not depend on the order of indices $i,j,k$, therefore we label contraction equations by unordered triplets of contraction indices — multisets of cardinality 3 — and contraction parameters by unordered pairs of grading indices. The set of all unordered triplets is denoted by $I^3_u$ and the set of all unordered pairs of grading indices is denoted by $I^2_u$. We collect all contraction equations in the set ${S}_{\Gamma}$, i.e. $$\label{general}
{S}_{\Gamma}={\left\{ e_w \, |\, w\in I^3_u \right\}}.$$
In general, the equations have three terms. In some cases of gradings, the equation reduces to two two-term equations. There are two approaches directed to reducing the equations to two terms while handling all Lie algebras with the same grading properties simultaneously.
(i) In the so called generic case, all contraction parameters are considered to be relevant [@WW4]. Moreover, parameters corresponding to the space $L_{000}=\{0\}$ are defined. This approach leads to the system of 224 two–term equations $$\label{extend}
{\varepsilon}_{jk}{\varepsilon}_{i,j+ k} = {\varepsilon}_{ki}{\varepsilon}_{j,k+i} = {\varepsilon}_{ij}{\varepsilon}_{k,i+
j}, \qquad i,j,k\in {\mathbb{Z}}_2^3.$$
(ii) The second less restrictive approach from [@PC1] takes the system of contraction equations in the form $$\label{ropa}
{\varepsilon}_{jk}{\varepsilon}_{i,j+ k} = {\varepsilon}_{ki}{\varepsilon}_{j,k+i} = {\varepsilon}_{ij}{\varepsilon}_{k,i+j} \qquad i,j,k\in I,$$ where all equations containing irrelevant parameters are omitted. This leads to the system consisting of 84 equations.
The existence of general conditions for equivalence of the system and the reduced systems , is still an open problem. Let us note that the systems and are equivalent in the case of the Pauli grading [@HN1] and the Cartan grading [@PC4] of ${\operatorname{sl}}(3, {\mathbb{C}})$. On the contrary, we will see later that — in our case of the Gell–Mann grading — no two of these three systems are equivalent.
Equivalence of solutions of ${S}_{\Gamma}$
------------------------------------------
Contraction parameters are usually written in the form of a symmetric square matrix ${\varepsilon}= ({\varepsilon}_{i,j})$ called a contraction matrix. The set of all contraction matrices, which solve the system ${S}_{\Gamma}$, is denoted by ${C}_{\Gamma}$. Each solution from ${C}_{\Gamma}$ then determines a Lie algebra called graded contraction of the Gell-Mann graded ${\operatorname{sl}}(3, {\mathbb{C}})$. An important example of a trivial graded contraction is given by a normalization matrix $\alpha=(\alpha_{ij})$, where $$\label{normat}
\alpha_{ij}=\frac{a_i a_j}{a_{i+j}}, \qquad a_k \in {\mathbb{C}}\setminus \{0\},\, k \in I.$$
We define an equivalence on the set of all contraction matrices in the following way. Two contraction matrices ${\varepsilon},{\widetilde}{\varepsilon}\in {C}_{\Gamma}$ are equivalent (${\varepsilon}\sim {\widetilde}{\varepsilon}$) if there exist a normalization matrix $\alpha$ and $A\in G$ such that $$\label{sequiv}
{\varepsilon}_{ij} = \frac{a_i a_j}{a_{i+j}}{\widetilde}{\varepsilon}_{iA,jA}$$ If $A = {\operatorname{Id}}$ then ${\varepsilon},{\widetilde}{\varepsilon}$ are strongly equivalent (${\varepsilon}\approx {\widetilde}{\varepsilon}$). It is shown in [@HN1] that two graded contractions given by two equivalent contraction matrices are isomorphic as Lie algebras. Moreover, taking any $A \in G$ and considering an action $$\label{subst}
{\varepsilon}_{ij} \mapsto {\varepsilon}_{iA,jA}$$ it can be easily verified (see [@HN1]) that the set of solutions ${C}_{\Gamma}$ is invariant with respect to this action. For this reason we determine the orbits of this action on relevant contraction parameters.
Taking the set of unordered pairs of indices $I^2_u$, this set splits under the action $(i\, j) \mapsto (iA,\, jA)$ of $A\in G$ into five orbits. Two of these orbits correspond to irrelevant contraction parameters: 6–point orbit represented by $((0,1,0)(0,1,0))$ and 1–point orbit represented by $((0,0,1)(0,0,1))$. The remaining three orbits are
- 12–point orbit represented by $((0,1,0)(1,0,0))$,
- 6–point orbit represented by $((0,1,0)(0,0,1))$, corresponding relevant contraction parameters will be marked by superscript ${+}$,
- 3–point orbit represented by $((0,1,0)(0,1,1))$, corresponding relevant contraction parameters will be marked by superscript ${-}$.
The set of pairs of indices formed by these three orbits corresponds to relevant contraction parameters and we denote it by ${\mathcal{I}}\subset I^2_u$. The general explicit form of the contraction matrix ${\varepsilon}$, with irrelevant parameters on the diagonal set to zero, is $$\renewcommand{\arraystretch}{1.2}
{\varepsilon}=
\begin{pmatrix}
0 & {\varepsilon}^{+}_{(001)(111)} & {\varepsilon}^{+}_{(001)(101)} & {\varepsilon}^{+}_{(001)(011)} &
{\varepsilon}^{+}_{(001)(110)} & {\varepsilon}^{+}_{(001)(010)} & {\varepsilon}^{+}_{(001)(100)} \\
{\varepsilon}^{+}_{(001)(111)} & 0 & {\varepsilon}_{(111)(101)} & {\varepsilon}_{(111)(011)} &
{\varepsilon}^{-}_{(111)(110)} & {\varepsilon}_{(111)(010)} & {\varepsilon}_{(111)(100)} \\
{\varepsilon}^{+}_{(001)(101)} & {\varepsilon}_{(111)(101)} & 0 & {\varepsilon}_{(101)(011)} &
{\varepsilon}_{(101)(110)} & {\varepsilon}_{(101)(010)} & {\varepsilon}^{-}_{(101)(100)} \\
{\varepsilon}^{+}_{(001)(011)} & {\varepsilon}_{(111)(011)} & {\varepsilon}_{(101)(011)} &0&
{\varepsilon}_{(011)(110)} & {\varepsilon}^{-}_{(011)(010)} & {\varepsilon}_{(011)(100)} \\
{\varepsilon}^{+}_{(001)(110)} & {\varepsilon}^{-}_{(111)(110)} & {\varepsilon}_{(101)(110)} & {\varepsilon}_{(011)(110)} &
0 & {\varepsilon}_{(110)(010)} & {\varepsilon}_{(110)(100)} \\
{\varepsilon}^{+}_{(001)(010)} & {\varepsilon}_{(111)(010)} & {\varepsilon}_{(101)(010)} &
{\varepsilon}^{-}_{(011)(010)} & {\varepsilon}_{(110)(010)} & 0 & {\varepsilon}_{(010)(100)} \\
{\varepsilon}^{+}_{(001)(100)} & {\varepsilon}_{(111)(100)} & {\varepsilon}^{-}_{(101)(100)} & {\varepsilon}_{(011)(100)} &
{\varepsilon}_{(110)(100)} & {\varepsilon}_{(010)(100)} & 0 \\
\end{pmatrix}.$$
System of contraction equations
===============================
Symmetries and reduction of the system ${S}_{\Gamma}$
-----------------------------------------------------
It is shown in [@HN1] that similarly to the invariance of the set of contraction matrices ${C}_{\Gamma}$ under the action of the symmetry group $G$, the set of contraction equations ${S}_{\Gamma}$ is invariant under the action of $A\in G$ $$e_{(i\, j\, k)}\mapsto e_{(iA,\, jA,\, kA)}.$$
The set $I^3_u$ of unordered triplets of grading indices, which label the equations from ${S}_{\Gamma}$, is decomposed into $11$ orbits with respect to the corresponding action on indices $(i\, j\, k)\mapsto(iA,\, jA,\, kA)$. The representatives of these orbits together with the corresponding number of their elements are summarized in Table \[orb\_gel3\].
----------------------- ---- ----------------------- --- ----------------------- ----
(0,1,0)(0,1,0)(1,0,0) 24 (0,1,0)(0,0,1)(0,0,1) 6 (0,1,0)(1,0,0)(0,1,1) 12
(0,1,0)(0,1,0)(0,1,0) 6 (0,1,0)(1,0,0)(1,1,0) 4 (0,1,0)(1,0,0)(0,0,1) 12
(0,1,0)(0,1,0)(0,1,1) 6 (0,1,0)(0,0,1)(0,1,1) 3 (0,1,0)(1,0,0)(1,1,1) 4
(0,1,0)(0,1,0)(0,0,1) 6 (0,0,1)(0,0,1)(0,0,1) 1
----------------------- ---- ----------------------- --- ----------------------- ----
: The representative elements of orbits of $I^3_u$ and the numbers of their points under the action of the symmetry group $G$.[]{data-label="orb_gel3"}
Since all grading subspaces of the Gell-Mann grading form commutative subalgebras of ${\operatorname{sl}}(3,{\mathbb{C}})$, only the three orbits in the last column in Table \[orb\_gel3\] lead to non-trivial contraction equations. Using the symmetry group $G$, we analyze and reduce the contraction equations in each of these three orbits. The reduction of the system ${S}_{\Gamma}$ is performed by equivalent rewriting and by linear operations on the equations, yielding an equivalent system with the same set of solutions. For convenient formulation of the resulting system, it is advantageous to extend the action , by acting on all variables simultaneously, to any equation containing contraction parameters. Thus, we obtain the following result.
The system of contraction equations ${S}_{\Gamma}$ is equivalent to the system generated by action on the equations $$\begin{aligned}
{\varepsilon}^{-}_{(101)(100)}{\varepsilon}_{(111)(010)} &=
{\varepsilon}^{-}_{(111)(110)}{\varepsilon}_{(010)(100)},\label{eq1}\\
{\varepsilon}^{+}_{(001)(010)}{\varepsilon}_{(011)(100)} &=
{\varepsilon}^{+}_{(001)(110)}{\varepsilon}_{(010)(100)},\label{eq2}\\
2{\varepsilon}^{+}_{(001)(100)}{\varepsilon}^{-}_{(011)(010)} &=
{\varepsilon}_{(111)(010)}{\varepsilon}_{(011)(100)}
+{\varepsilon}_{(011)(110)}{\varepsilon}_{(010)(100)}.\label{eq3}\end{aligned}$$ The equations and generate $32$ two–term equations and generates $12$ three–term equations.
We write the contraction equation $e_{(0,1,0)(1,0,0)(1,1,1)}$. Acting by $G$ on this contraction equation we generate 4 equations. Using the commutation relations we obtain $$\begin{aligned}
{\varepsilon}^{-}_{(011)(010)}{\varepsilon}_{(111)(100)}[e_7,[e_8,e_3]] & +
{\varepsilon}^{-}_{(101)(100)}{\varepsilon}_{(111)(010)}[e_8,[e_3,e_7]] + \\
& + {\varepsilon}^{-}_{(111)(110)}{\varepsilon}_{(010)(100)}[e_3,[e_7,e_8]] = 0,\end{aligned}$$ $${\varepsilon}^{-}_{(011)(010)}{\varepsilon}_{(111)(100)}(-2e_2) +
{\varepsilon}^{-}_{(101)(100)}{\varepsilon}_{(111)(010)}(2e_1+2e_2) +
{\varepsilon}^{-}_{(111)(110)}{\varepsilon}_{(010)(100)}(-2e_1) = 0.$$ Since $e_1$ and $e_2$ are linearly independent vectors, we obtain the following two–term equations $$\underbrace{{\varepsilon}^{-}_{(011)(010)}{\varepsilon}_{(111)(100)}}_a =
\underbrace{{\varepsilon}^{-}_{(101)(100)}{\varepsilon}_{(111)(010)}}_b =
\underbrace{{\varepsilon}^{-}_{(111)(110)}{\varepsilon}_{(010)(100)}}_c.$$ This comprises two independent equalities $a=b$, $b=c$ and one dependent $a=c$. Denoting the set of unordered pairs from ${\mathcal{I}}$ by ${\mathcal{I}}^2_u$ and considering the action of $G$ on ${\mathcal{I}}^2_u$, one can see that the indices of the terms $a,b,c$ lie in the same 12–point orbit. Moreover, the matrix $X=\left(
\begin{smallmatrix}
0 & 1 & 0\\
1 & 0 & 0\\
0 & 0 & 1\\
\end{smallmatrix} \right)$ transforms the equation $b=c$ into the equation $a=c$ and the matrix $Y=\left(
\begin{smallmatrix}
0 & 1 & 0\\
1 & 1 & 1\\
0 & 0 & 1\\
\end{smallmatrix} \right)$ transforms $b=c$ into $a=b$. Thus, the whole orbit of is generated from the equation $b=c$ by action of $G$. Since the stability subgroup $H=\{1,X,Y,Y^2,XY,XY^2\}$ of the point $((0,1,0)(1,0,0)(1,1,1))$ provides all six permutations of the terms $a,b,c$, each of four left cosets of $G$ — with respect to the group $H$ — generates six linearly dependent equations. Among these six equations only two are linearly independent, therefore, we have $8$ linearly independent equations.
The representative point $((0,1,0)(1,0,0)(0,0,1))$ contains the index of the 2–dimensional grading subspace $L_{001}$. Therefore, it leads to two equations $${\varepsilon}_{(101)(010)}{\varepsilon}^{+}_{(001)(100)}[e_7,[e_8,e_i]] +
{\varepsilon}_{(011)(100)}{\varepsilon}^{+}_{(001)(010)}[e_8,[e_i,e_7]] +
{\varepsilon}^{+}_{(001)(110)}{\varepsilon}_{(010)(100)}[e_i,[e_7,e_8]] = 0, \\$$ where $i=1,2$. Using the commutation relations we have $$\begin{aligned}
(-{\varepsilon}_{(101)(010)}{\varepsilon}^{+}_{(001)(100)}
-{\varepsilon}_{(011)(100)}{\varepsilon}^{+}_{(001)(010)} +
2{\varepsilon}^{+}_{(001)(110)}{\varepsilon}_{(010)(100)})e_3 = 0, \\
(-{\varepsilon}_{(101)(010)}{\varepsilon}^{+}_{(001)(100)}+
2{\varepsilon}_{(011)(100)}{\varepsilon}^{+}_{(001)(010)}
-{\varepsilon}^{+}_{(001)(110)}{\varepsilon}_{(010)(100)})e_3 = 0.\end{aligned}$$ By summing and subtracting these equations we obtain new two–term equations $$\underbrace{{\varepsilon}^{+}_{(001)(100)}{\varepsilon}_{(101)(010)}}_a =
\underbrace{{\varepsilon}^{+}_{(001)(010)}{\varepsilon}_{(011)(100)}}_b =
\underbrace{{\varepsilon}^{+}_{(001)(110)}{\varepsilon}_{(010)(100)}}_c.$$ Considering the action of $G$, the matrix $X=\left(
\begin{smallmatrix}
0 & 1 & 0\\
1 & 0 & 0\\
0 & 0 & 1\\
\end{smallmatrix} \right)$ transforms the term $a$ into the term $b$ while the term $c$ is unchanged. In fact, the index of the term $c$, i.e. \[(001)(110)\]\[(010)(100)\], lies in 12–point orbit in ${\mathcal{I}}^2_u$ while the indices of $a,b$ belong to the one 24–point orbit. The stability subgroup of $((0,1,0)(1,0,0)(0,0,1))$ is a cyclic group $\{1,X\}$ and, therefore, the whole orbit of consists of 24 linearly independent equations generated from $b=c$ by action of $G$.
The last representative point $((0,1,0)(1,0,0)(0,1,1))$ leads to three–term equation $$\begin{aligned}
\nonumber
{\varepsilon}_{(111)(010)}{\varepsilon}_{(011)(100)}[e_7,[e_8,e_5]] +
{\varepsilon}^{+}_{(001)(100)}{\varepsilon}^{-}_{(011)(010)}[e_8,[e_5,e_7]] &+
\\ \nonumber
+{\varepsilon}_{(011)(110)}{\varepsilon}_{(010)(100)}[e_5,[e_7,e_8]] &= 0, \\
\label{gel_eq3}
(2\underbrace{{\varepsilon}^{+}_{(001)(100)}{\varepsilon}^{-}_{(011)(010)}}_a-\underbrace{{\varepsilon}_{(111)(010)}{\varepsilon}_{(011)(100)}}_b
-\underbrace{{\varepsilon}_{(011)(110)}{\varepsilon}_{(010)(100)}}_c)e_4 &= 0.\end{aligned}$$ The indices of $b,c$ belong to the same 24–point orbit, while the index of $a$ belongs to 12–point orbit in ${\mathcal{I}}^2_u$. The matrix $Z=\left(
\begin{smallmatrix}
1 & 0 & 0\\
0 & 1 & 1\\
0 & 0 & 1\\
\end{smallmatrix} \right)\in G$ transforms $b$ into $c$ while $a$ is preserved. Since the stability subgroup of $((0,1,0)(1,0,0)(0,1,1))$ is $\{1,Z\}$, we get 12 linearly independent three–term equations by the action of $G$ on .
Finding the solution of ${S}_{\Gamma}$
--------------------------------------
An efficient algorithm, developed specially for solving contraction systems, was formulated in [@HN1]. It relies on the notion of the equivalence of solutions – the idea is not to evaluate all solutions but eliminate equivalent solutions during the solving process. Using this algorithm, we obtain the following explicit list of solutions.
\[solving\] For any solution of the contraction system ${\varepsilon}\in{C}_{\Gamma}$ there exist $a,\,b,\,c,\,d,\,e,\,f\in{\mathbb{C}}$ such that ${\varepsilon}$ is equivalent to some of the following solutions $$\begin{aligned}
{4}
{\varepsilon}^0_1 &=\left(\begin{smallmatrix}
0 & 1 & 1 & a & a & 1 & a\\
1 & 0 & b & c & cb & c & b\\
1 & b & 0 & 1 & b & 1 & b\\
a & c & 1 & 0 & ca & c & a\\
a & cb & b & ca & 0 & c & ba\\
1 & c & 1 & c & c & 0 & 1\\
a & b & b & a & ba & 1 & 0\\
\end{smallmatrix}\right),\,&
{\varepsilon}^1_1 &=\left(\begin{smallmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 1\\
0 & 0 & 0 & 0 & 1 & 1 & 1\\
0 & 0 & 0 & 0 & 0 & 0 & 1\\
0 & 0 & 0 & 0 & 1 & 1 & 1\\
0 & 1 & 0 & 1 & 0 & 0 & 1\\
0 & 1 & 0 & 1 & 0 & 0 & 1\\
1 & 1 & 1 & 1 & 1 & 1 & 0\\
\end{smallmatrix}\right),\, &
{\varepsilon}^1_2 &= \left(\begin{smallmatrix}
0 & 0 & 0 & 0 & 0 & a & 0\\
0 & 0 & 0 & 0 & 1 & 1 & 1\\
0 & 0 & 0 & 0 & a & a & 1\\
0 & 0 & 0 & 0 & 0 & 1 & 0\\
0 & 1 & a & 0 & 0 & a & 0\\
a & 1 & a & 1 & a & 0 & 1\\
0 & 1 & 1 & 0 & 0 & 1 & 0\\
\end{smallmatrix}\right),\, &
{\varepsilon}^1_3 &= \left(\begin{smallmatrix}
0 & a & 0 & 0 & 0 & 0 & 0\\ a & 0 & a & a & 1 & 1 & 1\\
0 & a & 0 & a & 0 & 0 & 1\\ 0 & a & a & 0 & 0 & 1 & 0\\
0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 1 & 0 & 0 & 1\\
0 & 1 & 1 & 0 & 0 & 1 & 0\\
\end{smallmatrix}\right), \\
{\varepsilon}^2_1 &=\left(\begin{smallmatrix}
0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & a\\
0 & 0 & 0 & 0 & 1 & b & \frac{1}{2}ab+\frac{1}{2}\\
0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0 & 0\\
1 & 0 & b & 0 & 0 & 0 & 1\\
0 & a & \frac{1}{2}ab+\frac{1}{2} & 0 & 0 & 1 & 0\\
\end{smallmatrix}\right),\,&
{\varepsilon}^2_2 &=\left(\begin{smallmatrix}
0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 1\\
0 & 0 & 0 & 0 & 1 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 1 & 0\\ 1 & -1 & 1 & 0 & 1 & 0 & -1\\
0 & 1 & 0 & 0 & 0 & -1 & 0\\
\end{smallmatrix}\right),\,&
{\varepsilon}^3_1 &=\left(\begin{smallmatrix}
0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & a & 0\\
0 & 0 & 0 & 0 & 0 & b & 0\\ 0 & 0 & 0 & 0 & 0 & c & 0\\
0 & 0 & 0 & 0 & 0 & d & 0\\ 1 & a & b & c & d & 0 & 1\\
0 & 0 & 0 & 0 & 0 & 1 & 0\\
\end{smallmatrix}\right),\, &
{\varepsilon}^3_2 &=\left(\begin{smallmatrix}
0 & a & 0 & 0 & 0 & 1 & b\\
a & 0 & 0 & 0 & 0 & c & d\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & c & 0 & 0 & 0 & 0 & 1\\ b & d & 0 & 0 & 0 & 1 & 0\\
\end{smallmatrix}\right),\\
{\varepsilon}^4_1 &=\left(\begin{smallmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 1\\
0 & -1 & 0 & 1 & -a & a & b\\ 0 & 0 & 1 & 0 & 0 & 0 & c\\
0 & 0 & -a & 0 & 0 & 0 & c\\ 0 & 0 & a & 0 & 0 & 0 & 1\\
0 & 1 & b & c & c & 1 & 0\\
\end{smallmatrix}\right),\,&
{\varepsilon}^4_2 &=\left(\begin{smallmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 1 & 0 & 1 & 1\\
0 & 1 & 0 & -1 & -a & a & 0\\ 0 & 1 & -1 & 0 & -b & 0 & b\\
0 & 0 & -a & -b & 0 & a & b\\ 0 & 1 & a & 0 & a & 0 & 1\\
0 & 1 & 0 & b & b & 1 & 0\\
\end{smallmatrix}\right)\,&
{\varepsilon}^5_1 &=\left(\begin{smallmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & a & 0 & b & c\\ 0 & 0 & a & 0 & 0 & d & e\\
0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & b & d & 0 & 0 & 1\\
0 & 0 & c & e & 0 & 1 & 0\\
\end{smallmatrix}\right),\,&
{\varepsilon}^5_2 &=\left(\begin{smallmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & a\\
0 & 0 & 0 & 0 & 0 & 0 & b\\ 0 & 0 & 0 & 0 & 0 & 0 & c\\
0 & 0 & 0 & 0 & 0 & 0 & d\\ 0 & 0 & 0 & 0 & 0 & 0 & 1\\
0 & a & b & c & d & 1 & 0\\
\end{smallmatrix}\right),\end{aligned}$$ $$\begin{aligned}
{4}
{\varepsilon}^5_3 &=\left(\begin{smallmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & a & b\\
0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & a & 0 & 0 & 0 & 0 & 1\\
0 & b & 0 & 0 & 0 & 1 & 0\\
\end{smallmatrix}\right),\, &
{\varepsilon}^5_4 &=\left(\begin{smallmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & a & 0\\
0 & 0 & 0 & 0 & 0 & b & 0\\ 0 & 0 & 0 & 0 & 0 & c & 0\\
0 & 0 & 0 & 0 & 0 & d & 0\\ 0 & a & b & c & d & 0 & 1\\
0 & 0 & 0 & 0 & 0 & 1 & 0\\
\end{smallmatrix}\right),\, &
{\varepsilon}^5_5 &=\left(\begin{smallmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & a & b\\ 0 & 0 & 0 & 0 & a & 0 & 1\\
0 & 0 & 0 & 0 & b & 1 & 0\\
\end{smallmatrix}\right),\, &
{\varepsilon}^5_6 &=\left(\begin{smallmatrix}
0 & a & 0 & 0 & 0 & 0 & 0\\ a & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & b & 0 & 0 & 0\\ 0 & 0 & b & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1\\
0 & 0 & 0 & 0 & 0 & 1 & 0\\
\end{smallmatrix}\right),\\
{\varepsilon}^6_1 &=\left(\begin{smallmatrix}
0 & a & b & c & d & e & f\\ a & 0 & 0 & 0 & 0 & 0 & 0\\
b & 0 & 0 & 0 & 0 & 0 & 0\\ c & 0 & 0 & 0 & 0 & 0 & 0\\
d & 0 & 0 & 0 & 0 & 0 & 0\\ e & 0 & 0 & 0 & 0 & 0 & 0\\
f & 0 & 0 & 0 & 0 & 0 & 0\\
\end{smallmatrix}\right),\, &
{\varepsilon}^6_2 &=\left(\begin{smallmatrix}
0 & 0 & a & 0 & 0 & 0 & b\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\
a & 0 & 0 & 0 & 0 & 0 & c\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\
b & 0 & c & 0 & 0 & 0 & 0\\
\end{smallmatrix}\right), \, &
{\varepsilon}^6_3 &=\left(\begin{smallmatrix}
0 & 0 & 0 & a & 0 & b & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0\\ a & 0 & 0 & 0 & 0 & c & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0\\ b & 0 & 0 & c & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0\\
\end{smallmatrix}\right), \, &
{\varepsilon}^6_4 &=\left(\begin{smallmatrix}
0 & a & 0 & 0 & b & 0 & 0\\ a & 0 & 0 & 0 & c & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\
b & c & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0\\
\end{smallmatrix}\right), \end{aligned}$$ $${\varepsilon}^6_5 =\left(\begin{smallmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & a & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & b\\ 0 & 0 & 0 & 0 & 0 & c & 0\\
0 & a & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & c & 0 & 0 & 0\\
0 & 0 & b & 0 & 0 & 0 & 0\\
\end{smallmatrix}\right).$$ Moreover, any two solutions ${\varepsilon}^m_k$, ${\varepsilon}^n_l$ with $m\neq n$ are not equivalent.
Repeatedly using Theorem 7 from [@HN1] we solve the system ${S}_{\Gamma}$ in seven steps:
1. The explicit solution of the system ${S}_{\Gamma}$ under the assumptions ${\varepsilon}_{(010)(100)}\neq 0$ and ${\varepsilon}^{+}_{(001)(101)}\neq
0$ is given as a matrix which is strongly equivalent to ${\varepsilon}^0_1$.
2. In order to eliminate solutions equivalent to those obtained in the previous step, the system generated from the equation ${\varepsilon}_{(010)(100)}{\varepsilon}^{+}_{(001)(101)}=0$ has to be satisfied. This non–equivalence system is denoted by ${\mathcal{S}}^0$. Solutions of ${S}_{\Gamma}$ and ${\mathcal{S}}^0$ under the assumptions ${\varepsilon}_{(010)(100)}\neq 0$ and ${\varepsilon}^{-}_{(111)(110)}\neq 0$ are described by $3$ parametric matrices which are strongly equivalent to the matrices ${\varepsilon}^1_1,\, {\varepsilon}^1_2$ and ${\varepsilon}^1_3$.
3. The solutions of ${S}_{\Gamma}\cup {\mathcal{S}}^0$ together with the non–equivalence system ${\mathcal{S}}^1$ generated by $${\varepsilon}_{(010)(100)}{\varepsilon}^{-}_{(111)(110)}=0$$ under the assumptions ${\varepsilon}_{(010)(100)}\neq 0, {\varepsilon}^{+}_{(001)(010)}\neq 0$, and ${\varepsilon}_{(101)(110)}\neq 0$ are given by two parametric matrices ${\varepsilon}^2_1$ and ${\varepsilon}^2_2$.
4. The solutions of ${S}_{\Gamma}\cup {\mathcal{S}}^0 \cup{\mathcal{S}}^1 $ together with the non–equivalence system ${\mathcal{S}}^2$ generated by $${\varepsilon}_{(010)(100)}{\varepsilon}^{+}_{(001)(010)}{\varepsilon}_{(101)(110)}=0$$ under the assumptions ${\varepsilon}_{(010)(100)}\neq 0,
{\varepsilon}^{+}_{(001)(010)}\neq 0$ are strongly equivalent to the solutions ${\varepsilon}^3_1$ and ${\varepsilon}^3_2$.
5. The solutions of ${S}_{\Gamma}\cup {\mathcal{S}}^0 \cup{\mathcal{S}}^1 \cup{\mathcal{S}}^2$ together with the non–equivalence system ${\mathcal{S}}^3$ generated by $${\varepsilon}_{(010)(100)}{\varepsilon}^{+}_{(001)(010)}=0$$ under the assumptions ${\varepsilon}_{(010)(100)}\neq 0$, ${\varepsilon}_{(111)(101)}\neq 0$ are strongly equivalent to the solutions ${\varepsilon}^4_1$ and ${\varepsilon}^4_2$.
6. The solutions of ${S}_{\Gamma}\cup {\mathcal{S}}^0 \cup{\mathcal{S}}^1 \cup{\mathcal{S}}^2\cup {\mathcal{S}}^3$ together with the non–equivalence system ${\mathcal{S}}^4$ generated by $${\varepsilon}_{(010)(100)}{\varepsilon}_{(111)(101)}=0$$ under the assumption ${\varepsilon}_{(010)(100)}\neq 0$ are strongly equivalent to the solutions ${\varepsilon}^5_1,\,{\varepsilon}^5_2,\,{\varepsilon}^5_3,\,{\varepsilon}^5_4,\,{\varepsilon}^5_5$ and ${\varepsilon}^5_6$.
7. Finally, the non–equivalence system ${\mathcal{S}}^5$ generated by ${\varepsilon}_{(010)(100)}=0$ enforces zero value of all contraction parameters in the orbit — these variables are not marked by ’$+$’ or ’$-$’ — and ensures fulfilment of all previous non–equivalence systems. Due to these $12$ zero contraction parameters, all two–term equations are satisfied and three–term equations are reduced to the system generated by ${\varepsilon}^{+}_{(001)(100)}{\varepsilon}^{-}_{(011)(010)}=0$. Corresponding solutions are strongly equivalent to ${\varepsilon}^6_1,\,{\varepsilon}^6_2,\,{\varepsilon}^6_3,\,{\varepsilon}^6_4$ and ${\varepsilon}^6_5$.
Even though any two solutions ${\varepsilon}^m_k$, ${\varepsilon}^n_l$ with $m\neq n$ are not equivalent in Proposition \[solving\], the list of solutions still contains equivalent solutions if $m=n$. Since equivalent solutions have the same number of zeros $\nu({\varepsilon})=|{\left\{ (i\,j) \in {\mathcal{I}}\, |\, {\varepsilon}_{ij}= 0 \right\}}|$, we discuss when elements of contraction matrices vanish and divide the solutions according to $\nu({\varepsilon})$. Besides all possible combinations of zero and non–zero parameters, the case $a=-1/b$ has to be also considered for the solution ${\varepsilon}^2_1$. After that, we collect all solutions with the same support $S({\varepsilon}) = {\left\{ (i\,j)\in {\mathcal{I}}\, |\, {\varepsilon}_{ij} \neq 0 \right\}}$. For example, solutions ${\varepsilon}^5_4$ with $a=b=c=0$ and ${\varepsilon}^5_5$ with $b=0$ have the same support $S({\varepsilon}^5_4)=S({\varepsilon}^5_5)=\{{(110)(010), (010)(100)} \}$ but they also represent the same solution — only with different notation of the parameters.
The notion of a projection $\hat{\varepsilon}$ of a solution ${\varepsilon}$ defined via $\hat{\varepsilon}_{ij}=\operatorname{sgn} |{\varepsilon}_{ij}|$ is useful for sorting the solutions of contraction equations. Since there is only one solution with the given support $S({\varepsilon})$ and thus $\hat{\varepsilon}^1 \sim \hat{\varepsilon}^2$ would imply ${\varepsilon}^1 \sim {\varepsilon}^2$, we eliminate those contraction matrices which have equivalent projections. Let us note that the solution without zeros, i.e. ${\varepsilon}^0_1$, where $a,b,c,d\neq0$, is strongly equivalent to the trivial solution ${\varepsilon}^{0,1}$ which has all relevant contraction parameters equal to 1. Thus, any contraction matrix without zeros has the form of the normalization matrix . Thus, by purging the list of solutions for overlaps and equivalencies, we obtain $89$ normalized representatives of equivalence classes of solutions of ${S}_{\Gamma}$.
It remains to answer the question of equivalence of the system of contraction equations with simplified two–term systems and . It can be verified directly that both systems and reduce the number of solutions of :
(i) among $89$ contraction matrices, there are 55 solutions which fulfill the system . On the contrary, there are 6 parametric solutions which fulfill this system only if all their parameters are equal to $1$ and $28$ non–parametric solutions which do not solve this system of two–term equations,
(ii) there are $74$ contraction matrices which satisfy the system . On the contrary, there is one parametric contraction matrix which fulfils this system only if its parameter is equal to $1$ and $14$ contraction matrices which do not fulfill this system of two–term equations at all.
The final list of all $89$ normalized representatives of equivalence classes of solutions, with the solutions which do not satisfy the simplified systems and marked, is located in Appendix A.1.
Continuous and discrete graded contractions
-------------------------------------------
A solution ${\varepsilon}\in{C}_{\Gamma}$ is called continuous if there exist continuous functions $$a_i:(0,1]
\rightarrow {\mathbb{C}}\setminus\{0\},\ i\in I$$ such that for all relevant contraction parameters $${\varepsilon}_{ij} = \lim\limits_{t\rightarrow 0} \frac{a_i(t)a_j(t)}{a_{i+j}(t)}$$ holds; otherwise it is called discrete. If the functions $a_i$ are of the form $a_i(t) = t^{n_i}$, $n_i\in {\mathbb{Z}}$ then a continuous graded contraction becomes a generalized Inönü–Wigner contraction [@WW2].
An efficient tool for distinguishing between continuous and discrete graded contractions was developed in [@WW2]. So called higher–order identities allow us to identify discrete graded contractions. Let us consider an equation where on both sides stand products of $r$ relevant contraction parameters. If this equation holds for all normalization matrices $\alpha_{ij}(t)=\frac{a_i(t)a_j(t)}{a_{i+j}(t)}$, it will also hold for their limit, i.e. for any continuous solution. Thus, any equation of the type $${\varepsilon}_{i_1}{\varepsilon}_{i_2}\dots{\varepsilon}_{i_r}={\varepsilon}_{j_1}{\varepsilon}_{j_2}\dots{\varepsilon}_{j_r},$$ where $r\in {\mathbb{N}}$ and $\{i_1,i_2,\ldots,i_r\}$, $\{j_1,j_2,\ldots,j_r\}$ are disjoint sets of relevant pairs of grading indices, is called higher–order identity of order $r$, if it holds for all normalization matrices, but is violated by some contraction matrix from $C_\Gamma$. Higher–order identities can be deduced from the identities which hold for the normalization matrix . For example the equation $$\label{gell_hoipr}
\alpha_{(001)(100)}\alpha_{(011)(010)} =
\frac{a_{(001)}a_{(100)}}{a_{(101)}}\frac{a_{(011)}a_{(010)}}{a_{(001)}}
=
\frac{a_{(011)}a_{(110)}}{a_{(101)}}\frac{a_{(010)}a_{(100)}}{a_{(110)}}
= \alpha_{(011)(110)}\alpha_{(010)(100)}$$ is evidently satisfied for any normalization matrix. However, considering the contraction matrix ${\varepsilon}^4_2$ we get $0=-b$ and thus is violated for any $b\neq 0$. Therefore, the equation represents second order identity and the solution ${\varepsilon}^4_2$ with $b\neq 0$ is discrete. Applying the symmetry group $G$ to , we can write the 24-point orbit of second order identities generated from the equation $${\varepsilon}^{+}_{(001)(100)}{\varepsilon}^{-}_{(011)(010)} =
{\varepsilon}_{(011)(110)}{\varepsilon}_{(010)(100)}.$$
In a similar way, all $57$ second order identities are found. These identities are divided into 5 orbits and their representatives and the number of the resulting identities under the action of $G$ are written in Table \[tab:hoig\]. For each solution of the system ${S}_{\Gamma}$, the set of second order identities allows us to distinguish whether the solution is continuous or discrete. Any discrete contraction violates at least one of the identities listed in Table \[tab:hoig\]. The remaining solutions are explicitly found as limits of $\alpha_{ij}(t)=t^{n_i+n_j-n_{i+j}}$, i.e. they correspond to generalized Inönü–Wigner contractions. Among $89$ solutions in Appendix A.1 there are $50$ continuous and $36$ discrete ones. The remaining 3 solutions are continuous only for a special value of their parameters, otherwise they are discrete.
Representative equation Number of equations
------------------------------------------------------------------------------------------------------------------------------- ---------------------
$ {\varepsilon}^{+}_{(001)(100)}{\varepsilon}^{-}_{(011)(010)} = {\varepsilon}_{(011)(110)}{\varepsilon}_{(010)(100)} $ $24$
$ {\varepsilon}^{+}_{(001)(100)}{\varepsilon}^{-}_{(101)(100)} = {\varepsilon}_{(110)(100)}{\varepsilon}_{(010)(100)}$ $12$
$ {\varepsilon}_{(111)(010)}{\varepsilon}_{(011)(100)} = {\varepsilon}_{(011)(110)}{\varepsilon}_{(010)(100)}$ $12$
$ {\varepsilon}_{(111)(100)}{\varepsilon}_{(011)(100)} = {\varepsilon}_{(110)(100)}{\varepsilon}_{(010)(100)}$ $6$
$ {\varepsilon}^{+}_{(001)(101)}{\varepsilon}^{+}_{(001)(100)} ={\varepsilon}^{+}_{(001)(111)}{\varepsilon}^{+}_{(001)(110)}$ $3$
: Orbits of the second order identities for the Gell-Mann grading of ${\operatorname{sl}(3,{\mathbb{C}})}$.[]{data-label="tab:hoig"}
The contracted Lie algebras
===========================
Identification of the contracted algebras
-----------------------------------------
All $89$ solutions of the system of contraction equations ${S}_{\Gamma}$ for the Gell–Mann graded ${\operatorname{sl}}(3,{\mathbb{C}})$ are divided into 14 groups according to the number of zeros $\nu$ among the 21 relevant contraction parameters. The numbers of contraction matrices in these groups are summarized in the following table: $$
Number of zeros $\nu$ 0 6 9 11 12 13 14 15 16 17 18 19 20 21
----------------------- --- --- --- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ----
Number of solutions 1 2 1 3 2 1 2 9 12 18 23 11 3 1
$$ Contraction matrices are denoted $\varepsilon^{\nu,i}$, where the second index $i$ is numbering solutions with the same number of zeros $\nu$. The contracted Lie algebra given by solution $\varepsilon^{\nu,i}$ is denoted ${\mathcal{G}}_{\nu,i}$.
There are two trivial solutions: $\varepsilon^{21,1}$ (with 21 zeros) corresponding to the 8–dimensional abelian Lie and $\varepsilon^{0,1}$ (without zeros) corresponding to the initial Lie algebra ${\operatorname{sl}}(3,{\mathbb{C}})$. Among the remaining 87 nontrivial solutions, 8 solutions depend on one nonzero complex parameter $a$ and two depend on two nonzero complex parameters $a$, $b$. The corresponding parametric families of Lie algebras — the parametric Lie algebras — are denoted by ${\mathcal{G}}_{\nu,i}(a)$, ${\mathcal{G}}_{\nu,i}(a,b)$. Each of these parametric Lie algebras will be counted as one algebra.
In order to classify our results we use an extension of the identification procedure from [@HN1]. We have to identify 87/10 Lie algebras — the number following the slash refers to the number of parametric families among all 87 algebras. Our extended algorithm for identification, applied to the Gell-Mann graded contractions, consists of the following steps (for more detailed calculation algorithm of steps 1,2 and 5 see [@ide]):
1. *Splitting of the maximal central component*\
Whenever the complement of the derived algebra $D({\mathcal{L}})=[{\mathcal{L}},{\mathcal{L}}]$ into the center $C({\mathcal{L}})$ of the Lie algebra ${\mathcal{L}}$ is nonzero, ${\mathcal{L}}$ can be decomposed into the direct sum ${\mathcal{L}}={\mathcal{L}}' \oplus k{\mathcal{A}}_1$, where $k=\dim C({\mathcal{L}})/(C({\mathcal{L}})\cap D({\mathcal{L}}))$, $k{\mathcal{A}}_1$ is an abelian algebra of dimension $k$ and non–abelian part ${\mathcal{L}}'$ fulfills $C({\mathcal{L}}')\subseteq D({\mathcal{L}}')$. The separation of maximal central component is possible in 66/8 cases of our contracted Lie algebras. Further, we proceed with non–abelian parts only.
2. *Decomposition into a direct sum of indecomposable ideals*\
A complex Lie algebra ${\mathcal{L}}$ is decomposable if and only if there exist an idempotent $0\neq E=E^2 \neq 1$ in the centralizer $C_R:={\left\{ x\in R \, |\, [x,y]=0,\ \forall y\in {\operatorname{ad}}({\mathcal{L}}) \right\}}$ of the adjoint representation of ${\mathcal{L}}$ in the ring $R={\mathbb{C}}^{n,n}$ of all $n\times n$ complex matrices. If so then ${\mathcal{L}}= {\mathcal{L}}_0 \oplus {\mathcal{L}}_1$ where ${\mathcal{L}}_0$ and ${\mathcal{L}}_1$ are eigen-subspaces of the idempotent $E$ corresponding to the eigenvalues $0,1$. There are only 7/1 decomposable Lie algebras among our results. All of them decompose into the sum of two indecomposable ideals. From now on, we proceed with indecomposable Lie algebras only. Thus, in further steps we deal with 94/10 indecomposable Lie algebras. These algebras are divided according to their dimensions as follows:
Dimension 3 5 6 7 8
-------------------- ------ ------ ---- ------ ------
Number of algebras 22/1 12/2 10 29/5 21/2
3. *Series of ideals*\
We calculate the derived series $ D^0({\mathcal{L}}) = {\mathcal{L}}, \ D^{k+1}({\mathcal{L}}) = [D^k({\mathcal{L}}),D^k({\mathcal{L}})]$, the lower central series ${\mathcal{L}}^1 = {\mathcal{L}},\ {\mathcal{L}}^{k+1} = [{\mathcal{L}}^k,{\mathcal{L}}]$ and the upper central series $C^1({\mathcal{L}}) = C({\mathcal{L}}), \ C^{k+1}({\mathcal{L}})/C^k({\mathcal{L}}) = C({\mathcal{L}}/C^k({\mathcal{L}}))$ for each Lie algebra. Dimensions of ideals in theses series are invariants of Lie algebra ${\mathcal{L}}$, therefore we divide all investigated Lie algebras into the classes according to these invariants. The results are 54/1 nilpotent, 33/9 solvable (non–nilpotent) and 7 non–solvable Lie algebras.
4. *$(\alpha,\beta,\gamma)$–derivations*\
Let $\alpha,\beta,\gamma$ be complex numbers. The dimension of the space of $(\alpha,\beta,\gamma)$–derivations of ${\mathcal{L}}$ $${\operatorname{der}}_{(\alpha,\beta,\gamma)}{\mathcal{L}}= {\left\{ A\in {\operatorname{gl}}({\mathcal{L}}) \, |\, \alpha A[x,y] = \beta[Ax,y] + \gamma[x,Ay], \ \forall x,y\in{\mathcal{L}}\right\}}$$ is an invariant of Lie algebra ${\mathcal{L}}$ [@HN5]. As a special case, the algebra of derivations is also obtained ${\operatorname{der}}{\mathcal{L}}={\operatorname{der}}_{(1,1,1)}{\mathcal{L}}$. We denote by $\dim_{(\alpha,\beta,\gamma)}{\mathcal{L}}$ the 6–tuple formed by the dimensions of the following spaces $$\quad\quad \dim_{(\alpha,\beta,\gamma)}{\mathcal{L}}= [{\operatorname{der}}{\mathcal{L}},\ {\operatorname{der}}_{(0,1,1)}{\mathcal{L}},\ {\operatorname{der}}_{(1,1,0)}{\mathcal{L}},\
{\operatorname{der}}{\mathcal{L}}\cap{\operatorname{der}}_{(0,1,1)}{\mathcal{L}},\
{\operatorname{der}}_{(1,1,-1)}{\mathcal{L}},\ {\operatorname{der}}_{(0,1,-1)}{\mathcal{L}}].$$ Values of these invariants divide our algebras into 28 classes of nilpotent, 17 classes of solvable and 4 classes of non–solvable Lie algebras.
5. *Determination of the radical, the Levi decomposition and the nilradical*\
There are 7 non–solvable Lie algebras for which we determine radicals according to $R({\mathcal{L}}) = {\left\{ x\in{\mathcal{L}}\, |\, {\operatorname{Tr}}({\operatorname{ad}}(x){\operatorname{ad}}(y))=0, \, \forall y\in D({\mathcal{L}}) \right\}}$ and find their Levi decompositions. The Levi decomposition is nontrivial only in 3 cases, two of them with abelian radical. For solvable Lie algebras we determine their nilradicals $N({\mathcal{L}})$.
6. *Casimir operators*\
The elements in the center of the universal enveloping Lie algebra $U({\mathcal{L}})$ of Lie algebra ${\mathcal{L}}$ – Casimir operators – can be found as follows [@Alonso]. Represent elements of basis $(e_1,\ldots,e_n)$ in ${\mathcal{L}}$ by the vector fields $$e_i \rightarrow \widehat x_i = \sum_{j,k=1}^n c_{ij}^k x_k
\frac{\partial}{\partial x_j},$$ which act on the space of continuously differentiable functions $F(x_1,\ldots,x_n)$ of $n$ variables. Find formal invariants i.e. such functions $F$ which solve the following linear system of first-order partial differential equations $\widehat{x_i} F(x_1,\ldots,x_n) =0 $ for all $i=1,\ldots,n$. The number of functionally independent solutions of this system $$\tau({\mathcal{L}}) = \dim({\mathcal{L}}) - \sup_{x_1,\ldots,x_n\in{\mathbb{C}}} \operatorname{rank}
M_{\mathcal{L}},$$ where $M_{\mathcal{L}}$ is skew–symmetric matrix with entries $(M_{\mathcal{L}})_{ij} = \sum_{k=1}^n c_{ij}^k x_k$, is an invariant of ${\mathcal{L}}$. Casimir operators correspond to the polynomial formal invariants. This correspondence is provided by symmetrization: any term $x_{k_1}\ldots x_{k_p}$ of polynomial $F(x_1,\ldots,x_n)$ in commuting variables $x_i$ is replaced by symmetric term in non–commuting basis elements $e_1,\ldots,e_n\in{\mathcal{L}}$ as follows $$x_{k_1}\ldots x_{k_p} \mapsto \frac{1}{p\, !}\sum_{\sigma\in S_p}
e_{k_{\sigma(1)}}\ldots e_{k_{\sigma(p)}}.$$ We determine the number of formal invariants $\tau({\mathcal{L}})$ for all investigated algebras. Since all independent formal invariants of complex nilpotent Lie algebras can be found in polynomial form, we also determine all Casimir operators for nilpotent Lie algebras.
7. *Isomorphisms*\
Having two $n$-dimensional Lie algebras ${\mathcal{L}}$ and ${\widetilde}{\mathcal{L}}$ with the same values of all above listed invariant characteristics, we search for isomorphisms explicitly. Thus, we solve the following system of $n^2(n-1)/2$ quadratic equations $$\sum_{r=1}^{n} c_{ij}^{r}A_{kr} = \sum_{\mu, \nu =1}^{n} \tilde
c_{\mu\nu}^k A_{\mu i}A_{\nu j}, \qquad i=1,\ldots,n-1,\quad j=
i,\ldots,n,\quad k\in\hat{n},$$ where $c_{ij}^k, \tilde
c_{ij}^k$ are structural constants of ${\mathcal{L}}, {\widetilde}{\mathcal{L}}$ and $A_{i,j}$ are components of $n\times n$ complex regular matrix representing the isomorphism. Solving this system for all Lie algebras in the same class we find: 26 isomorphic algebras in the classes of nilpotent, 16/4 isomorphic algebras in the classes of solvable and 3 isomorphic algebras in the classes of non–solvable Lie algebras. Omitting these algebras we get only one algebra in each class. Thus, all algebras are now identified up to ranges of parameters for the parametric algebras.
8. *Invariant functions*\
The invariants listed above are not able to distinguish Lie algebras within parametric families. Therefore we use the concept of invariant functions [@HN5; @HN6], which enables us to classify one–parametric families of Lie algebras. The invariant function $\psi_{\mathcal{L}}$ which arises from $(\alpha,\beta,\gamma)$–derivations is defined by $$\psi_{{\mathcal{L}}}(\alpha) = \dim({\operatorname{der}}_{(\alpha,1,1)}{\mathcal{L}}).$$ We calculate this invariant function for all one–parametric Lie algebras. Together with isomorphisms it allows us to determine the ranges of parameters for 3 one–parametric families. The invariant functions $\psi_{\mathcal{L}}$ of these three cases are listed in Appendix A.3.
The results of identification
-----------------------------
There are only 4 mutually non–isomorphic decomposable Lie algebras among the graded contractions of Gell–Mann graded ${\operatorname{sl}}(3,{\mathbb{C}})$:
------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------
Non–solvable (discrete contraction) ${\mathcal{G}}_{9,1} \cong {\mathcal{G}}'_{18,8} \oplus {\mathcal{G}}'_{18,8} \oplus 2{\mathcal{A}}_1$
Solvable (discrete contractions) ${\mathcal{G}}_{17,2}(a) \cong {\mathcal{G}}_{13,1} \cong {\mathcal{G}}'_{19,2} \oplus {\mathcal{G}}'_{19,2} \oplus 2{\mathcal{A}}_1$
${\mathcal{G}}_{18,2} \cong {\mathcal{G}}'_{19,2} \oplus {\mathcal{G}}'_{20,1} \oplus 2{\mathcal{A}}_1$
Nilpotent (continuous contractions) $ {\mathcal{G}}_{19,1} \cong {\mathcal{G}}_{19,11}
\cong{\mathcal{G}}_{17,18} \cong {\mathcal{G}}'_{20,1} \oplus {\mathcal{G}}'_{20,1} \oplus 2{\mathcal{A}}_1$
------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------
where ${\mathcal{A}}_1$ stands for one–dimensional abelian Lie algebra.
The list of all isomorphisms among the indecomposable graded contractions of the Gell–Mann graded ${\operatorname{sl}}(3,{\mathbb{C}})$ is presented in Table \[tab:iso\].
----------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------ -----------------------------------------------------
${\mathcal{G}}_{18,21} \cong {\mathcal{G}}_{18,8}$ ${\mathcal{G}}_{20,3} \cong {\mathcal{G}}_{20,2} \cong {\mathcal{G}}_{20,1}$ ${\mathcal{G}}_{19,10} \cong {\mathcal{G}}_{19,4}$
${\mathcal{G}}_{19,9} \cong {\mathcal{G}}_{19,7} \cong {\mathcal{G}}_{19,5} \cong {\mathcal{G}}_{19,3}$ ${\mathcal{G}}_{18,17} \cong {\mathcal{G}}_{18,16}$
${\mathcal{G}}_{19,8} \cong {\mathcal{G}}_{19,6} \cong {\mathcal{G}}_{19,2}$ ${\mathcal{G}}_{18,20} \cong {\mathcal{G}}_{18,19} \cong {\mathcal{G}}_{18,7}$ ${\mathcal{G}}_{18,3} \cong {\mathcal{G}}_{18,1}$
${\mathcal{G}}_{18,14} \cong{\mathcal{G}}_{18,13} \cong {\mathcal{G}}_{18,11} \cong {\mathcal{G}}_{18,9}$ ${\mathcal{G}}_{18,15} \cong {\mathcal{G}}_{18,12} \cong {\mathcal{G}}_{18,10} \cong {\mathcal{G}}_{18,6}$ ${\mathcal{G}}_{17,17} \cong {\mathcal{G}}_{17,13}$
${\mathcal{G}}_{17,10}\cong{\mathcal{G}}_{17,9} \cong {\mathcal{G}}_{17,7} \cong {\mathcal{G}}_{17,6}$ ${\mathcal{G}}_{18,22} \cong {\mathcal{G}}_{18,4} \cong {\mathcal{G}}_{16,12}$ ${\mathcal{G}}_{17,14} \cong {\mathcal{G}}_{17,12}$
${\mathcal{G}}_{17,8}(a) \cong {\mathcal{G}}_{17,11}(4a)$ ${\mathcal{G}}_{16,10} \cong {\mathcal{G}}_{16,8}$ ${\mathcal{G}}_{16,11} \cong {\mathcal{G}}_{16,7}$
${\mathcal{G}}_{16,3}(a) \cong {\mathcal{G}}_{16,5}(4a) \cong {\mathcal{G}}_{16,4}(4a)$
----------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------ -----------------------------------------------------
: Isomorphisms among the indecomposable ${\Gamma}-$graded contractions.[]{data-label="tab:iso"}
Note that isomorphic graded contractions are always of the same type, i.e. all discrete or all continuous. Among the resulting non-isomorphic indecomposable algebras, there are $28$ nilpotent, one of them being parametric, 17 solvable with 5 of them parametric and 4 non–solvable. These indecomposable Lie algebras are listed together with their invariant characteristics in Appendix A.2. All resulting decomposable Lie algebras can be written as their direct sums.
Including two trivial contractions we have obtained 55 non–isomorphic contracted Lie algebras as the graded contractions of the Gell–Mann graded Lie algebra ${\operatorname{sl}}(3,{\mathbb{C}})$. Among them there are 4 one–parametric and 2 two–parametric families of Lie algebras. From all these contracted Lie algebras 20 are discrete contractions, 32 continuous contractions and 3 parametric algebras represent continuous contractions for a special value of their parameter; otherwise they are discrete. Table \[tab:gellmann\] provides the overview of the numbers of contracted Lie algebras for the Gell–Mann graded ${\operatorname{sl}}(3,{\mathbb{C}})$. Lie algebras are divided there according to the dimension of their non–abelian parts and their types.
------------------ -------- ------ -------- ------- -------- ------ ----
Dimension of Total
non–abelian part Indec. Dec. Indec. Dec. Indec. Dec.
3 1 1 1 3
4
5 2 2 4
6 2 2 3 1 1 1 10
7 6 10 16
8 6 12 2 20
53
------------------ -------- ------ -------- ------- -------- ------ ----
: The numbers of the non–trivial graded contractions of the Gell-Mann graded ${\operatorname{sl}}(3,{\mathbb{C}})$.[]{data-label="tab:gellmann"}
It remains to compare our list of the graded contractions of the Gell-Mann graded ${\operatorname{sl}}(3,{\mathbb{C}})$ with the previous works which contain results for the Cartan grading [@PC4] and the Pauli grading [@HN1] of ${\operatorname{sl}}(3,{\mathbb{C}})$. Since there is a common coarsening for the Cartan and the Gell-Mann gradings, it is expectable that there will be also common Lie algebras among the corresponding non–trivial contractions. There is no common coarsening, however, for the Pauli and the Gell-Mann grading. Therefore, not all common results are the consequence of the existence of a common coarsening. The contracted Lie algebras ${\mathcal{G}}_{19,2},\ {\mathcal{G}}_{15,1}(a,b),\ {\mathcal{G}}_{11,2}$ and ${\mathcal{G}}_{18,6}$ appear as graded contractions of the Cartan grading, the algebras ${\mathcal{G}}_{15,2}({\textstyle\frac{-1+\sqrt{3}i}{8},\frac{-1-\sqrt{3}i}{2}}),\ {\mathcal{G}}_{19,4},\
{\mathcal{G}}_{18,7},\ {\mathcal{G}}_{17,13},\ {\mathcal{G}}_{18,18},\ {\mathcal{G}}_{17,5},\ {\mathcal{G}}_{18,5},\ {\mathcal{G}}_{15,6}({\textstyle\frac{-1+\sqrt{3}i}{2}})$ and ${\mathcal{G}}_{16,2}$ appear as graded contractions of the Pauli Grading and the algebras ${\mathcal{G}}_{18,8},\ {\mathcal{G}}_{6,1},\ {\mathcal{G}}_{15,1}(1,1),\ {\mathcal{G}}_{20,1},{\mathcal{G}}_{17,18},\ {\mathcal{G}}_{19,3},\ {\mathcal{G}}_{18,16},$ ${\mathcal{G}}_{16,12},$ ${\mathcal{G}}_{18,23},\ {\mathcal{G}}_{15,9}$ and ${\mathcal{G}}_{15,5}$ appear as graded contractions of both Cartan and Pauli gradings.
Concluding remarks
==================
- The richness of the outcome of the graded contraction procedure and its crucial dependence on the initial grading is illustrated by the fact that $16$ solvable algebras, including two–parametric continuum, which are not among the results neither from Pauli nor Cartan gradings, are obtained. Moreover, $12$ nilpotent including one–parametric continuum and three non-solvable algebras are also new among the graded contractions of ${\operatorname{sl}(3,{\mathbb{C}})}$.
- The physically important rigid rotor algebra $[\mathbb{R}^5]$so$(3)$, see [@Guise2] and references therein, is a real eight–dimensional algebra generated by five commuting quadrupole moments $Q_\nu,\,\nu=\pm2, \pm 1, 0$ and three angular moments $L_i, i=\pm 1, 0$. It is found among the results as a real form of the algebra ${\mathcal{G}}_{6,2}$ with commutation relations listed in Appendix A.2, where $Q_{-2}=e_1,\, Q_{-1}=e_2, \dots, Q_{2}=e_5$ and $L_{-1}=e_6,\, L_{0}=e_7,\, L_{1}=e_8 $ are identified.
- It remains to apply the graded contraction procedure to the last of the four fine gradings of ${\operatorname{sl}(3,{\mathbb{C}})}$, so called ${\Gamma}_d$ grading [@HPPT1]. In contrast to Cartan, Pauli and Gell-Mann gradings, this finest grading ${\Gamma}_d$ is known for Lie algebra ${\operatorname{sl}(3,{\mathbb{C}})}$ only — it decomposes ${\operatorname{sl}(3,{\mathbb{C}})}$ into eight one–dimensional grading subspaces determined by the matrices $$\begin{array}{llllllll}
\left(\begin{smallmatrix}
0&0&0\\
0&1&0\\
0&0&-1
\end{smallmatrix}\right), &
\left(\begin{smallmatrix}
0&1&0\\
0&0&0\\
-1&0&0
\end{smallmatrix}\right), &
\left(\begin{smallmatrix}
0&0&0\\
0&0&1\\
0&0&0
\end{smallmatrix}\right), &
\left(\begin{smallmatrix}
0&0&1\\
1&0&0\\
0&0&0\\
\end{smallmatrix}\right),&
\left(\begin{smallmatrix}
2&0&0\\
0&-1&0\\
0&0&-1
\end{smallmatrix}\right), &
\left(\begin{smallmatrix}
0&1&0\\
0&0&0\\
1&0&0
\end{smallmatrix}\right), &
\left(\begin{smallmatrix}
0&0&0\\
0&0&0\\
0&1&0
\end{smallmatrix}\right), &
\left(\begin{smallmatrix}
0&0&1\\
-1&0&0\\
0&0&0
\end{smallmatrix}\right).
\end{array}$$ Only non-solvable contracted algebras of ${\Gamma}_d$ grading are classified [@PN]. Since the symmetry group of ${\Gamma}_d$ has only four elements and a vast number of parametric solutions appears, a complete classification of the outcome of the graded contraction procedure is still beyond reach.
- For the study of the four one–parametric families of Lie algebras, the concept of generalized derivations [@HN5] is used and the corresponding invariant functions are calculated and tabulated in Appendix A.3. Even though this concept was extended to so called twisted cocycles in [@HN6] and is able to handle two–parametric continua of four–dimensional Lie algebras, the complexity of calculations in dimension eight still prevents reaching a classification of the two two–parametric continua in the present case. Thus, these two two–parametric algebras warrant further study.
- Even though the possibility of existence of a three–term contraction equation is brought forward in [@HN1], the solution ${\varepsilon}^{9,1}$ is the first solution to appear in the literature which would be lost if only two–term equations were considered. A detailed comparison of the results of the two simplifying approaches and , which both use two–term equations only, is located in Appendix A.1.
- The $33$ non–trivial IW contractions form a significant part among the solutions of the graded contraction procedure and are distinguished among the contraction matrices in Appendix A.1, as well as among the contracted algebras in Appendix A2. Thus, among a few others, this procedure is a very effective tool for detecting and explicit formulation of this physically important kind of relation among Lie algebras.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors gratefully acknowledge the support of this work by RVO68407700 and by the Ministry of Education of Czech Republic from the project MSM6840770039.
Appendix A.1: Contraction matrices {#appendix-a.1-contraction-matrices .unnumbered}
==================================
Non–equivalent solutions of contraction system for the Gell–Mann graded Lie algebra ${\operatorname{sl}(3,{\mathbb{C}})}$ are listed. These solutions are divided according to the number of zeros among the relevant contraction parameters. The solution ${\varepsilon}^{i,j}$ refers to the $j$–th solution in the relevant list of solutions with $i$ zeros. If not specified, the parameters $a,b$ in contraction matrices are arbitrary non–zero complex numbers; zeros in contraction matrices are shown as dots. The subscript $C$ or $D$ denotes continuous or discrete solution, respectively. The parametric solutions which satisfy only if all their parameters are equal to $1$ are marked by $V$ or $\overline{V}$. Non–parametric solutions which do not satisfy are marked by $W$ or $\overline{W}$. The parametric solutions which satisfy only if all their parameters are equal to $1$ are marked by $\overline{V}$. Non–parametric solutions which do not satisfy are marked by $\overline{W}$.
- Trivial solutions ${\varepsilon}^{0,1}, {\varepsilon}^{21,1}$ $$\left(
\begin{smallmatrix}
{\cdot}& 1 & 1 & 1 & 1 & 1 & 1 \\
1 & {\cdot}& 1 & 1 & 1 & 1 & 1 \\
1 & 1 & {\cdot}& 1 & 1 & 1 & 1 \\
1 & 1 & 1 & {\cdot}& 1 & 1 & 1 \\
1 & 1 & 1 & 1 & {\cdot}& 1 & 1 \\
1 & 1 & 1 & 1 & 1 & {\cdot}& 1 \\
1 & 1 & 1 & 1 & 1 & 1 & {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C} \left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}$$
- Solutions with 6 zeros ${\varepsilon}^{6,1}, {\varepsilon}^{6,2}$ $$\left(
\begin{smallmatrix}
{\cdot}& 1 & 1 & 1 & 1 & 1 & 1 \\
1 & {\cdot}& 1 & 1 & 1 & 1 & 1 \\
1 & 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
1 & 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
1 & 1 & 1 & 1 & {\cdot}& 1 & 1 \\
1 & 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
1 & 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& 1 & 1 & 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & 1 & 1 & {\cdot}& {\cdot}\\
1 & 1 & {\cdot}& 1 & 1 & 1 & 1 \\
1 & 1 & 1 & {\cdot}& 1 & 1 & 1 \\
1 & 1 & 1 & 1 & {\cdot}& 1 & 1 \\
{\cdot}& {\cdot}& 1 & 1 & 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & 1 & 1 & {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}$$
- Solution with 9 zeros ${\varepsilon}^{9,1}$ $$\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& -1 & 1 & {\cdot}& -1 & 1 \\
{\cdot}& -1 & {\cdot}& 1 & 1 & -1 & {\cdot}\\
{\cdot}& 1 & 1 & {\cdot}& 1 & {\cdot}& 1 \\
{\cdot}& {\cdot}& 1 & 1 & {\cdot}& 1 & 1 \\
{\cdot}& -1 & -1 & {\cdot}& 1 & {\cdot}& 1 \\
{\cdot}& 1 & {\cdot}& 1 & 1 & 1 & {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} \overline{W}}$$
- Solutions with 11 zeros ${\varepsilon}^{11,1}, {\varepsilon}^{11,2}, {\varepsilon}^{11,3}$ $$\left(
\begin{smallmatrix}
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& 1 & {\cdot}\\
1 & {\cdot}& 1 & 1 & 1 & 1 & 1 \\
1 & 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& 1 & {\cdot}\\
1 & 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C} \left(
\begin{smallmatrix}
{\cdot}& 1 & 1 & 1 & 1 & 1 & 1 \\
1 & {\cdot}& {\cdot}& 1 & {\cdot}& 1 & {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& 1 & {\cdot}& 1 & {\cdot}\\
1 & 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C} \left(
\begin{smallmatrix}
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & 1 & 1 & {\cdot}& {\cdot}\\
1 & 1 & {\cdot}& 1 & 1 & 1 & 1 \\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& 1 & {\cdot}\\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& 1 & {\cdot}\\
{\cdot}& {\cdot}& 1 & 1 & 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}$$
- Solutions with 12 zeros ${\varepsilon}^{12,1}, {\varepsilon}^{12,2}$ $$\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& -1 & 1 & 1 & -1 & 1 \\
{\cdot}& -1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & 1 & 1 & {\cdot}& 1 & 1 \\
{\cdot}& -1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} \overline{W}}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& -1 & -1 & {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& -1 & {\cdot}& 1 & -1 & 1 & {\cdot}\\
{\cdot}& -1 & 1 & {\cdot}& -1 & {\cdot}& 1 \\
{\cdot}& {\cdot}& -1 & -1 & {\cdot}& 1 & 1 \\
{\cdot}& {\cdot}& 1 & {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& 1 & 1 & {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} \overline{W}}$$
- Solution with 13 zeros ${\varepsilon}^{13,1}$ $$\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& -1 & 1 & {\cdot}& -1 & 1 \\
{\cdot}& -1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & 1 & {\cdot}& 1 & 1 \\
{\cdot}& -1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} \overline{W}}$$
- Solutions with 14 zeros ${\varepsilon}^{14,1}, {\varepsilon}^{14,2}$ $$\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& -1 & 1 & 1 & -1 & 1 \\
{\cdot}& -1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& 1 & {\cdot}\\
{\cdot}& -1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} \overline{W}}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & -1 & {\cdot}& {\cdot}& {\cdot}\\
1 & 1 & {\cdot}& 1 & 1 & 1 & {\cdot}\\
{\cdot}& -1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} \overline{W}}$$
- Solutions with 15 zeros ${\varepsilon}^{15,1},\ldots, {\varepsilon}^{15,9}$ $$\left(
\begin{smallmatrix}
{\cdot}& a & b & 1 & 1 & 1 & 1 \\
a & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
b & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} \ast}^{\hspace{-5pt} V}\left(
\begin{smallmatrix}
{\cdot}& a & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
a & {\cdot}& b & 1 & 1 & 1 & 1 \\
{\cdot}& b & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} \ast}^{\hspace{-5pt} V}\left(
\begin{smallmatrix}
{\cdot}& 1 & 1 & 1 & {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}\\
1 & 1 & {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}\\
1 & 1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& 1 & {\cdot}\\
1 & {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& 1 \\
1 & 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & 1 & 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& 1 & {\cdot}& {\cdot}& 1 \\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& 1 & {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}$$ $$\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& \frac{1}{2}(a + 1) & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& a & 1 & 1 & {\cdot}& {\cdot}\\
\frac{1}{2}(a+1) & a & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& 1 & {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} \dag}^{\hspace{-5pt} \overline{V}}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& -1 & 1 & {\cdot}& -1 & 1 \\
{\cdot}& -1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}\\
{\cdot}& -1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} \overline{W}}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & -1 & {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& 1 & 1 & 1 & {\cdot}\\
{\cdot}& -1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} \overline{W}} \left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& 1 & 1 & 1 & {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& 1 & 1 & {\cdot}\\
{\cdot}& 1 & {\cdot}& 1 & {\cdot}& 1 & {\cdot}\\
{\cdot}& 1 & {\cdot}& 1 & 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}$$ [$^\ast$ Continuous for $a=b=1$, otherwise discrete. ]{} [$^\dag$ $a\neq 0,-1$, continuous for $a=1$, otherwise discrete. ]{}
- Solutions with 16 zeros ${\varepsilon}^{16,1},\ldots, {\varepsilon}^{16,12}$ $$\left(
\begin{smallmatrix}
{\cdot}& a & 1 & 1 & 1 & 1 & {\cdot}\\
a & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} W} \left(
\begin{smallmatrix}
{\cdot}& 1 & 1 & 1 & {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}\\
1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& a & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
a & {\cdot}& 1 & 1 & 1 & 1 & {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} W}\left(
\begin{smallmatrix}
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& a & 1 & {\cdot}& 1 & 1 \\
{\cdot}& a & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} W}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& a & 1 & 1 & 1 & 1 \\
{\cdot}& a & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} W}$$ $$\left(
\begin{smallmatrix}
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}\\
1 & 1 & {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C} \left(
\begin{smallmatrix}
{\cdot}& {\cdot}& \frac{1}{2} & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & 1 & 1 & {\cdot}& {\cdot}\\
\frac {1}{2} & 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} \overline{W}}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& \frac{1}{2} & {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 & 1 & 1 \\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\frac{1}{2} & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} \overline{W}}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& 1 & {\cdot}& 1 \\
{\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}& 1 \\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& -1 & 1 & {\cdot}& {\cdot}& {\cdot}\\
1 & -1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} \overline{W}}$$ $$\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& -1 & 1 & 1 & {\cdot}& {\cdot}\\
{\cdot}& -1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& 1 & {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} \overline{W}}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& 1 & 1 & 1 & {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& 1 & {\cdot}& 1 & {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}$$
- Solutions with 17 zeros ${\varepsilon}^{17,1},\ldots, {\varepsilon}^{17,18}$ $$\left(
\begin{smallmatrix}
{\cdot}& 1 & 1 & 1 & 1 & {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} W}\left(
\begin{smallmatrix}
{\cdot}& a & {\cdot}& 1 & 1 & 1 & {\cdot}\\
a & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} V}\left(
\begin{smallmatrix}
{\cdot}& 1 & 1 & 1 & {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}\\
1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}\\
1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& 1 & 1 & 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} W}$$ $$\left(
\begin{smallmatrix}
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& 1 & 1 & {\cdot}& 1 & {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} W} \left(
\begin{smallmatrix}
{\cdot}& a & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
a & {\cdot}& 1 & {\cdot}& 1 & 1 & {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} V}\left(
\begin{smallmatrix}
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& 1 & 1 & 1 & {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} W}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & 1 & 1 & 1 & {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} W}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& a & 1 & {\cdot}& 1 & 1 \\
{\cdot}& a & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} V}\left(
\begin{smallmatrix}
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}$$ $$\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& \frac{1}{2} & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& 1 & 1 & {\cdot}& {\cdot}\\
\frac{1}{2} & {\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} \overline{W}}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& 1 & {\cdot}& 1 \\
{\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C} \left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& 1 & {\cdot}& 1 \\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 \\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& 1 \\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& 1 \\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& \operatorname{-1} & 1 & {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& \operatorname{-1} & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} \overline{W}}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& 1 & {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& 1 & {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}$$
- Solutions with 18 zeros ${\varepsilon}^{18,1},\ldots, {\varepsilon}^{18,23}$ $$\left(
\begin{smallmatrix}
{\cdot}& 1 & 1 & 1 & {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& 1 & 1 & {\cdot}& 1 & {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} W}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& 1 & 1 & 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& 1 & {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}$$ $$\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}\\
1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}\left(
\begin{smallmatrix}
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& 1 & {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} W}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & 1 & 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} W}\left(
\begin{smallmatrix}
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& 1 & {\cdot}& 1 & {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}$$ $$\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & 1 & {\cdot}& 1 & {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} W}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& 1 & 1 & {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}^{\hspace{-5pt} W}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& 1 & 1 & 1 & {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 \\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}$$ $$\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& 1 \\
{\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& 1 \\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 \\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 \\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}$$
- Solutions with 19 zeros ${\varepsilon}^{19,1},\ldots, {\varepsilon}^{19,11}$ $$\left(
\begin{smallmatrix}
{\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}\left(
\begin{smallmatrix}
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & 1 & {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}$$ $$\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& 1 & {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} D}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& 1 & {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 \\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 \\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}$$
- Solutions with 20 zeros ${\varepsilon}^{20,1}, {\varepsilon}^{20,2}, {\varepsilon}^{20,3}$ $$\left(
\begin{smallmatrix}
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}\left(
\begin{smallmatrix}
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& 1 & {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& 1 & {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
{\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}& {\cdot}\\
\end{smallmatrix}
\right)_{\hspace{-5pt} C}$$
Appendix A.2: Classification of graded contractions {#appendix-a.2-classification-of-graded-contractions .unnumbered}
====================================================
The lists of all contracted Lie algebras of the Gell–Mann graded ${\operatorname{sl}(3,{\mathbb{C}})}$ is presented. Indecomposable non–abelian parts of the contracted Lie algebras are tabulated. Algebras are divided into classes according to the dimensions of the derived series (DS), the lower central series (CS) and the upper central series (US). For each of the listed Lie algebras we give its nonzero commutation relations, dimensions of algebras of generalized derivations ${\dim_{(\alpha,\beta,\gamma)}}$, number of formal invariants $\tau$ and the type of the contraction (C–continuous, D–discrete). The Levi decomposition is given in the last column for non–solvable Lie algebras. For the non–solvable and the solvable non–nilpotent Lie algebras the nilradical is added. Casimir operators of the nilpotent Lie algebras are presented.
For specification of the ranges of parameters for one–parametric contractions, the following notations are used $$\renewcommand{{1.2}}{1.5}
\begin{array}{l}
{\mathbb{C}}_{10} = {\left\{ z\in {\mathbb{C}}\, |\, 0<|z|<1 \right\}}\cup {\left\{ z\in{\mathbb{C}}\, |\, |z|=1 \wedge
\operatorname{Im}(z)\geq 0 \right\}},\\
{\mathbb{C}}_{20} = {\left\{ z\in{\mathbb{C}}\, |\, 0<|z+1|<1 \wedge \operatorname{Re}(z)
\geq -\frac{1}{2} \right\}}\cup {\left\{ z\in{\mathbb{C}}\, |\, |z+1|=1 \wedge \operatorname{Re}(z)\geq
-\frac{1}{2} \wedge \operatorname{Im}(z)
> 0 \right\}}.
\end{array}$$ We use the superscript $\ast$ for any of the listed sets if there are no isomorphisms among Lie algebras corresponding to different parameters in the given set.
For low–dimensional Lie algebras the alternative name (AN) is assigned according to the list of algebras from [@PSWZ_Inv]. This name is also used for nilradicals and Levi decompositions. In order to display the structure of the contracted Lie algebras, their commutation relations are written in the basis $(e_1,e_2,\dots,e_n)$ which for each case begins with the vectors from the center $(e_1,\dots,e_k)$ and extends to the vectors from the nilradical $(e_1,\dots,e_l)$ and the radical $(e_1,\dots,e_m)$, $0 \leq k \leq l\leq m \leq n.$
[lllllccc]{}\
DS, CS, US & Name & Commutation relations & ${\dim_{(\alpha,\beta,\gamma)}}$ & $\tau$ & Nilradical & T & Levi dec.\
$(3)(3)(0)$ & ${\mathcal{G}}'_{18,8}$ & $[e_1,e_2]=e_3,\ [e_1,e_3]=e_2,\ [e_2,e_3]=e_1$ & $[3,0,1,0,0,1]$ & 1 & $\{0\}$ & $D$ & ${\operatorname{sl}(2,{\mathbb{C}})}$\
$(6)(6)(0)$ & ${\mathcal{G}}'_{12,2}$& $[e_1,e_4]=e_2,\ [e_1,e_6]=-e_3,\ [e_2,e_4]=-e_1,\ [e_2,e_5]=-e_3,$ & $[7,0,2,0,0,2]$ & 2 & $3{\mathcal{A}}_1$ & $D$ & $3{\mathcal{A}}_1\triangleleft {\operatorname{sl}(2,{\mathbb{C}})}$\
& & $[e_3,e_5]=e_2,\ [e_3,e_6]=e_1,\ [e_4,e_5]=e_6,\ [e_4,e_6]=-e_5,\ [e_5,e_6]=e_4$\
$(8)(8)(0)$ & ${\mathcal{G}}_{6,2}$& $[e_1,e_7]=2e_1,\ [e_1,e_8]=\sqrt{2}e_2,\ [e_2,e_6]=-\sqrt{2}e_1,\ [e_2, e_7]=e_2,$ & $[9,0,1,0,0,1]$ & 2 & $5{\mathcal{A}}_1$ & $C$ & $5{\mathcal{A}}_1 \triangleleft {\operatorname{sl}(2,{\mathbb{C}})}$\
& & $[e_2,e_8]=\sqrt{3}e_3,\ [e_3,e_6]=-\sqrt{3}e_2,\ [e_3,e_8]=\sqrt{3}e_4,\ [e_4,e_6]=-\sqrt{3}e_3,$\
& & $[e_4,e_7]=-e_4,\ [e_4,e_8]=\sqrt{2}e_5,\ [e_5,e_6]=-\sqrt{2}e_4,\ [e_5,e_7]=-2e_5,$\
& & $[e_6,e_7]=e_6,\ [e_6,e_8]=e_7,\ [e_7,e_8]=e_8 $\
$(87)(87)(0)$ & ${\mathcal{G}}_{6,1}$& $[e_1,e_5]=e_1,\ [e_1,e_6]=e_2,\ [e_1,e_8]=e_1,\ [e_2,e_5]=e_2,\ [e_2,e_7]=e_1, $ & $[9,0,1,0,0,1]$ & 2 & $4{\mathcal{A}}_1$ & $C $ & ${\mathcal{A}}_{5,7}^{(1,-1,-1)}\triangleleft {\operatorname{sl}(2,{\mathbb{C}})}$\
& & $[e_2,e_8]=-e_2,\ [e_3,e_5]=-e_3,\ [e_3, e_6]=e_4,\ [e_3,e_8]=e_3,$\
& & $[e_4,e_5]=-e_4,\ [e_4,e_7]=e_3,\ [e_4,e_8]=-e_4,\ [e_6,e_7]=e_8,$\
& & $[e_6,e_8]=-2e_6,\ [e_7,e_8]=2e_7$\
[llllllll]{}\
DS, CS, US & Name & Commutation relations & ${\dim_{(\alpha,\beta,\gamma)}}$ & $\tau$ & Casimir operators & T & AN\
$(310) (310) (13)$ & ${\mathcal{G}}'_{20,1}$ & $[e_2,e_3]=e_1$ & $[6,6,3,5,3,4]$ & $1$ & $e_1$ & C & ${\mathcal{A}}_{3,1} $\
$(510) (510) (15)$ & ${\mathcal{G}}'_{19,4}$ & $[e_2,e_4]=e_1,\ [e_3,e_5]=e_1$ & $[15,15,5,14,10,11]$ & $1$ & $e_1$ & C & ${\mathcal{A}}_{5,4} $\
$(520) (520) (25)$ & ${\mathcal{G}}'_{19,3}$ & $[e_3,e_5]=e_1,\ [e_4,e_5]=e_2$ & $[13,13,7,9,7,11]$ & $3$ & $e_1,\ e_2,\ e_2e_3-e_1e_4$ &C & ${\mathcal{A}}_{5,1} $\
$(620) (620) (26)$ & ${\mathcal{G}}'_{18,7}$ & $[e_3,e_5]=e_1,\ [e_4,e_6]=e_1,\ [e_5,e_6]=e_2$ & $[17,18,10,14,10,14]$ & $2$ & $e_1,\ e_2$ & C & ${\mathcal{A}}_{6,4} $\
$(630) (630) (36)$ & ${\mathcal{G}}'_{18,16}$ & $[e_4,e_5]=e_1,\ [e_4,e_6]=e_2,\ [e_5,e_6]=e_3$ & $[18,18,10,9,10,19]$ & $4$ & $e_1,\ e_2,\ e_3,\ e_1e_6-e_2e_5+e_3e_4$ & C & ${\mathcal{A}}_{6,3} $\
$(630) (6310) (136)$ & ${\mathcal{G}}'_{17,13}$ & $[e_2,e_5]=e_1,\ [e_3,e_6]=e_1,\ [e_4,e_5]=e_3,\ [e_4,e_6]=e_2$ & $[11,10,4,6,4,8]$ & $2$ & $e_1,\ e_2e_3-e_1e_4$ & D & ${\mathcal{A}}_{6,14}^{(1)} $\
$(710) (710) (17)$ & ${\mathcal{G}}'_{18,18}$ & $[e_2,e_5]=e_1,\ [e_3,e_6]=e_1,\ [e_4,e_7]=e_1$ & $[28,28,7,27,21,22]$ & $1$ & $e_1$ & C\
$(720) (720) (27)$ & ${\mathcal{G}}'_{17,5}$ & $[e_3,e_6]=e_1,\ [e_4,e_7]=e_2,\ [e_5,e_6]=e_2,\ [e_5,e_7]=e_1$ & $[19,19,11,15,11,15]$ & $3$ & $e_1,\ e_2,\ e_1^2e_4-e_1e_2e_5+e_2^2e_3$ & C\
& ${\mathcal{G}}'_{18,5}$ & $[e_3,e_6]=e_1,\ [e_4,e_7]=e_1,\ [e_5,e_7]=e_2$ & $[21,22,11,18,14,18]$ & $3$ & $e_1,\ e_2,\ e_1e_5-e_2e_4$ & C\
[lllllll]{}\
DS, CS, US & Name & Commutation relations & ${\dim_{(\alpha,\beta,\gamma)}}$ & $\tau$ & Casimir operators & T\
$(730) (730) (37)$ & ${\mathcal{G}}'_{17,16}$ & $[e_4,e_6]=e_1,\ [e_4,e_7]=e_2,\ [e_5,e_6]=e_2,\ [e_5,e_7]=e_3$ & $[19,24,13,15,13,22]$ & $3$ & $e_1,\ e_2,\ e_3$ & C\
& ${\mathcal{G}}'_{16,12}$ & $[e_4,e_6]=e_1,\ [e_5,e_7]=e_2,\ [e_6,e_7]=e_3$ & $[20,24,13,15,13,22]$ & $3$ & $e_1,\ e_2,\ e_3$ & C\
& ${\mathcal{G}}'_{17,12}$ & $[e_4,e_7]=e_1,\ [e_5,e_6]=e_1,\ [e_5,e_7]=e_2,\ [e_6,e_7]=e_3$ & $[22,24,13,15,13,22]$ & $3$ & $e_1,\ e_2,\ e_3$ & C\
& ${\mathcal{G}}'_{18,6}$ & $[e_4,e_7]=e_1,\ [e_5,e_7]=e_2,\ [e_6,e_7]=e_3$ & $[25,25,13,16,13,22]$ & $5$ & $e_1,\ e_2,\ e_3,\ e_1e_5-e_2e_4,\ e_1e_6-e_3e_4 $ & C\
$(730) (7310) (147)$ & ${\mathcal{G}}'_{16,8}$ & $[e_2,e_5]=e_1,\ [e_3,e_6]=e_1,\ [e_4,e_7]=e_1,\ [e_5,e_7]=e_3,$ & $[15,16,7,10,7,11]$ & $1$ & $e_1$ & D\
& & $[e_6,e_7]=e_2$\
$(740) (7410) (147)$ & ${\mathcal{G}}'_{15,6}(a)$ & $[e_2,e_5]=(a+1)e_1,\ [e_3,e_6]=e_1,\ [e_4,e_7]=e_1,$ & $[15,13,7,9,6,11]$ & $1$ & $e_1$ & D\
& & $[e_5,e_6]=-ae_4,\ [e_5,e_7]=e_3,\ [e_6,e_7]=e_2 \quad a\in{\mathbb{C}}_{20}^\ast$\
& & $a=-\frac{1}{2} \cong a=1$ & $[17,13,7,9,6,11]$ & & & C\
$(740) (7410) (247)$ & ${\mathcal{G}}'_{16,7}$ & $[e_3,e_6]=e_1,\ [e_4,e_7]=e_1,\ [e_5,e_6]=e_4,\ [e_5,e_7]=e_3,$ & $[15,17,8,9,7,16]$ & $3$ & $e_1,\ e_2,\ e_1e_5-e_3e_4$ & D\
& & $[e_6,e_7]=e_2$\
$(820) (820) (28)$ & ${\mathcal{G}}_{18,23}$ & $[e_3,e_6]=e_1,\ [e_4,e_7]=e_2,\ [e_5,e_8]=e_1+e_2$ & $[22,25,13,21,15,19]$ & $2$ & $e_1,\ e_2$ & C\
$(830) (830) (38)$ & ${\mathcal{G}}_{15,3}$ & $[e_4,e_6]=e_1,\ [e_4,e_8]=e_2,\ [e_5,e_7]=e_3,\ [e_5,e_8]=e_2,$ & $[19,24,16,15,16,25]$ & $4$ & $e_1,\ e_2,\ e_3,\ e_1(e_1e_5-e_2e_7+e_3e_8)$ & C\
& & $[e_6,e_7]=e_2,\ [e_6,e_8]=e_3,\ [e_7,e_8]=e_1$ & & & $+e_2^2(e_5-e_4)-e_2e_3e_6+e_3^2e_4$\
& ${\mathcal{G}}_{16,2}$ & $[e_4,e_6]=e_1,\ [e_4,e_8]=e_2,\ [e_5,e_7]=e_3,\ [e_5,e_8]=e_2,$ & $[20,24,16,15,16,25]$ & $4$ & $e_1,\ e_2,\ e_3,\ e_1(e_2e_7-e_3e_8) $& C\
& & $[e_6,e_7]=e_2,\ [e_6,e_8]=e_3$ & & & $+e_2^2(e_4-e_5)+e_2e_3e_6-e_3^2e_4$\
& ${\mathcal{G}}_{17,3}$ & $[e_4,e_6]=e_1,\ [e_5,e_7]=e_2,\ [e_5,e_8]=e_3,\ [e_6,e_8]=e_3,$ & $[21,25,16,16,16,25]$ & $4$ & $e_1,\ e_2,\ e_3,$ & C\
& & $[e_7,e_8]=e_1$ & & & $e_1(e_1e_5+e_2e_8-e_3e_7)+e_2e_3e_4$\
& ${\mathcal{G}}_{18,1}$ & $[e_4,e_7]=e_1,\ [e_5,e_7]=e_2,\ [e_5,e_8]=e_2,\ [e_6,e_8]=e_3$ & $[22,28,16,19,16,25]$ & $4$ & $e_1,\ e_2,\ e_3,\ e_1(e_2e_6-e_3e_5)+e_2e_3e_4$ & C\
& ${\mathcal{G}}_{16,6}$ & $[e_4,e_7]=e_1,\ [e_5,e_8]=e_2,\ [e_6,e_7]=e_2,\ [e_6,e_8]=e_1,$ & $[26,28,16,19,16,25]$ & $4$ & $e_1,\ e_2,\ e_3,\ e_1^2e_5-e_1e_2e_6+e_2^2e_4$ & C\
& & $[e_7,e_8]=e_3$\
& ${\mathcal{G}}_{17,4}$ & $[e_4,e_7]=e_1,\ [e_5,e_8]=e_2,\ [e_6,e_8]=e_1,\ [e_7,e_8]=e_3$ & $[27,30,17,21,17,26]$ & $4$ & $e_1,\ e_2,\ e_3,\ e_1e_5-e_2e_6$ & C\
$(840) (840) (48)$ & ${\mathcal{G}}_{15,9}$ & $[e_5,e_7]=e_1,\ [e_5,e_8]=e_2,\ [e_6,e_7]=e_3,\ [e_6,e_8]=e_4$ & $[24,33,17,17,17,33]$ & $4$ & $e_1,\ e_2,\ e_3,\ e_4$ & C\
& ${\mathcal{G}}_{16,9}$ & $[e_5,e_7]=e_1,\ [e_5,e_8]=e_2,\ [e_6,e_7]=e_2,\ [e_6,e_8]=e_3,$ & $[25,33,17,17,17,33]$ & $4$ & $e_1,\ e_2,\ e_3,\ e_4$ & C\
& & $[e_7,e_8]=e_4$\
& ${\mathcal{G}}_{17,15}$ & $[e_5,e_7]=e_1,\ [e_6,e_7]=e_2,\ [e_6,e_8]=e_3,\ [e_7,e_8]=e_4$ & $[27,33,17,17,17,33]$ & $4$ & $e_1,\ e_2,\ e_3,\ e_4$ & C\
$(840) (8410) (148)$ & ${\mathcal{G}}_{15,4}$ & $[e_2,e_6]=e_1,\ [e_3,e_7]=e_1,\ [e_4,e_8]=e_1,\ [e_5,e_7]=e_4,$ & $[18,16,6,10,7,12]$ & $2$ & $e_1,\ e_1e_5-e_3e_4$ & C\
& & $[e_5,e_8]=e_3,\ [e_6,e_8]=e_3,\ [e_7,e_8]=e_2$\
$(850) (8520) (258)$ & ${\mathcal{G}}_{15,5}$ & $[e_3,e_6]=e_1,\ [e_4,e_7]=e_1+e_2,\ [e_5,e_8]=e_2,$ & $[18,19,7,9,6,17]$ & $2$ & $e_1,\ e_2$ & C\
& & $[e_6,e_7]=e_5,\ [e_6,e_8]=e_4,\ [e_7,e_8]=e_3$\
[lllllcc]{}\
DS, CS, US & Name & Commutation relations AN & ${\dim_{(\alpha,\beta,\gamma)}}$ & $\tau$ & Nilradical & T\
$(320) (32) (0)$ & ${\mathcal{G}}'_{19,2}$ & $[e_1,e_3]=e_1,\ [e_2,e_3]=-e_2$ ${\mathcal{A}}_{3,4}$ & $[4,3,1,2,0,1]$ & 1 & $2{\mathcal{A}}_1$ & $D$\
$(530) (532) (12)$ & ${\mathcal{G}}'_{18,9}$ & $[e_2,e_5]=e_1,\ [e_3,e_5]=e_3,\ [e_4,e_5]=-e_4$ ${\mathcal{A}}_{5,8}^{(-1)}$ & $[8,9,3,5,2,6]$ & 3 & $4{\mathcal{A}}_1$ & $D$\
$(540) (54) (0)$ & ${\mathcal{G}}'_{17,11}(a)$ & $[e_1,e_5]=e_2,\ [e_2,e_5]=e_1,\ [e_3,e_5]=e_4,\ [e_4,e_5]=ae_3,\ a\in {\mathbb{C}}_{10}^\ast$ ${\mathcal{A}}_{5,17}^{(\sqrt{a},0,0)}$ & $[8,5,1,4,0,1]$ & 3 & $4{\mathcal{A}}_1$ & $D$\
& & $a=1$ & $[12,5,1,4,0,1]$\
$(640) (64) (0)$ & ${\mathcal{G}}'_{15,7}$ & $[e_1,e_6]=e_2,\ [e_2,e_6]=e_1,\ [e_3,e_5]=e_1,\ [e_3,e_6]=e_4,\ [e_4,e_5]=e_2,\ [e_4,e_6]=e_3$ & $[9,6,2,4,0,2]$ & 2 & ${\mathcal{A}}_{5,1} $ & $D$\
$(6510) (65) (1)$ & ${\mathcal{G}}'_{15,8}$ & $[e_2,e_4]=e_1,\ [e_2,e_6]=e_5,\ [e_3,e_5]=e_1,\ [e_3,e_6]=e_4,\ [e_4,e_6]=e_3,\ [e_5,e_6]=e_2$ & $[10,6,2,1,1,7]$ & 2 & ${\mathcal{A}}_{5,4}$ & $D$\
$(740) (742) (24)$ & ${\mathcal{G}}'_{17,6}$ & $[e_3,e_7]=e_1,\ [e_4,e_7]=e_2,\ [e_5,e_7]=e_6,\ [e_6,e_7]=e_5$ & $[16,19,7,10,6,15]$ & 5 & $6{\mathcal{A}}_1 $ & $D$\
$(750) (754) (1)$ & ${\mathcal{G}}'_{12,1}$ & $[e_2,e_6]=e_3,\ [e_2,e_7]=e_4,\ [e_3,e_6]=e_2,\ [e_3,e_7]=e_5,\ [e_4,e_6]=e_5,\ [e_4,e_7]=e_2,$ & $[10,12,3,6,2,8]$ & 3 & $5{\mathcal{A}}_1$ & $D$\
& & $[e_5,e_6]=e_4,\ [e_5,e_7]=e_3,\ [e_6,e_7]=e_1$\
& ${\mathcal{G}}'_{14,1}$ & $[e_2,e_7]=e_3,\ [e_3,e_7]=e_2,\ [e_4,e_6]=e_2,\ [e_4,e_7]=e_5,\ [e_5,e_6]=e_3,\ [e_5,e_7]=e_4,$ & $[11,12,3,6,2,8]$ & 3 & ${\mathcal{A}}_1 \oplus {\mathcal{A}}_{5,1}$ & $D$\
& & $[e_6,e_7]=e_1$\
$(750) (754) (12)$ & ${\mathcal{G}}'_{16,4}(a)$ & $[e_2,e_7]=ae_3,\ [e_3,e_7]=e_2,\ [e_4,e_7]=e_5,\ [e_5,e_7]=e_4,\ [e_6,e_7]=e_1,\ a\in{\mathbb{C}}_{10}^\ast$ & $[12,13,3,7,2,8]$ & 5 & $6{\mathcal{A}}_1$ & $D$\
& & $a=1$ & $[16,13,3,7,2,8]$\
$(7510) (75) (12)$ & ${\mathcal{G}}'_{14,2}$ & $[e_2,e_4]=e_1,\ [e_2,e_7]=e_5,\ [e_3,e_5]=e_1,\ [e_3,e_7]=e_4,\ [e_4,e_7]=e_3,\ [e_5,e_7]=e_2,$ & $[12,12,3,3,2,8]$ & $1$ & ${\mathcal{A}}_1 \oplus {\mathcal{A}}_{5,4}$ & $D$\
& & $[e_6,e_7]=e_1$\
$(760) (76) (0)$ & ${\mathcal{G}}'_{15,2}(a,b)$ & $[e_1,e_7]=4ae_4,\ [e_2,e_7]=be_5,\ [e_3,e_7]=e_6,\ [e_4,e_7]=e_1,\ [e_5,e_7]=e_2,$ & $[12,7,1,6,0,1]$ & $5$ & $6{\mathcal{A}}_1$ & $D$\
& & $[e_6,e_7]=e_3,\ a,b\neq 0$\
& & $a=\frac{1}{4}\ {\operatorname{xor}}\ b=1 \ {\operatorname{xor}}\ b=4a$ & $[16,7,1,6,0,1]$\
& & $a=\frac{1}{4} \wedge b=1$ & $[24,7,1,6,0,1]$\
& & $a=b=1$ & & & & $C$\
$(840) (842) (25)$ & ${\mathcal{G}}_{17,1}$ & $[e_3,e_5]=e_1,\ [e_3,e_8]=-e_1,\ [e_4,e_5]=e_2,\ [e_4,e_8]=e_2,\ [e_6,e_8]=e_7,\ [e_7,e_8]=e_6$ & $[16,21,9,12,8,17]$ & $4$ & $2{\mathcal{A}}_1 \oplus {\mathcal{A}}_{5,1}$ & $D$\
$(850) (854) (12)$ & ${\mathcal{G}}_{16,1}(a)$ & $[e_2,e_8]=ae_3,\ [e_3,e_8]=e_2,\ [e_4,e_7]=e_1,\ [e_4,e_8]=e_1,\ [e_5,e_7]=e_6,$ & $[13,14,4,8,3,9]$ & $4$ & $6{\mathcal{A}}_1$ & $D $\
& & $[e_6,e_7]=e_5,\ a\in {\mathbb{C}}_{10}$\
$(860) (86) (0)$ & ${\mathcal{G}}_{15,1}(a,b)$ & $[e_1,e_8]=ae_2,\ [e_2,e_8]=e_1,\ [e_3,e_7]=be_4,\ [e_4,e_7]=e_3,\ [e_5,e_7]=e_6,\ [e_5,e_8]=e_6,$ & $[12,7,1,6,0,1]$ & $4$ & $6{\mathcal{A}}_1$ & $D $\
& & $[e_6,e_7]=e_5,\ [e_6,e_8]=e_5,\ a,b \neq 0$\
& & $a=b=1$ & & & & $C$\
$(8620) (86) (0)$ & ${\mathcal{G}}_{11,2}$ & $[e_1,e_7]=e_1,\ [e_1,e_8]=e_1,\ [e_2,e_3]=e_1,\ [e_2,e_7]=e_2,\ [e_3,e_8]=e_3,\ [e_4,e_7]=-e_4,$ & $[10,2,1,2,0,1]$ & $2$ & ${\mathcal{A}}_{3,1} \oplus {\mathcal{A}}_{3,1}$ & $C$\
& & $[e_4,e_8]=-e_4,\ [e_5,e_6]=e_4,\ [e_5,e_7]=-e_5,\ [e_6,e_8]=-e_6 $\
& ${\mathcal{G}}_{11,1}$ & $[e_1,e_8]=e_2,\ [e_2,e_8]=e_1,\ [e_3,e_6]=e_1,\ [e_3,e_8]=e_5,\ [e_4,e_7]=e_2,\ [e_4,e_8]=e_5,$ & $[11,7,2,2,0,2]$ & $2$ & ${\mathcal{G}}'_{17,5}$ & $C$\
& & $[e_5,e_6]=e_2,\ [e_5,e_7]=e_1,\ [e_5,e_8]=2e_3+2e_4,\ [e_6,e_8]=-e_7,\ [e_7,e_8]=-e_6 $\
$(8730) (87) (1)$ & ${\mathcal{G}}_{11,3}$ & $[e_2,e_8]=2e_3,\ [e_3,e_8]=2e_2,\ [e_4,e_6]=e_1+e_3,\ [e_4,e_7]=e_2,\ [e_4,e_8]=e_5,$ & $[12,10,2,3,1,9]$ & $2$ & ${\mathcal{G}}'_{17,16} $ & $C$\
& & $[e_5,e_6]=e_2,\ [e_5,e_7]=e_3-e_1,\ [e_5,e_8]=e_4,\ [e_6,e_8]=e_7,\ [e_7,e_8]=e_6$\
Appendix A.3: Invariant Functions of one–parametric graded contractions {#appendix-a.3-invariant-functions-of-oneparametric-graded-contractions .unnumbered}
========================================================================
The blank space in the table stands for general complex number different from all previously listed in given table.
- ${\mathcal{G}}'_{17,11}(a),\ a
\neq 0$, ${\mathcal{G}}'_{17,11}(a) \cong
{\mathcal{G}}'_{17,11}\left(\frac{1}{a}\right)$ $\longrightarrow$ $a\in{\mathbb{C}}^\ast_{10}$
[|l||c|c|c|]{}\
$\alpha$ & -1 & 1 &\
$\psi(\alpha)$ & 13 & 12 & 5\
[|l||c|c|c|c|c|]{}\
$\alpha$ & $-i$ & $i$ & -1 & 1 &\
$\psi(\alpha)$ & 9 & 9 & 9 & 8 & 5\
[|l||c|c|c|c|c|c|c|]{}\
$\alpha$ & -1 & 1 & $\frac{1}{\sqrt{a}}$ & $-\frac{1}{\sqrt{a}}$ & $\sqrt{a}$ & $-\sqrt{a}$ &\
$\psi(\alpha)$ & 9 & 8 & 7 & 7 & 7 & 7 & 5\
- ${\mathcal{G}}'_{16,4}(a),\ a
\neq 0$, ${\mathcal{G}}'_{16,4}(a) \cong
{\mathcal{G}}'_{16,4}\left(\frac{1}{a}\right)$ $\longrightarrow$ $a\in{\mathbb{C}}^\ast_{10}$
[|l||c|c|c|c|]{}\
$\alpha$ & $-1$ & 1 & 0 &\
$\psi(\alpha)$ & 17 & 16 & 13 & 9\
[|l||c|c|c|c|c|c|]{}\
$\alpha$ & 0 & $-i$ & $i$ & $-1$ & 1 &\
$\psi(\alpha)$ & 13 & 13 & 13 & 13 & 12 & 9\
[|l||c|c|c|c|c|c|c|c|]{}\
$\alpha$ & 0 & $-1$ & 1 & $\frac{1}{\sqrt{a}}$ & $-\frac{1}{\sqrt{a}}$ & $\sqrt{a}$ & $-\sqrt{a}$ &\
$\psi(\alpha)$ & 13 & 13 & 12 & 11 & 11 & 11 & 11 & 9\
- ${\mathcal{G}}_{16,1}(a),\ a \neq
0$, ${\mathcal{G}}'_{16,1}(a)\cong {\mathcal{G}}'_{16,1}\left(\frac{1}{a}\right)$ $\longrightarrow$ $a\in{\mathbb{C}}_{10}$
- ${\mathcal{G}}'_{15,6}(a),\ a
\neq 0,-1$, $\longrightarrow$ $a\in{\mathbb{C}}^\ast_{20}$ $${\mathcal{G}}'_{15,6}(a) \cong {\mathcal{G}}'_{15,6}\left(\frac{1}{a}\right) \cong
{\mathcal{G}}'_{15,6}(-a-1)\cong {\mathcal{G}}'_{15,6}\left(\frac{-1}{a+1}\right) \cong
{\mathcal{G}}'_{15,6}\left(\frac{-a}{a+1}\right) \cong
{\mathcal{G}}'_{15,6}\left(\frac{a+1}{-a}\right)$$
[|l||c|c|c|c|]{}\
$\alpha$ & 1 & $-\frac{1}{2}$ & $-2$ &\
$\psi(\alpha)$ & 17 & 15 & 15 & 13\
[|l||c|c|c|c|]{}\
$\alpha$ & $-\frac{1 + \sqrt{3}i}{2}$ & $-\frac{1 - \sqrt{3}i}{2}$ & 1 &\
$\psi(\alpha)$ & 16 & 16 & 15 & 13\
[|l||c|c|c|c|c|c|c|c|]{}\
$\alpha$ & 1 & $a$ & $\frac{1}{a}$ & $-1-a$ & $-\frac{1}{1+a}$ & $-\frac{a}{1+a}$ & $-\frac{1+a}{a}$ &\
$\psi(\alpha)$ & 15 & 14 & 14 & 14 & 14 & 14 & 14 & 13\
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Spectroscopic analyses of hydrogen-rich WN5–6 stars within the young star clusters NGC 3603 and R136 are presented, using archival Hubble Space Telescope and Very Large Telescope spectroscopy, and high spatial resolution near-IR photometry, including Multi-Conjugate Adaptive Optics Demonstrator (MAD) imaging of R136. We derive high stellar temperatures for the WN stars in NGC 3603 ($T_{\ast} \sim$ 42$\pm$2 kK) and R136 ($T_{\ast} \sim$ 53$\pm$ 3 kK) plus clumping-corrected mass-loss rates of 2 – 5 $\times 10^{-5}$ M$_{\odot}$yr$^{-1}$ which closely agree with theoretical predictions from Vink et al. These stars make a disproportionate contribution to the global ionizing and mechanical wind power budget of their host clusters. Indeed, R136a1 alone supplies $\sim$7% of the ionizing flux of the entire 30 Doradus region. Comparisons with stellar models calculated for the main-sequence evolution of 85 – 500 M$_{\odot}$ accounting for rotation suggest ages of $\sim$1.5 Myr and initial masses in the range 105 – 170 M$_{\odot}$ for three systems in NGC 3603, plus 165 – 320 M$_{\odot}$ for four stars in R136. Our high stellar masses are supported by consistent spectroscopic and dynamical mass determinations for the components of NGC 3603 A1. We consider the predicted X-ray luminosity of the R136 stars if they were close, colliding wind binaries. R136c is consistent with a colliding wind binary system. However, short period, colliding wind systems are excluded for R136a WN stars if mass ratios are of order unity. Widely separated systems would have been expected to harden owing to early dynamical encounters with other massive stars within such a high density environment. From simulated star clusters, whose constituents are randomly sampled from the Kroupa initial mass function, both NGC 3603 and R136 are consistent with an tentative upper mass limit of $\sim$300 M$_{\odot}$. The Arches cluster is either too old to be used to diagnose the upper mass limit, exhibits a deficiency of very massive stars, or more likely stellar masses have been underestimated – initial masses for the most luminous stars in the Arches cluster approach 200 M$_{\odot}$ according to contemporary stellar and photometric results. The potential for stars greatly exceeding 150 M$_{\odot}$ within metal-poor galaxies suggests that such pair-instability supernovae could occur within the local universe, as has been claimed for SN 2007bi.'
author:
- |
Paul A. Crowther$^{1}$[^1], Olivier Schnurr$^{1,2}$, Raphael Hirschi$^{3, 4}$, Norhasliza Yusof$^{5}$, Richard J. Parker$^{1}$, Simon P. Goodwin$^{1}$, Hasan Abu Kassim$^{5}$\
$^{1}$Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, UK\
$^{2}$ Astrophysikalisches Institut Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany\
$^{3}$ Astrophysics Group, EPSAM, University of Keele, Lennard-Jones Labs, Keele, ST5 5BG, UK\
$^{4}$ Institute for the Physics and Mathematics of the Universe, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan\
$^{5}$ Department of Physics, University of Malaya, 50603 Kuala Kumpur, Malaysia
title: 'The R136 star cluster hosts several stars whose individual masses greatly exceed the accepted 150 M$_{\odot}$ stellar mass limit'
---
\[firstpage\]
binaries: general – stars: early-type – stars: fundamental parameters – stars: Wolf-Rayet – galaxies: star clusters: individual (R136, NGC 3603, Arches)
Introduction
============
Although the formation of very massive stars remains an unsolved problem of astrophysics (Zinnecker & Yorke 2007), the past decade has seen a shift from a belief that there is no observational upper stellar mass cutoff (e.g. Massey 2003) to the widespread acceptance of a limit close to 150 M$_{\odot}$ (Figer 2005; Koen 2006). If stars above this limit were to exist, they would be exclusive to the youngest, highest mass star clusters, which are very compact (Figer 2005). However, spatially resolved imaging of such clusters is currently exclusive to the Milky Way and its satellite galaxies, where they are very rare. In addition, the accurate determination of stellar masses generally relies upon spectroscopic and evolutionary models, unless the star is a member of an eclipsing binary system (Moffat 2008), something which occurs extremely rarely.
Most, if not all, stars form in groups or clusters (Lada & Lada 2003). An average star forms with an initial mass of $\sim$0.5 M$_{\odot}$ while the relative proportion of stars of higher and lower mass obeys an apparently universal initial mass function (IMF, Kroupa 2002). In addition, there appears to be a relationship between the mass of a cluster and its highest-mass star (Weidner & Kroupa 2006, Weidner et al. 2010). High mass stars ($>$8 M$_{\odot}$), ultimately leading to core-collapse supernovae, usually form in clusters exceeding 100 M $_{\odot}$ while stars approaching 150 M$_{\odot}$ have been detected in two 10$^{4}$ M$_{\odot}$ Milky Way clusters, namely the Arches (Figer 2005; Martins et al. 2008) and NGC 3603 (Schnurr et al. 2008a). Is this 150 M$_{\odot}$ limit statistical or physical?
The determination of stellar masses of very massive stars from colour-magnitude diagrams is highly unreliable, plus corrections to present mass estimates need to be applied to estimate initial masses. Studies are further hindered by severe spatial crowding within the cores of these star clusters (Maíz Apellániz 2008). In addition, sophisticated techniques are required to extract the physical parameters of early-type stars possessing powerful stellar winds from optical/infrared spectroscopy (Conti et al. 2008). In general, the incorporation of line blanketing has led to a reduction in derived temperatures for O stars from photospheric lines (Puls et al. 2008) while the reverse is true for emission lines in Wolf-Rayet stars (Crowther 2007).
Here we re-analyse the brightest members of the star cluster (HD 97950) responsible for NGC 3603, motivated in part by the case of A1 which is an eclipsing binary system, whose individual components have been measured by Schnurr et al. (2008a). We also re-analyse the brightest members of R136 (HD 38268) - the central ionizing cluster of the Tarantula nebula (30 Doradus) within the Large Magellanic Cloud (LMC). R136 is sufficiently young (Massey & Hunter 1998) and massive ($\leq$5.5 $\times$ 10$^{4}$ M$_{\odot}$, Hunter et al. 1995) within the Local Group of galaxies to investigate the possibility of a physical limit beyond 150 M$_{\odot}$ (Massey & Hunter 1998, Selman et al. 1999; Oey & Clarke 2005).
As recently as the 1980s, component ‘a’ within R136 was believed to be a single star with a mass of several thousand solar masses (Cassinelli et al. 1981; Savage et al. 1983), although others favoured a dense star cluster (Moffat & Seggewiss 1983). The latter scenario was supported by speckle interferometric observations (Weigelt & Bauer 1985) and subsequently confirmed by Hubble Space Telescope (HST) imaging (Hunter et al. 1995).
Star Instrument Grating Date Proposal/PI
------------ ------------- --------- ------------ --------------------
R136a1 HST/HRS G140L 1994 Jul 5157/Ebbets
R136a3 HST/HRS G140L 1994 Jul 5157/Ebbets
R136a1 HST/FOS G400H, 1996 Jan 6018/Heap
G570H
R136a2 HST/FOS G400H, 1996 Jan 6018/Heap
G570H
R136a3 HST/FOS G400H, 1996 Jan 6018/Heap
G570H
R136c HST/FOS G400H 1996 Nov 6417/Massey
R136a1+a2 VLT/SINFONI K 2005 Nov-, 076.D-0563/Schnurr
2005 Dec
R136a3 VLT/SINFONI K 2005 Nov- 076.D-0563/Schnurr
2005 Dec
R136c VLT/SINFONI K 2005 Nov- 076.D-0563/Schnurr
2005 Dec
NGC 3603A1 HST/FOS G400H 1994 Sep 5445/Drissen
NGC 3603B HST/FOS G400H 1994 Sep 5445/Drissen
NGC 3603C HST/FOS G400H 1994 Sep 5445/Drissen
NGC 3603A1 VLT/SINFONI K 2005 Apr- 075.D-0577/Moffat
2006 Feb
NGC 3603B VLT/SINFONI K 2005 Apr- 075.D-0577/Moffat
2006 Feb
NGC 3603C VLT/SINFONI K 2005 Apr- 075.D-0577/Moffat
2006 Feb
: Log of spectroscopic observations of R136 and NGC 3603 stars used in this study[]{data-label="log"}
Here, we present new analyses of the brightest sources of NGC 3603 and R136. In contrast to classical Wolf-Rayet stars, these WN stars are believed to be young, main-sequence stars, albeit possessing very strong stellar winds as a result of their high stellar luminosities (Crowther 2007). Section 2 describes archival HST and Very Large Telescope (VLT) observations, while the spectroscopic analysis is presented in Sect 3. Comparisons with contemporary evolutionary models are presented in Sect. 4, revealing spectroscopic masses in excellent agreement with dynamical masses of the components of NGC 3603 A1. For the R136 stars, exceptionally high initial masses of 165 – 320 M$_{\odot}$ are inferred. We consider the possibility that these stars are binaries in Sect 5, from archival X-ray observations of R136. In Sect. 6 we simulate star clusters to re-evaluate the stellar upper mass limit, and find that both R136 and NGC 3603 are consistent with a tentative upper limit of $\sim$ 300M$_{\odot}$. Sect. 7 considers the contribution of the R136 stars to the global properties of both this cluster and 30 Doradus. Finally, we consider the broader significance of an increased mass limit for stars in Sect. 8.
![ VLT MAD K$_{s}$-band 12 $\times$ 12 arcsec (3 $\times$ 3 parsec for the LMC distance of 49 kpc) image of R136 (Campbell et al. (2010) together with a view of the central 4 $\times$ 4 arcsec (1 $\times$ 1 parsec) in which the very massive WN5h stars discussed in this letter are labelled (component b is a lower mass WN9h star). Relative photometry agrees closely with integral field SINFONI observations (Schnurr et al. 2009).[]{data-label="MAD_K1"}](MAD_K1.eps){width="1.\columnwidth"}
------ ------------ ----------------- ----------------- ------------------ --------------
Name Sp Type m$_{K_{s}}$ A$_{K_{s}}$ M$_{K_{s}}^{a}$ Binary$^{b}$
(m$_{V}$) (A$_{V}$) (M$_{V}$)
A1 WN6h 7.42 $\pm$ 0.05 0.59 $\pm$ 0.03 –7.57 $\pm$ 0.12 Yes
(11.18) (4.91$\pm$0.25) (–8.13$\pm$0.27)
B WN6h 7.42 $\pm$ 0.05 0.56 $\pm$ 0.03 –7.54 $\pm$ 0.12 No?
(11.33) (4.70$\pm$0.25) (–7.77$\pm$0.27)
C WN6h$^{c}$ 8.28 $\pm$ 0.05 0.56 $\pm$ 0.03 –6.68 $\pm$ 0.12 Yes
(11.89) (4.66$\pm$0.25) (–7.17$\pm$0.27)
------ ------------ ----------------- ----------------- ------------------ --------------
: Stars brighter than M$_{\rm K_{s}} \sim$ –6 mag within 13 arcsec (0.5 parsec) of NGC 3603 A1, together with V-band photometry from Melena et al. (2008) in parenthesis.[]{data-label="ngc3603_photom"}
\(a) For a distance of 7.6 $\pm$ 0.35 kpc (distance modulus 14.4 $\pm$ 0.1 mag)\
(b) A1 is a 3.77 day double-eclipsing system, while C is a 8.9 day SB1 (Schnurr et al. 2008a)\
(c) An updated classification scheme for Of, Of/WN and WN stars (N.R. Walborn, & P.A. Crowther, in preparation) favours O3If\*/WN6 for NGC 3603 C
Observations
============
Our analysis of WN stars in NGC 3603 and R136 is based upon archival UV/optical HST and near-IR VLT spectroscopy, summarised in Table \[log\], combined with archival high spatial resolution near-IR imaging. We prefer the latter to optical imaging due to reduced extinction, efficient correction for severe spatial crowding (Schnurr et al. 2008a, 2009) and consistency with studies of other young, high mass clusters (e.g. Arches, Martins et al. 2008).
NGC 3603
--------
Our spectroscopic analysis of the three WN6h systems within NGC 3603 is based upon archival HST/FOS spectroscopy from Drissen et al. (1995) plus integral field VLT/SINFONI near-IR spectroscopy from Schnurr et al. (2008a) – see Table \[log\] for the log of observations. The spectral resolution of the SINFONI datasets is R$\sim$3000, with adaptive optics (AO) used to observe A1, B and C, versus R$\sim$1300 for FOS for which a circular aperture of diameter 0.26 arcsec was used.
Differential photometry from VLT/SINFONI integral field observations (Schnurr et al 2008a), were tied to unpublished VLT/ISAAC K$_{s}$-band acquisition images from 16 Jun 2002 (ESO Programme 69.D-0284(A), P.I. Crowther) using the relatively isolated star NGC 3603 C. The VLT/ISAAC frames are calibrated against 2MASS photometry (Skrutskie et al. 2006) using 6 stars in common ($\pm$0.05 mag). These are presented in Table \[ngc3603\_photom\] together with absolute magnitudes resulting from a distance of 7.6$\pm$0.35 kpc (distance modulus 14.4$\pm$0.1 mag) plus an extinction law from Melena et al. (2008) using $$A_{\rm V} = 1.1 R_{\rm V}^{\rm MW} + (0.29 - 0.35) R_{\rm V}^{\rm
NGC3603}$$ where R$_{\rm V}^{\rm MW}$ = 3.1 for the Milky Way foreground component and R$_{\rm V}^{\rm NGC3603}$ = 4.3 for the internal NGC 3603 component. We obtain A$_{K_{s}}$ = 0.12 A$_{V} \sim$ 0.56 – 0.59 mag from spectral energy distribution fits, which are consistent with recent determinations of E(B-V) = 1.39 mag from Melena et al. (2008). In their analysis, Crowther & Dessart (1998) used a lower overall extinction of E(B-V) = 1.23 mag, albeit a higher distance of 10 kpc (distance modulus of 15.0 mag) to NGC 3603.
{width="1.5\columnwidth"}
R136
----
Our UV/optical/infrared spectroscopic analysis of all four hydrogen-rich WN5h Wolf-Rayet (WR) stars in R136 (Crowther & Dessart 1998) is also based upon archival HST and VLT spectroscopy – see Table \[log\] for the log of observations.
Goddard High Resolution Spectrograph (GHRS) ultraviolet observations of R136a stars have been described by de Koter et al. (1997), achieving a spectral resolution of R$\sim$2000–3000 using the SSA aperture of 0.22$\times$0.22 arcsec. This prevented individual UV spectroscopy for R136a1 and a2 which are separated by $\sim$0.1 arcsec. For context, 0.1 arcsec subtends 5000 AU at the distance of the LMC. R136a2 UV spectroscopy was attempted by S. Heap (Programme 5297), but was assessed to be unreliable by de Koter et al. (1997) and so is also excluded here. Note also that R136c has not been observed with GHRS. Visual Faint Object Spectroscopy (FOS) datasets, with a circular aperture of diameter 0.26 arcsec achieved R$\sim$1300, although R136a1 and R136a2 once again suffer significant contamination from one another. It is solely at near-IR wavelengths (SINFONI) that R136a1 and a2 are spectrally separated, for which R136b served as an AO reference star (Schnurr et al. 2009). The spectral resolution of the SINFONI datasets is R$\sim$3000.
We employ high spatial resolution K$_{s}$-band photometry of R136. Differential K$_{s}$ photometry from AO assisted VLT/SINFONI integral field datasets are tied to identical spatial resolution wider field VLT Multi-Conjugate Adaptive Optics Demonstrator (MAD) imaging (Campbell et al. 2010) using the relatively isolated star R136b (WN9h). Three overlapping Fields were observed with VLT/MAD, for which Field 1 provided the highest quality in R136 (FWHM $\sim$ 0.1 arcsec), as shown in Fig. \[MAD\_K1\], itself calibrated using archival HAWK-I and 2MASS datasets (see Campbell et al. 2010). For 12 stars in common between MAD photometry and HST Near Infrared Camera and Multi Object Spectrometer (NICMOS) F205W imaging (Brandner et al. 2001) transformed into the CTIO K-band system, Campbell et al. (2010) find m$_{\rm K_{s}}$ (MAD) - m$_{\rm K}$ (HST) = – 0.04 $\pm$0.05 mag.
In Table \[r136\_photom\] we present K$_{s}$ aperture photometry and inferred absolute magnitudes of stars brighter than $M_{\rm K_{s}} \sim$ –6 mag within 20 arcsec (5 parsec) of R136a1. Of these only R136a, b and c components lie within a projected distance of 1 pc from R136a1. Spectral types are taken from Crowther & Dessart (1998) or Walborn & Blades (1997), although other authors prefer alternative nomenclature, e.g. O2If for both Mk39 and R136a5 according to Massey et al. (2004, 2005).
Interstellar extinctions for the R136 WN5h stars are derived from UV to near-IR spectral energy distribution fits (R136c lacks UV spectroscopy), adopting foreground Milky Way (LMC) extinctions of A$_{\rm K_{s}}$ = 0.025 (0.06) mag, plus variable internal 30 Doradus nebular extinction. The adopted extinction law follows Fitzpatrick & Savage (1984) as follows $$A_{\rm V} = 0.07 R_{\rm V}^{\rm MW} + 0.16 R_{\rm V}^{\rm LMC}
+ (0.25-0.45) R_{\rm V}^{\rm 30Dor}$$ where R$_{\rm V}^{\rm MW}$ = R$_{\rm V}^{\rm LMC}$ = 3.2 and R$_{\rm V}^{\rm 30Dor}$ = 4.0. We derive A$_{\rm K_{s}}
\sim$0.22 mag for the R136a stars and 0.30 mag for R136c and adopt A$_{\rm
K_{s}} =$ 0.20 mag for other stars except that Mk34 (WN5h) mirrors the higher extinction of R136c. From Table \[r136\_photom\], R136c is 0.5 mag fainter than R136a2 in the V-band (Hunter et al. 1995) but is 0.06 mag brighter in the K$_{s}$-band, justifying the higher extinction. An analysis based solely upon optical photometry could potentially underestimate the bolometric magnitude for R136c with respect to R136a2. The main source of uncertainty in absolute magnitude results from the distance to the LMC. The mean of 7 independent techniques (Gibson 2000) suggests an LMC distance modulus of 18.45 $\pm$ 0.06 mag. Here we adopt an uncertainty of $\pm$0.18 mag owing to systematic inconsistencies between the various methods.
-------- ----------------- ------------------ ----------------- ------------------ --------------
Name Sp Type m$_{K_{s}}$ A$_{K_{s}}$ M$_{K_{s}}^{a}$ Binary$^{b}$
(m$_{V}$) (A$_{V}$) (M$_{V}$)
R134 WN6(h) 10.91 $\pm$ 0.09 0.21 $\pm$ 0.04 –7.75 $\pm$ 0.20 No?
(12.89$\pm$0.08) (1.77$\pm$0.33) (–7.33$\pm$0.38)
R136a1 WN5h 11.10 $\pm$ 0.08 0.22 $\pm$ 0.02 –7.57 $\pm$ 0.20 No?
(12.84$\pm$0.05) (1.80$\pm$0.17) (–7.41$\pm$0.25)
R136c WN5h 11.34 $\pm$ 0.08 0.30 $\pm$ 0.02 –7.41 $\pm$ 0.20 Yes?
(13.47$\pm$0.08) (2.48$\pm$0.17) (–7.46$\pm$0.26)
R136a2 WN5h 11.40 $\pm$ 0.08 0.23 $\pm$ 0.02 –7.28 $\pm$ 0.20 No?
(12.96$\pm$0.05) (1.92$\pm$0.17) (–7.41$\pm$0.25)
Mk34 WN5h 11.68 $\pm$ 0.08 0.27 $\pm$ 0.04 –7.04 $\pm$ 0.20 Yes
(13.30$\pm$0.06) (2.22$\pm$0.33) (–7.37$\pm$0.38)
R136a3 WN5h 11.73 $\pm$ 0.08 0.21 $\pm$ 0.02 –6.93 $\pm$ 0.20 No?
(13.01$\pm$0.04) (1.72$\pm$0.17) (–7.16$\pm$0.25)
R136b WN9ha 11.88 $\pm$ 0.08 0.21 $\pm$ 0.04 –6.78 $\pm$ 0.20 No?
(13.32$\pm$0.04) (1.74$\pm$0.33) (–6.87$\pm$0.38)
Mk39 O2–3If/WN$^{c}$ 12.08 $\pm$ 0.08 0.18 $\pm$ 0.04 –6.55 $\pm$ 0.20 Yes
(13.01$\pm$0.08) (1.46$\pm$0.33) (–6.90$\pm$0.38)
Mk42 O2–3If/WN$^{c}$ 12.19 $\pm$ 0.08 0.17 $\pm$ 0.04 –6.43 $\pm$ 0.20 No?
(12.84$\pm$0.05) (1.38$\pm$0.33) (–6.99$\pm$0.38)
Mk37a O4If+$^{c}$ 12.39 $\pm$ 0.11 0.21 $\pm$ 0.04 –6.27 $\pm$ 0.21 No?
(13.57$\pm$0.05) (1.74$\pm$0.33) (–6.62$\pm$0.38)
Mk37Wa O4If+ 12.39 $\pm$ 0.11 0.19 $\pm$ 0.04 –6.25 $\pm$ 0.21 ?
(13.49$\pm$0.05) (1.62$\pm$0.33) (–6.58$\pm$0.38)
R136a5 O2–3If/WN$^{c}$ 12.66 $\pm$ 0.08 0.21 $\pm$ 0.04 –6.00 $\pm$ 0.20 No?
(13.93$\pm$0.04) (1.74$\pm$0.33) (–6.26$\pm$0.38)
-------- ----------------- ------------------ ----------------- ------------------ --------------
: Stars brighter than M$_{K_{s}} \sim$ –6 mag within 20 arcsec (5 parsec) of R136a1, together with V-band photometry from Hunter et al. (1995) in parenthesis. []{data-label="r136_photom"}
\(a) For a distance modulus of 18.45 $\pm$ 0.18 mag (49 $\pm$ 4 kpc)\
(b) R136c is variable and X-ray bright, Mk39 is a 92-day SB1 binary (Massey et al. 2002; Schnurr et al. 2008b) and Mk34 is a binary according to unpublished Gemini observations (O. Schnurr et al. in preparation)\
(c) An updated classification scheme for Of, Of/WN and WN stars (N.R. Walborn, & P.A. Crowther, in preparation) favours O2If$^{\ast}$ for Mk42 and R136a5, O3.5If for Mk37a and O2If/WN for Mk39
Spectroscopic Analysis
======================
For our spectroscopic study we employ the non-LTE atmosphere code CMFGEN (Hillier & Miller 1998) which solves the radiative transfer equation in the co-moving frame, under the additional constraints of statistical and radiative equilibrium. Since CMFGEN does not solve the momentum equation, a density or velocity structure is required. For the supersonic part, the velocity is parameterized with an exponent of $\beta$ = 0.8. This is connected to a hydrostatic density structure at depth, such that the velocity and velocity gradient match at this interface. The subsonic velocity structure is defined by a fully line-blanketed, plane-parallel TLUSTY model (Lanz & Hubeny 2003) whose gravity is closest to that obtained from stellar masses derived using evolutionary models, namely $\log g = 4.0$ for R136 stars and $\log g = 3.75$ for NGC 3603 stars. CMFGEN incorporates line blanketing through a super-level approximation, in which atomic levels of similar energies are grouped into a single super-level which is used to compute the atmospheric structure.
Stellar temperatures, T$_{\ast}$, correspond to a Rosseland optical depth 10, which is typically 1,000 K to 2,000 K higher than effective temperatures T$_{2/3}$ relating to optical depths of 2/3 in such stars.
Our model atom include the following ions: H I, He I-II, C III-IV, N III-V, O III-VI, Ne IV-V, Si IV, P IV-V, S IV-V, Ar V-VII, Fe IV-VII, Ni V-VII, totalling 1,141 super-levels (29,032 lines). Other than H, He, CNO elements, we adopt solar abundances (Asplund et al. 2009) for NGC 3603, which are supported by nebular studies (Esteban et al. 2005, Lebouteiller et al. 2008). LMC nebular abundances (Russell & Dopita 1990) are adopted for R136, with other metals scaled to 0.4 Z$_{\odot}$, also supported by nebular studies of 30 Doradus (e.g. Peimbert 2003, Lebouteiller et al. 2008). We have assumed a depth-independent Doppler profile for all lines when solving for the atmospheric structure in the co-moving frame, while in the observer’s frame, we have adopted a uniform turbulence of 50 km s$^{-1}$. Incoherent electron scattering and Stark broadening for hydrogen and helium lines are adopted. With regard to wind clumping (Hillier 1991), this is incorporated using a radially-dependent volume filling factor, $f$, with $f_{\infty}$ = 0.1 at $v_{\infty}$, resulting in a reduction in mass-loss rate by a factor of $\sqrt{(1/f)}$ $\sim$ 3.
![Rectified, spatially resolved near-IR (VLT/SINFONI) spectroscopy of NGC 3603 WN6h stars (Schnurr et al. 2008a, black), including A1 close to quadrature (A1a blueshifted by 330 km s$^{-1}$ and A1b redshifted by 433 km s$^{-1}$), together with synthetic spectra (red, broadened by 50 kms$^{-1}$). N[iii]{} 2.103 – 2.115$\mu$m and He[i]{} 2.112 – 2.113$\mu$m contribute to the emission feature blueward of Br $\gamma$[]{data-label="ngc3603_k"}](ngc3603.ps){width="0.6\columnwidth"}
NGC 3603
--------
Fig. \[ngc3603\_sed\] compares spectral energy distributions for the WN6h stars in NGC 3603 with reddened, theoretical models, including the individual components of A1. For this system we obtain M$_{\rm K_{s}}$ = –7.57 mag, such that we adopt $\Delta m$ = m$_{\rm A1a}$ – m$_{\rm A1b}$ = –0.43 $\pm$0.3 mag for the individual components (Schnurr et al. 2008a). We have estimated individual luminosities in two ways. First, on the basis of similar mean molecular weights $\mu$ for individual components, stellar luminosities result from L $\propto$ M$^{\alpha}$, with $\alpha \sim 1.5$ for ZAMS models in excess of 85 $M_{\odot}$. We derive L(A1a)/L(A1b) = 1.62$^{+0.6}_{-0.4}$ from the dynamical masses of 116 $\pm$ 31 M$_{\odot}$ and 89 $\pm$ 16 M$_{\odot}$ obtained by Schnurr et al. (2008a) for A1a and A1b, respectively. Alternatively, we have obtained a luminosity ratio for components within A1 from a fit to the NICMOS lightcurve (Moffat et al. 2004, and E. Antokhina, priv. comm), using the radii of the Roche lobes and lightcurve derived temperatures. We also obtain L(A1a)/L(A1b) = 1.62 from this approach.
Although we have matched synthetic spectra to near-IR photometry, we note that no major differences would be obtained with either HST Wide Field and Planetary Camera 2 (WFPC2) or Advanced Camera for Surveys (ACS) datasets. For example, our reddened spectral energy distribution for NGC 3603B implies B = 12.40 mag, which matches HST/ACS photometry to within 0.06 mag (Melena et al. 2008).
We have used diagnostic optical and near-IR lines, including N[iii]{} 4634-41, 2.103–2.115$\mu$m, N[iv]{} 3478-83, 4058, together with He[ii]{} 4686, 2.189$\mu$m plus Br$\gamma$. We are unable to employ helium temperature diagnostics since He[i]{} lines are extremely weak at optical and near-IR wavelengths. Spectroscopic comparisons with HST/FOS datasets are presented in Fig. \[ngc3603\_sed\]. Overall, emission features are well reproduced, although absorption components of higher Balmer-Pickering lines are too strong, especially for NGC 3603C. Fits are similar to Crowther & Dessart (1998), in spite of an improved TLUSTY structure within the photosphere. Fortunately, our spectral diagnostics are not especially sensitive to the details of the photosphere since they assess the inner wind conditions. Spectroscopically, we use VLT/SINFONI spectroscopy of A1a and A1b obtained close to quadrature (Schnurr et al. 2008a) to derive mass-loss rates and hydrogen contents. The ratio of He[ii]{} 2.189$\mu$m to Br$\gamma$ provides an excellent diagnostic for the hydrogen content in WN stars, except for the latest subtypes (approximately WN8 and later). As such, we are able to use primarily optical spectroscopy for the determination of stellar temperatures, with hydrogen content obtained from near-IR spectroscopy. Near-IR comparisons between synthetic spectra and observations are presented for each of the WN6h stars within NGC 3603 in Fig. \[ngc3603\_k\].
Name A1a A1b B C
------------------------------------------------- ----------------------- ------------------------ ---------------------- ----------------------
T$_{\ast}$ (kK)$^{a}$ 42 $\pm$ 2 40 $\pm$ 2 42 $\pm$ 2 44 $\pm$ 2
$\log$ (L/L$_{\odot}$) 6.39 $\pm$ 0.14 6.18 $\pm$ 0.14 6.46 $\pm$ 0.07 6.35 $\pm$ 0.07
R$_{\tau = 2/3}$ (R$_{\odot}$) 29.4$_{-4.3}^{+10.1}$ 25.9$_{-3.1}^{+7.2}$ 33.8$_{-2.5}^{+2.7}$ 26.2$_{-2.0}^{+2.1}$
N$_{\rm LyC}$ (10$^{50}$ s$^{-1}$) 1.6$_{-0.4}^{+0.8}$ 0.85$_{-0.23}^{+0.54}$ 1.9$_{-0.3}^{+0.3}$ 1.5$_{-0.3}^{+0.3}$
$\dot{M}$ (10$^{-5}$ M$_{\odot}$ yr$^{-1}$) 3.2$_{-0.6}^{+1.2}$ 1.9$_{-0.4}^{+0.9}$ 5.1$_{-0.6}^{+0.6}$ 1.9$_{-0.2}^{+0.2}$
$\log$ $\dot{M}$ - log $\dot{M}_{\rm Vink}^{c}$ +0.14 +0.24 +0.22 –0.04
V$_{\infty}$ (km s$^{-1}$) 2600 $\pm$ 150 2600 $\pm$ 150 2300 $\pm$ 150 2600 $\pm$ 150
X$_{H}$ (%) 60 $\pm$ 5 70 $\pm$ 5 60 $\pm$ 5 70 $\pm$ 5
M$_{\rm init}$ (M$_{\odot}$)$^{b}$ 148$_{-27}^{+40}$ 106$_{-20}^{+23}$ 166$_{-20}^{+20}$ 137$_{-14}^{+17}$
M$_{\rm current}$ (M$_{\odot}$)$^{b}$ 120$_{-17}^{+26}$ 92$_{-15}^{+16}$ 132$_{-13}^{+13}$ 113$_{-8}^{+11}$
M$_{K_{s}}$ (mag)$^{d}$ –7.0 $\pm$ 0.3 –6.6 $\pm$ 0.3 –7.5 $\pm$ 0.1 –6.7 $\pm$ 0.1
: Physical Properties of NGC 3603 WN6h stars.[]{data-label="ngc3603_params"}
\(a) Corresponds to the radius at a Rosseland optical depth of $\tau_{\rm
Ross}$ = 10\
(b) Component C is a 8.9 day period SB1 system (Schnurr et al. 2008a)\
(c) dM/dt$_{\rm Vink}$ relates to Vink et al. (2001) mass-loss rates for Z = Z$_{\odot}$\
(d) M$_{\rm K_{s}}$ = –7.57 $\pm$ 0.12 mag for A1, for which we adopt $\Delta$m = m$_{\rm A1a}$ - m$_{\rm A1b}$ = –0.43 $\pm$ 0.30 mag (Schnurr et al. 2008a). The ratio of their luminosities follows from their dynamical mass ratios together with $L \propto \mu M^{1.5}$ (and is supported by NICMOS photometry from Moffat et al. 2004).
{width="1.5\columnwidth"}
Physical properties for these stars in NGC 3603 are presented in Table \[ngc3603\_params\], including evolutionary masses obtained from solar-metallicity non-rotating models. Errors quoted in the Table account for both photometric and spectroscopic uncertainties, and represent the range of permitted values. However, for NGC 3603 C we assume that the primary dominates the systemic light since the companion is not detected in SINFONI spectroscopy (Schnurr et al 2008a). Should the companion make a non-negligible contribution to the integrated light, the derived properties set out in Table \[ngc3603\_params\] would need to be corrected accordingly. The improved allowance for metal line blanketing implies $\sim$10–20% higher stellar temperatures (T$_{\ast}$ $\sim$ 40,000 – 44,000 K) and, in turn, larger bolometric corrections (M$_{\rm Bol}$ – M$_{\rm V}$ $\sim$ –3.7 $\pm$ 0.2 mag) for these stars than earlier studies (Crowther & Dessart 1998), who adopted a different combination of E(B-V) and distance for NGC 3603. We note that Schmutz & Drissen (1999) have previously derived T$_{\ast}$ $\sim$ 46,000 K for NGC 3603 B, resulting in a similar luminosity ($\log L/L_{\odot}$ = 6.4) to that obtained here. Analysis of the NICMOS lightcurve (Moffat et al. 2004) yields T(A1a)/T(A1b) = 1.06, in support of the spectroscopically-derived temperature ratio, albeit 7% higher in absolute terms. In view of the underlying assumptions of the photometric and spectroscopic techniques, overall consistency is satisfactory.
For A1a and B, we adopt abundances of X$_{\rm C}$ = 0.008%, X$_{\rm N}$ = 0.8% and X$_{\rm
O}$ = 0.013% by mass, as predicted by evolutionary models for X$_{\rm H}$ $\sim$ 60%, versus X$_{\rm C}$ = 0.005%, X$_{\rm N}$ = 0.6% and X$_{\rm O}$ = 0.25% for A1b and C, for which X$_{\rm H}$ $\sim$ 70%.
R136
----
Theoretical spectral energy distributions of all four WN5h stars are presented in Fig. \[r136\_sed\]. Although we have matched synthetic spectra to VLT/MAD + SINFONI photometry, we note that no significant differences would be obtained from WFPC2 imaging (e.g. Hunter et al. 1995), providing these are sufficiently isolated. For example, our reddened spectral energy distribution for R136a3 implies V = 13.0 mag, in agreement with F555W photometry. VLT/MAD photometry is preferred to WFPC2 for the very crowded core of R136 (e.g. a1 and a2), although recent Wide Field Camera 3 (WFC3) imaging of R136 achieves a similar spatial resolution at visible wavelengths. Nevertheless, we adhere to VLT/MAD + SINFONI for our primary photometric reference since
1. Solely VLT/SINFONI spectrally resolves R136a1 from R136a2 (Schnurr et al. 2009). We also note the consistent line to continuum ratios between optical and near-IR diagnostics, ruling out potential late-type contaminants for the latter;
2. Uncertainties in dust extinction at K$_{s}$ are typically 0.02 mag versus $\sim$0.2 mag in the V-band, recalling $A_{K_{s}} \sim 0.12 A_{V}$. R136c is significantly fainter than R136a2 at optical wavelengths yet is brighter at K$_{s}$ and possesses a higher bolometric magnitude (contrast these results with the optical study of Rühling 2008);
3. We seek to follow a consistent approach to recent studies of the Arches cluster which necessarily focused upon the K$_{s}$-band for photometry and spectroscopy (e.g. Martins et al. 2008).
----------------------------------------------- ---------------------- ---------------------- ---------------------- ----------------------
Name a1 a2 a3 c
BAT99 108 109 106 112
T$_{\ast}$ (kK)$^{a}$ 53 $\pm$3 53 $\pm$ 3 53 $\pm$ 3 51 $\pm$ 5
$\log$ (L/L$_{\odot}$) 6.94 $\pm$ 0.09 6.78 $\pm$ 0.09 6.58 $\pm$ 0.09 6.75 $\pm$ 0.11
R$_{\tau = 2/3}$ (R$_{\odot}$) 35.4$_{-3.6}^{+4.0}$ 29.5$_{-3.0}^{+3.3}$ 23.4$_{-2.4}^{+2.7}$ 30.6$_{-3.7}^{+4.2}$
N$_{\rm LyC}$ (10$^{50}$ s$^{-1}$) 6.6$_{-1.3}^{+1.6}$ 4.8$_{-0.7}^{+0.8}$ 3.0$_{-0.4}^{+0.5}$ 4.2$_{-0.6}^{+0.7}$
$\dot{M}$ (10$^{-5}$ M$_{\odot}$ yr$^{-1}$) 5.1$_{-0.8}^{+0.9}$ 4.6$_{-0.7}^{+0.8}$ 3.7$_{-0.5}^{+0.7}$ 4.5$_{-0.8}^{+1.0}$
$\log \dot{M}$ - log $\dot{M}_{\rm Vink}^{c}$ +0.09 +0.12 +0.18 +0.06
V$_{\infty}$ (km s$^{-1}$) 2600 $\pm$ 150 2450 $\pm$ 150 2200 $\pm$ 150 1950 $\pm$ 150
X$_{H}$ (%) 40 $\pm$ 5 35 $\pm$ 5 40 $\pm$ 5 30 $\pm$ 5
M$_{\rm init}$ (M$_{\odot}$)$^{b}$ 320$_{-40}^{+100}$ 240$_{-45}^{+45}$ 165$_{-30}^{+30}$ 220$_{-45}^{+55}$
M$_{\rm current}$ (M$_{\odot}$)$^{b}$ 265$_{-35}^{+80}$ 195$_{-35}^{+35}$ 135$_{-20}^{+25}$ 175$_{-35}^{+40}$
M$_{K_{s}}$ (mag) –7.6 $\pm$ 0.2 –7.3 $\pm$ 0.2 –6.9 $\pm$ 0.2 –7.4 $\pm$ 0.2
----------------------------------------------- ---------------------- ---------------------- ---------------------- ----------------------
: Physical Properties of R136 WN5h stars.[]{data-label="r136_summary"}
\(a) Corresponds to the radius at a Rosseland optical depth of $\tau_{\rm
Ross}$ = 10\
(b) Component R136c is probably a colliding-wind massive binary. For a mass ratio of unity, initial (current) masses of each component would correspond to $\sim$160 M$_{\odot}$ ($\sim$130 M$_{\odot}$)\
(c) dM/dt$_{\rm Vink}$ relates to Vink et al. (2001) mass-loss rates for Z = 0.43 Z$_{\odot}$
![Rectified, ultraviolet (HST/GHRS), visual (HST/FOS) and near-IR (VLT/SINFONI) spectroscopy of the WN5h star R136a3 together with synthetic UV, optical and near-infrared spectra, for T$_{\ast}$ = 50,000 K (red) and T$_{\ast}$ =55,000 K (blue). Instrumental broadening is accounted for, plus an additional rotational broadening of 200 kms$^{-1}$.[]{data-label="r136a3_plot"}](r136a3_plot.ps){width="1.0\columnwidth"}
We estimate terminal wind velocities from optical and near-IR helium lines. Ultraviolet HST/GHRS spectroscopy has been obtained for the R136a stars (Heap et al. 1994; de Koter et al. 1997), although the severe crowding within this region results in the spectra representing blends of individual components for R136a1 and a2 (apparent in Fig. \[r136\_sed\]). As a result, we favour velocities from FOS and SINFONI spectroscopy for these cases.
In Fig. \[r136a3\_plot\] we present UV/optical/near-IR spectral fits for the WN5h star R136a3, which is sufficiently isolated that contamination from other members of R136a is minimal in each spectral region (in contrast to R136a1 and a2 at UV/optical wavelengths). Consistent optical and near-IR fits demonstrate that no underlying red sources contribute significantly to the K-band SINFONI datasets. Diagnostics include O[v]{} 1371, S[v]{} 1501, N[ iv]{} 3478-83, 4058, N[v]{} 4603-20, 2.10$\mu$m together with He[ii]{} 4686, 2.189$\mu$m plus H$\alpha$, H$\beta$, Br$\gamma$. For T$_{\ast}$ $\leq$ 50,000 K both O[v]{} 1371 and N[v]{} 2.11$\mu$m are underestimated for the R136a WN5h stars, while for T$_{\ast}$ $\geq$ 56,000 K, N[ iv]{} 3478-83 and 4057 become too weak, and S[v]{} 1501 becomes too strong. Therefore, we favour T$_{\ast} \sim$53,000 $\pm$ 3,000 K, except that T$_{\ast} \sim$ 51,000 $\pm$5,000 K is preferred for R136c since UV spectroscopy is not available and N[v]{} 2.100$\mu$m is weak/absent. Again, we are unable to employ helium temperature diagnostics since He[i]{} lines are extremely weak.
Fig. \[r136\_sed\] compares synthetic spectra with HST/FOS spectroscopy, for which emission features are again well matched, albeit with predicted Balmer-Pickering absorption components that are too strong. Fig. \[r136\] presents near infra-red spatially-resolved spectroscopy of R136 WN stars (Schnurr et al. 2009) together with synthetic spectra, allowing for instrumental broadening (100 kms$^{-1}$) plus additional rotational broadening of 200 kms$^{-1}$ for R136a2, a3, c. As for NGC 3603 stars, the ratio of He[ii]{} 2.189$\mu$m to Br$\gamma$ provides an excellent diagnostic for the hydrogen content for the R136 stars. A summary of the resulting physical and chemical parameters is presented in Table \[r136\_summary\], with errors once again accounting for both photometric and spectroscopic uncertainties, representing the range of permitted values, although the primary uncertainty involves the distance to the LMC.
With respect to earlier studies (Heap et al. 1994; de Koter et al. 1997; Crowther & Dessart 1998), the improved allowance for metal line blanketing also infers 20% higher stellar temperatures ($T_{\ast} \sim$53,000 K) for these stars, and in turn, larger bolometric corrections (M$_{\rm Bol}
- M_{\rm V}$ $\sim$ –4.6 mag). Such differences, with respect to non-blanketed analyses, are typical of Wolf-Rayet stars (Crowther 2007). Similar temperatures ($T_{\ast}$ = 50,000 or 56,000 K) to the present study were obtained for these WN stars by Rühling (2008) using a grid calculated from the Potsdam line-blanketed atmospheric code (for a summary see Ruehling et al. 2008), albeit with 0.1–0.2 dex lower luminosities ($\log L/L_{\odot}$ = 6.4 – 6.7) using solely optically-derived (lower) extinctions and absolute magnitudes.
To illustrate the sensitivity of mass upon luminosity, we use the example of R136a5 (O2–3If/WN) for which we estimate $T_{\ast} \sim$ 50,000 K using the same diagnostics as the WN5h stars, in good agreement with recent analyses of O2 stars by Walborn et al. (2004) and Evans et al. (2010b). This reveals a luminosity of $\log L/L_{\odot}$ = 6.35, corresponding to an initial mass in excess of 100 M$_{\odot}$, versus $T_{\ast} \sim$ 42,500 K, $\log L/L_{\odot}$ = 5.9 and $\sim$65 M$_{\odot}$ according to de Koter et al. (1997).
![Rectified, spatially resolved near-IR (VLT/SINFONI, Schnurr et al. 2009) spectroscopy of R136 WN5h stars (black) together with synthetic spectra, accounting for instrumental broadening (100 kms$^{-1}$) plus rotational broadening of 200 kms$^{-1}$ for R136a2, a3 and c. Consistent hydrogen contents are obtained from the peak intensity ratio of Br$\gamma$/HeII 2.189$\mu$m and optical (Pickering-Balmer series) diagnostics. Nitrogen emission includes N[v]{} 2.100$\mu$m and N[iii]{} 2.103 - 2.115$\mu$m.[]{data-label="r136"}](r136.ps){width="0.6\columnwidth"}
Regarding elemental abundances, we use the H$\beta$ to He[ii]{} 5412 and Br$\gamma$ to He[ii]{} 2.189$\mu$m ratios to derive (consistent) hydrogen contents, with the latter serving as the primary diagnostic for consistency with NGC 3603 stars (current observations do not include He[ii]{} 5412). We adopt scaled solar abundances for all metals other than CNO elements. Nitrogen abundances of X$_{N}$ = 0.35% by mass, as predicted by evolutionary models, are consistent with near-IR N[v]{} 2.100$\mu$m recombination line observations, while we adopt carbon and oxygen abundances of X$_{C}$ = X$_{O}$ = 0.004% by mass.
{width="1.5\columnwidth"}
Evolutionary Models
===================
We have calculated a grid of main-sequence models using the latest version of the Geneva stellar evolution code for 85, 120, 150, 200, 300 and 500 M$_{\odot}$. The main-sequence evolution of such high-mass stars does not suffer from stability issues. Although a detailed description is provided elsewhere (Hirschi et al. 2004), together with recent updates (Eggenberger et al. 2008), models include the physics of rotation and mass loss, which are both crucial to model the evolution of very massive stars. Although many details relating to the evolution of very massive stars remain uncertain, we solely consider the main-sequence evolution here using standard theoretical mass-loss prescription for O stars (Vink et al. 2001) for which Mokiem et al. (2007) provide supporting empirical evidence. In the models, we consider the onset of the Wolf-Rayet phase to take place when the surface hydrogen content X$_{\rm H} <$ 30% if T$_{\rm eff}$ $\geq$ 10,000 K, during which an empirical mass-loss calibration is followed (Nugis & Lamers 2000). The post-main sequence evolution is beyond the scope of this study and will be discussed elsewhere.
All the main effects of rotation are included in the calculations: centrifugal support, mass-loss enhancement and especially mixing in radiative zones (Maeder 2009), although predictions for both non-rotating and rotating models are considered here. For rotating models we choose an initial ratio of the velocity to critical (maximum) rotation of $v_{\rm init}/v_{\rm
crit}$ = 0.4, which corresponds to surface equatorial velocities of around 350 km s$^{-1}$ for the 85 M$_{\odot}$ model and around 450 km s$^{-1}$ for the 500 M$_{\odot}$ case.
We have calculated both solar (Z=1.4% by mass) and LMC (Z=0.6% by mass) metallicities. The evolution of 120 M$_{\odot}$ models in the Hertzsprung-Russell diagram is presented in Fig \[evol\]. The effects of rotation are significant. Due to additional mixing, helium is mixed out of the core and thus the opacity in the outer layers decreases. This allows rotating stars to stay much hotter than non-rotating stars. Indeed, the effective temperature of the rotating models stay as high as 45,000 - 55,000 K, whereas the effective temperature of non-rotating models decreases to 20,000 - 25,000 K. Therefore, rapidly rotating stars progress directly to the classical Wolf-Rayet phase, while slow rotators are expected to become $\eta$ Car-like Luminous Blue Variables (see Meynet & Maeder 2005).
Rotating models can reach higher luminosities, especially at very low metallicity (see Langer et al. 2007). This is explained by additional mixing above the convective core. Finally, by comparing models at solar metallicity and LMC metallicity, we can see that lower metallicity models reach higher luminosities. This is due to weaker mass-loss at lower metallicity.
Langer et al. (2007) provide 150 M$_{\odot}$ tracks at 0.2 and 0.05 Z$_{\odot}$ with high initial rotation velocities of 500 kms$^{-1}$. From these it is apparent that rapidly rotating very metal-deficient models can achieve high stellar luminosities. According to Langer et al. (2007) a 150 M$_{\odot}$ model reaches $\log L/L_{\odot} \sim$ 6.75 at 0.05 Z$_{\odot}$ and 6.5 at 0.2 Z$_{\odot}$. Could the R136 stars represent rapidly rotating stars of initial mass 150 M$_{\odot}$? We have calculated a SMC metallicity (0.14 Z$_{\odot}$) model for a 150 M$_{\odot}$ star initially rotating at $v_{\rm init}$ = 450 kms$^{-1}$, which achieves $\log L/L_{\odot} \approx$ 6.5 after 1.5 Myr and 6.67 after 2.5 Myr. Our models are thus compatible with Langer et al. (2007) described above. We nevertheless exclude the possibility that the R136 WN5 stars possess initial masses below 150 M$_{\odot}$ since:
1. the metallicity of 30 Doradus is a factor of three higher than the SMC (e.g. Peimbert 2003, Lebouteiller et al. 2008);
2. R136 has an age of less than 2 Myr (de Koter et al. 1998, Massey & Hunter 1998), since its massive stellar population is analogous to the young star clusters in Car OB1 (1–2 Myr, Walborn 2010) and NGC 3603 (this study);
3. Clumping-corrected mass-loss rates for the R136 WN stars agree well with LMC metallicity predictions (Vink et al. 2001), which are also supported by studies of O stars in the Milky Way, LMC and SMC (Mokiem et al. 2007).
{width="1\columnwidth"}
{width="1\columnwidth"}
The surface abundances of a majority of high mass stars can be well reproduced by models of single stars including the effects of rotation. However, the VLT FLAMES survey has highlighted some discrepancies between models and observations (Hunter et al. 2008), questioning the efficiency of rotation induced mixing. We are currently investigating this matter by comparing the composition of light elements like boron and nitrogen between models and observations and we find that models including rotation induced mixing reproduces the abundances of most stars well (Frischknecht et al. 2010).
We can nevertheless consider how a less efficient rotation-induced mixing (or absence of mixing) would affect the conclusions of this paper. If rotation induced mixing was less efficient than our models predict, the masses derived here would remain at least as high, and would usually be higher. Indeed, less efficient mixing prevents the luminosity to increase as much since less helium is mixed up to the surface and the mean molecular weight, $\mu$, remains lower ($L
\approx \mu M^{1.5}$). In particular, less efficient mixing would prevent stars with initial masses around or below 150 M$_{\odot}$ from reaching such high luminosities as predicted in Langer et al. (2007, see discussion in previous paragraph) and would bring further support to our claim that these stars are more massive than 150 M$_{\odot}$. The challenge would then be to explain the high effective temperature (T$_{\rm eff} \sim$ 50 kK) observed for the R136 stars, which is best explained by rotation induced mixing. Note that less efficient mixing would also make impossible the quasi-chemical evolution of fast rotating (single or binary) stars (Yoon et al. 2006) which is currently one of the best scenarios for long/soft Gamma Ray Bursts progenitors.
NGC 3603
--------
Fig. \[ngc3603\_evol\] compares various observational properties of NGC 3603 WN stars with solar metallicity evolutionary predictions. Non-rotating models imply current masses of 120$_{-17}^{+26}$ M$_{\odot}$ and 92$_{-15}^{+16}$ M$_{\odot}$ for A1a and A1b, respectively, at an age of $\sim$1.5 $\pm$ 0.1 Myr. These are in excellent agreement with dynamical mass determinations of 116 $\pm$ 31 M$_{\odot}$ and 89 $\pm$ 16 M$_{\odot}$ for the primary and secondary A1 components (Schnurr et al. 2008a). Initial and current stellar mass estimates are shown in Table \[ngc3603\_params\], and include a (high) initial mass of 166 $\pm$ 20 M$_{\odot}$ for NGC 3603 B. Independent age estimates using pre-main sequence isochrones of low mass stars also favour low 1$\pm$1 Myr ages (Sung & Bessell 2004), while Crowther et al. (2006) estimated 1.3$\pm$0.3 Myr for NGC 3603 from a comparison between massive O stars and theoretical isochrones (Lejeune & Schaerer 2001).
R136
----
Fig. \[r136\_fig1to4\] compares the derived properties of R136a1, a2, a3 and c with LMC metallicity evolutionary predictions, under the assumption that these stars are single. Initial stellar masses in the range 165 – 320 M$_{\odot}$ are implied, at ages of 1.7 $\pm$ 0.2 Myr, plus high initial rotational rates, in order that the observed surface hydrogen contents of 30 – 40% by mass are reproduced. Initial and current stellar mass estimates are included in Table \[r136\_summary\]. Differences in age estimates reflect variations in initial rotation rates. Nevertheless, equatorial rotation rates of $v_{e} \sim$ 200 (300) km s$^{-1}$ are predicted after $\sim$1.75 Myr (2.75 Myr) for a 300 (150) M$_{\odot}$ star.
In the absence of photospheric absorption features, the N[v]{} 2.100$\mu$m feature provides the best diagnostic of rotational broadening for the R136 stars. This recombination line is intrinsically narrow at the SINFONI spectral resolution (FWHM $\sim$ 15Å) because it is formed extremely deep in the stellar wind. Indeed, FWHM $\sim$ 15Å is observed for R136a1, as expected either for a non-rotating star, or one viewed pole-on (see Fig. \[r136\_nv\]). In contrast, a2 and a3 reveal FWHM $\sim$ 40Å, corresponding to $v_{e} \sin i \approx$ 200 kms$^{-1}$. We are unable to quantify FWHM for R136c since the N[v]{} feature is very weak, although it too is consistent with a large rotation rate. Therefore, a2, a3 and probably c show spectroscopic evidence for rapid rotation as shown in Fig. \[r136\_nv\], while a1 could either be a very slow rotator, or a rapid rotator viewed close to pole-on.
For comparison, Wolff et al. (2008) found that R136 lacks slow rotators among lower mass 6–30 $M_{\odot}$ stars. Wolff et al. derived $v_{e} \sin i$ = 189 $\pm$ 23 kms$^{-1}$ for eleven 15–30 $M_{\odot}$ stars within R136 versus $v_{e} \sin i$ = 129 $\pm$ 13 kms$^{-1}$ from equivalent mass field stars within the LMC. A much more extensive study of rotational velocities for O stars in 30 Doradus will be provided by the VLT-FLAMES Tarantula Survey (Evans et al. 2010a).
![Spectral comparison between VLT/SINFONI spectroscopy of N[ v]{} 2.10$\mu$m in R136 WN5 stars (Schnurr et al. 2009) and synthetic spectra (red), allowing for instrumental broadening plus rotational broadening of 200 kms$^{-1}$ for R136a2, a3 and c.[]{data-label="r136_nv"}](r136_nv.ps){width="0.6\columnwidth"}
Finally, although we defer the possibility that the WN5 stars in R136 are (equal mass) binaries until Sect. 5, it is necessary to remark upon the possibility of chance superpositions within the observed cluster. Line-of-sight effects should not play a significant role in our interpretation of bright systems such as R136a1 (Maíz Apellániz 2008). Chance superposition of other stars has been calculated to contribute at most $\sim$10 to 20% of the (visible) light (J. Maíz Apellániz, priv. comm.). For example, a 0.2 mag decrease in the absolute K-band magnitude of R136a1 arising from the contribution of lower mass stars along this sightline would lead to a 10% reduction in its initial (current) stellar mass, i.e. to 285 M$_{\odot}$ (235 $M_{\odot}$).
Synthetic spectra
-----------------
We have calculated synthetic spectra for each of the LMC metallicity models, both at the Zero Age Main Sequence (ZAMS) and ages corresponding to the surface hydrogen compositions reaching X$_{\rm H}$ = 30% for the rotating models. These are presented in Fig. \[synthetic\] in which mass-loss rates follow the theoretical mass-loss recipes from Vink et al. (2001), both for the case of radially-dependent [*clumped*]{} winds with a volume filling factor of $f_{\infty}$=0.1 at $v_{\infty}$ (solid) and [*smooth*]{} ($f_{\infty}$=1, dotted) winds. For clumped winds, these result in ZAMS synthetic O supergiant spectra which are equivalent to O3If (for 85 M$_{\odot}$) and O2-3If/WN5-6 (for 200 M$_{\odot}$) subtypes. Weaker He[ii]{} $\lambda$4686 emission would naturally be predicted if we were to instead adopt [*smooth*]{} winds, including O3V for the 85 M$_{\odot}$ case and O2III–If for the 200 M$_{\odot}$ case at the zero age main sequence.
Although specific details depend upon the degree of wind clumping in O star winds and validity of the smooth Vink et al. predictions, stars whose masses exceed a given threshold are expected to display a supergiant signature from the outset. Overall, these results suggest a paradigm shift in our understanding of early O dwarfs, which have hitherto been considered to represent ZAMS stars for the [*highest*]{} mass stars (Walborn et al. 2002).
For the NGC 3603 and R136 WN stars studied here, wind clumping is required to reproduce the electron scattering wings of He[ii]{} $\lambda$4686 (Hillier 1991). Overall, we find $\log \dot{M} - \log \dot{M}_{\rm Vink}$ = +0.13 $\pm$ 0.09, in close agreement with Martins et al. (2008) who found that spectroscopic mass-loss rates for the Arches WN7–9 stars exceeded predictions by +0.2 dex.
Very powerful winds naturally result from the dependence of the mass-loss rate upon the ratio of radiation pressure to gravity, $\Gamma_{e} \propto
L/M$. Since L $\propto M^{1.7}$ for ZAMS stars in the range 30 – 300 M$_{\odot}$ based upon evolutionary calculations discussed above, $\Gamma_{e} \propto M^{0.7}$. A 30 M$_{\odot}$ main-sequence star possesses an Eddington parameter of $\Gamma_{e} \sim$ 0.12 (Conti et al. 2008), therefore a 300 M$_{\odot}$ star will possess $\Gamma_{e} \sim 0.5$. The latter achieve $\Gamma_{e} \sim 0.7$ after 1.5 Myr, typical of the cluster age under investigation here. $\Gamma_{e}$ increases from 0.4 to 0.55 for stars of initial mass 150 M$_{\odot}$ after 1.5 Myr.
Therefore, the very highest mass stars in very young systems may never exhibit a normal O-type absorption line spectrum. Indeed, O2–3V stars may rather be limited to somewhat lower ZAMS stars, with a higher mass threshold at lower metallicity. Recall the relatively low dynamical mass of 57 M$_{\odot}$ for the O3 V primary in R136-38 (Massey et al. 2002), which currently represents the most massive O2–3 V star dynamically weighed, versus 83 M$_{\odot}$ (WR20a: Bonanos et al. 2004; Rauw et al. 2005), $\geq$87 M$_{\odot}$ (WR21a: Niemela et al. 2008) and 116 M$_{\odot}$ (NGC 3603 A1a: Schnurr et al. 2008a) for some of the most massive H-rich WN stars. OIf$^{\ast}$ or even WN-type emission line spectra may be expected for the highest mass stars within the very youngest Giant H[ii]{} regions, as is the case for R136 and NGC 3603 here, plus Car OB1 (Smith 2006) and W43 (Blum et al. 1999).
{width="1.0\columnwidth"}
Binarity
========
NGC 3603 A1 and C are confirmed binaries (Schnurr et al. 2008a), and R136c is a probable binary (Schnurr et al. 2009) while component B and R136a1, a2, a3 are presumed to be single. Of course, we cannot unambiguously confirm that these are genuinely single but if their mass ratios differ greatly from unity the derived physical properties reflect those of the primary. Therefore, here we shall focus upon the possibility that the R136 stars are massive binaries, whose ratio is close to unity (Moffat 2008).
If we adopt similar mean molecular weights, $\mu$, for individual components, current (evolutionary) masses of 150 + 150 M$_{\odot}$, 200 + 100 M$_{\odot}$ and 220 + 55 M$_{\odot}$ would be required to match the current properties of R136a1 for mass ratios of 1, 0.5 and 0.25 since L $\propto \mu M^{1.5}$. Of these possibilities, only near-equal mass binaries would contradict the high stellar masses inferred here. If the separation between putative binary components were small, radial velocity variations would be expected. There is no unambiguous evidence for binarity among the stars under investigation, although R136c does exhibit marginal radial velocity variability (Schnurr et al. 2009).
In contrast, the WN6h system A1 in the young Milky Way cluster NGC 3603 [*is* ]{} a short-period massive binary (Schnurr et al. 2008a). However, longer period systems with periods of months to years could have easily avoided spectroscopic detection. We shall employ anticipated properties of colliding wind systems and dynamical interactions to assess whether the R136a stars are realistically long-period binaries. This approach is especially sensitive to equal wind momenta (equal mass) systems since the X-ray emission is maximal for the case of winds whose momenta are equal. A1 in NGC 3603 is relatively faint in X-rays because the components are in such close proximity that their winds collide at significantly below maximum velocity.
-------- --------------- ------ ------ -------- ------------ ----------- ------------ ----------- ------------------
Source Sp $q$ $d$ Period $\chi_{1}$ $\Xi_{1}$ $\chi_{2}$ $\Xi_{2}$ $L_{\rm X, 1+2}$
Type AU year 10$^{34}$
ergs$^{-1}$
R136a1 2$\times$WN5 1.0 3 0.3 3.7 0.17 3.7 0.17 540
R136a1 2$\times$WN5 1.0 30 9.5 37 0.17 37 0.17 54
R136a1 2$\times$WN5 1.0 300 300 370 0.17 370 0.17 5.4
R136a1 WN5+O 0.25 3 0.3 3.7 0.02 4.5 0.44 88
R136a1 WN5+O 0.25 30 9.8 37 0.02 45 0.44 8.8
R136a1 WN5+O 0.25 300 310 370 0.02 450 0.44 0.9
R136a1 WN5 0.3
R136a1 WN5+O? 3.3
R136a1 2$\times$WN5? 16.3
R136a 2.4
-------- --------------- ------ ------ -------- ------------ ----------- ------------ ----------- ------------------
: Predicted X-ray luminosities from R136a1 for scenarios in which it is either a colliding wind binary using analytical predictions from Stevens et al. (1992) and Pittard & Stevens (2002), or single/multiple for various empirical $L_{\rm X}/L_{\rm Bol}$ values. Analytical calculations adopt component separations of $d$ = 3, 30, 300 AU, for mass ratios: (i) $q$ = 1 (150 + 150 $M_{\odot}$) and equal wind momenta, $\eta$ = 1 ($\dot{M}_{1} = 2.8 \times 10^{-5}$ $M_{\odot}$yr$^{-1}$, $v_{\infty, 1}$ = 2600 kms$^{-1}$); (ii) $q$ = 0.25 (220 + 55 $M_{\odot}$) and $\eta$ = 0.05 ($\dot{M}_{1} = 4.2 \times 10^{-5}$ $M_{\odot}$yr$^{-1}$, $v_{\infty, 1}$ = 2600 kms$^{-1}$, $\dot{M}_{2} = 3 \times 10^{-6}$ $M_{\odot}$yr$^{-1}$, $v_{\infty, 2}$ = 2000 kms$^{-1}$). The intrinsic X-ray luminosity of R136a is taken from Guerrero & Chu (2008).[]{data-label="r136a1_xray"}
X-rays
------
If both components in a binary system possessed similar masses, once outflow velocities of their dense stellar winds achieve asymptotic values, collisions would produce stronger X-ray emission than would be expected from a single star. Analytical estimates of X-ray emission from colliding wind systems are approximate (Stevens et al. 1992). Nevertheless, this approach does provide constraints upon orbital separations to which other techniques are currently insensitive and is especially sensitive to systems whose wind strengths are equal. According to Pittard & Stevens (2002) up to 17% of the wind power of the primary (and 17% of the secondary) can be radiated in X-rays for equal wind momenta systems, versus 0.4% of the primary wind power (56% of the secondary) for (unequal mass) systems whose wind momentum ratio is $\eta$ = 0.01.
To illustrate the diagnostic potential for X-ray observations, let us consider the expected X-ray emission from R136c under the assumption that it is single. The intrinsic X-ray luminosities of single stars can be approximated by L$_{\rm X}$/L$_{\rm Bol} \sim 10^{-7}$ (Chlebowski et al. 1989). From our spectroscopic analysis we would expect L$_{\rm X} \approx 2 \times 10^{33}$ ergs$^{-1}$, yet Chandra imaging reveals an intrinsic X-ray luminosity which is a factor of 30–50 times higher (Portegies Zwart et al. 2002; Townsley et al. 2006; Guerrero & Chu 2008), arguing for a colliding wind system in this instance. Analytical models favour equal mass components separated by $\sim$100 Astronomical Units (AU).
In contrast, the expected X-ray emission from the sum of R136a1, a2 a3 (and a5) – unresolved at Chandra resolution – is L$_{\rm X} \approx 7
\times 10^{33}$ ergs$^{-1}$ under the assumption that they are single. In this case, the intrinsic X-ray emission from R136a is observed to be only a factor of $\sim$3 higher (Portegies Zwart et al. 2002; Guerrero & Chu 2008) arguing against short period colliding wind systems from any R136a components. Multiple wind interactions (outside of the binaries but within the cluster) will also produce shocks and X-ray emission (see e.g. Reyes-Iturbide et al. 2009).
Empirically, colliding winds within O-type binaries typically exhibit L$_{\rm X}$/L$_{\rm Bol} \sim 10^{-6}$ (Rauw et al. 2002), although binaries comprising stars with more powerful winds (i.e. Wolf-Rayet stars) often possess stronger X-ray emission. Indeed, NGC 3603 C has an X-ray luminosity of $L_{\rm X} \gtsim 4 \times 10^{34}$ ergs$^{-1}$ (Moffat et al. 2002), corresponding to L$_{\rm X}$/L$_{\rm Bol} \gtsim 5 \times
10^{-6}$ based upon our spectroscopically derived luminosity (similar results are obtained for R136c). If we were to assume this ratio for the brightest components of R136a (a1, a2, a3 and a5) under the assumption that each were close binaries – i.e. disregarding predictions from colliding wind theory – we would expect an X-ray luminosity that is a factor of 15 times higher than the observed value. Therefore, the WN stars in R136a appear to be single, possess relatively low-mass companions or have wide separations.
If we make the reasonable assumption that equal mass stars would possess similar wind properties, we can consider the effect of wind collisions upon the production of X-rays. Let us consider a 150 + 150 M$_{\odot}$ binary system with a period of 100 days in a circular orbit with separation 3 AU, whose individual components each possess mass-loss rates of $\dot{M}$ = 2.8 $\times$ 10$^{-5}$ M$_{\odot}$ yr$^{-1}$ and wind velocities of $v_{\infty}$ = 2600 km s$^{-1}$. A pair of stars with such properties could match the appearance of R136a1, with individual properties fairly representative of R136a3.
Since the ratio of their wind momenta, $\eta$ is unity, the fraction, $\Xi$, of the wind kinetic power processed in the shock from each star is maximal ($\Xi \sim$1/6, Pittard & Stevens 2002). One must calculate the conversion efficiency, $\chi$, of kinetic wind power into radiation for each star. From Stevens et al. (1992) $$\chi \approx \frac{v_{8}^{4} d_{12}}{\dot{M}_{-7}},$$ where $v_{8}$ is the wind velocity in units of 10$^{8}$ cms$^{-1}$, $d_{12}$ is the distance to the interaction region in units of 10$^{12}$ cm (i.e. half the separation for equal wind momenta) and $\dot{M}_{-7}$ is the mass-loss rate in units of 10$^{-7} M_{\odot}$yr$^{-1}$. We set a lower limit of $\chi$ = 1 for cases in which the system radiates all of the collision energy. This allows an estimate of the intrinsic X-ray luminosity (J. Pittard, priv. comm.), $$L_{\rm X} \approx \frac{1}{2} \frac{\dot{M} v_{\infty}^2 \Xi}{\chi}.$$ For a 150 + 150 M$_{\odot}$ system with separation of 3 AU, we derive $\chi \sim$ 3.7 and predict L$_{\rm X}$ = 2.7 $\times$ 10$^{36}$ erg s$^{-1}$ for [*each*]{} component. The total exceeds the measured X-ray luminosity of R136a by a factor of $\sim$200 (Townsley et al. 2006; Guerrero & Chu 2008), as shown in Table \[r136a1\_xray\]. Recall that if R136a1 were to be a massive binary with a mass ratio of 0.25, the secondary would possess a mass of $\sim$55 M$_{\odot}$. This would contribute less than 10% of the observed spectrum, with a secondary to primary wind momentum ratio of $\eta\sim$0.05. Predictions for this scenario are also included in Table \[r136a1\_xray\], revealing X-ray luminosities a factor of $\sim$6 lower than the equal wind (equal mass) case.
Stevens et al. (1992) also provide an expression for the characteristic intrinsic column density in colliding wind binaries, $\bar{N}_{\rm H}$ (their Eqn. 11), from which $\bar{N}_{\rm H}$ $\sim$ 2.4 $\times 10^{22}$ cm$^{-2}$ would be inferred for R136a1 is it were an equal mass binary system, separated by 3 AU. This is an order of magnitude higher than the estimate for R136a (Townsley et al. 2006; Guerrero & Chu 2008), arising from a combination of foreground and internal components.
Of course, a significant fraction of the shock energy could produce relativistic particles rather than X-ray emission (e.g. Pittard & Dougherty 2006). Formally, an equal mass binary system whose components are separated by 600 AU is predicted to produce an X-ray luminosity of L$_{\rm X}$ = 2.7 $\times$ 10$^{34}$ erg s$^{-1}$, which is comparable to the total intrinsic X-ray luminosity of R136a (Guerrero & Chu 2008). If 90% of the shock energy were to contribute to relativistic particles, the same X-ray luminosity could arise from a system whose components were separated by 60 AU. Still, the relatively low X-ray luminosity of R136a suggests that if any of the R136a WN5 stars are composed of equal mass binaries, their separations would need to be [*in excess of*]{} 100–200 AU. How wide could very massive binaries be within such a dense cluster? To address this question we now consider dynamical interactions.
Dynamical interactions
----------------------
The binding energy of a ‘hard’ 100 M$_{\odot}$ + 100 M$_{\odot}$ binary with a separation of 100 AU is comparable to the binding energy of a massive cluster ( 10$^{41}$ J). Such a system will have frequent encounters with other massive stars or binaries. The collision rate for a system of separation, $a$, in a cluster of number density, $n$, and velocity dispersion, $\sigma$, is $$T_{\rm coll} \sim 30 n \sigma a^{2} (1 - \theta)$$ where $\theta$ is the Safranov number, indicating the importance of gravitational focusing (Binney & Tremaine 1987). Obviously, an encounter with a low mass star will not affect a very massive binary, so we assume only that encounters with stars in excess of 50 M$_{\odot}$ will significantly affect a very massive binary.
For a 100 + 100 M$_{\odot}$ binary with 100 AU separation in a cluster with a velocity dispersion of $\sigma$ = 5 km s$^{-1}$, initially containing a number density of 500 pc$^{-3}$ ($n$ = 1.5 x 10$^{-47}$ m$^{-3}$) then $\theta$ = 35. Each very massive binary should have an encounter every 1.8 Myr. For binaries wider than 100 AU the encounter rate will be even higher, since the encounter rate scales with the square of the separation, i.e. encounters every $\sim$0.2 Myr for separations of 300 AU or 0.05 Myr for separations of 600 AU.
Such close encounters will create an unstable multiple system that will rapidly decay by ejection of the lowest mass star (Anosova 1986). This would not necessarily destroy the binary. Instead, it will harden the system in order to gain the energy required to eject the other star. Thus very massive systems separated by more than $\sim$100 AU will reduce their separations through dynamical interactions with other high mass stars within the dense core of R136a.
{width="0.5\columnwidth"} {width="0.5\columnwidth"} {width="0.5\columnwidth"}
Should one of the high mass binaries in R136a elude hardening in the way outlined above, would it still escape detection if it remained at a large separation? One binary with components of 150 + 150 M$_{\odot}$ separated by $\sim$300 AU for which $\sim$30% of the shock energy contributes to the X-ray luminosity, together with intrinsic X-ray luminosities from single R136a members, obeying L$_{\rm
X}$/L$_{\rm Bol}$ $\sim 10^{-7}$ (Chlebowski et al. 1989) would indeed mimic the observed R136a X-ray luminosity (Guerrero & Chu 2008).
In summary, if we adopt similar ratios of X-ray to bolometric luminosities as for R136c and NGC 3603C, we would expect an X-ray luminosity from R136a that is a factor of 15 times higher than the observed value if a1, a2 and a3 were each colliding wind systems. Alternatively, we have followed the colliding wind theory of Stevens et al. (1992) and Pittard & Stevens (2002), and conservatively assume 30% of the shock energy contributes to the X-ray luminosity. If [*all*]{} the very massive stars within the Chandra field-of-view (R136a1, a2, a3 and a5) were members of equal mass binaries with separations of 300 AU, we would expect a X-ray luminosity that is over a factor of 2 higher than observed. In view of dynamical effects, at most one of the WN components of R136a might be a long period, large separation ($\geq$300 AU) equal mass binary. We cannot rule out short-period, highly unequal-mass binary systems of course, but such cases would have little bearing upon our derived stellar masses.
---------- -------------------------- ------------- ------ -------------------------- ----------------------------- ------
Name $M_{\rm cl}$ $\tau$ Ref M$_{\rm Max}^{\rm init}$ $M_{\rm Max}^{\rm current}$ Ref
$M_{\odot}$ Myr $M_{\odot}$ $M_{\odot}$
NGC 3603 10$^{4}$ $\sim$1.5 a, b 166$\pm$20 132$\pm$13 b
Arches 2$\times 10^{4}$ 2.5$\pm$0.5 c, d 120–150 $\geq$95–120 b, d
$\sim$1.8 b $\geq$185$^{+75}_{-45}$ $\geq$130$^{+45}_{-25}$ b
R136 $\leq$5.5$\times 10^{4}$ $\sim$1.7 e, b 320$^{+100}_{-40}$ 265$^{+80}_{-35}$ b
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: Comparison between the properties of the NGC 3603, Arches and R136 star clusters and their most massive stars, $M_{\rm Max}$.[]{data-label="clusters"}
\(a) Harayama et al. (2008); (b) This work; (c) Figer (2008); (d) Martins et al. (2008); (e) Hunter et al. (1995)
Cluster Simulations
===================
We now compare the highest mass stars in R136 and NGC 3603 with cluster predictions, randomly sampled from the stellar IMF, for a range of stellar mass limits. For R136, the stellar mass within a radius $\sim$4.7 pc from R136a1 has been inferred for $\geq$2.8 M$_{\odot}$ at 2.0 $\times$ 10$^{4}$ M$_{\odot}$ (Hunter et al. 1995). If the standard IMF (Kroupa 2002) is adopted for $<$ 2.8 M$_{\odot}$ a total cluster mass of $\leq$5.5 $\times$ 10$^{4}$ M$_{\odot}$ is inferred. Andersen et al. (2009) estimated a factor of two higher mass by extrapolating their measured mass down to 2.1 M$_{\odot}$ within 7 pc with a Salpeter slope to 0.5 M$_{\odot}$. These are upper limits to the mass of R136 itself since the mass function in the very centre of R136 cannot be measured (Maíz Apellániz 2008), such that they include the associated halo of older ($\geq$ 3 Myr) early-type stars viewed in projection – indeed Andersen et al. (2009) obtained an age of 3 Myr for this larger region. Examples of such stars include R136b (WN9h), R134 (WN6(h)) and Melnick 33Sb (WC5).
For NGC 3603, we adopt a cluster mass of 1.0 $\times 10^{4}$ M$_{\odot}$, the lower limit obtained by Harayama et al. (2008) from high resolution VLT NAOS/CONICA imaging. These are presented in Table \[clusters\], together with the Arches cluster, for which Figer (2008) estimated a cluster mass of 2 $\times 10^{4}$ M$_{\odot}$ based on the mass function of Kim et al. (2006).
Cluster populations
-------------------
We simulate a population of clusters and stars by randomly sampling first from a power-law cluster mass function (CMF), and then populating each cluster with stars drawn randomly from a stellar IMF (Parker & Goodwin 2007). Cluster masses are selected from a CMF of the form $N(M) \propto
M^{-\beta}$, with standard slope $\beta$ = 2 (Lada & Lada 2003) between cluster mass limits 50 M$_{\odot}$ and 2 $\times$ 10$^{5}$ M$_{\odot}$. These limits enable the full range of cluster masses to be sampled, including those with masses similar to that of R136. The total mass of clusters is set to 10$^{9}$ M$_{\odot}$ to fully sample the range of cluster masses. Each cluster is populated with stars drawn from a three-part IMF (Kroupa 2002) of the form
$$N(M) \propto \left\{ \begin{array}{ll} M^{+0.3} \hspace{0.4cm} m_0 < M/{\rm M_{\odot}} < m_1 \,, \\
M^{-1.3} \hspace{0.4cm} m_1 < M/{\rm M_{\odot}} < m_2 \,, \\
M^{-2.3} \hspace{0.4cm} m_2 < M/{\rm M_{\odot}} < m_3 \,,
\end{array} \right.$$
where $m_0 = 0.02$M$_\odot$, $m_1 = 0.1$M$_\odot$ and $m_2 =
0.5$M$_\odot$. We use three different values for $m_{3}$ in the simulations; 150 M$_{\odot}$, 300 M$_{\odot}$ and 1000 M$_{\odot}$. Stellar mass is added to the cluster until the total mass is within 2% of the cluster mass. If the final star added to the cluster exceeds this tolerance, then the cluster is entirely repopulated (Goodwin & Pagel 2005).
Our random sampling of the IMF allows low-mass clusters to be composed of one massive star and little other stellar material. This contravenes the proposed fundamental cluster mass-maximum stellar mass (CMMSM) relation (Weidner & Kroupa 2006, Weidner et al. 2010). However, the average maximum stellar mass for a given cluster mass closely follows the CMMSM relation (Parker & Goodwin 2007, Maschberger & Clarke 2008). The results for our three Monte Carlo runs are shown in Fig \[most\_massive\_star\]. They reveal that the average relation between the maximum stellar mass and cluster mass (in the 10$^{2}$ and 10$^{4}$ M$_{\odot}$ interval) is recovered (Weidner et al. 2010), without the constraint that the cluster mass governs the maximum possible stellar mass (Parker & Goodwin 2007). We also indicate 25% and 75% quartiles for the instances of a 10$^{4}$ M$_{\odot}$ (NGC 3603), 2 $\times$ 10$^{4}$ M$_{\odot}$ (Arches) and 5 $\times$ 10$^{4}$ M$_{\odot}$ (R136) cluster.
R136
----
If we assume that R136a1, a2 and c are either single or the primary dominates the optical/IR light, we obtain an average of 260 M$_{\odot}$ for their initial masses. In reality this value will be an upper limit due to binarity and/or line-of-sight effects. Nevertheless, from Fig. \[most\_massive\_star\], we would expect the average of the three most massive stars in a cluster of mass 5 $\times 10^{4}$ M$_{\odot}$ to be 150 M$_{\odot}$, 230, 500 M$_{\odot}$ for an adopted upper mass limit of $m_{3}$ = 150, 300 and 1000 M$_{\odot}$, respectively. Therefore, an upper limit close to 300 M$_{\odot}$ is reasonably consistent with the stellar masses derived (see also Oey & Clarke 2005).
In the lower panel of Figure \[mass\_function\] we therefore present typical mass functions for a cluster of mass 5 $\times 10^{4}$ M$_{\odot}$ for an adopted upper limit of m$_{3}$ = 300 M$_{\odot}$. From a total of 1.05–1.10 $\times$ 10$^{5}$ stars we would expect $\sim$14 initially more massive than 100 M$_{\odot}$ to have formed within 5 parsec of R136a1, since this relates to the radius used by Hunter et al. (1995) to derive the cluster mass. Indeed, of the 12 brightest near-infrared sources within 5 parsec of R136a1, we infer initial evolutionary masses in excess of 100 M$_{\odot}$ for 10 cases (all entries in Table \[r136\_photom\] except for R136b and R139). Of course, this comparison should be tempered by (a) contributions of putative secondaries to the light of the primary in case of binarity; (b) dynamical ejection during the formation process (e.g. Brandl et al. 2007). Indeed, Evans et al. (2010b) propose that an O2III-If$^{\ast}$ star from the Tarantula survey is a potential high velocity runaway from R136, in spite of its high stellar mass ($\sim 90 M_{\odot}$).
Overall, R136 favours a factor of $\approx$2 higher stellar mass limit than is currently accepted. We now turn to NGC 3603 to assess whether it supports or contradicts this result.
NGC 3603
--------
On average, the highest initial stellar mass expected in a 10$^{4}$ M$_{\odot}$ cluster would be $\sim$ 120 M$_{\odot}$ with an upper mass limit of m$_{3}$ = 150 M$_{\odot}$, $\sim$ 200 M$_{\odot}$ for m$_{3}$ = 300 M$_{\odot}$ and $\sim$ 400 M$_{\odot}$ for m$_{3}$ = 10$^{3}$ M$_{\odot}$. Since we infer an initial mass of 166$_{-20}^{+20}$ M$_{\odot}$ for component B, a stellar limit intermediate between 150 and 300 M$_{\odot}$ might be expected, if it is either single or dominated by a single component[^2]. However, no stars initially more massive than 150 M$_{\odot}$ would be expected in 25% of 10$^{4}$ M$_{\odot}$ clusters for the case of m$_{3}$ = 300 M$_{\odot}$ (recall middle panel of Fig. \[most\_massive\_star\]).
In the upper panel of Figure \[mass\_function\] we present typical mass functions for a cluster of mass 1 $\times 10^{4}$ M$_{\odot}$ for an adopted upper mass limit of m$_{3}$ = 300 M$_{\odot}$. We would expect 3 stars initially more massive than 100 M$_{\odot}$, versus 4 observed in NGC 3603 (Table \[ngc3603\_photom\]). If we were to extend the upper limit of the IMF to m$_{3}$ = 1000 M$_{\odot}$, then the average of the three most massive stars of a 10$^{4}$ M$_{\odot}$ cluster would be 230 M$_{\odot}$ which is not supported by NGC 3603 (150 M$_{\odot}$).
Therefore, the very massive stars inferred here for both R136 and NGC 3603 are fully consistent with a mass limit close to m$_{3}$ = 300 M$_{\odot}$, both in terms of the number of stars initially exceeding 100 M$_{\odot}$ and the most massive star itself. Before we conclude this section, let us now consider the Arches cluster.
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Name Sp Type m$_{\rm K_{s}}$ m$_{\rm H}$ - m$_{\rm K_{s}}$ Ref A$_{\rm K_{s}}$ Ref (m-M)$_{0}$ Ref M$_{\rm K_{s}}$ BC$_{\rm Ref $\log L$ M$_{\rm init}$ M$_{\rm current}$ Ref
K_{s}}^{\ast}$
mag mag mag mag mag mag $L_{\odot}$ M$_{\odot}$ M$_{\odot}$
F6 WN8–9h 10.37$^{\ddag}$ 1.68$^{\ddag}$ a 2.8 $\pm$ 0.1 b 14.4 $\pm$ 0.1 c –6.83 $\pm$ 0.14 –4.0 $\pm$ 0.35 d 6.25 $\pm$ 0.15 $\geq$115$^{+30}_{-25}$ $\geq$90$^{+17}_{-13}$ h
10.07 1.96 e 3.1 $\pm$ 0.2 f 14.5 $\pm$ 0.1 g –7.53 $\pm$ 0.22 –4.0 $\pm$ 0.35 d 6.53 $\pm$ 0.16 $\geq$185$^{+75}_{-45}$ $\geq$130$^{+45}_{-25}$ h
F9 WN8–9h 10.77$^{\ddag}$ 1.67$^{\ddag}$ a 2.8 $\pm$ 0.1 b 14.4 $\pm$ 0.1 c –6.43 $\pm$ 0.14 –4.4 $\pm$ 0.35 d 6.25 $\pm$ 0.15 $\geq$115$^{+30}_{-25}$ $\geq$92$^{+19}_{-14}$ h
10.62 1.78 e 3.1 $\pm$ 0.2 f 14.5 $\pm$ 0.1 g –6.98 $\pm$ 0.22 –4.4 $\pm$ 0.35 d 6.47 $\pm$ 0.16 $\geq$165$^{+60}_{-40}$ $\geq$120$^{+35}_{-25}$ h
------ --------- ----------------- ------------------------------- ----- ----------------- ----- ---------------- ----- ------------------ ----------------- ----- ----------------- ------------------------- ------------------------- -----
\(a) Figer et al. (2002); (b) Stolte et al. (2002); (c) Eisenhauer et al. (2005); (d) Martins et al. (2008); (e) Espinoza et al. (2009); (f) Kim et al. (2006); (g) Reid (1993); (h) This work\
$\ddag$: NICMOS F205W and F160W filters; $\ast$: Includes –0.25 mag offset to bolometric correction relative to F. Martins (priv. comm.), as reported in Clark et al. (2009).
Arches
------
The upper mass limit of $\sim$150 M$_{\odot}$ from Figer (2005) was obtained for the Arches cluster from photometry. This was confirmed from spectroscopic analysis by Martins et al. (2008) who estimated initial masses of 120 – 150 M$_{\odot}$ range for the most luminous (late WN) stars. How can these observations be reconciled with our results for NGC 3603 and R136?
Let us consider each of the following scenarios:
1. the Arches cluster is sufficiently old (2.5 $\pm$ 0.5 Myr, Martins et al. 2008) that the highest mass stars have already undergone core-collapse,
2. the Arches is unusually deficient in very massive stars for such a high mass cluster, or
3. Previous studies have underestimated the masses of the highest mass stars in the Arches cluster.
We have compared the derived properties for the two most luminous WN stars in the Arches cluster (F6 and F9) from Martins et al. (2008) to the solar metallicity grids from Sect. 4. These are presented in Table \[arches\], revealing initial masses of $\geq$115$^{+30}_{-25}$ $M_{\odot}$ from comparison with non-rotating models. Uncertainties from solely K-band spectroscopy are significantly higher than our (UV)/optical/K-band analysis of WN stars in NGC 3603 and R136. Masses represent lower limits since Martins et al. (2008) indicated that the metallicity of the Arches cluster is moderately super-solar.
Core hydrogen exhaustion would be predicted to occur after $\sim$2.5 Myr for the non-rotating 150 M$_{\odot}$ solar metallicity models, with core-collapse SN anticipated prior to an age of 3 Myr. However, comparison between the properties of F6 and F9 – the most luminous WN stars in the Arches cluster – with our solar metallicity evolutionary models suggest ages of $\sim$2 Myr. On this basis, it would appear that scenario (i) is highly unlikely.
Let us therefore turn to scenario (ii), on the basis that the initial mass of the most massive star within the Arches cluster was 150 $M_{\odot}$. On average, the highest mass star in a 2 $\times 10^{4}$ M$_{\odot}$ cluster would be expected to possess an initial mass in excess of 200 $M_{\odot}$ for an upper mass limit of $m_{3}$ = 300 $M_{\odot}$. From our simulations, the highest mass star within such a cluster spans a fairly wide range (recall Fig. \[most\_massive\_star\]). However, an upper limit of 120 (150) $M_{\odot}$ would be anticipated in only 1% (5%) of cases if we were to adopt $m_{3}$ = 300 $M_{\odot}$. Could the Arches cluster be such a statistical oddity?
Our simulations indicate that a 2 $\times 10^{4}$ M$_{\odot}$ cluster would be expected to host [*six*]{} stars initially more massive than 100 $M_{\odot}$ for $m_{3}$ = 300 $M_{\odot}$. Our solar metallicity evolutionary models combined with spectroscopic results from Martins et al. (2008) suggest that the 5 most luminous WN stars (F1, F4, F6, F7, F9), with $\log
L/L_{\odot} \geq$ 6.15, are consistent with initial masses of $\geq$100 $M_{\odot}$, providing they are single or one component dominates their near-IR appearance.
Finally, let us turn to scenario (iii), namely that the stellar masses of the Arches stars have been underestimated to date. Martins et al. (2008) based their analysis upon HST/NICMOS photometry from Figer et al. (2002) together with a (low) Galactic Centre distance of 7.6 kpc from Eisenhauer et al. (2005) plus a (low) foreground extinction of $A_{\rm
K_{s}}$ = 2.8 mag from Stolte et al. (2002). Let us also consider the resulting stellar parameters and mass estimates on the basis of the standard Galactic Centre distance of 8 kpc (Reid 1993), more recent (higher) foreground extinction of $A_{\rm K_{s}}$ = 3.1 $\pm$ 0.19 mag from Kim et al. (2006), plus VLT/NACO photometric results from Espinoza et al. (2009)[^3]. These yield absolute K-band magnitudes that are 0.55–0.7 mag brighter than Figer et al. (2002), corresponding to 0.22–0.28 dex higher bolometric luminosities. With respect to the hitherto 150 M$_{\odot}$ stellar mass limit identified by Figer (2005), these absolute magnitude revisions would conspire to increasing the limit to $\geq$200 M$_{\odot}$.
In Table \[arches\] we provide the inferred properties of F6 and F9 on the basis of these photometric properties plus the stellar temperatures and K-band bolometric corrections derived by Martins et al. (2008). Our solar metallicity non-rotating models indicate initial masses of $\geq$185$^{+75}_{-45}$ M$_{\odot}$ and $geq$165$^{+60}_{-40}$ M$_{\odot}$ for F6 and F9, respectively. These are lower limits to the actual initial masses, in view of the super-solar metallicity of the Arches cluster (Martins et al. 2008). In total, $\sim$5 stars are consistent with initial masses in excess of $\simeq$150 M$_{\odot}$ (those listed above), plus a further 5 stars for which initial masses exceed $\sim$100 M$_{\odot}$ (F3, F8, F12, F14, F15).
Overall, [*ten*]{} stars initially more massive than $\sim$100 would suggest either that the mass of the Arches cluster approaches 3 $\times 10^{4}$ M$_{\odot}$ or several cases are near-equal mass binaries (Lang et al. 2005, Wang et al. 2006). As such, we would no longer require that the Arches cluster is a statistical oddity, since the highest mass star would not be expected to exceed 200 M$_{\odot}$ in 25% of 2 $\times 10^{4}$ M$_{\odot}$ clusters.
In conclusion, we have attempted to reconcile the properties of the highest mass stars in Arches cluster with an upper mass limit of $m_{3} =
300$ M$_{\odot}$. Based upon the Martins et al. (2008) study, comparison with evolutionary models suggests that the initial masses of the most massive stars are $\geq$115$^{+30}_{-25}$ M$_{\odot}$ with ages of $\sim$2 Myr. We find that an upper mass of 120 M$_{\odot}$ would be expected in only 1% of 2$\times 10^{4}$ M$_{\odot}$ clusters for which $m_{3} = 300$ M$_{\odot}$. However, use of contemporary near-IR photometry and foreground extinctions towards the Arches cluster, together with the standard 8 kpc Galactic Centre distance reveal an initial mass of $\geq$185$^{+75}_{-45}$ M$_{\odot}$ for the most massive star based on solar metallicity evolutionary models, with 4–5 stars consistent with initial masses $\geq$150 M$_{\odot}$. An upper mass of 200 M$_{\odot}$ would be expected in 25% of cases. Robust inferences probably await direct dynamical mass determinations of its brightest members, should they be multiple.
![Theoretical mass functions for clusters with stellar masses of (upper) 10$^{4}$ M$_{\odot}$ and (lower) 5 $\times 10^{4}$ M$_{\odot}$, for the scenario with an upper mass limit of $m_{3}$ = 300 M$_{\odot}$. 3 and 14 stars with initial masses in excess of 100 M$_{\odot}$ are anticipated, respectively. Our Monte Carlo approach (Parker & Goodwin 2007) breaks down for very massive stars in the upper panel due to small number statistics.[]{data-label="mass_function"}](mass_dist_104_150_b.ps "fig:"){width="0.7\columnwidth"} ![Theoretical mass functions for clusters with stellar masses of (upper) 10$^{4}$ M$_{\odot}$ and (lower) 5 $\times 10^{4}$ M$_{\odot}$, for the scenario with an upper mass limit of $m_{3}$ = 300 M$_{\odot}$. 3 and 14 stars with initial masses in excess of 100 M$_{\odot}$ are anticipated, respectively. Our Monte Carlo approach (Parker & Goodwin 2007) breaks down for very massive stars in the upper panel due to small number statistics.[]{data-label="mass_function"}](mass_dist5104_150_b.ps "fig:"){width="0.7\columnwidth"}
Global properties of R136
=========================
We will re-evaluate the ionizing and mechanical wind power resulting from all hot, luminous stars in R136 and NGC 3603 elsewhere (E. Doran et al. in preparation). Here, we shall consider the role played by the very massive WN stars in R136. We have updated the properties of early-type stars brighter than M$_{\rm V}$ = –4.5 mag within a radius of $\sim$5 parsec from R136a1 (Crowther & Dessart 1998) to take account of contemporary T$_{\rm eff}$–spectral type calibrations for Galactic stars (Conti et al. 2008), and theoretical mass-loss rates (Vink et al. 2001) for OB stars. Until a census of the early-type stars within R136 is complete, we adopt O3 subtypes for those stars lacking spectroscopy (Crowther & Dessart 1998). Aside from OB stars and the WN5h stars discussed here, we have included the contribution of other emission-line stars, namely O2–3If/WN stars for which we adopt identical temperatures to those of the WN5 stars, one other WN5h star (Melnick 34), a WN6(h) star (R134) for which we estimate T$_{\ast} \approx$ 42,000 K and a WN9h star (R136b) for which we estimate T$_{\ast} \approx$ 35,000 K from its HST/FOS and VLT/SINFONI spectroscopy. Finally, a WC5 star (Melnick 33Sb) lies at a projected distance of 2.9 pc for which we adopt similar wind properties and ionizing fluxes to single WC4 stars (Crowther et al. 2002).
![(Upper panel) Incremental Lyman continuum ionizing radiation from early-type stars within 5 parsec from R136a1 (Crowther & Dessart 1998) either following the Galactic O subtype-temperature calibration (solid black), or systematically increasing the temperature calibration by 2,000 K for LMC stars (dotted line). The four very luminous WN5h stars discussed here (dashed red line) contribute 43 - 46% of the total; (Lower panel) As above, except for the mechanical wind power from early-type stars within 5 parsec of R136a1, based upon theoretical wind prescriptions (Vink et al. 2001) to which the four WN5h stars contribute 34% of the total.[]{data-label="sum"}](power.eps){width="0.9\columnwidth"}
Fig. \[sum\] shows the integrated Lyman continuum ionizing fluxes and wind power from all early-type stars in R136, together with the explicit contribution of R136a1, a2, a3 and c, amounting to, respectively, 46% and 35% of the cluster total. If we were to adopt a 2,000 K systematically higher temperature calibration for O-type stars in the LMC, as suggested by some recent results (Mokiem et al. 2007, Massey et al. 2009), the contribution of WN5h stars to the integrated Lyman continuum flux and wind power is reduced to 43% and 34%, respectively.
We have compared our empirical results with population synthesis predictions (Leitherer et al. 1999) for a cluster of mass 5.5 $\times$ 10$^{4}$ M$_{\odot}$ calculated using a standard IMF (Kroupa 2002) and evolutionary models up to a maximum limit of 120 M$_{\odot}$. These stars alone approach the mechanical power and ionizing flux predicted for a R136-like cluster at an age of $\sim$1.7 Myr (Leitherer et al. 1999). Indeed, R136a1 alone provides the Lyman continuum output of seventy O7 dwarf stars (Conti et al. 2008), supplying 7% of the radio-derived $\sim$10$^{52}$ photon s$^{-1}$ ionizing flux from the entire 30 Doradus region (Mills et al. 1978, Israel & Koornneef 1979). Improved agreement with synthesis models would be expected if evolutionary models allowing for rotational mixing were used (Vazquez et al. 2007), together with contemporary mass-loss prescriptions for main-sequence stars.
Discussion and conclusions
==========================
If very massive stars – exceeding the currently accepted 150 $M_{\odot}$ limit – were to exist in the local universe, they would be:
1. Located in high mass ($\geq 10^{4}$ $M_{\odot}$), very young ($\leq$ 2 Myr) star clusters;
2. Visually the brightest stars in their host cluster, since L $\propto
M^{1.5}$ for zero age main sequence stars above 85 M$_{\odot}$[^4], and surface temperatures remain approximately constant for the first 1.5 Myr
3. Possess very powerful stellar winds, as a result of the mass dependence of the Eddington parameter, $\Gamma_{e} \propto L/M
\approx M^{0.5}$ for such stars. 150 – 300 M$_{\odot}$ zero age main sequence stars, for which $\Gamma_{e} \approx$ 0.4 – 0.55, would likely possess an O [*supergiant*]{} morphology, while a Wolf-Rayet appearance would be likely to develop within the first 1–2 Myr, albeit with significant residual surface hydrogen.
In this study we have presented spectroscopic analyses of bright WN stars located within R136 in the LMC and NGC 3603 in the Milky Way that perfectly match such anticipated characteristics. The combination of line blanketed spectroscopic tools and contemporary evolutionary models reveals excellent agreement with dynamical mass determinations for the components of A1 in NGC 3603, with component B possessing a higher initial mass of $\sim$170 M$_{\odot}$, under the assumption that it is single. Application to the higher temperature (see also Rühling 2008) brighter members of R136 suggests still higher initial masses of 165 – 320 M$_{\odot}$. Owing to their dense stellar winds, they have already lost up to 20% of their initial mass with the first $\sim$1.5 Myr of their main sequence evolution (see also de Koter et al. 1998). The R136 WN5 stars are moderately hydrogen depleted, to which rotating models provide the best agreement, whereas the NGC 3603 WN6 stars possess normal hydrogen contents, as predicted by non-rotating models at early phases. These differences may arise from different formation mechanisms (e.g. stellar mergers). Wolff et al. (2008) found a deficit of slow rotators within a sample of R136 early-type stars, although statistically significant results await rotational velocities from the VLT-FLAMES Tarantula Survey (Evans et al. 2010a).
We have assessed the potential that each of the R136 stars represent unresolved equal-mass binary systems. SINFONI spectroscopic observations argue against close, short period binaries (Schnurr et al. 2009). R136c possesses an X-ray luminosity which is a factor of $\sim$100 times greater than that which would be expected from a single star and is very likely a massive binary. In contrast, X-ray emission from R136a exceeds that expected from single stars by only a factor of $\sim$3, to which multiple wind interactions within the cluster will also contribute. If the R136a stars possessed similar X-ray properties to R136c and NGC 3603C, its X-ray emission would be at least a factor of 15 times higher than the observed luminosity. At most, only one of the WN sources within R136a might be a very long period, large separation ($\sim$300 AU), equal-mass binary system if as little as 30% of the shock energy contributes to the X-ray luminosity. Dynamical effects would harden such systems on $\ll$Myr timescales. We cannot rule out shorter-period, unequal-mass binary systems of course, but such cases would have little bearing upon our mass limit inferences.
------------ --------- ----------------- ----------------- ------------- ---------- ------------------ --------------- ----------------
Star Sp Type m$_{\rm K_{s}}$ A$_{\rm K_{s}}$ (m-M)$_{0}$ M$_{\rm BC$_{\rm K_{s}}$ M$_{\rm Bol}$ M$_{\rm init}$
K_{s}}$
mag mag mag mag mag mag M$_{\odot}$
R136a1 WN5h 11.1 0.2 18.45 –7.6 –5.0 –12.6 320
R136a2 WN5h 11.4 0.2 18.45 –7.3 –4.9 –12.2 240
R136c WN5h 11.3 0.3 18.45 –7.4 –4.7 –12.1 220$^{\ast}$
Mk34 WN5h 11.7 0.3 18.45 –7.1 –4.8: –11.9: 190$^{\ast}$
Arches F6 WN8–9h 10.1 3.1 14.5 –7.5 –4.0 –11.5 $\geq$185
NGC 3603 B WN6h 7.4 0.6 14.4 –7.5 –3.9 –11.4 166
R136a3 WN5h 11.7 0.2 18.45 –6.9 –4.8 –11.7 165
Arches F9 WN8–9h 10.6 3.1 14.5 –7.0 –4.4 –11.4 $\geq$165
Arches F4 WN7–8h 10.2 3.1 14.5 –7.4 –3.9 –11.3 $\geq$150:
NGC 3603 WN6h 8.0 0.6 14.4 –7.0 –4.2 –11.2 148
A1a
Arches F7 WN8–9h 10.3 3.1 14.5 –7.3 –4.0 –11.3 $\geq$148:
Arches F1 WN8–9h 10.35 3.1 14.5 –7.25 –4.0 –11.25 $\geq$145:
------------ --------- ----------------- ----------------- ------------- ---------- ------------------ --------------- ----------------
: Compilation of stars within R136/30 Dor, NGC 3603 and the Arches cluster whose initial masses exceed $\approx$150 M$_{\odot}$ according to evolutionary models presented here. Photometry, extinctions and bolometric corrections are presented in this study, with the exception of the Arches cluster for which we follow Espinoza et al. (2009), Kim et al. (2006) and Martins et al. (2008), respectively. Known or suspected binaries are marked with $\ast$.[]{data-label="vms"}
In Table \[vms\] we present a compilation of stars in R136 (30 Doradus), NGC 3603 and the Arches cluster whose initial masses challenge the currently accepted upper mass limit of $\sim$150 M$_{\odot}$. Although the formation of high mass stars remains an unsolved problem in astrophysics (Zinnecker & Yorke 2007), there are no theoretical arguments in favour of such a limit at 150 M$_{\odot}$ (e.g. Klapp et al. 1987) – indeed Massey & Hunter (1998) argued against an upper stellar limit based on R136 itself. Observations of the Arches cluster provide the primary evidence for such a sharp mass cutoff (Figer 2005). However, as Table \[vms\] illustrates, contemporary photometry and foreground extinction towards the cluster coupled with the spectroscopy results from Martins et al. (2008) suggest 4–5 stars initially exceed $\approx$150 M$_{\odot}$, with an estimate of $\geq$185 M$_{\odot}$ for the most luminous star. Recall Martins et al. (2008) used identical spectroscopic tools to those employed in the current study, plus near-IR spectroscopic observations which also form a central component of our study. On this basis the Arches cluster would no longer be a statistical oddity.
Monte Carlo simulations for various upper stellar mass limits permit estimates of the revised threshold from which $\sim$300 M$_{\odot}$ is obtained. It may be significant that both NGC 3603 and R136 are consistent with an identical upper limit, in spite of their different metallicities. Oey & Clarke (2005) obtained an upper limit of $\ll$500 M$_{\odot}$, primarily from a maximum stellar mass of 120 – 200 M$_{\odot}$ inferred for stars within R136a itself at that time.
Would there be any impact of an upper mass limit of order $\sim$300 M$_{\odot}$ on astrophysics? Population synthesis studies are widely applied to star-forming regions within galaxies, for which an upper mass limit of $\sim$ 120 M$_{\odot}$ is widely adopted (Leitherer et al. 1999). A number of properties are obtained from such studies, including star formation rates, enrichment of the local interstellar medium (ISM) through mechanical energy through winds, chemical enrichment and ionizing fluxes. A higher stellar mass limit would increase their global output, especially for situations in which rapid rotation leads to stars remaining at high stellar temperatures.
Could the presence of very high mass stars, such as those discussed here, be detected in spatially unresolved star clusters? The high luminosities of such stars inherently leads to the development of very powerful stellar winds at early evolutionary phases (1–2 Myr). Therefore the presence of such stars ensures that the integrated appearance of R136a (and the core of NGC 3603) exhibits broad He[ii]{} $\lambda$4686 emission, which at such early phases would not be the case for a lower stellar limit. Other high mass clusters, witnessed at a sufficiently early age, would also betray the presence of very massive stars through the presence of He[ii]{} emission lines at $\lambda$1640 and $\lambda$4686. Hitherto, such clusters may have been mistaken for older clusters exhibiting broad helium emission from classical Wolf-Rayet stars.
Finally, how might such exceptionally massive stars end their life? This question has been addressed from a theoretical perspective (Heger et al. 2003), suggesting Neutron Star remnants following Type Ib/c core-collapse supernovae close to solar metallicities, with weak supernovae and Black Hole remnants at LMC compositions. Rapid rotation causes evolution to proceed directly to the classical Wolf-Rayet phase, whereas slow rotators, such as the NGC 3603 WN6h stars, will likely produce a $\eta$ Car-like Luminous Blue Variable phase. Extremely metal-deficient stars exceeding $\sim$140 M$_{\odot}$ may end their lives prior to core-collapse (Bond et al. 1984). They would undergo an electron-positron pair-instability explosion during the advanced stages that would trigger the complete disruption of the star (Heger & Woosley 2002).
Until recently, such events have been expected from solely metal-free (Population III) stars, since models involving the collapse of primordial gas clouds suggest preferentially high characteristic masses as large as several hundred M$_{\odot}$ (Bromm & Larson 2004). Langer et al. (2007) have demonstrated that local pair-instability supernovae could be produced either by slow rotating moderately metal-poor ($\leq$ 1/3 Z$_{\odot}$) yellow hypergiants with thick hydrogen-rich envelopes – resembling SN 2006gy (Ofek et al. 2007; Smith et al. 2007) – or rapidly rotating very metal-deficient ($\leq 10^{-3}$ Z$_{\odot}$) Wolf-Rayet stars. Therefore it is unlikely any of the stars considered here are candidate pair-instability supernovae. Nevertheless, the potential for stars initially exceeding 140 M$_{\odot}$ within metal-poor galaxies suggests that such pair-instability supernovae could occur within the local universe, as has recently been claimed for SN 2007bi (Gal-Yam et al. 2009, see also Langer 2009).
Finally, close agreement between our spectroscopically derived mass-loss rates and theoretical predictions allows us to synthesise the ZAMS appearance of very massive stars. We find that the very highest mass progenitors will possess an emission-line appearance at the beginning of their main-sequence evolution due to their proximity to the Eddington limit. As such, spectroscopic dwarf O2–3 stars are not anticipated with the very highest masses. Of course the only direct means of establishing stellar masses is via close binaries. For the LMC metallicity, Massey et al. (2002) have obtained dynamical masses of a few systems, although spectroscopic and photometric searches for other high-mass, eclipsing binaries in 30 Doradus are in progress through the VLT-FLAMES Tarantula Survey (Evans et al. 2010a) and other studies (O. Schnurr et al. in prep.). Nevertheless, it is unlikely that any other system within 30 Doradus, or indeed the entire Local Group of galaxies will compete in mass with the brightest components of R136 discussed here.
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to C. J. Evans for providing VLT MAD images of R136 and photometric zero-points in advance of publication, and E. Antokhina, J. Maíz Apellániz, J. M. Pittard and I. R. Stevens for helpful discussions. We thank C. J. Evans and N. R. Walborn and an anonymous referee for their critical reading of the manuscript. Based on observations made with ESO Telescopes at the Paranal Observatory during MAD Science Demonstration runs SD1 and SD2, plus programme ID’s 69.D-0284 (ISAAC), 075.D-0577 (SINFONI), 076-D.0563 (SINFONI). Additional observations were taken with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Institute. STScI is operated by the association of Universities for Research in Astronomy, Inc. under the NASA contract NAS 5-26555. Financial support was provided to O. Schnurr and R.J. Parker by the Science and Technology Facilities Council. N. Yusof and H.A. Kassim gratefully acknowledge the University of Malaya and Ministry of Higher Education, Malaysia for financial support, while R. Hirschi acknowledges support from the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. A visit by N. Yusof to Keele University was supported by UNESCO Fellowships Programme in Support of Programme Priorities 2008-2009.
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\[lastpage\]
[^1]: [email protected]
[^2]: We do not attempt to claim that NGC 3603 B is single, even though it is photometrically and spectroscopically stable and is not X-ray bright (Moffat et al. 2004; Schnurr et al. 2008a).
[^3]: Indeed, if we adopt (H–K$_{s})_{0}$ = –0.11 mag (Crowther et al. 2006) for F6 and F9, $A_{\rm K_{s}}$ = 3.2 $\pm$ 0.2 mag is implied from the Espinoza et al. (2009) photometry.
[^4]: The exponent flattens further at the highest masses, such that L $\propto M^{1.3}$ for zero age main sequence stars between 300–500 M$_{\odot}$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We calculate the in-medium masses of the bottomonium states ($\Upsilon(1S)$, $\Upsilon(2S)$, $\Upsilon(3S)$ and $\Upsilon(4S)$) in isospin asymmetric strange hadronic matter at finite temperatures. The medium modifications of the masses arise due to the interaction of these heavy quarkonium states with the gluon condensates of QCD. The gluon condensates in the hot hadronic matter are computed from the medium modification of a scalar dilaton field within a chiral SU(3) model, introduced in the hadronic model to incorporate the broken scale invariance of QCD. There is seen to be drop in the masses of the bottomonium states and mass shifts are observed to be quite considerable at high densities for the excited states. The effects of density, isospin asymmetry, strangeness as well as temperature of the medium on the masses of the $\Upsilon$-states are investigated. The effects of the isospin asymmetry as well as strangeness fraction of the medium are seen to be appreciable at high densities and small temperatures. The density effects are the most dominant medium effects which should have observable consequences in the compressed baryonic matter (CBM) in the heavy ion collision experiments in the future facility at FAIR, GSI. The study of the $\Upsilon$ states will however require access to energies higher than the energy regime planned at CBM experiment. The density effects on the bottomonium masses should also show up in the dilepton spectra at the SPS energies, especially for the excited states for which the mass drop is observed to quite appreciable.'
author:
- Amruta Mishra
- Divakar Pathak
title: Bottomonium states in hot asymmetric strange hadronic matter
---
\#1
Introduction
============
The study of properties of hadrons in hot and dense matter is an important topic of contemporary research in strong interaction physics. The medium modifications of the hadrons affect the experimental observables from the strongly interacting matter created in the relativistic heavy ion collision experiments. The in-medium properties of the light vector mesons [@wambach; @hatsuda; @klinglnpa] are of relevance for observables like dilepton spectra, whereas modifications of the kaons and antikaons modify the production and propagation of these particles. The modifications of the masses of the charm mesons, $D$ ($\bar D$) mesons as well as of the charmonium states in the medium can modify the yield of the open charm mesons and of the charmonium states. A larger drop of the masses of the $D$ mesons as compared to the mass drop of the charmonium states can open decay channels of the excited states of charmonium to $D\bar D$ in the medium, which are not accessible in vacuum. This can reduce the production of $J/\psi$ from the decay of these excited charmonium states and thereby lead to J/$\psi$ suppression in heavy ion collision experiments. In high energy nuclear collisions, $J/\psi$ suppression could also be due to formation of quark gluon plasma [@blaiz; @satz]. In the presence of quark gluon plasma, the quark antiquark potential is screened leading to decrease in the binding energy of the quarkonium states and when the distance between the quark and antiquark becomes larger than the inverse of the Debye mass, the $Q\bar Q$ can no longer exist as a bound state. This can lead to step-wise melting of the quarkonium states from the hot deconfined matter and hence cause suppression of the ground states $J/\psi$ and $\Upsilon(1S)$ as the feed-down from the excited quarkonium states will no longer be possible at temperatures above the dissociation temperatures of the excited states [@satz2013]. At still higher temperatures, the quarkonium ground state can also dissociate due to color screening. The suppression of the heavy quarkonium states in the deconfined matter [@mocsy2013], can also be due to the processes of gluo-dissociation [@satzgluodiss] and inelastic parton scattering [@rapppartscatt]. In the former process, the quarkonium state breaks up due to interaction with a hard thermal gluon which changes the color singlet quarkonium state to an unbound color octet $Q\bar Q$ pair. In the inelastic parton scattering, the quarkonium state interacts with a parton (quark or gluon) through exchange of gluons, leading to dissociation of the heavy quarkonium. An effective field theory describing the quarkonia systems through potentials, namely pNRQCD (potential nonrelativistic QCD) has been extensively used in the literature [@brambilla1]. Within this framework, two mechanisms contributing to the decay width of the quarkonium were identified as singlet-to-octet thermal breakup and Landau damping. In the leading order, these correspond to gluo-dissociation and dissociation of quarkonium due to inelastic parton scattering [@brambilla2].
The masses of the $D$ ($\bar D$) mesons, which consist of a heavy charm quark (antiquark) and a light antiquark (quark), within the QCD sum rule approach, are modified largely due to their interaction with the light quark condensates in the hadronic medium [@haya1]. On the contrary, the masses of the heavy quarkonium states, e.g., the charmonium states, are modified in the leading order due to their interaction with the gluon condensates in the hadronic medium [@leeko; @leeko2; @kimlee]. These gluon condensates can be written in terms of the color electric and color magnetic fields. The modifications of the masses of the charmonium states, due to changes in the gluon condensates in the nuclear medium have been studied in the linear density approximation [@leeko; @leeko2], using the leading order QCD formula [@pes1]. The formula for the charmonium mass shift becomes proportional to $\langle \frac{\alpha_s}{\pi} {\vec E}^2\rangle$, similar to the second order Stark effect [@leeko; @leeko2], since the Wilson coefficient for the operator $\langle \frac{\alpha_s}{\pi}
{\vec B}^2\rangle$ vanishes in the non-relativistic limit. These studies [@leeko; @leeko2] show the drop of the $J/\psi$ mass at the nuclear matter saturation density to be quite small, whereas the masses of the excited charmonium states, e.g. $\psi(3686)$ and $\psi(3770)$ are observed to have appreciable drop in the nuclear medium. The charmonia masses in hot strange hadronic matter have been studied using the leading order QCD formula, due to changes in the gluon condensates in the hadronic medium calculated in a chiral effective model [@amepja]. The gluon condensate of QCD is simulated by a scalar dilaton field introduced in the effective hadronic model to incorporate scale symmetry breaking of QCD. The medium modifications of the decay widths of the charmonium states to $D\bar D$ have been studied arising from the mass shifts of the $D$ mesons in the medium [@frimanlee] as well as of the charmonium states [@amepja] accounting for the internal quark structure of the charmonium as well as $D(\bar D)$ mesons within the so called $^3P_0$ model. The charmonium decay widths in the medium have also been studied recently using a field theoretical model for composite hadrons with constituent quarks [@amspmchm]. Within the QCD sum rule approach, the mass modifications of the charmonium states, $J/\psi$ as well as $\eta_c$ have also been studied [@amchqsr] as arising from the medium modifications of the scalar gluon condensate and the twist-2 gluon operator, computed from the medium modification of the dilaton field in the chiral effective model. The mass shift of the D meson at finite density was studied using the QCD sum rule approach [@haya1] using condensates upto dimension 4 in the operator product expansion and the dominant contribution to the D meson mass drop comes from the term $m_c\langle \bar q q \rangle$, which turns out to be appreciable due to the large value of $m_c$. The mass shifts and the splitting of the $D$-$\bar D$ (as well as $B$-$\bar B$) meson masses have been studied within the QCD sum rule approach by considering condensate operators upto dimension 5 in the operator product expansion. This includes also the contributions from the gluon condensates as well as the mixed quark-gluon condensate [@hilgerprc; @hilgerjpg], with the dominant contribution to the mass modifications of the $D(B)$ mesons arising from the term $m_c \langle \bar q q \rangle$ ($m_b \langle \bar q q \rangle$) in the operator product expansion. In the recent past, the discovery of many hadrons including the heavy charm [@physreplee2010] and bottom quarks, as well as, the observed suppression of the heavy quarkonia at SPS, RHIC and LHC, which could indicate the formation of deconfined matter, has added further motivation to study the properties of the heavy quarkonium states in the hot and dense matter resulting from heavy ion collision experiments. The charmonium suppression has been observed at SPS energies [@spsexpt] and at RHIC [@rhicjpsisupp]. The bottomonium suppression has been observed at RHIC for Au+Au collision at $\sqrt s$=200 GeV [@starcoll] as well as at LHC in Pb+Pb collision at $\sqrt s$= 2.76 TeV [@cms]. The quarkonia production have been studied using a rate equation [@rappchmbott] for SPS (for Pb-Pb collision at 158 AGeV ($\sqrt {s}$=17.3 GeV)), RHIC (Au+Au collisions at $\sqrt s$=200 GeV) and LHC (Pb+Pb collision at $\sqrt s$=2.76 TeV) energies, including their production from initial nucleon nucleon hard scattering and their regeneration from produced QGP, as well as accounting for the in-medium effects of quarkonia [@kochmbott]. At RHIC and LHC, the medium effects on the quarkonia production are due to the thermal effects, whereas the density effects on the heavy quarkonia production should be important at SPS, CERN as well as at the Compressed baryonic matter (CBM) experiment at FAIR project at the future facility at GSI [@gsifuture]. At the CBM experiment at FAIR, one of the areas of focus for research will be the study of rare probes, e.g., the heavy quarkonia. The measurements of the production of the heavy quarkonia will be possible due to the high beam intensities and long running times to be used in these experiments. The experimental facility will be using heavy ion beams in fixed target mode with beam energy of about 10 to 45 AGeV ($\sqrt s$=4.5 to 10 GeV). These energies will make the study of charmonia possible quite extensively, whereas, a study of bottomonia production will require access to higher energies, as the top energy of CBM is about the threshold energy when $b$ and $\bar b$ can be pair produced which can later combine to form the bottomonium state. The density effects, which seem to the important medium effects on the bottomonia masses in the present investigation, should also be possible to study at SPS energies, where the yields of the $\Upsilon$ states have already been measured in pA collisions at incident energy of 450 AGeV ($\sqrt s$=29.1GeV) by the NA50 Collaboration [@pANA50bottomonium]. These measurements can provide a baseline for the $\Upsilon$ production studies to be carried out in the ion-ion collisions at higher centre of mass energies.
In the present work, we study the in-medium masses of the bottomonium states, $\Upsilon(1S)$, $\Upsilon(2S)$, $\Upsilon(3S)$ and $\Upsilon(4S)$ in hot asymmetric strange hadronic matter due to the interaction with the gluon condensates using the leading order QCD formula. The medium modification of the gluon condensate in the hadronic medium is calculated from the medium modification of a scalar dilaton field introduced within a chiral SU(3) model [@papa] to incorporate scale symmetry breaking of QCD. The model has been used to study the in-medium properties of the vector mesons [@hartree; @kristof1], kaons and antikaons [@isoamss; @isoamss2]. The model has then been generalized to chiral SU(4) to derive the interactions of the charm $D(\bar D)$ mesons with the light hadron sector and study the effects of isospin asymmetry [@amarind], strangeness [@amepja] and temperature [@amdmeson] on the mass modifications of these mesons in the hadronic medium. The mass shifts of the charmonium states have been calculated due to the changes in the gluon condensates in the medium [@amepja; @amchqsr] obtained from the change of a scalar dilaton field, which mimics the scale symmetry breaking of QCD, in the chiral effective model. In the present investigation, we study the in-medium masses of bottomonium states, obtained from the dilaton field, $\chi$, calculated for the asymmetric strange hadronic matter at finite temperatures.
The outline of the paper is as follows : In section II, we discuss briefly the chiral $SU(3)$ model which has been used to investigate the mass modification of the bottomonium states in the present work. The medium modifications of the bottomonium masses arise from the medium modification of a scalar dilaton field introduced in the hadronic model to incorporate broken scale invariance of QCD leading to QCD trace anomaly. In section III, we summarize the results obtained in the present investigation.
The hadronic chiral $SU(3) \times SU(3)$ model
==============================================
We use a chiral $SU(3)$ model [@papa; @weinberg; @coleman; @bardeen], which incorporates the scale symmetry breaking of QCD through introduction of a scalar dilaton field [@sche1; @ellis] within the hadronic model. The modification of the gluon condensates in the hadronic matter is obtained from the medium modification of the scalar dilaton field, which then is used to investigate the in-medium masses of the bottomonium states in the hot isospin asymmetric strange hadronic matter. The effective hadronic chiral Lagrangian density contains the following terms: $${\cal L} = {\cal L}_{kin}+\sum_{W=X,Y,V,A,u} {\cal L}_{BW} +
{\cal L}_{vec} + {\cal L}_{0} + {\cal L}_{scalebreak}+ {\cal L}_{SB}
\label{genlag}$$ In Eq. (\[genlag\]), ${\cal L}_{kin}$ is kinetic energy term, ${\cal L}_{BW}$ is the baryon-meson interaction term in which the baryon-spin-0 meson interaction term generates the vacuum baryon masses. ${\cal L}_{vec}$ describes the dynamical mass generation of the vector mesons via couplings to the scalar mesons and contain additionally quartic self-interactions of the vector fields. ${\cal L}_{0}$ contains the meson-meson interaction terms inducing the spontaneous breaking of chiral symmerty, ${\cal L}_{scalebreak}$ is the scale invariance breaking logarithmic term. ${\cal L}_{SB}$ describes the explicit chiral symmetry breaking. To study the hadron properties at finite temperature and densities in the present investigation, we use the mean field approximation, where all the meson fields are treated as classical fields. In this approximation, only the scalar and the vector fields contribute to the baryon-meson interaction, ${\cal L}_{BW}$ since for all the other mesons, the expectation values are zero. The scale breaking term [@sche1; @ellis] of the Lagrangian density $${\cal L}_{scalebreak}= -\frac{1}{4} \chi^{4}
{\rm ln} \frac{\chi^{4}}{\chi_{0}^{4}} + \frac{d}{3} \chi^{4}
{\rm ln} \Bigg( \frac{\left( \sigma^{2} - \delta^{2}\right)\zeta }
{\sigma_{0}^{2} \zeta_{0}} \Big( \frac{\chi}{\chi_{0}}\Big) ^{3}\Bigg),
\label{scalebreak}$$ where $\chi$, $\sigma$, $\zeta$ and $\delta$ are the scalar dilaton field, non-strange scalar field, strange scalar field and the scalar-isovector field respectively. The effect of these logarithmic terms is to break the scale invariance, which leads to the trace of the energy momentum tensor as [@heide1] $$\theta_{\mu}^{\mu} = \chi \frac{\partial {\cal L}}{\partial \chi}
- 4{\cal L}
= -(1-d)\chi^{4}.
\label{tensor1}$$
The trace of the energy momentum tensor in QCD is given as $$\theta_{\mu}^{\mu} = \langle \frac{\beta_{QCD}}{2g}
G_{\mu\nu}^{a} G^{\mu\nu a} \rangle + \sum_i m_i \bar {q_i} q_i ,
\label{tensorquark}$$ where the second term in the trace accounts for the finite quark masses, with $m_i$ as the current quark mass for the quark of flavor, $i=u,d,s$. The one loop QCD $\beta$ function is given as $$\beta_{\rm {QCD}} \left( g \right) = -\frac{11 N_{c} g^{3}}{48 \pi^{2}}
\left( 1 - \frac{2 N_{f}}{11 N_{c}} \right),
\label{qcdbeta}$$ where $N_c=3$ is the number of colors and $N_f$ is the number of quark flavors. Comparing the trace of the energy momentum tensor in the chiral effective model given by (\[tensor1\]) to that of QCD given by (\[tensorquark\]) and using equation (\[qcdbeta\]), we obtain the scalar gluon condensate related to the dilaton field as, $$\langle \frac{\alpha_{s}}{\pi}
G_{\mu\nu}^{a} G^{\mu\nu a} \rangle = \frac{24}{(33-2N_f)}\left[ (1-d)\chi^{4}
+ \sum_i m_i \bar {q_i} q_i\right].
\label{chiglu}$$ The second term, $\sum_i m_i \bar q_i q_i$ can be identified to be the negative of the explicit chiral symmetry breaking term ${\cal L}_{SB}$ of equation (\[genlag\]) [@amchqsr] and is given as $$\begin{aligned}
\sum_i m_i \bar {q_i} q_i &=& - {\cal L} _{SB} =
\Big[ m_{\pi}^{2}
f_{\pi} \sigma
+ \left( \sqrt{2} m_{k}^{2}f_{k} - \frac{1}{\sqrt{2}}
m_{\pi}^{2} f_{\pi} \right) \zeta
\Big].
\label{lsb}\end{aligned}$$ We thus see from the equation (\[chiglu\]) that the scalar gluon condensate $\left\langle \frac{\alpha_{s}}{\pi} G_{\mu\nu}^{a}
G^{\mu\nu a}\right\rangle$ is proportional to the fourth power of the dilaton field, $\chi$, in the limiting situation of massless quarks [@amepja].
The coupled equations of motion for the non-strange scalar field $\sigma$, strange scalar field $ \zeta$, scalar-isovector field $ \delta$ and dilaton field $\chi$, are derived from the Lagrangian density. These equations of motion are solved to obtain the density and temperature dependent values of the scalar fields ($\sigma$, $\zeta$ and $\delta$) and the dilaton field, $\chi$, in the isospin asymmetric hot strange hadronic medium. The values of the scalar fields and the dilaton field, for baryon density, $\rho_B$ and temperature, T, are calculated for given values of the strangeness fraction of the medium, $f_s= \frac{\sum_{i} s_{i} \rho_{i}}{\rho_{B}}$, and isospin asymmetry parameter, $\eta= -\frac{\sum_{i} I_{3i} \rho_{i}}{\rho_{B}}$, where $s_{i}$ is the number of strange quarks of baryon $i$ (i=p,n,$\Lambda$, $\Sigma ^{\pm,0}$, $\Xi^{-,0}$) and $I_{3i}$ is the third component of isospin of the $i$-th baryon. The in-medium masses of heavy quarkonium states are modified due to the modifications of the scalar gluon and the twist-2 gluon condensates [@haya1; @leeko] in the hadronic medium. These gluon condensates can be written in terms of the color electric and color magnetic fields, $\langle \frac{\alpha_s}{\pi} {\vec E}^2\rangle$ and $\langle \frac{\alpha_s}{\pi} {\vec B}^2\rangle$ [@david] and as has already been mentioned, for heavy quarkonium states, the contribution from the magnetic field part vanishes. Hence the mass shift for these states in the hadronic medium arise due to the change in the electric field part, $\frac{\alpha_s}{\pi}
\langle {\vec E}^2\rangle$, similar to the second order Stark effect [@leeko]. In the leading order mass shift formula derived in the large bottom quark mass limit [@pes1], the shift in the mass of the bottomonium state is given as [@leeko] $$\Delta m_{\Upsilon} = -\frac{1}{9} \int dk^{2} \vert
\frac{\partial \psi (k)}{\partial k} \vert^{2} \frac{k}{k^{2}
/ m_{b} + \epsilon} \bigg (
\left\langle \frac{\alpha_{s}}{\pi} E^{2} \right\rangle-
\left\langle \frac{\alpha_{s}}{\pi} E^{2} \right\rangle_{0}
\bigg ).
\label{massl}$$ In the above, $m_b$ is the mass of the bottom quark, $m_\Upsilon$ is the vacuum mass of the bottomonium state and $\epsilon = 2 m_{b} - m_{\Upsilon}$ represents the binding energy of the bottomonium state. $\psi (k)$ is the wave function of the bottomonium state in the momentum space, normalized as $\int\frac{d^{3}k}{(2\pi)^{3}}
\vert \psi(k) \vert^{2} = 1 $ [@leetemp]. It may be noted here that the leading order perturbative formula (\[massl\]) for the quarkonium ($Q\bar Q$) state has been derived [@pes1] in the limit of large quark mass, so that the inverse of the radius of the quarkonium state is much larger than its binding energy, $\epsilon$. Furthermore, the $Q\bar Q$ binding in the heavy quarkonium state is approximated to be one-gluon exchange Coulomb potential, where the gluon energy is small compared to the binding energy of the quarkonium state. The wave function of the quarkonium state in this approximation is Coulombic. However, the Coulombic wave function for the quarkonium state may not be realistic for the excited states of the quarkonium state [@leeko]. The mass modifications of the charmonium states due to their interaction with the gluon condensates were studied in Ref. [@leeko; @amepja], assuming the wave functions to be harmonic oscillator type. In the present work, we also use the harmonic oscillator wave functions to study of the medium modifications of the masses of the bottomonium states, due to their interactions with the gluon condensates. In the non-relativistic limit, due to vanishing of the contribution from the magnetic field, the expectation value of the scalar gluon condensate can be expressed in terms of the color electric field and hence the mass shift formula for the bottomonium states can be written in terms of the difference in the value of the scalar gluon condensate in the medium and in the vacuum as [@amepja] $$\begin{aligned}
\Delta m_{\Upsilon} &=& \frac{1}{18} \int dk^{2} \vert
\frac{\partial \psi (k)}{\partial k} \vert^{2} \frac{k}{k^{2}
/ m_{b} + \epsilon}
\bigg (
\left\langle \frac{\alpha_{s}}{\pi}
G_{\mu\nu}^{a} G^{\mu\nu a}\right\rangle -
\left\langle \frac{\alpha_{s}}{\pi}
G_{\mu\nu}^{a} G^{\mu\nu a}\right\rangle _{0}
\bigg ).
\label{mass1}\end{aligned}$$
In the present investigation, the wave functions for the bottomonium states are taken to be Gaussian and are given as [@frimanlee] $$\psi_{N, l} = {\tilde {\rm N}} Y_{l}^{m} (\theta, \phi)
(\beta^{2} r^{2})^{\frac{l}{2}} e^{-\frac{1}{2} \beta^{2} r^{2}}
L_{N - 1}^{l + \frac{1}{2}} \left( \beta^{2} r^{2}\right)
\label{wavefn}$$ where $\beta^{2} = M \omega / \hbar$ is the parameter describing the strength of the harmonic oscillator potential, $M = m_{b}/2$ is the reduced mass of the bottom quark and bottom anti-quark system, $L_{p}^{k} (z)$ is the associated Laguerre polynomial. $\tilde {\rm N}$ is the normalization constant determined from $\int |\psi_{N,l}({\bf r})|^2 d^3 r=1$. The oscillator constant $\beta$ is calculated from the decay widths of the bottomonium state to $e^+e^-$ given by the formula [@vanroyen; @spmmesonspect]
$$\Gamma_{\Upsilon \rightarrow e^+e^-}=\frac{16 \pi \alpha^2}{9M_\Upsilon^2}
|\psi (\bf 0)|^2,
\label{leptonicdw}$$
where, $\alpha=\frac{1}{137}$ is the fine structure constant and $\psi(\bf 0)$ is the wave function of the bottomonium state at the origin. Knowing the wave functions of the bottomonium states and calculating the medium modification of the scalar gluon condensate from the in-medium dilaton field and the scalar fields, we obtain the mass shift of the bottomonium states. In the next section we discuss the results for these in-medium bottomonium masses in hot asymmetric strange hadronic matter obtained in the present work.
Results and Discussions
=======================
![(Color online) The scalar gluon condensate, $G_0\equiv\langle (\alpha_s/{\pi})G_{\mu \nu}^aG^{\mu \nu a}\rangle$ is plotted as a function of the baryon density, $\rho_B$ in units of nuclear matter saturation density, $\rho_0$. This is shown for different values of the temperature for typical values of the strangeness fraction of the medium, $f_s$, in subplots (a), (c) and (e) for the isospin symmetric matter and in subplots (b), (d) and (f) for the isospin asymmetric matter with $\eta$=0.5. []{data-label="ggscdens"}](fig1.eps){width="16cm" height="16cm"}
![(Color online) The mass shift of $\Upsilon(1S)$ plotted as a function of the baryon density in units of nuclear matter saturation density at given temperatures, for different values of the strangeness fraction for the isospin symmetric ($\eta$=0) as well as isospin asymmetric ($\eta$=0.5) hadronic matter. []{data-label="dmupsln1s"}](fig2.eps){width="16cm" height="16cm"}
![(Color online) The mass shift of $\Upsilon(2S)$ plotted as a function of the baryon density in units of nuclear matter saturation density at given temperatures, for different values of the strangeness fraction for the isospin symmetric ($\eta$=0) as well as isospin asymmetric ($\eta$=0.5) hadronic matter. []{data-label="dmupsln2s"}](fig3.eps){width="16cm" height="16cm"}
![(Color online) The mass shift of $\Upsilon(3S)$ plotted as a function of the baryon density in units of nuclear matter saturation density at given temperatures, for different values of the strangeness fraction for the isospin symmetric ($\eta$=0) as well as isospin asymmetric ($\eta$=0.5) hadronic matter. []{data-label="dmupsln3s"}](fig4.eps){width="16cm" height="16cm"}
![(Color online) The mass shift of $\Upsilon(4S)$ plotted as a function of the baryon density in units of nuclear matter saturation density at given temperatures, for different values of the strangeness fraction for the isospin symmetric ($\eta$=0) as well as isospin asymmetric ($\eta$=0.5) hadronic matter. []{data-label="dmupsln4s"}](fig5.eps){width="16cm" height="16cm"}
![(Color online) The rms radius of $\Upsilon(1S)$ plotted as a function of the baryon density in units of nuclear matter saturation density at given temperatures, for different values of the strangeness fraction for the isospin symmetric ($\eta$=0) as well as isospin asymmetric ($\eta$=0.5) hadronic matter. []{data-label="rmsupsln1s"}](fig6.eps){width="16cm" height="16cm"}
![(Color online) The rms radius of $\Upsilon(2S)$ plotted as a function of the baryon density in units of nuclear matter saturation density at given temperatures, for different values of the strangeness fraction for the isospin symmetric ($\eta$=0) as well as isospin asymmetric ($\eta$=0.5) hadronic matter. []{data-label="rmsupsln2s"}](fig7.eps){width="16cm" height="16cm"}
![(Color online) The rms radius of $\Upsilon(3S)$ plotted as a function of the baryon density in units of nuclear matter saturation density at given temperatures, for different values of the strangeness fraction for the isospin symmetric ($\eta$=0) as well as isospin asymmetric ($\eta$=0.5) hadronic matter. []{data-label="rmsupsln3s"}](fig8.eps){width="16cm" height="16cm"}
![(Color online) The rms radius of $\Upsilon(4S)$ plotted as a function of the baryon density in units of nuclear matter saturation density at given temperatures, for different values of the strangeness fraction for the isospin symmetric ($\eta$=0) as well as isospin asymmetric ($\eta$=0.5) hadronic matter. []{data-label="rmsupsln4s"}](fig9.eps){width="16cm" height="16cm"}
In this section, we investigate the effects of density and temperature on the masses of the bottomonium states for given isospin-asymmetry and strangeness of the hadronic medium. Within a chiral SU(3) model, the scalar gluon condensate in the hadronic matter has been calculated from the change in the dilaton field and scalar fields. Using the leading order QCD formula for the mass shift of heavy quarkonium state, the medium modification of the charmonium states $J/\psi$, $\psi(3686)$ and $\psi (3770)$ were studied from the modification of the gluon condensate in the hadronic matter [@leeko; @amepja]. In the present work, we investigate the medium modification of the $\Upsilon$-states using equation (\[mass1\]) from the value of the scalar gluon condensate calculated in the chiral SU(3) model. The value of the mass of the b-quark is taken to be $m_b$=5.36 GeV [@repko] in the present investigation. This value of $m_b$ is similar to the value of 5.1 GeV [@bhanotpeskin9] and 5.005 GeV [@spmmesonspect] used in the potential model calculations for the study of the bottomonium spectroscopy. As has already been mentioned the wave functions of the $\Upsilon$-states have been taken to be harmonic oscillator eigenfunctions as given by equation (\[wavefn\]) and the values of the harmonic oscillator strength, $\beta$ for the states are determined from the experimental values of the leptonic decay widths of the bottomonium states, given by equation (\[leptonicdw\]). The values of the decay widths ($\Upsilon \rightarrow e^+e^-$) of 1.34 keV, 0.612 keV, 0.443 keV and 0.272 keV for the $\Upsilon$-states, $\Upsilon (1S)$, $\Upsilon (2S)$, $\Upsilon (3S)$ and $\Upsilon (4S)$ [@pdg2012] give the values of the harmonic oscillation strength, $\beta$ as 1309.2, 915.4, 779.75 and 638.6 MeV respectively. These values of the parameter $\beta$ may be compared with the calculations of the heavy quarkonium systems using a quark model [@spmmesonspect] with a linear confining potential, calculated as 1291 and 953.46 MeV for the $\Upsilon (1S)$ and $\Upsilon (2S)$ states.
In figure \[ggscdens\], the density dependence of the scalar gluon condensate, $G_0=\langle (\alpha_s/{\pi})G_{\mu \nu}^aG^{\mu \nu a}\rangle$ is shown for temperatures, T=0, 100 and 150 MeV, for the isospin symmetric matter in subplots (a), (c) and (e), for values of the strangeness fraction, $f_s$ as 0, 0.3 and 0.5. These are compared with the values for the isospin asymmetric matter with $\eta$=0.5 plotted in (b), (d) and (f). The scalar gluon condensate is calculated in the chiral SU(3) model, using equations (\[chiglu\]) and (\[lsb\]), from the in-medium values of the dilaton field, $\chi$ and the scalar fields, $\sigma$, $\zeta$ and $\delta$ in the isospin asymmetric strange hadronic medium, obtained by solving the coupled equations of motion of these fields. For the isospin symmetric nuclear matter, the value of the scalar gluon condensate changes from the vacuum value of 1.9387$\times$10$^{-2}$GeV$^4$ to 1.9$\times$10$^{-2}$GeV$^4$ at the nuclear matter saturation density, and to 1.822$\times$10$^{-2}$GeV$^4$ and 1.7155$\times$10$^{-2}$GeV$^4$ at densities 2$\rho_0$ and 4$\rho_0$ respectively. In the presence of hyperons in the system, the dilaton field is observed to have a larger drop in the medium. When the quark masses are neglected, the second term of the trace of the energy momentum tensor does not contribute and the scalar gluon condensate becomes proportional to the fourth power of the dilaton field. However, the contributions from the quark condensate term in the trace of the energy momentum tensor expressed in terms of the scalar fields, $\sigma$ and $\zeta$, modifies the value of the scalar gluon condensate in the medium. At zero temperature, for symmetric hadronic matter, the modification arising from the finite strangeness fraction is observed to be small, with the value of the scalar gluon condensate amounting to 1.9075$\times$10$^{-2}$GeV$^4$, 1.8336$\times$10$^{-2}$GeV$^4$, and 1.7223$\times$10$^{-2}$GeV$^4$, for $f_s$=0.3 and 1.9117$\times$10$^{-2}$GeV$^4$, 1.8412$\times$10$^{-2}$GeV$^4$, and 1.7248$\times$10$^{-2}$GeV$^4$, for $f_s$=0.5, for densities of $\rho_0$, 2$\rho_0$ and 4$\rho_0$ respectively. At T=0, the effect from the finite $f_s$ is observed to be larger for the isospin asymmetric case as compared to the symmetric situation. The difference in the in-medium gluon condensate from the vacuum value is observed to decrease as the temperature is increased, due to the fact that the drop in the magnitude in the dilaton and the scalar fields, $\sigma$ and $\zeta$ decrease with increase in temperature.
In figures \[dmupsln1s\], \[dmupsln2s\], \[dmupsln3s\] and \[dmupsln4s\], we show the density dependence of the mass shifts of the upsilon states, $\Upsilon(1S)$, $\Upsilon(2S)$, $\Upsilon(3S)$ and $\Upsilon(4S)$ respectively, in isospin asymmetric strange hadronic matter. The mass of $\Upsilon (1S)$ is observed to drop from its vacuum value of 9460.3 MeV to 9459.94 MeV (9458.12 MeV) at density of $\rho_0$ (4$\rho_0$) for symmetric nuclear matter at zero temperature. For the asymmetric nuclear matter ($\eta$=0.5), the in-medium mass is changed to 9459.956 MeV (9458.36 MeV) at density $\rho_0$ (4$\rho_0$). The strangeness fraction in the medium is seen to lead to marginal modification for the symmetric matter, whereas it is larger for the asymmetric situation for T=0. However, the value of the mass shift remains small, of the order of about 3 MeV at a density of six times nuclear matter density. With increase in temperature, the drop is seen to be even less (of the order of 2.5 MeV at a density of 6$\rho_0$). For $\Upsilon (2S)$ plotted in figure \[dmupsln2s\], the mass is observed to be modified to 10019.83 MeV (10002.49 MeV) at $\rho_0$ (4$\rho_0$) from its vacuum value of 10023.26 MeV. However, the effects from temperature, strangeness and isospin asymmetry remain small as compared to the modification of the mass from density effects. For $\Upsilon (3S)$ shown in figure \[dmupsln3s\], the mass shift is observed to be 12.21 MeV (73.916 MeV) from its vacuum value of 10355.2 MeV, at density of $\rho_0$ (4$\rho_0$) for symmetric nuclear matter at T=0. The dependence on $f_s$ is seen to be larger for the asymmetric case at high densities. The mass shift of the $\Upsilon$ state is proportional to the difference in the value of the scalar gluon condensate from the vacuum value, with the constant of proportionality determined by the integral of equation (\[mass1\]) whose integrand is given in terms of the wave function of the specific $\Upsilon$ state. For given values of the temperature, isospin asymmetry, strangeness fraction and density of the medium, the ratio of the magnitudes of the mass shifts for the $\Upsilon(1S)$, $\Upsilon (2S)$, $\Upsilon (3S)$ and $\Upsilon (4S)$ states turns out to be the ratio of the magnitudes of the integral calculated from their respective wave functions. These mass shifts are observed to be in the ratio $\Delta m_{\Upsilon (1s)}$:$\Delta m_{\Upsilon (2s)}$:$\Delta m_{\Upsilon (3s)}$:$\Delta m_{\Upsilon (4s)}$ = 1$\;$:$\;$9.53$\;$:$\;$33.9$\;$:$\;$137.8. The mass drop of the excited states are thus observed to be larger for the excited states and for $\Upsilon (4S)$, the drop in the mass is seen to be about 49.64 MeV (300.4 MeV) from the vacuum mass of 10579.4 MeV, at a density of $\rho_0$(4$\rho_0$) for symmetric nuclear matter at zero temperature. We might note here that the mass of the $\Upsilon$ state has initially an increase with density, as the contribution from the finite quark mass term dominates over the first term in the expression for the scalar gluon condensate given by equation (\[chiglu\]). The rise is observed to be about 3.4 MeV at a density of about 0.2$\rho_0$ for $\Upsilon (4S)$, followed by a drop in the mass as the density is further increased. In the case when the finite quark mass term in the trace of the energy momentum tensor in QCD is neglected, there is seen to be a monotonic drop of the $\Upsilon$ states with increase in density, since the dilaton field dereases with density.
With the harmonic oscillator parameter, $\beta$ of the wave function of the bottomonium states as fitted from their leptonic decay widths, the root mean square radii for the states, $\Upsilon(1S)$, $\Upsilon(2S)$, $\Upsilon (3S)$ and $\Upsilon (4S)$ are obtained as 0.1843, 0.4027, 0.5928 and 0.8466 fermis respectively in the present investigation. These values are similar to the values for the rms radii obtained from using a Cornell potential for the bottomonium bound state of Ref. [@eichten80], of 0.2, 0.48, 0.72 and 0.92 fermis for these bottomonium states and, 0.1869 and 0.3865 fermis for the $\Upsilon (1S)$ and $\Upsilon (2S)$ states calculated in a quark model [@spmmesonspect] using a confining linear potential. In the hadronic medium, due to the mass drop in the bottomonium states, the strength of the harmonic oscillator wave function, $\beta$ is modified from which we can obtain an estimate for the size of the bottomonium state in the hadronic medium. For the states, $\Upsilon(1S)$, $\Upsilon (2S)$, $\Upsilon (3S)$ and $\Upsilon(4S)$, the change in the strength of the quarkonium wave function is obtained [@leeko; @amepja] from $\Delta \beta^2=\frac{ 2}{3}M_\Upsilon\Delta M_{\Upsilon}$, $\Delta \beta^2=\frac{ 2}{7}M_\Upsilon\Delta M_{\Upsilon}$, $\Delta \beta^2=\frac{ 2}{11}M_\Upsilon\Delta M_{\Upsilon}$, $\Delta \beta^2=\frac{ 2}{15}M_\Upsilon\Delta M_{\Upsilon}$, for the states $\Upsilon(1S)$, $\Upsilon(2S)$, $\Upsilon(3S)$, and $\Upsilon(4S)$ respectively. The rms radii of these states in the isospin asymmetric hot strange hadronic matter are plotted in figures \[rmsupsln1s\], \[rmsupsln2s\], \[rmsupsln3s\] and \[rmsupsln4s\]. There is seen to be increase in the rms radii of these states with density and this rise is seen to be especially prominent for $\Upsilon(4S)$ state. These can have consequences on the scattering cross-sections of these states by the baryons in the medium. This is because the leading order QCD calculations [@pes1; @oh] show that this scattering cross-section is proportional to the root mean square radii, $r_{rms}^2$ [@frimanlee]. Hence the increase in the size of the $\Upsilon$-states in the medium can enhance their decay widths.
Summary
=======
In the present work, we have investigated the mass modification of the bottomonium states ($\Upsilon (1S)$, $\Upsilon (2S)$, $\Upsilon (3S)$ and $\Upsilon (4S)$), using the leading order QCD mass formula, from the medium modification of the scalar gluon condensate. The gluon condensate in the isospin asymmetric strange hadronic matter is calculated in a chiral SU(3) model. The broken scale invariance of QCD is incorporated into the hadronic model by introducing a scalar dilaton field. In the limit of massless quarks, the scalar gluon condensate is proportional to the fourth power of the dilaton field. However, in the case of finite quark masses, there is contribution to the scalar gluon condensate from the quark condensates, which are determined in the chiral SU(3) model from the values of the scalar fields, $\sigma$, $\zeta$ and $\delta$ in the hadronic medium. The mass shifts of the $\Upsilon$-states are observed to be larger for the excited states and are of the order of few hundred MeV for $\Upsilon (4S)$ state. The density effect is the dominant medium effect as compared to the effects from strangeness fraction, isospin asymmetry and the temperature of the hadronic matter. These mass shifts can possibly show in the dilepton spectra arising from the compressed baryonic matter in the future facility at GSI, when there is access to higher energies as compared to the planned energy range at CBM. The density effects of the mass modifications of the bottomonium states should also show up in the dilepton spectra at SPS. The $\Upsilon$ states have already been measured in pA collisions at incident energy of 450 AGeV ($\sqrt s$=29.1GeV) by the NA50 Collaboration [@pANA50bottomonium], which can provide a baseline for the $\Upsilon$ production studies to be carried out in the ion-ion collisions at higher centre of mass energies at SPS. In the present work, the medium modifications of rms radii of the $\Upsilon$ states have been calculated due to the change in the strength in the harmonic oscillator wave functions of these states in the hadronic medium. There is seen to be an increase in the rms radii of these states with density in the present investigation. This can lead to appreciable contribution to the decay widths of the bottomonium states due to scattering from the nucleons in the hadronic medium.
AM would like to thank S.P.Misra and H. Mishra for discussions and Department of Science and Technology, Government of India (project number SR/S2/HEP-031/2010) for financial support. DP acknowledges financial support from University of Grants Commission, India (Sr. No. 2121051124, Ref. No. 19-12/2010(i)EU-IV).
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abstract: 'Neuronal variability plays a central role in neural coding and impacts the dynamics of neuronal networks. Unreliability of synaptic transmission is a major source of neural variability: synaptic neurotransmitter vesicles are released probabilistically in response to presynaptic action potentials and are recovered stochastically in time. The dynamics of this process of vesicle release and recovery interacts with variability in the arrival times of presynaptic spikes to shape the variability of the postsynaptic response. We use continuous time Markov chain methods to analyze a model of short term synaptic depression with stochastic vesicle dynamics coupled with three different models of presynaptic spiking: one model in which the timing of presynaptic action potentials are modeled as a Poisson process, one in which action potentials occur more regularly than a Poisson process and one in which action potentials occur more irregularly. We use this analysis to investigate how variability in a presynaptic spike train is transformed by short term depression and stochastic vesicle dynamics to determine the variability of the postsynaptic response. We find that regular presynaptic spiking increases the average rate at which vesicles are released, that the number of vesicles released over a time window is more variable for smaller time windows than larger time windows and that fast presynaptic spiking gives rise to Poisson-like variability of the postsynaptic response even when presynaptic spike times are non-Poisson. Our results complement and extend previously reported theoretical results and provide possible explanations for some trends observed in recorded data.'
author:
- Steven Reich
- Robert Rosenbaum
title: 'The impact of short term synaptic depression and stochastic vesicle dynamics on neuronal variability[^1] '
---
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Introduction {#intro}
============
Variability of neural activity plays an important role in population coding and network dynamics [@Faisal08]. Random fluctuations in the number of action potentials emitted by a population of neurons affects the firing rate of downstream cells [@Shadlen98; @Shadlen98b]. In addition, spike count variability over both short and long timescales can impact the reliability of a rate-coded signal [@Dayan01]. It is therefore important to understand how this variability is shaped by synaptic and neuronal dynamics.
Several studies examine the question of how intrinsic neuronal dynamics interact with variability in presynaptic spike timing to determine the statistics of a postsynaptic neuron’s spiking response, but many of these studies do not account for dynamics and variability introduced at the synaptic level by short term synaptic depression and stochastic vesicle dynamics. Synapses release neurotransmitter vesicles probabilistically in response to presynaptic spikes and recover released vesicles stochastically over a timescale of several hundred milliseconds [@Zucker02; @Fuhrmann02]. The dynamics and variability introduced by short term depression and stochastic vesicle dynamics alter the response properties of a postsynaptic neuron [@Vere1966; @Abbott97; @Chance98; @Markram98; @Goldman99; @Senn01; @Goldman02; @Hanson02; @Rocha04; @Rocha05; @Rothman09; @Branco09; @RosenbaumPLoS12] and therefore play an important role in information transfer [@Zador98; @Rocha02; @Goldman04; @Merkel10; @Rotman11], neural coding [@Tsodyks97; @Cook03; @Abbott04; @Grande05; @Rocha08; @Lindner09; @Oswald12] and network dynamics [@Tsodyks98; @Galarreta98; @Bressloff99; @Wang99; @Tsodyks00; @Barbieri08]. Understanding how variability in presynaptic spike times interact with short term depression and stochastic vesicle dynamics to determine the statistics of the postsynaptic response is therefore an important goal.
In this study, we use a model of short term synaptic depression with stochastic vesicle dynamics to examine how variability in a presynaptic input is transferred to variability in the synaptic response it produces. We use the theory of continuous-time Markov chains to construct exact analytical methods for calculating the the statistics of the postsynaptic response to three different presynaptic spiking models: one model with Poisson spike arrival times, one with more regular spike arrival times, and one with more irregular spike arrival times. We find that depressing synapses shape the timescale over which neuronal variability occurs: the number of neurotransmitter vesicles released over a time interval is highly variable for shorter time windows, but less variable for longer time windows when variability is quantified using Fano factors. Additionally, we find that when presynaptic inputs are highly irregular (Fano factor greater than 1), synaptic dynamics cause a reduction in Fano factor, consistent with previous studies [@Goldman99; @Goldman02; @Rocha02; @Rocha05]. On the other hand, when presynaptic input is more regular (Fano factor less than 1), synaptic dynamics often cause an increase in Fano factor. This observation suggests a mechanism through which irregular and Poisson-like variability can be sustained in spontaneously spiking neuronal networks [@Tolhurst83; @Softky93; @Britten93; @Buracas98; @Mcadams99; @Churchland10], which complements previously proposed mechanisms [@Vanreeswick96; @Vanreeswijk98; @Stevens98; @Harsch00; @Kumar12].
Methods
=======
We begin by introducing the synapse model used throughout this study. We then proceed by analyzing the statistics of the synaptic response to three different input models.
Synapse model
-------------
A widely used model of depressing synapses [@Tsodyks97; @Abbott97; @Tsodyks98; @Markram98; @Senn01] does not capture stochasticity in vesicle recovery and release. As a result, this model underestimates the variability of the synaptic response [@Rocha05; @RosenbaumPLoS12]. For this reason, we use a more detailed synapse model that takes stochastic recovery times and probabilistic release into account [@Vere1966; @Wang99; @Fuhrmann02; @RosenbaumPLoS12].
We consider a presynaptic neuron with spike train $I(t)=\sum_j \delta(t-t_j)$ that makes $M$ functional contacts onto a postsynaptic cell. Here, $t_j$ is the time of the $j$th presynaptic action potential. Define $m(t)$ to be the number of contacts with a readily releasable neurotransmitter vesicle at time $t$ (so that $0\le m(t)\le M$). For simplicity, we assume that each contact can release at most one neurotransmitter vesicle in response to a presynaptic spike. When a presynaptic spike arrives, each contact with a releasable vesicle releases its vesicle independently with probability ${p_r}$. After releasing a vesicle, a synaptic contact enters a refractory period during which it is unavailable to release a vesicle again until it recovers by replacing the released vesicle. The recovery time at a single contact is modeled as a Poisson process with rate $1/\tau_u$. Equivalently, the duration of the refractory period is exponentially distributed with mean $\tau_u$.
Define $w_j$ to be the number of contacts that release a vesicle in response to the presynaptic spike at time $t_j$ (so that $0\le w_j\le m(t_j^-)\le M$ where $m(t_j^-)=\lim_{t\to t_j^-}m(t)$). The synaptic response is quantified by the marked point process $$x(t)=\sum_j w_j \delta(t-t_j).$$ Since the signal observed by the postsynaptic cell is determined by $x(t)$, we quantify synaptic response statistics in terms of the statistics of $x(t)$ in our analysis. The process $x(t)$ can be convolved with a post-synaptic response kernel to obtain the conductance induced on the postsynaptic cell [@RosenbaumPLoS12]. The effects of this convolution on response statistics is well understood [@Tetzlaff08], so we do not consider it here.
This model can be described more precisely using the equation [@RosenbaumPLoS12] $$%\label{E:Jaime}
dm(t)=-dN_x(t)+dN_u(t)$$ where $dN_u(t)=u(t)dt$ is the increment of an inhomogeneous Poisson process with instantaneous rate that depends on $m(t)$ through $\langle dN_u(t)\rangle\,|\, m(t)\rangle=dt(M-m(t))/\tau_u$ (here, $\langle\cdot\,|\,\cdot\rangle$ denotes conditional expectation), $N_x(t)=\int_0^t x(s)ds$ is the number of vesicles released up to time $t$, and each $w_j$ is a binomial random variable with mean ${p_r} m(t_j)$ and variance $m(t_j){p_r}(1-{p_r})$.
Statistical measures of the presynaptic spike train and the synaptic response
-----------------------------------------------------------------------------
We focus on steady state statistics in this article, and therefore assume that the presynaptic spike trains are stationary and that the synapses have reached statistical equilibrium. The intensity of a presynaptic spike train is quantified by the mean presynaptic firing rate, $$r_{\textrm{in}}=\langle I(t)\rangle=\langle N_I(T)/T\rangle$$ where $\langle \cdot \rangle$ denotes the expected value and $$N_I(T)=\int_0^T I(s)ds$$ represents the number of spikes in the time interval $[0,T]$. Temporal correlations in the presynaptic spike times are quantified by the auto-covariance, $$R_{\textrm{in}}(\tau)={\textrm{cov}}(I(t),I(t+\tau)),$$ and the variability in the presynaptic spike train is quantified by its Fano factor, $$%\label{E:FanoDef}
F_{\textrm{in}}(T)=\frac{{\textrm{var}}(N_I(T))}{Tr_{\textrm{in}} }.$$ For much of this work, we will focus on Fano factors over large time windows which, through a slight abuse of notation, we denote by $F_{\textrm{in}}=\lim_{T\to\infty}F_{{\textrm{in}}}(T)$. To compute Fano factors, we will often exploit their relationship to auto-covariance functions, $$\label{E:VarInt}
F_{\textrm{in}}(T)=\frac{1}{r_{\textrm{in}}}\int_{-T}^T R_{{\textrm{in}}}(\tau)(1-|\tau|/T)d\tau$$ and $$\label{E:VarIntInf}
F_{\textrm{in}}=\frac{1}{r_{\textrm{in}}}\int_{-\infty}^\infty R_{{\textrm{in}}}(\tau)d\tau.$$
The statistics of the synaptic response, $x(t)$, are defined analogously to the statistics of $I(t)$. The steady state rate of vesicle release is defined as $$r_x=\langle x(t)\rangle=\langle N_x(T)/T\rangle$$ where $N_x(T)=\int_0^T x(s)ds$ represents the number of vesicles released in the time interval $[0,T]$. Temporal correlations in the synaptic response are quantified by the auto-covariance, $$R_x(\tau)={\textrm{cov}}(x(t),x(t+\tau))$$ and response variability is quantified by the Fano factor of the number of vesicles released, $$F_x(T)=\frac{{\textrm{var}}(N_x(T))}{Tr_x}.$$ As above, we define $F_x=\lim_{T\to\infty}F_x(T)$ and note that $$\label{E:VarIntx}
F_x(T)=\frac{1}{r_{\textrm{in}}}\int_{-T}^T R_x(\tau)(1)d\tau \,\textrm{ and }\, F_x=\int_{-\infty}^\infty R_x(\tau)d\tau.$$
Model analysis with Poisson presynaptic inputs
----------------------------------------------
We first consider a homogeneous Poisson input, $I(t)$, with rate $r_{\textrm{in}}$. The input auto-covariance for this model is given by $R_{{\textrm{in}}}(\tau)=r_{\textrm{in}} \delta(\tau)$ and the Fano factor is given by $F_{\textrm{in}}(T)=1$ for any $T> 0$. The mean rate of vesicle release for this model is given by $$%\label{E:rxPoisson}
r_x=\frac{M {p_r} r_{\textrm{in}}}{{p_r} r_{\textrm{in}} \tau _u+1}$$ which saturates to $M/\tau_u$ for large presynaptic rates, $r_{\textrm{in}}$. A closed form approximations to the auto-covariance function of the response for this Poisson input model are derived in [@RosenbaumPLoS12; @Merkel10] (see also [@Rocha04]) and consist of a sum of a delta function and an exponential, $$\label{E:RxPoisson}
R_x(\tau)=Dr_x\delta(\tau)-Er_xe^{-|\tau|/\tau_0},$$ where the mass of the delta function is given by $$\label{E:D}
D=\frac{2 {p_r} \left(r_{\textrm{in}} \tau _u+M-1\right)+2-{p_r}^2 r_{\textrm{in}} \tau _u}{(2-{p_r}) {p_r} r_{\textrm{in}} \tau _u+2}>0,$$ the timescale of the exponential decay is given by $$\tau_0=\frac{\tau_u}{1+{p_r} r_{\textrm{in}}\tau_u},$$ and the peak of the exponential is given by $$\label{E:E}
E=\frac{{p_r} r_{\textrm{in}} ((M-2) {p_r}+2) \tau _u+2 (M-1) {p_r}+2}{M (2-{p_r}) {p_r} r_{\textrm{in}} \tau _u+2 M}\; r_x.$$ It can easily be checked that $E>0$ whenever $M\ge 1$, $0\le {p_r}\le1$, $r_{\textrm{in}}>0$, and $\tau_u>0$ so that the peak of the exponential in is negative. For finite $T$, the Fano factor, $F_x(T)$, is given by $$\label{E:FxPoisson}
F_x(T)=D-2 E \tau _0-\left(e^{-\frac{T}{\tau _0}}-1\right)\frac{2 E \tau _0^2}{T}$$ and, in the limit of large $T$, $$\label{E:FxInfPoisson}
F_x=D-2E\tau_0.$$
To test the accuracy of these approximations, exact solutions can be found numerically using standard methods for the analysis of continuous-time Markov chains, as described for alternate input models below. This analysis is a special case of the analysis for the “regular” input model described below that is achieved by taking $\theta=1$. Alternatively, exact numerical results can be achieved by taking $r_s=r_f$ for the “irregular” input model. In figures showing results for the Poisson input model, we plot the closed form approximations described above along with exact numerical results obtained using the regular input model with $\theta=1$.
Model analysis with irregular presynaptic inputs
------------------------------------------------
Spike trains measured in vivo often exhibit irregular spiking statistics indicated by Fano factors larger than 1 [@Bair94; @Dan96; @Baddeley97; @Churchland10]. To describe the synaptic response to irregular inputs, we use a model of presynaptic spiking in which the instantaneous rate of the presynaptic spike train, $I(t)$, randomly switches between two values, $r_s$ and $r_f>r_s$, representing a slow spiking state and a fast spiking state. The time spent in the slow state before transitioning to the fast state is exponentially distributed with mean $\tau_s$. Likewise, the amount of time spent in the fast state before switching to the slow state is exponentially distributed with mean $\tau_f$. Transition times are independent from one another and from the spiking activity. Between transitions, spikes occur as a Poisson process.
To find $r_{\textrm{in}}$, $R_{\textrm{in}}(\tau)$, and $F_{\textrm{in}}$, we represent this model as a doubly stochastic Poisson process. Define $r(t)\in\{r_s,r_f\}$ to be the instantaneous firing rate at time $t$. Then $r(t)$ is a continuous time Markov chain [@Karlin1] on the state space $\Gamma=(r_s,r_f)$ with infinitesimal generator matrix $$A=\left[\begin{array}{cc} -1/\tau_s & 1/\tau_s\\ 1/\tau_f & -1/\tau_f\end{array}\right].$$
Clearly, $r(t)$ spends a proportion $\tau_s/(\tau_s+\tau_f)$ of its time in the slow state (defined by $r(t)=r_s$) and a proportion $\tau_f/(\tau_s+\tau_f)$ of its time in the fast state (defined by $r(t)=r_f$). This gives a steady-state mean firing rate of $$r_\textrm{in}=\frac{r_s \tau_s+r_f\tau_f}{\tau_s+\tau_f}.$$
At non-zero lags ($\tau\ne 0$), the auto-covariance of a doubly stochastic Poisson process is the same as the auto-covariance of $r(t)$ [@RosenbaumPLoS12], which we can compute using techniques for analyzing continuous time Markov chains. For $\tau>0$, we have $$\begin{aligned}
\langle r(t)r(t+\tau)\rangle&=r_s\Pr(r(t)=r_s)\langle r(t+\tau)\,|\,r(t)=r_s\rangle\notag\\
&\;\;+r_f\Pr(r(t)=r_f)\langle r(t+\tau)\,|\,r(t)=r_f\rangle \notag\\
&=\frac{r_s\tau_s}{\tau_s+\tau_f}\langle r(t+\tau)\,|\,r(t)=r_s\rangle\label{E:Err}\\
&\;\;+\frac{r_f\tau_f}{\tau_s+\tau_f}\langle r(t+\tau)\,|\,r(t)=r_f\rangle\notag\end{aligned}$$ where $\langle \cdot |\cdot\rangle$ denotes conditional expectation and $$\begin{aligned}
&\langle r(t+\tau)\,|\,r(t)=r_s\rangle=r_s \Pr(r(t+\tau)=r_s\,|\,r(t)=r_s)\\
&\quad+r_f (1-\Pr(r(t+\tau)=r_s\,|\,r(t)=r_s)).\end{aligned}$$ The probability in this expression can be written in terms of an exponential of the generator matrix, $A$, and then calculated explicitly to obtain $$\begin{aligned}
\Pr(r(t+\tau)=r_s\,|\,r(t)=r_s)&=
\left[e^{A^T \tau}\left(\begin{array}{c}1\\0\end{array}\right)\right]_1\\
&=\frac{\tau _f e^{-\frac{\tau }{\tau _f}-\frac{\tau }{\tau _s}}+\tau _s}{\tau _f+\tau_s}\end{aligned}$$ where $[v]_k$ denotes the $k$th component of a vector, $v$. An identical calculation can be performed to obtain an analogous expression for $\langle r(t+\tau)\,|\,r(t)=r_f\rangle$. Combining these with Eq. gives $$\langle r(t)r(t+\tau)\rangle=\frac{\tau _f \tau _s \left(r_f-r_s\right)^2 }{\left(\tau _f+\tau _s\right)^2}e^{-\frac{\tau}{\tau_s}-\frac{\tau}{\tau_f}}+r_{\textrm{in}}^2.$$ For positive $\tau$, we have $R_{\textrm{in}}(\tau)=\langle r(t)r(t+\tau)\rangle-r_{\textrm{in}}^2$. As with all stationary point processes $R_{\textrm{in}}(\tau)=R_{\textrm{in}}(-\tau)$ and $R_{\textrm{in}}(\tau)$ has a Dirac delta function with mass $r_{\textrm{in}}$ at the origin [@Cox]. Thus, the auto-covariance of $I(t)$ is given by $$\label{E:RinBurst}
R_{\textrm{in}}(\tau)=r_{\textrm{in}}\delta(\tau)+\frac{\tau _f \tau _s \left(r_f-r_s\right)^2 }{\left(\tau _f+\tau _s\right)^2}e^{-\frac{|\tau|}{\tau_s}-\frac{|\tau|}{\tau_f}}.$$ For finite $T$, the Fano factor, $F_{\textrm{in}}(T)$, can be computed using Eqs. and . In the limit of large $T$, we can use Eqs. and to obtain a closed form expression, $$\label{E:FinBurst}
F_{\textrm{in}}=1+\frac{2 \tau _f^2 \tau _s^2 \left(r_f-r_s\right)^2}{\left(\tau _f+\tau _s\right)^2
\left(r_f \tau _f+r_s \tau _s\right)}.$$
Poisson spiking is recovered by setting $r_f=r_s$, $\tau_f=0$, or $\tau_s=0$. For any other parameter values (i.e., when $r_f\ne r_s$ and $\tau_f,\tau_s>0$), it follows from Eq. that $F_{\textrm{in}}(T)> 1$ for any $T$. Therefore this input model, hereafter referred to as the “irregular spiking” model, represents spiking that is more irregular than a Poisson process.
The analysis in [@RosenbaumPLoS12] used to derive closed form expressions for the response statistics with Poisson inputs cannot easily be generalized to derive expressions with non-Poisson inputs like those considered here. Instead, we analyze the synaptic response for the irregular input model using techniques for analyzing continuous time Markov chains. First note that the process $b(t)=(m(t),r(t))$ is a continuous-time Markov chain on the discrete state space $\{0,1,\ldots,M\}\times \{r_s,r_f\}$. Here, $m(t)$ denotes the size of readily releasable pool and $r(t)$ represents the instantaneous presynaptic rate (which switches between $r_s$ and $r_f$). We enumerate all $2(M+1)$ elements of this state space and denote the $j$th element of this enumeration as $\Gamma_j=(m_j,r_j)$ for $j=1,\ldots,2(M+1)$.
The infinitesimal generator, $B$, of $b(t)$ is a $2(M+1)\times 2(M+1)$ matrix with off-diagonal terms defined by the instantaneous transition rates, $$\label{E:B}
B_{j,k}=\lim_{h\to 0}\frac{1}{h}\Pr(b(t+h)=\Gamma_k\,|\,b(t)=\Gamma_j),\;\; j\ne k$$ and with diagonal terms chosen so that the rows sum to zero: $B_{j,j}=-\sum_{k\ne j}B_{j,k}$ [@Karlin1].
To fill the matrix $B$, we consider each type of transition that the process $b(t)$ undergoes. Vesicle recovery events occur at the instantaneous rate $(M-m(t))/\tau_u$ and increment the value of $m(t)$ by one vesicle. Therefore $$\lim_{h\to 0}\frac{1}{h}\Pr(b(t+h)=(m+1,r)\,|\,b(t)=(m,r))=\frac{M-m}{\tau_u}$$ for $m\in\{0,\ldots,M-1\}$ and $r\in\{r_s,r_f\}$. Vesicle release events occur at the instantaneous rate $r(t)$ and decrement the value of $m(t)$ by a random amount $k$ with a binomial distribution so that $$\begin{aligned}
\lim_{h\to 0}&\frac{1}{h}\Pr(b(t+h)=(m-k,r)\,|\,b(t)=(m,r))=\\
&r\, \frac{m!}{(m-k)!}{p_r}^k(1-{p_r})^{m-k}\end{aligned}$$ for $m\in\{1,\ldots,M\}$, $k\in \{0,\ldots,m\}$, and $r\in\{r_s,r_f\}$. The value of $r(t)$ switches from $r_s$ to $r_f$ with instantaneous rate $1/\tau_s$ so that $$\lim_{h\to 0}\frac{1}{h}\Pr(b(t+h)=(m,r_f)\,|\,b(t)=(m,r_s))=\frac{1}{\tau_s}$$ and, similarly, $$\lim_{h\to 0}\frac{1}{h}\Pr(b(t+h)=(m,r_s)\,|\,b(t)=(m,r_f))=\frac{1}{\tau_f}.$$ These four transition types account for all of the transitions that $b(t)$ undergoes. They can be used to fill the off-diagonal terms of the matrix $B$. The diagonal terms are then filled to make the rows sum to zero, as discussed above.
Once a the infinitesimal generator matrix, $B$, is obtained, the probability distribution of $b(t)$ given an initial distribution $p(0)$ is given by $$p(t)=e^{t B^T}p(0).$$ The stationary distribution, $p_0$, of $b(t)$ is given by the vector in the one-dimensional null space of $B$ with elements that sum to one [@Karlin1]. The instantaneous rate of vesicle release, conditioned on the current state of $r(t)$ and $m(t)$, is given by $$\langle x(t)\,|\, r(t)=r,\,m(t)=m\rangle=r{p_r} m.$$ Averaging over $r$ and $m$ in the steady state gives $$r_x=\sum_{j=1}^{2(M+1)}[p_0]_j r_j {p_r} m_j$$ where $[\cdot]_j$ denotes the $j$th element. The auto-covariance, $R_x(\tau)$, has a Dirac delta function at $\tau=0$. We separate this delta function from the continuous part by writing $
R_x(\tau)=A_x \delta(\tau)+R^+_x(\tau)
$ where $R^+_x(\tau)$ is a continuous function. The area of the delta function can be found by conditioning on the current state of $r(t)$ in the steady state to get $$\begin{aligned}
A_x&=\lim_{t\to\infty}\langle x(t)^2dt\rangle\notag\\
&=\lim_{t\to \infty}\langle dN_x^2(t)/dt\rangle\notag\\
&=\sum_{j=1}^{2(M+1)}r_j[p_0]_j \lim_{k\to\infty} \langle w_k^2\,|\, m(t_k^-)=m_j\rangle\label{E:Ax1}\end{aligned}$$ where $w_k$ is the number of vesicles released by the $k$th presynaptic spike. Conditioned on the size, $m(t_k^-)$, of the readily releasable pool immediately before the presynaptic spike arrives, $w_k$ has a binomial distribution with second moment, $$\lim_{k\to\infty}\langle w_k^2 \,|\, m(t_k^-)=m_j\rangle=m_j{p_r} (1-{p_r} )+m_j^2{p_r}^2$$ which can be substituted into Eq. to calculate $A_x$.
All that remains is to calculate the continuous part, $R^+_x(\tau)$, of $R_x(\tau)$. First note that, for $\tau>0$, $$\begin{aligned}
&\lim_{t\to\infty}\langle x(t)x(t+\tau)\rangle\notag\\
&=\sum_{i,j=1}^{2(M+1)}[p_0]_i \Pr(b(t+\tau)=\Gamma_j\,|\,b(t)=\Gamma_i)\label{E:xxBurst}\\
&\times \langle dN_x(t)dN_x(t+\tau)\,|\,b(t)=\Gamma_i,b(t+\tau)=\Gamma_j\rangle/dt^2.\notag\end{aligned}$$ The second term in Eq. can be computed as $$\Pr(b(t+\tau)=\Gamma_j\,|\,b(t)=\Gamma_i)=[e^{\tau B^T} {\textbf e}_{i}]_j=[e^{\tau B^T}]_{j,i}$$ where ${\bf e}_i$ is the $2(M+1)\times 1$ vector whose $i$th element is 1 and all other elements are zero, which represents an initial distribution concentrated at $\Gamma_i$. The last term in Eq. is given by $$\begin{aligned}
&\langle dN_x(t)dN_x(t+\tau)\,|\,b(t)=\Gamma_i,b(t+\tau)=\Gamma_j\rangle/dt^2=\\
&r_i r_j {p_r}^2 m_im_j.\end{aligned}$$ Finally, $R_{x}^+(\tau)=\lim_{t\to\infty}\langle x(t)x(t+|\tau|)\rangle-r_x^2$ for $\tau\ne 0$ so that $$\begin{aligned}
&R_x(\tau)=\\
&A_x\delta(\tau)-r_x^2+\sum_{i,j=1}^{2(M+1)} [p_0]_i\left[e^{|\tau| B^T}\right]_{j,i}r_i r_j {p_r}^2m_im_j\end{aligned}$$ which can be computed efficiently using matrix multiplication. The response Fano factor, $F_x$, can then be found by integrating $R_x(\tau)$ according to Eqs. and .
Model analysis with regular presynaptic inputs
----------------------------------------------
We now consider a spiking model that gives Fano factors smaller than 1 and therefore spike trains that are more regular than Poisson processes. We achieve this by defining a renewal process with gamma-distributed interspike intervals (ISIs). Such a process can be obtained by first generating a Poisson process, $\sum_k\delta(t-s_k)$ with rate $r=\theta\, r_{\textrm{in}}$ for some positive integer $\theta$, then keeping only every $\theta$th spike to build the spike train $I(t)$. More precisely, the first spike of the gamma process is obtained by choosing an integer, $k$, uniformly from the set $\{1,\ldots,\theta\}$ and defining defining $t_1=s_k$. The remaining spikes are defined by $t_{j+1}=s_{j\theta +k}$ to obtain the stationary renewal process, $I(t)=\sum_j \delta(t-t_j)$ [@CoxRenewal].
Clearly, this process has rate $r_{\textrm{in}}$ since the original Poisson process has rate $\theta r_{\textrm{in}}$ and a proportion $1/\theta$ of these spikes appear in $I(t)$. The auto-covariance is given by [@RosenbaumThesis] $$\label{E:RinGamma}
R_{\textrm{in}}(\tau)=r_{\textrm{in}} \delta(\tau)+r_{\textrm{in}}\left(\sum_{k=1}^\infty f_k(\tau)-r_{\textrm{in}}\right).$$ where $$f_k(t)=\frac{t^{k\theta-1}(\theta r_{\textrm{in}})^{k\theta} e^{-\theta r_{\textrm{in}} t}}{(k\theta-1)!}$$ is the density of the waiting time between the first spike and the $(k+1)$st spike (i.e., the duration of $k$ consecutive ISIs).
For finite $T$, the Fano factor, $F_{\textrm{in}}(T)$, can be computed using Eqs. and . In the limit of large $T$, we can use Eq. or use the fact that, for renewal processes, $F_{\textrm{in}}={\textrm{var}}(\textrm{ISI})/\langle \textrm{ISI}\rangle^2$ where ${\textrm{var}}(\textrm{ISI})=1/(r_{\textrm{in}}^2\theta)$ is the variance and $\langle \textrm{ISI}\rangle=1/r_{\textrm{in}}$ is the mean of the gamma distributed ISIs [@CoxRenewal]. This gives $$%\label{E:FinGamma}
F_{\textrm{in}}=\frac{1}{\theta}.$$ Poisson spiking is recovered by setting $\theta=1$. When $\theta>1$, we have that $F_{\textrm{in}}(T)< 1$ for any $T$. Therefore this model, hereafter referred to as the “regular” input model, represents spiking that is more regular than a Poisson process.
The synaptic response with the regular input model can be analyzed using methods similar to those used for the irregular model. We introduce an auxiliary process, $q(t)$, that transitions sequentially through the state space $\{1,\ldots,\theta\}$. Once reaching $\theta$, $q(t)$ transitions back to state 1. Transitions occur as a Poisson process with rate $\theta r_{\textrm{in}}$. The waiting times between transitions from $q=\theta$ to $q=1$ are gamma distributed. Thus, to recover the regular input model, we specify that each transition from $1=\theta$ to $q=1$ represents a single presynaptic spike. The process $g(t)=(m(t),q(t))$ is then a continuous time Markov chain on the discrete state space $\{1,\ldots,\theta\}\times \{0,\ldots,M\}$. We enumerate all $\theta(M+1)$ elements of this space and denote the $j$th element as $\Gamma_j=(m_j,q_j)$ for $j=1,\ldots,\theta (M+1)$.
The infinitesimal generator, $G$, which is a $\theta (M+1)\times \theta (M+1)$ matrix is defined analogously to the matrix $B$ in Eq. above. The elements of $G$ can be filled using the following transition probabilities. As for the irregular input model, vesicle recovery occurs as a Poisson process with rate $(M-m(t))/\tau_u$ so that $$\lim_{h\to 0}\frac{1}{h}\Pr(g(t+h)=(m+1,q)\,|\,g(t)=(m,q))=\frac{M-m}{\tau_u}$$ for $m=0,\ldots,M$ and $q=1,\ldots,\theta$. Transitions that increment $q(t)$ occur with instantaneous rate, $\theta r_{\textrm{in}}$ so that $$\lim_{h\to 0}\frac{1}{h}\Pr(g(t+h)=(m,q+1)\,|\,g(t)=(m,q))=\theta r_{\textrm{in}}$$ for $q=1,\ldots,\theta-1$ and $m=0,\ldots,M$. The only other transitions are those from $q(t)=\theta$ to $q(t)=1$, which represent a presynaptic spike and are therefore accompanied by a release of vesicles. The transitions contribute the following, $$\begin{aligned}
&\lim_{h\to 0}\frac{1}{h}\Pr(g(t+h)=(1,m-k)\,|\,g(t)=(\theta,m))=\\
&\theta r_{\textrm{in}} \frac{m!}{(m-k)!}{p_r}^k(1-{p_r} )^{m-k}\end{aligned}$$ for $m\in\{0,1,\ldots,M\}$ and $k\in \{0,\ldots,m\}$. These transition rates can be used to fill the off-diagonal terms of the matrix $G$. The diagonal terms are then filled so that the rows sum to zero. The stationary distribution, $p_0$, of $g(t)=(m(t),q(t))$ is given by the vector in the one-dimensional null space of $G$ with elements that sum to one.
A proportion $[p_0]_{\gamma(k)}$ of time is spent in state $(m(t),q(t))=(k,\theta)$ where $\gamma(k)$ represents the index of the element $(k,\theta)$ in the enumeration chosen for $\Gamma$ (i.e., the index, $j$, at which $\Gamma_j=(k,\theta)$). In that state, the transition to $q(t)=1$ occurs with instantaneous rate $\theta r_{\textrm{in}}$ and releases average of ${p_r} m(t)$ vesicles. Thus, the mean rate of vesicle release is given by $$r_x=\sum_{k=1}^M \theta r_{\textrm{in}} {p_r} k \,[p_0]_{\gamma(k)}. %=\sum_{j=1}^M r_{\textrm{in}} {p_r} k \,[p_{0|\theta}]_k$$
As above, we separate the auto-covariance into a delta function and a continuous part by writing $
R_x(\tau)=A_x \delta(\tau)+R^+_x(\tau)
$ where $R^+_x(\tau)$ is a continuous function. The area of the delta function at the origin is given by $$A_x=\sum_{k=1}^M \theta r_{\textrm{in}} [p_0]_{\gamma(k)} \left(k{p_r} (1-{p_r} )+k^2{p_r}^2\right)$$ by an argument identical to that used for the irregular input model above. Also by a similar argument used for the irregular input model, we have that $$\begin{aligned}
%\label{E:Rxgamma}
&R_x(\tau)=\\
&A_x\delta(\tau)-r_x^2+\theta^2r_{\textrm{in}}^2\sum_{k,l=1}^{M} [p_0]_{\gamma(k)}\left[e^{|\tau|G^T}\right]_{\gamma(l),\gamma(k)} {p_r}^2 k l.\end{aligned}$$
Parameters used in figures
--------------------------
Theoretical results are obtained for arbitrary parameter values, but for all figures we use a set of parameter values that are consistent with experimental studies. For synaptic parameters, we use $\tau_u=700$ ms and ${p_r} =0.5$ consistent with measurements of short term depression in pyramidal-to-pyramidal synapses in the rat neocortex [@Tsodyks97; @Fuhrmann02]. We also choose $M=5$ which is within the range observed in several cortical areas [@Branco09].
The Poisson presynaptic input model is determined completely by its firing rate and the regular input model is determined completely by its firing rate and Fano factor. Presynaptic firing rates and Fano factors are reported on the axes or captions of each figure. The irregular input model has four parameters that determine the firing rate and Fano factor. In all figures, we set $\tau_b=\tau_s=1.315/c$, $r_b=37c$, and $r_s=3c$ which gives a Fano factor of $F_{\textrm{in}}=20.0017\approx 20$ for any value of $c$ (from Eq. ). Changing $c$ effectively scales the timescale of presynaptic spiking, hence scaling $r_{\textrm{in}}$, without changing $F_{\textrm{in}}$.
Results
=======
We analyze the synaptic response to different patterns of presynaptic inputs using a stochastic model of short term synaptic depression in which a presynaptic neuron makes $M$ functional contacts onto a postsynaptic neuron [@Vere1966; @Fuhrmann02; @Goldman04; @Rocha05]. The input to the presynaptic neuron is a spike train denoted by $I(t)$. Neurotransmitter vesicles are released probabilistically in response to each presynaptic spike. Specifically, a contact with a readily available vesicle releases this vesicle with probability ${p_r} $ in response to a single presynaptic spike. After a synaptic contact has released its neurotransmitter vesicle, it enters a refractory state where it is unable to release again until the vesicle is replaced. The duration of this refractory period is an exponentially distributed random variable with mean $\tau_u$, so that vesicle recovery is Poisson in nature.
We are interested in how the statistics of the presynaptic spike train determine the statistics of the synaptic response. The presynaptic statistics are quantified using the presynaptic firing rate, $r_{\textrm{in}}$, the presynaptic auto-covariance function, $R_{\textrm{in}}(\tau)$, and the Fano factor, $F_{\textrm{in}}(T)$, of the number of presynaptic spikes during a window of length $T$. Similarly, we quantify the statistics of the synaptic response using the mean rate of vesicle release, $r_x$, the auto-covariance of vesicle release, $R_x(\tau)$, and the Fano factor, $F_x(T)$, of the number of vesicles released during a window of length $T$. We will especially focus on Fano factors over large time windows and define $F_{\textrm{in}}=\lim_{T\to\infty}F_{\textrm{in}}(T)$, $F_x=\lim_{T\to\infty}F_x(T)$ accordingly. See Methods for more details.
We begin by considering the effect of $F_{\textrm{in}}$ on the mean rate of vesicle release, $r_x$. We then examine the dependence of $F_x(T)$ on the length, $T$, of the time window over which vesicle release events are counted. Finally, we show that short term synaptic depression promotes Poisson-like responses to non-Poisson presynaptic inputs.
Irregular presynaptic spiking reduces the rate at which neurotransmitter vesicles are released
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![[**Rate of vesicle release as a function of presynaptic firing rate for various presynaptic Fano factors.**]{} The rate of vesicle release, $r_x$, is an increasing function of presynaptic firing rate, $r_{\textrm{in}}$. Vesicle release is slower for the irregular spiking model than for the Poisson and regular spiking model.[]{data-label="F:rinvsrx"}](Figure1.eps){width="84mm"}
{width="174mm"}
We first briefly investigate the dependence of the rate of vesicle release, $r_x$, on the rate and variability of the presynaptic spike train, as measured by $r_{\textrm{in}}$ and $F_{\textrm{in}}$ respectively. Vesicle release rate generally increases with $r_{\textrm{in}}$, but saturates to $r_x=M/\tau_u$ whenever ${p_r} r_{\textrm{in}}\gg 1/\tau_u$ since synapses are depleted in this regime (Fig. \[F:rinvsrx\]).
When presynaptic spike times occur as a Poisson process (so that $F_{\textrm{in}}=1$), the mean rate of vesicle release is given by $r_x=M{p_r} r_{\textrm{in}}/({p_r} r_{\textrm{in}} \tau_u+1)$ [@Fuhrmann02; @Rocha05; @RosenbaumPLoS12]. Interestingly, vesicle release is slower for more irregular presynaptic spiking and faster for more regular presynaptic spiking even when presynaptic spikes arrive at the same mean rate (Fig. \[F:rinvsrx\], also see [@Rocha02]). This can be understood by noting that, for the irregular input model, spikes arrive in bursts of higher firing rate followed by durations of lower firing rate. Vesicles are depleted by the first few spikes in a burst and subsequent spikes in that burst are ineffective and therefore essentially “wasted” spikes (Fig. \[F:TracesBG\]A). When presynaptic spikes arrive more regularly, more vesicles are released on average (Fig. \[F:TracesBG\]B).
Variability in the number of vesicles released in a time window decreases with window size {#S:FT}
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We now consider how the the variability of the synaptic response to a presynaptic input depends on the timescale over which this variability is measured. We quantify the variability of the synaptic response using the Fano factor, $F_x(T)$, which is defined to be the variance-to-mean ratio of the number of vesicles released in a time window of length $T$ (see Methods) and can be calculated from an integral of the auto-covariance function, $R_{x}(\tau)$, using Eq. .
{width="174mm"}
The auto-covariance of a Poisson presynaptic spike train is simply a delta function at the origin, $R_{\textrm{in}}(\tau)=r_{\textrm{in}} \delta(\tau)$, and the Fano factor over any window size is therefore equal to one, $F_{\textrm{in}}(T)=1$ (Figs. \[F:ACGs\]C and \[F:FvsT\]C). The auto-covariance of the synaptic response when presynaptic inputs are Poisson consists of a delta function at the origin surrounded by a double-sided exponential with a negative peak (see Eq. and Fig. \[F:ACGs\]D) that decays with timescale $\tau_0={\tau_u}/{(1+{p_r} r_{\textrm{in}}\tau_u)}$. The fact that the auto-covariance is negative away from $\tau=0$ implies that the Fano factor, $F_x(T)$, is monotonically decreasing in the window size, $T$ (see Eq. and Fig. \[F:FvsT\]D). For small $T$, the mass of the delta function at the origin dominates the integral in Eq. so that the Fano factor is approximately equal to the ratio of this mass to the mean rate, $r_x$, at which vesicles are released. As $T$ increases, the negative mass of the exponential peak subtracts from the positive contribution of the delta function and decreases the Fano factor. In particular, $F_x(T)\approx D-ET+\mathcal O(T^2)$ where $Dr_x$ is the mass of the delta function in $R_x(\tau)$ and $-Er_x$ is the peak of the exponential in $R_x(\tau)$ (see Eqs. and ). As $T$ continues to increase, $F_x(T)$ monotonically decreases towards its limit, $F_x:=\lim_{T\to\infty}F_x(T)=D-2E\tau_0$. Thus, short term synaptic depression converts a Fano factor that is constant with respect to window size into one that decreases with window size (Fig. \[F:FvsT\]C,D).
When presynaptic spike times are not Poisson, the statistics of the postsynaptic response cannot be derived analytically using the methods utilized for the Poisson input model. Instead, we use the fact that the synapse model can be represented using a continuous time Markov chain, which can be analyzed to derive expressions for the response statistics in terms of an infinitesimal generator matrix (see Methods).
Irregular presynaptic spiking (i.e., inputs with $F_{\textrm{in}}>1$) is achieved by varying the rate of presynaptic spiking randomly in time to produce a doubly stochastic Poisson process (see Methods). For this model, the input auto-covariance is a delta function at the origin surrounded by an exponential peak (see Eq. \[E:RinBurst\] and Fig. \[F:ACGs\]A). The input Fano factor therefore increases with window size (see Eq. \[E:FinBurst\] and Fig. \[F:FvsT\]A). The positive temporal correlations exhibited in the input auto-covariance function are canceled by the temporal de-correlating effects of short term synaptic depression [@Goldman99; @Goldman02; @Goldman04]. For the parameters chosen in this study, this de-correlation outweighs the positive presynaptic correlations so that the auto-covariance function of the response is negative away from $\tau=0$ (Fig. \[F:ACGs\]B), although parameters can also be chosen so that temporal correlations in the response are small and positive [@Goldman02]. As with the Poisson input model, negative temporal correlations cause the response Fano factor to decrease with window size (Fig. \[F:FvsT\]B). Thus short term synaptic depression and stochastic vesicle dynamics can convert a presynaptic Fano factor that *increases* with window size into one that *decreases*.
Regular presynaptic spiking is achieved by generating a renewal process with gamma-distributed interspike-intervals. The input auto-covariance function for this model exhibits temporal oscillations (Eq. and Fig. \[F:ACGs\]E) and the Fano factor generally decreases with window size (Fig. \[F:FvsT\]E). Perhaps unsurprisingly, the auto-covariance function of the synaptic response exhibits oscillations and the response Fano factor decreases with window size (Figs. \[F:ACGs\]F and \[F:FvsT\]F).
{width="174mm"}
For all three input models, the variability of the synaptic response is larger over shorter time windows and smaller over larger time windows. A postsynaptic neuron that is in an excitable regime will generally respond most effectively to inputs that exhibit more variability over short time windows [@Salinas00; @Salinas02; @MorenoBote08; @MorenoBote10]. In addition, rate coding is often more efficient when spike counts over larger time windows are less variable [@Zohary94]. Thus, the dependence of $F_x(T)$ on window size is especially efficient for the neural transmission of rate-coded information [@Goldman04].
In addition to the temporal dependence of $F_x(T)$ introduced by short term depression, note that the response Fano factor for the irregular input model is substantially smaller than the input Fano factor (Fig. \[F:FvsT\]). Conversely, the response Fano factor for the regular input model is larger than the input Fano factor (Fig. \[F:FvsT\]). For both models, the response Fano factor is substantially nearer to 1 than the input Fano factor. We explain this phenomenon next.
Depleted synapses exhibit Poisson-like variability even when presynaptic inputs are highly non-Poisson
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![[**Response Fano factor as a function of presynaptic firing rate for three input models.**]{} Response Fano factors calculated over large windows for [**A)**]{} the irregular input model [**B)**]{} Poisson input model and [**C)**]{} regular input model. Fano factors approach 1 at high presynaptic firing rates regardless of the presynaptic Fano factor (triangle on right is placed at $F_x=1$). At low presynaptic firing rates, response Fano factors approach the value given in Eq. (indicated by triangle on left). Dotted line in (B) is from closed form approximation in Eq. and dashed line is from the expansion given in Eq. . []{data-label="F:rinvsFx"}](Figure5.eps){width="84mm"}
We now investigate the dependence of the variability in synaptic response on the rate and variability of the presynaptic input. Since we have already discussed the dependence of $F_x(T)$ on $T$ above, we will focus here on the Fano factor calculated over long time windows, $F_x=\lim_{T\to\infty}F_x(T)$.
We first consider parameter regimes where the effective rate of presynaptic inputs is much slower than the rate of vesicle recovery (${p_r} r_{\textrm{in}}\ll 1/\tau_u$). In such a regime, each contact is likely to recover between two consecutive presynaptic spikes and therefore all $M$ contacts are likely to have a vesicle ready to release when each spike arrives (Fig. \[F:TracesSlowFast\]A). In this limit, the number of vesicles released by each spike is an independent binomial variable with mean $\langle w_j\rangle ={p_r} M$ and variance ${\textrm{var}}(w_j)=M{p_r} (1-{p_r} )$. The number, $N_x(T)$, of vesicles released in a time window of length $T$ can then be represented as a sum of $N_{\textrm{in}}(T)$ independent binomial random variables (i.e., a random sum). The mean of this sum is given by $\langle N_x(T)\rangle=\langle N_{\textrm{in}}(T)\rangle \langle w_j\rangle$, which implies that $r_x=M{p_r} r_{\textrm{in}}$ in this limit. Similarly, the variance of this sum is given by [@Karlin1] ${\textrm{var}}(N_x(T))=\langle N_{\textrm{in}}(T)\rangle {\textrm{var}}(w_j)+\langle w_j\rangle^2{\textrm{var}}(N_{\textrm{in}}(T))$, which implies $$\begin{aligned}
\label{E:FxSlow}
\lim_{r_{\textrm{in}}\to 0}F_x(T)&=\langle w_j\rangle F_{\textrm{in}}(T)+{\textrm{var}}(w_j)/\langle w_j\rangle\\
&=1+{p_r} (MF_{\textrm{in}}(T)-1).\notag\end{aligned}$$ Eq. is verified for the Poisson input model by taking $r_{\textrm{in}}\to 0$ in Eqs. . For the irregular and regular input models, Eq. should be interpreted heuristically, as it was derived heuristically. A counterexample to Eq. for the irregular input model can be constructed by fixing $r_f$ and $\tau_f$, then letting $\tau_s\to\infty$ and $r_s\to 0$ to achieve the $r_{\textrm{in}}\to 0$ limit. In this case, our assumption that each contact is increasingly likely to recover between two consecutive spikes is violated and Eq. is not valid (not pictured). Regardless, we verify numerically that Eq. is accurate when $r_{\textrm{in}}$ is decreased toward zero while keeping $F_{\textrm{in}}$ fixed (Fig. \[F:rinvsFx\]).
{width="174mm"}
We now discuss the statistics of the postsynaptic response when the effective presynaptic spiking is much faster than vesicle recovery (${p_r} r_{\textrm{in}}\gg 1/\tau_u$). In such a regime, incoming spikes occur much more frequently than recovery events and synapses becomes depleted. As a result, the number of vesicles released over a long time window is determined predominantly by the number of recovery events in that time window and largely independent from the number of presynaptic spikes (Fig. \[F:TracesSlowFast\]B) [@Rocha05; @RosenbaumPLoS12]. The synaptic response therefore inherits the Poisson statistics of the recovery events so that $$\lim_{r_{\textrm{in}}\to\infty}F_x(T)=1.$$ For the Poisson input model, this limit can be made more precise in the $T\to\infty$ limit by expanding Eq. in terms of the parameter $\alpha=1/({p_r} r_{\textrm{in}}\tau_u)$ to obtain $$\label{F:Falpha}
F_x=1-2\alpha+4\alpha^2+\mathcal O(\alpha^3)$$ which converges to 1 as $r_{\textrm{in}} \tau_u\to\infty$. For the irregular and regular input models, we verify in Fig. \[F:rinvsFx\] that $F_x\to 1$ when $r_{\textrm{in}}$ is increased while keeping $F_{\textrm{in}}$ fixed.
The time constant, $\tau_u$, at which a synapse recovers from short term depression has been measured in a number of experimental studies and is often found to be several hundred milliseconds [@Tsodyks97; @Varela97; @Markram98; @Galarreta98; @Fuhrmann02; @Hanson02; @RavAcha05]. Therefore, for even moderate presynaptic firing rates, synapses are often in a highly depleted state. As discussed above, this promotes Poisson-like variability in the synaptic response. This provides one possible mechanism through which irregular Poisson-like firing can be sustained in neuronal populations [@Churchland10].
Discussion
==========
We used continuous time Markov chain methods to derive the response statistics of a stochastic model of short term synaptic depression with three different presynaptic input models. We then used this analysis to understand how the mean presynaptic firing rate and the variability of presynaptic spiking interact with synaptic dynamics to determine the mean rate of vesicle release and variability in the number of vesicles released. This analysis revealed a number of fundamental, qualitative dependencies of response statistics on presynaptic spiking statistics. Some of the dependencies have been previously noted in the literature and some have not.
The number of vesicles released over a time window is smaller for irregular inputs than for more regular inputs (Figs. \[F:rinvsrx\] and \[F:TracesBG\]) given the same number of presynaptic spikes. Thus, regular presynaptic spiking is more efficient at driving synapses. This mechanism competes with a well-known property of excitable cells: that they are driven more effectively by irregular, positively correlated synaptic input currents [@Salinas00; @Salinas02; @MorenoBote08; @MorenoBote10]. In addition, a *population* of presynaptic spike trains drives a postsynaptic neuron more efficiently when the population-level activity is more irregular, for example due to pairwise correlations [@Rocha05]. Together, these results suggest that a postsynaptic neuron is most efficiently driven by presynaptic populations that exhibit small or negative auto-correlations, but positive pairwise cross-correlations.
Our model predicts that the de-correlating effects of short term depression and stochastic vesicle dynamics can produce negative temporal auto-correlations in the synaptic response even when presynaptic spiking is temporally uncorrelated or positively correlated, in agreement with previous studies [@Goldman02; @Rocha05]. This yields a response Fano factor that decreases with window size, as observed in some recorded data [@Kara00]. We note, though, that some parameter choices can yield positively a correlated synaptic response when presynaptic inputs are positively correlated [@Goldman02] and neuronal membrane dynamics can introduce positive correlations to a postsynaptic spiking response even when synaptic currents are not positively correlated in time [@MorenoBote06]. This is consistent with several studies showing positive temporal correlations in recorded spike trains [@Bair94; @Dan96; @Baddeley97; @Churchland10].
We predict that moderate or high firing rates can induce a Poisson-like synaptic response even when presynaptic inputs are non-Poisson (Fig. \[F:rinvsFx\] and [@Rocha02]). This is because even moderate firing rates can deplete synapses and depleted synapses inherit the Poisson-like variability of synaptic vesicle recovery (Fig. \[F:TracesSlowFast\]B and [@Rocha05; @RosenbaumPLoS12]). At lower firing rates, short term depression and synaptic variability can increase or decrease Fano factor. For example, in Fig. \[F:rinvsFx\]B, the response Fano factor is larger than the presynaptic Fano factor ($F_{\textrm{in}}=1$) at low firing rates, decreases at higher firing rates, then approaches $F_x=1$ at higher firing rates. This complex dependence of firing rate on Fano factor might be related to the stimulus dependence of Fano factors observed in several cortical brain regions [@Churchland10].
Our conclusion that fast presynaptic spiking causes Poisson-like variability in the synaptic response relied on the assumption that vesicle recovery times are exponentially distributed. The exponential distribution is a justifiable choice for recovery times only if recovery times obey a memoryless property: having already waited $t$ units of time for a recovery event, the probability of waiting an additional $s$ units of time does not depend on $t$. The precise mechanics of vesicle re-uptake and docking determine whether this is an appropriate assumption. If recovery times have a different probability distribution, then the synaptic response will inherit the properties of this distribution at high presynaptic firing rates instead of inheriting the Poisson-like nature of exponentially distributed recovery times.
Previous methods have been developed to analyze the synaptic response of the model used here. In [@Rocha02; @Goldman04], the model restricted to the $M=1$ case is analyzed for presynaptic spike trains that are renewal processes. This includes the Poisson and the regular input model discussed here, but excludes the irregular input model in which the spike train is a non-renewal inhomogeneous Poisson process. In [@RosenbaumPLoS12] approximations are obtained for the case where the presynaptic spike train is an inhomogeneous Poisson process, but the approximation is only valid when the rate-modulation of the Poisson process is small compared to the average firing rate. Thus, these approximations are only valid for the irregular input model when $r_f-r_s\ll r_s$. Other studies [@Lindner09; @Merkel10] use a deterministic synapse model that implicitly treats the number of available vesicles as a continuous rather than a discrete quantity. This deterministic model represents the trial average of the model considered here and can vastly underestimate the variability of a synaptic response [@RosenbaumPLoS12].
A more detailed synapse model allows for multiple docking sites at a single contact [@Wang99; @Rocha05]. This model can yield different response properties than the model used here in certain parameter regimes [@Rocha05]. Even though this more detailed model can be represented as a continuous-time Markov chain, the analysis of this model would be significantly more complex than the analysis considered here since it would be necessary to keep track of the number of readily releasable vesicles at each contact separately. This would result in a Markov chain with $K\times N^M$ states where $M$ is the number of contacts, $N$ is the number of docking sites per contact and $K$ is the number of states used for the presynaptic input model ($K=1$ for the Poisson input model, $K=2$ for the irregular input model, and $K=\theta$ for the regular input model).
To quantify the synaptic response to a presynaptic spike train, we focused on the statistics of the number of vesicles released in a time window. Postsynaptic neurons observe changes in synaptic conductance in response to presynaptic spikes. The synaptic conductance are often modeled in such a way that they can be easily derived from our process $x(t)$ through a convolution: $g(t)=\int_0^t x(t-s)\alpha(s)ds$ where $g(t)$ is the synaptic conductance elicited by a presynaptic spike train and $\alpha(s)$ is a kernel representing the characteristic postsynaptic conductance elicited by the release of a single neurotransmitter vesicle. Since this mapping is linear, the statistics of $g(t)$ can easily be derived in terms of the statistics of $x(t)$ [@Tetzlaff08; @RosenbaumPLoS12].
[^1]: This work supported by National Foundation of Science grant NSF-EMSW21-RTG0739261 and National Institute of Health grant NIH-1R01NS070865-01A1.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract:
- 'In 1980, Albert Fathi asked whether the group of area-preserving homeomorphisms of the $2$-disc that are the identity near the boundary is a simple group. In this paper, we show that the simplicity of this group is equivalent to the following fragmentation property in the group of compactly supported, area preserving diffeomorphisms of the plane: *there exists a constant $m$ such that every element supported on a disc $D$ is the product of at most $m$ elements supported on topological discs whose area are half the area of $D$.*'
- 'En 1980, Albert Fathi pose la question de la simplicité du groupe des homéomorphismes du disque qui préservent l’aire et sont l’identité près du bord. Dans cet article, nous montrons que la simplicité de ce groupe est équivalente à une propriété de fragmentation dans le groupe des difféomorphismes du plan, préservant l’aire et à support compact, à savoir : *il existe une constante $m$ telle que tout élément à support dans un disque $D$ est le produit d’au plus $m$ éléments dont les supports sont inclus dans des disques topologiques dont l’aire est la moitié de l’aire de $D$.*'
author:
- |
Frédéric Le Roux[^1]\
Laboratoire de mathématiques, UMR 8628\
Université Paris Sud, Bat. 425\
91405 Orsay Cedex FRANCE
title: 'Simplicity of $\homeo({{\mathbb{D}}}^2, \partial {{\mathbb{D}}}^2 , \mathrm{Area})$ and fragmentation of symplectic diffeomorphisms'
---
**AMS classification:** 37E30, 57S99, 28D15. **Keywords:** simple group; surface homeomorphism; hamiltonian dynamics; fragmentation; symplectic diffeomorphisms.
This paper is concerned with the algebraic study of the group $$G = \homeo({{\mathbb{D}}}^2, \partial {{\mathbb{D}}}^2 , \mathrm{Area})$$ of area-preserving homeomorphisms of the $2$-disc that are the identity near the boundary. The central open question is the following.
\[ques.fathi\] Is $G$ a simple group?
The study of the simplicity of groups of homeomorphisms goes back as far as 1935. Indeed in the famous Scottish Book ([@ScottishBook]), S. Ulam asked if the identity component in the group of homeomorphisms of the $n$-sphere is a simple group. This question was answered in the affirmative by Anderson and Fisher in the late fifties ([@Anderson58; @Fisher60]). In the seventies lots of (smooth) transformation groups were studied by D. Epstein, M. Herman, W. Thurston, J. Mather, A. Banyaga, and proved to be simple (see the books [@Banyaga97] or [@Bounemoura08]). Let us give some details on the group $G^\diff = \diffeo({{\mathbb{D}}}^2, \partial {{\mathbb{D}}}^2 , \mathrm{Area})$, which is the smooth analog of our group $G$. This group is not simple, since there exists a morphism from $G^\diff$ to ${{\mathbb{R}}}$, called the *Calabi invariant*. But Banyaga proved that the kernel of the Calabi invariant coincides with the subgroup $[G^\diff, G^\diff]$ generated by commutators, and is a simple group. Thus the normal subgroups of $G^\diff$ are exactly the inverse images of the subgroups of ${{\mathbb{R}}}$ under the Calabi morphism. The analog of question \[ques.fathi\] is also solved in higher dimensions. Indeed A. Fathi proved the simplicity of the group of volume preserving homeomorphisms of the $n$-ball which are the identity near the boundary, when $n \geq 3$. However, Question \[ques.fathi\] remains unsolved (see [@Fathi80]).
Actually some normal subgroups of $G$ have been defined by E. Ghys ([@Ghys07], see [@Bounemoura08]), and by S. Müller and Y.-G. Oh ([@MullerOh07]). But so far no one has been able to prove that these are proper subgroups: they might turn out to be equal to $G$. In this text, I propose to define still another family of normal subgroups $\{N_{\varphi}\}$ of $G$. I have not been able to prove that these subgroups are proper, but we can prove that they are good candidates.
\[theo.good-candidates\] If some normal subgroup of the family $\{N_{\varphi}\}$ is equal to $G$, then $G$ is simple.
The present work has its origin in Fathi’s proof of the simplicity in higher dimensions. Fathi’s argument has two steps. The first step is a fragmentation result: any element of the group can be written as a product of two elements, each of which is supported on a topological ball whose volume is $\frac{3}{4}$ of the total volume. The second step shows how this fragmentation property, let us call it $(P_{1})$, implies the perfectness (and simplicity) of the group. While the second step is still valid in dimension $2$, the first one fails. In the sequel we propose to generalise the fragmentation property $(P_{1})$ by considering a family of fragmentation properties $(P_{\rho})$ depending on the parameter $\rho \in (0,1]$ (a precise definition is provided in section \[sec.fragmentation-metric\]). A straightforward generalisation of Fathi’s second step will prove that if the property $(P_{\rho})$ holds for some $\rho$, then $G$ is simple (Lemma \[lem.pis\] below). On the other hand, we notice that if none of the properties $(P_{\rho})$ holds, then the subgroups $N_{\varphi}$ are proper, and thus $G$ is not simple (Lemma \[lem.npins\]). Thus we see firstly that Theorem \[theo.good-candidates\] holds, and secondly that Question \[ques.fathi\] is translated into a fragmentation problem, namely the existence of some $\rho$ such that property $(P_{\rho})$ holds. Christian Bonatti drew my attention to the possibility of formulating this fragmentation problem in terms of a single property $(P_{0})$. This property, which may be seen as the limit of the properties $(P_{\rho})$ as $\rho$ tends to zero, is the following: *there exists a constant $m$ such that any homeomorphism of the plane, supported on a disc having area equal to one, is the composition of $m$ homeomorphisms supported on some topological discs having area equal to one half.*
This discussion is summarised by the next theorem.
\[theo.fragmentation-homeo\] The following properties are equivalent:
1. the group $G$ is simple,
2. there exists some $\rho \in (0,1]$ such that the property $(P_{\rho})$ holds,
3. property $(P_{0})$ holds.
Furthermore we will prove that the simplicity of $G$ is also equivalent to the similar fragmentation property on the smooth subgroup $G^\diff$ (see Lemma \[lem.P0smooth\] and Theorem \[theo.fragmentation-diffeo\] in section \[sec.diffeo\] below). We will see in section \[sec.quasi-morphisms\] that Entov-Polterovich quasi-morphisms, coming from Floer homology, implies that the fragmentation property $(P_{\rho})$ do not hold for $\rho \in (\frac{1}{2},1]$. Whether it holds or not for $\rho \in [0,\frac{1}{2}]$ remains an open question.
The definitions and precise statments are given in section \[sec.fragmentation-metric\], as well as the links between properties $(P_{0})$ and $(P_{\rho})$ for $\rho>0$. The proofs of Theorem \[theo.good-candidates\] and \[theo.fragmentation-homeo\] are given in sections \[sec.npins\] and \[sec.pis\]. Sections \[sec.diffeo\] and \[sec.profile-diffeo\] provide the link with diffeomorphisms. Some more remarks, in particular the connection with other surfaces, are mentionned in section \[sec.remarks\]. Sections \[sec.profile-diffeo\], \[sec.quasi-morphisms\] and \[sec.remarks\] are independant.
#### Acknowledgments
I am pleased to thank Etienne Ghys for having introduced the problem to me (in La bussière, 1997); Albert Fathi, Yong-Geun Oh and Claude Viterbo for having organised the 2007 Snowbird conference that cast a new light on the subject; the “Symplexe” team for the excellent mathematical atmosphere, and especially Vincent Humilière, Emmanuel Opshtein and Pierre Py for the Parisian seminars and lengthy discussions around the problem; Pierre Py again for his precious commentaries on the text; Christian Bonatti for his “je transforme ton emmental en gruyère” trick; and Sylvain Crovisier and François Béguin for the daily morning coffees, with and without normal subgroups.
The fragmentation norms {#sec.fragmentation-metric}
=======================
In the whole text, the disc ${{\mathbb{D}}}^2$ is endowed with the normalised Lebesgue measure, denoted by $\mathrm{Area}$, so that $\mathrm{Area}({{\mathbb{D}}}^2)=1$. The group $G$ is endowed with the topology of uniform convergence (also called the $C^0$ topology), that turns it into a topological group. We recall that $G$ is arcwise connected: an elementary proof is provided by the famous Alexander trick ([@Alexander23]). We will use the term *topological disc* to denote any image of a euclidean closed disc under an element of the group $G$. As a consequence of the classical theorems by Schönflies and Oxtoby-Ulam, any Jordan curve of null area bounds a topological disc (see [@OxtobyUlam41]). Remember that the *support* of some $g \in G$ is the closure of the set of non-fixed points. For any topological disc $D$, denote by $G_{D}$ the subgroup of $G$ consisting of the elements whose support is included in the interior of $D$. Then each group $G_{D}$ is isomorphic to $G$, as shown by the following “re-scaling” process. Let $\Phi \in G$ be such that $D=\Phi^{-1}(D_{0})$ where $D_{0}$ is a euclidean disc. Then the map $g \mapsto \Phi g \Phi^{-1}$ provides an isomorphism between the groups $G_{D_{0}}$ and $G_{D}$. We may now choose a homothecy $\Psi$ that sends the whole disc ${{\mathbb{D}}}^2$ onto $D_{0}$, and similarly get an isomorphism $g \mapsto \Psi g \Psi ^{-1}$ between $G$ and $G_{D_{0}}$.
Definition of the fragmentation metrics {#definition-of-the-fragmentation-metrics .unnumbered}
---------------------------------------
Let $g$ be any element of $G$. We define the *size* of $g$ as follows: $$\mathrm{Size}(g) = \inf \{\mathrm{Area} (D), D \mbox{ is a topological disc that contains the support of } g \}.$$ Let us emphasize the importance of the word *disc*: an element $g$ which is supported on an annulus of small area surrounding a disc of large area has a large size. Also note that if $g$ has size less than the area of some disc $D$, then $g$ is conjugate to an element supported in $D$.
The following proposition says that the group $G$ is generated by elements of arbitrarily small size. It is an immediate consequence of Lemma 6.5 in [@Fathi80] (where the size is replaced by the diameter).
\[prop.fragmentation-homeo\] Let $g \in G$, and $\rho \in (0,1]$. Then there exists some positive integer $m$, and elements $g_{1}, \dots g_{m} \in G$ of size less than $\rho$, such that $$g = g_{m} \cdots g_{1}.$$
We now define the family of “fragmentation norms”.[^2] For any element $g \in G$ and any $\rho \in (0,1]$, we consider the least integer $m$ such that $g$ is equal to the product of $m$ elements of size less than $\rho$. This number is called the *$\rho$-norm* of $g$ and is denoted by $||g||_{\rho}$. The following properties are obvious.
\[prop.properties\] $$||h g h^{-1}||_{\rho} = ||g||_{\rho}, \ \
||g^{-1}||_{\rho} = ||g||_{\rho} , \ \
||g_{1} g_{2}||_{\rho} \leq ||g_{1}||_{\rho} + ||g_{2}||_{\rho}.$$
As a consequence, the formula $$d_{\rho}(g_{1}, g_{2}) = ||g_{1} g_{2}^{-1}||_{\rho}$$ defines a bi-invariant metric on $G$.
The normal subgroups $N_{\varphi}$ {#the-normal-subgroups-n_varphi .unnumbered}
----------------------------------
Given some element $g \in G$, we consider the $\rho$-norm of $g$ as a function of the size $\rho$: $$\rho \mapsto ||g||_{\rho},$$ and call it the *complexity profile* of $g$. Let $\varphi : (0,1] \to {{\mathbb{R}}}^+$ be any non-increasing function. We define the subset $N_{\varphi}$ containing those elements of $G$ whose complexity profile is essentially bounded by $\varphi$: $$N_{\varphi} = \{g \in G, ||g||_{\rho} = O(\varphi(\rho))\}$$ where the notation $\psi(\rho) = O(\varphi(\rho))$ means that there exists some $K>0$ such that $\psi(\rho) < K \varphi(\rho)$ for every small enough $\rho$. The following is an immediate consequence of proposition \[prop.properties\].
\[prop.normal\] For any non-increasing function $\varphi : (0,1] \to {{\mathbb{R}}}^+$, the set $N_{\varphi}$ is a normal subgroup of $G$.
The reader who wants some examples where we can estimate the complexity profile may jump to section \[sec.profile-diffeo\], where we will see that commutators of diffeomorphisms have a profile equivalent to the function $\varphi_{0} : \rho \mapsto \rho^{-1}$. This will imply that $N_{\varphi_{0}}$ is the smallest non-trivial subgroup of our family $\{N_{\varphi}\}$.
The fragmentation properties $(P_{\rho})$ {#the-fragmentation-properties-p_rho .unnumbered}
-----------------------------------------
Let $\rho \in (0,1]$. We now define our fragmentation property $(P_{\rho})$ by asking for a uniform bound in the fragmentation of elements of size less than $\rho$ into elements of a smaller given size.
- There exists some number $s \in (0,\rho)$, and some positive integer $m$, such that any $g\in G$ of size less than $\rho$ satisfies $||g||_{s} \leq m$.
Here are some easy remarks. Let us denote by $P(\rho,s)$ the property that there exists a bound $m$ with $||g||_{s} \leq m$ for every element $g$ of size less than $\rho$. Fix some $\rho \in (0,1]$ and some ratio $k \in (0,1)$. Assume that property $P(\rho,k\rho)$ holds. Then by re-scaling we get that property $P(\rho',k\rho')$ also holds for any $\rho' < \rho$ (with the same bound $m$). In particular we can iterate the fragmentation to get, for every positive $n$, property $P(\rho, k^n\rho)$ (with the bound $m^n$). This shows that property $P(\rho,s)$ implies property $P(\rho,s')$ for every $s' < s$. The converse is clearly true, so that property $P(\rho,s)$ depends only on $\rho$ and not on $s$. In particular we see that property $(P_{\rho})$ is equivalent to the existence of a number $m$ such that every $g$ of size less than $\rho$ satisfies $$||g||_{\frac{\rho}{2}} \leq m.$$ Also note that property $P(\rho_{0})$ implies property $P(\rho_{1})$ if $\rho_{1} < \rho_{0}$ (again by re-scaling). Thus property $P_{\rho}$ is more and more likely to hold as $\rho$ decreases from $1$ to $0$.
The fragmentation property $(P_{0})$ {#the-fragmentation-property-p_0 .unnumbered}
------------------------------------
In this paragraph we introduce the fragmentation property $(P_{0})$, and prove that it is equivalent to the existence of some $\rho>0$ such that property $(P_{\rho})$ holds. Consider, just for the duration of this section, the group $$\homeo_{c}({{\mathbb{R}}}^2, \mathrm{Area})$$ of compactly supported, area preserving homeomorphisms of the plane. Any image of a euclidean closed disc under some element of this bigger group will again be called a topological disc. We also define the size of an element of the group as in $G$. Property $(P_{0})$ is as follows.
- There exists some positive integer $m$ such that any $g\in \homeo_{c}({{\mathbb{R}}}^2, \mathrm{Area})$ of size less than $1$ is the composition of at most $m$ elements of $\homeo_{c}({{\mathbb{R}}}^2, \mathrm{Area})$ of size less than $\frac{1}{2}$.
Since each piece of the fragmentation provided by property $(P_{0})$ is supported on a disc with area $\frac{1}{2}$, the union of the supports has area at most $\frac{m}{2}$, but this gives no bound on the area of a topological disc containing this union. However, if the union of the supports surround a region with big area, then we may find a new fragmentation by “bursting the bubble”, *i.e.* conjugating the situation by a map that contracts the areas of the surrounded regions and preserves the area everywhere else. This is the key observation, due to Christian Bonatti, to the following lemma.
\[lem.P0\] Property $(P_{0})$ holds if and only if there exists some $\rho \in (0,1]$ such that property $(P_{\rho})$ holds.
Let us prove the easy part. Suppose $(P_{\rho})$ holds for some $\rho>0$, let $m$ be a bound for $||g||_{\frac{\rho}{2}}$ for those $g \in G$ of size less than $\rho$. Let $g \in \homeo_{c}({{\mathbb{R}}}^2, \mathrm{Area})$ of size less than $1$. Choose some element of the group that sends the support of $g$ into the euclidean unit disc ${{\mathbb{D}}}^2$, and compose it with the homothecy that sends ${{\mathbb{D}}}^2$ onto a disc of area $\rho$ included in ${{\mathbb{D}}}^2$; we denote by $\Psi$ the resulting map. Then $\Psi g \Psi^{-1}$ is an element of $G$ of size less than $\rho$. According to hypothesis $(P_\rho)$, we may write this element as a composition of $m$ elements of $G$ of size less than ${\frac{\rho}{2}}$. We may conjugate these elements by $\Psi^{-1}$ and take the composition to get a fragmentation of $g$ into $m$ elements of size $\frac{1}{2}$. Thus $(P_{0})$ holds.
Now assume that $(P_{0})$ holds, and let $m \geq 2$ be given by this property. We will prove that property $(P_{\rho})$ holds for $\rho = \frac{2}{m}$. Consider, in the plane, a euclidean disc $D$ of area $\frac{1}{\rho}$. By the same re-scaling trick as before, it suffices to prove that any $g \in \homeo_{c}({{\mathbb{R}}}^2, \mathrm{Area})$ with size less than $1$ and supported in the interior of $D$ may be fragmented as a product of $m$ elements of $\homeo_{c}({{\mathbb{R}}}^2, \mathrm{Area})$ with size less than one half and supported in the interior of $D$. Property $(P_{0})$ provides us with a fragmentation $g = g'_{m} \circ \cdots \circ g'_{1}$ by elements of size less than one half, but maybe not supported in $D$. Now comes the “bursting the bubbles” trick. Let $D'$ be a topological disc whose interior contains all the supports of the $g'_{i}$’s. The union of the supports has area less than $\frac{1}{\rho}$. Thus we may find some topological discs $K'_{1}, \dots K'_{\ell}$, included in the interior of $D'$, that are pairwise disjoint and disjoint from the supports of the $g'_{i}$’s, such that $$\mathrm{Area}\left (D' \setminus \bigcup_{j=1}^\ell K'_{j} \right) < \frac{1}{\rho}.$$ Denote by $D_{0}$ the support of our original map $g$. Note that $D_{0}$ is included in the union of the supports of the $g'_{i}$’s, thus it is disjoint from the $K'_{j}$’s. Since $D$ has area $\frac{1}{\rho}$, the previous inequality ensures the existence of some pairwise disjoint discs $K_{1}, \dots K_{\ell}$ in the interior of $D$, disjoint from $D_{0}$, such that $$\mathrm{Area}\left (D \setminus \bigcup_{j=1}^\ell K_{j} \right) = \mathrm{Area}\left (D' \setminus \bigcup_{j=1}^\ell K'_{j} \right).$$ Using Schönflies and Oxtoby-Ulam theorems, we can construct a homeomorphism $\Psi$ of the plane satisfying the following properties:
1. $\Psi$ is the identity on $D_{0}$,
2. $\Psi(D') = D$, $\Psi(K'_{j})= K_{j}$ for each $j$,
3. the restriction of $\Psi$ to the set $D' \setminus \cup_{j=1}^\ell K'_{j}$ preserves the area. [^3]
The first item shows that $\Psi g \Psi^{-1} = g$. Now for each $i$ we define $g_{i} = \Psi g'_{i} \Psi^{-1}$. Then the second item guarantees that the $g_{i}$’s are supported in $D$, and the third item entails that they preserve area and have size less than one half. The product of the $g_{i}$’s is equal to $g$, which provides the desired fragmentation.
Simplicity implies fragmentation {#sec.npins}
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\[lem.npins\] Assume that none of the properties $(P_{\rho}),\rho \in (0,1]$ holds. Let $\varphi : (0,1] \to {{\mathbb{R}}}^+$ be any function. Then the normal subgroup $N_{\varphi}$ is proper, *i. e.* it is not equal to $G$. In this case the group $G$ is not simple.
If we consider any element $f \neq \mathrm{Id}$ in $G$, and the function $\varphi_{f} : \rho \mapsto ||f||_{\rho}$, then the normal subgroup $N_{\varphi_{f}}$ contains $f$ and thus is not equal to $\{\mathrm{Id}\}$. Hence the non-simplicity of $G$ will be a consequence of the non-triviality of the subgroups $N_{\varphi}$.
According to the easy remarks following the definition of property $(P_{\rho})$, the hypothesis of the lemma reads the following way:
- for every $\rho \in (0,1]$ and every positive integer $m$ there exists some element $g$ of size less than $\rho$ such that $ ||g||_{\frac{\rho}{2}} > m$.
We fix any function $\varphi : (0,1] \to {{\mathbb{R}}}^+$, and we will construct some element $g$ in $G$ that does not belong to $N_{\varphi}$. Let us define $D_{0}= {{\mathbb{D}}}^2$. We pick two sequences of discs $(C_{i})_{i \geq 1}$ and $(D_{i})_{i \geq 1}$ converging to a point, such that for every $i$ (see figure \[fig.sequence1\]),
- $C_{i}$ and $D_{i}$ are disjoint and included in $D_{i-1}$,
- the area of $D_{i}$ is less than half the area of $C_{i}$.
![Construction of $g$[]{data-label="fig.sequence1"}](sequence1.eps)
We denote the area of $C_{i}$ by $\rho_{i}$. We will construct a sequence $(g_{i})_{i \geq 1}$, with each $g_{i}$ supported in the interior of $C_{i}$, and then $g$ will be defined as the (infinite) product of the $g_{i}$’s. Note that since the discs $C_{i}$’s are pairwise disjoint this product has a meaning, and since the sequence $(C_{i})$ converges to a point it actually defines an element of $G$. Since all the $g_{j}$’s with $j>i$ will be supported in the interior of the disc $D_{i}$ whose area is less than $\rho_{i}/2$ we will get $$\begin{aligned}
||g||_{\frac{\rho_{i}}{2}} & \geq & ||g_{i} \dots g_{1}||_{\frac{\rho_{i}}{2}}-1.\end{aligned}$$ The sequence $(g_{i})$ is constructed by induction. Assume $g_{1}, \dots , g_{i-1}$ have been constructed. Using hypothesis $(\star)$, we may choose $g_{i}$ supported on $C_{i}$ such that $
||g_{i}||_{\frac{\rho_{i}}{2}}$ is arbitrarily high, more precisely we demand the following inequality: $$\begin{aligned}
||g_{i}||_{\frac{\rho_{i}}{2}} & \geq & \frac{1}{\rho_{i}} \varphi\left(\frac{\rho_{i}}{2}\right) \quad + \quad
||g_{i-1} \dots g_{1}||_{\frac{\rho_{i}}{2}}
\quad + \quad 1.\end{aligned}$$ Using inequality (1), the triangular inequality and inequality (2) we get $$\begin{aligned}
||g||_{\frac{\rho_{i}}{2}} &\geq & ||g_{i} \dots g_{1}||_{\frac{\rho_{i}}{2}}-1 \\
& \geq & ||g_{i}||_{\frac{\rho_{i}}{2}}
\quad - \quad
||g_{i-1} \dots g_{1}||_{\frac{\rho_{i}}{2}} -1 \\
& \geq & \frac{1}{\rho_{i}} \varphi\left(\frac{\rho_{i}}{2}\right).\end{aligned}$$ This proves that the complexity profile of $g$ is not equal to $O(\varphi)$. In other words $g$ does not belong to $N_{\varphi}$.
Fragmentation implies simplicity {#sec.pis}
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\[lem.pis\] Assume that property $(P_{\rho})$ holds for some $\rho \in (0,1]$. Then $G$ is simple.
This lemma is just a slight generalisation of Fathi’s argument showing that, under property $(P_{1})$, the group $G$ is perfect: any element decomposes as a product of commutators. Then perfectness implies simplicity: this is due to “Thurston’s trick”, for completeness the argument is included in the proof below.
We assume that there exists a number $\rho \in (0,1]$ and a positive integer $m$ such that any element of size less than $\rho$ may be written as the product of $m$ elements of size less than $\frac{\rho}{2}$.
Let $C_{1}$ be a small disc. By usual fragmentation (proposition \[prop.fragmentation-homeo\]), any element of $G$ is a product of elements supported in a disc of area less than that of $C_{1}$, and any such element is conjugate to an element supported in the interior of $C_{1}$. Thus to prove perfectness it is enough to consider some element $g$ supported in the interior of $C_{1}$ and to prove that $g$ is a product of commutators.
Let us first prove that such a $g$ is a product of two commutators when considered in the group $ \homeo({{\mathbb{D}}}^2, \partial {{\mathbb{D}}}^2 )$, that is, let us forget for a while about the area (this is a “pedagogical” step). Choose two sequences of discs $(C_{i})_{i \geq 1}$ and $(D_{i})_{i \geq 1}$ converging to a point, such that (see figure \[fig.sequence2\])
- the interior of $D_{i}$ contains both $C_{i}$ and $C_{i+1}$,
- the $C_{i}$’s are pairwise disjoint,
- the $D_{2i}$’s (resp. the $D_{2i+1}$’s) are pairwise disjoint.
![The sequences $(C_{i})_{i \geq 1}$ and $(D_{i})_{i \geq 1}$[]{data-label="fig.sequence2"}](sequence2.eps)
For any $i\geq 1 $ choose some $h_{i} \in\homeo({{\mathbb{D}}}^2, \partial {{\mathbb{D}}}^2 )$, supported on $D_{i}$, that sends $C_{i}$ onto $C_{i+1}$. We let $g_{1}:= g$, thus $g_{1}$ is supported on $C_{1}$, and define inductively $g_{i+1} := h_{i} g_{i} h_{i }^{-1}$ ; thus $g_{i}$ is a “copy” of $g$, supported on $C_{i}$, and the $g_{i}$’s are pairwise commuting. Let $$K := g_{2} g_{3}^{-1} g_{4} g_{5}^{-1} \cdots , \ \ K' := g_{1} g_{2}^{-1} g_{3} g_{4} ^{-1} \cdots$$ so that $KK'=K'K=g$. The map $K = [g_{2},h_{2} ][ g_{4}, h_{4}] \cdots $ may be seen as an infinite product of commutators, but we need a finite product. Now define $$G := g_{2} g_{4} \dots , \ \ H:= h_{2} h_{4} \cdots , \ \ G' := g_{1} g_{3} \cdots , \ \ H' := h_{1} h_{3} \cdots$$ and observe that $K = [G,H]$ and $K'=[G',H']$: indeed these equalities may be checked independently on each disc $D_{i}$. Thus $g = [G,H][G',H']$ is a product of two commutators in $\homeo({{\mathbb{D}}}^2, \partial {{\mathbb{D}}}^2 )$.
Now let us take care of the area. We will use sequences $(C_{i})$ and $(D_{i})$ as before, and we will get around the impossibility of shrinking $C_{i}$ onto $C_{i+1}$ inside the group $G$ by using the fragmentation hypothesis. We may assume, for every $i$, the equality $$\area(C_{i+1}) = \frac{1}{2} \area(C_{i}).$$ Moreover, by fragmentation, we may assume this time that $g$ is supported in the interior of a disc $C'_{1} \subset C_{1}$ of area $\rho\area (C_{1})$. We use the fragmentation hypothesis re-scaled on $C_{1}$ to write $$g = f_{1,1} \dots f_{1,m}$$ (see figure \[fig.perfect\]) with each $f_{1,j}$ supported in the interior of a topological disc included in $C_{1}$ and whose area is $$\frac{\rho}{2}\area (C_{1}) = \rho \area (C_{2}).$$ We choose a disc $C'_{2} \subset C_{2}$ whose area also equals $\rho \area (C_{2})$ and, for each $j=1, \dots , m$, some $h_{1,j} \in G$ supported on $D_{1}$ and sending the support of $f_{1,j}$ inside $C'_{2}$. We define $$g_{2} := \prod_{j=1, \dots , m} h_{1,j} f_{1,j} h_{1,j}^{-1}$$ which is supported on $C'_{2}$ (see figure \[fig.perfect\]).
![Fragment and push every piece inside the small disc...[]{data-label="fig.perfect"}](fragmentation.eps)
We apply recursively the (re-scaled) fragmentation hypothesis to get a sequence $(g_{i})_{i \geq 1}$ with each $g_{i}$ supported in the interior of a disc $C'_{i} \subset C_{i}$ having area $\rho \area(C_{i})$ and sequences $(f_{i,j})_{i \geq 1,j=1, \dots , m}$ and $(h_{i,j})_{i \geq 1,j=1, \dots , m}$ with $f_{i,j}$ supported on $C_{i}$ and $h_{i,j}$ supported on $D_{i}$, such that $$g_{i} = \prod_{j=1, \dots , m} f_{i,j} \mbox{ and }
g_{i+1} = \prod_{j=1, \dots , m} h_{i,j} f_{i,j} h_{i,j}^{-1}.$$ Obviously $g_{i}$ and $g_{i+1}$ are equal up to a product of commutators *whose number of terms depends only on $m$*. More precisely, we have $$\begin{aligned}
g_{i} g_{i+1} ^{-1} & = & \prod_{j=1, \dots , m} f_{i,j} \prod_{j=m, \dots , 1} h_{i,j} f_{i,j}^{-1} h_{i,j}^{-1} \\
& = & \left[ f_{i,1} , P \right] \left( \prod_{j=2, \dots , m} f_{i,j} \prod_{j=m, \dots , 2} h_{i,j} f_{i,j} ^{-1} h_{i,j}^{-1} \right)
\left[f_{i,1}, h_{i,1} \right]\end{aligned}$$ where $P$ is equal to the term between parentheses, and we see recursively that $g_{i} g_{i+1} ^{-1}$ is a product of $2m$ commutators of elements supported in $D_{i}$; we write $$g_{i} g_{i+1} ^{-1} = \prod_{j=1, \dots , 2m} [s_{i,j}, t_{i, j}].$$ It remains to define the infinite commutative products $$K := g_{2} g_{3}^{-1} g_{4} g_{5}^{-1} \cdots , \ \ K' := g_{1} g_{2}^{-1} g_{3} g_{4} ^{-1} \cdots$$ $$S_{j} := s_{2,j} s_{4,j} \dots , \ \ T_{j}:= t_{2,j} t_{4,j} \cdots , \ \ S'_{j} := s_{1,j} s_{3,j} \cdots , \ \ T'_{j} := t_{1,j} t_{3,j} \cdots$$ and to check that $$K= \prod_{j = 1, \dots 2m} [S_{j}, T_{j}], \quad
K' = \prod_{j = 1, \dots 2m} [S'_{j}, T'_{j}], \quad
\mbox{ and } g = KK'$$ is a product of $4m$ commutators. This proves that $G$ is perfect.
Let us recall briefly, according to Thurston, how perfectness implies simplicity. Let $D$ be a disc and $g,h\in G$ be such that the discs $D, g(D), h(D)$ are pairwise disjoint. Let $u,v\in G$ be supported in $D$. In this situation the identity $$[u,v] = [[u,g],[v,h]]$$ may easily be checked, and shows that $[u,v]$ belongs to the normal subgroup generated by $g$. Now given any $g\neq \mathrm{Id}$ in $G$, one can find an $h \in G$ and a disc $D$ such that the above situation takes place. If $G$ is perfect then so is the isomorphic group $G_{D}$, hence every $f$ supported in $D$ is a product of commutators supported in $D$, and by the above equality such an $f$ belongs to the normal subgroup generated by $g$. By fragmentation this subgroup is thus equal to $G$. This proves that $G$ is simple, and completes the proof of the lemma.
Fragmentation of diffeomorphisms {#sec.diffeo}
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Here we further translate Question \[ques.fathi\] into the diffeomorphisms subgroup $G^\diff$.
Let $G^\diff = \diffeo({{\mathbb{D}}}^2, \partial {{\mathbb{D}}}^2 , \mathrm{Area})$ be the group of elements of $G$ that are $C^\infty$-diffeomorphisms. Note that for every topological disc $D$ the group of elements supported in the interior of $D$, $$G_{D}^\diff := G^\diff \cap G_{D},$$ is isomorphic to $G^\diff$, even when $D$ is not smooth: indeed we may use Riemann conformal mapping theorem and Moser’s lemma to find a smooth diffeomorphism $\Phi$ between the interiors of ${{\mathbb{D}}}^2$ and $D$ with constant Jacobian, and the conjugacy by $\Phi$ provides an isomorphism (see [@GreeneShiohama79] for the non-compact version of Moser’s lemma). As in the continuous case, we define the $\rho$-norm $||g||^\diff_{\rho}$ of any element $g \in G^\diff$ as the minimum number $m$ of elements $g_{1}, \dots , g_{m}$ of $G^\diff$, having size less than $\rho$, whose composition is equal to $g$. The fragmentation properties $(P^\diff_{\rho})$ are defined as in the continuous setting.
- (for $\rho \in (0,1]$) There exists some number $s \in (0,\rho)$, and some positive integer $m$, such that any $g\in G^\diff$ of size less than $\rho$ satisfies $||g||^\diff_{s} \leq m$.
- There exists some positive integer $m$ such that any $g\in \diffeo_{c}({{\mathbb{R}}}^2, \mathrm{Area})$ of size less than $1$ is the composition of at most $m$ elements of $\diffeo_{c}({{\mathbb{R}}}^2, \mathrm{Area})$ of size less than $\frac{1}{2}$.
Here $\diffeo_{c}({{\mathbb{R}}}^2, \mathrm{Area})$ is the group of $C^\infty$-diffeomorphisms of the plane that are compactly supported and preserve the area. The smooth version of Lemma \[lem.P0\] holds, with the same proof (using footnote \[foot.smooth\]).
\[lem.P0smooth\] Property $(P^\diff_{0})$ holds if and only if there exists some $\rho \in (0,1]$ such that property $(P^\diff_{\rho})$ holds.
We now turn to the equivalence between fragmentation properties for homeomorphisms and diffeomorphisms.
\[theo.fragmentation-diffeo\] For any $\rho \in (0,1]$, the properties $(P_{\rho})$ and $(P^\diff_{\rho})$ are equivalent.
The proof of this equivalence requires two ingredients. The first one is the density of $G^\diff$ in $G$; this is a classical result, see for example [@Sikorav07]. The second one is the uniformity of the fragmentation in $G$, and in $G^\diff$, inside some $C^0$-neighbourhood of the identity. This is provided by the following proposition, which is proved below.
\[prop.fragmentation-locale\] For any $\rho \in (0, 1)$, there exists a neighbourhood $\cV_{\rho}$ of the identity in $G$ with the following properties.
- Any $g$ in $\cV_{\rho}$ satisfies $||g||_{\rho} \leq \frac{2}{\rho}$;
- any $g$ in $\cV_{\rho} \cap G^\diff$ satisfies $||g||^\diff_{\rho} \leq \frac{2}{\rho}$.
As a consequence of this proposition we get the following comparison between the norms $||.||^\diff_{\rho}$ and $||.||_{\rho}$ on $G^\diff$.
\[coro.comparison-metrics\] Any $g \in G^\diff$ satisfies $$||g||_{\rho} \leq ||g||^\diff_{\rho} \leq ||g||_{\rho} + \frac{2}{\rho}.$$
The first inequality is clear. To prove the second one consider some $g \in G^\diff$, and let $m = ||g||_{\rho}$. By definition there exists some elements $g_{1}, \dots , g_{m}$ in $G$ of size less than $\rho$ such that $g = g_{m} \cdots g_{1}$. Since $G^\diff$ is $C^0$-dense in $G$, for any topological disc $D$ the subgroup $G^\diff_{D}$ is also $C^0$-dense in $G_{D}$. Thus we can find elements $g'_{1}, \dots , g'_{m}$ in $G^\diff$, still having size less than $\rho$, whose product $g' = g'_{m} \cdots g'_{1}$ is a diffeomorphism arbitrarily $C^0$-close to $g$. According to the second item in proposition \[prop.fragmentation-locale\] we get $$||g'g^{-1}||^\diff_{\rho} \leq \frac{2}{\rho}.$$ Since $||g'||^\diff_{\rho} \leq m$ the triangular inequality gives $||g||^\diff_{\rho} \leq m + \frac{2}{\rho}$, as wanted.
Let us fix $\rho \in (0,1]$. The fact that $(P_{\rho})$ implies $(P_{\rho}^\diff)$ immediatly follows from the corollary. We prove the converse implication. Assume that $(P_{\rho}^\diff)$ holds: there exists some $s \in (0,\rho)$ and $m>0$ such that the quantity $||g'||^\diff_{s}$ is bounded by $m$ on the elements $g'$ of $G^\diff$ of size less than $\rho$. Choose any $g \in G$ of size less than $\rho$. According to the first item in proposition \[prop.fragmentation-locale\], and using the density of $G^\diff$ in $G$, we get some $g' \in G^\diff$ having size less than $\rho$, and sufficiently close to $g$ so that $||g'g^{-1}||_{s} \leq \frac{2}{s}$. Since $(P_{\rho}^\diff)$ holds we also have $||g'||^\diff_{s} \leq m$, and thus $||g'||_{s} \leq m$, from which we get $||g||_{s} \leq m + \frac{2}{s}$. Thus $(P_{\rho})$ also holds.
We now turn to the proof of proposition \[prop.fragmentation-locale\]. The classical proof of fragmentation for diffeomorphisms relies on the inverse mapping theorem and would only gives uniformity in a $C^1$-neighbourhood of the identity (see [@Banyaga97; @Bounemoura08]). Thus we will rather try to mimic the proof of the fragmentation for homeomorphisms.
The proof of both items (uniformity of local fragmentation for homeomorphisms and diffeomorphisms) are very similar; we will only provide details for the diffeomorphisms case.
We choose an integer $m$ bigger than $\frac{2}{\rho}$, and we cut the disc into $m$ strips of area less than $\frac{\rho}{2} $: more precisely, we choose $m$ topological discs $D_{1}, \cdots, D_{m}$ such that (see figure \[fig.Vrho\])
1. $\mathrm{Area}(D_{i}) \leq \frac{\rho}{2}$,
2. ${{\mathbb{D}}}^2 = D_{1} \cup \cdots \cup D_{m}$,
3. $D_{i} \cap D_{j} = \emptyset $ if $| j-i| > 1$,
4. $D_{i} \cup D_{i+1} \cup \cdots \cup D_{j}$ is a topological disc for every $i \leq j$, and the intersection of $D_{1} \cup \cdots \cup D_{i}$ and $D_{i} \cup \cdots \cup D_{m}$ with the boundary of ${{\mathbb{D}}}^2$ is non-empty and connected for every $i$.
Now we define the following set $\cV_{\rho}$: $$\cV_{\rho} = \left\{g \in G \mbox{ such that } g(D_{i}) \cap D_{j} = \emptyset \mbox{ for every } i,j \mbox{ with } | j-i| > 1\right\}.$$
![The discs $D_{i}$ and the action of some $g$ in $\cV_{\rho}$[]{data-label="fig.Vrho"}](Vrho.eps)
Note that, due to item 3, $\cV_{\rho}$ is a $C^0$-neighbourhood of the identity in $G$. We will prove that each element of $\cV_{\rho} \cap G^\diff$ can be written as a product of $m-1$ elements of $G_{\rho}^\diff$.
Let $g \in \cV_{\rho} \cap G^\diff$. By hypothesis $D_{1}$ and $g(D_{1})$ are both disjoint from the topological disc $D_{3} \cup \cdots \cup D_{m}$. By the classical Lemma \[lem.classical\] below, we can find $\Psi_{1} \in G^\diff$ such that
- $\Psi_{1} = g$ on some neighbourhood of $D_{1}$,
- $\Psi_{1}$ is the identity on some neighbourhood of $D_{3} \cup \cdots \cup D_{m}$.
The diffeomorphism $\Psi_{1}$ is supported in the interior of the topological disc $D_{1} \cup D_{2}$ whose area is less than or equal to $\rho$. Let $g_{1} := \Psi_{1}^{-1} g$, thus $g_{1}$ is supported in the interior of $D_{2} \cup \cdots \cup D_{m}$, and we easily check that this diffeomorphism is still in $\cV_{\rho}$. In particular $D_{1} \cup D_{2}$ and its image under $g_{1}$ are both disjoint from $D_{4} \cup \cdots \cup D_{m}$. We apply again the lemma to get some $\Psi_{2}\in G$ such that
- $\Psi_{2} = g_{1}$ on some neighbourhood of $D_{1}\cup D_{2}$,
- $\Psi_{2}$ is the identity on some neighbourhood of $D_{4} \cup \cdots \cup D_{m}$.
Thus $\Psi_{2}$ is supported in the interior of $D_{2} \cup D_{3}$. Let $g_{2} := \Psi_{2}^{-1} g_{1}$; this diffeomorphism is in $\cV_{\rho}$ and is supported in the interior of $D_{3} \cup \cdots \cup D_{m}$. In the same way we construct diffeomorphisms $\Psi_{1}, \dots ,\Psi_{m-1}$, such that each $\Psi_{i}$ is supported in the interior of $D_{i} \cup D_{i+1}$, and such that $g = \Psi_{1} \circ \cdots \circ \Psi_{m-1}$. This completes the proof for the diffeomorphisms case.
\[lem.classical\] Let $D'_{1}, D'_{2}$ be two disjoint topological discs in ${{\mathbb{D}}}^2$, and assume that the intersection of $D'_{1}$ (resp. $D'_{2}$) with the boundary of $\partial {{\mathbb{D}}}^2$ is non-empty and connected (and thus ${{\mathbb{D}}}^2 \setminus \inte(D'_{1} \cup D'_{2})$ is again a topological disc). Let $\Phi \in G^\diff$, and suppose that $\Phi(D'_{1})$ is disjoint from $D'_{2}$. Then there exists $\Psi \in G^\diff$ such that $\Psi = \Phi$ on some neighbourhood of $D'_{1}$ and $\Psi = \mathrm{Id}$ on some neighbourhood of $D'_{2}$.
By Smale’s theorem ([@Smale59]) and Moser’s lemma (see for example [@Banyaga97] or [@Bounemoura08]), the group $G$ is arcwise connected. Let $(\Phi_t)_{t \in [0,1]}$ be a smooth isotopy from the identity to $\Phi$ in $G$. It is easy to find another smooth isotopy $(g_t)_{t \in [0,1]}$ (that does not preserve area), supported in the interior of ${{\mathbb{D}}}^2$, such that
- for every $t$, $g_t (\Phi_t (D'_{1}))$ is disjoint from $D'_{2}$,
- $g_0 = g_1= \mathrm{Id}$.
The isotopy $(\Phi'_{t}) = (g_t \Phi_t)$ still goes from the identity to $\Phi$. Consider the vector field tangent to this isotopy, and multiply it by some smooth function that is equal to $1$ on some neighbourhood of $\cup_{t} \Phi'_{t}(D'_{1})$ and vanishes on $D'_{2}$. By integrating this truncated vector field we get another isotopy $(\Psi_t)$ such that
- on some neighbourhood of $D'_{1}$ we have $\Psi_t = g_t \Phi_t$ for every $t$, and in particular $\Psi_1 = \Phi$,
- the support of $\Psi_{1}$ is disjoint from $D'_{2}$.
Thus $\Psi_{1}$ satisfies the conclusion of the lemma, except that it does not preserve the area. Let $\omega_{0}$ be the Area form on ${{\mathbb{D}}}^2$, and $\omega_{1}$ be the pre-image of $\omega_{0}$ under $\Psi_{1}$. Then $\omega_{1} = \omega_{0}$ on some neighbourhood of $\partial {{\mathbb{D}}}^2 \cup D'_{1} \cup D'_{2}$. By Moser’s lemma we may find some $\Psi_{2}\in G$, whose support is disjoint from $D'_{1}$ and $D'_{2}$, and that sends $\omega_{1}$ to $\omega_{0}$. The diffeomorphism $\Psi = \Psi_{1} \Psi_{2}^{-1}$ suits our needs.
Note that this lemma has a $C^0$-version, which is proved by replacing Smale’s theorem by Alexander’s trick, the truncation of vector fields by Schönflies’s theorem, and Moser’s lemma by Oxtoby-Ulam’s theorem.
Profiles of diffeomorphisms {#sec.profile-diffeo}
===========================
One can wonder what the complexity profile looks like for a diffeomorphism, both inside the group $G^\diff$ and inside the group $G$. The following proposition only partially solves this problem.
\[prop.profile-diffeo\]
- For any $g \in G^\diff$ we have $||g||^\diff_{\rho} =O(\frac{1}{\rho^2})$.
- For any $g$ in the commutator subgroup $[G^\diff, G^\diff]$ we have $||g||^\diff_{\rho} =O(\frac{1}{\rho})$.
If the support of $g \in G$ has area $A$, then clearly for any $\rho$ we need at least $\frac{A}{\rho}$ elements of $G_{\rho}$ or $G^\diff_{\rho}$ to get $g$. Thus, according to the second point, the profile of any $g$ in $[G^\diff, G^\diff]$ is bounded from above and below by multiples of the function $\varphi_{0} : \rho \mapsto \frac{1}{\rho}$; and this holds both in $G$ and $G^\diff$. In particular we see that $N_{\varphi_{0}}$ is the smallest non-trivial subgroup of our family $\{N_{\varphi}\}$. I have not been able to decide whether the first point is optimal, nor whether $G^\diff \subset N_{\varphi_{0}}$ or not (if not, of course, then $G$ is not simple). It might also happens that $G^\diff$ is included in $N_{\varphi_{0}}$ but not in the analog smooth group $N^\diff_{\varphi_{0}}$.
We also notice that *every non trivial normal subgroup of $G$ contains the commutator subgroup $[G^\diff, G^\diff]$*, and thus the normal subgroup of $G$ generated by $[G^\diff, G^\diff]$ is the only minimal non trivial normal subgroup of $G$. This fact is an immediate consequence of Thurston’s trick (see the last paragraph of section \[sec.pis\]) and Banyaga’s theorem. Indeed let $g$ be a non trivial element in $G$, choose $h \in G$ and a disc $D$ such that $D$, $g(D)$, $h(D)$ are pairwise disjoint. Thurston’s trick shows that the normal subgroup $N(g)$ generated by $g$ in $G$ contains some non trivial commutator of diffeomorphisms, let us denote it by $\Phi$. By Banyaga’s theorem any element of $[G^\diff, G^\diff]$ is a product of conjugates (in $G^\diff$) of $\Phi$ and $\Phi^{-1}$, and thus $[G^\diff, G^\diff]$ is included in $N(g)$.
Let $g \in G^\diff$, and fix some smooth isotopy $(g_{t})_{t \in [0,1]}$ from the identity to $g$ in $G^\diff$. Let $M>0$ be such that every trajectory of the isotopy has speed bounded from above by $M$.
We now fix some $\rho>0$, and let $m$ be the smallest integer such that $m \geq \frac{2}{\rho}$. We consider some discs $D_{1}, \dots D_{m}$ as in the proof of proposition \[prop.fragmentation-locale\]; if we choose the $D_{i}$’s to be horizontal slices, then for every $i,j$ with $| i -j | > 1$ we get $$d(D_{i},D_{j}) > \frac{C}{m}$$ where $d$ is the euclidean metric of the unit disc and $C$ is some constant (maybe $C=\frac{\pi}{2}$). Due to the definition of $M$, within any interval of time less than $\frac{C}{mM}$, no point moves a distance more than $\frac{C}{m}$: for every $t,t'\in [0,1]$, for every $x \in {{\mathbb{D}}}^2$, $$| t- t'| < \frac{C}{mM} \quad \Longrightarrow \quad d(g_{t}(x), g_{t'}(x)) < \frac{C}{m}.$$ In particular the topological disc $g_{t'}g_{t}^{-1}(D_{i})$ remains disjoint from $D_{j}$ for every $|i-j| >1$; that is, the diffeomorphism $g_{t'}g_{t}^{-1}$ belongs to the neighbourhood $\cV_{\rho}$ defined in the proof of proposition \[prop.fragmentation-locale\]. Thus we can write $g$ as the product of at most $\frac{mM}{C}+1$ elements of $\cV_{\rho}$, $$g =g_{1} = \left(g_{1} g_{1-\frac{1}{k}}^{-1}\right) \left(g_{1-\frac{1}{k}} g_{1-\frac{2}{k}}^{-1}\right) \cdots \left(g_{\frac{1}{k}} g_{0}^{-1}\right)$$ (where $k$ is the integer part of $\frac{mM}{C}+1$). Each element in $\cV_{\rho}$ is the product of at most $m-1$ elements whose sizes are less than $\rho$, thus we get the estimate $$||g||^\diff_{\rho} \leq (m-1) \left(\frac{mM}{C}+1\right).$$ When $\rho$ tends to $0$, the right-hand side quantity is equivalent to $\frac{4M}{C}\frac{1}{\rho^2}$, which proves the first point of the proposition.
We turn to the second point. We first prove the result for some special commutator. Let $D$ be any displaceable disc, say of area $\frac{1}{3}$, and let $\Phi$ be any non trivial element of $G^\diff$ supported in the interior of $D$. Choose some $\Psi\in G^\diff$ such that $\Psi(D)$ is disjoint from $D$. Let us define $g := [\Phi,\Psi]$. We claim that for any $\rho$ we have $$||g||^\diff_{\rho} \leq \frac{4}{3\rho}.$$ To prove the claim fix some positive $\rho$. It is easy to find, almost explicitly, some $\Psi_{\rho} \in G^\diff$ which is a product of less than $\frac{2}{3\rho}$ elements of size less than $\rho$ and that moves $D$ disjoint from itself (see figure \[fig.moves\]). Then $$||\ [\Phi,\Psi_{\rho}] \ ||^\diff_{\rho} \leq ||\Phi\Psi_{\rho} \Phi^{-1}||^\diff_{\rho} + ||\Psi_{\rho}^{-1}||^\diff_{\rho} = 2||\Psi_{\rho}||^\diff_{\rho} \leq \frac{4}{3\rho}.$$ We now notice that the map $g = [\Phi,\Psi]$ is conjugate to $[\Phi,\Psi_{\rho}]$: indeed we may find some $\Theta \in G^\diff$ that is the identity on $D$ and equals $\Psi_{\rho} \Psi^{-1}$ on $\Psi(D)$ (this uses a variation on Lemma \[lem.classical\]), and such a $\Theta$ provides the conjugacy. Since the fragmentation norm is a conjugacy invariant, this proves the claim.
![How to move a disc disjoint from itself within $\frac{2}{3\rho}$ modifications of size less than $\rho$: after each modification, the area of the intersection of $D$ with its image has decreased by almost $\frac{\rho}{2}$[]{data-label="fig.moves"}](moves.eps)
We end the proof of the proposition by using Banyaga’s theorem. Since the commutator subgroup $[G^\diff, G^\diff]$ is simple, the normal subgroup of $G^\diff$ generated by $g$ in $G^\diff$ is equal to $[G^\diff, G^\diff]$. As in the $C^0$ case the set of elements $g'\in G^\diff$ satisfying $||g'||^\diff_{\rho} =O(\frac{1}{\rho})$ is a normal subgroup, since it contains $g$ it has to contain $[G^\diff, G^\diff]$. This proves the second point of the proposition.
Solution of the fragmentation problem for $\rho > \frac{1}{2}$ {#sec.quasi-morphisms}
==============================================================
In [@EntovPolterovichPy08] the authors describe some quasi-morphisms on the group of symplectic diffeomorphisms on various surfaces that are continuous with respect to the $C^0$-topology and, as a consequence, extend continuously to the group of area-preserving homeomorphisms. It turns out that their family of quasi-morphisms on the disc provides a solution to the “easiest” half of our fragmentation problems: properties $(P_{\rho})$ and $(P^\diff_{\rho})$ do not hold when $\rho > \frac{1}{2}$. We discuss this briefly.
Remember that a map $\phi : G^\diff \to {{\mathbb{R}}}$ is a *quasi-morphism* if the function $$| \phi(gh) - \phi(g) -\phi(h) |$$ is bounded on $G^\diff \times G^\diff$ by some quantity $\Delta(\phi)$ called the *defect* of $\phi$. A quasi-morphism is called *homogeneous* if it satisfies $\phi(g^n) = n \phi(g)$ for every $g$ and every integer $n$. The construction of the continuous quasi-morphisms uses the quasi-morphisms of Entov and Polterovich on the $2$-sphere ([@EntovPolterovich03]). This quasi-morphism is a *Calabi* quasi-morphism, that is, it coincides with the Calabi morphism when restricted to those diffeomorphims supported on any displaceable disc. By embedding ${{\mathbb{D}}}^2$ inside the two-sphere as a non-displaceable[^4] disc ([@EntovPolterovich03], Theorem 1.11 and section 5.6), and subtracting the Calabi morphism ([@EntovPolterovichPy08]), one can get a family $(\phi_{\rho'})_{\rho' \in (\frac{1}{2},1]}$ of homogeneous quasi-morphisms on $G^\diff$. (As these quasi-morphisms extend to $G$, see [@EntovPolterovichPy08], we could alternatively choose to work with $G$ and $(P_{\rho})$ instead of $G^\diff$ and $(P_{\rho}^\diff)$.) They satisfy the following properties.
1. $\phi_{\rho'}(g') = 0$ for any $g' \in G^\diff$ whose size is less than $\rho'$,
2. for every $\rho>\rho'$ there exists some $g \in G^\diff$ of size less than $\rho$ with $\phi_{\rho'}(g) \neq 0$.
This entails that for every $\rho \in (\frac{1}{2},1]$, property $(P_{\rho}^\diff)$ do not hold. To see this, given some $\rho \in (\frac{1}{2},1]$, we fix some $\rho' \in (\frac{1}{2},\rho)$, and we search for some $g \in G^\diff$ having size less than $\rho$ and whose $\rho'$-norm is arbitrarily large (see the easy remarks at the end of section \[sec.fragmentation-metric\]). The first property of $\phi_{\rho'}$ entails, for every $g \in G^\diff$, $$\phi_{\rho'}(g) \leq (||g||^\diff_{\rho'}-1)\Delta(\phi_{\rho'}).$$ On the other hand the second property provides some $g$ with size less than $\rho$ and such that $\phi_{\rho'}(g)\neq0$. Since $\phi_{\rho'}$ is homogeneous, the sequence $(\phi_{\rho'} (g^n))_{n \geq 0}$ is unbounded. Thus the sequence $(||g^n||^\diff_{\rho'})_{n \geq 0}$ is also unbounded, which proves that $(P_{\rho}^\diff)$ do not hold.
We are naturally led to the following question.
Does there exist, for every $\rho' \in (0,\frac{1}{2}]$, some homogeneous quasi-morphism $\phi_{\rho'}$ satisfying properties 1 and 2 as expressed above?
A positive answer would imply a negative answer to Question \[ques.fathi\].
Some more remarks {#sec.remarks}
=================
“Lots of” normal subgroups (if any!) {#lots-of-normal-subgroups-if-any .unnumbered}
------------------------------------
The proof of Lemma \[lem.npins\] can easily be modified to show that, if none of the properties $(P_{\rho})$ holds, then there exists an uncountable family $\cF$ of functions $\varphi$ such that the corresponding family of normal subgroups $(N_{\varphi})_{\varphi \in \cF}$ is totally ordered by inclusion. The following is another attempt to express that if $G$ is not simple, then it has to contain “lots of” normal subgroups.
Assume $G$ is not simple. Then every compact subset $K$ of $G$ is included in a proper normal subgroup of $G$.
Note that the situation is radically different for the diffeomorphisms group $G^\diff$, since (by Banyaga’s theorem [@Banyaga97], and since the centralizer of $G^\diff$ is trivial) any one-parameter subgroup of diffeomorphisms that is not included in the commutator subgroup $[G^\diff,G^\diff]$ normally generates $G^\diff$. However these are not purely algebraic statements since they involve the topology of the groups $G$ and $G^\diff$.
Consider some $\rho \in (0,1]$. Let $\cV_{\rho}$ be the neighbourhood of the identity given by proposition \[prop.fragmentation-locale\]: we have $||g||_{\rho} < \frac{2}{\rho}$ for every $g \in \cV_{\rho}$. By compactness we may find a finite family $g_{1}, \dots , g_{k}$ such that the sets $g_{i}.\cV_{\rho}$ cover $K$. Thus the fragmentation is also uniform on $K$, in other words the set $K$ is bounded with respect to the norm $||g||_{\rho}$. Define $$\varphi_{K}(\rho):=\sup\left\{||g||_{\rho}, g \in K \right\}.$$ This defines a non-increasing function, and clearly $K$ is included in the normal subgroup $N_{\varphi_{K}}$. According to Theorem \[theo.good-candidates\], if $G$ is not simple then $N_{\varphi_{K}}$ is a proper subgroup of $G$, which completes the proof of the corollary.
Other surfaces {#other-surfaces .unnumbered}
--------------
Let $S$ be any compact surface equipped with an area form. Consider the group of homeomorphisms that preserves the measure associated to the area form, and denote by $G_{0}(S)$ the (normal) subgroup generated by the homeomorphisms that are supported inside a topological disc. For example, $G_{0}({{\mathbb{S}}}^2)$ is just the group of orientation and area preserving homeomorphisms of the sphere, and $G_{0}(\bbT^2)$ is the group of orientation and area preserving homeomorphisms of the torus with zero mean rotation vector. The group $G_{0}(S)$ may also be seen as the closure of the group of hamiltonian diffeomorphisms of $S$ inside the group of homeomorphisms. For every surface $S$, it is an open question whether the group $G_{0}(S)$ is simple or not.
Exactly as before, on $G_{0}(S)$ we may define the size of an element supported in a topological disc, the family of fragmention norms $||.||^S_{\rho}$, the family of normal subgroups $N^S_{\varphi}$, and the fragmentation properties $(P^S_{\rho})$. Then Lemma \[lem.npins\] still holds, with the same proof: the failure of all the properties $(P^S_{\rho})$ would entail that every normal subgroup $N^S_{\varphi}$ is proper, and that $G_{0}(S)$ is not simple. On the other hand the proof of Lemma \[lem.pis\] shows that each property $(P^{{{\mathbb{D}}}^2}_{\rho})$ forces the simplicity of $G_{0}(S)$ (alternatively, we could use the original Lemma \[lem.pis\] and Thurston trick to see that the simplicity of $G=G_{0}({{\mathbb{D}}}^2)$ implies that of $G_{0}(S)$). But property $(P^{S}_{\rho})$ is *a priori* weaker than $(P^{{{\mathbb{D}}}^2}_{\rho})$, because on a general surface one has more space than in the disc to perform the fragmentation. This prevents us from fully translating the simplicity question into a fragmentation problem on the other surfaces.
Using a different approach, one might hope to recover this equivalence by adapting to the $C^0$ context the homology machinery introduced by Thurston in the smooth category (see [@Banyaga97] or [@Bounemoura08], section 2.2).
[Br12]{} Alexander, James W. *On the deformation of an $n$-cell*, Proc. Nat. Acad. Sci. 9 (1923), 406–407.
R. D. Anderson. The Algebraic Simplicity of Certain Groups of Homeomorphisms. *Amer. J. Math* 80 (1958), 955–963.
Banyaga, Augustin. *The structure of classical diffeomorphism groups*. Mathematics and its Applications, 400. Kluwer Academic Publishers Group, Dordrecht, 1997.
Bounemoura, Abed. *Simplicité des groupes de transformations de surfaces*. [Ensaios Matemáticos]{} 14 (2008).
Burago, Dmitri; Ivanov, Sergei; Polterovich, Leonid. Conjugation-invariant norms on groups of geometric origin. <http://arxiv.org/abs/0710.1412v1>.
Entov, Michael; Polterovich, Leonid. Calabi quasimorphism and quantum homology. *Int. Math. Res. Not.* 30 (2003), 1635–1676.
Entov, Michael; Polterovich, Leonid ; Py, Pierre. *On continuity of quasi-morphisms for area-preserving maps*, in preparation.
Fathi, Albert. Structure of the group of homeomorphisms preserving a good measure on a compact manifold. *Ann. Sci. École Norm. Sup.* 13 (1980), 45–93.
G.M. Fisher. On the group of all homeomorphisms of a manifold. *Trans. of the Amer. Math. Soc.* 97 (1960), 193–212.
Ghys, Étienne. Private communication, 1996, and communication at the Symplectic Topology and Measure-Preserving Dynamical Systems Conference, Snowbird, July 2007.
Greene, Robert E.; Shiohama, Katsuhiro. Diffeomorphisms and volume-preserving embeddings of noncompact manifolds. *Trans. Amer. Math. Soc.* 255 (1979), 403–414.
Oh, Yong-Geun; Müller, Stefan. The group of Hamiltonian homeomorphisms and $C\sp 0$-symplectic topology. *J. Symplectic Geom.* 5 (2007), 167–219.
Oxtoby, John C.; Ulam, Stanislas M. Measure-preserving homeomorphisms and metrical transitivity. *Ann. of Math*. 42 (1941), 874–920.
Sikorav, Jean-Claude. Approximation of a volume-preserving homeomorphism by a volume-preserving-diffeomorphism. Disponible sur [http://www.umpa.ens-lyon.fr/ symplexe](http://www.umpa.ens-lyon.fr/~symplexe).
Smale, Stephen. Diffeomorphisms of the $2$-sphere. *Proc. Amer. Math. Soc.* **10** (1959), 621–626.
Ulam, Stanislaw M. *The Scottish Book: A Collection of Problems*, Los Alamos, 1957.
[^1]: This work was partially supported by the ANR Grant “Symplexe” BLAN 06-3-137237. However, the author does not support the French research policy represented by the ANR, which promotes post-doctoral positions at the expense of permanent positions and project funding at the expense of long-term funding.
[^2]: The definition of the fragmentation norm is not new, see example 1.24 in [@BuragoIvanovPolterovich07].
[^3]: \[foot.smooth\] Having in mind the smooth case (Lemma \[lem.P0smooth\] below), we notice that we may further demand that the map $\Psi$ is a $C^\infty$-diffeomorphism on the interior of $D' \setminus \cup_{j=1}^\ell K'_{j}$. Actually, we may even choose the sets $D'$, $K_{j}$ and $K'_{j}$ to be smooth discs, and then the map $\Psi$ may be chosen to be a $C^\infty$-diffeomorphism of the plane.
[^4]: Note that when the area of the disc tends to the total area of the sphere, the parameter $\rho'$ below tends to $\frac{1}{2}$; the bigger the disc, the more useful the quasi-morphism, as far as our problem is concerned.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study the tripartite entanglement for a class of mixed states defined by the mixture of GHZ and W states, $\rho=p|GHZ\rangle\langle GHZ|+(1-p)|W\rangle\langle W|$. Based on the Caratheodory theorem and the periodicity assumption, the possible optimal decomposition of the states has been derived, which is not independent on the detailed measure of entanglement. We find that, according to $p$, there are two different decompositions containing 3 or 4 quantum states in the decomposition respectively. When the decomposition contains 3 quantum states, the tripartite entanglement of the mixed state is simply the entanglement of superposition states of GHZ and W. When the decomposition contains 4 quantum states, the tripartite entanglement of the mixed state is a liner function of $p$. We also study the relations between the three-tangle and three-$\pi$. It is shown that the three-tangle is smaller than the three-$\pi$. Moreover, the three-$\pi$ has a minimal point in the interval 0 and 1, while the three-tangle is a non decreasing function of $p$.'
address:
- |
Department of Physics, Capital normal University,\
Beijing, 100048, China\
E-mail: [email protected]
- |
School of Mathematical Sciences, Capital Normal University,\
Beijing, 100048, China\
Max-Planck-Institute for Mathematics in the Sciences,\
Leipzig, 04103, Germany\
E-mail: [email protected]
author:
- Teng ma
- 'Shao-Ming Fei'
title: 'Three-tangle and Three-$\pi$ for a class of tripartite mixed states'
---
Introduction {#aba:sec1}
============
Quantum entangled states are the key resources in quantum computation and quantum information processing [@Nielsen2000]. Computation and detection of quantum entanglement are the essential subjects in the theory of quantum entanglement. For bipartite, in particular, lower dimensional systems, there are already many useful results, such as entanglement of formation[@Bennett1996; @H2001], concurrece[@con], negativity[@G.Vidal],relative entropy[@vv], PPT criterion[@A.P1996] and Bell inequalities[@Bell1996].
Tripartite entanglement is more complicated than bipartite entanglement. Investigation of tripartite entanglement is the basis of studying multipartite entanglement. However, although tripartite entanglement is well defined, it is formidably difficult to compute the tripartite entanglement analytically. Up to now, there is no general formulation to calculate tripartite entanglement, only a few class of tripartite entanglement can be calculated efficiently[@lohmayer2006; @eltschka2007].
In this article, we study the entanglement of a class of tripartite mixed stats $\rho = p\left| {GHZ} \right\rangle \left\langle {GHZ}
\right| + (1 - p)\left| W \right\rangle \left\langle W \right|$. We give a more detailed impossible optimal decomposition than , moreover our formulation can be used not only three-tangle[@coffman], but also other entanglement measure. In section 3, we study two important tripartite entanglement measure and make a comparison between them.
The main formulations
=====================
Consider the tripartite state $\rho=p|GHZ\rangle\langle
GHZ|+(1-p)|W\rangle\langle W|$, where$|GHZ\rangle=1/\sqrt{2}(|000\rangle+|111\rangle)$, $|W\rangle=1/\sqrt{3}(|001\rangle+|010\rangle+|100\rangle)$ and $|0\rangle$, $|1\rangle$ represents the two dimensions of every partite . If the entanglement of pure states is $E(\Psi_{i})$, then the entanglement of mixed state is $E(\rho)=min\sum
p_{i}E(\Psi_{i})$[@Bennett1996], where $\rho=\sum p_{i}\Psi_{i}$ and the $min$ is taking all the possible decompositions of $\rho$. The decomposition taking the minimum $\sum p_{i}E(\Psi_{i})$ is call the optimal decomposition, and if we know the optimal decomposition of $\rho$, then we can get the entanglement $E(\rho)$. One can proof that our mixed state $\rho$ can be disassembled by the pure state $|q,\theta\rangle=\sqrt{q}|GHZ\rangle-\sqrt{1-q}e^{i\theta}|W\rangle$.
To get the optimal decomposition of a state, we need to answer two questions: first, what is number of pure states of the optimal decomposition? second, which is those pure states? To the first question, Caratheodory’s theorem[@4H] said that 4 pure states are sufficient to minimize the entanglement for rank-2 states. Hence we need to investigate decompositions with 2,3, or 4 pure states.
To answer the second question and consider the symmetry properties of $|GHZ\rangle$ and $|W\rangle$ to three parties, we assume that the entanglement of $|q,\theta\rangle$ is a periodical function of $\theta$ with period $2\pi/3$. By the assumption that the entanglement of $|q,\theta\rangle$ is a periodical function of $\theta$ with period $2\pi/3$ for a fixed $q$, we assume $E(|q,\theta\rangle)$ get minimal value when $\theta_{n}=\theta_{\ast}+2\pi n/3, n=... -2, -1, 0, 1, 2 ...$. So $\sum_{i}a_{i}E(q_{i},\theta_{i}\rangle)\geq\sum_{i}a_{i}E(q_{i},\theta_{ni}\rangle)\nonumber$, then the possible pure state of optimal decomposition becomes $|q,\theta_{n}\rangle=\sqrt{q}|GHZ\rangle-\sqrt{1-q}e^{i\theta_{n}}|W\rangle$.
We first investigate the optimal decomposition contains 3 pure states $$\rho_{op3}=a|q_{1},\theta_{n1}\rangle\langle
q_{1},\theta_{n1}|+b|q_{2},\theta_{n2}\rangle\langle
q_{2},\theta_{n2}|+c|q_{3},\theta_{n3}\rangle\langle
q_{3},\theta_{n3}|$$ where $a,b,c\in[0,1]$ and $a+b+c=1$. In fact, the $\rho_{op3}$ also investigates the situation that the optimal decomposition contains 2 pure stats, because $a,b,c$ can be zero. Under the bases $\{|GHZ\rangle,|W\rangle\}$, $\rho_{op3}$ and $\rho$ can be expressed as follows: $$\begin{aligned}
&\rho_{op3}=\left({\begin{array}{c}
aq_{1}+bq_{2}+cq_{3} \\
-a\sqrt{q_{1}(1-q_{1})}e^{i\theta_{n1}} -b\sqrt{q_{2}(1-q_{2})}e^{i\theta_{n2}} -c\sqrt{q_{3}(1-q_{3})}e^{i\theta_{n3}}
\end{array}}\right.\\
&\left.{\begin{array}{c}
-a\sqrt{q_{1}(1-q_{1})}e^{-i\theta_{n1}} -b\sqrt{q_{2}(1-q_{2})}e^{-i\theta_{n2}} -c\sqrt{q_{3}(1-q_{3})}e^{-i\theta_{n3}}\\
1-(aq_{1}+bq_{2}+cq_{3})
\end{array}}
\right)\nonumber
\end{aligned}$$ $$\rho_{op3}=\left(
\begin{array}{cc}
p & 0 \\
0 & 1-p \\
\end{array}
\right)\nonumber$$ Obversely, $\rho_{op3}=\rho$ must hold, we have $$\left\{ \begin{array}{l}
aq_1 + bq_2 + cq_3 = p \\
a\sqrt {q_1 (1 - q_1 )} e^{ - i\theta _{n1} } + b\sqrt {q_2 (1 - q_2 )} e^{ - i\theta _{n2} } + c\sqrt {q_3 (1 - q_3 )} e^{ - i\theta _{n3} } = 0 \\
a + b + c = 1 \\
\end{array} \right.$$
If $\rho_{op3}$ is the optimal decomposition, then $E(\rho )= aE(|
q_1 ,\theta _{n1}\rangle ) + bE(| q_2 ,\theta _{n2} \rangle ) +
cE(|q_3 ,\theta _{n3}\rangle)$. Because $ aE(|q_1 ,\theta _{n1}
\rangle ),bE(|q_2 ,\theta _{n2}\rangle )$ and $cE(|q_3 ,\theta _{n3}
\rangle) \ge 0$, we have $$\begin{aligned}
aE(|q_1 ,\theta _{n1}\rangle ) + bE(|q_2 ,\theta _{n2}\rangle) +
cE(|q_3 ,\theta _{n3}\rangle) \\ \geq 3\sqrt[3]{aE(|q_1 ,\theta
_{n1}\rangle )bE(|q_2 ,\theta _{n2}\rangle )cE(|q_3 ,\theta
_{n3}\rangle )}\end{aligned}$$ The equality hold if and only if $ aE(|q_1
,\theta _{n1}\rangle ) = bE(|q_2 ,\theta _{n2}\rangle ) = cE(|q_3
,\theta _{n3}\rangle )$, and the definition of $E(\rho)$ demand the equality of (3) must hold, then we have $$aE(|q_1, \theta _{n1}\rangle) = bE(|q_2 ,\theta _{n2}\rangle ) =
cE(|q_3 ,\theta _{n3}\rangle)$$ Equation (2) and (4) is the conditions that $ a,b,c,q_1 ,q_2 ,q_3$ must satisfy, and we fond (2) and (4) can be satisfied if $$\left\{{\begin{array}{*{20}c}
{a = b = c} \\
{q_1 = q_2 = q_3 } \\
{\theta _{n1} = \theta _* } \\
{\theta _{n2} = \theta _* + \frac{{2\pi }}{3}} \\
{\theta _{n3} = \theta _* + \frac{{4\pi }}{3}}
\end{array}}\right.$$ By (2), (4) and (5), equation (1) becomes $$\begin{aligned}
&\rho _{opt3} = \frac{1}{3}\left| {p,\theta _* } \right\rangle
\left\langle {p,\theta _* } \right| + \frac{1}{3}\left| {p,\theta _*
+ \frac{{2\pi }}{3}} \right\rangle \left\langle {p,\theta _* +
\frac{{2\pi }}{3}} \right| \\&+ \frac{1}{3}\left| {p,\theta _* +
\frac{{4\pi }}{3}} \right\rangle \left\langle {p,\theta _* +
\frac{{4\pi }}{3}} \right|
\end{aligned}$$ This is the possible optimal decomposition containing 2 and 3 pure states.
Let us investigate the optimal decomposition that containing 4 pure states. The optimal decomposition should be (6) adding one pure state $$\begin{aligned} &\rho _{opt4} = a\left| {q',\theta _n ^\prime } \right\rangle
\left\langle {q',\theta _n ^\prime } \right| + b(\left| {q,\theta
_* } \right\rangle \left\langle {q,\theta _* } \right| + \left|
{q,\theta _* + \frac{{2\pi }}{3}} \right\rangle \left\langle
{q,\theta _* + \frac{{2\pi }}{3}} \right| \\&+ \left| {q,\theta _*
+ \frac{{4\pi }}{3}} \right\rangle \left\langle {q,\theta _* +
\frac{{4\pi }}{3}} \right|)
\end{aligned}$$ where $a,b\in[0,1]$ and $a+3b=1$. Under the bases $ \{ \left| {GHZ}
\right\rangle ,\left| W \right\rangle \}$, $\rho_{opt4}$ can be expressed as $$\begin{aligned} \rho _{opt4} =
a\left( {\begin{array}{*{20}c}
{q'} & { - \sqrt {q'(1 - q')} e^{ - i\theta _n } } \\
{ - \sqrt {q'(1 - q')} e^{i\theta _n } } & {1 - q'} \\
\end{array}} \right) + b\left( {\begin{array}{*{20}c}
{3q} & 0 \\
0 & {3(1 - q)} \\
\end{array}} \right)\nonumber
\end{aligned}$$ Because $\rho _{opt4} = \rho$ must hold, we have $$\left\{ \begin{array}{l}
aq' + 3bq = p \\
a\sqrt {q'(1 - q')} e^{ - i\theta _n } = 0 \\
a + 3b = 1 \\
\end{array} \right.$$ When $\rho_{op4}$ contains four pure states, $a\neq0$, so $\sqrt
{q'(1 - q')} = 0$, that is $q' = 0$ or $ q' = 1$, so the adding pure state is $|W\rangle$ or $|GHZ\rangle$. When $q' = 0$, by (8), we get $b=p/3q$, a=(q-p)/q, due to $a, b\geq0$ and $q>0$, $p$ have an range $0 \le p \le q$. The corresponding optimal decomposition is $ \rho
_{opt40} = \frac{{q - p}}{q}\left| W \right\rangle \left\langle W
\right| + \frac{p}{{3q}}(\left| {q,\theta _* } \right\rangle
\left\langle {q,\theta _* } \right| + \left| {q,\theta _* +
\frac{{2\pi }}{3}} \right\rangle \left\langle {q,\theta _* +
\frac{{2\pi }}{3}} \right| + \left| {q,\theta _* + \frac{{4\pi
}}{3}} \right\rangle \left\langle {q,\theta _* + \frac{{4\pi }}{3}}
\right|)$, and the corresponding entanglement is $E_{opt40} =
\frac{{q - p}}{q}E\left( {\left| W \right\rangle } \right) +
\frac{p}{q}E\left( {\left| {q,\theta _* } \right\rangle } \right)$. Note that for a giving $p$, $E_{opt40}$ can vary due to $q$, so $q$ must take a fixed value $q_{*0}$ to make $E_{opt40}$ minimal, then the final $\rho_{opt40}$ is $$\begin{aligned}
&\rho _{opt40} = \frac{{q_{*0} - p}}{{q_{*0} }}\left| W
\right\rangle \left\langle W \right| + \frac{p}{{3q_{*0} }}(\left|
{q_{*0} ,\theta _* } \right\rangle \left\langle {q_{*0} ,\theta _* }
\right| \\&+ \left| {q_{*0} ,\theta _* + \frac{{2\pi }}{3}}
\right\rangle \left\langle {q_{*0} ,\theta _* + \frac{{2\pi }}{3}}
\right| + \left| {q_{*0} ,\theta _* + \frac{{4\pi }}{3}}
\right\rangle \left\langle {q_{*0} ,\theta _* + \frac{{4\pi }}{3}}
\right|)\end{aligned}$$ where $0 \le p \le q$ and $q_{*0}$ is the minimal point of $E_{opt40}$. When $q' = 1$, by the same method we get $E_{opt41} =
\frac{{p - q}}{{1 - q}}E\left( {\left| {GHZ} \right\rangle } \right)
+ \frac{{1 - p}}{{1 - q}}E\left( {\left| {q,\theta _* }
\right\rangle } \right)$, and $$\begin{aligned}
&\rho _{opt41} = \frac{{p - q_{*1} }}{{1 - q_{*1} }}\left| {GHZ}
\right\rangle \left\langle {GHZ} \right| + \frac{1}{3}\frac{{1 -
p}}{{1 - q_{*1} }}(\left| {q_{*1} ,\theta _* } \right\rangle
\left\langle {q_{*1} ,\theta _* } \right| \\&+ \left| {q_{*1}
,\theta _* + \frac{{2\pi }}{3}} \right\rangle \left\langle {q_{*1}
,\theta _* + \frac{{2\pi }}{3}} \right| + \left| {q_{*1} ,\theta _*
+ \frac{{4\pi }}{3}} \right\rangle \left\langle {q_{*1} ,\theta _* +
\frac{{4\pi }}{3}} \right|)
\end{aligned}$$ where $q_{*1} \le p \le 1$ and $q_{*1}$ is the minimal point of $E_{opt41}$.
Up to now we have investigate all the possible situations, $\rho_{opt3}$ is the possible decomposition containing two and three pure states and $\rho_{opt40}$ and $\rho_{opt41}$ is the possible decomposition containing four pure states. By equations (6),(9),(10) we can get the corresponding entanglement $E_{opt3}$, $E_{opt40}$, and $E_{opt41}$, and by the definition of entanglement of formation[@Bennett1996] we have our main formulation $$E\left( \rho \right) = \min \{ E_{opt3} ,E_{opt40} ,E_{opt41} \}$$ where $E_{opt3} = E\left( {\left| {p,\theta _* } \right\rangle }
\right)$, when $0 \le p \le 1$, $E_{opt40} = \frac{{q_{*0} -
p}}{{q_{*0} }}E\left( {\left| W \right\rangle } \right) +
\frac{p}{{q_{*0} }}E\left( {\left| {q_{*0} ,\theta _* }
\right\rangle } \right)$, when $0 \le p \le q_{*0}$, $E_{opt41} =
\frac{{p - q_{*1} }}{{1 - q_{*1} }}E\left( {\left| {GHZ}
\right\rangle } \right) + \frac{{1 - p}}{{1 - q_{*1} }}E\left(
{\left| {q_{*1} ,\theta _* } \right\rangle } \right)$, when $q_{*1}
\le p \le 1$, and $q_{*0}$, $q_{*1}$ is the minimal point of $E_{op40}$, $E_{op41}$ respectively.
If we know the entanglement of pure state $|q,\theta\rangle$, and $E(|q,\theta\rangle)$ is a periodical function of $\theta$ with period $2\pi/3$, then by equation (11) we can calculate the entanglement of mixed state $\rho$. If $E_{opt40}$ and $E_{opt41}$ is a differentiable function on $q\in(0,1)$, we get $$\begin{aligned}
&\frac{{\partial E_{opt40} }}{{\partial q}} = p(\frac{1}{{q^2
}}E(\left| W \right\rangle - \frac{1}{{q^2 }}E(\left| {q,\theta _*
} \right\rangle ) + q\frac{{dE(\left| {q,\theta _* } \right\rangle
}}{{dq}})\\
&\frac{{\partial E_{opt41} }}{{\partial q}} = (1 - p)( -
\frac{1}{{(1 - q)^2 }}E(\left| {GHZ} \right\rangle ) + \frac{1}{{(1
- q)^2 }}E(\left| {q,\theta _* } \right\rangle )\\&+ \frac{1}{{1 -
q}}\frac{{dE(\left| {q,\theta _* } \right\rangle }}{{dq}})
\end{aligned}$$ If $q_{*0}$, $q_{*1}$ is the minimal point of $E_{opt40}$, $E_{opt41}$ respectively, then $\left. {\frac{{\partial E_{opt40}
}}{{\partial q}}} \right|_{q = q_{*0} } = 0$, $\left.
{\frac{{\partial E_{opt41} }}{{\partial q}}} \right|_{q = q_{*1} } =
0$, by (12) we known $q_{*0}$, $q_{*1}$ do not depend on $p$. If $E_{opt40}$ and $E_{opt41}$ have few singular points on $q\in(0,1)$, we can compare those point with the differentiable points to get the minimal point.
The study of three-tangle and three-$\pi$ measurement
=====================================================
Three-tangle[@coffman] and Three-$\pi$[@yongchengou2007] is two different important tripartite entanglement measure, in this section we study these two measurements using our former formulations. First, let us study the three-tangle. For state $\left| {q,\theta } \right\rangle = \sqrt q \left| {GHZ}
\right\rangle - \sqrt {1 - q} e^{i\theta } \left| W \right\rangle$, the three-tangle is[@lohmayer2006] $$\tau \left( {\left| {q,\theta } \right\rangle } \right) = \left|
{q^2 - \frac{8}{9}e^{i3\theta } \sqrt {6q(1 - q)^3 } } \right|$$ Obversely, $\tau(|q,\theta\rangle)$ is a periodical function of $\theta$ with periodic $2\pi/3$, and when $\theta _n = \frac{{2\pi
}}{3}n$, $n\in\mathbb{Z}$, the three-tangle of state $|q,\theta\rangle$ get minimal for a fixed $q$, that is $\theta_{*}=0$. Note that $\tau_{opt3}$ have a singular point $q_{*}=\frac{4\sqrt[3]{2}}{3+4\sqrt[3]{2}}\doteq0.627$ on $q\in(0,1)$, and by (12), we can see that it is also the singular point of $\tau_{opt40}$ and $\tau_{opt41}$. With the equation (11) we have $$\tau \left( \rho \right) = \min \{ \tau _{opt3} ,\tau _{opt40}
,\tau _{opt41} \}$$ where $\tau _{opt3} = \tau \left( {\left| {p,0} \right\rangle }
\right)$, when $0 \le p \le 1$; $\tau _{opt40} = \frac{p}{{q_{*0}
}}\tau \left( {\left| {q_{*0} ,0} \right\rangle } \right)$, when $0
\le p \le q_{*0}$; $\tau _{opt41} = \frac{{p - q_{*1} }}{{1 -
q_{*1} }} + \frac{{1 - p}}{{1 - q_{*1} }}\tau \left( {\left| {q_{*1}
,0} \right\rangle } \right)$, when $q_{*1} \le p \le 1$. To get the minimal point of $\tau_{opt40}$, we first consider the singular point $q_{*}$. When $q=q_{*}$, we get $\tau _{opt40} =
\frac{p}{{q_* }}\tau \left( {\left| {q_* ,0} \right\rangle } \right)
= 0$ and $\tau _{opt41} = \frac{{p - q_* }}{{1 - q_* }} + \frac{{1
- p}}{{1 - q_* }}\tau \left( {\left| {q_* ,0} \right\rangle }
\right) = \frac{{p - q_* }}{{1 - q_* }}$. $\tau_{opt40}$ has already get minimal, so $q_{*0} = q_*$. For $\tau_{opt41}$, we only consider the interval $q\in(q_{*},1)$ due to when $0 \le p \le 1$, $\tau_{opt40}$ is already the optimal decomposition. By (12) and $0<q<1$, we have $2q - 1 > 0$ and $155q^2 - 155q + 32 = 0$, then get the possible minimal point $q_{*1} ^\prime = \frac{1}{2} +
\frac{3}{{310}}\sqrt {465} \buildrel\textstyle\over\doteq 0.709$. Comparing with the singular point $\tau _{opt41} \left( {q_* }
\right) > \tau _{opt41} \left( {q_{*1} ^\prime } \right)$, so the minimal point of $\tau_{opt41}$ is $q_{*1} = q_{*1} ^\prime =
\frac{1}{2} + \frac{3}{{310}}\sqrt {465}
\buildrel\textstyle\over\doteq 0.709$. By comparing $\tau _{opt3} ,
\tau _{opt40}$ and $\tau _{opt41}$, we get when $0 \le p \le
q_{*0}$, $\tau _{opt40} \le \tau _{opt3}$, if and only if $p = 0$ or $p = q_{*0}$ equality holds, when $q_{*1} \le p \le 1$, $\tau
_{opt41} \le \tau _{opt3}$, if and only if $p = q_{*0}$ or $p = 1$ equality holds, when $q_{*0} < p < q_{*1}$ there is only $\tau_{opt3}$. Then by (14) we get $$\begin{array}{cc}
\tau \left( \rho \right) = \left\{ \begin{array}{l}
\frac{p}{{q_{*0} }}( - q_{*0} ^2 + \frac{8}{9}\sqrt {6q_{*0} (1 - q_{*0} )^3 } ) \\
p^2 - \frac{8}{9}\sqrt {6p(1 - p)^3 } \\
\frac{{p - q_{*1} }}{{1 - q_{*1} }} + \frac{{1 - p}}{{1 - q_{*1} }}(q_{*1} ^2 - \frac{8}{9}\sqrt {6q_{*1} (1 - q_{*1} )^3 } ) \\
\end{array} \right.
&
\begin{array}{l}
0 \le p \le q_{*0} \\
q_{*0} < q < q_{*1} \\
q_{*1} \le p \le 1 \\
\end{array}
\end{array}$$ where $q_{*0} = {\raise0.7ex\hbox{${4\sqrt[3]{2}}$}
\!\mathord{\left/
{\vphantom {{4\sqrt[3]{2}} {(3 + 4\sqrt[3]{2})}}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{${(3 + 4\sqrt[3]{2})}$}}
\buildrel\textstyle.\over= 0.627$, $q_{*1}= \frac{1}{2} +
\frac{3}{{310}}\sqrt {465} \buildrel\textstyle.\over= 0.709$, and (6), (9), (10) is the corresponding optimal decomposition.
To study three-$\pi$, we first need getting the expression of three-$\pi$ of the pure state $|q,\theta\rangle$. The definition of three-$\pi$ of a pure state is[@yongchengou2007] $$\pi = \frac{1}{3}(\pi _a + \pi _b + \pi _c )$$ where $\pi _a = N_{a(bc)} ^2 - N_{ab} ^2 - N_{ac} ^2$, $\pi _b =
N_{b(ac)} ^2 - N_{ba} ^2 - N_{bc} ^2$, and $\pi _c = N_{c(ab)} ^2
- N_{ca} ^2 - N_{cb} ^2$. $N$ is a kind of bipartite entanglement measure called negativity[@4H1998][@G.Vidal], the definition is $N_{ab} = \left\| {\rho _{ab} ^{Ta} } \right\| - 1$, $\|\rho\|$ is trace norm, it equals the sum of modulus of eigenvalues of $\rho$. $\rho _{ab} ^{Ta}$ is the partial transpose of $\rho _{ab}$, satisfy $(\rho _{ab} ^{Ta} )_{ij,kl} = (\rho _{ab} )_{kj,il}$. By pure state $|q,\theta\rangle$, under the natural base, we get $$\rho _{ab}= \left( {\begin{array}{*{20}c}
{\frac{q}{2} + \frac{{1 - q}}{3}} & { - \sqrt {\frac{{q(1 - q)}}{6}} e^{ - i\theta } } & { - \sqrt {\frac{{q(1 - q)}}{6}} e^{ - i\theta } } & { - \sqrt {\frac{{q(1 - q)}}{6}} e^{i\theta } } \\
{ - \sqrt {\frac{{q(1 - q)}}{6}} e^{i\theta } } & {\frac{{1 - q}}{3}} & {\frac{{1 - q}}{3}} & 0 \\
{ - \sqrt {\frac{{q(1 - q)}}{6}} e^{i\theta } } & {\frac{{1 - q}}{3}} & {\frac{{1 - q}}{3}} & 0 \\
{ - \sqrt {\frac{{q(1 - q)}}{6}} e^{ - i\theta } } & 0 & 0 & {\frac{q}{2}} \\
\end{array}} \right)$$ $$\rho _{ab} ^{Ta} = \left( {\begin{array}{*{20}c}
{\frac{q}{2} + \frac{{1 - q}}{3}} & { - \sqrt {\frac{{q(1 - q)}}{6}} e^{ - i\theta } } & { - \sqrt {\frac{{q(1 - q)}}{6}} e^{i\theta } } & {\frac{{1 - q}}{3}} \\
{ - \sqrt {\frac{{q(1 - q)}}{6}} e^{i\theta } } & {\frac{{1 - q}}{3}} & { - \sqrt {\frac{{q(1 - q)}}{6}} e^{ - i\theta } } & 0 \\
{ - \sqrt {\frac{{q(1 - q)}}{6}} e^{ - i\theta } } & { - \sqrt {\frac{{q(1 - q)}}{6}} e^{i\theta } } & {\frac{{1 - q}}{3}} & 0 \\
{\frac{{1 - q}}{3}} & 0 & 0 & {\frac{q}{2}} \\
\end{array}} \right)$$ and $$\rho _a= \left( {\begin{array}{*{20}c}
{\frac{q}{2} + \frac{{2(1 - q)}}{3}} & {\sqrt {\frac{{q(1 - q)}}{6}} e^{ - i\theta } } \\
{\sqrt {\frac{{q(1 - q)}}{6}} e^{i\theta } } & {\frac{q}{2} + \frac{{(1 - q)}}{3}} \\
\end{array}} \right)$$ By calculate $\rho_{bc}$, $\rho_{ac}$, $\rho_{b}$, $\rho_{c}$, we can see that due to the symmetry property of $|GHZ\rangle$ and $|W\rangle$ to every partite, $\rho _{ab} = \rho _{bc} = \rho
_{ac}$, $\rho _a = \rho _b = \rho _c$, then we have $N_{ab} =
N_{ac} = N_{bc}$, $\pi _a = \pi _b = \pi _c$, $\pi = \pi _a$. One can prove for three qubit pure states, $N_{a(bc)} = C_{a(bc)}$, where $C_{a(bc)}$ is the $Concurrence$[@con] between systems $a$ and $b,c$. Then by (19) we get $N_{ab}=C_{a(bc)} = \sqrt {2(1 -
Tr\rho _a ^2 )} = \sqrt {\frac{5}{9}q^2 - \frac{4}{9}q +
\frac{8}{9}}$. By (18) we get the characteristic equation of matrix $\rho _{ab} ^{Ta}$ $$\begin{array}{l}
\lambda ^4 - \lambda ^3 + (\frac{5}{{36}}q^2 - \frac{q}{9} + \frac{2}{9})\lambda ^2 \\ + [\frac{(q(1 -
q))^{3/2}}{3\sqrt 6 }\cos{3\theta} - \frac{7}{{27}}q^3 + \frac{7}{{18}}q^2 - \frac{q}{6}+ \frac{1}{{27}}]\lambda
\\+ [ - \frac{q(q(1 -
q))^{3/2}}{6\sqrt 6 }\cos{3\theta} - \frac{{41}}{{648}}q^4 + \frac{{149}}{{648}}q^3 - \frac{{13}}{{54}}q^2 + \frac{7}{{81}}q - \frac{1}{{81}}] = 0 \\
\end{array}$$ Then the three-$\pi$ of pure state $|q,\theta\rangle$ can be expressed as $$\begin{array}{l}
\pi = \pi _a = N_{a(bc)} ^2 - N_{ab} ^2 - N_{ac} ^2 = C_{a(bc)} ^2 - 2N_{ab} ^2 \\
= \frac{5}{9}q^2 - \frac{4}{9}q + \frac{8}{9} - 2(\sum\limits_{i = 1}^4 {\left| {\lambda _i \left( {q,Cos3\theta } \right)} \right|} - 1)^2 \\
\end{array}$$
where $ i = 1,2,3,4$ and $\lambda _i \left( {q,Cos3\theta } \right)$ is the solutions of (20). From fig1 and (21) we can see that three-$\pi$ is a periodical function of $\theta$ with period $2\pi/3$ and when $\theta=2\pi n/3$, $n\in\mathbb{Z}$, the three-$\pi$ get minimal, that is $\theta_{*}=0$, so we can use our formal formulations. Using our formulation (11) and the method just used for three-tangle we finial get $$\pi \left( \rho \right) = \left\{ \begin{array}{l}
\frac{{q_{*0} - p}}{{q_{*0} }}\frac{4}{9}(\sqrt 5- 1) +\frac{p}{{q_{*0} }}[\frac{5}{9}q_{*0} ^2 - \frac{4}{9}q_{*0}\\ \qquad\qquad +\frac{8}{9}- 2(\sum\limits_{i = 1}^4 {\left| {\lambda _i \left( {q_{*0} ,1} \right)} \right|} - 1)^2 ] \qquad 0 \le p \le q_{*0}\\
\frac{5}{9}p^2 - \frac{4}{9}p + \frac{8}{9} - 2(\sum\limits_{i = 1}^4 {\left| {\lambda _i \left( {p,1} \right)} \right|} - 1)^2 \qquad q_{*0} \le q \le q_{*1}\\
\frac{{p - q_{*1} }}{{1 - q_{*1} }} +\frac{{1 - p}}{{1 - q_{*1}
}}[\frac{5}{9}q_{*1} ^2 - \frac{4}{9}q_{*1} + \frac{8}{9}\\
\qquad\qquad\qquad - 2(\sum\limits_{i = 1}^4 {\left| {\lambda _i
\left( {q_{*1} ,1} \right)} \right|} - 1)^2 ] \qquad q_{*1} \le p
\le 1
\end{array} \right.$$ where $q_{*0} = {\rm{0}}{\rm{.564}}...$, $ q_{*1} =
{\rm{0}}{\rm{.963}}...$, $ i = 1,2,3,4$, $\lambda _i \left({q,1}
\right)$ is the solutions of (20), and (6), (9), (10) is the corresponding optimal decomposition.
Let us make a comparison between three-tangle and three-$\pi$. We first consider pure state. For $m\otimes n$, $m\leq n$ bipartite mixed states, we have $\sqrt {\frac{2}{{m(m - 1)}}} (\left\| {\rho
_{ab} ^{Ta} } \right\| - 1) \le C(\rho _{ab} )$[@chenkai], for $2\otimes2$ system $N(\rho _{ab} ) \le C(\rho _{ab} )$. For three qubit pure states we have $N_{a(bc)} = C_{a(bc)}$, then for three qubit pure states we have $$\begin{array}{l}
\pi = \frac{1}{3}(\pi _a + \pi _b + \pi _c ) = \frac{1}{3}(N_{a(bc)} ^2 - N_{ab} ^2 - N_{ac} ^2 + N_{b(ac)} ^2 - N_{ba} ^2 - N_{bc} ^2 \\ + N_{c(ab)} ^2 - N_{ca} ^2 - N_{cb} ^2 )
\ge \frac{1}{3}(C_{a(bc)} ^2 - C_{ab} ^2 - C_{ac} ^2 + C_{b(ac)} ^2 - C_{ba} ^2 - C_{bc} ^2 \\ + C_{c(ab)} ^2 - C_{ca} ^2 - C_{cb} ^2 ) = \frac{1}{3}(\tau _a + \tau _b + \tau _c ) = \tau
\end{array}$$ Therefore for three qubit pure states, three-$\pi$ is great or equal than three-tangle. Fig1 shows the entanglement of pure stats $|q,\theta\rangle$ under these two entanglement measure.
For rank-2 mixed states $\rho = p\left| {GHZ} \right\rangle
\left\langle {GHZ} \right| + (1 - p)\left| W \right\rangle
\left\langle W \right|$, by (15) and (22) we have fig2. From fig2 we can see that for this rank-2 class states three-tangle is smaller or equal then three-$\pi$, and the two measurements show different trend when $p$ increases. An interest thing is that on $p=q_{*0}=0.564...$ three-$\pi$ get minimal $0.50103...$ which contrary to our intuition that the increase of weight of maximal entangled state[@Gisin] $|GHZ\rangle$ means a lager entanglement. While, three-tangle is a nondecreasing function of $p$.
Conclusions
===========
We study the the entanglement for a class of rank-2 mixed states $\rho = p\left| {GHZ} \right\rangle \left\langle {GHZ} \right| + (1
- p)\left| W \right\rangle \left\langle W \right|$. Base on $Caratheodory$ theorem and the periodicity assumption, the possible optimal decomposition has been derived. Our optimal decomposition does not depend on the kinds of entanglement measure if the entanglement measure satisfy our assumptions. We also apply our formulation to study two important tripartite entanglement measure three-tangle and three-$\pi$. We find three-tangle is always smaller or equal than three-$\pi$, and three-tangle is a nondecreasing function of $p$ while three-$\pi$ has a minimal point on $p\in(0,1)$, and this show that different entanglement measure can have different trend for the same state. Our study of this class of tripartite mixed rank-2 states may be useful for studying other tripartite quantum states and even some multipartite higher dimensional states and explore the essence the quantum entanglement and quantum mechanic.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author would like to thank Shao-Ming Fei for the discussions.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We have analysed the angular clustering of X-ray selected active galactic nuclei (AGN) in different flux-limited sub-samples of the [*Chandra*]{} Deep Field North (CDF-N) and South (CDF-S) surveys. We find a strong dependence of the clustering strength on the sub-sample flux-limit, a fact which explains most of the disparate clustering results of different XMM and [*Chandra*]{} surveys. Using Limber’s equation, we find that the inverted CDF-N and CDF-S spatial clustering lengths are consistent with direct spatial clustering measures found in the literature, while at higher flux-limits the clustering length increases considerably; for example, at $f_{x,{\rm limit}}\sim 10^{-15}$ erg s$^{-1}$ cm$^{-2}$ we obtain $r_0\simeq 17 \pm 5$ and $18\pm 3 \; h^{-1}$ Mpc, for the CDF-N and CDF-S, respectively. We show that the observed flux-limit clustering trend hints towards an X-ray luminosity dependent clustering of X-ray selected, $z\sim 1$, AGNs.'
author:
- 'M. Plionis,, M. Rovilos, S. Basilakos, I. Georgantopoulos, F. Bauer'
title: 'Luminosity Dependent X-ray AGN Clustering ?'
---
Introduction
============
X-ray selected AGNs provide a relatively unbiased census of the AGN phenomenon, since obscured AGNs, largely missed in optical surveys, are included in such surveys. Furthermore, they can be detected out to high redshifts and thus trace the distant density fluctuations providing important constraints on supermassive black hole formation, the relation between AGN activity and Dark Matter (DM) halo hosts, the cosmic evolution of the AGN phenomenon (eg. Mo & White 1996, Sheth et al. 2001), and on cosmological parameters and the dark-energy equation of state (eg. Basilakos & Plionis 2005; 2006, Plionis & Basilakos 2007).
Until quite recently our knowledge of X-ray AGN clustering came exclusively from analyses of ROSAT data ($\le 3$keV) (eg. Boyle & Mo 1993; Vikhlinin & Forman 1995; Carrera et al. 1998; Akylas, Georgantopoulos, Plionis, 2000; Mullis et al. 2004). These analyses provided conflicting results on the nature of high-$z$ AGN clustering. Vikhlinin & Forman (1995), using the angular correlation approach and inverting to infer the spatial correlation length, found a strong amplitude of $\bar{z}\sim 1$ sources ($r_{0}\simeq 9 \; h^{-1}$ Mpc), which translates into $r_{0}\simeq 12 \; h^{-1}$ Mpc for a $\Lambda$CDM cosmology and a luminosity driven density evolution (LDDE) luminosity function (eg. Hasinger et al. 2005). Carrera et al. (1998), however, using spectroscopic data, could not confirm such a large correlation amplitude. With the advent of the XMM and [*Chandra*]{} X-ray observatories, many groups have attempted to settle this issue. Recent determinations of the high-$z$ X-ray selected AGN clustering, in the soft and hard bands, have provided again a multitude of conflicting results, intensifying the debate (eg. Yang et al. 2003; Manners et al. 2003; Basilakos et al. 2004; Gilli et al. 2005; Basilakos et al 2005; Yang et al. 2006; Puccetti et al. 2006; Miyaji et al. 2007; Gandhi et al. 2006; Carrera et al. 2007).
In this letter we investigate these clustering differences by re-analysing the CDF-N and CDF-S surveys, using the Bauer et al. (2004) classification to select only AGNs in the 0.5-2 and 2-8 keV bands. To use all the available sources, and not only those having spectroscopic redshifts, we work in angular space and then invert the angular correlation function using Limber’s equation. Hereafter, we will be using $h\equiv H_{\circ}/100$ km $s^{-1}$ Mpc$^{-1}$.
The X-ray source catalogues
============================
The 2Ms CDF-N and 1Ms CDF-S [*Chandra*]{} data represent the deepest observations currently available at X-ray wavelengths (Alexander et al. 2003, Giaconni et al. 2001). The CDF-N and CDF-S cover an area of 448 and 391 arcmin$^{2}$, respectively. We use the source catalogues of Alexander et al. (2003) for both CDF-N and CDF-S. The flux limits that we use for the CDF-N are $3\times 10^{-17}$ and $2\times 10^{-16}$ $\rm erg~cm^{-2}~s^{-1}$ in the soft and hard band, while for the CDF-S the respective values are $6\times 10^{-17}$ and $5\times 10^{-16}$ erg cm$^{-2}$ s$^{-1}$. Note that sensitivity maps were produced following the prescription of Lehmer et al. (2005) and in order to produce random catalogues in a consistent manner to the source selection, we discard sources which lie below our newly determined sensitivity map threshold, at their given position. Our final CDF-N catalogues contain 383 and 263 sources in the soft (0.5-2 keV) and hard band (2-8 keV), respectively, out of which 304 and 255 are AGNs, according to the “pessimistic” Bauer et al. (2004) classification. The corresponding CDF-S catalogues contain 257 and 168 sources in the same bands, out of which 227 and 165 are AGNs. A number of sources (roughly half) have spectroscopic redshift determinations (mostly taken from Barger et al. 2003; Szokoly et al. 2004; Vanzella et al. 2005; Vanzella et al. 2006; Le Févre et al. 2004; Mignoli et al. 2005).
Correlation function analysis
=============================
The angular correlation
-----------------------
The clustering properties of the X-ray AGNs are estimated using the two-point angular correlation function, $w(\theta)$, estimated using $w(\theta)=f(N_{DD}/N_{DR})-1$, where $N_{DD}$ and $N_{DR}$ is the number of data-data and data-random pairs, respectively, within separations $\theta$ and $\theta+d\theta$. The normalization factor is given by $f = 2 N_R /(N_D-1)$, where $N_D$ and $N_R$ are the total number of data and random points respectively. The Poisson uncertainty in $w(\theta)$ is estimated as $\sigma_{w}=\sqrt{(1+w(\theta))/N_{DR}}$ (Peebles 1973).
The random catalogues are produced to account for the different positional sensitivity and edge effects of the surveys. To this end we generated 1000 Monte Carlo random realizations of the source distribution, within the CDF-N and CDF-S survey areas, by taking into account the local variations in sensitivity. We also reproduce the desired $\log N - \log S$ distribution, either the Kim et al. (2007) or the one recovered directly from the CDF data (however our results remain mostly unchanged using either of the two). Random positioned sources with fluxes lower than that corresponding to the particular position of the sensitivity map are removed from our final random catalogue.
We apply the correlation analysis evaluating $w(\theta)$ in the range $[5^{''}, 900^{''}]$ in 10 logarithmic intervals with $\delta \log
\theta= 0.226$. We find statistically significant signals for all bands and for both CDF-N and CDF-S. As an example we provide in Table 1 the integrated signal to noise ratios, given for two different flux-limits and two different angular ranges. The significance appears to be low only for the CDF-S hard-band, but for the lowest flux-limit.
The angular correlation function for two different flux-limits are shown in Figure 1, with the lines corresponding to the best-fit power law model: $w(\theta)=(\theta_{0}/\theta)^{\gamma-1}$, using $\gamma=1.8$ and the standard $\chi^{2}$ minimization procedure. Note that for the CDF-N, we get at some $\theta$’s very low or negative $w(\theta)$ values. These, however, are taken into account in deriving the integrated signal, presented in Table 1.
Applying our analysis for different flux-limited sub-samples, we find that the clustering strength increases with increasing flux-limit, in agreement with the CDF-S results of Giacconi et al (2001). In Figure 2 we plot the angular clustering scale, $\theta_0$, derived from the power-law fit of $w(\theta)$, as a function of different sample flux-limits. The trend is true for both energy bands and for both CDF-N and CDF-S, although for the latter is apparently stronger.
We also find that at their lowest respective flux-limits the clustering of CDF-S sources is stronger than that of CDF-N (more so for the soft-band), in agreement with the spatial clustering analysis of Gilli et al. (2005). This difference has been attributed to cosmic variance, in the sense that there are a few large superclusters present in the CDF-S (Gilli et al. 2003). However, selecting CDF-N and CDF-S sources at the same flux-limit reduces this difference, which remains strong only for the highest flux-limited sub-samples (see Fig.2).
It is worth mentioning that our results could in principle suffer from the so-called [*amplification bias*]{}, which can enhance artificially the clustering signal due to the detector’s PSF smoothing of source pairs with intrinsically small angular separations (see Vikhlinin & Forman 1995; Basilakos et al. 2005). However, we doubt whether this bias can significantly affect our results because at the median redshift of the sources ($z\sim 1$) the [*Chandra*]{} PSF angular size of $\sim 1^{''}$ corresponds to a rest-frame spatial scale of only $\sim 5$ $h^{-1}$ kpc (even at large off-axis angles, where the PSF size increases to $\sim 4^{''}$, the corresponding spatial scale is only $\sim 20$ $h^{-1}$ kpc). In any case, and ignoring for the moment the additional effect of the variable PSF size, the above imply that only the $w(\theta)$ of the lowest flux-limited samples could in principle be affected, but in the direction of reducing (and not inducing) the observed $\theta_0-f_{x, {\rm limit}}$ trend (since the uncorrected $\theta_0$ values are, if anything, artificially larger than the true underlying one).
However, the variability of the PSF size through-out the [*Chandra*]{} field can have an additional effect, and possibly enhance or even produce the observed $\theta_0-f_{x, {\rm limit}}$ trend. To test for this we have repeated our analysis, restricting the data to a circular area of radius 6$^{'}$ around the center of the Chandra fields, where we expect to have a relatively small variation of the PSF size. This choice of radius was dictated as a compromise between excluding as much external area as possible but keeping enough sources (${\raise
-3.truept\hbox{\rlap{\hbox{$\sim$}}\raise4.truept\hbox{$<$}\ }}$50% of original) to perform the clustering analysis. The results show that indeed the $\theta_0-f_{x, {\rm limit}}$ trend is present and qualitatively the same as when using all the sources, implying that the previously mentioned biases do not create the observed trend.
Comparison with other $w(\theta)$ results
-----------------------------------------
We investigate here whether the large span of published X-ray AGN clustering results can be explained by the derived $\theta_0-f_{x,{\rm
limit}}$ trend. To this end we attempt to take into account the different survey area-curves, by estimating a characteristic flux for each survey, $f_{x}(\frac{1}{2} AC)$, corresponding to half its area-curve (easy to estimate from the different survey published area-curves).
In Figure 3 we plot the corresponding values of $\theta_0$ (for fixed $\gamma=1.8$) as a function of $f_{x}(\frac{1}{2} AC)$ (for both hard and soft bands) for the [*Chandra*]{} Large AREA Synoptic X-ray survey (CLASXS) (Yang et al. 2003), the XMM/2dF (Basilakos et al. 2004; 2005), XMM-COSMOS (Miyaji et al. 2007), XMM-ELAIS-S1 (Puccetti et al. 2006), XMM-LSS (Gandhi et al. 2006) and AXIS (Carrera et al. 2007) surveys. With the exception of the Yang et al. (2003) and the Carrera et al. (2007) hard-band results, the rest are consistent with the general flux-dependent trend. Note also that Gandhi et al. (2006) do not find any significant clustering of their hard-band sources. Of course, cosmic variance is also at work (as evidenced also by the clustering differences between the CDF-N and CDF-S; see Fig. 2 and Gilli et al. 2005) which should be responsible for the observed scatter around the main trend (see also Stewart et al. 2007).
We would like to stress that the CDF surveys have a large flux dynamical range which is necessary in order to investigate the $f_{x,{\rm limit}}-\theta_o$ correlation. This is probably why this effect has not been clearly detected in other surveys, although recently, a weak such effect was found also in the CLASXS survey (Yang et al. 2006).
The spatial correlation length using $w(\theta)$
------------------------------------------------
We can use Limber’s equation to invert the angular clustering and derive the corresponding spatial clustering length, $r_0$ (eg. Peebles 1993). To do so it is necessary to model the spatial correlation function as a power law and to assume a clustering evolution model, which we take to be that of constant clustering in comoving coordinates (eg. de Zotti et al. 1990; Kundić 1997). For the inversion to be possible it is necessary to know the X-ray source redshift distribution, which can be determined by integrating the corresponding X-ray source luminosity function above the minimum luminosity that corresponds to the particular flux-limit used. To this end we use the Hasinger et al. (2005) and La Franca et al. (2005) LDDE luminosity functions for the soft and hard bands, respectively.
We perform the above inversion in the framework of the [ *concordance*]{} $\Lambda$CDM cosmological model ($\Omega_{\rm m}=1-\Omega_{\Lambda}=0.3$) and the comoving clustering paradigm. The resulting values of the spatial clustering lengths, $r_0$, show the same dependence on flux-limits, as in Fig.2.
We can compare our results with direct determinations of the spatial-correlation function from Gilli et al. (2005), who used a smaller ($\sim 50\%$) spectroscopic sample from the CDF-N and CDF-S. They found a significant difference between the CDF-S and CDF-N clustering, with $r_0=10.3 \pm 1.7 \; h^{-1}$ Mpc and $r_0=5.5 \pm 0.6 \; h^{-1}$ Mpc, respectively (note also that the corresponding slopes were quite shallow, roughly $\gamma\simeq 1.4-1.5$). Since, Gilli et al. used sources from the full (0.5-8 keV) band, we compare their results with our soft-band results which, dominate the total-band sources. This comparison is possible, because as we have verified using a Kolmogorov-Smirnov test, the flux distributions of the sub-samples that have spectroscopic data are statistically equivalent with those of the whole samples. Our inverted clustering lengths, for the lowest flux-limit used, are: $r_0=10.3 \pm 2 \; h^{-1}$ Mpc and $r_0=6.4 \pm 2.5 \; h^{-1}$ Mpc (fixing $\gamma=1.8$) for the CDF-S and CDF-N respectively, in good agreement with the Gilli et al. (2005) direct 3D determination[^1].
Our values can be also compared with the re-calculation of the CDF-N spatial clustering by Yang et al. (2006), who find $r_0\simeq 4.1 \pm 1.1 \; h^{-1}$ Mpc.
We return now to the strong trend between $\theta_0$ (or the corresponding $r_0$) and the sample flux-limit (see Fig.2 and 3), which could be due to two possible effects (or the combination of both). Either the different flux-limits correspond to different intrinsic luminosities, ie., a luminosity-clustering dependence (see also hints in the CLASXS and CDF-N based Yang et al. 2006 results; while for optical data see Porciani & Norberg 2006) or a redshift-dependent effect (ie., different flux-limits correspond to different redshifts traced). Using the sources which have spectroscopic redshift determinations we have derived their intrinsic luminosities, in each respective band, from their count rates using a spectral index $\Gamma=1.9$ and the [*concordance*]{} $\Lambda$CDM cosmology. We have also applied an absorption correction by assuming a power-law X-ray spectrum with an intrinsic $\Gamma=1.9$, obscured by an optimum column density to reproduce the observed hardness ratio. We then derive, for each flux-limit used, the median redshift and median luminosity of the corresponding sub-sample. We find relatively small variations and no monotonic change of the median redshift with subsample flux-limit. For example, the median spectroscopic redshift for the soft and hard bands, at the lowest flux-limit used, is ${\bar z}\sim 0.8$ and $\sim 0.95$, respectively, while its mean variation between the different flux-limits used is $\langle \delta z/z \rangle\simeq -0.11$ and -0.03 for the CDF-N and $\langle \delta z/z \rangle \simeq -0.27$ and 0.05 for the CDF-S soft and hard-bands, respectively. The large redshift variation of the soft-band CDF-S data should be attributed to the presence of a few superclusters at $z\sim 0.7$; see Gilli et al. 2003).
In Figure 4 we present the correlation between the subsample median X-ray luminosity and the corresponding subsample clustering length, as provided by Limber’s inversion. Although the CDF luminosity dynamical range is limited, it is evident that the median X-ray luminosity systematically increases with increasing sample flux-limit and it is correlated to $r_0$ (as expected from Fig.2). It should be noted that the correlation length of the highest-flux limited CDF-S soft-band subsample is by far the largest ever found ($\sim 30 h^{-1}$ Mpc), but one has to keep in mind that the CDF-S appears not to be a typical field, as discussed earlier (see Gilli et al. 2003). The CDF-N high-flux results appear to converge to a value of $r_0 \sim 18h^{-1}$ Mpc, similar to that of some other surveys (eg. Basilakos et al. 2004; 2005 and Puccetti et al. 2007). We therefore conclude that not only are there indications for a luminosity dependent clustering of X-ray selected high-$z$ AGNs, but also that they are significantly more clustered than their lower-$z$ counterparts, which have $r_0 \sim 7-8 \;
h^{-1}$ Mpc (eg. Akylas et al. 2000; Mullis et al. 2004). This is a clear indication of a strong bias evolution (eg. Basilakos, Plionis & Ragone-Figueroa 2007).
Conclusions
===========
We have analysed the angular clustering of the CDF-N and CDF-S X-ray AGNs and find:
\(1) A dependence of the angular clustering strength on the sample flux-limit. Most XMM and [*Chandra*]{} clustering analyses provide results that are consistent with the observed trend; a fact which appears to lift the confusion that arose from the apparent differences in their respective clustering lengths.
\(2) Within the concordance cosmological model, the comoving clustering evolution model and the LDDE luminosity function, our angular clustering results are in good agreement with direct estimations of the CDF-N and CDF-S spatial clustering, which are based however on roughly half the total number of sources, for which spectroscopic data were available.
\(3) The apparent correlation between clustering strength and sample flux-limit transforms into a correlation between clustering strength and intrinsic X-ray luminosity, since no significant redshift-dependent trend was found.
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[lcccc]{} Sample & &\
& $<400^{''}$ & $<900^{''}$ & $<400^{''}$ & $<900^{''}$\
CDF-N soft & $2.1\sigma$ & $0.0\sigma$ & $2.1\sigma$ & $1.5\sigma$\
CDF-N hard & $3.3\sigma$ & $2.6\sigma$ & $6.7\sigma$ & $5.5\sigma$\
CDF-S soft & $4.2\sigma$ & $2.6\sigma$ & $3.4\sigma$ & $3.7\sigma$\
CDF-S hard & $0.3\sigma$ & $1.3\sigma$ & $4.1\sigma$ & $2.7\sigma$\
\
\
[^1]: Leaving both $r_{0}$ and $\gamma$ as free parameters in the fit, we obtain $\gamma$’s quite near their nominal value ($\gamma\sim 1.6 - 1.8$)
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider using a secret key and a noisy quantum channel to generate noiseless public communication and noiseless private communication. The optimal protocol for this setting is the *publicly-enhanced private father protocol*. This protocol exploits random coding techniques and piggybacking of public information along with secret-key-assisted private codes. The publicly-enhanced private father protocol is a generalization of the secret-key-assisted protocol of Hsieh, Luo, and Brun and a generelization of a protocol for simultaneous communication of public and private information suggested by Devetak and Shor.'
author:
- 'Min-Hsiu Hsieh'
- 'Mark M. Wilde'
bibliography:
- 'Ref.bib'
title: Public and private communication with a quantum channel and a secret key
---
Introduction
============
The qualitative connection between secrecy of information and the ability to maintain quantum correlations has long been a part of quantum information theory. The connection comes about from the observation that a maximally entangled ebit state, shared between two parties named Alice and Bob, has no correlations with the rest of the universe—in this sense, the ebit is *monogamous* [@T04]. We can represent the global state of the ebit and the rest of the universe as$$\Phi^{AB}\otimes\sigma^{E},$$ where Alice and Bob share the ebit $\Phi^{AB}$, and$$\begin{aligned}
\Phi^{AB} & \equiv\left\vert \Phi\right\rangle \left\langle \Phi\right\vert
^{AB},\\
\left\vert \Phi\right\rangle ^{AB} & \equiv\frac{1}{\sqrt{2}}(\left\vert
0\right\rangle ^{A}\left\vert 0\right\rangle ^{B}+\left\vert 1\right\rangle
^{A}\left\vert 1\right\rangle ^{B}),\end{aligned}$$ and $\sigma^{E}$ is some state of Eve, a third party representing the rest of the universe. Eve’s state $\sigma^{E}$ is independent of Alice and Bob’s ebit. The relation to a secret key comes about when Alice and Bob perform local measurements of the ebit in the computational basis. The resulting state is$$\overline{\Phi}^{AB}\otimes\sigma^{E},$$ where $\overline{\Phi}^{AB}$ is the maximally correlated state:$$\overline{\Phi}^{AB}\equiv\frac{1}{2}\left( \left\vert 0\right\rangle
\left\langle 0\right\vert ^{A}\otimes\left\vert 0\right\rangle \left\langle
0\right\vert ^{B}+\left\vert 1\right\rangle \left\langle 1\right\vert
^{A}\otimes\left\vert 1\right\rangle \left\langle 1\right\vert ^{B}\right) .$$ In this setting, the cryptographic setting, we consider Eve as a potential eavesdropper. She is no longer the rest of the universe, because some party now holds the purification of the dephased state $\overline{\Phi}^{AB}$.
The body of literature on the privacy/quantum-coherence connection has now grown substantially. Some of the original exploitations of this connection were the various quantum key distribution protocols [@BB84; @E91; @BBM92]. These protocols establish a shared secret key with the help of a noisy quantum channel or noisy entanglement. The subsequent proofs [@SP00; @LD07QKD] for the security of these protocols rely on the formal mathematical equivalence between entanglement distillation [@BDSW96] and key distillation. Schumacher and Westmoreland explored the connection with an information-theoretical study [@SW98]—they established a simple relation between the capacity of a quantum channel for transmitting quantum information and its utility for quantum key distribution. Collins and Popescu [@CP02] and Gisin *et al*. [@GRW02] initiated the formal study of the connections between entanglement and secret key. Since then, researchers have determined a method for mapping an entangled state to a probability distribution with secret correlations [@AG05] and have continued to extend existing quantum results [@CVDC03] to analogous results for privacy [@bae:032304].
The connection has also proven fruitful for quantum Shannon theory, where we study the capabilities of a large number of independent uses of a noisy quantum channel or a large number of copies of a noisy bipartite state. The first step in this direction was determining the capacity of a quantum channel for transmitting a private message or establishing a shared secret key [@Devetak03; @CWY04]. Devetak further showed how *coherently* performing each step of a private protocol leads to a code that achieves the capacity of a quantum channel for transmitting quantum information [@Devetak03]. Since these initial insights, we have seen how the seemingly different tasks of distilling secret key, distilling entanglement, transmitting private information, and transmitting quantum information all have connections [@DW03b]. Oppenheim *et al*. have determined a merging protocol for private correlations [@OSW05], based on the quantum state merging protocol [@nature2005horodecki; @cmp2007HOW]. Additionally, the secret-key-assisted private capacity of a quantum channel [@HLB08SKP] is analogous to its entanglement-assisted quantum capacity [@DHW03; @DHW05RI].
The connection is only qualitative because the Horodeckis and Oppenheim have observed that there exist *bound entangled* states [@H3O05]. These bound entangled states are entangled, yet have no distillable entanglement (one cannot extract ebits from them), but they indeed have distillable secret key. The dynamic equivalent of this state is an entanglement binding channel [@H300; @horodecki:110502; @H3LO08]. This channel has no ability to transmit quantum information. The loss of the privacy-coherence connection here is not necessarily discomforting. In fact, it is more interesting because it leads to the superactivation effect [@science2008smith]—the possibility of combining two zero-capacity channels to form a quantum channel with non-zero quantum capacity. Additionally, the private analog of this scenario exhibits some unexpected behavior [@smith:010501].
In this paper, we continue along the privacy-coherence connection and detail the publicly-enhanced private father protocol. This protocol exploits a secret key and a large number of independent uses of a noisy quantum channel to generate noiseless public communication and noiseless private communication. This protocol is the public-private analog of the classically-enhanced father protocol [@HW08GFP], and might lead to further insights into the privacy-coherence connection. The publicly-enhanced private father protocol combines the coding techniques of the suggested protocol in Section 4 of Ref. [@DS03] (originally proven for the classical wiretap channel [@CK67]) with the recent secret-key-assisted private communication protocol [@HLB08SKP].
We structure this work as follows. The next section establishes the definition of a noiseless public channel, a noiseless private channel, noiseless common randomness, and a perfect secret key. We then clarify a small point with the protocol for private communication [@Devetak03; @CWY04]—specifically, we address the apparent ability of that protocol to transmit public information in addition to private information. Section \[sec:main-theorem\] describes the publicly-enhanced private father protocol and states our main theorem (Theorem \[thm:PEPFP\]). This theorem gives the capacity region for the publicly-enhanced private father protocol. We proceed with the proof of the corresponding converse theorem in Section \[sec:converse-theorem\] and the proof of the corresponding direct coding theorem in Section \[sec:direct-coding-theorem\]. Section \[sec:children\] shows that the suggested protocol from Ref. [@DS03] is a child of the publicly-enhanced private father protocol. We then conclude with some remaining open questions.
Definitions and Notation
========================
We first introduce the notion of a noiseless public channel, a noiseless private channel, and a noiseless secret key as resources. Our communication model includes one sender Alice, a receiver Bob, and an eavesdropper Eve. Alice chooses classical messages $k$ from a set $\left[ K\right]
\equiv\left\{ 1,\ldots,K\right\} $. She encodes these messages as quantum states $\{\left\vert k\right\rangle \left\langle k\right\vert ^{A}\}_{k\in\left[ K\right] }$. We assume that each party is in a local, secret facility that does not leak information to the outside world. For example, Eve cannot gain any information about a state that Alice or Bob prepares locally. We consider two dynamic resources, public classical communicaton and private classical communication, and two static resources, common randomness and secret key.
A noiseless public channel id$_{\text{pub}}^{A\rightarrow B}$ from Alice to Bob implements the following map for $k\in\left[ K\right] $:$$\text{id}_{\text{pub}}^{A\rightarrow B}:\left\vert k\right\rangle \left\langle
k\right\vert ^{A}\rightarrow\left\vert k\right\rangle \left\langle
k\right\vert ^{B}\otimes\sum_{k^{\prime}\in\left[ K\right] }p_{K^{\prime}|K}\left( k^{\prime}|k\right) \rho_{k^{\prime}}^{E},
\label{eq:noiseless-public}$$ where $p_{K^{\prime}|K}\left( k^{\prime}|k\right) $ is some conditional probability distribution and $\rho_{k^{\prime}}^{E}$ is a state on Eve’s system. The above definition of a noiseless public channel captures the idea that Bob receives the classical information perfectly, but Eve receives only partial information about Alice’s message. Eve has perfect correlation with Alice’s message if and only if her conditional distribution $p_{K^{\prime}|K}\left( k^{\prime}|k\right) $ is $\delta_{k^{\prime},k}$ and her states $\rho_{k^{\prime}}^{E}=\left\vert k^{\prime}\right\rangle \left\langle
k^{\prime}\right\vert ^{E}$ for all $k^{\prime}$. We make no distinction between a noiseless public channel where Eve receives partial information and one where Eve receives perfect information because we are only concerned with the rate at which Alice can communicate to Bob—we are not concerned with the more general scenario of broadcast communication where Eve is an active party in the communication protocol [@YHD2006]. We represent the noiseless public channel symbolically as the following resource:$$\left[ c\rightarrow c\right] _{\text{pub}}.$$
The resource inequality framework [@DHW05RI] uses the notation $\left[
c\rightarrow c\right] $ to represent one noiseless bit of classical communication. We require a symbol different from $\left[ c\rightarrow
c\right] $ because that symbol does not distinguish between public and private communication. For example, the superdense coding protocol [@BW92] actually produces two private classical bits, but the notation $\left[ c\rightarrow c\right] $ does not indicate this fact.
A noiseless private channel is the following map:$$\text{id}_{\text{priv}}^{A\rightarrow B}:\left\vert k\right\rangle
\left\langle k\right\vert ^{A}\rightarrow\left\vert k\right\rangle
\left\langle k\right\vert ^{B}\otimes\sigma^{E},$$ where $\sigma^{E}$ is a constant state on Eve’s system, independent of what Bob receives. A private channel appears as a special case of a public channel where random variable $K^{\prime}$ that represents Eve’s knowledge is independent of random variable $K$. The definition in (\[eq:noiseless-public\]) reduces to that of a private channel if we set the probability distribution in (\[eq:noiseless-public\]) to $p_{K^{\prime}|K}\left( k^{\prime}\right) $. But we define a private channel as the case when $K^{\prime}$ and $K$ are independent. Otherwise, the channel is public. This difference is the distinguishing feature of a noiseless private channel. We represent the noiseless private channel symbolically as the following resource:$$\left[ c\rightarrow c\right] _{\text{priv}}.$$ The above definitions of a public classical channel and private classical channel are inspired by definitions in Refs. [@HLB08SKP; @LS08].
*Common randomness* is the static analog of a noiseless public channel [@AC93CR; @AC93II; @DW03a]. In fact, Alice can actually use a public channel to implement common randomness. Alice first prepares a local maximally mixed state $\pi^{A}$ where$$\pi^{A}\equiv\frac{1}{\left\vert K\right\vert }\sum_{k\in\left[ K\right]
}\left\vert k\right\rangle \left\langle k\right\vert ^{A}.$$ She makes an exact copy of the random state locally to produce the following state:$$\overline{\Phi}^{AA^{\prime}}\equiv\frac{1}{\left\vert K\right\vert }\sum_{k\in\left[ K\right] }\left\vert k\right\rangle \left\langle
k\right\vert ^{A}\otimes\left\vert k\right\rangle \left\langle k\right\vert
^{A^{\prime}}. \label{eq:copied-state}$$ She sends the $A^{\prime}$ system through the noiseless public channel. The resulting state represents common randomness shared between Alice and Bob, about which Eve may have partial information:$$\frac{1}{\left\vert K\right\vert }\sum_{k\in\left[ K\right] }\left\vert
k\right\rangle \left\langle k\right\vert ^{A}\otimes\left\vert k\right\rangle
\left\langle k\right\vert ^{B}\otimes\sum_{k^{\prime}\in\left[ K\right]
}p_{K^{\prime}|K}\left( k^{\prime}|k\right) \rho_{k^{\prime}}^{E}$$
A noiseless secret key is the static analog of a noiseless private channel. Alice again prepares the state $\pi^{A}$ and makes a copy of it to an $A^{\prime}$ system. She sends the $A^{\prime}$ system through a noiseless private channel, generating the following resource:$$\frac{1}{K}\sum_{k\in\left[ K\right] }\left\vert k\right\rangle \left\langle
k\right\vert ^{A}\otimes\left\vert k\right\rangle \left\langle k\right\vert
^{B}\otimes\sigma^{E}=\overline{\Phi}^{AB}\otimes\sigma^{E}.$$ Alice and Bob share perfect common randomness, but this time, Eve has no knowledge of this common randomness. This resource is a secret key. A perfect secret key resource has two requirements [@Renner:2005:thesis]:
1. The key should have a uniform distribution.
2. Eve possesses no correlations with the secret key.
We denote the resource of a shared secret key as follows:$$\left[ cc\right] _{\text{priv}}.$$
Note that a noiseless public channel alone cannot implement a noiseless private channel, and a noiseless private channel alone cannot implement a noiseless public channel. This relation is different from the corresponding relation between a noiseless quantum channel and a noiseless classical channel [@HW09T3] because a noiseless quantum channel alone can implement a noiseless classical channel, but a noiseless classical channel alone cannot implement a noiseless quantum channel.
Relative Resource in Private Communication {#sec:relative-resource-priv}
==========================================
We would like to clarify one point with the protocol for private communication [@Devetak03; @CWY04] before proceeding to our main theorem. By inspecting the proof of the direct coding theorem in Ref. [@Devetak03], one might think that Alice could actually transmit public information at an additional rate of $I\left( X;E\right) $. The following sentence from Ref. [@Devetak03] may lead one to arrive at such a conclusion:
> By construction, Bob can perform a measurement that correctly identifies the pair $\left( k,m\right) $, and hence $k$, with probability $\geq1-\sqrt[4]{\epsilon}$.
But this conclusion is incorrect because the random variable $M$ representing the public message $m$ must have a uniform distribution. This random variable $M$ serves the purpose of randomizing Eve’s knowledge of the private message $k$ [@igor09]. The protocol would not operate as intended if random variable $M$ had a distribution other than the uniform distribution. The size of the message set for the random variable $M$ must be at least $2^{nI\left( X;E\right) }$. The rate $I\left( X;E\right) $ of randomization further confirms the role of the mutual information as the minimum amount of noise needed to destroy one’s correlations with a random variable [@GPW05] (see Refs. [@B08; @B09] for further explorations of this idea). It is thus not surprising that the mutual information $I\left( X;E\right) $ arises in the protocol for private communication because Alice would like to destroy Eve’s correlations with her private message $k$.
The resource inequality [@DHW05RI] for the protocol for private communication is as follows:$$\begin{gathered}
\left\langle \mathcal{N}\right\rangle \geq I\left( X;E\right) \left[
c\rightarrow c:\pi\right] _{\text{pub}}+\\
\left( I\left( X;B\right) -I\left( X;E\right) \right) \left[
c\rightarrow c\right] _{\text{priv}},\label{eq:devetak-priv}$$ where the mutual information quantities are with respect to the following classical-quantum state:$$\sum_{x\in\mathcal{X}}p_{X}\left( x\right) \left\vert x\right\rangle
\left\langle x\right\vert ^{X}\otimes U_{\mathcal{N}}^{A^{\prime}\rightarrow
BE}(\sigma_{x}^{A^{\prime}}),$$ corresponding to the channel input ensemble $\{p_{X}\left( x\right)
,\sigma_{x}^{A^{\prime}}\}_{x\in\mathcal{X}}$. The meaning of the resource inequality is that Alice can transmit $nI\left( X;E\right) $ bits of public information (with the requirement that Alice’s random variable has a uniform distribution) and $n\left( I\left( X;B\right) -I\left( X;E\right)
\right) $ bits of private information by using a large number $n$ of independent uses of the noisy quantum channel $\mathcal{N}$. The resource $\left[ c\rightarrow c:\pi\right] _{\text{pub}}$ is not an absolute resource, but is rather a *relative resource* [@DHW05RI; @D06; @A06], meaning that the protocol only works properly if Alice’s public variable has a uniform distribution, or equivalently, is equal to the maximally mixed state $\pi$. This public information must be completely random because Alice uses it to randomize Eve’s knowledge of the private message.
The resource inequality in (\[eq:devetak-priv\]) leads to a simpler way of implementing the direct coding theorem of the secret-key-assisted private communication protocol [@HLB08SKP]. Suppose that Alice has public information in a random variable $M$. If she combines this random variable with a secret key, the resulting random variable has a uniform distribution because the secret key randomizes the public variable. This variable can then serve as the input needed to implement the relative resource of public communication. Alice can transmit an extra $nI\left( X;E\right) $ of private information by combining this public communication with the secret key resource, essentially implementing a one-time pad protocol [@V26; @S49]. We phrase the above argument with the theory of resource inequalities:$$\begin{aligned}
& \left\langle \mathcal{N}\right\rangle +I\left( X;E\right) \left[
cc\right] _{\text{priv}}\\
& \geq I\left( X;E\right) \left[ c\rightarrow c:\pi\right] _{\text{pub}}+I\left( X;E\right) \left[ cc\right] _{\text{priv}}+\\
& \left( I\left( X;B\right) -I\left( X;E\right) \right) \left[
c\rightarrow c\right] _{\text{priv}}\\
& \geq I\left( X;E\right) \left[ c\rightarrow c\right] _{\text{priv}}+\left( I\left( X;B\right) -I\left( X;E\right) \right) \left[
c\rightarrow c\right] _{\text{priv}}\\
& =I\left( X;B\right) \left[ c\rightarrow c\right] _{\text{priv}}.\end{aligned}$$ This resource inequality is equivalent to that obtained in Ref. [@HLB08SKP].
Public and Private Transmission with a Secret Key {#sec:main-theorem}
=================================================
We begin by defining our publicly-enhanced private father protocol (PEPFP) for a quantum channel $\mathcal{N}^{A^{\prime
}\rightarrow B}$ from a sender Alice to a receiver Bob. The channel has an extension to an isometry $U_{\mathcal{N}}^{A^{\prime}\rightarrow BE}$, defined on a bipartite quantum system $BE$, where Bob has access to system $B$ and Eve has access to system $E$. Alice’s task is to transmit, by some large number $n$ uses of the channel $\mathcal{N}$, one of $K$ public messages and one of $M$ private messages to Bob. The goal is for Bob to identify the messages with high probability and for Eve to receive no information about the private message. In addition, Alice and Bob have access to a private string (a secret key), picked uniformly at random from the set $\left[ S\right] $, before the protocol begins.
An $(n,R,P,R_{S},\epsilon)$ *secret-key-assisted private channel code* consists of six steps: preparation, encryption, channel coding, transmission, channel decoding, and decryption. We detail each of these steps below.
**Preparation.** Alice prepares a public message $k$ in a register $K$ and private message $m$ in a register $M$. Each of these has a uniform distribution:$$\begin{aligned}
\pi^{K} & \equiv\frac{1}{K}\sum_{k=1}^{K}\left\vert k\right\rangle
\left\langle k\right\vert ^{K},\\
\pi^{M} & \equiv\frac{1}{M}\sum_{m=1}^{M}\left\vert m\right\rangle
\left\langle m\right\vert ^{M}.\end{aligned}$$ Alice also shares the maximally correlated secret key state $\overline{\Phi
}^{S_{A}S_{B}}$ with Bob:$$\overline{\Phi}^{S_{A}S_{B}}\equiv\frac{1}{S}\sum_{s=1}^{S}\left\vert
s\right\rangle \left\langle s\right\vert ^{S_{A}}\otimes\left\vert
s\right\rangle \left\langle s\right\vert ^{S_{B}}.$$ The overall state after preparation is$$\pi^{K}\otimes\pi^{M}\otimes\overline{\Phi}^{S_{A}S_{B}}.$$
**Encryption.** Alice exploits an encryption map$$f:\left[ M\right] \times\left[ S\right] \rightarrow\left[ M\right] .$$ The encryption map $f$ computes an encrypted variable $f(m,s)$ that depends on the private message $m$ and the secret key $s$. Furthermore, the encryption map $f$ satisfies the following conditions:
1. For all $s_{1},s_{2}\in\left[ S\right] $ where $s_{1}\neq s_{2}$:$$f(m,s_{1})\neq f(m,s_{2}).$$
2. For all $m_{1},m_{2}\in\left[ M\right] $ where $m_{1}\neq m_{2}$:$$f(m_{1},s)\neq f(m_{2},s).$$
The encryption map $f$ corresponds physically to a CPTP map$\ \mathcal{F}^{MS_{A}\rightarrow P}$. The state after the encryption map is$$\begin{gathered}
\mathcal{F}^{MS_{A}\rightarrow P}(\pi^{K}\otimes\pi^{M}\otimes\overline{\Phi
}^{S_{A}S_{B}})=\\
\pi^{K}\otimes\frac{1}{MS}\sum_{m,s}\left\vert f\left( m,s\right)
\right\rangle \left\langle f\left( m,s\right) \right\vert ^{P}\otimes\left\vert s\right\rangle \left\langle s\right\vert ^{S_{B}}.\end{gathered}$$
**Channel Encoding.** Alice prepares the codeword state $\sigma
_{k,f\left( m,s\right) }^{A^{\prime n}}$ based on the public message $k$ and the encrypted message $f(m,s)$. This encoding corresponds physically to some CPTP map $\mathcal{E}^{KP\rightarrow A^{\prime n}}$. The state after the encoding map is$$\frac{1}{KMS}\sum_{k,m,s}\sigma_{k,f\left( m,s\right) }^{A^{\prime n}}\otimes\left\vert s\right\rangle \left\langle s\right\vert ^{S_{B}}.$$
**Transmission.** Alice sends the state $\sigma_{k,f\left( m,s\right)
}^{A^{\prime n}}$ through the channel $U_{\mathcal{N}}^{A^{\prime
n}\rightarrow B^{n}E^{n}}$, generating the state$$\frac{1}{KMS}\sum_{k,m,s}\sigma_{k,f\left( m,s\right) }^{B^{n}E^{n}}\otimes\left\vert s\right\rangle \left\langle s\right\vert ^{S_{B}},$$ where$$\sigma_{k,f\left( m,s\right) }^{B^{n}E^{n}}\equiv U_{\mathcal{N}}^{A^{\prime
n}\rightarrow B^{n}E^{n}}(\sigma_{k,f\left( m,s\right) }^{A^{\prime n}}).$$
**Channel Decoding.** Bob receives the above state from the channel and would like to decode the messages. He exploits a decoding positive-operator-valued measure (POVM) that acts on his system $B^{n}$. The elements of this POVM are$$\{\Lambda_{k,f\left( m,s\right) }^{B^{n}}\}_{k\in\left[ K\right] ,f\left(
m,s\right) \in\left[ M\right] }.$$ Bob places the measurement results $k$ and $f\left( m,s\right) $ in the respective registers $\hat{K}$ and $\hat{P}$. The ideal output state after Bob’s decoding operation is$$\sum_{k,m,s}\sigma_{k,f\left( m,s\right) }^{B^{n}E^{n}}\otimes\left\vert
s\right\rangle \left\langle s\right\vert ^{S_{B}}\otimes|k\rangle\langle
k|^{\hat{K}}\otimes|f\left( m,s\right) \rangle\langle f\left( m,s\right)
|^{\hat{P}},$$ where it is understood that the normalization factor is $1/\left( KMS\right)
$.
**Decryption.** The final step is for Bob to decrypt the encrypted message $f\left( m,s\right) $. He employs a decryption function $g$, where$$g:\left[ M\right] \times\left[ S\right] \rightarrow\left[ M\right] .$$ The decryption function $g$ satisfies the following property:$$\forall~s,m\ \ \ \ \ g(f(m,s),s)=m.$$ This decryption function allows Bob to recover Alice’s private message as $m=g(f(m,s),s)$ based on the encrypted message $f\left( m,s\right) $ and the secret key $s$. Physically, this operation corresponds to a CPTP map $\mathcal{G}^{S_{B}\hat{P}\rightarrow\hat{M}}$. The state after this decryption map is$$\frac{1}{KMS}\sum_{k,m,s}\sigma_{k,f\left( m,s\right) }^{B^{n}E^{n}}\otimes\left\vert s\right\rangle \left\langle s\right\vert ^{S_{B}}\otimes|k\rangle\langle k|^{\hat{K}}\otimes|m\rangle\langle m|^{\hat{M}}.$$ Figure \[fig:PEPFP\] depicts all of the above steps in a general publicly-enhanced private father code.
\[ptb\]
[public-enhanced-private-father.pdf]{}
The conditions for a good publicly-enhanced secret-key-assisted private code are that Bob be able to decode the public message $k$ and encrypted message $p=f\left( m,s\right) $ with high probability:$$\forall k,p\ \ \ \ \ \text{Tr}\{\Lambda_{k,p}^{B^{n}}\sigma_{k,p}^{B^{n}}\}\geq1-\epsilon.$$ It is sufficient to consider the above criterion because Bob can determine the private message $m$ with high probability if he can determine the encrypted message $p$ with high probability. Also, the following inequality is our security criterion:$$\forall k,m\ \ \left\Vert \sum_{s}\sigma_{k,f\left( m,s\right) }^{E^{n}}\otimes\left\vert s\right\rangle \left\langle s\right\vert ^{S_{B}}-\sigma_{k}^{E^{n}}\otimes\pi^{S_{B}}\right\Vert _{1}\leq\epsilon.
\label{eq:security-criterion}$$ This criterion ensures that Eve’s state is independent of the key and the private message $m$.
A rate triple $(R,P,R_{S})$ is *achievable* if there exists an $(n,R-\delta,P-\delta,R_{S}+\delta,\epsilon)$ publicly-enhanced private father code for any $\epsilon,\delta>0$ and sufficiently large $n$. The capacity region $C_{\text{PEPFP}}(\mathcal{N})$ is a three-dimensional region in the $(R,P,R_{S})$ space with all possible achievable rate triples $(R,P,R_{S})$.
\[thm:PEPFP\] The capacity region $C(\mathcal{\mathcal{N}})$ of a secret-key-assisted quantum channel $\mathcal{N}$ for simultaneously transmitting both public and private classical information is equal to the following expression:$$C(\mathcal{N})=\overline{\bigcup_{l=1}^{\infty}\frac{1}{l}C^{(1)}(\mathcal{N}^{\otimes l})},\label{pgf}$$ where the overbar indicates the closure of a set. The one-shot region $C^{(1)}(\mathcal{N})$ is the set of all $R,P,R_{S}\geq0$, such that$$\begin{aligned}
R & \leq I(X;B)_{\sigma},\label{pgf1}\\
P & \leq R_{S}+I\left( Y;B|X\right) _{\sigma}-I\left( Y;E|X\right)
_{\sigma},\label{pgf2}\\
P & \leq I(Y;B|X)_{\sigma}.\label{pgf3}$$ The above entropic quantities are with respect to a one-shot quantum state $\sigma^{XYBE}$, where $$\sigma^{XYBE}\equiv\sum_{x}p(x)|x\rangle\langle x|^{X}\otimes\rho_{x}^{YBE},\label{eq:maximization-state}$$ and the states $\rho_{x}^{YBE}$ are of the form$$\rho_{x}^{YBE}=\sum_{y}p(y|x)|y\rangle\langle y|^{Y}\otimes U_{\mathcal{N}}^{A^{\prime}\rightarrow BE}(\rho_{x,y}^{A^{\prime}}),\label{eq:state-private}$$ for some density operator $\rho_{x,y}^{A^{\prime}}$ and $U_{\mathcal{N}}^{A^{\prime}\rightarrow BE}$ is an isometric extension of $\mathcal{N}$. It is sufficient to consider $|\mathcal{X}|\leq\min\{|A^{\prime}|,|B|\}^{2}+1$ by the method in Ref. [@YHD05ieee].
The proof of the above capacity theorem consists of two parts. The first part that we show is the *converse theorem*. The converse theorem shows that the rates in the above theorem are optimal—any given coding scheme that has asymptotically good performance cannot perform any better than the above rates. We prove the converse theorem in the next section. The second part that we prove is the *direct coding theorem*. The proof of the direct coding theorem gives a coding scheme that achieves the limits given in the above theorem.
Proof of the Converse Theorem {#sec:converse-theorem}
=============================
We outline the proof strategy of the converse before delving into its details. Consider that a noiseless public channel can generate common randomness and a noiseless private channel can generate a secret key. Let $K(\mathcal{N})$ denote the capacity of a quantum channel $\mathcal{N}$ for generating common randomness, generating a secret key, while consuming a secret key at respective rates $\left( R,P,R_{S}\right) $. The capacity region $K(\mathcal{N})$ contains the capacity region $C(\mathcal{N})$ of Theorem \[thm:PEPFP\] ($C(\mathcal{N})\subseteq K(\mathcal{N})$) because of the aforementioned one-way relation between a noiseless public channel and common randomness and that between a noiseless private channel and a secret key. It thus suffices to prove the converse for a secret-key-assisted common randomness generation and secret key generation protocol. We consider the most general such protocol when proving the converse and show that the capacity region in (\[pgf1\]-\[pgf3\]) bounds the capacity region $K(\mathcal{N})$. The result of the converse theorem is then that $K(\mathcal{N})\subseteq C(\mathcal{N})$ and thus that $K(\mathcal{N})=C(\mathcal{N})$.
\[ptb\]
[public-enhanced-private-father-converse.pdf]{}
\[Converse\] Suppose Alice creates the maximally correlated state $\pi
^{MM_{A}^{\prime}}$ locally, where$$\overline{\Phi}^{MM_{A}^{\prime}}\equiv\frac{1}{M}\sum_{m=1}^{M}\left\vert
m\right\rangle \left\langle m\right\vert ^{M}\otimes\left\vert m\right\rangle
\left\langle m\right\vert ^{M_{A}^{\prime}}.$$ (the protocol should be able to transmit the correlations in state $\overline{\Phi}^{MM_{A}^{\prime}}$ with $\epsilon$-accuracy while keeping them secret). Alice shares the maximally correlated secret key state $\overline{\Phi}^{S_{A}S_{B}}$ with Bob:$$\overline{\Phi}^{S_{A}S_{B}}\equiv\frac{1}{S}\sum_{s=1}^{S}\left\vert
s\right\rangle \left\langle s\right\vert ^{S_{A}}\otimes\left\vert
s\right\rangle \left\langle s\right\vert ^{S_{B}}.$$ Alice prepares a state $\overline{\Phi}^{KK_{A}^{\prime}}$ for common randomness generation:$$\overline{\Phi}^{KK_{A}^{\prime}}\equiv\frac{1}{K}\sum_{k=1}^{K}\left\vert
k\right\rangle \left\langle k\right\vert ^{K}\otimes\left\vert k\right\rangle
\left\langle k\right\vert ^{K_{A}^{\prime}}.$$ Alice combines her states $\overline{\Phi}^{KK_{A}^{\prime}}$, $\overline
{\Phi}^{MM_{A}^{\prime}}$, and $\overline{\Phi}^{S_{A}S_{B}}$. The most general encoding operation that she can perform on her three registers $K_{A}^{\prime}$, $M_{A}^{\prime}$, and $S_{A}$ is a conditional quantum encoder $\mathcal{E}^{K_{A}^{\prime}M_{A}^{\prime}S_{A}\rightarrow A^{\prime
n}}$ consisting of a collection $\{\mathcal{E}_{k}^{M_{A}^{\prime}S_{A}\rightarrow A^{\prime n}}\}_{k}$ of CPTP maps [@HW08GFP]. Each element $\mathcal{E}_{k}^{M_{A}^{\prime}S_{A}\rightarrow A^{\prime n}}$ of the conditional quantum encoder consists of an encryption with the secret key and the mapping to channel codewords. Each element $\mathcal{E}_{k}^{M_{A}^{\prime}S_{A}\rightarrow A^{\prime n}}$ produces the following state:$$\omega_{k}^{MS_{B}A^{\prime n}}\equiv\mathcal{E}_{k}^{M_{A}^{\prime}S_{A}\rightarrow A^{\prime n}}(\overline{\Phi}^{MM_{A}^{\prime}}\otimes\overline{\Phi}^{S_{A}S_{B}}).$$ The average density operator over all public messages is then as follows:$$\frac{1}{K}\sum_{k}\left\vert k\right\rangle \left\langle k\right\vert
^{K}\otimes\omega_{k}^{MS_{B}A^{\prime n}}.$$ Alice sends the $A^{\prime n}$ system through the noisy channel $U_{\mathcal{N}}^{A^{\prime n}\rightarrow B^{n}E^{n}}$, producing the following state:$$\omega^{KMS_{B}B^{n}E^{n}}\equiv\frac{1}{K}\sum_{k}\left\vert k\right\rangle
\left\langle k\right\vert ^{K}\otimes U_{\mathcal{N}}^{A^{\prime n}\rightarrow
B^{n}E^{n}}(\omega_{k}^{MS_{B}A^{\prime n}}).$$ Define the systems $Y\equiv MS_{B}$ and $X\equiv K$ so that the above state is a particular $n^{\text{th}}$ extension of the state in the statement of the public-private secret-key-assisted capacity theorem. The above state is the state at time $t$ in Figure \[fig:converse\]. Bob receives the above state and performs a decoding instrument $\mathcal{D}^{B^{n}S_{B}\rightarrow
K_{B}^{\prime}M_{B}^{\prime}}$ [@HW08GFP] (each element $\mathcal{D}_{k}^{B^{n}S_{B}\rightarrow M_{B}^{\prime}}$ of the instrument consists of a channel decoding and a decryption). The protocol ends at time $t_{f}$ (depicted in Figure \[fig:converse\]). Let $\left( \omega^{\prime
}\right) ^{KMK_{B}^{\prime}M_{B}^{\prime}E^{n}}$ be the state at time $t_{f}$ after Bob processes $\omega^{KMS_{B}B^{n}E^{n}}$ with the decoding instrument $\mathcal{D}^{B^{n}S_{B}\rightarrow K_{B}^{\prime}M_{B}^{\prime}}$.
Suppose that an $\left( n,R-\delta,P-\delta,R_{S}+\delta,\epsilon\right) $ secret-key-assisted protocol as given above exists. In particular, the following information-theoretic security conditions follow from the security criterion in (\[eq:security-criterion\]):$$\begin{aligned}
I\left( M;E^{n}|K\right) _{\omega} & \leq\epsilon
,\label{eq:private-correlations-pub}\\
I\left( S_{B};E^{n}|K\right) _{\omega} & \leq\epsilon
,\label{eq:secret-key-pub}$$ by the application of the Alicki-Fannes inequality [@0305-4470-37-5-L01] and evaluating the conditional mutual informations of the ideal state $\sigma_{k}^{E^{n}}\otimes\pi^{S_{B}}$ in (\[eq:security-criterion\]). These conditions imply that Eve learns nothing about the secret correlations in system $M$ and Eve learns nothing about the secret key $S_{B}$ (at time $t$) even if she knows the public variable $K$. We prove that the following bounds apply to the elements of the protocol’s rate triple $\left(
R-\delta,P-\delta,R_{S}+\delta\right) $,$$\begin{aligned}
R-\delta & \leq\frac{I(X;B^{n})_{\omega}}{n},\label{eq:pub-bound}\\
P-\delta & \leq\frac{I(Y;B^{n}|X)_{\omega}}{n},\label{eq:priv-bound-1}\\
P-\delta & \leq R_{S}+\frac{I(Y;B^{n}|X)_{\omega}-I(Y;E^{n}|X)_{\omega}}{n},\label{eq:priv-bound-2}\\
R_{S}+\delta & \geq\frac{I(Y;E^{n}|X)_{\omega}}{n},\label{eq:sec-bound}$$ for any $\epsilon,\delta>0$ and all sufficiently large $n$.
In the ideal case, the ideal private channel acts on system $M$ to produce the maximally correlated and secret state $\pi^{MM^{\prime}}$. So, for our case, the inequality$$\left\Vert \left( \omega^{\prime}\right) ^{MM_{B}^{\prime}E^{n}}-\overline{\Phi}^{MM_{B}^{\prime}}\otimes\sigma^{E^{n}}\right\Vert _{1}\leq\epsilon\label{eq:converse-good-private-comm}$$ holds because the protocol is $\epsilon$-good for private communication. The state $\sigma^{E^{n}}$ is some constant state on Eve’s system.
The lower bound in (\[eq:sec-bound\]) is the most straightforward to prove. Consider the following chain of inequalities:$$\begin{aligned}
& n\left( R_{S}+\delta\right) +2\epsilon\\
& \geq I\left( M;E^{n}|K\right) _{\omega}+I\left( S_{B};E^{n}|K\right)
_{\omega}+H\left( S_{B}|K\right) _{\omega}\\
& =H\left( M|K\right) _{\omega}+H\left( E^{n}|K\right) -H\left(
ME^{n}|K\right) _{\omega}+\\
& I\left( S_{B};E^{n}|K\right) _{\omega}+H\left( S_{B}|K\right) _{\omega
}\\
& \geq H\left( M|S_{B}K\right) _{\omega}+H\left( E^{n}|S_{B}K\right)
-H\left( ME^{n}|K\right) _{\omega}+\\
& I\left( S_{B};E^{n}|K\right) _{\omega}+H\left( S_{B}|K\right) _{\omega
}\\
& \geq H\left( M|S_{B}K\right) _{\omega}+H\left( E^{n}|S_{B}K\right)
-H\left( ME^{n}S_{B}|K\right) _{\omega}+\\
& I\left( S_{B};E^{n}|K\right) _{\omega}+H\left( S_{B}|K\right) _{\omega
}\\
& =I\left( M;E^{n}|S_{B}K\right) _{\omega}+I\left( S_{B};E^{n}|K\right)
_{\omega}\\
& =I\left( MS_{B};E^{n}|K\right) _{\omega}\\
& =I\left( Y;E^{n}|X\right) _{\omega}$$ The first inequality follows by combining the equality $n\left( R_{S}+\delta\right) =H\left( S_{B}\right) =H\left( S_{B}|K\right) $ and the security criteria in (\[eq:private-correlations-pub\]-\[eq:secret-key-pub\]). The first equality follows from the definition of mutual information. The second inequality follows because $H\left( M\right)
_{\omega}=H\left( M|S_{B}K\right) _{\omega}$ ($M$, $S_{B}$, and $K$ are independent) and conditioning does not increase entropy $H\left(
E^{n}|K\right) \geq H\left( E^{n}|S_{B}K\right) $. The third inequality follows because the addition of a classical system can increase entropy $H\left( ME^{n}|K\right) _{\omega}\leq H\left( ME^{n}S_{B}|K\right)
_{\omega}$. The second equality follows from the definition of conditional mutual information. The third equality follows from the chain rule of mutual information, and the last equality follows from the definitions $Y\equiv
MS_{B}$ and $X\equiv K$.
We next prove the upper bound in (\[eq:priv-bound-1\]) on the private communication rate:$$\begin{aligned}
& n(P-\delta)\\
& =H\left( M\right) \\
& =I\left( M;M_{B}^{\prime}\right) _{\omega^{\prime}}+H\left(
M|M_{B}^{\prime}\right) \\
& \leq I(M;M_{B}^{\prime}K)_{\omega^{\prime}}+n\delta^{\prime}\\
& \leq I(M;B^{n}S_{B}K)_{\omega}+n\delta^{\prime}\\
& =I(M;B^{n}K|S_{B})_{\omega}+n\delta^{\prime}\\
& =H\left( M|S_{B}\right) +H(B^{n}K|S_{B})_{\omega}-\\
& H\left( MB^{n}S_{B}K\right) +H\left( S_{B}\right) +n\delta^{\prime}\\
& =H\left( MS_{B}|K\right) -H\left( S_{B}|K\right) +H(B^{n}K|S_{B})_{\omega}-\\
& H\left( MB^{n}S_{B}K\right) +H\left( S_{B}|K\right) +n\delta^{\prime}\\
& =H\left( MS_{B}|K\right) +H(B^{n}K|S_{B})_{\omega}-\\
& H\left( MB^{n}S_{B}K\right) +n\delta^{\prime}\\
& \leq H\left( MS_{B}|K\right) +H(B^{n}K)_{\omega}-H\left( MB^{n}S_{B}K\right) +\\
& H\left( K\right) -H\left( K\right) +n\delta^{\prime}\\
& =I\left( MS_{B};B^{n}|K\right) _{\omega}+n\delta^{\prime}\\
& =I\left( Y;B^{n}|X\right) _{\omega}+n\delta^{\prime}$$ The first equality follows by evaluating the entropy for the state $\overline{\Phi}^{M}$ and noting that $H\left( M\right) =H\left(
M|K\right) $. The second equality follows by standard entropic relations. The first inequality follows from (\[eq:converse-good-private-comm\]), Fano’s inequality [@CT91], and conditioning does not increase entropy. The second inequality is from quantum data processing. The third equality follows from the chain rule for mutual information and $I\left( M;S_{B}\right) =0$ because $M$ and $S_{B}$ are independent. The fourth equality follows by expanding the conditional mutual information. The fifth and sixth equalities follow from standard entropic relations. The last inequality follows because conditioning does not increase entropy $H(B^{n}K|S_{B})_{\omega}\leq
H(B^{n}K)_{\omega}$. The fifth equality follows by the definition of mutual information, and the last equality follows from the definitions $Y\equiv
MS_{B}$, $X\equiv K$, and $\delta^{\prime}\equiv\frac{1}{n}+\epsilon P$.
The second bound in (\[eq:priv-bound-2\]) on the private communication rate follows from adding the bound in (\[eq:priv-bound-1\]) to the bound in (\[eq:sec-bound\]).
We can use a proof by contradiction to get the bound on the public rate $R$. Suppose that we have secret key available at some rate $>I(X;E^{n})_{\omega
}/n$. Then one could combine the public communication at rate $R$ with the extra secret key in a one-time pad protocol in order to generate private communication at a rate $R+P$. The resulting protocol consumes secret key at a rate greater than $I\left( YX;E^{n}\right) $ because$$\frac{I\left( Y;E^{n}|X\right) _{\omega}}{n}+\frac{I\left( X;E^{n}\right)
_{\omega}}{n}=\frac{I\left( YX;E^{n}\right) }{n}.$$ The state $\omega$ is of the form given by the secret-key-assisted capacity theorem [@HLB08SKP]. The total amount of private communication that a secret-key-assisted protocol can generate cannot be any larger than $I\left(
YX;B^{n}\right) /n$ [@HLB08SKP]. The chain rule also applies to the mutual information $I\left( YX;B^{n}\right) /n$:$$\frac{I\left( Y;B^{n}|X\right) _{\omega}}{n}+\frac{I\left( X;B^{n}\right)
_{\omega}}{n}=\frac{I\left( YX;B^{n}\right) }{n}.$$ If the public rate $R$ were to exceed $I\left( X;B^{n}\right) _{\omega}/n$, then this public rate would contradict the optimality of the secret-key-assisted protocol from Ref. [@HLB08SKP]. Thus, the public rate $R$ must obey the bound in (\[eq:pub-bound\]).
Proof of the Direct Coding Theorem {#sec:direct-coding-theorem}
==================================
The direct coding theorem is the proof of the following *publicly-enhanced private father protocol resource inequality* (See Refs. [@DHW03; @DHW05RI] for the theory of resource inequalities):$$\begin{gathered}
\left\langle \mathcal{N}\right\rangle +I\left( Y;E|X\right) _{\sigma}\left[
cc\right] _{\text{priv}}\geq\\
I\left( Y;B|X\right) _{\sigma}\left[ c\rightarrow c\right] _{\text{priv}}+I\left( X;B\right) _{\sigma}\left[ c\rightarrow c\right] _{\text{pub}}.
\label{eq:PEPFP-resource-ineq}$$ The resource inequality has an interpretation as the following statement. For any $\epsilon,\delta>0$ and sufficiently large $n$, there exists a protocol that consumes $nI\left( Y;E|X\right) _{\sigma}$ bits of secret key and $n$ independent uses of the noisy quantum channel $\mathcal{N}$ to generate $nI\left( Y;B|X\right) _{\sigma}$ bits of private communication and $nI\left( X;B\right) _{\sigma}$ bits of public communication with $\epsilon$ probability of error. In addition, Eve’s state is $\epsilon$-close to a state that is independent of the private message and the secret key. The entropic quantities are with respect to the state $\sigma^{XYBE}$ in (\[eq:maximization-state\]).
The proof of the direct coding theorem proceeds similarly to the proof of the direct coding theorem for the classically-enhanced father protocol from Ref. [@HW08GFP]. There are some subtle differences between the two proofs, and we highlight only the parts of the proof that are different from the proof of the classically-enhanced father protocol. The proof begins by showing how to construct a *random private father code*, similar to the notion of a random father code [@HW08GFP] or a random quantum code [@Devetak03]. We introduce the *channel input density operator* for a random private father code and show that it is possible to make it close to a tensor-product state. We then show how to associate a classical string to a random private father code by exploiting the code pasting technique from Ref. [@DS03]. The proof proceeds by applying the HSW theorem [@Hol98; @SW97] to show that Bob can decode the public information first. Based on the public information, Bob decodes the private information. The details of the proof involve showing how the random publicly-enhanced private father code has low probability of error for decoding the public information and the private information. Finally, we employ the standard techniques of derandomization and expurgation to show that there exists a particular publicly-enhanced private father code that achieves the rates given in Theorem \[thm:PEPFP\].
Random Private Coding
---------------------
We first recall the secret-key-assisted private communication capacity theorem (also known as the private father capacity theorem) [@HLB08SKP].
\[pf\_2\]The secret-key-assisted private channel capacity region $C_{\text{SKP}}(\mathcal{N})$ is given by $$C_{\text{\emph{SKP}}}(\mathcal{N})=\overline{\bigcup_{l=1}^{\infty}\frac{1}{l}\widetilde{C}_{\text{\emph{SKP}}}^{(1)}(\mathcal{N}^{\otimes l})},
\label{cp2}$$ where the overbar indicates the closure of a set, and $\widetilde
{C}_{\text{\emph{SKP}}}^{(1)}(\mathcal{N})$ is the set of all $R_{S}\geq0$, $P\geq0$ such that $$\begin{aligned}
P & \leq I(Y;B)_{\rho}-I(Y;E)_{\rho}+R_{S}\label{thm2cond1}\\
P & \leq I(Y;B)_{\rho}, \label{thm2cond2}$$ where $R_{S}$ is the secret key consumption rate and $\rho$ is a state of the form $$\rho^{YBE}\equiv\sum_{y}p(y)|y\rangle\langle y|^{Y}\otimes U_{\mathcal{N}}^{A^{\prime}\rightarrow BE}(\rho_{y}^{A^{\prime}}), \label{eq:private-state}$$ for some ensemble $\{p(y),\rho_{y}^{A^{\prime}}\}$ and $U_{\mathcal{N}}^{A^{\prime}\rightarrow BE}$ is an isometric extension of $\mathcal{N}$.
The *channel input density operator* $\rho^{A^{\prime n}}\left(
\mathcal{C}\right) $ for a private father code $\mathcal{C}\equiv\{\rho
_{m}^{A^{\prime n}}\}_{m\in\left[ M\right] }$ is a uniform mixture of all the private codewords $\rho_{m}^{A^{\prime n}}$ in code $\mathcal{C}$:$$\rho^{A^{\prime n}}\left( \mathcal{C}\right) \equiv\frac{1}{M}\sum_{m=1}^{M}\rho_{m}^{A^{\prime n}}.$$
We cannot say much about the channel input density operator $\rho^{A^{\prime
n}}\left( \mathcal{C}\right) $ for a particular private father code $\mathcal{C}$. But we can say something about the expected channel input density operator of a *random private father code* $\mathcal{C}$ (where $\mathcal{C}$ itself becomes a random variable).
A *random private father code* is an ensemble $\{p_{\mathcal{C}},\mathcal{C}\}$ of codes where each code $\mathcal{C}$ occurs with probability $p_{\mathcal{C}}$. The *expected channel input density operator* $\overline{\rho}^{A^{\prime n}}$ is as follows: $$\overline{\rho}^{A^{\prime n}}\equiv\mathbb{E}_{\mathcal{C}}\left\{
\rho^{A^{\prime n}}\left( \mathcal{C}\right) \right\} .
\label{eq:def_expected_state}$$ A random private father code is $\rho$-like if the expected channel input density operator is close to a tensor power of some state $\rho$:$$\left\Vert \overline{\rho}^{A^{\prime n}}-\rho^{\otimes n}\right\Vert _{1}\leq\epsilon.$$
We now state a version of the direct coding theorem that applies to random private father codes. The proof shows that we can produce a random secret-key-assisted private code with an expected channel input density operator close to a tensor power state.
\[thm:random-private-code\]For any $\epsilon,\delta>0$ and all sufficiently large $n$, there exists a random $\rho^{A^{\prime}}$-like secret-key-assisted private code for a channel $\mathcal{N}^{A^{\prime
}\rightarrow B}$ such that$$\left\Vert \overline{\rho}^{A^{\prime n}}-(\rho^{A^{\prime}})^{\otimes
n}\right\Vert _{1}\leq2\epsilon+4\sqrt[4]{\epsilon},$$ where $\overline{\rho}^{A^{\prime n}}$ is defined in (\[eq:def\_expected\_state\]). The random private code has private communication rate $I(Y;B)_{\rho}-\delta$ and secret key consumption rate $I(Y;E)_{\rho}+\delta$. The entropic quantities are with respect to the state in (\[eq:private-state\]) and the state $\rho^{A^{\prime}}\equiv\sum
_{y}p\left( y\right) \rho_{y}^{A^{\prime}}$.
\[AP\_RG0\]The proof of Proposition \[thm:random-private-code\] is an extension of the development in Appendix D of Ref. [@DS03] and the development in Ref. [@HLB08SKP].
Consider the density operator $\rho^{A^{\prime}}$ where$$\rho^{A^{\prime}}=\sum_{y\in\mathcal{Y}}p\left( y\right) \rho_{y}^{A^{\prime}}.$$ The $n^{\text{th}}$ extension of the above state as a tensor power state is as follows: $$\rho^{A^{\prime n}}\equiv(\rho^{A^{\prime}})^{\otimes n}=\sum_{y^{n}\in\mathcal{Y}^{n}}p^{n}\left( y^{n}\right) \rho_{y^{n}}^{A^{\prime n}},$$ where$$\rho_{y^{n}}^{A^{\prime n}}\equiv\rho_{y_{1}}^{A^{\prime}}\otimes\rho_{y_{2}}^{A^{\prime}}\otimes\cdots\otimes\rho_{y_{n}}^{A^{\prime}}.$$ We define the pruned distribution $p^{\prime n}$ as follows:$$p^{\prime n}\left( x^{n}\right) \equiv\left\{
\begin{array}
[c]{ccc}p^{n}\left( y^{n}\right) /\sum_{y^{n}\in T_{\delta}^{Y^{n}}}p^{n}\left(
y^{n}\right) & : & y^{n}\in T_{\delta}^{Y^{n}}\\
0 & : & \text{else},
\end{array}
\right.$$ where $T_{\delta}^{Y^{n}}$ denotes the $\delta$-typical set of sequences with length $n$. Let $\widetilde{\rho}^{A^{\prime n}}$ denote the following pruned state:$$\widetilde{\rho}^{A^{\prime n}}\equiv\sum_{y^{n}\in T_{\delta}^{Y^{n}}}p^{\prime n}\left( y^{n}\right) \rho_{y^{n}}^{A^{\prime n}}.
\label{eq:pruned-state}$$ For any $\epsilon>0$ and sufficiently large $n$, the state $\rho^{A^{\prime
n}}$ is close to $\widetilde{\rho}^{A^{\prime n}}$ by the gentle measurement lemma [@Winter99] and because the probability for sequences outside the typical set is small:$$\left\Vert \rho^{A^{\prime n}}-\widetilde{\rho}^{A^{\prime n}}\right\Vert
_{1}\leq2\epsilon.$$
For any density operator $\rho^{A^{\prime}}$, it is possible to construct a secret-key-assisted private code that achieves the private communication rate and secret key consumption rate in Proposition \[thm:random-private-code\].
Let $[M]$ denote a set of size $2^{n[I(Y;B)-c\delta]}$ for some constant $c$ and let $U_{m}$ denote $2^{n[I(Y;B)-c\delta]}$ random variables that we choose according to the pruned distribution $p^{\prime n}(y^{n})$. The realizations $u_{m}$ of the random variables $U_{m}$ are sequences in $\mathcal{Y}^{n}$ and are the basis for constructing a secret-key-assisted private code $\mathcal{C}$ with the following codeword ensemble:$$\mathcal{C}=\{p^{\prime n}(u_{m}),\rho_{u_{m}}^{A^{n}}\}_{m}.$$ We then perform a decoding positive operator-valued measure (POVM) with elements $\{\Lambda_{m}\}_{m\in\left[ M\right] }$ and decryption map $g$, resulting in failure with probability $4\epsilon+20\sqrt{\epsilon}$ by the arguments in Ref. [@HLB08SKP].
Suppose that we choose a particular secret-key-assisted private code $\mathcal{C}$ according to the above prescription. Its code density operator is$$\rho^{A^{\prime n}}(\mathcal{C})=\frac{1}{M}\sum_{m=1}^{M}\rho_{u_{m}}^{A^{\prime n}}.$$
Suppose we now consider the secret-key-assisted private code chosen according to the above prescription as a *random* code $\mathcal{C}$ (where $\mathcal{C}$ is now a random variable). Let $\rho^{\prime A^{\prime n}}\left( \mathcal{C}\right) $ be the channel input density operator for the random code before expurgation and $\rho^{A^{\prime n}}\left( \mathcal{C}\right) $ its channel input density operator after expurgation:$$\begin{aligned}
\rho^{\prime A^{\prime n}}(\mathcal{C}) & \equiv\frac{1}{M^{\prime}}\sum_{m=1}^{M^{\prime}}\rho_{U_{m}}^{A^{\prime n}},\\
\rho^{A^{\prime n}}(\mathcal{C}) & \equiv\frac{1}{M}\sum_{m=1}^{M}\rho_{U_{m}}^{A^{\prime n}},\end{aligned}$$ where the primed rates are the rates before expurgation and the unprimed rates are those after expurgation (they are slightly different but identical for large $n$). Let $\overline{\rho}^{\prime A^{\prime n}}$ and $\overline{\rho
}^{A^{\prime n}}$ denote the expectation of the above channel input density operators:$$\begin{aligned}
\overline{\rho}^{\prime A^{\prime n}} & \equiv\mathbb{E}_{\mathcal{C}}\left\{ \rho^{\prime A^{\prime n}}\left( \mathcal{C}\right) \right\} ,\\
\overline{\rho}^{A^{\prime n}} & \equiv\mathbb{E}_{\mathcal{C}}\left\{
\rho^{A^{\prime n}}\left( \mathcal{C}\right) \right\} .\end{aligned}$$
Choosing our code in the particular way that we did leads to an interesting consequence. The expectation of the density operator corresponding to Alice’s codeword $\rho_{U_{m}}^{A^{\prime n}}$ is equal to the pruned state in (\[eq:pruned-state\]): $$\mathbb{E}_{\mathcal{C}}\left\{ \rho_{U_{m}}^{A^{\prime n}}\right\}
=\sum_{y^{n}}p^{\prime n}(y^{n})\rho_{y^{n}}^{A^{\prime n}},$$ because we choose the codewords $\rho_{y^{n}}^{A^{\prime n}}$ randomly according to the pruned distribution $p^{\prime n}(y^{n})$. Then the expected channel input density operator $\overline{\rho}^{\prime A^{\prime n}}$ is as follows: $$\begin{aligned}
\overline{\rho}^{\prime A^{\prime n}} & =\mathbb{E}_{\mathcal{C}}\left\{
\rho^{\prime A^{\prime n}}\left( \mathcal{C}\right) \right\} \\
& =\frac{1}{M^{\prime}}\sum_{m=1}^{M^{\prime}}\mathbb{E}_{\mathcal{C}}\left\{ \rho_{U_{m}}^{A^{\prime n}}\right\} \\
& =\sum_{y^{n}}p^{\prime n}(y^{n})\rho_{y^{n}}^{A^{\prime n}}.\end{aligned}$$ Then we know that the following inequality holds for $\overline{\rho}^{\prime
A^{\prime n}}$ and the tensor power state $\rho^{A^{\prime n}}$$$\left\Vert \overline{\rho}^{\prime A^{\prime n}}-\rho^{A^{\prime n}}\right\Vert _{1}\leq2\epsilon\label{eq:typical-close}$$ by the typical subspace theorem and the gentle measurement lemma. The expurgation of any secret-key-assisted private code $\mathcal{C}$ has a minimal effect on the resulting channel input density operator [@DS03]:$$\left\Vert \rho^{\prime A^{\prime n}}\left( \mathcal{C}\right)
-\rho^{A^{\prime n}}\left( \mathcal{C}\right) \right\Vert _{1}\leq
4\sqrt[4]{\epsilon}.$$ The above inequality implies that the following one holds for the expected channel input density operators $\overline{\rho}^{\prime A^{\prime n}}$ and $\overline{\rho}^{A^{\prime n}}$$$\left\Vert \overline{\rho}^{\prime A^{\prime n}}-\overline{\rho}^{A^{\prime
n}}\right\Vert _{1}\leq4\sqrt[4]{\epsilon},\label{eq:expected-close}$$ because the trace distance is convex. The following inequality holds$$\left\Vert \overline{\rho}^{A^{\prime n}}-\rho^{A^{\prime n}}\right\Vert
_{1}\leq2\epsilon+4\sqrt[4]{\epsilon}$$ by applying the triangle inequality to (\[eq:typical-close\]) and (\[eq:expected-close\]). Therefore, the random secret-key-assisted private code is $\rho$-like.
Associating a Random Private Code with a Classical String
---------------------------------------------------------
Suppose that we have an ensemble $\{p(x),\rho_{x}\}_{x\in\mathcal{X}}$ of quantum states. The density operator $\rho_{x}$ arises as the expected density operator of another ensemble $\left\{ p\left( y|x\right) ,\rho
_{x,y}\right\} $. Let $x^{n}\equiv x_{1}\cdots x_{n}$ denote a classical string generated by the density $p(x)$ where each symbol $x_{i}\in\mathcal{X}$. Then there is a density operator $\sigma_{x^{n}}$ corresponding to the string $x^{n}$ where$$\rho_{x^{n}}\equiv\bigotimes_{i=1}^{n}\rho_{x_{i}}.$$ Suppose that we label a random private code by the string $x^{n}$ and let $\overline{\rho}_{x^{n}}^{A^{\prime n}}$ denote its expected channel input density operator.
A random private code is $(\rho_{x^{n}})$-like if the expected channel input density operator $\overline{\rho}_{x^{n}}^{A^{\prime n}}$ is close to the state $\rho_{x^{n}}$:$$\left\Vert \overline{\rho}_{x^{n}}^{A^{\prime n}}-\rho_{x^{n}}\right\Vert
_{1}\leq\epsilon.$$
\[prop:random-grandfather\] Suppose we have an ensemble as above. Consider a quantum channel $\mathcal{N}^{A^{\prime}\rightarrow B}$ with its isometric extension $U_{\mathcal{N}}^{A^{\prime}\rightarrow BE}$. Then there exists a random $(\rho_{x^{n}})$-like secret-key-assisted private code for the channel $\mathcal{N}^{A^{\prime}\rightarrow B}$ for any $\epsilon,\delta>0$, for all sufficiently large $n$, and for any classical string $x^{n}$ in the typical set $T_{\delta}^{X^{n}}$ [@CT91]. Its private communication rate is $I(Y;B|X)-c^{\prime}\delta$, and its secret key consumption rate is $I(Y;E|X)-c^{\prime\prime}\delta\ $for some constants $c^{\prime},c^{\prime\prime}$ where the entropic quantities are with respect to the state in (\[eq:maximization-state\]). The state $\rho_{x}$ is the restriction of the following state$$\rho_{x}^{YA^{\prime}}=\sum_{y}p(y|x)|y\rangle\langle y|^{Y}\otimes\rho
_{x,y}^{A^{\prime}}$$ to the $A^{\prime}$ system.
\[Proposition \[prop:random-grandfather\]\] The proof of this theorem proceeds exactly as the proof of Proposition 3 in Ref. [@HW08GFP] and the proof of Proposition 5 in Ref. [@DS03].
Publicly-enhanced secret-key-assisted private code
--------------------------------------------------
\[HSW Coding Theorem [@Hol98; @SW97]\]\[prop:HSW\]Consider an input ensemble $\{p(x),\rho_{x}^{A^{\prime}}\}$ that gives rise to a classical-quantum state $\sigma^{XB}$, where$$\sigma^{XB}\equiv\sum_{x\in\mathcal{X}}p(x)|x\rangle\langle x|^{X}\otimes\mathcal{N}^{A^{\prime}\rightarrow B}(\rho_{x}^{A^{\prime}}).$$ Let $R=I(X;B)_{\sigma}-c^{\prime}\delta$ for any $\delta>0$ and for some constant $c^{\prime}$. Then for all $\epsilon>0$ and for all sufficiently large $n$, there exists a classical encoding map$$h:\left[ 2^{nR}\right] \rightarrow T_{\delta}^{X^{n}},$$ and a decoding POVM$$\{\Lambda_{k}^{B^{n}}\}_{k\in\lbrack2^{nR}]},$$ that allows Bob to decode any classical message $k\in\lbrack2^{nR}]$ with high probability: $$\operatorname{Tr}\{\tau_{k}^{B^{n}}\Lambda_{k}^{B^{n}}\}\geq1-\epsilon.$$ The density operators $\tau_{k}^{B^{n}}$ are the channel outputs$$\tau_{k}^{B^{n}}\equiv\mathcal{N}^{A^{\prime n}\rightarrow B^{n}}(\rho
_{h(k)}^{A^{\prime n}}),\label{eq:channel-outputs}$$ and the channel input states $\rho_{x^{n}}^{A^{\prime n}}$ are a tensor product of states in the ensemble:$$\rho_{x^{n}}^{A^{\prime n}}\equiv{\bigotimes\limits_{i=1}^{n}\ }\rho_{x_{i}}^{A^{\prime}}.$$
We are now in a position to prove the direct coding part of the publicly-enhanced private father capacity theorem. The proof is similar to that in Ref. [@DS03; @HW08GFP].
\[Direct Coding Theorem\] Define the public message set $[2^{nR}]$, the classical encoding map $h$, the channel output states $\tau_{k}^{B^{n}}$, and the decoding POVM $\{\Lambda_{k}^{B^{n}}\}_{k\in2^{nR}}$ as in Proposition \[prop:HSW\]. We label each public message $k\in\lbrack2^{nR}]$ where $R=I(X;B)-c^{\prime}\delta$.
Invoking Proposition \[prop:random-grandfather\], there exists a random $(\rho_{h(k)}^{A^{\prime n}})$-like private code $\mathcal{C}_{k}$ with probability density $p_{\mathcal{C}_{k}}$ because each input to the channel $\rho_{h(k)}^{A^{\prime n}}$ is a tensor product of an ensemble $\{p(x),\rho
_{x}^{A^{\prime}}\}$. The random private code $\mathcal{C}_{k}$ has encryption-decryption pair $(f_{\mathcal{C}_{k}},g_{\mathcal{C}_{k}})$ and encoding-decoding pair $\left( \mathcal{E}_{\mathcal{C}_{k}},\mathcal{D}_{\mathcal{C}_{k}}\right) $ for each of its realizations. We label the combined operations simply as the pair $(\mathcal{E}_{\mathcal{C}_{k}}^{MS_{A}\rightarrow A^{\prime n}},\mathcal{D}_{\mathcal{C}_{k}}^{B^{n}S_{B}\rightarrow M})$. It transmits $n[I(Y;B|X)+c^{\prime}\delta]$ private bits, provided Alice and Bob share at least $n[I(Y;E|X)+c^{\prime\prime}\delta]$ secret key bits.
Let $\mathcal{C}$ denote the *random publicly-enhanced secret-key-assisted private code* that is the collection of random private codes $\{\mathcal{C}_{k}\}_{k\in\lbrack2^{nR}]}$. We first prove that the expectation of the error probability for public message $k$ is small. The expectation is with respect to the random private code $\mathcal{C}_{k}$. Let $\tau_{\mathcal{C}_{k}}^{B^{n}}$ denote the *channel output density operator* corresponding to the private code $\mathcal{C}_{k}$: $$\tau_{\mathcal{C}_{k}}^{B^{n}}\equiv\text{Tr}_{S_{B}}\left\{ \mathcal{N}^{A^{\prime n}\rightarrow B^{n}}(\mathcal{E}_{\mathcal{C}_{k}}^{MS_{A}\rightarrow A^{\prime n}}(\pi^{M}\otimes\overline{\Phi}^{S_{A}S_{B}}))\right\} .$$ Let $\overline{\tau}_{k}^{B^{n}}$ denote the *expected channel output density operator* of the random father code $\mathcal{C}_{k}$:$$\overline{\tau}_{k}^{B^{n}}\equiv\mathbb{E}_{\mathcal{C}_{k}}\left\{
\tau_{\mathcal{C}_{k}}^{B^{n}}\right\} =\sum_{\mathcal{C}_{k}}p_{\mathcal{C}_{k}}\tau_{\mathcal{C}_{k}}^{B^{n}}.$$ The following inequality holds$$\left\Vert \overline{\rho}_{h(k)}^{A^{\prime n}}-\rho_{h(k)}^{A^{\prime n}}\right\Vert _{1}\leq\left\vert \mathcal{X}\right\vert \epsilon$$ because the random private code $\mathcal{C}_{k}$ is $(\rho_{h(k)}^{A^{\prime
n}})$-like. Then the expected channel output density operator $\overline{\tau
}_{k}^{B^{n}}$ is close to the tensor product state $\tau_{k}^{B^{n}}$ in (\[eq:channel-outputs\]):$$\left\Vert \overline{\tau}_{k}^{B^{n}}-\tau_{k}^{B^{n}}\right\Vert _{1}\leq\left\vert \mathcal{X}\right\vert \epsilon,
\label{eq:channel-product-closeness}$$ because the trace distance is monotone under the quantum operation $\mathcal{N}^{A^{\prime n}\rightarrow B^{n}}$. It then follows that the POVM element $\Lambda_{k}^{B^{n}}$ has a high probability of detecting the expected channel output density operator $\overline{\tau}_{k}^{B^{n}}$:$$\begin{aligned}
\operatorname{Tr}\{\Lambda_{k}^{B^{n}}\overline{\tau}_{k}^{B^{n}}\} & \geq\operatorname{Tr}\{\Lambda
_{k}^{B^{n}}\tau_{k}^{B^{n}}\}-\left\Vert \overline{\tau}_{k}^{B^{n}}-\tau
_{k}^{B^{n}}\right\Vert _{1}\nonumber\\
& \geq1-\epsilon-\left\vert \mathcal{X}\right\vert \epsilon.
\label{eq:average-output-op-good}$$ The first inequality follows from the following lemma that holds for any two quantum states $\rho$ and $\sigma$ and a positive operator $\Pi$ where $0\leq\Pi\leq I$:$$\text{Tr}\left\{ \Pi\rho\right\} \geq\text{Tr}\left\{ \Pi\sigma\right\}
-\left\Vert \rho-\sigma\right\Vert _{1}.$$ The second inequality follows from Proposition \[prop:HSW\] and (\[eq:channel-product-closeness\]). Let $p_{e,\text{pub}}(\mathcal{C}_{k})$ denote the public message error probability for each public message $k$ of the publicly-enhanced father code $\mathcal{C}$:$$p_{e,\text{pub}}(\mathcal{C}_{k})\equiv1-\Pr\{K^{\prime}=k\ |\ K=k\}.$$ Then by the above definition, and (\[eq:average-output-op-good\]), it holds that the expectation of the error probability $p_{e,\text{pub}}(\mathcal{C}_{k})$ for public message $k$ with respect to the random private code $\mathcal{C}_{k}$ is low:$$\begin{aligned}
\mathbb{E}_{\mathcal{C}_{k}}\left\{ p_{e,\text{pub}}(\mathcal{C}_{k})\right\} & =1-\operatorname{Tr}\{\Lambda_{k}^{B^{n}}\overline{\tau}_{k}^{B^{n}}\}\label{eq:classical-error}\\
& \leq\left( 1+|\mathcal{X}|\right) \epsilon.\end{aligned}$$
We now show that the private error is small. Input the state $\pi^{M}\otimes\overline{\Phi}^{S_{A}S_{B}}$ to the encoder $\mathcal{E}_{\mathcal{C}_{k}}^{MS_{A}\rightarrow A^{\prime n}}$, followed by the channel $\mathcal{N}^{A^{\prime n}\rightarrow B^{n}}$. The resulting state is an extension $\Omega_{\mathcal{C}_{k}}^{S_{B}B^{n}}$ of $\tau_{\mathcal{C}_{k}}^{B^{n}}$:$$\Omega_{\mathcal{C}_{k}}^{S_{B}B^{n}}\equiv\mathcal{N}^{A^{\prime
n}\rightarrow B^{n}}\left( \mathcal{E}_{\mathcal{C}_{k}}^{MS_{A}\rightarrow
A^{\prime n}}(\pi^{M}\otimes\overline{\Phi}^{S_{A}S_{B}})\right) .$$ Let $\overline{\Omega}_{k}^{S_{B}B^{n}}$ denote the expectation of $\Omega_{\mathcal{C}_{k}}^{S_{B}B^{n}}$ with respect to the random code $\mathcal{C}_{k}$:$$\overline{\Omega}_{k}^{S_{B}B^{n}}\equiv\mathbb{E}_{\mathcal{C}_{k}}\left\{
\Omega_{\mathcal{C}_{k}}^{S_{B}B^{n}}\right\} .$$ It follows that $\overline{\Omega}_{k}^{S_{B}B^{n}}$ is an extension of $\overline{\tau}_{k}^{B^{n}}$. The following inequality follows from (\[eq:average-output-op-good\]):$$\operatorname{Tr}\{\overline{\Omega}_{k}^{S_{B}B^{n}}\Lambda_{k}^{B^{n}}\}\geq
1-(1+|\mathcal{X}|)\epsilon.$$ The above inequality is then sufficient for us to apply a modified version of the gentle measurement lemma (See Appendix C of Ref. [@HW08GFP]) so that the following inequality holds$$\begin{aligned}
& \mathbb{E}_{\mathcal{C}_{k}}\left\{ \left\Vert \sqrt{\Lambda_{k}^{B^{n}}}\Omega_{\mathcal{C}_{k}}^{S_{B}B^{n}}\sqrt{\Lambda_{k}^{B^{n}}}-\Omega_{\mathcal{C}_{k}}^{S_{B}B^{n}}\right\Vert _{1}\right\} \nonumber\\
& \leq\sqrt{8(1+|\mathcal{X}|)\epsilon}.\label{gm}$$ We define a decoding instrument $\mathcal{D}_{\mathcal{C}}^{B^{n}S_{B}\rightarrow KM}$ for the random publicly-enhanced private father code $\mathcal{C}$ as follows [@Yard05a; @HW08GFP]:$$\begin{aligned}
& \mathcal{D}_{\mathcal{C}}^{B^{n}S_{B}\rightarrow KM}\left( \rho
^{B^{n}S_{B}}\right) \\
& \equiv\sum_{k}\mathcal{D}_{\mathcal{C}_{k}}^{B^{n}S_{B}\rightarrow
M}\left( \sqrt{\Lambda_{k}^{B^{n}}}\rho^{B^{n}S_{B}}\sqrt{\Lambda_{k}^{B^{n}}}\right) \otimes\left\vert k\right\rangle \left\langle k\right\vert ^{K},\end{aligned}$$ where $\mathcal{D}_{\mathcal{C}_{k}}^{B^{n}S_{B}\rightarrow M}$ is the decoder for the private father code $\mathcal{C}_{k}$ and each map $\mathcal{D}_{\mathcal{C}_{k}}^{B^{n}S_{B}\rightarrow M}(\sqrt{\Lambda_{k}^{B^{n}}}\rho^{B^{n}S_{B}}\sqrt{\Lambda_{k}^{B^{n}}})$ is trace-reducing. The induced quantum operation corresponding to this instrument is as follows:$$\mathcal{D}_{\mathcal{C}}^{B^{n}S_{B}\rightarrow M}\left( \rho\right)
=\text{Tr}_{K}\left\{ \mathcal{D}_{\mathcal{C}}^{B^{n}S_{B}\rightarrow
KM}\left( \rho\right) \right\} .$$ Monotonicity of the trace distance gives an inequality for the trace-reducing maps of the quantum decoding instrument:$$\begin{aligned}
& \mathbb{E}_{\mathcal{C}_{k}}\left\{ \left\Vert
\begin{array}
[c]{c}\mathcal{D}_{\mathcal{C}_{k}}^{B^{n}S_{B}\rightarrow M}\left( \sqrt
{\Lambda_{k}^{B^{n}}}\Omega_{\mathcal{C}_{k}}^{S_{B}B^{n}}\sqrt{\Lambda
_{k}^{B^{n}}}\right) -\\
\mathcal{D}_{\mathcal{C}_{k}}^{B^{n}S_{B}\rightarrow M}\left( \Omega
_{\mathcal{C}_{k}}^{S_{B}B^{n}}\right)
\end{array}
\right\Vert _{1}\right\} \nonumber\\
& \leq\sqrt{8(1+|\mathcal{X}|)\epsilon}.\label{eq:epsilon-sqrt-map}$$ The following inequality also holds$$\begin{aligned}
& \mathbb{E}_{\mathcal{C}_{k}}\left\{ \left\Vert
\begin{array}
[c]{c}\mathcal{D}_{\mathcal{C}}^{B^{n}S_{B}\rightarrow M}\left( \Omega
_{\mathcal{C}_{k}}^{S_{B}B^{n}}\right) -\\
\mathcal{D}_{\mathcal{C}_{k}}^{B^{n}S_{B}\rightarrow M}\left( \sqrt
{\Lambda_{k}^{B^{n}}}\Omega_{\mathcal{C}_{k}}^{S_{B}B^{n}}\sqrt{\Lambda
_{k}^{B^{n}}}\right)
\end{array}
\right\Vert _{1}\right\} \nonumber\\
& \leq\mathbb{E}_{\mathcal{C}_{k}}\left\{ \sum_{k^{\prime}\neq k}\left\Vert
\mathcal{D}_{\mathcal{C}_{k^{\prime}}}^{B^{n}S_{B}\rightarrow M}\left(
\sqrt{\Lambda_{k^{\prime}}^{B^{n}}}\Omega_{\mathcal{C}_{k}}^{S_{B}B^{n}}\sqrt{\Lambda_{k^{\prime}}^{B^{n}}}\right) \right\Vert _{1}\right\}
\nonumber\\
& =\mathbb{E}_{\mathcal{C}_{k}}\left\{ \sum_{k^{\prime}\neq k}\left\Vert
\sqrt{\Lambda_{k^{\prime}}^{B^{n}}}\Omega_{\mathcal{C}_{k}}^{S_{B}B^{n}}\sqrt{\Lambda_{k^{\prime}}^{B^{n}}}\right\Vert _{1}\right\} \nonumber\\
& =\mathbb{E}_{\mathcal{C}_{k}}\left\{ \sum_{k^{\prime}\neq k}\text{Tr}\left\{ \Lambda_{k^{\prime}}^{B^{n}}\Omega_{\mathcal{C}_{k}}^{S_{B}B^{n}}\right\} \right\} \nonumber\\
& =1-\text{Tr}\left\{ \Lambda_{k}^{B^{n}}\overline{\Omega}_{k}^{S_{B}B^{n}}\right\} \nonumber\\
& \leq(1+|\mathcal{X}|)\epsilon.\label{eq:epsilon-diff-maps}$$ The first inequality follows by definitions and the triangle inequality. The first equality follows because the trace distance is invariant under isometry. The second equality follows because the operator $\Lambda_{k}^{B^{n}}\Omega_{\mathcal{C}_{k}}^{S_{B}B^{n}}$ is positive. The third equality follows from some algebra, and the second inequality follows from (\[eq:average-output-op-good\]). The private communication for all public messages $k$ and codes $\mathcal{C}_{k}$ is good $$\left\Vert \mathcal{D}_{\mathcal{C}_{k}}^{B^{n}S_{B}\rightarrow M}\left(
\Omega_{\mathcal{C}_{k}}^{S_{B}B^{n}}\right) -\pi^{M}\right\Vert _{1}\leq\epsilon,$$ because each code $\mathcal{C}_{k}$ in the random private father code is good for private communication. It then follows that$$\mathbb{E}_{\mathcal{C}_{k}}\left\{ \left\Vert \mathcal{D}_{\mathcal{C}_{k}}^{B^{n}S_{B}\rightarrow M}\left( \Omega_{\mathcal{C}_{k}}^{S_{B}B^{n}}\right) -\pi^{M}\right\Vert _{1}\right\} \leq\epsilon
.\label{eq:good-q-comm}$$ Application of the triangle inequality to (\[eq:good-q-comm\]), (\[eq:epsilon-diff-maps\]), and (\[eq:epsilon-sqrt-map\]) gives the following bound on the expected private error probability:$$\mathbb{E}_{\mathcal{C}_{k}}\left\{ p_{e,\text{priv}}\left( \mathcal{C}_{k}\right) \right\} \leq\epsilon^{\prime}\label{eq:private-error}$$ where$$\epsilon^{\prime}\equiv(1+|\mathcal{X}|)\epsilon+\sqrt{8(1+|\mathcal{X}|)\epsilon}+2\sqrt{\epsilon},$$ and where we define the private error $p_{e,\text{priv}}\left( \mathcal{C}_{k}\right) $ of the code $\mathcal{C}_{k}$ as follows:$$p_{e,\text{priv}}\left( \mathcal{C}_{k}\right) \equiv\left\Vert
\mathcal{D}_{\mathcal{C}}^{S_{B}\rightarrow M}\left( \Omega_{\mathcal{C}_{k}}^{S_{B}B^{n}}\right) -\pi^{M}\right\Vert _{1}.$$
The above random publicly-enhanced secret-key-assisted private code relies on Alice and Bob having access to a source of common randomness. We now show that they can eliminate the need for common randomness and select a good publicly-enhanced secret-key-assisted private code $\mathcal{C}$ that has a low public error $p_{e,\text{pub}}(\mathcal{C}_{k})$ and low private error $p_{e,\text{priv}}(\mathcal{C}_{k})$ for all public messages in a large subset of $[2^{nR}]$. By the bounds in (\[eq:classical-error\]) and (\[eq:private-error\]), the following bound holds for the expectation of the averaged summed error probabilities: $$\mathbb{E}_{\mathcal{C}_{k}}\left\{ \frac{1}{2^{nR}}\sum_{k}p_{e,\text{pub}}(\mathcal{C}_{k})+p_{e,\text{priv}}(\mathcal{C}_{k})\right\} \leq
\epsilon^{\prime}+(1+|\mathcal{X}|)\epsilon.$$ If the above bound holds for the expectation over all random codes, it follows that there exists a particular publicly-enhanced private father code $\mathcal{C}=\left\{ \mathcal{C}_{k}\right\} _{k\in\left[ 2^{nR}\right] }$ with the following bound on its averaged summed error probabilities:$$\frac{1}{2^{nR}}\sum_{k}p_{e,\text{pub}}(\mathcal{C}_{k})+p_{e,\text{priv}}(\mathcal{C}_{k})\leq\epsilon^{\prime}+(1+|\mathcal{X}|)\epsilon.$$ We fix the code $\mathcal{C}$ and expurgate the worst half of the private father codes—those private father codes with public messages $k$ that have the highest value of $p_{e,\text{pub}}(\mathcal{C}_{k})+p_{e,\text{priv}}(\mathcal{C}_{k})$. This derandomization and expurgation yields a publicly-enhanced private father code that has each public error $p_{e,\text{pub}}(\mathcal{C}_{k})$ and each private error $p_{e,\text{priv}}(\mathcal{C}_{k})$ upper bounded by $2\left( \epsilon^{\prime}+(1+|\mathcal{X}|)\epsilon\right) $ for the remaining public messages $k$. This expurgation decreases the public rate by a negligible factor of $\frac
{1}{n}$.
Child Protocols {#sec:children}
===============
Two simple protocols for the public-private setting are *secret key distribution* and the *one-time pad* [@V26; @S49]. Secret key distribution is a protocol where Alice creates the state $\overline{\Phi}^{AA^{\prime}}$ locally and sends the system $A^{\prime}$ through a noiseless private channel. The protocol creates a secret key and corresponds to the following resource inequality:$$\left[ c\rightarrow c\right] _{\text{priv}}\geq\left[ cc\right]
_{\text{priv}}.$$ The one-time pad protocol exploits a secret key and a noiseless public channel to create a noiseless private channel. It admits the following resource inequality:$$\left[ c\rightarrow c\right] _{\text{pub}}+\left[ cc\right] _{\text{priv}}\geq\left[ c\rightarrow c\right] _{\text{priv}}.$$
We now consider some protocols that are child protocols of the publicly-enhanced private father protocol. Consider the resource inequality in (\[eq:PEPFP-resource-ineq\]). We can combine the protocol with secret key distribution, and we recover the protocol suggested in Section 4 of Ref. [@DS03]:$$\begin{aligned}
& \left\langle \mathcal{N}\right\rangle +I\left( Y;E|X\right) _{\sigma
}\left[ cc\right] _{\text{priv}}\\
& \geq I\left( Y;B|X\right) _{\sigma}\left[ c\rightarrow c\right]
_{\text{priv}}+I\left( X;B\right) _{\sigma}\left[ c\rightarrow c\right]
_{\text{pub}}.\\
& \geq\left( I\left( Y;B|X\right) _{\sigma}-I\left( Y;E|X\right)
_{\sigma}\right) \left[ c\rightarrow c\right] _{\text{priv}}+\\
& I\left( Y;E|X\right) _{\sigma}\left[ c\rightarrow c\right]
_{\text{priv}}+I\left( X;B\right) _{\sigma}\left[ c\rightarrow c\right]
_{\text{pub}}\\
& \geq\left( I\left( Y;B|X\right) _{\sigma}-I\left( Y;E|X\right)
_{\sigma}\right) \left[ c\rightarrow c\right] _{\text{priv}}+\\
& I\left( Y;E|X\right) _{\sigma}\left[ cc\right] _{\text{priv}}+I\left(
X;B\right) _{\sigma}\left[ c\rightarrow c\right] _{\text{pub}}$$ By cancellation of the secret key term, we are left with the following resource inequality:$$\begin{aligned}
\left\langle \mathcal{N}\right\rangle +o\left[ cc\right] _{\text{priv}} &
\geq\left( I\left( Y;B|X\right) _{\sigma}-I\left( Y;E|X\right) _{\sigma
}\right) \left[ c\rightarrow c\right] _{\text{priv}}\\
& +I\left( X;B\right) _{\sigma}\left[ c\rightarrow c\right] _{\text{pub}},\end{aligned}$$ where $o\left[ cc\right] _{\text{priv}}$ represents a sublinear amount of secret key consumption.
We can combine the publicly-enhanced private father protocol with the one-time pad:$$\begin{aligned}
& \left\langle \mathcal{N}\right\rangle +I\left( Y;E|X\right) _{\sigma
}\left[ cc\right] _{\text{priv}}+I\left( X;B\right) _{\sigma}\left[
cc\right] _{\text{priv}}\nonumber\\
& \geq I\left( Y;B|X\right) _{\sigma}\left[ c\rightarrow c\right]
_{\text{priv}}+I\left( X;B\right) _{\sigma}\left[ c\rightarrow c\right]
_{\text{pub}}\nonumber\\
& +I\left( X;B\right) _{\sigma}\left[ cc\right] _{\text{priv}}\\
& \geq I\left( Y;B|X\right) _{\sigma}\left[ c\rightarrow c\right]
_{\text{priv}}+I\left( X;B\right) _{\sigma}\left[ c\rightarrow c\right]
_{\text{priv}}\nonumber\\
& =I\left( XY;B\right) _{\sigma}\left[ c\rightarrow c\right]
_{\text{priv}}\label{eq:combine-OTP}$$ This protocol is one for secret-key-assisted transmission of private information. It is not an efficient protocol because the optimal secret-key-assisted protocol [@HLB08SKP] implements the following resource inequality:$$\left\langle \mathcal{N}\right\rangle +I\left( XY;E\right) _{\sigma}\left[
cc\right] _{\text{priv}}\geq I\left( XY;B\right) _{\sigma}\left[
c\rightarrow c\right] _{\text{priv}}$$ For a channel with non-zero private capacity so that $I\left( X;B\right)
_{\sigma}-I\left( X;E\right) _{\sigma}>0$, the protocol in (\[eq:combine-OTP\]) is not efficient because it uses more secret key than necessary. This inefficiency is similar to the inefficiency that we found for combining the classically-enhanced father protocol with teleportation (See Section VII of Ref. [@HW08GFP]). It is not surprising that this inefficiency occurs because the publicly-enhanced private father protocol is the public-private analog of the classically-enhanced father protocol and the one-time pad protocol is the public-private analog of the teleportation protocol [@CP02].
Conclusion
==========
We have introduced an optimal protocol, the publicly-enhanced private father protocol, that exploits a secret key and a large number of independent uses of a noisy quantum to transmit public and private information. Several protocols in the literature are now special cases of this protocol.
A few open questions remain. It remains to determine the capacity regions of a multiple-access quantum channel [@YHD05ieee; @itit2008hsieh] and a broadcast channel [@YHD2006] for transmitting public and private information while consuming a secret key. One might also consider the five-dimensional region corresponding to the scenario where Alice and Bob consume secret key, entanglement, and a noisy quantum channel to produce quantum communication, public classical communication, and private classical communication. This scenario might give more insight into the privacy/coherence correspondence. It remains open to determine the full triple trade-off for the use of a quantum channel in connection with public communication, private communication, and secret. We have made initial progress on this problem by exploiting techniques developed in Ref. [@HW09T3]. Before completing this work, we need to determine a publicly-assisted private mother protocol, the analog of the classically-assisted mother protocol in Refs. [@DHW05RI; @HW09T3]. This protocol should then allow us to determine the full triple trade-off for both the dynamic setting and the static setting.
The authors thank Igor Devetak for a private discussion regarding the issue in Section \[sec:relative-resource-priv\] with the protocol for private communication. MMW acknowledges partial support from an internal research and development grant SAIC-1669 of Science Applications International Corporation.
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\*\*\* Withdrawn Message \*\*\*
===============================
Please note that this paper was presented in July during DARK98-Second International Conference on Dark Matter in Astro and Particle Physics (Heidelberg, Germany); nowadays there are more details and crucial developments in the last OPERA paper: CERN/SPSC 98-25, LNGS-LOI 8/97, Add.1 Addendum to the Letter of Intents, Oct 9, 1998, that you can also find in http://www1.na.infn.it/wsubnucl/accel/neutrino/opera.html.
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---
abstract: |
Comparing two geometric graphs embedded in space is important in the field of transportation network analysis. Given street maps of the same city collected from different sources, researchers often need to know how and where they differ. However, the majority of current graph comparison algorithms are based on structural properties of graphs, such as their degree distribution or their local connectivity properties, and do not consider their spatial embedding. This ignores a key property of road networks since the similarity of travel over two road networks is intimately tied to the specific spatial embedding. Likewise, many current algorithms specific to street map comparison either do not provide quality guarantees or focus on spatial embeddings only. Motivated by road network comparison, we propose a new path-based distance measure between two planar geometric graphs that is based on comparing sets of travel paths generated over the graphs. Surprisingly, we are able to show that using paths of bounded link-length, we can capture global structural and spatial differences between the graphs.
We show how to utilize our distance measure as a local signature in order to identify and visualize portions of high similarity in the maps. And finally, we give an experimental evaluation of our distance measure and its local signature on street map data from Berlin, Germany and Athens, Greece.
author:
- 'Mahmuda Ahmed Brittany Terese Fasy Kyle S. Hickmann Carola Wenk'
bibliography:
- 'gs\_tsas.bib'
title: 'A Path-Based Distance for Street Map Comparison'
---
This work is supported by the National Science Foundation, under grant CCF-1301911.
Author’s addresses: M. Ahmed, Computer Science Department, University of Texas at San Antonio; B. T. Fasy [and]{} C. Wenk, Computer Science Department, Tulane University; K. Hickmann, Center for Computational Science, Tulane University
Introduction
============
Street Map Graphs {#sec-background}
=================
Path-Based Distance {#sec-distance}
===================
Experimental Results {#sec-experiments}
====================
Conclusion and Future Work
==========================
This work has been supported by the National Science Foundation grants CCF-0643597 and CCF-1216602. We thank Dieter Pfoser for providing the TeleAtlas maps, Sophia Karagiorgou for helping with data conversion, James Biagioni for providing his code, and the anonymous referees for providing thoughtful feedback.
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---
abstract: 'In order to probe a possible time variation of the fine-structure constant ($\alpha$), we propose a new method based on Strong Gravitational Lensing and Type Ia Supernovae observations. By considering a class of dilaton runaway models, where $\frac{\Delta \alpha}{\alpha}= - g \ln{(1+z)}$ ($g$ captures the physical properties of the model), we obtain constraints on $\frac{\Delta \alpha}{\alpha}$ at the level of $g \approx 10^{-2}$. Since the data set covers the redshift range $0.075 \leq z \leq 2.2649$, the constraints derived here provide independent bounds on a possible time variation of $\alpha$ at low, intermediate and high redshifts.'
author:
- 'L. R. Colaço$^{1}$'
- 'R. F. L. Holanda$^{1}$'
- 'R. Silva$^{1,2}$'
title: 'Probing variation of the fine-structure constant using the strong gravitational lensing'
---
Introduction
============
The Hypothesis of Large Numbers (HLN), proposed a long time ago by Paul Dirac in Ref. [@Dirac:1938mt], has opened possible approaches associated with a variation of the constants of nature. An early investigation addressed the gravitational constant ($G$), as main result, a possible temporal dependence of $G$ was ruled out in Ref. [@Teller:1948zz]. he HLN has gained a lot of attention with the advance of the many experimental breakthroughs and, in the last decades, the Dirac’s hypothesis has been tested in many physical contexts, e.g. by using geological evidence, no variation in $G$ was also found by investigating the effects on the evolution and asteroseismology of the low-mass star KIC 7970740 [@Bellinger:2019lnl], considering the Earth-Moon system, experiments have provided an upper bound, such as $\dot{G}/G=0.2 \pm 0.7 \mbox{x}10^{-12}$ per year [@Muller]. From the string theory and other theories of modified gravity standpoint, on the other hand, $G$ assumes a variable gravitational constant, being a derived parameter [@Uzan:2010pm; @Chiba:2011bz]. Moreover, due to the possibility of dynamical constants, including other fundamental constants, some theories based on extra dimensions have also been discussed [@Chodos:1979vk; @Kolb:1985sj; @Nath:1999fs]. It is important to stress that the General Relativity discards a dynamical fundamental constant due to the violation of the Equivalence Principle [@Bekenstein:1982eu].
Yet, some observational measurements have been considered to investigate a possible variation of the fine-structure constant ($\alpha = e^2/\hbar c$, where $e$ the elementary charge, $\hbar$ the Planck’s constant, and $c$ the speed of the light, respectively). The absorption spectra of quasars, for instance, has been used to explore a possible cosmological time variation of $\alpha$ [@11; @12; @99; @03; @2930; @Martins:2017yxk]. The constraints on the magnitude of a possible time variation of $\alpha$ are also obtained through other observational techniques, e.g., the rare-earth element abundance data from Oklo [@Damour:1996zw]. More recently, by using the physics of the cosmic microwave background (CMB), it was possible to use CMB anisotropies in order to test models with varying $\alpha$. From the Planck satellite [@Planck1; @Planck2] data, experiments of South Pole Telescope [@SPT1; @SPT2] and Atacama Cosmology Telescope [@ACT], it was obtained that the difference between the $\alpha$ today and at recombination is $\delta \alpha/ \alpha \leq 7.3 \times 10^{-3}$ at $68 \%$ Confidence Level [@Avelino:2001nr; @Martins:2003pe; @Rocha:2003gc; @Ichikawa:2006nm; @Menegoni:2009rg; @Galli:2010it; @Menegoni:2012tq; @Ade:2014zfo; @deMartino:2016tbu; @Hart:2017ndk]. However, this limit obtained from the CMB physics is inferred considering a specific cosmological model (flat $\Lambda$CDM), being weakened by opening up the parameter space to variations of the number of relativistic species or the helium abundance. Indeed, the effects of a spatial variation of $\alpha$ on the CMB spectrum can be found (see e.g. [@Tristan] and references therein). A possible time variation of the fine structure constant also can be explored during the Big Bang nucleosynthesis (BBN)[@Mosquera], as well as in the context of a supermassive black hole in the Galactic Center with a high gravitational potential, which used the data of late-type evolved giant stars from the S-star [@Hees2020].
Particularly, the low-energy string theory models predict the existence of a scalar field called dilaton, a spin-2 graviton scalar partner [@damour1; @damour2]. In this context, the runaway of the dilaton towards strong coupling can lead to temporal variations of $\alpha$. This possible variation/evolution of the fine-structure constant at low and intermediate redshifts can be given by [@Martins:2017yxk]
$$\frac{\Delta \alpha}{\alpha} \approx -\frac{1}{40}\beta_{had,0} \phi_{0}^{'} \ln{(1 +z)} \approx -g \ln{(1 +z)},$$
where $\beta_{had,0}$ is the current value of the coupling between dilaton and hadronic matter, and $\phi'_0$ is defined by $\partial \phi /\partial \ln{(a)}$ measured today. The dilaton runaway models and chameleon models has not been completely ruled out by the experiments that test violations on the weak equivalence principle [@Khoury; @Brax; @Mota; @Martins:2017yxk].
Constraints on the dilaton Runaway Model by using Galaxy clusters measurements have been proposed in order to probe a possible temporal variation in $\alpha$ (see also the Ref.[@martins2015] for other contexts). The Ref. [@holanda1], for instance, introduced a method capable of probing a possible variation in $\alpha$ only by using Galaxy Cluster (GC) gas mass fraction measurements. The constraint obtained in $g \equiv \frac{1}{40}\beta_{had,0}\phi_{0}^{'} $ was $g = 0.065 \pm 0.095$ ($1\sigma$ c.l.). By using angular diameter distance of GC and luminosity distance of type Ia supernovae, a possible temporal variation in $\alpha$ was also investigated, being obtained $g = -0.037 \pm 0.0157$ at $1 \sigma$ c.l. [@holanda2]. Several other tests capable of probing $\alpha$ with galaxy cluster data have been emerging since then (see e.g. [@holanda3; @martinsgc] and references therein).
In this work, by assuming a flat universe, it is discussed by the first time the role of the Strong Gravitational Lensing (SGL) on a possible temporal variation of the fine-structure constant. The method is performed by using combined measurements of SGL systems and Type Ia Supernovae (SNe Ia). This new approach used 95 pair of observations (SGL-SNe Ia) covering the redshift ranges $0.075\leq z_l \leq 0.722$ and $0.2551 \leq z_s \leq 2.2649$. These data are considered in order to put limits on the $g$ parameter, considering dilaton runaway models (see Eq.1). The approach developed here offers new limits on the $g$ parameter using observations in higher redshifts than those from galaxy clusters ($z\approx 1$). In the following, the work is organized as follows: in section II we describe the method developed to probe $\Delta \alpha /\alpha$ and other peculiarities; in section III we describe the data used for the purpose, while in section IV contains the analysis and discussions; and finally, in session V, the conclusions of this paper.
Methodology
===========
Strong Gravitational Lensing Systems
------------------------------------
Strong gravitational Lensing systems, one of the predictions of GR [@lentes], have recently become a powerful astrophysical tool capable of investigating gravitational and cosmological theories, measuring various cosmological parameters and investigating fundamental physics. For example, time-delay measurements of gravitational lensings can be used to measure the Hubble constant [@kocha], and the Cosmic Diameter Distance Relation (CDDR) [@rana]. Other statistical properties of SGL can restrict the deceleration parameter of the universe [@gott], space-time curvature [@jzqi; @ranacurv], also departures of CDDR [@czruan; @holg], the cosmological constant [@fuku], the speed of light [@luz], and others. It is a purely gravitational phenomenon that occurs when the source ($s$), lens ($l$), and observer ($o$) are at the same signal line forming a structured ring called the Einstein radius ($\theta_E$) [@sef]. In the cosmological scenario, a lens can be a foreground galaxy or cluster of galaxies positioned between a source-—Quasar, as a lens, the multiple-image separation from the source only depends on the lens and source angular diameter distance.
However, the system depends on a model for mass distribution. On the assumption of the singular isothermal sphere (SIS) model, the Einstein radius $\theta_E$ is given by [@lentes]
$$\theta_E = 4\pi \frac{D_{A_{ls}}}{D_{A_{s}}} \frac{\sigma_{SIS}^{2}}{c^2},$$
where $D_{A_{ls}}$ is the angular diameter distance of the lens to the source, $D_{A_{s}}$ the angular diameter distance of the observer to the source, $c$ the speed of light, and $\sigma_{SIS}$ the velocity dispersion caused by the lens mass distribution. It is important to note here that $\sigma_{SIS}$ is not exactly equal to the observed stellar velocity dispersion ($\sigma_0$) due to a strong indication, via X-ray observations, that dark matter halos are dynamically hotter than luminous stars. Thus, taking this fact into account, a purely phenomenological free parameter is introduced: $f_e$[^1], that is, $\sigma_{SIS}^{2} = f_e \sigma_{0}^{2}$, where $\sqrt{0.8}<f_e<\sqrt{1.2}$ (see details in [@ofek]). However, $f_e$ is treated as a free parameter in our analysis.
A method developed by Holanda et al. (2017) [@Hol] provided a powerful test for the CDDR using SGL systems and SNe Ia. The method is based on equation (2) for lenses and an observational quantity defined by
$$D \equiv \frac{D_{A_{ls}}}{D_A} = \frac{ \theta_E c^2}{4\pi \sigma_{SIS}^{2}}.$$
By assuming a flat universe with the comoving distance between the lens and the observer as $r_{ls} = r_s-r_l$ [@bartel], and using the relations $r_s = (1 + z_s) D_{A_s}$, $r_l = (1 + z_l) D_{A_l}$, $r_{ls} = (1 + z_s) D_{A_{ls}}$, it is possible to obtain
$$D = 1-\frac{(1+z_l)}{(1+z_s)}\frac{D_{A_l}}{D_{A_s}}.$$
By assuming possible departures of CDDR through $D_LD_{A}^{-1}(1+z)^{-2}=\eta (z)$, the previous equation can be rewritten by
$$D=1-\frac{(1+z_s)}{(1+z_l)} \frac{D_{L_l}}{D_{L_s}}\frac{\eta (z_s)}{\eta (z_l)}.$$
However, it is important to stress that the method provided by [@Hol] did not investigate a possible variation of the fine structure constant on SGL observations. Furthermore, as shown in [@hees; @Minazzoli; @hees2], for some class of models a variation of $\alpha$ necessarily leads to a violation of the CDDR.
Here, we extend the method proposed in [@Hol] in order to investigate the effect of varying $\alpha$ and the departure of CDDR. Thus, by the definition of the fine structure constant, $\alpha = e^2/\hbar c$, the equation (3) becomes
$$D\equiv \frac{D_{A_{ls}}}{D_{A_{s}}}=\frac{e^4\theta_E}{4\pi \alpha^2 \hbar^2 \sigma_{SIS}^{2}}.$$
Theoretical Model
-----------------
In the modified gravity theories, which is associated with the presence of a scalar field with non-minimal multiplicative coupling to the usual electromagnetic Lagrangian, the entire electromagnetic sector is modified [@hees; @Minazzoli; @hees2]. Hence, the fine-structure constant and the cosmic distance duality relation should change with cosmological time, and both are intimately and unequivocally related to each other by
$$\frac{\Delta \alpha}{\alpha} \equiv \frac{\alpha(z)-\alpha_0}{\alpha_0} = \eta^2(z) - 1.$$
If we consider $\alpha (z)= \alpha_0 \phi (z) $, where $\alpha_0$ is the current value of the fine-structure constant, and $\phi (z)$ a scalar field that controls the variation of $\alpha$, the relation (7) gives $\phi (z) = \eta^2 (z)$. Thus, the equations (5) and (6) can be rewritten, respectively, by:
$$D=1-\frac{(1+z_s)}{(1+z_l)} \frac{D_{L_l}}{D_{L_s}} \Bigg( \frac{\phi (z_s)}{\phi (z_l)} \Bigg)^{1/2},$$
and $$D = \frac{e^4\theta_E}{4\pi\alpha_{0}^{2}\hbar^2 \sigma_{SIS}^{2} }\phi^{-2}(z_s) = D_0\phi^{-2}(z_s),$$ where $D_0 \equiv e^4\theta_E /4 \pi\alpha_{0}^{2}\hbar^2 \sigma_{SIS}^{2}$. The procedure makes $D$ much more homogeneous for the lensing sample located at different redshifts. Therefore, combining the equations (8) and (9) it is possible to obtain
$$D_0\phi^{-2} (z_s)= 1-\frac{(1+z_s)}{(1+z_l)} \frac{D_{L_l}}{D_{L_s}} \Bigg( \frac{\phi (z_s)}{\phi (z_l)} \Bigg)^{1/2}.$$
Luminosity Distance
-------------------
Now, let us consider the pair of luminosity distances for each SGL system, which is obtained from the SNe Ia sample called Pantheon [@pantheon]. It is worth to mention that this is the most recent wide refined sample of SNe Ia observations found in the literature. The compilation consists of 1049 spectroscopically confirmed SNe Ia and covers a redshift range of $0.01\leq z \leq 2.3$. However, to perform the appropriate tests, it must be used SNe Ia at the same (or approximately) redshift of the lens system. Thus, for each lens system, it is necessary to make a selection of SNe Ia according to the criterion: $| z_s-z_{SNe}|
\leq 0.005$ and $| z_l-z_{SNe}| \leq 0.005$. Then, we perform the weighted average by for each system by [@Hol]:
$$\bar{\mu} = \frac{\sum_i \mu_i/\sigma_{\mu_i}^{2}}{\sum_i 1/\sigma_{\mu_i}^{2}},$$
$$\sigma_{\bar{\mu_i}}^2 = \frac{1}{\sum_i 1/ \sigma_{\mu_i}^{2}},$$
where $\mu_i (z)$ is the distance module of SNe. Hence, the luminosity distance follows $D_L (z) = 10^{(\bar{\mu}-25)/5}$ $[Mpc]$, and its error is given by error propagation, $\sigma_{D_L}^{2} = ( \partial D_L / \partial \bar{\mu} )^2\sigma_{\bar{\mu}}^{2}$ [@AA] (see Figure 1).
![Luminosity distances of spectroscopically confirmed SNe Ia from Pantheon compilation.[]{data-label="fig4"}](figura2.pdf)
Samples
=======
For our analysis, we consider a subsample built from two compilation of SGL systems. The first compilation is composed of 118 SGL from the Sloan Lens ACS, BOSS Emission-Line Lens Survey (BELLS), Lenses Structure and Dynamics Survey (LSD), and Strong Legacy Survey SL2S [@cao2015]. The lens mass is assumed to be spherically and symmetrically distributed, covering redshift intervals of $0.075 \leq z_l \leq 1.004$ and $0.20 \leq z_s\leq 3.60$. The second compilation consists of 34 new SGL systems pre-selected by [@zltu]. In fact, this sub-sample is obtained through a careful analysis: 7 reliable BELLS data by [@shu]. Actually, there are 17 SGL systems provided by [@shu], but ten systems were excluded for presenting unknown systematic errors; and 27 SLACS data by [@shu2]. There are 40 new SGL systems provided by Ref. [@shu2], but only 27 systems have been carefully studied with $\chi_{red}^{2} \approx 1$ achieved (see Table 1 in [@zltu]).
![The $D_0$ data related to the source redshift from SGL samples for each system considered.[]{data-label="fig3"}](figura1.pdf)
We consider the general approach to describe the lensing systems: the one with spherically symmetric mass distribution in lensing galaxies in favor of power-law index $\gamma$, $\rho \propto r^{-\gamma}$ (Power law model - PLAW model henceforward). This kind of model is important since recent several studies have shown that slopes of density profiles of individual galaxies show a non-negligible scatter from the SIS model [@elilensing]. Under this assumption, part of the equation (9) is written by:
$$D_0 = \frac{e^4 \theta_E}{\alpha_{0}^{2} \hbar^2 4\pi \sigma_{ap}^{2}}f(\theta_E, \theta_{ap}, \gamma),$$
where $f (\theta_E, \theta_{ap}, \gamma)$ is a complex function which depends on the Einstein’s radius $\theta_E$, the angular aperture $\theta_{ap}$, used by certain gravitational lensing surveys, and the power-law index $\gamma$. If $\gamma = 2$, it resumes the SIS model. In this paper, the factor $\gamma$ is approached as a free parameter [^2]The uncertainty related to quantity (13) is given by:
$$\sigma_{D_0} = D_0\sqrt{4\Bigg( \frac{\sigma_{\sigma_{ap}}}{\sigma_{ap}} \Bigg)^2 + (1-\gamma)^2 \Bigg( \frac{\sigma_{\theta_E}}{\theta_E} \Bigg)^2}.$$
Following the approach taken by [@grillo], Einstein’s radius uncertainties follow $\sigma_{\theta_E} = 0.05 \theta_E$ ($5\%$ for all systems) and $\sigma_{SIS} = f_e \sigma_{ap}$.
Therefore, the complete sample consists of 152 SGL systems covering a wide range of redshift. However, not all the SGL systems have the corresponding pair of luminosity distances via SNe Ia that obey the criteria $| z_s-z_{SNe}| \leq 0.005$ and $| z_l-z_{SNe}| \leq 0.005$. By excluding these systems, we ended up with 95 pairs of observations (SGL-SNe Ia) in our analysis, which cover ranges of redshifts $0.075\leq z_l \leq 0.722$ and $0.2551 \leq z_s \leq 2.2649$ (see Figure 2).
Analysis and Discussions
========================
Data set $ g $
-------------------------------------------------------- -----------------------------
Gas Mass Fractions$^*$ [@holanda1] $+0.065 \pm 0.095$
Angular Diameter Distance$^*$ plus SNe Ia [@holanda2] $-0.037 \pm 0.157$
Gas Mass Fractions$^*$ plus SNe Ia [@holanda3] $+0.008\pm 0.035$
Gas Mass Fractions$^*$ plus SNe Ia [@holanda3] $+0.018\pm 0.032$
Gas Mass Fractions$^*$ plus SNe Ia [@holanda3] $+0.010 \pm 0.030$
Gas Mass Fractions$^*$ plus SNe Ia [@holanda3] $+0.030\pm 0.033$
$Y_{SZ}D_{A}^{2}/Y_X$ scaling-relation$^*$ [@leonardo] $-0.15\pm 0.10$
This work $- 0.013_{- 0.09}^{+ 0.08}$
: A summary of current constraints on a possible time evolution of $\alpha$ for a class of dilaton runaway models ($\Delta \alpha / \alpha = -g \ln{(1+z)}$) by using galaxy cluster observations and SNe Ia measurements jointly with that obtained in this work. The symbol \* denotes galaxy cluster data.
![Posterior probability distribution of free parameters $g$, $\gamma$, and $f_e$ for PLAW model. The red color represents analyzes without intrinsic error and the blue color represents analyzes with an intrinsic error.[]{data-label="fig2"}](statistics.pdf)
We used Markov Chain Monte Carlo (MCMC) methods to calculate the posterior probability distribution functions (pdf) of free parameters [@foreman]. The analysis consists on the general case, PLAW model, where the free parameter space is $\Theta = (g, \gamma, f_e)$. Thus, the likelihood distribution function is given by:
$$\mathcal{L} (Data|\Theta) = \prod \frac{1}{\sqrt{2\pi} \sigma_{\mu}} exp \Bigg( -\frac{1}{2} \chi^2 \Bigg),$$
where
$$\chi^2 =\Bigg( \frac{D_0 - \phi^{2}(z_s)Y}{\sigma_T} \Bigg)^2,$$
$$Y \equiv 1-\frac{(1+z_s)}{(1+z_l)}\frac{D_{L_l}}{D_{L_s}} \Bigg( \frac{\phi(z_s)}{\phi(z_l)} \Bigg)^{1/2},$$
and $$\sigma_T = (\sigma_{D_0}^{2} + \sigma_{Y}^{2})^{1/2}$$ the associated error. The pdf posteriori is proportional to the product between the likelihood and the prior,
$$P(\Theta |Data) \propto \mathcal{L} (Data|\Theta)XP_0(\Theta).$$
In this analysis, we assume an informative prior: $-1.0 \leq g \leq +1.0$; $1.5 \leq \gamma \leq 2.5$; $\sqrt{0.8} \leq f_e \leq \sqrt{1.2}$.
In Figure 3, the results considering PLAW Model are plotted. In this case, not only the fine-structure constant is constrained, but the power-law index $\gamma$, and the parameter $f_e$ are restricted as well. We obtain at $1\sigma$ ($68.3\%$): $g = -0.067_{- 0.07}^{+ 0.06}$, $\gamma = 2.003_{- 0.06}^{+ 0.06}$, $f_e = 1.012_{- 0.03}^{+ 0.03}$ with $\chi^{2}_{red}\approx 2.215$ (see red contours). This $\chi^{2}_{red}$ indicates the presence of an unknown intrinsic error. We estimate this intrinsic error to be approximately $15\%$. Hence, taking it into consideration, it is possible to obtain: $g = - 0.013_{- 0.09}^{+ 0.08}$, $\gamma = 1.915_{- 0.07}^{+ 0.08}$, and $f_e = 1.057_{- 0.05}^{+ 0.05}$ at $1\sigma$ with $\chi^{2}_{red}\approx 1$ (see blue contours).
Table I shows bounds on $g$ derived in this paper along with other recent constraints obtained from galaxy clusters and SNe Ia observations. As one may see, our results are in full agreement with the previous ones from galaxy clusters plus SNe Ia analyses and indicate no significant variation of the fine structure constant $\alpha$.
Conclusions
===========
The search and understanding of a possible temporal or spatial variation of the fundamental constants of nature have become important in recent years due to the possibility of a new and fascinating physics waiting to be discovered. In this paper, a new technique was proposed to investigate a possible time variation of the fine structure constant, such as $\alpha(z)=\alpha_0 \phi(z)$ by using recent measurements of SGL systems and SNe Ia observations. A possible time variation of $\alpha$ was investigated in a class of runaway dilaton models, where $\phi(z)=1-g\ln(1+z)$.
As we have already discussed, considering the general spherical isothermal model with power-law index ($\rho \propto r^{-\gamma}$) describing the mass distribution in lensing galaxies in the SGL systems, the following constraints have been performed, i.e, $g = -0.067_{- 0.07}^{+ 0.06}$ and $\gamma = 2.003_{- 0.06}^{+ 0.06}$ with $\chi^{2}_{red} \approx 2.215$. In addition, by using an intrinsic error on the SGL data ($\approx 15 \%$), it was possible to obtain: $g = - 0.013_{- 0.09}^{+ 0.08}$ and $\gamma = 1.915_{- 0.07}^{+ 0.08}$ with $\chi^{2}_{red} \approx 1$. These results are in full agreement with the standard cosmology. Finally, although SGL systems data not to be competitive with the limits imposed by quasar absorption systems, the constraints imposed in this paper provide new and independent limits on a possible time variation of the fine structure constant.
Acknowledgments
===============
The authors thank Brazilian scientific and financial support federal agencies, CAPES, and CNPq. RS thanks CNPq (Grant No. 303613/2015-7) for financial support. This work was supported by the High-Performance Computing Center (NPAD)/UFRN.
[99]{}
.
\[lastpage\]
[^1]: This parameter takes into account systematic errors, and/or even unknown intrinsic errors for addressing $\sigma_0$ instead of $\sigma_{SIS}$.
[^2]: This method is widely used in the literature [@qxia; @zli; @xli].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Explicit formulae are given for the Airy and Bessel bispectral involutions, in terms of Calogero-Moser pairs. Hamiltonian structure of the motion of the poles of the operators is discussed.'
address: |
Department of Mathematics\
University of Georgia\
Athens, GA 30602. Temporary Address: Department of Mathematics\
Harvard University\
Cambridge, MA\
02138
author:
- Mitchell Rothstein
title: 'Calogero-Moser pairs and the Airy and Bessel bispectral involutions'
---
\[section\] \[Def\][Proposition]{} \[Def\][Lemma]{} \[Def\][Remark]{} \[Def\][Example]{} \[Def\][Theorem]{} \[Def\][Corollary]{}
¶[P]{}
[^1]
Introduction
============
This paper follows upon the study of the Airy bispectral involution made in [@KR]. There we gave an analogue, for arbitrary rank, of the rank-one bispectral involution developed by Wilson [@W1]. Recently, [@W2], Wilson has established a relationship between the rank-one bispectral involution and the complex analogue of the Calogero-Moser phase space. This relationship leads to explicit formulae for the Baker function and the corresponding involution, which make many important features manifest. As shown below, similar results hold for bispectral algebras obtained from generalized Airy and Bessel operators.
Given a positive integer $n$, define $C_n$ to be the quotient, under conjugation by $Gl(n,\C)$, of the space of pairs ([*Calogero-Moser pairs*]{}) of $n\times n$ complex matrices, $(P,Q)$, such that $$\label{cm}
\text{rank}([P,Q]-I)=1\ .$$ This is the complex analogue of the definition in [@KKS], in which $P$ and $Q$ are taken to be hermitian, and $I$ is replaced by $iI$. Define $\grad$ (see [@W1; @W2]) to be the subspace of the Sato grassmannian ([@DJKM; @Sa; @SW]) corresponding to Krichever data whose spectral curve is rational and unicursal (no nodes). The Baker function for such data is always of the form $e^{xz}p(x,z)$, where $p(x,z)$ is rational and separable. By separable we mean its denominator is a product $q(z)\tau(x)$. Wilson has proved
There is a one-to-one correspondence $$\label{correspondence}
\cup_{n=0}^{\infty}C_n\leftrightarrow \grad\ ,$$ such that a point $W\in \grad$ corresponds to a point $(P,Q)$ if and only if its Baker function, $\psi_W$ is given by $$\label{baker formula}
\psi_W(x,z)=e^{xz}\text{Det}(I-(zI-Q)^{-1}(xI-P)^{-1})\ .$$
Remark: The “generic” rational unicursal curve has simple cusps. These correspond to $Q$ with distinct eigenvalues. In the hermitian case this is automatic. An important aspect of Wilson’s theorem is the observation that in the complex case, nonsemisimple $Q$ correspond to nonsimple cusps, or in physical terms, collision of the Calogero-Moser particles, which are now moving in $\Bbb C$ rather than $\Bbb R$.
The spectral algebra $\cal R_W$ is the algebra of differential operators $L(x,\d)$ such that $L\psi_W=f(z)\psi_W$ for some function $f(z)$. Wilson’s result proves that the spectral algebra of any point $W\in\grad$ is bispectral in the sense of [@DG]. That is, the spectral algebra of $\psi_W(z,x)$ is also nontrivial.
It proves much more, for it says that $\psi_W(z,x)$ is also a Baker function, namely the one corresponding to $(Q^T,P^T)$. The involution $(P,Q)\mapsto (Q^T,P^T)$ is clearly antisymplectic with respect to the symplectic form $$\omega=tr(dP\wedge dQ)\ .$$ This symplectic structure is an important example of “unreduction” [@KKS]. Namely, the Calogero-Moser hierarchy [@AMM] is a completely integrable hamiltonian system defined on the quotient space $C_n$. The hamiltonians are rather complicated in the reduced variables, but on the level of matrices $(P,Q)$ they are given simply by the hamiltonians are $h_n=\tr(P^n)$. Moreover, the involution $((P,Q))\too((Q^T,P^T))$ is the linearizing map for the Calogero-Moser particle system [@AMM; @KKS]. Thus, Wilson’s result gives the best proof that this linearizing map and the bispectral involution are one and the same (cf. [@Ka]). As we shall see, the corresponding involutions, in terms of an auxilliary monic polynomial $\rho(t)$, are given in the Airy and Bessel cases respectively by $$\begin{aligned}
(P,Q)&\mapsto (P^T,\rho(P^T)-Q^T)\ ,\\
(P,Q)&\mapsto (
(QP\rho(QP)^{-1} Q)^T,(Q^{-1}\rho(QP))^T)\ .\end{aligned}$$
Acknowledgments: The author wishes to acknowledge valuable discussions with J. Harnad, A. Kasman, R. Varley and G. Wilson.
The Airy case {#airy case}
=============
The rank of a commutative algebra $\cal R$ of ordinary differential operators is defined to be the greatest common divisor of the orders of its elements. The true rank of $\cal R$ is the rank of its centralizer [@LP; @PW]. For instance, fix a monic $r$th order polynomial $\rho(t)$, and define $$\lair=\rho(\d)-x\ \ \ \ \ \text{(Airy)}\label{airyop}$$ Then $\Bbb C[\lair]$ has true rank $r$ [@KR]. Moreover, this algebra is bispectral. Indeed, for any $f\in ker(\lair)$, define $$\fair(x,z)=f(x+z)\ .$$ Then $$\lair(\fair)=z\fair \ ,
\ \lair(z,\d_z)(\fair)=x\fair\ .\label{airy is bispectral}$$
To generalize Wilson’s formula to this case, introduce a [*Baker functional*]{}, $\psiair$, defined on $ker \lair$, of the form $$\Psi_{\text{Ai}}(f)=\sum_{i=0}^{r-1}k_i(x,z)\d^i(\fair)
\ .\label{airy functional}$$ To specify its properties, it is useful to introduce the dual description of $\grad$ (cf. [@W1; @KR]). As Wilson shows, every point of $W\in\grad$ arises from a homogeneous, finite dimensional space of finitely supported distributions in the complex plane. That is, there should be complex numbers $\lambda_1, \dots ,
\lambda_n$ and polynomials over $\Bbb C$, $\ell_1, \dots , \ell_n$, such that $$W=\frac 1{q(z)}\{\ p(z)\in\Bbb C[z]\ |\ c_i(p)=0\ \}\ ,$$ where $$q(z)=\Pi_i(z-\lambda_i)$$ and $$c_i=\delta_{\lambda_i}\circ\ell(\d_z)\ .$$
Now define $\psiair$ by the following properties. Let $C$ denote the span of $c_1,...,c_n$.
[ Property 1a:]{} The functions $q(z) k_i(x,z)$ are polynomial in $z$. [ Property 2a:]{} For all $f\in Ker(L)$, $q_C(z)\psiair(f)$ is annihilated by all $c\in C$. [ Property 3a:]{} $\lim\limits_{z\to\infty}(k_0,...,k_{r-1})=(1,0,...,0)$. Fixing the operator $\lair$, set $$\label{airy vector}
\vec{k}_W=\left\lgroup\begin{matrix} k_0\\ \vdots
\\k_{r-1}\end{matrix}\right\rgroup.$$ It is important to remark that the map $C\mapsto W$ is not one-to-one. The equivalence relation on $C$ induced by this map is the one generated by $$\label{equivalence relation}
C + \Bbb C\delta_{\lambda} \sim C \circ
(z -
\lambda)\ .$$ In particular, the properties of $\psiair$ depend only on $W$ and not on the representative $C$.
Choose a space of conditions $C$ defining $W$, and set $k_i(x,z)=\frac 1{q_C(z)}\sum_jk_{i,j}(x)z^j$. Consider the differential operator $$K_{\text{Ai},C}=\sum_{i,j} k_{i,j}(x)\d^i\lair^j\ .$$ Defining $$\label{airy vector f}
\vecair(x,z)=(\fair,\d(\fair),...,\d^{r-1}(\fair))\ ,$$ one has $$K_{\text{Ai},C}(\fair)=q_C(z) \vecair\cdot\vec{k}_W\ .$$ The asympotics of $\vec{k}_W$ imply that $K_{\text{Ai},C}$ is a monic operator of order $rn$. Property 2a implies that $K_{\text{Ai},C}$ annihilates $c(\fair)$ for all the distributions $c\in C$ and all $f\in Ker(\lair)$. Thus $K_{\text{Ai},C}$ is unique operator with the two properties just stated. It then follows (cf. [@KR]) that for any polynomial $p\in R_W$, the pseudodifferential operator $M_p=K_{\text{Ai},C}p(\lair)K_{\text{Ai},C}^{-1}$ is a differential operator, and $$M_p(K_{\text{Ai},C}(\fair))=p(z)K_{\text{Ai},C}(\fair)$$ for all $f\in Ker(\lair)$. Note that $M_p$ could also have been obtained by conjugating $p(\lair)$ by the monic $0$th order pseudodifferential operator $K_{\text{Ai},C}q(\lair)^{-1}$. The latter operator is independent of the space $C$ representing $W$, and is the analogue of the Sato operator in this theory.
Thus one has a rank $r$ commutative algebra of differential operators $$\label{airy algebra}
\cal R_{\text{Ai},W}=\{\ M_p\ |\ p\in\ R_W\ \}\ ,$$ with an $r$-dimensional space of eigenfunctions $\psiair(f)$, $f\in Ker(L)$. To accomodate the usual normalization of spectral algebras, namely that the subprincipal symbol should vanish, take $\rho$ of the form $\rho(t)=t^r+O(t^{r-2})$.
The task is to obtain a formula for $\vec{k}_W$ in terms of the matrices $P$ and $Q$ corresponding to the point $W$. The following lemma expresses property 2a in terms of covariant differentiation of $\vec {k}_W$. Set $$\label{first order airy system}
\bair(x,z)=
\left[\begin{matrix}
0&&\dots&&0&x+z\\
1&\ddots&&&&a_1\\
0&\ddots&&&\vdots&a_2\\
\vdots&\ddots&&&&\vdots\\
&&&&0&a_{r-2}\\
0&&\dots&0&1&0
\end{matrix}\right]\ ,$$ where the $a_i$’s are the coefficients of $\rho$, and set $$\label{nabla}
\nabair=\frac{\d}{\d z}+\bair(x,z)\ .$$
\[covariant airy\] Let $c$ be a distribution of the form $c=\delta_{\lambda}\circ p(\d_z)$. Let $\vec{g}=
\left[\begin{matrix} g_0(x,z)\\
\vdots\\ g_{r-1}(x,z)\end{matrix}\right]$ be a vector of polynomials in $z$ with coefficients in $\C(x)$. Then $c(\vecair\cdot\vec{g})=0$ for all $f\in Ker(L)$ if and only if $\vec{g}$ is annihilated by $\delta_{\lambda}\circ p(\nabair)$.
Given $f\in Ker(L)$, $$\d_z(\vecair)=\vecair\cdot\bair(x,z)\ .$$ Thus, for all $j$, $$\delta_{\lambda}\circ\d_z^j(\vecair\cdot\vec{g})=
\vecair(x,\lambda)\cdot(\d_z+\bair(x,\lambda))^j(\vec{g})\ .$$ This proves the lemma, since there is no differential equation of order less than $r$ satisfied by $f(x+\lambda)$ for all $f\in Ker(L)$.
Now consider the involutions on each $C_n$ defined by $$\begin{aligned}
\label{airy involution}
C_n&\overset{\beta_{\text{Ai}}}\too C_n\notag\\
((P,Q))&\mapsto((\hat{P},\hat{Q}))\ ,\ \ \ \text{where}\notag\\
\hat{P}=P^T;\ \ \ &\hat{Q}=\rho(P^T)-Q^T\ .\end{aligned}$$
\[airy formula\] Let $W \in \grad$ correspond to a point $((P, Q)) \in C_n$. Let $$[P, Q] = I - w_1w^T_2$$ where $w_1$ and $w_2$ are column vectors. For $j = 0, \dots ,
r - 1$, let $$\rho_j(t) = t^{r-1-j} - \sum^{r-2}_{i=j+1} a_it^{i-1-j} .$$ Then the components of $\vec{k}_W$ are $$\label{k formula}
k_j(x, z) = \delta_{0,j} - w^T_2(zI -
Q)^{-1}\rho_{j}(P)(x -\hat{Q}^T)^{-1}w_1 \ .$$
It suffices to consider the generic case, in which $W$ is defined by conditions $c_i = \delta_{\lambda_i}
\circ(\partial - \alpha_i)$, $i = 1, \dots , n$, for distinct $\lambda$’s. Set $$\label{gamma}
\gamma_i = \alpha_i - \sum_{j \ne i} \frac{1}{\lambda_i -
\lambda_j} .$$ Then $W$ corresponds under Wilson’s theorem to the Calogero-Moser pair $$Q =
\left[
\begin{matrix}
\lambda_1 &&0\\ &\ddots& \\ 0 & & \lambda_n
\end{matrix}
\right]$$ $$P =
\left[
\begin{matrix}
\gamma_1 & \frac{1}{\lambda_1 - \lambda_2} & \dots &
\frac{1}{\lambda_1 - \lambda_n} \\
\frac{1}{\lambda_2 - \lambda_1} & & & \frac{1}{\lambda_2 -
\lambda_n} \\
\vdots&&\ddots&\vdots \\
\frac{1}{\lambda_n - \lambda_1} & \dots & & \gamma_n
\end{matrix}
\right]\ .$$
Let $(e_j)_{j=1,\dots , r}$ be the standard basis for $\Bbb C^r$. Then $$\label{k form}
\vec{k}_W= e_1 + \sum^n_{i=1} \frac{\vec{v}_i(x)}{(z - \lambda_i)}\ ,$$ for some vectors $\vec{v}_i(x) \in \Bbb C^r$. Applying $\delta_{\lambda_i} \circ \left(\d_z + \bair(x,
\lambda_i) - \alpha_i\right)$ to $q(z)\vec{k}_W$, one obtains the set of equations $$\begin{aligned}
0 &= \bair(x , \lambda_i)\vec{v}_i(x) \prod_{\ell\ne i}
(\lambda_i -
\lambda_{\ell}) + \notag \\ &\prod_{\ell\ne i} (\lambda_i - \lambda_{\ell})e_1 +
\sum^n_{m=1}
\vec{v}_m(x) \sum_{\ell\ne m} \prod_{s\ne \ell, m} (\lambda_i -
\lambda_s) \notag \\ &- \alpha_i \vec{v}_i(x) \prod_{\ell\ne i} (\lambda_i -
\lambda_{\ell}) ,\end{aligned}$$ $i = 1, \dots , n$. Dividing by $\prod_{\ell\ne i} (\lambda_i -
\lambda_{\ell})$ and using , $$\label{separate equations} e_1 = -\bair(x ,
\lambda_i)\vec{v}_i(x) +
\gamma_i\vec{v}_i(x) -
\sum_{\ell\ne i} \frac{\vec{v}_{\ell}(x)}{\lambda_i -
\lambda_{\ell}} .$$ Let $(u_i)_{i=1,\dots , n}$ be the standard basis for $\Bbb C^n$, and let $$v(x) = \sum u_i \otimes \vec{v}_i(x) \in \Bbb C^n \otimes \Bbb C^r .$$ Let $w = \sum u_i$. Write $\bair(x , \lambda_i) =
\bair(x,0) +
\lambda_i\Delta$, where $$\Delta =
\left(
\begin{matrix} 0 & \cdots & 0 & 1 \\ 0&&0& 0 \\
\vdots&\cdots&\vdots& \vdots \\ 0 &\cdots&0& 0
\end{matrix}
\right) .$$ Then is encoded as a single equation $$w \otimes e_1 = (-I \otimes B(x) + P \otimes I - Q \otimes
\Delta)v(x) .$$ Altogether, becomes $$\vec{k}_W = e_1 - (w^T \otimes I) \circ ((zI - Q)^{-1} \otimes I) \circ A^{-1}
(w
\otimes e_1) ,$$ where $$A = I \otimes \bair(x,0) - P \otimes I + Q \otimes \Delta \in
End(\Bbb C^n \otimes \Bbb C^r) .$$ Thinking of $\Bbb C^n \otimes \Bbb C^r$ as $rn$-tuples in blocks of length $n$, $$w \otimes e_1 =
\left[
\begin{matrix} 1 \\
\vdots \\ 1 \\ 0 \\
\vdots \\ 0
\end{matrix}
\right]
\begin{matrix}
\biggr\}n \\
\phantom{} \\
\phantom{} \\
\phantom{}
\end{matrix}$$ and $$\label{airy matrix} A =
\left[
\begin{matrix} -P & 0 & \cdots & 0 & xI + Q \\ I & -P &&& a_1I \\ 0 & I & \ddots &&
\vdots \\
\vdots & 0 &&& a_{r-1}I \\ 0 & 0 & 0 & I & -P
\end{matrix}
\right] .$$ One checks quite easily that $$A
\left[
\begin{matrix}
\rho_0(P) \\
\vdots \\
\vdots \\
\rho_{r-1}(P)
\end{matrix}
\right] =
\left[
\begin{matrix} xI + Q - \rho(P) \\ 0 \\
\vdots \\ 0
\end{matrix}
\right] .$$ Then $$\begin{aligned}
&(w^T \otimes I) \circ ((zI - Q)^{-1} \otimes I) \circ A^{-1}(w
\otimes e_1) \notag \\ &= \left[
\begin{matrix} w^T&&0 \\ &\ddots& \\ 0 && w^T
\end{matrix}
\right]
\left[
\begin{matrix} (zI - Q)^{-1}&&0 \\ &\ddots& \\ 0 && (zI - Q)^{-1}
\end{matrix}
\right]
\left[
\begin{matrix}
\rho_0(P) \\
\vdots \\
\vdots \\
\rho_{r-1}(P)
\end{matrix}
\right] (xI + Q - \rho(P))^{-1}w . \notag \\ &= \left[
\begin{matrix} w^T(zI - Q)^{-1}\rho_0(P)(xI-\hat{Q}^T )^{-1}w \\
\vdots \\ w^T(zI - Q)^{-1}\rho_{r-1}(P)(xI-\hat{Q}^T )^{-1}w
\end{matrix}
\right] .\end{aligned}$$ This proves the theorem, since $$[P,Q] = I - ww^T .$$
For any generalized Airy operator $\lair$, and for all $W\in\grad$, $$\label{xz symmetry}
\vec{k}_W(z,x)=\vec{k}_{\beta_{\text{Ai}}(W)}(x,z)\ .$$ In particular, the algebra $\cal R_{\text{Ai},W}$, , is bispectral, with an $r$-dimensional space of joint eigenfunctions $\psiair(f)$, $f\in
Ker(\lair)$.
Formula follows immediately from , for if $[P,Q]=I-w_1w_2^T$, then $$[\hat{P},\hat{Q}]=[P^T,\rho(P^T)-Q^T]=[P,Q]^T=I-w_2w_1^T\ .$$ The rest of the corollary is immediate.
The Bessel Case
===============
The Bessel case works in much the same way. Consider again a polynomial $\rho(t)$, now normalized so that $a_{r-1}=\binom r2$. Set $$\lbess=x^{-r}\rho(D)\ \ \ \ \ \ \ \text{(Bessel)}\
,\label{besselop}$$ where $$\d=\frac d{dx}\ ,\ D=x\d\ .$$ Consider $Ker(\lbess-1)$, which should now be thought of as a sheaf rather than a space. For $f\in Ker(\lbess-1)$, define $$\fbess(x,z)=f(xz)\ .$$ Then $$\lbess(\fbess)=z^r\fbess \ ,
\ \lbess(z,\d_z)(\fbess)=x^r\fbess\ .\label{bessel is
bispectral}$$ Assume now that the matrix $Q$ is invertible. Define, as the analogue of , $$\label{bessel vector f}
\vecbess(x,z)=(\fbess,D(\fbess),...,D^{r-1}(\fbess))\ .$$ Then $$\label{bessel exchange}
D_z(\vecbess)=\vecbess\bbes(x^r,z^r)\ ,$$ where $$\label{first order bessel system}
\bbes(x,u)=
\left[\begin{matrix} 0&&\dots&&0&a_0+xu\\ 1&\ddots&&&&a_1\\
0&\ddots&&&\vdots&a_2\\
\vdots&\ddots&&&&\vdots\\ &&&&0&a_{r-2}\\
0&&\dots&0&1&a_{r-1}
\end{matrix}\right]\ .$$ Accordingly, one expects a Baker functional of the form $$\label{bessel baker functional}
\psibess(f)=\vecbess\cdot\vec{k}_W(x^r,z^r)\ \ ,\ f\in Ker(\lbess-1)\ .$$
To state the properties of $\psibess$, introduce the functions $$\mu(x,z)=(x^r,z^r)\ ;\ \nu(z)=z^r\ .$$ Denote by $\nu^*$ the action of $\nu$ on the space of finitely supported distributions in $\C^*$. Let $$\nabbess=D_z+ \frac 1r \bbes(x,z)\ .$$ Given a distribution $c=\delta_{\lambda}\circ p(D)$, define $$c_{\nabbess}=\delta_{\lambda}\circ p(\nabbess)\ ,$$ acting on vector valued functions of $z$.
\[covariant bessel\] Let $c$ be a distribution of the form $c=\delta_{\lambda}\circ p(D)$. Let $\vec{g}=
\left[\begin{matrix} g_0(x,z)\\
\vdots\\ g_{r-1}(x,z)\end{matrix}\right]$ be a vector of polynomials in $z$ with coefficients in $\C(x)$. Then $c(\vecbess\cdot\mu^*(\vec{g}))=0$ for all $f\in Ker(\lbess-1)$ if and only if $\nu^*(c)_{\nabbess}(\vec{g})=0$.
By virtue of the identity $$D\circ \nu^*=\nu^*\circ r D\ ,$$ one has $$\nu^*(c)=\delta_{\lambda^r}\circ p(rD)\ .$$ By , $$\begin{aligned}
c\circ\vecbess\cdot\mu^*(\vec{g})&
=\delta_{\lambda}\circ\vecbess\cdot p(D+\bbes(x^r,z^r))\circ
\mu^*(\vec{g})\notag\\
&=\vecbess(x,\lambda)\cdot
\delta_{\lambda}\circ \mu^*\circ p(rD+\bbes(x,z))
(\vec{g})\notag\\
&=\vecbess(x,\lambda)\cdot\nu_x^*\circ
\delta_{\lambda^r}\circ p(r\nabbess)
(\vec{g})\notag\\
&=\vecbess(x,\lambda)\cdot\nu_x^*\circ \nu^*(c)_{\nabbess} (\vec{g})\ ,\end{aligned}$$ where $\nu_x$ is $\nu$ acting in the $x$-variable. The lemma now follows as in lemma \[covariant airy\].
In light of the preceding lemma, it makes sense to impose the following properties on $\psibess$.
[ Property 1b:]{} The functions $q_C(z) k_i(x,z)$ are polynomial in $z$. [ Property 2b:]{} Let $C'$ be any space of distributions such that $\nu^*(C')=C$. Then for all $f\in Ker(\lbess-1)$, $q_C(z)\psibess(f)$ is annihilated by all $c\in C'$. [ Property 3b:]{} $\lim\limits_{z\to\infty}\vec{k}_W=e_1$. As in the Airy case, one reconstructs a differential operator $K_{\text{Be},C}$, but now $$K_{\text{Be},C}(\fbess)=q_C(z^r)
\vecbess\cdot\vec{k}_W(x^r,z^r)\ .$$ Then for any polynomial $p\in R_W$, the pseudodifferential operator $M_p=K_{\text{Be},C}p(\lbess)K_{\text{Be},C}^{-1}$ is a differential operator, and $$M_p(K_{\text{Be},C}(\fbess))=p(z^r)K_{\text{Be},C}(\fbess)$$ for all $f\in Ker(\lbess-1)$. Define $\cal R_{\text{Be},W}$ to be the algebra of the all the $M_p$’s.
Everything now proceeds as before. Assume that the matrix $Q$ is invertible. We have $n$ distributions $c_i = \delta_{\lambda_i}
\circ(\partial_z - \alpha_i)$. Note that $\delta_{\lambda_i}
\circ(\partial_z - \alpha_i)=\frac 1{\lambda_i} \delta_{\lambda_i}
\circ(D_z - \lambda_i\alpha_i)$. Thus, according to lemma \[covariant bessel\], property 2b imposes the $n$ conditions $$0=\delta_{\lambda_i}
\circ(\partial_z +\frac 1{r \lambda_i}\bbes(x,\lambda_i)-
\alpha_i)(q_C(z)\vec{k}_W(x,z))\ .$$ Setting $$\label{bessel k form}
\vec{k}_W = e_1 + \sum^n_{i=1} \frac{\vec{v}_i(x)}{(z - \lambda_i)} \ ,$$ one now finds $$\label{separate bessel
equations} e_1 = -\frac 1{r\lambda_i} \bbes(x,\lambda_i)\vec{v}_i(x) +
\gamma_i\vec{v}_i(x) -
\sum_{\ell\ne i} \frac{\vec{v}_{\ell}(x)}{\lambda_i -
\lambda_{\ell}} .$$ This time, $$\frac 1{ru}\bbes(x,u) = \frac 1{ru}\Delta_1 + \frac xr\Delta_2\ ,$$ where $$\begin{aligned}
\Delta_1&=
\left[
\begin{matrix} 0 & \cdots && 0 & a_0 \\
1&\ddots&&\vdots& \vdots \\
0&\ddots&&0&\vdots\\
&\ddots&&&\\
0 &\cdots&0&1& a_{r-1}
\end{matrix}
\right] \ ,\\
&\notag\\
\Delta_2&=\left[
\begin{matrix}
0&\cdots&0&1\\
\vdots&&\vdots&0\\
\vdots&&\vdots&\vdots\\
0&\cdots&0&0
\end{matrix}
\right] \ .\end{aligned}$$ Thus $$w \otimes e_1 =
-(\frac xr I \otimes \Delta_2 - P \otimes I + \frac 1r Q^{-1} \otimes
\Delta_1)v(x) \ .$$ Then $$\vec{k}_W = e_1 - (w^T \otimes I) \circ ((zI - Q)^{-1} \otimes I) \circ A^{-1}
(w
\otimes e_1) ,$$ where $A$ is now given in block matrix form by $$\label{bessel matrix}
A =
\left[
\begin{matrix}
-P & 0 & \cdots & 0 & \frac xr I + \frac{a_0}r Q^{-1} \\
\frac 1r Q^{-1} & \ddots &&\vdots& \frac{a_1}r Q^{-1} \\
0 &\ddots & &0&\vdots \\
\vdots & \ddots &&-P& \frac{a_{r-2}}r Q^{-1} \\
0 & \cdots & 0 & \frac 1r Q^{-1} & -P+\frac{a_{r-1}}r Q^{-1}
\end{matrix}
\right] .$$ One obtains the following result.
\[bessel formula\] Let $W \in \grad$ correspond to a point $((P, Q))
\in C_n$. Let $$[P, Q] = I - w_1w^T_2$$ where $w_1$ and $w_2$ are column vectors. Writing the $r$th order Bessel operator $\lbess$ in the form $\lbess=x^{-r}\rho(D)$, let $$\rho_j(t) = t^{r-1-j} - \sum^{r-1}_{i=j+1} a_it^{i-1-j}$$ for $j = 0, \dots , r - 1$. Then the components of $\vec{k}_W$ are $$\label{bessel k formula}
k_{j}(x, z) = \delta_{0,j} -
r w^T_2(zI - Q)^{-1}\rho_{j}(r QP)(x
-\hat{Q}^T)^{-1}w_1 ,$$ where $$\hat{Q}=(Q^{-1}\rho(r QP))^T\ .$$
One checks now that with $A$ given by , $$A
\left[
\begin{matrix}
r \rho_0(r QP) \\
\vdots \\
r \rho_{r-1}(r QP)
\end{matrix}
\right] =
\left[
\begin{matrix} xI - Q^{-1} \rho(r QP) \\ 0 \\
\vdots \\ 0
\end{matrix}
\right] .$$ The result then follows as in theorem \[airy formula\].
Theorem \[bessel formula\] suggests the definition $$\begin{aligned}
\hat{P}&=\hat{Q}^{-1}P^TQ^T\notag\\
&=(QP\rho(QP)^{-1} Q)^T \ .\end{aligned}$$ Then $$\begin{aligned}
\hat{\hat{Q}}&=(\hat{Q}^{-1}\rho(r \hat{Q}\hat{P}))^T\notag\\
&=(\hat{Q}^{-1}\rho(r P^TQ^T))^T\notag\\
&=\rho(r QP)\rho(r QP)^{-1} Q=Q\ ,\end{aligned}$$ and $$\begin{aligned}
\hat{\hat{P}}&=\hat{\hat{Q}}^{-1}\hat{P}^T \hat{Q}^T\notag\\
&=Q^{-1}QP(\hat{Q}^{-1})^T\hat{Q}^T=P\ .\end{aligned}$$ Moreover, $$\begin{aligned}
[\hat{P},\hat{Q}]&=[\hat{Q}^{-1}P^TQ^T,\hat{Q}]\notag\\
&=\hat{Q}^{-1}P^TQ^T\hat{Q}-P^TQ^T\notag\\
&=Q^T(\rho(rQP)^{-1})^T(QP)^T\rho(rQP)^T(Q^{-1})^T-P^TQ^T\notag\\
&=Q^TP^T-P^TQ^T=[P,Q]^T\ .\end{aligned}$$ So again one has an involution (densely defined) on $C_n$, $$\begin{aligned}
\label{bessel involution}
C_n&\overset{\beta_{\text{Be}}}{--\rightarrow} C_n\notag\\
(P,Q)&\mapsto(\hat{P},\hat{Q})\ ,\ \ \ \text{where}\notag\\
\hat{P}=\hat{Q}^{-1}P^TQ^T\ ;
\ \ \ &\hat{Q}=
(Q^{-1}\rho(r QP))^T
\ .\end{aligned}$$
For any generalized Bessel operator $\lbess$, $$\label{bessel xz symmetry}
\vec{k}_W(z,x)=\vec{k}_{\beta_{\text{Be}}(W)}(x,z)\ .$$ In particular, the algebra $\cal R_{\text{Be},W}$ is bispectral, with a rank $r$ joint eigensheaf $\psibess(f)$, $f\in Ker(\lbess-1)$.
Dynamics
========
It is well-known that the Calogero-Moser particle system is a completely integrable hamiltonian system on the symplectic manifold $C_n$. The symplectic form is $$\omega=\tr(dP dQ)\ ,$$ and the hamiltonians are $h_n=\tr(P^n)$ (cf. [@KKS]).
It is pleasing then that the Airy and Bessel involutions are antisymplectic on each $C_n$. In the Airy case, $$\hat{P}=P^T;\ \ \ \hat{Q}=\rho(P^T)-Q^T\ ,$$ this follows from the fact that $$\tr(dP d P^n)=\sum_{i+j=n-1}\tr(dP P^i dP P^j) =0 \ .$$ (The basic trace identity for form-valued matrices is $$\tr(XY)=(-1)^{\text{deg}(X)\text{deg}(Y)}\tr(YX)\ .)$$
The Bessel case is slightly more involved.
Let $\sigma$ be a polynomial over $\C$. Let $$\hat{Q}= (Q^{-1}\sigma(QP))^T\ ;\ \hat{P}=\hat{Q}^{-1}P^TQ^T\ .$$ Then $$\tr(d\hp d\hq)=-\tr(dP dQ)\ .$$
Let $\tq=Q^{-1}\sigma(QP)$, $\tp=QP\sigma(QP)^{-1} Q=\hp^T$. Then $$\tr(d\hp d\hq)=\tr(d \tp d\tq)\ .$$ Set $R=QP$. Then $$\tr(dP dQ)=\dtr (Q^{-1} R dQ)\ ,$$ while $$\begin{aligned}
\tr(d\tp d\tq) &= \dtr(R\sigma^{-1} Qd(Q^{-1}\sigma)) \nonumber \\ &=
\dtr(R\sigma^{-1}d\sigma - R\sigma^{-1}dQ Q^{-1}\sigma)
\nonumber \\ &= \dtr(R\sigma^{-1}d\sigma - Q^{-1} R dQ) \ .\end{aligned}$$ So it must be proved that $$\dtr(R\sigma^{-1}d\sigma) = 0 \ .\label{to prove}$$ If $\sigma = \sigma_1\sigma_2$ and $\sigma_2$ commutes with $R$, then $$\begin{aligned}
\dtr(R\sigma^{-1}d\sigma) &=
\dtr(R\sigma^{-1}_2\sigma^{-1}_1(d\sigma_1\sigma_2 + \sigma_1 d\sigma_2))
\nonumber \\ &= \dtr(R\sigma^{-1}_1d\sigma_1) +
\dtr(R\sigma^{-1}_2d\sigma_2)\ . \label{additivity}\end{aligned}$$ Also, $$\dtr(\sigma^{-1}d\sigma) = -\tr(\sigma^{-1}d\sigma
\sigma^{-1}d\sigma) = 0 .$$ This last identity implies that one can replace $R$ by $R-(const) I$ in , and reduce to the case that $\sigma =
R\sigma_1(R)$, $\sigma_1$ polynomial.
Then by $$\begin{aligned}
\dtr(R\sigma^{-1}d\sigma) &= \dtr(dR + R\sigma^{-1}_1 d\sigma_1) \nonumber \\
&= \dtr(R\sigma^{-1}_1 d\sigma_1) \ . \end{aligned}$$ Since the result is obvious when $\sigma$ is a constant, the proposition follows by induction on the degree of $\sigma$.
Now introduce time dependence into the Baker functionals of the previous sections in a manner generalizing the standard procedure in the rank-one case [@SW]. Fix a positive integer $m$. The time-dependent Baker function in the rank one case is the function $$e^{xz+tz^m}p(x,z,t)$$ satisfying properties 1,2 and 3 of section \[airy case\] with a fixed space of conditions $C$. On the other hand, one may introduce time-dependence into the conditions by defining $C_t=C\circ e^{tz^m}$. Then the function $e^{xz}p(x,z,t)$ satisfies properties 1,2 and 3 for the variable conditions $C_t$.
If $c=\delta_{\lambda}\circ(\d-\alpha)$, then $c\circ e^{t z^m}=\delta_{\lambda}\circ(\d+t m \lambda^{m-1}-\alpha)$. In other words, the flow $C_t$ is seen on the level Calogero-Moser pairs as $$\label{cm flows} Q_t=Q_0\ ;\ P_t=P_0-tmQ^{m-1}\ .$$ This is the flow of the completely integrable hamiltonion $$h_m=\tr(Q^m)\ .$$ This hamiltonian is the Calogero-Moser hamiltonian with the roles of $P$ and $Q$ reversed. Finally, the Baker function gives rise in a standard way to a solution of the KP-hierarchy, with poles in $x$ the same as those of the Baker function. Thus, Wilson’s formula makes it immediately clear that the poles in $x$ of such a KP-solution move as a Calogero-Moser particle system (cf. [@Kr; @Sh; @Ka]).
To carry this over to the Airy and Bessel cases, define $\vec{k}_{W,t}$ to be the vector , for the variable space of conditions $C_t$. We are led to solutions of a subhierarchy of the KP-heirarchy, in the following way. Let $\kair$ be the monic $0^{th}$-order pseudodifferential operator such that $$\kair\d^r\kair^{-1}=\lair\ .$$ Let $\tilde{K}_t$ be the monic $0^{th}$-order pseudodifferential operator such that $$\vecair\cdot\vec{k}_{W,t}=\tilde{K}_t(\fair)\ .$$ Then the argument in [@SW] shows that $$\label{pre dressing}
\d_t(\tilde{K})\tilde{K}^{-1}+(\tilde{K}\lair^m\tilde{K}^{-1})_-=0\ .$$ Now let $$K_t=\tilde{K}_t\kair\ .$$ Then $$\label{dressing}
\d_t(K)K^{-1}+(K\d^{rm}K^{-1})_-=0\ .$$ It now follows as in [@SW] that the operator $$M_t=K_t\d K_t^{-1}$$ satisfies the $rm^{th}$ term of the KP-hierarchy, $$\d_t(M)=[M^{rm}_+,M]\ .$$ From formula , $M$ has coefficients in $\C(t)[x,\frac 1{\tau_t(x)}]$, where $$\begin{aligned}
\tau_t(x)&=Det(x-\hat{Q}_t)\ ,\\
\hat{Q}_t&=(\rho(P_t)-Q_t)^T\notag\\ &=(\rho(P_0-tmQ_0^{m-1})-Q_0)^T\ .\end{aligned}$$ Thus we have constructed a solution $M_t$, of the $rm^{th}$ term of the KP-hierarchy, whose poles in $x$ move according to the completely integrable hamiltonian $\tr(\hq^m)$ on the Calogero-Moser phase space.
In the Bessel case, exactly the same analysis holds, except that $M$ has coefficients in $\C(t)[x,\frac 1x,\frac 1{\tau_t(x^r)}]$. Thus $M_t$ has a fixed pole at $x=0$, with the motion of the remaining poles being governed by the hamiltonian $\tr(\hq^m)$.
With several particles it becomes quite cumbersome to write out these hamiltonians explicitly. The lowest rank cases are $$\begin{aligned}
\lair=\d^2-x\ &;\ \rho(t)=t^2\ ,\\
\lbess=\d^2-x^{-2}\ &;\ \rho(t)=t^2-t-1\ .\end{aligned}$$ Note that the first hamiltonian, $h_1=\tr(\hq)$, is already non-linear. In the rank-two Bessel case with one particle, for instance, $h_1(\lambda,\gamma)=\lambda^{-1}\rho(\lambda\gamma)$. This gives the equations of motion $$\begin{aligned}
\dot{\lambda}&=8\gamma\lambda-2\\
\dot{\gamma}&=-4\gamma^2-\lambda^{-2}\ .\end{aligned}$$ These equations are solved by applying the Bessel involution and changing $t$ to $-t$, i.e. by setting $$\hat{\lambda}=c_1\ ;\ \hat{\gamma}=c_2+t\ .$$ After some calculation, $$\lambda(t)= 4 c_1 t^2 + ( 8 c_1 c_2-2) t -\frac 1{c_1} - 2 c_2 + 4 c_1
c_2^2\ .$$ With two particles and a second order Airy operator, the first hamiltonian is $${\gamma_1^2} + {\gamma_2^2} -\lambda_1 - \lambda_2 - {2\over
{{{\left( -\lambda_1 + \lambda_2 \right) }^2}}}\ .$$ With two particles and a second order Bessel operator, the first hamiltonian is $$-{{\lambda_1 + \lambda_2}\over {\lambda_1\,\lambda_2}} - \gamma_1 +
\lambda_1\,{\gamma_1^2} -
\gamma_2 +
\lambda_2\,{\gamma_2^2} +
{{2\,\left( \lambda_1\,\gamma_1 -
\lambda_2\,\gamma_2 \right) }\over {-\lambda_1 + \lambda_2}}\ .$$
[99999999]{} H. Airault, H.P. McKean and J. Moser, [*Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem*]{}, Comm. Pure Appl. Math., [**30**]{} (1977), 95–148 B. Bakalov, E. Horozov and M. Yakimov, [*Highest Weight Modules of $W_{1+\infty}$, Darboux transformations and the bispectral problem,*]{} q-alg/9601017 B. Bakalov, E. Horozov and M. Yakimov, [*Bispectral algebras of commuting ordinary differential operators*]{} q-alg/9602011 B. Bakalov, E. Horozov and M. Yakimov, [*General methods for construction bispectral operators*]{} q-alg/9605011 J.J. Duistermaat and F.A. Grünbaum, [*Differential Equations in the Spectral Parameter,*]{} Communications in Mathematical Physics [**103**]{} (1986), 177–240 E. Date, M. Jimbo, M. Kashiwara, T. Miwa, [*Transformation groups for soliton equations.*]{} Proc. RIMS Symp. [*Nonlinear integrable systems - Classical and Quantum theory*]{} (Kyoto 1981), M. Jimo, T. Miwa(eds.), 39–111, Singapore: World Scientific, 1983 A. Kasman, [*Bispectral KP Solutions and Linearization of Calogero-Moser Particle Systems,*]{} Comm. Math. Phys. [**172**]{} (1995) 427–448 D. Kazhdan, B. Kostant and S. Sternberg, [*Hamiltonian group actions and dynamical systems of Calogero type*]{}, Comm. Pure Appl. Math. [**31**]{} (1978), 481–507 I. Krichever, [*On rational solutions of the Kadomtsev-Petviashvili equation and integrable systems of N particles on the line*]{}, Funct. Anal. Appl. [**12:1**]{} (1978), 76–78 (Russian), 59–61 (English) A. Kasman and M. Rothstein, [*Bispectral Darboux transformations: the generalized Airy case*]{}, Physica D (to appear), q-alg/9606018 G. Latham and E. Previato, [*Higher rank Darboux transformations*]{}, MSRI preprint 05229-21. Proc. NATO ARW Nonsingular Limits of Dispersive Waves (Plenum Press, New York, to appear). E. Previato and G. Wilson, [*Vector bundles over curves and solutions of the KP equations*]{}, Proc. /sympos. Pure Math. [**49**]{}, (1989), 553–569 M. Sato, [*Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds.*]{} RIMS Kokyuroku [**439**]{}, (1981) 30–40 T. Shiota, [*Calogero-Moser hierarchy and KP hierarchy*]{}, J. Math. Phys. [**35**]{}(11), (1994) 5844–5849 G. Segal and G. Wilson, [*Loop Groups and Equations of KdV Type*]{} Publications Mathematiques de l’Institut des Hautes Etudes Scientifiques [**61**]{} (1985), 5–65 G. Wilson, [*Bispectral Commutative Ordinary Differential Operators*]{}, J. reine angew. Math. [**442**]{} (1993) 177–204 G. Wilson, [*Collisions of Calogero-Moser particles and an adelic Grassmannian*]{}, preprint, Imperial College, London, 1996
[^1]: Research supported by NSF Grant No. 58-1353149
|
{
"pile_set_name": "ArXiv"
}
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[Type II and heterotic one loop string effective actions in four dimensions]{}
Filipe Moura
[*Security and Quantum Information Group - Instituto de Telecomunicações\
Instituto Superior Técnico, Departamento de Matemática\
Av. Rovisco Pais, 1049-001 Lisboa, Portugal*]{}
[[email protected]]{}
.2in
[**Abstract** ]{}
> We analyze the reduction to four dimensions of the ${\cal R}^4$ terms which are part of the ten-dimensional string effective actions, both at tree level and one loop. We show that there are two independent combinations of ${\cal R}^4$ present, at one loop, in the type IIA four dimensional effective action, which means they both have their origin in M-theory. The $d=4$ heterotic effective action also has such terms. This contradicts the common belief that there is only one ${\cal R}^4$ term in four-dimensional supergravity theories, given by the square of the Bel-Robinson tensor. In pure ${\cal N}=1$ supergravity this new ${\cal R}^4$ combination cannot be directly supersymmetrized, but we show that, when coupled to a scalar chiral multiplet (violating the U(1) $R$-symmetry), it emerges in the action after elimination of the auxiliary fields.
Introduction
============
String theories require higher order in $\a$ corrections to their corresponding low energy supergravity effective actions. The leading type II string corrections are of order $\a^3$, and include ${\cal R}^4$ terms (the fourth power of the Riemann tensor), both at tree level and one loop [@Gross:1986iv; @Grisaru:1986px]. These ${\cal R}^4$ corrections are also present in the type I/heterotic effective actions [@Gross:1986mw] and in M-theory [@Green:1997as].
These string corrections to supergravity theories should obviously be supersymmetric. Unfortunately there is still no known way to compute these corrections in a manifestly supersymmetric way, although important progresses have been achieved. The supersymmetrization of these higher order string/M-theory terms has been a topic of research for a long time [@Peeters:2000qj; @deRoo:1992zp].
After compactification to four dimensions, one obtains a supergravity theory, whose number $\mathcal{N}$ of supersymmetries and different matter couplings depend crucially on the manifold where the compactification is taken. Most of the times, in four dimensions the higher order terms are studied as part of the supergravity theories, either simple [@Deser:1977nt; @Moura:2001xx; @Moura:2002ft] or extended [@Kallosh:1980fi; @Howe:1980th; @Moura:2002ip; @Deser:1978br], and are therefore considered only from a supergravity point of view. These theories are believed to be divergent, and those are candidate counterterms. Their possible stringy origin, as higher order terms in string/M theory after compactification from ten/eleven dimensions, is often neglected. One of the reasons for that criterion is chronological: the study of the quantum properties of four dimensional supergravity theories started several years before superstring theories were found to be free of anomalies and taken as the main candidates to a unified theory of all the interactions. In higher dimensions the procedure has been different: the low-energy limits of superstring theories are the different ten-dimensional supergravity theories. People have studied higher order corrections to these theories most of the times in the context of string theory, which requires them to be supersymmetric.
Tacitly one makes the natural assumption that, when compactified, these higher order terms also emerge as corrections to the corresponding four-dimensional supergravity theories. But this does not necessarily need to be the case. The quantum behavior of these theories is still an active topic of research, and recent works claim that the maximal ${\cal N}=8$ theory may even be ultraviolet finite [@Bern:2006kd; @Green:2006gt]. If that is the case, the ${\cal N}=8$ higher order terms will not be necessary from a supergravity point of view, although they will still appear in the ${\cal N}=8$ theory we obtain when we compactify type II superstrings on a six-dimensional torus. All the higher order terms considered are, from a supergravity point of view, *candidate* counterterms; it has never been explicitly shown that they indeed appear in the quantum effective actions with nonzero coefficients. Even in ${\cal N}<8$ theories, it may eventually happen that some of these counterterms are not necessary as supergravity counterterms, but are needed as compactified string corrections.
From the known bosonic terms in the different $\a$-corrected string effective actions in ten dimensions, one should therefore determine precisely which terms should emerge in four dimensions for each compactification manifold, not worrying if they are needed in $d=4$ supergravity. This is the goal of the present article, but here we restrict ourselves mainly to the order $\a^3$ ${\cal R}^4$ terms. We will also be mainly (but not strictly) concerned with the simplest toroidal compactifications; the reason is that the terms one gets are “universal”, i.e. they must be present (possibly together with other moduli-dependent terms) no matter which compactification manifold we take.
The article is organized as follows. In section 2 we review the purely gravitational parts in the effective actions, up to order $\a^3$, of type IIA, IIB and heterotic strings, at tree level and one loop. In section 3 we analyze their dimensional reduction to $d=4$. We show that there are two independent ${\cal R}^4$ terms in the four dimensional superstring effective action, although a classical result tells us that, of these terms, only the one which was previously known can be directly supersymmetrized. The supersymmetrization of the new ${\cal R}^4$ term gives rise to a new problem, which we address in ${\mathcal N}=1$ supergravity in section 4 by considering the coupling of the new ${\cal R}^4$ term to a chiral multiplet in superspace.
String effective actions to order $\a^3$ in $d=10$
==================================================
The Riemann tensor admits, in $d$ spacetime dimensions, the following decomposition in terms of the Weyl tensor ${\cal
W}_{mnpq}$, the Ricci tensor ${\cal R}_{mn}$ and the Ricci scalar ${\cal R}$: $$\begin{aligned}
{\cal R}_{mnpq}&=&{\cal W}_{mnpq}-\frac{1}{d-2} \left(g_{mp} {\cal
R}_{nq} - g_{np} {\cal R}_{mq} + g_{nq} {\cal R}_{mp} - g_{mq}
{\cal R}_{np} \right) \nonumber \\ &+&\frac{1}{(d-1)(d-2)}
\left(g_{mp} g_{nq} - g_{np} g_{mq}\right) {\cal R}.
\label{riemann}\end{aligned}$$
As proven in [@Fulling:1992vm], in $d=10$ dimensions, the critical dimension of superstring theories, there are seven independent real scalar polynomials made from four powers of the irreducible components of the Weyl tensor, which we label, according to [@Peeters:2000qj], as $R_{41}, \dots, R_{46}, A_7$. These polynomials are given by $$\begin{aligned}
R_{41} &=& {\cal W}_{mnpq} {\cal W}^{nrqt} {\cal W}_{rstu} {\cal
W}^{smup}, \nonumber \\ R_{42} &=& {\cal W}_{mnpq} {\cal W}^{nrqt}
{\cal W}^{ms}_{\ \ \ tu} {\cal W}_{sr}^{\ \ up}, \nonumber
\\ R_{43} &=& {\cal W}_{mnpq} {\cal W}_{rs}^{\ \ pq} {\cal
W}^{mn}_{\ \ \ tu} {\cal W}^{rstu}, \nonumber
\\ R_{44} &=& {\cal W}_{mnpq} {\cal W}^{mnpq} {\cal W}_{rstu} {\cal
W}^{rstu}, \nonumber \\ R_{45} &=& {\cal W}_{mnpq} {\cal W}^{nrpq}
{\cal W}_{rstu} {\cal W}^{smtu}, \nonumber
\\ R_{46} &=& {\cal W}_{mnpq} {\cal W}_{rs}^{\
\ pq} {\cal W}^{mr}_{\ \ \ tu} {\cal W}^{nstu}, \nonumber \\ A_7
&=& {\cal W}_{mn}^{\ \ \ pq} {\cal W}^{mt}_{\ \ \ pu} {\cal
W}_{tr}^{\ \ ns} {\cal W}_{\ \ qs}^{ur}. \label{r47}\end{aligned}$$
The superstring $\a^3$ effective actions are given in terms of two independent bosonic terms, from which two separate superinvariants are built [@Peeters:2000qj; @Tseytlin:1995bi]. These terms are given, at linear order in the NS-NS gauge field $B_{mn}$, by: $$\begin{aligned}
I_X&=&t_8 t_8 {\cal R}^4 + \frac{1}{2} \varepsilon_{10} t_8 B
{\cal R}^4, \nonumber \\ I_Z&=&-\varepsilon_{10} \varepsilon_{10}
{\cal R}^4 +4 \varepsilon_{10} t_8 B {\cal R}^4. \label{ixiz}\end{aligned}$$ Each $t_8$ tensor has eight free spacetime indices. It acts in four two-index antisymmetric tensors, as defined in [@Gross:1986iv; @Grisaru:1986px], where one can also find the precise index contractions. In terms of the seven fundamental polynomials $R_{41}, \dots, R_{46}, A_7$ from (\[r47\]), the purely gravitational parts of $I_X$ and $I_Z$, which we denote by $X$ and $Z$ respectively, are given by [@Peeters:2000qj]: $$\begin{aligned}
X := t_8 t_8 {\cal W}^4 &=& 192 R_{41} + 384 R_{42} + 24 R_{43} +
12 R_{44} -192 R_{45} -96 R_{46}, \nonumber \\ \frac{1}{8} Z := -
\frac{1}{8} \varepsilon_{10} \varepsilon_{10} {\cal W}^4 &=& X
+192 R_{46} - 768 A_7. \label{b14}\end{aligned}$$
For the heterotic string two extra terms $Y_1$ and $Y_2$ appear at order $\a^3$ at one loop level [@Peeters:2000qj; @deRoo:1992zp; @Tseytlin:1995bi], the pure gravitational parts of which being given respectively by $$\begin{aligned}
Y_1 := t_8 \left(\mbox{tr} {\cal W}^2\right)^2 &=& -4 R_{43} -2
R_{44} +16 R_{45} +8 R_{46}, \nonumber \\ Y_2 := t_8 \mbox{tr}
{\cal W}^4 &=& 8 R_{41} + 16 R_{42} -4 R_{45} -2 R_{46}.
\label{y1y2}\end{aligned}$$ with $\mbox{tr} {\cal W}^2= {\cal W}_{mnpq} {\cal W}_{rs}^{\ \
qp}$, etc. Only three of these four invariants are independent because, as one may see, one has the relation $X=24 Y_2 -6 Y_1.$
To be precise, let’s review the form of the purely gravitational superstring and heterotic effective actions in the string frame up to order $\a^3$. The perturbative terms occur at string tree and one loop levels; there are no higher loop contributions [@Green:1997as; @Tseytlin:1995bi; @Green:1997tv; @Iengo:2002pr].
The effective action of type IIB theory must be written, because of its well known SL$(2,{\mathbb Z})$ invariance, as a product of a single linear combination of order $\a^3$ invariants and an overall function of the complexified coupling constant $\Omega= C^0 + i
e^{-\phi},$ $C^0$ being the axion. This function accounts for perturbative (loop) and non-perturbative (D-instanton [@Green:1997tv; @Green:1997di]) string contributions. The perturbative part is given in the string frame by $$\left. \frac{1}{\sqrt{-g}} {\mathcal L}_{\mathrm{IIB}}
\right|_{\a^3} = -e^{-2 \phi} \a^3 \frac{\zeta(3)}{3 \times 2^{10}}
\left(I_X - \frac{1}{8} I_Z \right) - \a^3 \frac{1}{3 \times 2^{16}
\pi^5} \left(I_X - \frac{1}{8} I_Z \right). \label{2bea}$$
Type IIA theory has exactly the same term of order $\a^3$ as type IIB at tree level, but at one loop the sign in the coefficient of $I_Z$ is changed when compared to type IIB: $$\left. \frac{1}{\sqrt{-g}} {\mathcal L}_{\mathrm{IIA}}
\right|_{\a^3} = -e^{-2 \phi} \a^3 \frac{\zeta(3)}{3 \times 2^{10}}
\left(I_X - \frac{1}{8} I_Z \right) - \a^3 \frac{1}{3 \times 2^{16}
\pi^5} \left(I_X + \frac{1}{8} I_Z \right). \label{2aea}$$ The reason for this sign flip is that at one string loop the relative GSO projection between the left and right movers is different for type IIA and type IIB, since these two theories have different chirality properties [@Kiritsis:1997em; @Antoniadis:1997eg].
Type II superstring theories only admit $\a^3$ and higher corrections because the corresponding sigma model is two and three-loop finite, as shown in [@Grisaru:1986px]: ten dimensional ${\mathcal N}=2$ supersymmetry prevents these corrections. Heterotic string theories have ${\mathcal N}=1$ supersymmetry in ten dimensions, which allows corrections to the sigma model already at order $\a$, including ${\mathcal R}^2$ corrections. These corrections come both from three-graviton scattering amplitudes and anomaly cancellation terms (the Green-Schwarz mechanism). The effective action is then given in the string frame, up to order $\a^3$ and neglecting the contributions of gauge fields, by $$\begin{aligned}
\left. \frac{1}{\sqrt{-g}} {\mathcal L}_{\mathrm{heterotic}}
\right|_{\a+ \a^3} &=& e^{-2 \phi} \left[\frac{1}{16} \a \mbox{tr}
{\cal R}^2 +\frac{1}{2^9} \a^3 Y_1 - \frac{\zeta(3)}{3 \times
2^{10}} \a^3 \left(I_X - \frac{1}{8} I_Z \right) \right] \nonumber
\\ &-& \a^3 \frac{1}{3 \times 2^{14} \pi^5} \left(Y_1+ 4 Y_2
\right). \label{hea}\end{aligned}$$ For the type IIB theory only the combination $I_X - \frac{1}{8}
I_Z$ is present in the effective action. For the type IIA and heterotic theories different combinations show up. The supersymmetrization of these terms has been the object of study in many articles [@Peeters:2000qj; @deRoo:1992zp], although a complete understanding of the full supersymmetric effective actions is still lacking. Here we are more concerned with the number of independent superinvariants they would belong to. Because in every theory the $I_X - \frac{1}{8} I_Z$ term includes a transcendental factor $\zeta(3)$ (which is not shared by any other bosonic term at the same order in $\a$), it cannot be related to other bosonic terms by supersymmetry and requires its own superinvariant. This way in type IIA and heterotic string theories one then needs at least one ${\mathcal R}^4$ superinvariant for the tree level terms and another one for one loop.
Type IIA theory comes from compactification of M-theory on ${\mathbb
S}^1$, but its tree level $\a^3$ terms vanish on the eleven-dimensional limit, as shown in [@Green:1997as]. Therefore the one-loop type IIA ${\mathcal R}^4$ term is the true compactification of the $d=11$ ${\mathcal R}^4$ term. In M-theory, there is only one ${\mathcal R}^4$ superinvariant. The existence of this term was shown in [@Howe:2003cy], using spinorial cohomology, and its coefficient was fixed using anomaly cancellation arguments. The full calculation, using pure spinor BRST cohomology, was carried out in [@Anguelova:2004pg], where it was shown that this term is indeed unique and its coefficient can be directly determined without using the anomaly cancellation argument.
For a more detailed review of the present knowledge of ${\mathcal
R}^4$ terms in M-theory and supergravity, including a discussion of their supersymmetrization and related topics, see [@Howe:2004pn].
String effective actions to order $\a^3$ in $d=4$
=================================================
In this section we analyze the reduction to four dimensions of the effective actions considered in the previous section.
${\cal R}^4$ terms in $d=4$ from $d=10$
---------------------------------------
It is interesting to check how many independent superinvariants one still has in four dimensions. In this case, the Weyl tensor can still be decomposed in its self-dual and antiself-dual parts[^1]: $${\cal W}_{\mu \nu \rho \sigma}= {\cal W}^+_{\mu \nu \rho \sigma} +
{\cal W}^-_{\mu \nu \rho \sigma}, {\cal W}^{\mp}_{\mu \nu \rho
\sigma} :=\frac{1}{2} \left({\cal W}_{\mu \nu \rho \sigma} \pm
\frac{i}{2} \varepsilon_{\mu \nu}^{\ \ \ \lambda \tau} {\cal
W}_{\lambda \tau \rho \sigma} \right), \label{wpm}$$ which have the following properties: $${\cal W}^+_{\mu \nu \rho \sigma} {\cal W}^{- \ \rho \sigma}_{\tau
\lambda} =0, {\cal W}^\pm_{\mu \nu \rho \sigma} {\cal W}^{\pm \nu
\rho \sigma}_\tau =\frac{1}{4} g_{\mu \tau} {\cal W}^2_\pm.
\label{wprop}$$ Besides the usual Bianchi identities, the Weyl tensor in four dimensions obeys Schouten identities like this one: $${\cal W}^{\mu \nu}_{\ \ \ \rho \tau} {\cal W}_{\mu \nu \sigma
\lambda}= \frac{1}{4} \left(g_{\rho \sigma} g_{\tau \lambda} -
g_{\rho \lambda} g_{\tau \sigma} \right) {\cal W}^2 +2 \left({\cal
W}_{\rho \mu \nu \sigma} {\cal W}^{\ \mu \nu}_{\lambda \ \ \ \tau}
- {\cal W}_{\tau \mu \nu \sigma} {\cal W}^{\ \mu \nu}_{\lambda \ \
\ \rho} \right). \label{schouten4}$$ Because of the given properties, the Bel-Robinson tensor, which can be shown to be totally symmetric, is given in four dimensions by $${\cal W}^+_{\mu \rho \nu \sigma} {\cal W}^{- \rho \ \sigma}_{\tau
\ \lambda}.$$ In the van der Warden notation, using spinorial indices, the decomposition (\[wpm\]) is written as [@Penrose:1985jw] $${\cal W}_{A \dot A B \dot B C \dot C D \dot D}= -2
\varepsilon_{\dot A \dot B} \varepsilon_{\dot C \dot D} {\cal
W}_{ABCD} -2 \varepsilon_{AB} \varepsilon_{CD} {\cal W}_{\dot A
\dot B \dot C \dot D} \label{wpms}$$ with the totally symmetric ${\cal W}_{ABCD}, {\cal W}_{\dot A \dot
B \dot C \dot D}$ being given by (in the notation of [@Moura:2002ft]) $${\cal W}_{ABCD}:=-\frac{1}{8} {\cal W}^+_{\mu \nu \rho \sigma}
\sigma^{\mu \nu}_{\underline{AB}} \sigma^{\rho
\sigma}_{\underline{CD}}, \, {\cal W}_{\dot A \dot B \dot C \dot
D}:=-\frac{1}{8} {\cal W}^-_{\mu \nu \rho \sigma} \sigma^{\mu
\nu}_{\underline{\dot A \dot B}} \sigma^{\rho
\sigma}_{\underline{\dot C \dot D}}.$$ Using this notation, calculations involving the Weyl tensor become much more simplified. The Bel-Robinson tensor is simply given by ${\cal W}_{ABCD} {\cal W}_{\dot A \dot B \dot C \dot D}$.
In reference [@Fulling:1992vm] it is also shown that, in four dimensions, there are only two independent real scalar polynomials made from four powers of the Weyl tensor. Like in [@Moura:2002ft], these polynomials can be written, using the previous notation, as $$\begin{aligned}
{\cal W}_+^2 {\cal W}_-^2 &=& {\cal W}^{ABCD} {\cal W}_{ABCD}
{\cal W}^{\dot A \dot B \dot C \dot D} {\cal W}_{\dot A \dot B
\dot C \dot D}, \label{r441}\\ {\cal W}_+^4+{\cal W}_-^4 &=&
\left({\cal W}^{ABCD} {\cal W}_{ABCD}\right)^2 + \left({\cal
W}^{\dot A \dot B \dot C \dot D} {\cal W}_{\dot A \dot B \dot C
\dot D}\right)^2. \label{r442}\end{aligned}$$ In particular, the seven polynomials $R_{41}, \dots, R_{46}, A_7$ from (\[r47\]), when computed directly in four dimensions (i.e. replacing the ten dimensional indices $m,n, \ldots$ by the four dimensional indices $\mu, \nu, \ldots$) should be expressed in terms of them. That is what we present in the following. For that we wrote each polynomial in the van der Warden notation, using (\[wpms\]), and we used some properties of the four dimensional Weyl tensor, like (\[wprop\]) and (\[schouten4\]). This way we have shown that, *in four dimensions*, $$\begin{aligned}
R_{41} &=& \frac{1}{24} {\cal W}_+^4 + \frac{1}{24} {\cal W}_-^4
-\frac{5}{8} {\cal W}_+^2 {\cal W}_-^2, \nonumber \\ R_{42} &=&
\frac{1}{12} {\cal W}_+^4 + \frac{1}{12} {\cal W}_-^4 +\frac{11}{8}
{\cal W}_+^2 {\cal W}_-^2, \nonumber \\ R_{43} &=& \frac{1}{6} {\cal
W}_+^4 + \frac{1}{6} {\cal W}_-^4 -4 {\cal W}_+^2 {\cal W}_-^2,
\nonumber \\ R_{44} &=& {\cal W}_+^4 + {\cal W}_-^4 +2 {\cal W}_+^2
{\cal W}_-^2, \nonumber \\ R_{45} &=& \frac{1}{4} {\cal W}_+^4 +
\frac{1}{4} {\cal W}_-^4 +\frac{1}{2} {\cal W}_+^2 {\cal W}_-^2,
\nonumber \\ R_{46} &=& -\frac{1}{6} {\cal W}_+^4 - \frac{1}{6}
{\cal W}_-^4 -\frac{3}{2} {\cal W}_+^2 {\cal W}_-^2, \nonumber \\
A_7 &=& -\frac{1}{24} {\cal W}_+^4 - \frac{1}{24} {\cal W}_-^4
-\frac{1}{4} {\cal W}_+^2 {\cal W}_-^2.\end{aligned}$$
Using the definitions (\[b14\]), we have then $$\begin{aligned}
X &=& 24 \left({\cal W}_+^4 + {\cal W}_-^4 \right) +384 {\cal
W}_+^2 {\cal W}_-^2, \\ \frac{1}{8} Z&=& 24 \left({\cal W}_+^4 +
{\cal W}_-^4 \right) +288 {\cal W}_+^2 {\cal W}_-^2, \nonumber\end{aligned}$$ or $$\begin{aligned}
X - \frac{1}{8} Z &=& 96 {\cal W}_+^2 {\cal W}_-^2, \\ X +
\frac{1}{8} Z &=& 48 \left({\cal W}_+^4 + {\cal W}_-^4 \right)
+672 {\cal W}_+^2 {\cal W}_-^2.\end{aligned}$$ $X - \frac{1}{8} Z$ is the only combination of $X$ and $Z$ which in $d=4$ does not contain (\[r442\]), i.e. which contains only the square of the Bel-Robinson tensor (\[r441\]). We find it extremely interesting that exactly this very same combination (or, to be precise, $I_X - \frac{1}{8} I_Z$) is, from (\[ixiz\]), the only one which does not depend on the ten dimensional field $B^{mn}$ and, therefore, due to its gauge invariance, is the only one that can appear in string theory at arbitrary loop order. This combination is indeed present at string tree level in every superstring theory, multiplied by a transcendental factor $\zeta(3)$, as we have seen in the previous section.
From (\[y1y2\]) one also derives in $d=4:$ $$\begin{aligned}
Y_1&=& 8 {\cal W}_+^2 {\cal W}_-^2, \\Y_1 + 4 Y_2 = \frac{X}{6} +
2 Y_1 &=& 80 {\cal W}_+^2 {\cal W}_-^2 + 4 \left({\cal W}_+^4 +
{\cal W}_-^4 \right).\end{aligned}$$
As seen in the previous section, for the type IIB theory only the combination $I_X - \frac{1}{8} I_Z$ (or ${\cal W}_+^2 {\cal
W}_-^2$ in $d=4$) is present in the effective action (\[2bea\]). For the type IIA and heterotic theories different combinations show up. In these two cases, ${\cal W}_+^4 + {\cal W}_-^4$ shows up at string one loop level in the effective actions (\[2aea\]) and (\[hea\]) of these theories when they are compactified to four dimensions. At string tree level, though, for all these theories in $d=4$ only ${\cal W}_+^2 {\cal W}_-^2$ shows up. This fact is quite remarkable, particularly for the heterotic theory, if we consider that the two different contributions $I_X -
\frac{1}{8} I_Z$ and $Y_1$ in (\[hea\]) have completely different origins.
Moduli-independent terms in $d=4$ effective actions
---------------------------------------------------
All the terms we have been considering, when taken in the Einstein frame (which is the right frame for a supergravity analysis to be performed), are multiplied by an adequate power of $\exp(\phi).$ To be precise, consider an arbitrary term $I_i({\cal R, M})$ in the string frame lagrangian in $d$ dimensions. $I_i({\cal R, M})$ is a function, with conformal weight $w_i$, of any given order in $\a$, of the Riemann tensor $\cal R$ and any other fields - gauge fields, scalars, and also fermions - which we generically designate by $\cal M$. To pass from the string to the Einstein frame, we redefine the metric through a conformal transformation involving the dilaton, given by $$\begin{aligned}
g_{mn}
&\rightarrow& \exp \left( \frac{4}{d-2} \phi \right) g_{mn},
\nonumber \\ {{\cal R}_{mn}}^{pq} &\rightarrow& \exp
\left(-\frac{4}{d-2} \phi \right) {\widetilde{{\cal R}}_{mn}}^{\ \
\ pq}, \label{rsre}\end{aligned}$$ with ${\widetilde{{\cal R}}_{mn}}^{\ \ \ pq}={{\cal R}_{mn}}^{pq}
- {\delta_{\left[m\right.}}^{\left[p\right.} \nabla_{\left.n
\right]} \nabla^{\left.q \right]} \phi.$ The transformation above takes $I_i({\cal R,M})$ to $e^{\frac{4}{d-2} w_i \phi} I_i({\cal
\widetilde{R}, M}).$ After considering all the dilaton couplings and the effect of the conformal transformation on the metric determinant factor $\sqrt{-g},$ the string frame lagrangian $$\label{esf} \frac{1}{2}
\sqrt{-g}\ \mbox{e}^{-2 \phi} \Big( -{\cal R} + 4 \left(
\partial^m \phi \right) \partial_m \phi + \sum_i I_i({\cal R,M}) \Big)$$ is converted into the Einstein frame lagrangian $$\label{eef} \frac{1}{2} \sqrt{-g} \left( -{\cal R} - \frac{4}{d-2}
\left( \partial^m \phi \right) \partial_m \phi + \sum_i
\mbox{e}^{\frac{4}{d-2} \left( 1 + w_i \right) \phi} I_i({\cal
\widetilde{R},M}) \right).$$
We finish this section by writing, for later reference, the effective actions (\[2bea\]), (\[2aea\]), (\[hea\]) in four dimensions, in the Einstein frame (considering only terms which are simply powers of the Weyl tensor, without any other fields except their couplings to the dilaton, and introducing the $d=4$ gravitational coupling constant $\kappa$): $$\begin{aligned}
\left. \frac{\kappa^2}{\sqrt{-g}} {\mathcal L}_{\mathrm{IIB}}
\right|_{{\cal R}^4} &=& - \frac{\zeta(3)}{32} e^{-6 \phi} \a^3
{\cal W}_+^2 {\cal W}_-^2 - \frac{1}{2^{11} \pi^5} e^{-4 \phi}\a^3
{\cal W}_+^2 {\cal W}_-^2, \label{2bea4} \\ \left.
\frac{\kappa^2}{\sqrt{-g}} {\mathcal L}_{\mathrm{IIA}}
\right|_{{\cal R}^4} &=& - \frac{\zeta(3)}{32} e^{-6 \phi} \a^3
{\cal W}_+^2 {\cal W}_-^2 \nonumber \\ &-& \frac{1}{2^{12} \pi^5}
e^{-4 \phi}\a^3 \left[\left({\cal W}_+^4 + {\cal W}_-^4 \right)
+224 {\cal W}_+^2 {\cal W}_-^2 \right], \label{2aea4}
\\ \left. \frac{\kappa^2}{\sqrt{-g}} {\mathcal L}_{\mathrm{het}}
\right|_{{\cal R}^2 + {\cal R}^4} &=& -\frac{1}{16} e^{-2 \phi} \a \left({\cal
W}_+^2 + {\cal W}_-^2 \right) +\frac{1}{64} \left(1-2 \zeta(3)
\right) e^{-6 \phi} \a^3 {\cal W}_+^2 {\cal W}_-^2 \nonumber \\
&-& \frac{1}{3\times2^{12} \pi^5} e^{-4 \phi}\a^3
\left[\left({\cal W}_+^4 + {\cal W}_-^4 \right) +20 {\cal W}_+^2
{\cal W}_-^2 \right]. \label{hea4}\end{aligned}$$ Here one must refer that these are only the moduli-independent terms of these effective actions. Strictly speaking these are not moduli-independent terms, since they are all multiplied by the volume of the compactification manifold (a factor we omitted for simplicity). But they are always present, no matter which compactification is taken. The complete action, for every different compactification manifold, includes many moduli-dependent terms which we do not consider here.
A complete study of the heterotic string moduli dependent terms, but only for $\a=0$ and for a ${\mathbb T}^6$ compactification, can be seen in [@Sen:1994fa]. The tree level and one loop contributions to the four graviton amplitude, for a compactification on an $n$-dimensional torus ${\mathbb T}^n$ of ten dimensional type IIA/IIB string theories, can be found in [@Green:1997di].
A detailed study of these moduli-dependent ${\cal R}^4$ terms, at string tree level and one loop, for type IIA and IIB superstrings, for several compactification manifolds preserving different ammounts of supersymmetry, is available in [@Giusto:2004xm]. In many cases one must consider extra contributions to the effective action coming from string winding modes and worldsheet instantons. For the particularly simple but illustrative case of an ${\mathbb S}^1$ compactification (presented in detail in [@Green:1997di; @Giusto:2004xm]), the tree level terms for both type IIA and IIB theories are trivial: they are simply multiplied by the volume $2 \pi R.$ At one loop level, one gets terms proportional to the compactification radius $R$; by applying $T$-duality to these terms, one gets other terms proportional to $\frac{\a}{R}.$ This way one gets the term $X+\frac{1}{8}Z$, in $d=9$, even for type IIB effective action (in this case, only at a higher order in $\a$). The same is true in $d=4$, for more complicated compactification manifolds.
To conclude, for any $d=4$ compactification of heterotic or superstring theories one has, in the respective effective action, the two different $d=4$ ${\cal R}^4$ terms (\[r441\]) and (\[r442\]), multiplied by a corresponding dilaton factor and maybe some moduli terms. This is the most important result for the rest of this paper. From now on we will be concerned with the supersymmetrization of these terms.
${\cal R}^4$ terms and $d=4$ supersymmetry
==========================================
Up to now, we have only been considering bosonic terms for the effective actions, but we are interested in their full supersymmetric completion in $d=4.$ In general each superinvariant consists of a leading bosonic term and its supersymmetric completion, given by a series of terms with fermions. In this work we are particularly focusing on ${\cal R}^4$ terms.
Some known results
------------------
It has been known for a long time that the square of the Bel-Robinson tensor ${\cal W}_+^2 {\cal W}_-^2$ can be made supersymmetric, in simple [@Deser:1977nt; @Moura:2001xx] and extended [@Kallosh:1980fi; @Moura:2002ip; @Deser:1978br] four dimensional supergravity. For the term ${\cal W}_+^4 + {\cal
W}_-^4$ there is a “no-go theorem”, based on ${\mathcal N}=1$ chirality arguments [@Christensen:1979qj]: for a polynomial $I({\cal W})$ of the Weyl tensor to be supersymmetrizable, each one of its terms must contain equal powers of ${\cal W}^+_{\mu \nu
\rho \sigma}$ and ${\cal W}^-_{\mu \nu \rho \sigma}$. The whole polynomial must then vanish when either ${\cal W}^+_{\mu \nu \rho
\sigma}$ or ${\cal W}^-_{\mu \nu \rho \sigma}$ do. The only exception is ${\cal W}^2 = {\cal W}_+^2 + {\cal W}_-^2$, which in $d=4$ is part of the Gauss-Bonnet topological term and is automatically supersymmetric.
But the new term (\[r442\]) is part of the heterotic and type IIA effective actions at one loop which must be supersymmetric, even after compactification to $d=4$. One must then find out how this term can be made supersymmetric, circumventing the ${\mathcal
N}=1$ chirality argument from [@Christensen:1979qj]. That is our main goal in this paper.
One must keep in mind the assumptions in which it was derived, namely the preservation by the supersymmetry transformations of $R$-symmetry which, for ${\mathcal N}=1$, corresponds to U(1) and is equivalent to chirality. That is true for pure ${\mathcal N}=1$ supergravity, but to this theory and to most of the extended supergravity theories (except ${\mathcal N}=8$) one may add matter couplings and extra terms which violate U(1) $R$-symmetry and yet can be made supersymmetric, inducing corrections to the supersymmetry transformation laws which do not preserve U(1) $R$-symmetry.
Since the article [@Christensen:1979qj] only deals with the term (\[r442\]) by itself, one can consider extra couplings to it and only then try to supersymmetrize. These couplings could eventually (but not necessarily) break U(1) $R$-symmetry. This procedure is very natural, taking into account the scalar couplings that multiply (\[r442\]) in the actions (\[2aea4\]), (\[hea4\]).
Considering couplings to other multiplets and breaking U(1) may be possible in ${\mathcal N}=4$ supergravity, for ${\mathbb T}^6$ compactifications of heterotic strings, but ${\mathcal N}=1$ supergravity has the advantage of being much less restrictive than its extended counterparts. To our purposes, the simplest and most obvious choice of coupling is to ${\mathcal N}=1$ chiral multiplets. That is what we do in the following subsection.
${\cal W}_+^4 + {\cal W}_-^4$ in ${\mathcal N}=1$ matter-coupled supergravity
-----------------------------------------------------------------------------
The ${\mathcal N}=1$ supergravity multiplet is very simple. What also makes this theory easier is the existence of several different full off-shell formulations. We work in standard “old minimal” supergravity, having as auxiliary fields a vector $A_{A\dot A}$, a scalar $M$ and a pseudoscalar $N$, given as $\theta=0$ components of superfields $G_{A\dot A}, R,
\overline{R}:$[^2] $$\left. G_{A\dot A}\right| =\frac{1}{3} A_{A\dot A},
\left. \overline{R}\right| = 4 \left( M+iN \right),
\left. R \right| = 4 \left( M-iN \right).$$ Besides there is a chiral superfield $W_{ABC}$ and its hermitian conjugate $W_{\dot A \dot B \dot C}$, which together at $\theta=0$ constitute the field strength of the gravitino. The Weyl tensor shows up as the first $\theta$ term: in the notation of (\[wpms\]), at the linearized level, $$\left.\nabla _{\underline{D}}W_{\underline{ABC}}\right|={\cal
W}_{ABCD} + \ldots \label{wabcd}$$ ${\cal W}_+^4 + {\cal W}_-^4$ is proportional to the $\theta=0$ term of $\left( \nabla^2 W^2 \right)^2 +\mathrm{h.c.},$ which cannot result from a superspace integration. This whole term itself is U(1) $R$-symmetric, like $\nabla _{\underline{D}}W_{\underline{ABC}}$; indeed, the components of the Weyl tensor are U(1) $R$-neutral, according to the weights [@Moura:2002ft] $$\nabla_A \mapsto +1,
R \mapsto +2, G_m \mapsto 0, W_{ABC} \mapsto -1.$$
This way, as expected, one needs some extra coupling to (\[r442\]) in order to break U(1) $R$-symmetry. We can use the fact that there are many more matter fields with its origin in string theory and many different matter multiplets to which one can couple the ${\mathcal N}=1$ supergravity multiplet in order to build superinvariants. This way we hope to find some coupling which breaks U(1) $R$-symmetry and simultaneously supersymmetrizes (\[r442\]), which could result from the elimination of the matter auxiliary fields.
Having this in mind, we consider a chiral multiplet, represented by a chiral superfield $\mathbf{\Phi}$ (we could take several chiral multiplets $\Phi_i$, but we restrict ourselves to one for simplicity), and containing a scalar field $\Phi = \left.
\mathbf{\Phi} \right|$, a spin$-\frac{1}{2}$ field $\left. \nabla_A
\mathbf{\Phi} \right|$, and an auxiliary field $F=-\frac{1}{2}
\left. \nabla^2 \mathbf{\Phi} \right|$. This superfield and its hermitian conjugate couple to ${\mathcal N}=1$ supergravity in its simplest version through a superpotential $$P\left(\mathbf{\Phi}\right)=d + a \mathbf{\Phi} + \frac{1}{2} m
\mathbf{\Phi}^2 + \frac{1}{3} g \mathbf{\Phi}^3 \label{p}$$ and a Kähler potential $K\left(\mathbf{\Phi},
\overline{\mathbf{\Phi}} \right)=-\frac{3}{\kappa^2} \ln
\left(-\frac{\Omega\left(\mathbf{\Phi}, \overline{\mathbf{\Phi}}
\right)}{3} \right),$ with $\Omega\left(\mathbf{\Phi},
\overline{\mathbf{\Phi}} \right)$ given by $$\Omega\left(\mathbf{\Phi}, \overline{\mathbf{\Phi}} \right)=-3+
\mathbf{\Phi} \overline{\mathbf{\Phi}} + c \mathbf{\Phi} +
\overline{c} \overline{\mathbf{\Phi}}. \label{o}$$
In order to include the term (\[r442\]), we take the following effective action: $$\begin{aligned}
{\cal L}&=&-\frac{1}{6 \kappa^2} \int E
\left[\Omega\left(\mathbf{\Phi}, \overline{\mathbf{\Phi}} \right) +
\a^3 \left(b \mathbf{\Phi} \left(\nabla^2 W^2\right)^2 +
\overline{b} \overline{\mathbf{\Phi}} \left(\overline{\nabla}^2
\overline{W}^2\right)^2 \right) \right] d^4\theta \nonumber
\\ &-&\frac{2}{\kappa^2} \left(\int \epsilon
P\left(\mathbf{\Phi}\right) d^2\theta + \mathrm{h.c.} \right)
\nonumber \\ &=& \frac{1}{4 \kappa^2} \int \epsilon \left[ \left(
\overline{\nabla}^2 +\frac{1}{3} \overline{R} \right) \left(
\Omega\left(\mathbf{\Phi}, \overline{\mathbf{\Phi}} \right) + \a^3
\left(b \mathbf{\Phi} \left(\nabla^2 W^2\right)^2 + \overline{b}
\overline{\mathbf{\Phi}} \left(\overline{\nabla}^2
\overline{W}^2\right)^2 \right) \right) \right. \nonumber \\ && \,
\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,
- \left. 8 P\left(\mathbf{\Phi}\right) \right] d^2\theta +
\mathrm{h.c.}. \label{r421}\end{aligned}$$ $E$ is the superdeterminant of the supervielbein; $\epsilon$ is the chiral density. The $\Omega\left(\mathbf{\Phi},
\overline{\mathbf{\Phi}} \right)$ and $P\left(\mathbf{\Phi}\right)$ terms represent the most general renormalizable coupling of a chiral multiplet to pure supergravity [@Cremmer:1978hn]; the extra terms represent higher-order corrections. Of course (\[r421\]) is meant as an effective action and therefore does not need to be renormalizable.
The component expansion of this action may be found using the explicit $\theta$ expansions for $\epsilon$ and $\nabla^2 W^2$ given in [@Moura:2002ft]. From (\[wabcd\]), we have $$\left. \nabla^2 W^2 \right| = -2{\cal W}_+^2 + \ldots \label{d2w2}$$
It is well known that an action of this type in pure supergravity (without the higher-order corrections) will give rise, in $x$-space, to a leading term given by $\frac{1}{6 \kappa^2} e
\left. \Omega \right| {\mathcal R}$ instead of the usual $-\frac{1}{2 \kappa^2} e {\mathcal R}.$[^3] In order to remove the extra $\Phi {\mathcal R}$ terms in $\frac{1}{6 \kappa^2} e \left. \Omega
\right| {\mathcal R}$, one takes a $\Phi,
\overline{\Phi}$-dependent conformal transformation [@Cremmer:1978hn]; if one also wants to remove the higher order $\Phi {\mathcal R}$ terms, this conformal transformation must be $\a$-dependent. Here we are only interested in obtaining the supersymmetrization of ${\cal W}_+^4 + {\cal W}_-^4$; therefore we will not be concerned with the Ricci terms of any order.
If one expands (\[r421\]) in components, one does not directly get (\[r442\]), but one should look at the auxiliary field sector. Because of the presence of the higher-derivative terms, the auxiliary field from the original conformal supermultiplet $A_m$ also gets higher derivatives in its equation of motion, and therefore it cannot be simply eliminated [@Moura:2001xx; @Moura:2002ip]. Here we only consider the much simpler terms which include the chiral multiplet auxiliary field $F$. Take the superfields $$\mathbf{\tilde{C}}= c + \a^3 b \left(\nabla^2 W^2\right)^2,
\widetilde{\Omega}\left(\mathbf{\Phi}, \overline{\mathbf{\Phi}},
\mathbf{\tilde{C}}, \overline{\mathbf{\tilde{C}}} \right)= -3 +
\mathbf{\Phi} \overline{\mathbf{\Phi}} + \mathbf{\tilde{C}}
\mathbf{\Phi} + \overline{\mathbf{\tilde{C}}}
\overline{\mathbf{\Phi}}, \label{ctil}$$ so that the action (\[r421\]) becomes $$\frac{1}{4 \kappa^2} \int \epsilon \left[ \left( \overline{\nabla}^2
+\frac{1}{3} \overline{R} \right)
\widetilde{\Omega}\left(\mathbf{\Phi}, \overline{\mathbf{\Phi}},
\mathbf{\tilde{C}}, \overline{\mathbf{\tilde{C}}} \right)- 8
P\left(\mathbf{\Phi}\right) \right] d^2\theta + \mathrm{h.c.}$$ and all the $\a^3$ corrections considered in it become implicitly included in $\widetilde{\Omega}\left(\mathbf{\Phi},
\overline{\mathbf{\Phi}}, \mathbf{\tilde{C}},
\overline{\mathbf{\tilde{C}}} \right)$ through $\mathbf{\tilde{C}},
\overline{\mathbf{\tilde{C}}}.$ We also define $\tilde{C} = \left.
\mathbf{\tilde{C}} \right|$ and the functional derivative $P_{\mathbf{\Phi}}=\partial P/
\partial \mathbf{\Phi}.$ From now on, we will work in $x$-space and assume there is no confusion between the superfield functionals $\widetilde{\Omega}\left(\mathbf{\Phi}, \overline{\mathbf{\Phi}},
\mathbf{\tilde{C}}, \overline{\mathbf{\tilde{C}}} \right)$, $P\left(\mathbf{\Phi}\right)$, $P_{\mathbf{\Phi}}$ and their corresponding $x$-space functionals $\widetilde{\Omega}\left(\Phi,
\overline{\Phi}, \tilde{C}, \overline{\tilde{C}} \right)$, $P\left(\Phi\right)$, $P_{\Phi}.$ The terms we are looking for are given by [@Cremmer:1978hn] $$\begin{aligned}
\kappa^2 {\cal L}_{F, \overline{F}}&=& \frac{1}{9}e
\widetilde{\Omega}\left(\Phi, \overline{\Phi}, \tilde{C},
\overline{\tilde{C}} \right) \left| M-i N
-\frac{3}{\widetilde{\Omega}\left(\Phi, \overline{\Phi}, \tilde{C},
\overline{\tilde{C}} \right)} \left(\Phi + \overline{\tilde{C}}
\right) F\right|^2 \nonumber \\ &-& e \frac{3+ \tilde{C}
\overline{\tilde{C}}}{\widetilde{\Omega}^2\left(\Phi,
\overline{\Phi}, \tilde{C}, \overline{\tilde{C}} \right)} F
\overline{F} +e \tilde{P}_{\Phi} F +e
\overline{\tilde{P}}_{\overline{\Phi}} \overline{F}. \label{ptil}\end{aligned}$$ This equation would be exact, with $\tilde{P}_{\Phi}=P_{\Phi}$ and $\overline{\tilde{P}}_{\overline{\Phi}} =
\overline{P}_{\overline{\Phi}}$, if we were only considering the $\theta=0$ components of $\mathbf{\tilde{C}},
\overline{\mathbf{\tilde{C}}}$. But, of course (as it is clear from (\[r421\])), coupled to $F$ we will have $\nabla_{\dot A}
\left(\nabla^2 W^2\right)^2$ and $\overline{\nabla}^2 \left(\nabla^2
W^2\right)^2$ terms (and $\nabla_A \left(\overline{\nabla}^2
\overline{W}^2\right)^2$ and $\nabla^2 \left(\overline{\nabla}^2
\overline{W}^2\right)^2$ terms coupled to $\bar{F}$). These terms will not play any role for our purpose (which is to show that there exists a supersymmetric lagrangian which contains (\[r442\]), and not necessarily to compute it in full), and therefore we do not compute them explicitly. We write them in (\[ptil\]) because we include them in $\tilde{P}_{\Phi}$, through the definition (analogous for $\overline{\tilde{P}}_{\overline{\Phi}}$) $$\tilde{P}_{\Phi}=P_{\Phi} +\left(\nabla_{\dot A}
\mathbf{\tilde{C}} + \overline{\nabla}^2 \mathbf{\tilde{C}} \,
\mathrm{terms} \right).$$
The first term in (\[ptil\]) contains the well known term $-\frac{1}{3} e \left(M^2 + N^2 \right)$ from “old minimal” supergravity. Because the auxiliary fields $M, N$ belong to the chiral compensating multiplet, their field equation should be algebraic, despite the higher derivative corrections [@Moura:2001xx; @Moura:2002ip]. That calculation should still require some effort; plus, those $M, N$ auxiliary fields should not generate *by themselves* terms which violate U(1) $R$-symmetry: these terms should only occur through the elimination of $F,
\bar{F}.$ This is why we will only be concerned with these auxiliary fields, which therefore can be easily eliminated through their field equation $$\left(\frac{\left(\overline{\Phi} + \tilde{C} \right) \left(\Phi
+ \overline{\tilde{C}} \right)}{\widetilde{\Omega} \left(\Phi,
\overline{\Phi}, \tilde{C}, \overline{\tilde{C}} \right)} - \frac{3+
\tilde{C} \overline{\tilde{C}}}{\widetilde{\Omega}^2 \left(\Phi,
\overline{\Phi}, \tilde{C}, \overline{\tilde{C}} \right)} \right) F=
- \overline{\tilde{P}}_{\overline{\Phi}} - \frac{1}{3}
\left(\overline{\Phi} + \tilde{C} \right) \left( M-i N \right).$$ Replacing $F, \bar{F}$ in ${\cal L}_{F, \overline{F}}$, one gets $$\kappa^2 {\cal L}_{F, \overline{F}} =- e \frac{ \tilde{P}_{\Phi}
\overline{\tilde{P}}_{\overline{\Phi}} \widetilde{\Omega}^2
\left(\Phi, \overline{\Phi}, \tilde{C}, \overline{\tilde{C}}
\right)}{\left(\overline{\Phi} + \tilde{C} \right) \left(\Phi +
\overline{\tilde{C}} \right) \widetilde{\Omega} \left(\Phi,
\overline{\Phi}, \tilde{C}, \overline{\tilde{C}} \right) - \left(
\tilde{C} \overline{\tilde{C}} +3 \right)} + M, N\,\,
\mathrm{terms.}$$ This is a nonlocal, nonpolynomial action. Since we take it as an effective action, we can expand it in powers of the fields $\Phi,
\overline{\Phi}$, but also in powers of $\tilde{C},
\overline{\tilde{C}}.$ These last fields contain both the couplings of $\mathbf{\Phi}$ to supergravity $c$ and the string parameter $\a;$ expanding in these fields is equivalent to expanding in a certain combination of these parameters. Here one should notice that we are only considering up to $\a^3$ terms. If we wanted to consider higher (than $\a^3$) order corrections, together with these we should also have included *a priori* in (\[r421\]) the leading higher order corrections, which should be independently supersymmetrized. Considering solely the higher than $\a^3$ order corrections coming directly from the elimination of (any of) the auxiliary fields from the $\a^3$ effective action (\[r421\]) would be misleading. The correct expansion of (\[r421\]) to take, in the first place, is in $\a^3$. That is what we do in the following, after replacing $\tilde{C}, \overline{\tilde{C}}$ by their explicit superfield expressions given by (\[ctil\]) and taking $\theta=0$. We also exclude the $M, N$ contributions and the higher $\theta$ terms from $\mathbf{\tilde{C}}, \mathbf{\overline{\tilde{C}}}$ in $\tilde{P}_{\Phi}, \overline{\tilde{P}}_{\overline{\Phi}}$, for the reasons mentioned before: they are not significant for the term we are looking for. The resulting lagrangian we get (which we still call ${\cal L}_{F, \overline{F}}$ to keep its origin clear, although it is not anymore the complete lagrangian resulting from the elimination of $F, \bar{F}$) is $$\begin{aligned}
\kappa^2 {\cal L}_{F, \overline{F}}&=& - e \frac{ P_{\Phi}
\overline{P}_{\overline{\Phi}} \Omega^2 \left(\Phi,
\overline{\Phi}\right)}{\left(\overline{\Phi} + c \right) \left(\Phi
+ \overline{c} \right) \Omega \left(\Phi, \overline{\Phi}\right) -
\left( c \overline{c} +3 \right)}
\\ &+& \a^3 \frac{e P_{\Phi} \overline{P}_{\overline{\Phi}} \Omega
\left(\Phi, \overline{\Phi} \right)}{\left(\left(\overline{\Phi} + c
\right) \left(\Phi + \overline{c} \right) \Omega \left(\Phi,
\overline{\Phi} \right) - \left( c \overline{c} +3 \right)\right)^2}
\left[-2 \left(
b \Phi \left(\nabla^2 W^2\right)^2\right| \right.\nonumber \\
&+& \left. \left. \overline{b} \overline{\Phi}
\left(\overline{\nabla}^2 \overline{W}^2\right)^2 \right| \right)
\left(\left(\overline{\Phi} + c \right) \left(\Phi + \overline{c}
\right) \Omega \left(\Phi, \overline{\Phi} \right) - \left( c
\overline{c} +3 \right) \right)\nonumber \\ &+& \Omega \left(\Phi,
\overline{\Phi} \right) \left( -b \overline{c} \Phi
\left.\left(\nabla^2 W^2\right)^2\right| - \overline{b} c
\overline{\Phi} \left.\left(\overline{\nabla}^2
\overline{W}^2\right)^2 \right| \right.\nonumber \\
&+& \left(\overline{\Phi} + c \right) \left(\Phi + \overline{c}
\right) \left(b \Phi \left.\left(\nabla^2 W^2\right)^2\right| +
\left. \overline{b} \overline{\Phi} \left(\overline{\nabla}^2
\overline{W}^2\right)^2 \right| \right)\nonumber \\ &+& \left.
\left. \Omega \left(\Phi, \overline{\Phi} \right) \left( b \left(
\overline{c} +\Phi \right)\left.\left(\nabla^2 W^2\right)^2\right| +
\overline{b} \left(c+ \overline{\Phi} \right)
\left.\left(\overline{\nabla}^2 \overline{W}^2\right)^2 \right|
\right) \right) \right] + \ldots \nonumber\end{aligned}$$ If we look at the last line of the previous equation, we can already identify the term we are looking for. This is still a nonlocal, nonpolynomial action, which we expand now in powers of the fields $\Phi, \overline{\Phi}$ coming from the denominators and the $P_{\Phi} \overline{P}_{\overline{\Phi}}$ factors. We obtain $$\begin{aligned}
\kappa^2 {\cal L}_{F, \overline{F}}&=& -15 e \frac{ \left(3 + c
\overline{c}\right)}{\left(3 + 4 c \overline{c}\right)^2} \left(m
\overline{a} \Phi + \overline{m} a \overline{\Phi} \right) \left(c
\Phi + \overline{c} \overline{\Phi} \right) \nonumber \\ &+& e
\frac{2 c^3 \overline{c}^3 + 60 c^2 \overline{c}^2 + 117 c
\overline{c}-135}{\left(3 + 4 c \overline{c}\right)^3} a
\overline{a} \Phi \overline{\Phi} - 36 \a^3 e \left( b
\overline{c} \left(\nabla^2 W^2\right)^2\right| \nonumber \\ &+&
\overline{b} c \left. \left. \left(\overline{\nabla}^2
\overline{W}^2\right)^2 \right| \right) \frac{a \overline{a} + m
\overline{a} \Phi + \overline{m} a \overline{\Phi} + g
\overline{a} \Phi^2 + \overline{g} a \overline{\Phi}^2 + m
\overline{m} \Phi \overline{\Phi} }{\left(3 + 4 c
\overline{c}\right)^2} \nonumber
\\ &-& 3 \a^3 a\overline{a} \frac{74 c^2 \overline{c}^2 + 192 c
\overline{c}-657}{\left(3 + 4 c \overline{c}\right)^4} \Phi
\overline{\Phi} \left( b \overline{c} \left(\nabla^2
W^2\right)^2\right| + \overline{b} c \left. \left.
\left(\overline{\nabla}^2 \overline{W}^2\right)^2 \right| \right)
\nonumber \\ &+& 15 \a^3 e \frac{a \overline{a} + m \overline{a}
\Phi + \overline{m} a \overline{\Phi}}{\left(3 + 4 c
\overline{c}\right)^3} \left[ \left( \overline{c}^2 \left(21 + 4 c
\overline{c}\right) \overline{\Phi} + \left(-9 + 6 c
\overline{c}\right) \Phi \right) b \left(\nabla^2
W^2\right)^2\right| \nonumber \\ &+& \left. \left( c^2 \left(21 +
4 c \overline{c}\right) \Phi + \left(-9 + 6 c \overline{c}\right)
\overline{\Phi} \right) \overline{b} \left.
\left(\overline{\nabla}^2 \overline{W}\right)^2\right| \right] +
\ldots \label{r42s}\end{aligned}$$ This way we are able to supersymmetrize ${\cal W}_+^4 + {\cal
W}_-^4$, although we had to introduce a coupling to a chiral multiplet. These multiplets show up after $d=4$ compactifications of superstring and heterotic theories and truncation to ${\mathcal
N}=1$ supergravity [@Cecotti:1987nw]. Since from (\[d2w2\]) the factor in front of ${\cal W}_+^4$ (resp. ${\cal W}_-^4$) in (\[r42s\]) is given by $\frac{72 b \overline{c} a \overline{a}}{\left(3 + 4 c
\overline{c}\right)^2}$ (resp. $\frac{72 \overline{b} c a
\overline{a}}{\left(3 + 4 c \overline{c}\right)^2}$), for this supersymmetrization to be effective, the factors $a$ from $P\left(\Phi\right)$ in (\[p\]) and $c$ from $\Omega \left(\Phi,
\overline{\Phi}\right)$ in (\[o\]) (and of course $b$ from (\[r421\])) must be nonzero.
The action (\[r42s\]) includes the ${\mathcal N}=1$ supersymmetrization of ${\cal W}_+^4 + {\cal W}_-^4$, but without any coupling to a scalar field or only with couplings to powers of the scalar field from the chiral multiplet, which may be seen as compactification moduli. But, as one can see from (\[2aea4\]), (\[hea4\]), this term should be coupled to powers of the dilaton. It is well known [@Cecotti:1987nw] that in ${\mathcal
N}=1$ supergravity the dilaton is part of a linear multiplet, together with an antisymmetric tensor field and a Majorana fermion. One must then work out the coupling to supergravity of the linear and chiral multiplets. As usual one starts from conformal supergravity and obtain Poincaré supergravity by coupling to compensator multiplets which break superconformal invariance through a gauge fixing condition. When there are only chiral multiplets coupled to supergravity [@Cremmer:1978hn], this gauge fixing condition can be generically solved, so that a lagrangian has been found for an arbitrary coupling of the chiral multiplets. In the presence of a linear multiplet, there is no such a generic solution of the gauge fixing condition, which must be solved case by case. Therefore, there is no generic lagrangian for the coupling of supergravity to linear multiplets. We shall not consider this problem here, like we did not in [@Moura:2001xx; @Moura:2002ft]. In both cases we were only interested in studying the ${\mathcal N}=1$ supersymmetrization of the two different $d=4$ ${\cal R}^4$ terms. The coupling of a linear multiplet to these terms can be determined following the procedure in [@Derendinger:1994gx].
${\cal W}_+^4 + {\cal W}_-^4$ in extended supergravity
------------------------------------------------------
${\cal W}_+^4 + {\cal W}_-^4$ must also arise in extended $d=4$ supergravity theories, for the reasons we saw, but the “no-go” result of ([@Christensen:1979qj]) should remain valid, since it was obtained for ${\mathcal N}=1$ supergravity, which can always be obtained by truncating any extended theory. For extended supergravities, the chirality argument should be replaced by preservation by supergravity transformations of U(1), which is a part of $R$-symmetry.
${\mathcal N}=2$ supersymmetrization of ${\cal W}_+^4 + {\cal W}_-^4$ should work in a way similar to what we saw for ${\mathcal N}=1$. ${\mathcal N}=2$ chiral superfields must be Lorentz and SU(2) scalars but they can have an arbitrary U(1) weight, which allows supersymmetric U(1) breaking couplings.
A similar result should be more difficult to implement for ${\mathcal N} \geq 3$, because there are no generic chiral superfields. Still, there are other multiplets than the Weyl, which one can consider in order to couple to ${\cal W}_+^4 + {\cal W}_-^4$ and allow for its supersymmetrization. The only exception is ${\mathcal N}=8$ supergravity, which only allows for the Weyl multiplet. ${\mathcal N}=8$ supersymmetrization of ${\cal W}_+^4 + {\cal W}_-^4$ should therefore be a very difficult problem, which we expect to study in a future work.
Related to this is the issue of possible finiteness of ${\mathcal
N}=8$ supergravity, which has been a recent topic of research. A linearized three-loop candidate (the square of the Bel-Robinson tensor) has been presented in [@Kallosh:1980fi]. But recent works [@Bern:2006kd] show that there is no three-loop divergence (which includes the two ${\cal R}^4$ terms). Power-counting analysis from unitarity cutting-rule techniques predicted the lowest counterterm to appear at least at five loops [@Bern:1998ug]. An improved analysis based on harmonic superspace power-counting improved this lower limit to six loops [@Howe:2002ui]. In [@Howe:1980th] a seven loop counterterm was proposed, but in [@Green:2006gt] it is proposed from string perturbation theory arguments that the four graviton amplitude may be eight-loop finite. The claim in [@Bern:2006kd] is even stronger: ${\mathcal N}=8$ supergravity may have the same degree of divergence as ${\mathcal
N}=4$ super-Yang-Mills theory and may therefore be ultraviolet finite. But no definitive calculations have been made yet to prove that claim; up to now, there is no firmly established example of a counterterm which does not arise in the effective actions but would be allowed by superspace non-renormalization theorems.
Because of all these open problems, we believe that higher order terms in ${\mathcal N}=8$ supergravity definitely deserve further study.
Conclusions
===========
In this paper, we analyzed in detail the reduction to four dimensions of the purely gravitational higher-derivative terms in the string effective actions, up to order $\a^3$, for heterotic and type IIA/IIB superstrings. From this analysis we have shown that in the four dimensional heterotic and type IIA string effective actions there must exist, besides the usual square of the Bel-Robinson tensor ${\cal W}_+^2 {\cal W}_-^2$, a new ${\cal
R}^4$ term given in terms of the Weyl tensor by ${\cal W}_+^4 +
{\cal W}_-^4$. This new term results from the dimensional reduction of the order $\a^3$ effective actions, at one string loop, of these theories. By requiring four dimensional supersymmetry, this term must be, like any other, part of some superinvariant, but it had been shown, under some assumptions (conservation of chirality), that such a superinvariant could not exist by itself in pure ${\mathcal N}=1$ supergravity. But, by taking a specific (chirality-breaking) coupling of this term to a chiral multiplet in ${\mathcal N}=1$ supergravity, we were indeed able to obtain the desired superinvariant. The ${\cal W}_+^4 +
{\cal W}_-^4$ term appeared after elimination of its auxiliary fields, by itself, without any couplings to the chiral multiplet fields.
To summarize, we have demonstrated the existence of a new ${\cal
R}^4$ superinvariant in $d=4$ supergravity, a result that many people would find unexpected. The supersymmetrization of this new ${\cal R}^4$ term in extended supergravity remains an open problem, but we found it in ${\mathcal N}=1$ supergravity. As we concluded from our analysis of the dimensional reduction of order $\a^3$ gravitational effective actions, this new ${\cal R}^4$ term has its origin in the dimensional reduction of the corresponding term in M-theory, a theory of which there is still a lot to be understood. We believe therefore that the complete study of this term and its supersymmetrization deserves further attention in the future.
Acknowledgments {#acknowledgments .unnumbered}
===============
I wish to thank Pierre Vanhove for very important discussions, suggestions and comments on the manuscript. I also wish to thank Paul Howe for very useful correspondence and Martin Roček for nice suggestions and for having persuaded me to consider the ${\mathcal N}=1$ case. It is a pleasure to acknowledge the excellent hospitality of the Service de Physique Théorique of CEA/Saclay in Orme des Merisiers, France, where some parts of this work were completed.
This work has been supported by Fundação para a Ciência e a Tecnologia through fellowship BPD/14064/2003 and Centro de Lógica e Computação (CLC).
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[^1]: In the previous section, we used latin letters - $m,n, \ldots$ - to represent ten dimensional spacetime indices. From now on we will be only working with four dimensional spacetime indices which, to avoid any confusion, we represent by greek letters $\mu, \nu, \ldots$
[^2]: The ${\mathcal N}=1$ superspace conventions are exactly the same as in [@Moura:2001xx; @Moura:2002ft].
[^3]: As usual in supergravity theories we work with the vielbein and not with the metric. Therefore, here we write $e$, the determinant of the vielbein, instead of $\sqrt{-g}.$
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
Dino Giovannoni$^{1}$[^1], Jeff Murugan$^{1,2}$[^2] and Andrea Prinsloo$^{1,2}$[^3]\
$^{1}$Astrophysics, Cosmology & Gravity Center and\
Department of Mathematics and Applied Mathematics,\
University of Cape Town,\
Private Bag, Rondebosch, 7700,\
South Africa.\
$^{2}$National Institute for Theoretical Physics,\
Private Bag X1,\
Matieland, 7602,\
South Africa.
title: 'The giant graviton on $AdS_{4}\times\mathbb{CP}^{3}$ - another step towards the emergence of geometry'
---
[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'A. Kandus, M.J. Vasconcelos,'
- 'A.H. Cerqueira'
date: 'Received 29 August 2005 / Accepted 7 March 2006'
title: On the mean field dynamo with Hall effect
---
Introduction
============
The origin and evolution of magnetic fields observed in all objects of the universe is one of the main problems in astrophysics. The basic physical process assumed to create them is a dynamo, which needs two basic ingredients, a seed field and an amplifying mechanism, each of them constituting at present an independent line of research (e.g., Grasso & Rubinstein [@rev-dg], Widrow [@rev-lw], Giovannini [@rev-mg]). An amplifying mechanism usually considered is the so called turbulent Mean Field Dynamo (MFD), as turbulence is normally present in astrophysical environments. In this mechanism it is assumed that turbulence is excited at a small scale $\ell_s$ and that as a consequence a magnetic field is induced at a larger scale $\ell_L$. This theory has been a useful framework for modeling local origin of large scale magnetic fields in stars and galaxies.
In the Universe there are very different astrophysical environments: compact stars, low density and low temperature plasmas, accretion disks around stars and in AGN’s, etc. The plasma in each of those ambients has a different composition and therefore different physical processes may be relevant: in low ionized plasmas as the interstellar medium, ambipolar difussion is important (Zweibel [@zweibel]); in the high-temperature intracluster gas ohmic dissipation plays a major role, and Hall effect can be relevant in accretion disks (Sano & Stone [@stone], Wardle [@wardle99], Balbus & Terquem [@balb-terq]) as well as in the early universe (Tajima et al. [@tajima]). Turbulent dynamo operation may therefore be affected by the composition of the plasma: if we consider a plasma formed by, e.g. protons, electrons and neutrals, then the different interactions among these constituents can be expressed as a generalized Ohm’s law (Spitzer [@spitzer], Priest [@priest]).
One of the main steps in the development of a turbulent mean field (or large scale) dynamo theory was the recognition of the pivotal role played by magnetic helicity (e.g., Pouquet et al. [@pfl], Blackman & Field [@fb2000-1; @fb2000-2], Brandenburg [@bran-2001]). In the absence of resistive dissipation, and for boundary conditions such that total divergencies vanish, this quantity is globally conserved, independently of any assumption about the turbulent state of the system. Its evolution does not explicitly depend on the non-linear backreaction due to Lorentz force, it merely depends on the induction equation, providing therefore a strong constraint on the nonlinear evolution of the large scale magnetic field.
As stated above, mean field dynamo amounts to split the fields into large scale mean fields ${\bf U}_0$, ${\bf B}_0$, ${\bf A}_0$ and small scale turbulent fields ${\bf u}$, ${\bf b}$, ${\bf a}$. This small scale fields represent the “waste product” of turbulence, and they can be very intense in spite of their small coherence length [^1]. In this theory the evolution equation for ${\bf B}_0$ can be cast as $\partial
{\bf B}_0/\partial t = {\bf \nabla} \times \left({\bf U}_0\times {\bf
B}_0 + {\bf \varepsilon} - \eta {\bf J}_0\right)$, where ${\bf J}_0$ is the mean electric current, $\eta$ the resisitivity and ${\bf \varepsilon}
= \langle {\bf u}\times {\bf b}\rangle_0$ the turbulent electromotive force (e.m.f.). In the two scale approach it is assumed that ${\bf
\varepsilon}$ can be expanded in powers of the gradients of ${\bf B}_0$ in the rather general form $ \varepsilon_i = \alpha_{ij}\left(\hat {\bf g},
\hat{\bf \Omega},{\bf B}_0,...\right) B_{0j} + \eta_{ijk}\left(\hat{\bf
g},\hat {\bf \Omega},{\bf B}_0,...\right) \partial B_{0j} /\partial x_k$, where the functions $\alpha_{ij}$ and $\eta_{ijk}$ are called turbulent transport coefficients. They depend on the stratification $\hat {\bf g}$, angular velocity $\hat {\bf \Omega }$ and mean magnetic field ${\bf
B}_0$.They may also depend on correlators involving the small scale magnetic field in the form, for example, of small scale current helicity.
The simplest way of calculating the turbulent transport coefficients consists of linearizing the equations for the small scale quantities, ignoring quadratic terms that would lead to triple correlations in the expressions for the quadratic terms. In other words, the backreaction of mean field ${\bf B}_0$ on the correlation tensor of the turbulence is taken into account, while neglecting the effect of the small scale fields. In this way the electromotive force can be written as (Krause & Rädler [@krause]) ${\bf \varepsilon} = \alpha {\bf
B}_0 - \beta {\bf J}_0$ with $\alpha \simeq -(1/3)\tau_{corr}\langle
{\bf u}\cdot {\bf \nabla}\times {\bf u}\rangle_0$ and $\beta \simeq
(1/3)\tau_{corr}\langle u^2\rangle_0$, with $\tau_{corr}$ the correlation time. These modifications of the turbulent transport coefficients have been calculated about thirty years ago, and the approximation is known as [*First order smoothing approximation*]{}, or FOSA (see also Moffat [@moffat2], Rüdiger [@rudiger] Parker [@parker]; Moffat [@moffat]; Zel’dovich, Ruzmaikin & Sokoloff [@zeldovich]).
There remains to incorporate the modifications to ${\bf \varepsilon}$ that involve the small scale, fluctuating fields. They arise when calculating $\langle {\bf u} \times {\bf b}\rangle_0$ from terms involving the nonlinear terms and the Lorentz force in the evolution equations for ${\bf b}$ and ${\bf u}$ respectively. As a consequence, the $\alpha$ term written above gets renormalized in the nonlinear regime by the addition of a term proportional to the current helicity $\langle {\bf b}\cdot \left( {\bf \nabla}\times {\bf b}\right)$ of the fluctuating field, which in turn is related to the magnetic helicity of the small scale magnetic field. The $\beta$ term on the other side is not affected by the backreaction of the small scale fields (Pouquet et all [@pfl], Subramanian & Brandenburg [@sub-bran], Brandemburg & Subramanian [@report-bs]). In this way we have $\alpha \simeq -(1/3)
\tau_{corr} \left( \langle {\bf u}\cdot \left({\bf \nabla} \times {\bf
u}\right)\rangle_0 - \langle {\bf b} \cdot \left( {\bf \nabla}\times
{\bf b}\right)\rangle_0\right)$.
In this paper we investigate how Hall effect modifies the process of quenching of the e.m.f. ${\bf \varepsilon}$, described in the previous paragraphs. For this purpose we use a closure scheme recently introduced by Blackman & Field ([@fb2002-2; @fb2004]), that permits to study dinamically the backreaction of both the large and small scale fields on ${\bf \varepsilon}$. This closure, also named the “minimal $\tau$ approximation”, consists in finding the evolution equation for the electromotive force instead of finding ${\bf \varepsilon}$ itself. In it, three point correlations of the generated small scale fields, $\bar {\bf T}$, are not neglected, but their sum is assumed to be a negative multiple of the second order correlator, i.e. $\bar {\bf
T}=-{\bf \varepsilon}/\tau$. This assumption produces results that are in very good agreement with numerical simulations (Brandenburg & Subramanian [@report-bs]).
Hall effect is taken into account by considering a corresponding term in Ohm’s law (see Spitzer [@spitzer], Priest [@priest]). The parameter that meassures its intensity is the [*Hall length*]{}, that in alfvénic units is defined as $\ell_H = \left( 4\pi\rho\right)^{1/2}/n_e
e$, with $\rho$ the mass density and $n_e$ the electronic numerical density. Of interest is the ratio of this length, to the Ohmic dissipation length, $\ell_{\eta}$, and to the scale of the flow, $\ell_u$. For $\ell_H
\lesssim l_{\eta}$, the Hall-MHD equations reduce to those of standard MHD, as ohmic dissipation erases any other interaction. In several astrophysical problems, such as accretion disks, protoplanetary disks, the early universe plasma and the magnetopause (Birn et al [@birn], Balbus & Terquem [@balb-terq], Sano & Stone [@stone], Tajima et al [@tajima]), the Hall scale is larger than the Ohmic scale, but it can be smaller or larger than $\ell_u$.
Under the hypothesis that large scale fields are force-free (which is a reasonable assumption in many astrophysical environments) we obtain evolution equations for the mean magnetic field and for the large and small scale magnetic helicities, that are formally identical to those obtained in absence of Hall effect, provided that we redefine the electromotive force by ${\bf \varepsilon}_H = \langle {\bf u}_e\times {\bf
b}\rangle$ with $u_e = u - \ell_H \langle \left( {\bf \nabla}\times {\bf
b}\right)\times {\bf b}\rangle_0$. This means that the component of ${\bf
\varepsilon}_H$ along the mean field ${\bf B}_0$ governs the evolution of magnetic helicity, and in turn magnetic helicity influences the growth of ${\bf B}_0$ ( Parker [@parker], Ji [@ji-99]). We study a system of coupled evolution equations for ${\bf \varepsilon}_H$ and for the large and small scale magnetic helicities. As mentioned above, the evolution equations for the helicities are formally identical to the ones in absence of Hall effect. In contrast, the equation for ${\bf
\varepsilon}_H$ presents substantial differences in comparison to the standard MHD case: [*(i)*]{} In the $\alpha$ term, proportional to ${\bf
B}_0$, the fluid helicity is replaced by the electronic fluid helicity and there appears an extra term, explicitly dependent on $\ell_H$ that couples ${\bf b}$ with ${\bf u}_e$. [*(ii)*]{} In the turbulent diffusion term, proportional to ${\bf \nabla}\times {\bf B}_0$, the $\beta$ term, the fluid kinetic energy term $\langle u^2\rangle_)$ is replaced by $\langle
{\bf u}\cdot {\bf u}_e\rangle_0$ and there also appears a correction explicitly dependent on $\ell_H$, that couples ${\bf b}$ with ${\bf
u}$. All these modifications render this term not positive definite, a fact that could produce a transfer of energy from small scales toward large scales or, in other words an inverse cascade of energy (Mininni et al. [@pablito4]). [*(iii)*]{} There appears a new term, proportional to $\nabla^2 {\bf B}_0$, that again couples the mentioned velocities. The coupling of ${\bf u}_e$ with ${\bf b}$ indicates that Hall effect acts by transferring energy between these two fields in a non-trivial way.
In order to illustrate how Hall effect affects the quenching process of the mean field dynamo, we applied the obtained equations to a specific physical situation in which we considered that turbulence of maximal kinetic helicity is excited at a certain scale $\ell_s$ and that the large scale magnetic field is generated at a scale $\ell_L = 5\ell_s$. As for the Hall effect, we considered $\ell_s < \ell_H <\ell_L$ and treated it as a perturbation. We find that for this situation the overall effect is a quenching of the e.m.f. stronger than in standard MHD, acompanied by a supression of magnetic helicity inverse cascade. Our results are in qualitative agreement with recent numerical simulations performed by Mininni et al ([@pablito3]).
As the aim of this paper is to understand conceptually how Hall effect acts on a MFD, we did not apply our results to a concrete astrophysical object. We leave this issue for future work, after a deeper understanding of the mechanism is attained. The paper is organized as follows: in section §2 we present the main equations and deduce the evolution equations for large and small scale fields. In section §3 we deduce the dynamo equations, i.e., the ones for the stochastic electromotive force and for the large and small scale magnetic helicities. In section §4 we implement a two scale approximation and numerically integrate the system of equations and discuss the results. Finally in section §5 we sumarize our conclusions.
Main Equations
==============
In Magnetohydrodynamics, Hall effect can be taken into account through the generalized Ohm’s law as (e.g., Spitzer [@spitzer], Priest [@priest]):
$${\bf E}+{\bf U}\times {\bf B}=\frac{1}{n_e e}{\bf J}\times {\bf B}
+ \eta {\bf \nabla} \times {\bf B}\, ,
\label{a1}$$
with ${\bf J}={\bf \nabla }\times {\bf B}$, $n_e$ the electron number density, $e$ the modulus of the fundamental electric charge and $\eta$ the Ohmic diffusion coefficient. We need the magnetic field induction and the Navier Stokes equations. We use units in which the magnetic field has dimensions of velocity. To simplify the calculations and the comparison with previous works we shall consider an incompressible fluid, i.e., $\nabla \cdot {\bf U} = 0$[^2]. This condition is fulfilled in several astrophysical environments. The Navier-Stokes equation is then written as:
$$\frac{\partial {\bf U}}{\partial t} = -{\bf \bar P}\left[
\left( {\bf U}\cdot{\bf \nabla }\right) {\bf U}
- \left( {\bf B}\cdot {\bf \nabla}\right) {\bf B} \right]
- \nu {\bf \nabla}\times\left({\bf \nabla}\times {\bf U}\right)\, ,
\label{a2}$$
where $ {\bf \bar P} \equiv {\bf I} - {{\bf \nabla} {\bf
\nabla}\cdot}/ {\nabla^2}$ is the projector operator onto the subspace of solutions of this equation that satisfy the condition of incompressibility (McComb [@macomb]) and $\nu$ the kinematic viscosity. The induction equation reads:
$$\frac{\partial {\bf B}}{\partial t}={\bf \nabla }\times \left\{ {\bf U}
\times {\bf B}-\ell_H \left( {\bf \nabla }\times {\bf B}\right) \times
{\bf B}-\eta {\bf \nabla }\times {\bf B}\right\}\, , \label{a3}$$
where we defined the [*Hall length*]{}, $\ell_H$, as $\ell_H =
{\left( 4\pi\rho\right)^{1/2}}/{n_e e}$. We also need the equation for the vector potential ${\bf A}$. If we choose to work with the Coulomb gauge, i.e., ${\bf \nabla}\cdot {\bf A}=0$, it reads:
$$\frac{\partial {\bf A}}{\partial t}= {\bf \bar P} \left[
{\bf U} \times {\bf B}-\ell_H \left({\bf \nabla }\times {\bf B}\right)
\times {\bf B}\right]
-\eta {\bf \nabla}\times {\bf B}\, , \label{a4}$$
where ${\bf \bar P}$ is the previously defined projector, but now projecting onto the space of functions that satisfy the chosen gauge. When $\eta = 0$, Equation (\[a3\]) represents the freezing of the magnetic field to the electron flux. To see this, let us write ${\bf U}_e
= {\bf U} - \ell_H {\bf \nabla}\times{\bf B}$, which when substituted in eqs (\[a3\]) and (\[a4\]) transforms them in equations formally identical to the ones without Hall effect.
Large and Small Scale Fields
----------------------------
As we are interested in studying mean field dynamo, we split the fields ${\bf U}$, ${\bf B}$ and ${\bf A}$ as ${\bf U} = {\bf u}$, ${\bf B} =
{\bf B}_0 + {\bf b}$ and ${\bf A} = {\bf A}_0 + {\bf a}$. Upper case and subindex $0$ denote large scale fields, i.e. vector quantities whose value may vary in space but whose direction and sense are almost uniform or vary very smoothly. Technically speaking, they represent *local spatial averages*. Lowercase denotes small scale, *stochastic* fields, i.e. fields whose amplitude may be large, but that have a very small coherence length. We assume that any average of stochastic quantities is zero. Observe that we assumed ${\bf U}_0=0$, i.e. no large scale flows.
### Evolution equation for the mean fields
To derive the evolution equations for the large scale fields, we replace the previous decomposition into eqs. (\[a3\]) and (\[a4\]) and take *local spatial averages* that we denote as $\langle ... \rangle_0$ [^3]. If besides we demand large scale fields to be force-free, we obtain:
$$\frac{\partial {\bf B}_0}{\partial t} =
{\bf \nabla }\times {\bf \varepsilon}_H +\eta \nabla^2 {\bf B}_0\, ,
\label{b1}$$
$$\frac{\partial {\bf A}_0}{\partial t} =
{\bf \bar P} {\bf \varepsilon}_H +\eta \nabla^2 {\bf A}_0\, ,
\label{b2}$$
where with the aid of Reynolds rules (McComb [@macomb]) we have interchanged derivatives with averages. We have also defined a [*Hall turbulent electromotive force*]{} as:
$${\bf \varepsilon}_H = \langle {\bf u}\times {\bf b}\rangle_0
- \ell_H \langle \left( {\bf \nabla }\times {\bf b}\right)
\times{\bf b}\rangle_0 \equiv \langle {\bf u}_e \times {\bf b}\rangle_0\, .
\label{b3}$$
### Evolution equations for the small scale fields
The evolution equations for the small scale fields are obtained by replacing the decomposition of fields into global averages and stochastic component, into eqs. (\[a2\]), (\[a3\]) and (\[a4\]) and substracting from them the equations for the mean fields. Thus we have:
$$\begin{aligned}
\frac{\partial {\bf u}}{\partial t} &=&
{\bf \bar P}\left[ \left( {\bf b}\cdot{\bf \nabla }\right) {\bf B}_0
+ \left( {\bf B}_0\cdot{\bf \nabla }\right) {\bf b} \right]
+ {\bf \bar P}\left[ \left( {\bf b}\cdot{\bf \nabla }\right) {\bf b}
- \left( {\bf u}\cdot{\bf \nabla } \right) {\bf u} \right] \nonumber\\
&-& \nu {\bf \nabla}\times\left({\bf \nabla}\times
{\bf u}\right)\, ,
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\label{b4}\end{aligned}$$
$$\begin{aligned}
\frac{\partial {\bf b}}{\partial t} &=&
{\bf \nabla }\times \left( {\bf u}_e\times {\bf B}_0\right)
- \ell_H {\bf \nabla }\times \left[\left( {\bf \nabla }\times {\bf B}_0\right)
\times{\bf b}\right]
+ {\bf \nabla }\times \left( {\bf u}_e\times {\bf b}\right) \nonumber\\
&-& \langle {\bf \nabla }\times
\left( {\bf u}_e\times {\bf b}\right)\rangle_0
- \eta {\bf \nabla }\times \left({\bf \nabla }\times {\bf b}\right)\, ,
\label{b5}\end{aligned}$$
and:
$$\begin{aligned}
\frac{\partial {\bf a}}{\partial t} &=&
{\bf \bar P} \left( {\bf u}_e \times {\bf B}_0\right)
- \ell_H {\bf \bar P} \left[\left({\bf \nabla }\times
{\bf B}_0\right) \times {\bf b}\right]
+ {\bf \bar P} \left( {\bf u}_e \times {\bf b}\right) \nonumber\\
&-& \langle{\bf \bar P} \left( {\bf u}_e \times {\bf b}\right) \rangle_0
-\eta {\bf \nabla}\times \left( {\bf \nabla}\times {\bf a}\right)\, .
\label{b6}\end{aligned}$$
Dynamo equations
================
Now that we have the complete set of evolution equations for large and small scale quantities, we can proceed to derive the evolution equations for the electromotive force and the magnetic helicity.
Magnetic helicity evolution equation
------------------------------------
Magnetic helicity is defined as the *global average*, or *average over the entire volume* of ${\bf A}\cdot {\bf B}$, that we denote by $H^M_T=\langle {\bf A}\cdot {\bf B}\rangle_{vol}$ (Biskamp [@biskamp]). These quantities do not vary in space, they depend only on time. Call $H^M \equiv \langle {\bf A}_0\cdot {\bf B}_0
\rangle_{vol}$ and $h^M \equiv \langle {\bf a}\cdot {\bf b}\rangle_{vol}$. By taking the time derivative of these quantities with respect to time and using eqs. (\[a3\]), (\[a4\]), (\[b5\]) and (\[b6\]) we obtain
$$\begin{aligned}
\frac{\partial H^M}{\partial t}
&=& 2\langle {\bf \varepsilon}_H \cdot {\bf B}_0 \rangle_{vol}
- 2 \eta \langle{\bf B}_0 \cdot\left({\bf \nabla} \times {\bf B}_0\right)
\rangle_{vol}\nonumber\\
&+& \langle {\bf \nabla}\cdot \left[ {\bf \varepsilon}_H \times {\bf A}_0
- \eta \left( {\bf \nabla}\times {\bf B}_0 \right)
\times {\bf A}_0\right] \rangle_{vol}\, ,
\label{c1}\end{aligned}$$
and:
$$\begin{aligned}
\frac{\partial h^M}{\partial t}
&=& -2\langle {\bf \varepsilon}_H \cdot
{\bf B}_0\rangle_{vol} - 2\eta \langle \langle \left( {\bf \nabla}
\times {\bf b}\right) \cdot {\bf b}\rangle_0\rangle_{vol}\nonumber\\
&+& \langle {\bf \nabla} \cdot \left\{ \left[ {\bf u}_e \times {\bf B}_0
-\ell_H \left( {\bf \nabla} \times {\bf B}_0\right) \times {\bf b}
\right] \times {\bf a}\right\}\rangle_{vol}\nonumber\\
&+& \langle {\bf \nabla} \cdot \left\{ \left[ {\bf u}_e \times {\bf b}
-\eta {\bf \nabla} \times {\bf b} \right] \times {\bf a}
\right\} \rangle_{vol}\, , \label{c2}\end{aligned}$$
To deal with the operator ${\bf \bar P}$ we followed the procedure deviced by Gruzinov & Diamond ([@gruzinov-2]), that consists in transforming Fourier the equations before taking averages, and make a development to first order in $k_L/k_s$ with $k_L$ the scale of $B_0$ and $k_s$ the scale of the small scale fields. We also made some simple algebraic manipulation to put the total divergencies in evidence. If we add up eqs (\[c1\]) and (\[c2\]) we see that total magnetic helicity is conserved, except for the divergencies and the dissipative terms. This means that the term $\langle {\bf \varepsilon}_H\cdot
{\bf B}_0\rangle_{vol}$ transforms magnetic helicity between mean and fluctuating fields. In what follows we consider boundary conditions such that the divergencies in eqs. (\[c1\]) and (\[c2\]) vanish. This selection is debatable, however, in view of the fact that such conditions may not be quite general, or easily attainable in practice. Nevertheless they have two advantages: First, the resulting magnetic helicity is gauge invariant and second, they are widely used in numerical simulations, a fact that will facilitate comparisons with those works. The effect of boundary conditions on the evolution and gauge invariance of magnetic helicity is discussed in Berger & Field ([@berger]), Ji ([@ji-99]), Vishniac & Cho ([@vish-cho]), and Subramanian & Brandenburg ([@sub-bran]).
Evolution equation for ${\bf \varepsilon}_H^{\parallel}$
--------------------------------------------------------
According to its definition, eq. (\[b3\]), ${\bf \varepsilon}_H$ is the combination of two terms. So we need to find evolution equations for each term and then join them into one equation. The derivation is sketched in Appendix [**A**]{}, eq. (\[apa5b\]), and here we quote the final result, namely:
$$\begin{aligned}
\frac{\partial {\bf \varepsilon}_H}{\partial t}
&=& \frac13 \left\{
- \langle {\bf u}_e \cdot \left( {\bf \nabla}\times {\bf u}_e\right) \rangle_0
+\langle \left( {\bf \nabla } \times {\bf b}\right)\cdot {\bf b}\rangle_0
+ \ell_H \langle {\bf b}\cdot \nabla^2 {\bf u}_e\rangle_0 \right\}{\bf B}_0\nonumber\\
&-& \frac13 \left[\langle {\bf u}\cdot {\bf u}_e\rangle_0
+ \ell_H \langle {\bf b}\cdot \left( {\bf \nabla}\times {\bf u}\right) \rangle_0
\right] \left( {\bf \nabla} \times {\bf B}_0 \right) \nonumber\\
&+& \frac13\ell_H \langle {\bf u}_e\cdot {\bf b}\rangle_0 \nabla^2 {\bf B}_0
\nonumber\\
&+& \eta \left[\langle {\bf u}_e \times \nabla^2 {\bf b} \rangle_0
-\ell_H \langle \left( {\bf \nabla} \times \nabla^2
{\bf b} \right) \times {\bf b} \rangle_0 \right]\nonumber\\
&+& \nu \langle \nabla^2 {\bf u} \times {\bf b} \rangle_0 + {\bf \bar T}\, ,
\label{d20}\end{aligned}$$
with ${\bf \bar T}$ representing the small scale field, three-point correlations and given in Appendix [**A**]{} by eq. (\[apa6\]). The term proportional to ${\bf B}_0$ is similar to the “$\alpha$” term that appears in the kinematic dynamo, except that now it has the *electronic kinetic helicity* ($1^{st}$ term inside braces) instead of the fluid kinetic helicity of the ordinary dynamo. Besides this term, there is a current helicity term ($2^{nd}$ term inside braces, also present in standard MHD dynamo equation) that is due to the small scale magnetic field and that can be cast in terms of the small scale magnetic helicity. Finally there is a new term that is an explicit Hall modification ($3^{rd}$ term in braces), that couples the small scale magnetic field to the electronic velocity field.
The term proportional to $\left( {\bf \nabla}\times {\bf B}_0\right)$, named “$\beta$ term” is also strongly modified: the first term turns out to be the scalar product of the fluid and kinetic velocities, and there appears a second term, explicitly dependent on $\ell_H$ that couples ${\bf u}$ to ${\bf b}$. All these modifications made this term not positive definite anymore. A negative value of this coefficient represents non-local transfer from small scale turbulent fields to the large scale magnetic field (Mininni et al [@pablito4]). Finally there appears a new term, proportional to $\nabla^2 {\bf B}_0$.
From eqs. (\[c1\]) and (\[c2\]) we see that the important quantity in the mean field dynamo operation is the component of ${\bf \varepsilon}_H$ parallel to ${\bf B}_0$ (Parker [@parker]), which can be written as ${\bf \varepsilon}_H^{\parallel} = {\bf \varepsilon}_H\cdot {\bf
B}_0/\vert {\bf B}_0\vert$ and whose evolution equation is then given by ${\partial {\bf \varepsilon}_H^{\parallel}}/{\partial t} = \left({\partial
{\bf \varepsilon}_H } /{\partial t} \right) \left( {{\bf B}_0}/{\vert
{\bf B}_0\vert}\right) + {\bf \varepsilon}_H \left[ \partial \left( {\bf
B}_0/\vert {\bf B}_0\vert\right) /{\partial t}\right]$. To numerically integrate the resulting equation it is more convenient to write ${\bf
u}_e$ back in terms of ${\bf u}$ and ${\bf b}$. The physical reason is that ${\bf u}$ is the velocity that can be externally excited or prescribed. Thus we shall work with (see eq. \[apa5a\] of Appendix.)
$$\begin{aligned}
\frac{\partial {\bf \varepsilon}_H^{\parallel}}{\partial t} &=&
\frac13 \left\{
\langle \left( {\bf \nabla } \times {\bf b}\right)\cdot {\bf b} \rangle_0
- \langle {\bf u} \cdot \left( {\bf \nabla}\times {\bf u}\right) \rangle_0
- \ell_H \langle {\bf u} \cdot \nabla^2 {\bf b}\rangle_0 \right.\nonumber\\
&+& \ell_H \langle {\bf b}\cdot \nabla^2 {\bf u}\rangle_0
+ \ell_H \langle \left( {\bf \nabla}\times {\bf b}\right)
\cdot \left( {\bf \nabla}\times {\bf u}\right)\rangle_0 \nonumber\\
&-& \ell_H^2 \langle {\bf b} \cdot \left[ {\bf \nabla}\times
\nabla^2 {\bf b}\right]\rangle_0
+\ell_H^2 \langle \left( {\bf \nabla} \times {\bf b}\right) \cdot
\nabla^2 {\bf b}\rangle_0 \left.\right\} \vert{\bf B}_0\vert \nonumber\\
&-& \frac13 \left[\langle v^2\rangle_0 + \ell_H \langle
{\bf \nabla} \cdot \left( {\bf u}\times {\bf b}\right)\rangle_0 \right]
\frac{\left( {\bf \nabla} \times {\bf B}_0 \right)\cdot {\bf B}_0}
{\vert {\bf B}_0\vert} \nonumber\\
&+& \frac13 \ell_H \left[ \langle {\bf u}\cdot {\bf b}\rangle_0
-\ell_H \langle \left( {\bf \nabla}\times {\bf b}\right) \cdot {\bf b}
\rangle_0 \right]
\frac{\left(\nabla^2 {\bf B}_0\right) \cdot {\bf B}_0}{\vert {\bf B}_0 \vert}
\nonumber\\
&-& \zeta_H^{\parallel} {\bf \varepsilon}_H^{\parallel}\, .
\label{e2}\end{aligned}$$
where the last term represents the dissipative terms and the more important the three-point correlations of the generated small scale fields.
Solving the System
==================
Further approximations and numerical integration
------------------------------------------------
To numerically integrate the equations, we assume that full helical turbulence is excited at a certain scale $\ell_S$ smaller than the system’s size, and that the large scale magnetic field is induced at a larger scale $\ell_L = 5\ell_s$, that can be the system’s size. This assumption enables us to consider that spectra of small scale quantities peak at wavenumber $k_S = 2\pi/\ell_S$ while large scale quantities do so at $k_L = 2\pi/\ell_L$. This assumption in based on the work of Pouquet, Frish & Leorat ([@pfl]), who several years ago showed that when helical turbulence is induced at the scale $k_s$, large scale quantities peaked at a smaller $k_L$ (see also Maron & Blackman [@maron]). Therefore we write the different terms of equations (\[c1\]), (\[c2\]) and (\[e2\]) as: $\langle {\bf a.b}\rangle_{vol} = h^M_s$, $\langle {\bf A}_0.{\bf B}_0\rangle_{vol} = H^M_L$, $\langle {\bf u.b}\rangle_{vol} = h^C$, $\langle {\bf b}.\left({\bf \nabla} \times {\bf b} \right) \rangle_0 = k_S^2 h^M_s$, $\langle {\bf u}.\left( {\bf \nabla} \times {\bf u}\right) \rangle_0 = h^u$, $\langle {\bf u}. \nabla^2{\bf b}\rangle_0 = -k_S^2 h^C$, $\langle {\bf b}. \nabla^2{\bf u}\rangle_0 = -k_S^2 h^C$, $\langle \left( {\bf \nabla}\times {\bf b}\right) . \left( {\bf \nabla}\times {\bf
u}\right) \rangle_0 = k_S^2 h^C$, $\langle \left( {\bf \nabla} \times {\bf b} \right) . \nabla^2 {\bf b}\rangle_0 =
-k_S^4 h^M_s$, $\langle {\bf b}. \left( {\bf \nabla} \times \nabla^2 {\bf b}\right)
\rangle_0 = -k_S^4 h^M_s$, $\langle {\bf \nabla}\cdot \left( {\bf u}
\times {\bf b}\right) \rangle_0 = \pm k_S \vert \varepsilon_0\vert $, $\nabla^2 {\bf B}_0 = -k_L^2 {\bf B}_0$. Besides we write $\langle {\bf u.u}\rangle_0 = 2 e^u$. We see that in this case the the interaction between ${\bf u}$ and ${\bf b}$ is described by the small scale cross-helicity $\langle {\bf u}\cdot
{\bf b}\rangle_0 \equiv h^C$. For ${\bf B}_0$, as it is force-fee, we have $\vert {\bf B}_0\vert = k_L^{1/2} \vert h^M_L\vert^{1/2}$, and $\left ({\bf \nabla}\times {\bf B}_)\right) . {\bf B}_0 = k_L^2 h^M_L$. When we replace these expressions in eqs. (\[c1\]) and (\[c2\]) and (\[e2\]) we obtain:
$$\begin{aligned}
\frac{\partial \varepsilon_H^{\parallel}}{\partial t}
&=& \frac13 \left\{ k_S^2 h^M_s - h^u + \ell_H k_S^2 h^C
\right\} k_L^{1/2}\vert h^M_L\vert^{1/2}\nonumber\\
&-& \frac23 \left(e^u \pm \ell_H k_S \vert \varepsilon_0\vert\right)
k_L^{3/2} \frac{h^M_L}{\vert h^M_L\vert^{1/2}}\nonumber\\
&-& \frac{\ell_H}3\left[ h^C - \ell_H k_s^2 h^M_s\right] k_L^{5/2}\vert
h^M_L\vert^{1/2} - \zeta_H^{\parallel} {\bf \varepsilon}_H^{\parallel}\, , \label{f3}\end{aligned}$$
$$\frac{\partial}{\partial t} h^M_L
= 2 k_L^{1/2}\varepsilon_H^{\parallel}\vert h^M_L\vert^{1/2}
- 2 \eta k_L^2h^M_L\, \label{f6}$$
and
$$\frac{\partial}{\partial t} h^M_s
= - 2 k_L^{1/2}\varepsilon_H^{\parallel}\vert h^M_L\vert^{1/2}
- 2 \eta k_s^2h^M_s\, \label{f7}$$
$h^C$ is not an ideal invariant in Hall-MHD, as can be seen from its evolution equation. It is obtained by deriving $\langle {\bf u}\cdot
{\bf b}\rangle_{vol}$ with respect to time, and using eqs. (\[b4\]) and (\[b5\]), and reads
$$\begin{aligned}
\frac{\partial \langle {\bf u}\cdot {\bf b}\rangle_{vol}}{\partial t}
&=&
- \ell_H \langle \left( {\bf \nabla}\times {\bf u} \right)
\times \left( {\bf \nabla}\times {\bf b}\right) \rangle_0
\cdot {\bf B}_0 \nonumber\\
&-& \ell_H \langle {\bf b} \times \left( {\bf \nabla}\times {\bf u}\right)
\rangle_0
\cdot \left( {\bf \nabla}\times {\bf B}_0 \right) \nonumber\\
&+& \left( \nu + \eta \right) \langle \left( {\bf \nabla}\times {\bf u}\right)\cdot
\left( {\bf \nabla}\times {\bf b}\right)\rangle_{vol}\, . \label{f4}\end{aligned}$$
In order to close our equation system, we could try to make in eq. (\[f4\]) the same approximations used in eq. (\[c1\]), (\[c2\]) and (\[e2\]). However they would produce expressions for which new equations should be deduced. These new equations in turn would produce new terms and so on, thus resulting in a system difficult to integrate and hard to interpret physically. Therefore we shall proceed as follows. The presence of Hall effect implies that the magnetic field must satisfy ${\bf \nabla}\times {\bf b}\not\propto {\bf b}$, i.e. it cannot be force-free. Assuming ${\bf \nabla}\times {\bf b} \propto {\bf b}$ means two things: on one side that we are in the standard case (i.e., without Hall effect), and on the other, that the small scale magnetic field is in an equilibrium state (i.e., no Lorentz force is excerted on the stochastic electric currents). Therefore to use ${\bf \nabla}\times
{\bf b} \propto k_s {\bf b}$ in eq. (\[f4\]) means to consider a leading order in a perturbative expansion of the different terms of eq. (\[f3\]), around the Hall-free state, but it does not mean that we are expanding around a force-free state, as the approximation is used only in eq. (\[f4\]). There remains the issue of the sign. As we shall be interested in a situation in which small scale magnetic helicity grows to negative values, we choose ${\bf \nabla}\times {\bf b} \simeq - k_s
{\bf b}$, to guarantee that condition. For the factor ${\bf \nabla}\times
{\bf v}$, we shall assume maximal negative helicity and thus write it as ${\bf \nabla}\times {\bf v} \simeq - k_s {\bf v}$. We then write the first and second terms in the r.h.s. of eq. (\[f4\]) as $\langle \left(
{\bf \nabla}\times {\bf u} \right) \times \left( {\bf \nabla}\times {\bf
b}\right) \cdot {\bf B}_0 \rangle_{vol} \simeq k_S^2 \langle {\bf u}
\times {\bf b} \rangle_{vol}\cdot {\bf B}_0$ and $\langle {\bf b} \times
\left( {\bf \nabla}\times {\bf u}\right) \cdot \left( {\bf \nabla} \times
{\bf B}_0 \right) \rangle_{vol} \simeq k_S \langle {\bf b} \times {\bf
u}\rangle_{vol}\cdot \left( {\bf \nabla}\times {\bf B}_0\right)$. Under this approximation, we note that the second term becomes smaller than the first one by a factor $k_L/k_S$, and that therefore can be discarded if $k_L/k_S \ll 1$. From the remaining expression, we see that we would also need the evolution equation for ${\bf \varepsilon}_0\cdot {\bf
B}_0$. However as we are treating Hall effect as a perturbation, we make a negligible error if we use ${\bf \varepsilon}_H\cdot {\bf B}_0$ instead of ${\bf \varepsilon}_0\cdot {\bf B}_0$. We use the same reasoning to write $\vert \varepsilon_H\vert$ instead of $\vert \varepsilon_0\vert$ in the second term between brackets in the r.h.s. of eq. (\[f3\]). We are then left with the following equation for the cross helicity:
$$\frac{\partial h^C}{\partial t} =
-\ell_H k_S^2 \varepsilon_H^{\parallel} k_L^{1/2}\vert H^M\vert^{1/2}
- \left( \nu+\eta\right)k_s^2 h^C\, , \label{f5}$$
and our equation system consists of eqs. (\[f3\]), (\[f6\]), (\[f7\]) and (\[f5\]). In order to numerically integrate it and to correctly devise the perturbative treatement of the Hall effect, we need to make the equations nondimensional. We then define the following dimensionless quantities: $\tau = uk_L t$, $G^M = h^M_L k_L/u^2$, $g^M = h^M_s k_s/u^2$, $g^C = h^C /u^2$, $g^u = h^u/\left(k_Lu^2\right)$, $Q_H^{\parallel} = \varepsilon_H^{\parallel} /u^2$, $\lambda_H = \ell_H k_L$, $\xi = \zeta / k_L u$, $f^u = e^u/u^2$, $R_M = u/\left( k_s\eta\right)$, $r = k_S/k_L$. This scheme of normalization is similar to the one of Blackman & Field ([@fb2002-2]), except that we use $k_L$ instead of $k_S$. Besides we shall consider magnetic Prandtl number $\nu /\eta = 1$ and thus $\left(\eta + \nu\right) = 2\eta$. When we replace these quantities in eqs. (\[f3\]) and (\[f7\])-(\[f5\]) we obtain the following system:
$$\begin{aligned}
\frac{\partial Q_H^{\parallel}}{\partial \tau}
&=&
\frac13 \left[ g^M - \frac{g^u}{r^2} + \lambda_Hg^C\right]
r^2 \vert G^M\vert ^{1/2} \nonumber\\
&-& \frac23 \left( f^u \pm \lambda_H r \vert Q_H^{\parallel}\vert\right)
\frac{G^M}{\vert G^M\vert^{1/2}} \nonumber\\
&-&\lambda_H \left[ g^C -\lambda_H r^2 g^M\right]\vert G^M\vert^{1/2}
- \xi_H^{\parallel} Q_H^{\parallel}\, ,
\label{f8}\end{aligned}$$
$$\frac{\partial G^M}{\partial \tau} = 2 Q_H^{\parallel} \vert G^M\vert^{1/2}
- \frac{2}{R_M r} G^M\, , \label{f9}$$
$$\frac{\partial g^M}{\partial \tau} = - 2 Q_H^{\parallel} \vert G^M\vert^{1/2}
- \frac{2r}{R_M} g^M\, , \label{f10}$$
$$\frac{\partial g^C}{\partial \tau} = -\lambda_H r^2 Q_H^{\parallel}
\vert G^M\vert^{1/2} - \frac{2r}{RM} g^C\, . \label{f11}$$
![Electromotive force for $\zeta = 1$, i.e. strong three-point correlations.[]{data-label="Figure1"}](fig1.ps){width="8.5cm"}
In order to have a simple picture of how ${\bf \varepsilon}_H$ evolves, we can reason as follows. For high $R_M$ assume a prescribed, negative value for $g^u$ in the $\alpha$ term of eq. (\[f8\]) (i.e, the first term between square brackets), and that the term $\lambda_H g^C$ is negligible as well as the initial value of $g^M$. In that situation $G^M$ will initially grow toward positive values (due to the first term of eq. \[\[f9\]\]) and $g^M$ toward negative values (due to the first term of eq. \[\[f10\]\]). This will cause the $\alpha$ term to go to zero at a certain instant, and hence to the end of the kinetic regime, i.e., the period during which the growth of ${\bf B}_0$ is exponential. The presence of the $\lambda_H g^C$ term drastically modifies this scenario: were this term negative, then it would take the $g^M$ term a shorter time to cancel the other two terms, i.e., we would have a shorter kinetic phase. Were it positive, then the opposite situation would occur: the kinetic phase would last longer[^4]. This simple picture is even more modified by the fact that now the $\beta$ term in eq. (\[f8\]) (the second term between brackets) is not possitive definite, a fact that could act in favour or against of the two situations described above. We can conclude that the operation of a Hall-MHD dynamo is far more subtle and complicated than the standard MHD one.
Discussion
----------
To integrate the system we used a 4$^{th}$ order Runge-Kutta method with variable step and considered the following values for the different parameters that enter in the equations: $r=5$, $g^u=-5$ (this value is equivalent to the $g^u=-1$ of Blackman & Field [@fb2002-2] with the normalization they used), $f^u = 1$, $R_M=2000$ and $\lambda_H = 0$ and $0.4$. The second value of $\lambda_H$ corresponds to a Hall length almost twice the turbulent scale, but shorter than the coherence large scale. The high value of $R_M$ is easily found in astrophysical environments. For the three-point correlations we considered two cases: $\zeta = 1$ (strong correlations) and $\zeta = 2/R_M$ (weak correlations). As initial conditions we assumed $Q_{H0}^{\parallel} = 0$, $g^M_0 = 0.001 = G^M_0$ (i.e., an initial state with small magnetic helicity. Other initial conditions do not give qualitative different results) and $g^C=0$. We also considered the two possible signs in the $\beta$ term, namely $\beta_{\pm} = f^u \pm \lambda_H r \vert Q_H^{\parallel}\vert$ (see eq. (\[f8\])).
![ Magnetic helicities for $\zeta = 1$, i.e. strong three-point correlations. Upper curves correspond to large scale MH, and lower curves to small scale MH. []{data-label="Figure2"}](fig2.ps){width="8.5cm"}
In Fig.\[Figure1\] we plotted $Q_H^{\parallel}$ as a function of time, for strong nonlinearities, i.e. $\zeta = 1$. We see that for $\lambda_H
\not= 0$, $Q_H^{\parallel}$ is damped faster than for $\lambda_H =
0$. For $\beta_+$ this process is in turn slightly stronger than for $\beta_-$. The saturation value however is the same for all cases, i.e. is not modified by Hall effect. The rise of the first oscilation in the transitory regime corresponds to the kinetic regime, in which the large scale magnetic field would grow exponentially. We see that the instant at which this rise stops is slightly smaller than the one at which stops the standard MHD curve, while the amplitudes of the curves are substantially smaller. This instant is independet of either $\beta_+$ or $\beta_-$. This behaviour can be interpreted as that the Hall dynamo is less efficient than its standard MHD counterpart to generate large scale fields. In Fig. \[Figure2\] we plotted small scale magnetic helicity $g^M$ (lower curves) and large scale magnetic helicity $G^M$ (upper curves), also for $\zeta = 1$. We see that for $\lambda_H \not= 0$ the saturation value of $G^M$ is substantially smaller than for $\lambda_H
= 0$, and is the same for both possible $\beta$’s. This means that the inverse cascade of magnetic helicity is suppressed compared to standard MHD, for the considered parameters. Consistently with Fig. \[Figure1\], we see again that the rise of the first peak takes slightly less time for $\lambda_H \not= 0$ than for $\lambda_H = 0$, and the amplitudes in the former case are much smaller than in the latter case.
In Fig. \[Figure3\] we plotted the e.m.f. $Q_{H0}^{\parallel}$ for $\zeta = 2/R_M$, i.e. weak non-linearities. The smaller amplitudes of the Hall-MHD curves means that the e.m.f. is more quenched than for standard MHD, as in the case of strong non-linearities. Again in this case the quenching due to $\beta_+$ is slightly stronger than the one produced by $\beta_-$. In this case the action of Hall effect in the e.m.f. is manifested for all times, as no saturation value is attained. Again here the rise of the first peak corresponds to the kinetic regime, and similar features as for $\zeta=1$ are found: durantion slightly shorter and amplitude significantly smaller, showing that in this case again the Hall-dynamo is less efficient than the standard MHD one. In Fig. \[Figure4\] we plotted the small scale magnetic helicity $g^M$ (lower curves) and the large scale magnetic helicity $G^M$ (upper curves). We can appreciate more clearly the quenching produced by Hall effect, as the amplitude of the Hall dynamo is about 5 times shorter than the standard MHD. The comments about the durantion of the kinetic phase are the same as for the $\zeta=1$ case.
![Electromotive force for $\zeta = 2/RM$, i.e. weak three-point correlations.[]{data-label="Figure3"}](fig3.ps){width="8.5cm"}
In all figures, the main features (oscillations for the $\zeta = 2/R_M$ and saturation for $\zeta = 1$) are determined by the $\alpha$ term, i.e. by the interplay between kinetic and current small scale helicites, and for the Hall dynamo also by the coupling between small scale magnetic and velocity fields: Given a prescribed negative value of $g^u$ in the $\alpha$ term of eq. (\[f8\]), then $g^M$ grows negative, thus leading to a cancellation of them and to a consequent supression of the growth of $Q_{H0}^{\parallel}$ (this is the end of the kinetic regime). In the Hall-dynamo, in the approximation we work with, the $\alpha$ term gets an extra term proportional to $g^C$, which in principle can be positive or negative. For the parameters we considered in this work it is negative and consequently reinforces the action of $g^M$, thus leading to suppression of the $\alpha$ term faster than in the non-Hall case. Were it positive, then the opposite would occur: it would reinforce $-g^u$ and thus it would take longer for the (negative) small scale magnetic helicity to catch up with the other (positive) terms in the $\alpha$ term. As a result the kinetic regime would last longer.
The results quoted in this paper, namely a stronger quenching of the Hall-MHD electromotive force for $\ell_{turbulence} < \ell_{Hall} <
\ell_{system}$, agree with numerical simulations performed by Mininni et al (Mininni, Gomez & Mahajan [@pablito3]).
Conclusions
===========
In this paper we studied semianalitically how Hall effect modifies the quenching process of the electromotive force in Mean Field Dynamo theory.We used a dynamical closure scheme named minimal $\tau$ approximation (Blackman & Field [@fb2002-2], Brandenburg & Subramanian [@report-bs]), that takes into account the back-reaction of the small scale fields generated by the turbulence. As we considered helical turbulence, those small scale fields are not to be considered as the result of a small scale dynamo, but as a waste product, that nevertheless strongly quenches the e.m.f.
![Magnetic helicities for $\zeta = 2/RM$, i.e. weak three-point correlations.[]{data-label="Figure4"}](fig4.ps){width="8.5cm"}
Considering force-free large scale magnetic fields, we found that Hall effect modifies the evolution equation of the e.m.f. in several ways: the main driving term, the so-called $\alpha$ term, proportional to $\vert
{\bf B}_0\vert$ now depends on the electronic velocity $u_e$ instead of the fluid velocity. Besides there appears a third term, explicitly dependent on the Hall parameter $\ell_H$, that couples the magnetic field to this electronic velocity. The diffusive term, also known as $\beta$ term, proportional to $\vert {\bf \nabla}\times {\bf B}_0\vert$ also depends on the electronic velocity but besides it acquired a new term that couples the small scale magnetic field to the electronic velocity, and that renders it not positive definite. A negative value of this coefficient represents non local transfer from small scale turbulent fields to the large scale magnetic field. Finally there appears a new term proportional to $\nabla^2 {\bf B}_0$. In our case this term plays no significant role because it is substantially smaller than the others due to the perturbative scheme we use.
To give a concrete numerical example, we considered Hall effect as a perturbation of characteristic scale larger than the turbulent scale, but shorter than the large scale magnetic field. This situation can be found in e.g. accretion disks and in the early universe plasma (Sano & Stone [@stone], Tajima et al. [@tajima]). After implementing the two scale approximation, we numerically integrated the resulting evolution equations for the e.m.f., the large and small scale magnetic helicities, and the cross-helicity which is the quantity that in this approximation mimics the coupling between the small scale velocity and magnetic fields. The overall effect is that in the presence of Hall effect, the e.m.f. is more strongly quenched than in the case of standard MHD dynamo. This fact is acompanied by a damping in the inverse cascade of magnetic helicity.
As the main scope of this paper is to understand conceptually how Hall effect acts on a MFD, we did not apply our results to a concrete astrophysical object. We leave this issue for future work, after a deeper understanding of the mechanism is attained. Besides this point, this work can be improved in several aspects. A very important one is to extend the perturbative expansion to higher orders (or even to device a non-perturbative scheme) in order to attain other regimes, where dynamo action might be enhanced by Hall effect. This extension might also mean that we should abandon the hypothesis that ${\bf B}_0$ is force-free. Next, to work with more general boundary conditions, that permit to relax the conservation of total magnetic helicity. Finally, a more detailed study of the three-point correlations is in order, as well as to consider non-helical turbulence. We are working on some of these topics at present.
We are grateful to E. G. Blackman who kindly and patiently clarified several conceptual and technical aspects of his work and of dynamo theory, and carefully read this manuscript. We also thank D. Gomez for carefully reading and commenting this manuscript. A.K. acknowledges financial support from FAPESB under grant APR0125/2005. M.J.V. thanks PRODOC/UFBA (project n. 108). The work of AHC was partially supported by a post-doctoral fellowship from the Brazilian agency CAPES (process BEX 0285/05-6). AHC and MJV would like also to acknowledge A. Raga and P. Velázquez (ICN-UNAM), for their kind and warm hospitality during our visit to the México City, where part of this work was developed.
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Evolution Equation for the Electromotive Force
==============================================
The evolution equation for $\varepsilon_H^{\parallel}$ is obtained by calculating
$$\begin{aligned}
\frac{\partial {\bf \varepsilon}_H}{\partial t} &=&
\langle \frac{\partial {\bf u}}{\partial t}\times {\bf b}\rangle_0
+ \langle {\bf u}\times\frac{\partial {\bf b}}{\partial t} \rangle_0 \nonumber\\
&-& \ell_H \left[\langle \frac{\partial \left( {\bf \nabla} \times {\bf b}\right) }{\partial t}
\times {\bf b}\rangle_0 +\langle \left( {\bf \nabla} \times {\bf b}\right)
\times \frac{\partial {\bf b}}{\partial t} \rangle_0\right]\, ,
\label{apa1}\end{aligned}$$
and the corresponding equation for ${\bf
\varepsilon}_H^{\parallel} = {\bf \varepsilon}_H\cdot \left({\bf
B}_0/\vert {\bf B}_0\vert \right)$ by doing ${\partial {\bf
\varepsilon}_H^{\parallel}}/{\partial t} = \left({\partial {\bf
\varepsilon}_H } /{\partial t} \right) \left( {{\bf B}_0}/{\vert {\bf
B}_0\vert}\right) + {\bf \varepsilon}_H \left[ \partial \left( {\bf
B}_0/\vert {\bf B}_0\vert\right) /{\partial t}\right]$. Replacing eqs. (\[b4\]) and (\[b5\]), considering the development of operator ${\bf \bar P}$ to first order in the terms linear in ${\bf B}_0$, as is done in Gruzinov & Diamond ([@gruzinov-2]), and Blackman & Field ([@fb2002-2]), and assuming homogeneous and isotropic turbulence, we obtain:
$$\begin{aligned}
\frac{\partial {\bf \varepsilon}_H}{\partial t}
&=&
\frac13 \left\{
\langle
\left( {\bf \nabla } \times {\bf b} \right)
\cdot {\bf b} \rangle_0 - \langle {\bf u} \cdot
\left( {\bf \nabla}\times {\bf u}\right)
\rangle_0 - \ell_H \langle {\bf u} \cdot
\nabla^2 {\bf b}\rangle_0
\right.
\nonumber\\
&+&
\ell_H \langle {\bf b}\cdot \nabla^2 {\bf u}\rangle_0
+ \ell_H \langle
\left( {\bf \nabla}\times {\bf b} \right)
\cdot
\left( {\bf \nabla}\times {\bf u}\right)
\rangle_0
\nonumber\\
&-&
\left.
\ell_H^2 \langle {\bf b} \cdot
\left[ {\bf \nabla}\times \nabla^2 {\bf b}\right]
\rangle_0 +\ell_H^2 \langle
\left( {\bf \nabla} \times {\bf b} \right)
\cdot \nabla^2 {\bf b}\rangle_0
\right\}
{\bf B}_0 \nonumber\\
&-& \frac13
\left[
\langle v^2\rangle_0 +
\ell_H \langle {\bf \nabla} \cdot
\left( {\bf u}\times {\bf b} \right)
\rangle_0
\right]
\left( {\bf \nabla} \times {\bf B}_0 \right)
\nonumber\\
&+& \frac13
\left[
\langle {\bf u}\cdot {\bf b}\rangle_0 - \ell_H \langle
\left({\bf \nabla}\times {\bf b}\right)
\cdot {\bf b}\rangle_0
\right]
\nabla^2{\bf B}_0 \nonumber\\
&+& \eta
\left[\langle {\bf u}_e \times \nabla^2 {\bf b} \rangle_0
-\ell_H \langle
\left( {\bf \nabla} \times \nabla^2 {\bf b} \right)
\times {\bf b} \rangle_0
\right]
\nonumber\\
&+& \nu \langle \nabla^2 {\bf u} \times {\bf b}
\rangle_0 + {\bf \bar T}\, ,
\label{apa5a}\end{aligned}$$
where by ${\bf \bar T}$ we denote the non linear terms, i.e.:
$$\begin{aligned}
{\bf \bar T}
&=&
\langle {\bf u} \times \left[ {\bf \nabla }\times
\left( {\bf u}\times {\bf b}\right) \right] \rangle_0
+
\langle {\bf \bar P}\left(\left[ {\bf u}\times \left({\bf \nabla }\times
{\bf u} \right) \right] \times {\bf b}\right) \rangle_0
\nonumber\\
&+&
\langle {\bf \bar P}\left( \left[ \left( {\bf \nabla } \times
{\bf b}\right) \times {\bf b}\right] \times {\bf b}\right) \rangle_0
- \ell_H \langle{\bf u} \times\left\{
{\bf \nabla }\times \left[\left( {\bf \nabla }\times
{\bf b}\right) \times{\bf b}\right]\right\} \rangle_0\nonumber\\
&-& \ell_H \langle \left( {\bf \nabla} \times {\bf b}\right)\times
\left[ {\bf \nabla } \times \left( {\bf u}\times {\bf b}\right) \right] \rangle_0
- \ell_H \langle \left\{ {\bf \nabla} \times\left[ {\bf \nabla }\times
\left( {\bf u} \times {\bf b}\right) \right]\right\} \times {\bf b} \rangle_0
\nonumber\\
&+& \ell_H^2 \langle \left( {\bf \nabla} \times {\bf b}\right)\times \left\{
{\bf \nabla } \times \left[\left( {\bf \nabla }\times {\bf b}\right)
\times{\bf b}\right]\right\} \rangle_0
\nonumber\\
&+& \ell_H^2 \langle \left[ {\bf \nabla} \times \left\{ {\bf \nabla }\times
\left[\left( {\bf \nabla }\times {\bf b}\right) \times{\bf b}\right]
\right\} \right] \times {\bf b} \rangle_0\, . \label{apa6}\end{aligned}$$
Recalling that ${\bf u}_e = {\bf u} -\ell_H {\bf \nabla}\times {\bf b}$, we can write eq. (\[apa5a\]) in a form that shows explicitly that now it is the electronic flow the driver of the dynamo:
$$\begin{aligned}
\frac{\partial {\bf \varepsilon}_H}{\partial t}
&=&
\frac13 \left\{ \langle \left( {\bf \nabla } \times {\bf b}\right)\cdot
{\bf b} \rangle_0 - \langle {\bf u}_e \cdot
\left( {\bf \nabla}\times {\bf u}_e\right) \rangle_0
+ \ell_H \langle {\bf b}\cdot \nabla^2 {\bf u}_e\rangle_0 \right\}{\bf B}_0
\nonumber\\
&-& \frac13 \left[\langle {\bf u}\cdot {\bf u}_e\rangle_0
+ \ell_H \langle {\bf b}\cdot \left( {\bf \nabla}\times {\bf u}\right) \rangle_0
\right] \left( {\bf \nabla} \times {\bf B}_0 \right) \nonumber\\
&+& \frac13 \langle {\bf u}_e\cdot {\bf b}\rangle_0 \nabla^2{\bf B}_0
+ \eta \left[\langle {\bf u}_e \times \nabla^2 {\bf b} \rangle_0
-\ell_H \langle \left( {\bf \nabla} \times \nabla^2
{\bf b} \right) \times {\bf b} \rangle_0 \right]\nonumber\\
&+&\nu \langle \nabla^2 {\bf u} \times {\bf b} \rangle_0 + {\bf \bar T}\, .
\label{apa5b}\end{aligned}$$
[^1]: In order to generate large scale fields, helical turbulence is needed. The generation of these small scale fields is not to be considered as a result of a [*small scale dynamo*]{}. Those dynamos require turbulence to be non-helical (see e.g., Zel’dovich et al [@zeldovich]).
[^2]: This condition implies that the time that a sound signal takes to travel through a given distance $l$ must be small compared to the time $\tau$ during which the flow changes appreciably, i.e., $\tau \gg l/c_s$, so that the propagation of interactions in the fluid may be regarded as instantaneous (e.g., Landau & Lifshitz [@landau-fluids])
[^3]: In the absence of large scale flows, i.e., if ${\bf U}_0 = 0$, we obtain from eq. (\[a2\]) the following constraint: $\langle{\bf \bar
P}\left[ \left({\bf U}\cdot {\bf \nabla }\right) {\bf U} \right]\rangle_0
- \langle{\bf \bar P}\left[ \left({\bf b}\cdot {\bf \nabla }\right) {\bf
b}\right]\rangle_0 = 0$, that must be satisfied in order to guarantee the vanishing of ${\bf U}_0$ for all time.
[^4]: From the negative sign of the first term in eq. (\[f11\]) we see that in this case, the first situation will occur, i.e. an earlier quenching of the dynamo
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'J. Rodriguez'
- 'Ph. Durouchoux'
- 'I. F. Mirabel'
- 'Y. Ueda'
- 'M. Tagger'
- 'K. Yamaoka'
date: 'Received date; Accepted date'
title: Energy dependence of a Low Frequency QPO in GRS 1915+105
---
1915[GRS $1915$+$105$]{} 1550[XTE J$1550$–$564$]{} 1655[GRO J$1655$–$40$]{} \#1[[**\#1**]{}]{}
Introduction
============
X-ray binaries exhibit strong X-ray emission, from the soft ($\sim 0.1$ $keV$) to the hard X-rays (up to a few hundred keV), sometimes up to the MeV domain. The emission processes are thought to occur in the close vicinity of a stellar-mass compact object (either a Neutron Star or a Black Hole), the soft part of the spectrum being usually taken as the thermal emission of an accretion disk, whereas the hard part is thought to be the manifestation of an inverse compton scattering of the soft photons, with relativistic electrons present in a hot coronal medium. The sources may be distinguished by several characteristics, such as the companion mass, whenever this latter is known, the shape of their spectra, or by the presence of strong collimated ejecta. In the latter case, the similarity with AGN led to the definition of microquasars (Mirabel , 1992), some of them known to be sources with superluminal jets (Mirabel & Rodríguez, 1999).\
1915 has first been discovered as a Soft X-ray Transient by WATCH on board [*[GRANAT]{}*]{} (Castro-Tirado , 1992), and then identified as the first Galactic source to have ejections with apparent superluminal motion (Mirabel & Rodríguez, 1994). The distance to the source has been estimated to 12.5 kpc, its inclination $\sim 70^{\circ}$, and the velocity of the jet 0.92c (Mirabel & Rodríguez, 1994). Since then, the source has been observed with many X-ray satellites, and its spectrum is typical of that of Black Hole Candidates (BHC), such as 1655. Only recently, however, the spectral type of the companion has been identified as a K–M III star (Greiner , 2001), classifying the source as a low mass X-ray binary. The mass of the primary has been estimated to $14\pm4$ $M_{\odot}$ (Greiner, Cuby & McCaughrean, 2001), confirming the black hole nature of the compact object.\
With the launch of the Rossi X-ray Timing Experiment (RXTE), and the excellent timing capacities of both its pointed instruments, the [ *[Proportional Counter Array]{}*]{} (PCA) and the [*[High Energy X-ray Timing Experiment]{}*]{} (HEXTE), many X-ray Binaries and 1915 in particular, have been discovered to exhibit Quasi Periodic Oscillations (QPOs), in several ranges of frequency (a few mHz up to hundred, and kilohertz in the case of neutron star primary). Though no physical explanation has yet been widely accepted, the QPOs are thought to occur in the close vicinity of the compact object.\
Furthermore, it has been pointed out by Psaltis (1999), that the QPOs could represent the same type of variability in both neutron stars and black hole systems, constraining the theoretical models, and giving important clues to the physics of these phenomena. In particular the study of QPOs should give important informations on the accretion flow, and thus on the physics of the disk.\
The detection of several types of QPOs can be attributed to different mechanisms, depending in particular on the source spectral state.\
We will only focus here on the strong $\sim 0.5-10$ $Hz$ QPO, present during the low/hard spectral state of 1915, often called “ubiquitous”, since it is nearly always present in that state and often observed in other Black Hole Binaries (e.g. 1550, or 1655). In that case, several authors have pointed out correlations between the frequency of the oscillations and some of the spectral parameters, such as the flux (Swank , 1997; Markwardt , 1999), the temperature of the disk (Muno , 1999), and the disk color radius (Rodriguez , 2002).\
All these correlations constrained the location of the QPO in or close to the disk, and the systematic study of the QPO parameters should lead to a better understanding of the accretion and ejection mechanisms, thought to occur in this region.\
Recently a new mechanism has been proposed by Tagger & Pellat (1999), to extract energy and angular momentum from the inner regions of the disk (permitting, thus the accretion) and transport them toward the corotation radius of the spiral wave formed in the disk, where they can be emitted directly toward the corona (Tagger & Pellat, 1999; Varnière & Tagger, 2001).\
It has been shown by Rodriguez (2001), and Varnière (2002) , that this model could explain the different frequency vs. radius correlations observed in 1655 compared to 1915 or (as had been found by Sobczak , 1999) 1550.\
This model could also explain the correlations found by Mirabel (1998), Eikenberry (1998), Ueda (2002, our observations being part of this latter work) during the $\sim 30$ min cycle (Tagger, 1999 for a possible scenario), between X-ray light curves and the infrared and radio emissions, considered as the synchrotron signatures of an expanding ejected blob of material, relating then the energy needed to accelerate those blobs, to the one extracted from the accretion.\
Date MJD Obs Id Interval $\#$ Time start (UT) Time stop (UT) PCUs “On”
------------ --------- ------------------ --------------- ----------------- ---------------- -----------
04 17 2000 $51651$ $50405-01-01-00$ $1$ $12h52m15s$ $13h42m55s$ $0-4$
$ $ $ $ $2$ $14h27m43s$ $15h18m39s$ $0-4$
04 22 2000 $51656$ $50405-01-02-00$ $1$ $09h21m35s$ $10h15m27s$ $0-4$
$ $ $50405-01-02-01$ $2$ $10h55m59s$ $11h38m07s$ $0,2-4$
$ $ $50405-01-02-02$ $3$ $12h31m59s$ $13h14m07s$ $0,2-4$
04 23 2000 $51657$ $50405-01-03-00$ $1$ $07h40m31s$ $08h35m27s$ $0,2,3$
$ $ $ $ $2$ $09h16m15s$ $09h59m59s$ $0,2,3$
We present here observations of the source taken as a RXTE Target of Opportunity, in April 2000. In section \[sec:data\] we present the data reduction and analysis methods used; in section \[sec:17th\], we examine the first of the three observations, which is the most variable one, and focus then on the dynamical properties of the source, observed in different energy ranges. In section \[sec:22-23\] we study the data of the following observations, where the source is much more steady, and thus, more adapted to extract the QPO parameters with high accuracy; we will interpret our observations in the last part of this paper.
Data reduction and analysis {#sec:data}
===========================
The source has been observed on April $17^{th}$, $22^{nd}$ and $23^{rd}$, 2000 as a target of opportunity. We have reduced and analyzed the processed data using the FTOOLS package (update 5.04). Observations IDs, exact time intervals, and dates are shown in table \[tab:ref\].\
We first extracted, for the three observations, lightcurves covering the entire PCA energy range, from binned data with $2^{-7}$ s = $7.8125$ ms resolution, and event data with $2^{-16}$ s = $15.25878$ $\mu$s, which were rebinned during the extraction process to $7.8125$ ms.\
In all cases, lightcurves were extracted from all the PCUs that were simultaneously turned “on” over a single interval (5 on Apr. $17^{th}$, and $22^{nd}$ first interval, four during the two following intervals that day, and three on Apr. $23^{rd}$). We combined all PCUs and all layers to get the most possible incoming flux. The exact PCA configuration over each interval is given in table \[tab:ref\].\
“Good Time Intervals” (GTIs) were defined when the elevation angle was above $10^{\circ}$, the offset pointing less than $0.02^{\circ}$, and we also excluded the data taken while crossing the SAA.\
Background lightcurves were generated using the PCABACKEST tool, from standard2 data, and subtracted from the raw lightcurves. We then generated power spectra and dynamical power spectra (hereafter DPS) using POWSPEC 1.0, calculating each FFT over $\sim4$ s time intervals (2048 bins in each intervals), and averaging then the result over 4 intervals. The resultant DPS has a resultant time bin $\sim 16$ s, comparable to the time resolution of the standard 2 lightcurves. To follow the evolution of the QPOs parameters with the energy, we extracted, in the same standard way, lightcurves in five PCA energy channels: absolute channel $0-11$ (in Matrix epoch4 corresponding to $<2-4.99$ keV), channel $12-29$ ($4.99-12.68$ keV), channel $30-46$ ($12.68-20.06$ keV), channel $47-89$ ($20.06-39.29$ keV), channel $90-174$ ($39.29-80.04$ keV). We then produced DPS and power spectra, as explained above, in each energy range.
First Observation : on April $17^{th}$ {#sec:17th}
======================================
We extracted from both instruments standard lightcurves with 16 s time resolution, using the standard PCA and HEXTE reduction steps, for this observation; they are plotted on figure \[fig:bothlite\].\
The source is in a [[*[$\alpha$ state ]{}*]{}]{} as defined by Belloni (2000). PCA dynamical power spectra, covering the entire PCA energy range ($\sim2-100$ $keV$), are shown on figure \[fig:dynpow17-04\] together with the PCA lightcurves.\
The source presents large flux variations on short time scales ($\sim 100$ s), together with a single QPO whose frequency has a similar behavior (figure \[fig:dynpow17-04\]). Then around time $\sim 600$ s (first interval), and $\sim 6000$ s (second), a large $\sim 1000$s dip occurs (figure \[fig:bothlite\]). During that time, the QPO frequency varies from $9$ Hz to $2.25$ Hz, and a strong second QPO appears with a frequency $\sim$ twice that of the fundamental, following the same frequency variations (figure \[fig:dynpow17-04\]). Then around relative time 2032 s (first interval), and 7716 s (second interval) , a sudden and large soft X-ray spike, reaching $\sim 4.8 \times$ (respectively $\sim 6.4 \times$) the dip minimum flux, for the first (respectively second) interval, occurs and the source returns to a state similar to the one before the dip. Here the harmonic disappears, while the fundamental returns to a larger frequency and behaves as before the dip.
In addition we show in fig. \[fig:rangedynpow\], DPS in the five energy ranges defined in section \[sec:data\], together with the corresponding lightcurves. One can immediately see that above 20 keV the harmonic is absent or very faint, and that above 40 keV (probably due to the high noise) the QPO disappears.
We also see on figure \[fig:rangedynpow\] the evolution of the flux variations with the energy; the large dip seems to be smoothed with the energy.\
We extracted from the soft lightcurves the relative time and the value of the flux of the peak occuring just before the dip (relative time $554$ s, for the first interval, and $6064$, for the second one); we then re-did the same procedure for the minimum of the dip (relative time $954$ s for the first interval, and $6368$ s for the second one), and we thus could estimate the relative amplitude of the variation of the flux, at the time where, also, the fundamental QPO sees its frequency varying from 9 to 2.25 Hz. We did this in each energy range, at the same times (allowing a maximum of two bins ($\sim \pm 32$ s) of difference between each range). Results are shown in table \[table:var\].
Energy Range (keV) Variation Rate Interval \# $1$ ($\%$) Variation Rate Interval \# $2$ ($\%$)
-------------------- --------------------------------------- ---------------------------------------
$2$–$5$ $72.95 \pm 0.63$ $71.19 \pm 0.66$
$5$–$13$ $70.24 \pm 0.61$ $69.61 \pm 0.63$
$13$–$20$ $47.84 \pm 1.78$ $42.37 \pm 1.88$
$20$–$40$ $22.18 \pm 4.26$ $21.29 \pm 4.07$
Note that the soft spike corresponds in the higher energy range (above 20 keV) to a sudden decrease of the flux, indicating the cooling, or the disappearance of a part of the corona (multi-wavelength results can be found in Ueda , 2002 ).
Second and Third Observation : April $22^{nd}$ and $23^{rd}$ {#sec:22-23}
============================================================
As the lightcurves and dynamical power spectra did not present variations as strong as on the previous date, we did not focus here on the dynamical evolution of the QPO, but we just tried to correlate the QPO parameters with the energy range. Figure \[fig:22dynpow\] shows the lightcurves with the dynamical power spectra from all the GTIs of both observations. The source is in a [[*[$\chi$ state]{}*]{}]{} of Belloni , 2000, characterized by a steady flux.
Power spectra covering the entire PCA range,shown on figure \[fig:powspec22\], are fitted with a model consisting of two broad lorentzians (continuum) , plus sharper ones, modeling the QPO features. When the presence of the QPOs was not obvious, we estimated the parameters by freezing the lorentzian centroid frequency to the value found in the other energy ranges, and allowing both the width and the power to vary. In the case of the $40-80$ keV range, since the statistics from single interval was poor, we choosed to merged the observations were the QPO frequency was found to be close, i.e. intervals $\#1$ and $\#2$ from April 22, and intervals $\#1$ and $\#2$ from April 23; interval $\#3$ from April 22 was fitted alone. Results from the fits for all the energy ranges defined in section \[sec:data\] are shown in table \[table:paramqpo\].
No variations similar to those of April 17 are present here; the flux remains fairly constant around a mean value 1050 cts/s/PCU-on, for the April 22 two first intervals, rising slowly to $\sim 1100$ cts/s/PCU-on, for the April 22 third interval, and reaching $\sim 1200$ cts/s/PCU-on, on April 23. As expected, in the same time intervals the fundamental QPO sees its frequency slowly increase with time from $\sim 2.14$ Hz (on Apr. 22) to $\sim 2.9$ Hz (on Apr. 23) (figure \[fig:22dynpow\], and table \[table:paramqpo\]). The harmonic is still present during the five intervals, with a frequency varying from $\sim4.3$ Hz (on Apr. 22), to $\sim 5.8$ Hz on April 23 first interval.\
We plotted in figure \[fig:correl\] the evolution of the QPO power vs. energy range for the five GTIs. The upper points represent the behavior of the fundamental QPO, and the lower that of the harmonic; we can see that the power of the fundamental increases up to $40$ keV, and them seems to decrease, whereas that of the harmonic seems to peak between the $5-13$ and $13-20$ keV ranges.
Figure \[fig:width\] represents the evolution of the QPOs width vs. their frequencies. Both distributions of points can be well fitted by lines of slopes $0.390$ for the fundamental, and $0.389$ for the harmonic. The zero abscissa values are found to be $-0.526052$ for the fundamental, and $-0.932459$ for the harmonic (although their physical meaning is not clear) . It is clearly visible on the plot that both QPOs are tightly correlated, the width of the harmonic being $\sim$ twice that of the fundamental (resulting thus in a Q value ($=\frac{frequency}{FWHM}$) similar for both).
Results and Interpretation
==========================
The April 17 observation confirms and expands the conclusion of Markwardt (1999) and Muno (1999), that the QPO frequency is better correlated with the soft flux, but seems stronger in the higher energy bands (which is confirmed by the following dates).\
In addition a precise study of the lightcurve of the same date shows that the $\sim 30$ min dips are smoothed with the energy, and that the sudden increase of the soft flux (the spike) is anti correlated with the hard flux; indeed the spike, in both interval, corresponds to a major decrease of the flux in the $20-40$ keV, and $40-80$ keV bands, usually considered to be emitted by the corona. The soft spike marks here the transition from the low hard state (C state of Belloni , 2000), to a soft high state (A-B states). Within the interpretation in terms of disk states, this transition and the rapid variations following (interpreted as rapid transitions through A B C states (Belloni , 2000)) can be seen as a succession of rapid replenishments and disappearances of the innermost parts of the disk (Belloni , 1997). The behaviour of the corona may appear difficult to understand, since the abrupt cutoff of the hard X-rays could either be the manifestation of a sudden cooling of the relativistic electrons by the re-emergence of a high soft flux, or the disappearance of the corona (by advection or ejection).\
Thanks to a large number of multi-wavelength observations, the radio and infra red behaviors of 1915 have now been widely studied for years. In particular, former studies such as the one presented in Mirabel , 1998, or Eikenberry , 1998 had linked the soft X-ray spike (transition from low hard to soft high state) with radio and infra red flares. Dhawan (2000) have shown that indeed superluminal ejections took place during abrupt change in the X-ray state of the source. More recently, Klein-Wolt (2001) have found a strong correlation between radio events (radio oscillations, compact jets, large radio flares), and state C properties (duration, transition to other states). It is, however, to be noted that Klein-Wolt , did not find any simultaneous radio - alpha state observations. Furthermore, “The Largest Multi-wavelength Campaign” on 1915 presented in Ueda (2002), shows that the state transitions on Apr. $17^{th}$ are followed by radio flares consistent with an ejection of material starting at the state transition. This leads us to suggest that the abrupt cutoff of the hard X-rays is more probably related to the disappearance of a part of the corona, blown away under the form of a synchrotron emitting blob of material detected in the infra red, and radio domains (fig. 1 and 2 in Ueda , 2002).\
On the other hand, the behavior of the QPO and its harmonic at high energies poses severe constraints on theoretical models. The decrease of the QPO power above $40$ keV may indicate that not all the corona is affected. The decrease of the harmonic above $\sim 20$ keV also raises very challenging questions.\
These could find an explanation in the context of the Accretion-Ejection Instability (Tagger and Pellat, 1999), which has been shown to form a rotating spiral structure in the disk, similar to galactic ones but driven by magnetic stresses rather than by self-gravity. The spiral arms should be expected to heat as well as compress the gas in the disk, and thus to appear as a rotating spiral or hot spot. The harmonic would then be a signature of the non-linear behavior of the spiral, just as the gas form shocks (and thus strong harmonics of the underlying 2-armed spiral) along galactic spiral arms. The high-energy cutoff of the fundamental could, then, favor an interpretation where most or all of the quasi-periodic modulation at high energies comes, not from the comptonized corona as usually assumed, but from a hot point in the optically thick disk. This would be consistent with the previous result (Rodriguez , 2001; Rodriguez , 2002) that the anomalously small color radius of the disk, often observed in some Black-Hole Binaries, could actually be interpreted by the black-body emission of a small area hot point in the disk. We could in principle have an estimate of its physical size, by adding a blackbody model in the spectral fits (such as the [*[BBODYRAD]{}*]{} model of XSPEC), one of the parameters being the normalized area of the emitting region, (since the black body luminosity is proportionnal to the area). But the limited sensitivity and spectral resolution of the present data do not allow any realistic fit. We expect that future instruments will provide better constraints on this problem .\
It would be very tempting to consider the width of the QPO as a measure of the size (due for example to the differential rotation acting between the inner and outer edges of the spot). But the fact that we are dealing with a QPO probably rules out this explanation, since it has to result from a quasi-stationary feature in the disk. This is precisely the case for the AEI, where a standing spiral wave results in a quasi-stationary feature rotating at a single frequency. In this context the width of the QPO would correspond to the coherence time of this pattern, fixed either by non-linear effects or by variations in the background disk equilibrium, [*e.g.*]{} the inner disk radius or other disk parameters (temperature, magnetization, etc.)..\
The spot physical properties (its temperature) may also depend on a number of external parameters, hard to deduce from the observations, such as the $\beta$ ratio (the ratio between thermal and magnetic pressure), which drives the instability (see for example Varnière , 2002, for a discussion on the effects of this parameter), or even the efficiency of the instability. Indeed, in a non linear regime for example, the amount of energy deposited in the disk (under the form of shocks) would be much greater, and would locally warm it up much more than in the linear case.\
Further observational and theoretical work should, however, allow to test this hypothesis: by producing, from numerical simulations of the instability (such as Caunt and Tagger, 2001), synthetic light curves of the QPO, and by fitting the observed energy dependence of the modulated light curve by a high-temperature, hotter black body over a small area of the disk rather than the usual power-law of the coronal emission.
The authors would like to thank S. Corbel, M. Muno, P. Varnière, T.Foglizzo, and the anonymous referee for usefull discussions and comments which allowed to improve the quality of the paper.\
IFM acknowledges partial support from Fundacíon Antorchas.\
We also thank the [[*[Athena help]{}*]{}]{} at GSFC for appreciable help on the RXTE data reduction processes.\
This research has made use of data obtain through the High Energy Astrophysics Science Archive Center Online Service, provided by the NASA/Goddard Space Flight Center.
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Date $\#$ Energy range (keV) $f_{QPO1} (Hz)$ $Q_1$ $\% RMS_1$ $f_{QPO2} (Hz)$ $Q_2$ $\% RMS_2$ $\chi^2$ (d.o.f.)
------------------ -------------- -------------------- --------------------------- ---------- ------------------------- --------------------------- ---------- ------------------------ ------------------- --
$04$ $22$ $2000$ $1$ PCA$^*$ $2.148_{-0.006}^{+0.006}$ $6.97$ $12.47_{-0.57}^{+0.62}$ $4.242_{-0.019}^{+0.019}$ $5.58$ $6.31_{-0.42}^{+0.48}$ $69.13 (62)$
$2-5$ keV $2.137_{-0.006}^{+0.008}$ $7.12$ $9.92_{-0.46}^{+0.46}$ $4.270_{-0.02}^{+0.022}$ $5.34$ $6.26_{-0.4}^{+0.42}$ $89.2 (62)$
$5-13$ keV $2.142_{-0.005}^{+0.007}$ $6.88$ $14.64_{-0.61}^{+0.65}$ $4.249_{-0.018}^{+0.019}$ $6.85$ $6.72_{-0.46}^{+0.50}$ $67.62 (62)$
$13-20$ keV $2.151_{-0.006}^{+0.007}$ $6.68$ $16.16_{-0.73}^{+0.73}$ $4.270_{-0.038}^{+0.038}$ $7.89$ $4.63_{-0.76}^{+0.87}$ $62.11 (62)$
$20-40$ keV $2.143_{-0.008}^{+0.01}$ $6.00$ $16.89_{-1.03}^{+1.10}$ $4.27$ [*[frozen]{}*]{} $>6.1$ $<4.17$ $30.19 (36)$
$2$ PCA $2.161_{-0.006}^{+0.007}$ $6.79$ $12.27_{-0.75}^{+0.78}$ $4.305_{-0.024}^{+0.024}$ $6.43$ $5.39_{-0.51}^{+0.57}$ $102.5 (62)$
$2-5$ keV $2.152_{-0.007}^{+0.008}$ $7.24$ $9.70_{-0.66}^{+0.72}$ $4.326_{-0.025}^{+0.025}$ $6.21$ $5.72_{-0.52}^{+0.63}$ $108.6 (62)$
$5-13$ keV $2.162_{-0.007}^{+0.006}$ $6.90$ $14.53_{-0.78}^{+0.79}$ $4.286_{-0.023}^{+0.024}$ $7.10$ $6.24_{-0.59}^{+0.64}$ $80.10 (62)$
$13-20$ keV $2.169_{-0.009}^{0.008}$ $5.60$ $16.95_{-1.26}^{+1.28}$ $4.333_{-0.092}^{+0.102}$ $5.82$ $4.71_{-1.35}^{+1.75}$ $87.52 (42)$
$20-40$ keV $2.182_{-0.012}^{+0.012}$ $6.28$ $15.62_{-1.53}^{+1.66}$ $4.30$ [*[frozen]{}*]{} $>6.14$ $<4.12$ $46.24 (47)$
$1-2$ Merged $40-80$ keV $2.15$ [*[Frozen]{}*]{} $>15$ $<14.21$ $7.69 (10)$
$3$ PCA $2.378_{-0.008}^{+0.008}$ $5.8$ $12.31_{-0.68}^{+0.70}$ $4.654_{-0.028}^{+0.029}$ $5.28$ $5.94_{-0.50}^{+0.55}$ $87.98 (62)$
$2-5$ keV $2.361_{-0.008}^{+0.009}$ $5.84$ $9.67_{-0.5}^{+0.53}$ $4.690_{-0.025}^{+0.026}$ $5.39$ $6.07_{-0.43}^{+0.46}$ $94.8 (62)$
$5-13$ keV $2.382_{-0.007}^{+0.008}$ $5.71$ $14.78_{-0.81}^{+0.86}$ $4.691_{-0.031}^{+0.03}$ $5.54$ $6.95_{-0.66}^{+0.7}$ $70.54 (62)$
$13-20$ keV $2.391_{-0.009}^{+0.009}$ $5.97$ $15.78_{-1.16}^{+1.26}$ $4.558_{-0.1}^{+0.127}$ $6.16$ $4.94_{-1.42}^{+1.49}$ $100.1 (62)$
$20-40$ keV $2.376_{-0.011}^{+0.013}$ $5.78$ $15.83_{-1.42}^{+1.52}$ $4.65$ [*[Frozen]{}*]{} $>5.8$ $<4.48$ $43.58 (36)$
$40-80$ keV $2.35$ [*[Frozen]{}*]{} $>10.21$ $<16.08$ $19.29 (28)$
$04$ $23$ $2000$ $1$ PCA $2.901_{-0.007}^{+0.009}$ $4.77$ $12.07_{-0.47}^{+0.51}$ $5.706_{-0.036}^{+0.036}$ $4.42$ $5.23_{-0.35}^{+0.43}$ $87.98 (62)$
$2-5$ keV $2.871_{-0.009}^{+0.009}$ $5.31$ $9.30_{- 0.44}^{+0.48}$ $5.832_{-0.039}^{+0.041}$ $5.12$ $4.80_{-0.41}^{+0.44}$ $83.09 (62)$
$5-13$ keV $2.905_{-0.007}^{+0.008}$ $5.12$ $14.14_{-0.65}^{+0.68}$ $5.752_{-0.033}^{+0.034}$ $5.70$ $5.71_{-0.48}^{+0.52}$ $79.32 (62)$
$13-20$ keV $2.921_{0.009}^{+0.01}$ $5.35$ $15.53_{-0.78}^{+0.84}$ $5.535_{-0.146}^{+0.132}$ $4.22$ $5.48_{-1.16}^{+1.71}$ $122.8 (62)$
$20-40$ keV $2.925_{-0.015}^{+0.015}$ $4.91$ $17.64_{-1.29}^{+1.38}$ $5.940_{-0.16}^{+0.2}$ $7.36$ $5.60_{-2.06}^{+3.10}$ $76.89 (59)$
$2$ PCA $2.882_{-0.008}^{+0.01}$ $5.34$ $11.47_{-0.54}^{+0.64}$ $5.627_{-0.051}^{+0.051}$ $3.91$ $5.41_{-0.47}^{+0.50}$ $100.1 (62)$
$2-5$ keV $2.866_{-0.012}^{+0.012}$ $5.83$ $8.88_{-0.57}^{+0.64}$ $5.714_{-0.065}^{+0.067}$ $3.19$ $6.11_{-0.71}^{+0.85}$ $79.77 (62)$
$5-13$ keV $2.883_{-0.009}^{+0.01}$ $5.49$ $13.86_{-0.72}^{+0.74}$ $5.640_{-0.063}^{+0.062}$ $3.90$ $6.33_{-0.68}^{+0.72}$ $94.63 (62)$
$13-20$ keV $2.899_{-0.01}^{+0.013}$ $5.71$ $14.97_{-0.87}^{+1.31}$ $5.65$ [*[Frozen]{}*]{} $4.92$ $4.58_{-1.3}^{+2.16}$ $63.84 (42)$
$20-40$ keV $2.901_{-0.019}^{+0.019}$ $4.98$ $16.96_{-1.59}^{+1.73}$ $5.65$ [*[Frozen]{}*]{} $>17.65$ $<3.65$ $36.86 (29)$
$1-2$ Merged $40-80$ keV $2.85$ [*[Frozen]{}*]{} $>7.5$ $<12.41$ $50.04 (41)$
|
{
"pile_set_name": "ArXiv"
}
|
**Persistent magnetization at neutrino pair emission**
M. Yoshimura
Research Institute for Interdisciplinary Science, Okayama University\
Tsushima-naka 3-1-1 Kita-ku Okayama 700-8530 Japan
[**ABSTRACT**]{}
Measurement of parity violating magnetization is proposed as a means to determine neutrino properties such as Majorana/Dirac distinction and absolute neutrino masses. The process we use is radiative neutrino pair emission from a collective and coherent body of lanthanoid ions doped in host crystals. A vector-component of electron spin flip parallel to photon direction emitted from ion excited state generates this type of magnetization which is stored for a long time in crystals till spin relaxation time.
Keywords Majorana neutrino, neutrino mass, parity violation, neutrino pair emission, magnetization of lanthanoid ions, SQUID
**Introduction**
================
Measurement of parity odd quantity in radiative neutrino pair emission (RENP) [@renp; @overview] is of great use to distinguish the process involving weak interaction from purely QED processes [@pv; @ysu]. An interesting quantity of this nature is magnetization parallel to the emitted photon direction along with neutrino pair emission: expectation value in the final state $| f\rangle $, $2 \mu_{B} \langle f| \hat{ k}\cdot \vec{S}| f \rangle $, where $\hat k$ is the unit vector along the emitted photon momentum and $\vec{S}$ is the electron spin operator with $\mu_B$ the Bohr magneton. This quantity is parity odd and time reversal (T) even. The generated magnetization persists till spin relaxation time.
Parity odd quantity may emerge from interference term of parity even and odd operators in the weak hamiltonian $H_W$ of neutrino pair emission. Writing the neutrino pair emission hamiltonian in terms of the neutrino mixing matrix elements, $U_{ei}\,, i=1,2,3\,,
\nu_e = \sum_i U_{ei} \nu_i $, the hamiltonian is given by $$\begin{aligned}
&&
H_W =
\frac{G_F}{\sqrt{2}}\, \Sigma_{ij}\,
\bar{\nu}_i \gamma^{\alpha} (1 - \gamma_5)\nu_j \,
\bar{e}\left( \gamma_{\alpha} c_{ij} - \gamma_{\alpha} \gamma_5 b_{ij}
\right) e
\equiv
\frac{G_F}{\sqrt{2}}\left( {\cal N}_c ^{\alpha}
\bar{e} \gamma_{\alpha} e - {\cal N}_c ^{\alpha}
\bar{e}\gamma_{\alpha} \gamma_5 e
\right)
\,,
\label{weak cc}
\\ &&
b_{ij} = U_{ei}^* U_{ej} - \frac{1}{2}\delta_{ij}
\,, \hspace{0.5cm}
c_{ij} = U_{ei}^* U_{ej} - \frac{1}{2} ( 1- 4 \sin^2 \theta_w) \delta_{ij}
= b_{ij} + 2 \sin^2 \theta_w \delta_{ij}
\,,\end{aligned}$$ where $\nu_i$ denotes a neutrino mass eigenstate of mass $m_i$. The dominant interference term of parity violation arises from product of spatial part of axial vector current of electron, the spin operator $\bar{e} \vec{\gamma}\gamma_5 e \sim \langle \vec{S}
\rangle $ (in the non-relativistic limit), and spatial part of vector current, the velocity operator $\bar{e} \vec{\gamma} e \sim \langle \vec{v} \rangle = \langle \vec{p}/m_e \rangle$. Roughly, magnetization caused by $(ij)$ neutrino-pair emission is proportional to $\Re (b_{ij} c_{ij})$, assuming CP conservation, while RENP event rate is proportional to $\Re (b_{ij}^2)$. Experimentally, the weak mixing angle is given by $1 - 4\sin^2 \theta_W \sim 0.046 \pm 0.00 64$. Matrix elements of spin $\langle \vec{S} \rangle $ and velocity operator $\langle \vec{v} \rangle $ are usually of order unity vs a number less than $10^{-3}$, hence interference terms are at least smaller by $10^{-3}$ than rate, a parity even quantity. We shall quantify this ratio for lanthanoid ion we use as target.
Using an effective hamiltonian $H_W'$ of RENP, one calculates generated magnetization, $$\begin{aligned}
&&
\sum_{\nu} \langle f | \hat{k} \cdot \vec{S} | f \rangle
| \langle f | H_W' | e\rangle |^2
= \sum_{\nu} \langle e | H_W' | f \rangle
\langle f | \hat{k} \cdot \vec{S} | f \rangle
\langle f | H_W' | e\rangle
\,,
\label {mag matrix el}\end{aligned}$$ where neutrino variables, their helicities and momenta, are summed over. Hamiltonian $H_W$, hence $ H_W'$ as well, contains both parity even and parity odd operators due to weak interaction of neutrino pair emission. One of hamiltonian matrix elements in eq.(\[mag matrix el\]) must be parity even and the other parity odd, requiring interference of these. Furthermore, in order to have a non-vanishing $ \langle f | \hat{k} \cdot \vec{S} | f \rangle $, the state $| f \rangle$, an energy eigenstate of unperturbed parity even hamiltonian $H_0$, must contain both parity even and odd components. This peculiar situation, which never occurs in isolated atoms, arises when ions are placed in crystals, since crystal field acting on target ions provides parity mixing. In other words, host crystals provide environmental parity violation. Lanthanoid ions of 4f$^n$ ($n=11$ for Er$^{3+}$) system gives rise to what is called forced electric dipole, as pointed out in [@van; @vleck] and its calculation method formulated in [@judd-ofelt]. This is why we adopt lanthanoid ions doped in host crystals as targets. Moreover, 4f electrons in lanthanoid ions are insensitive to environment of host crystals due to filled 5s and 5p electrons in outer shells, which give large spin relaxation time, very important to magnetization measurement. Lanthanoid ions doped with low concentration exhibit paramagnetic property at room temperature.
There have been two proposed methods that use a collective and coherent body of atoms to study still unknown neutrino properties [@renp; @overview], [@ranp]. The method we propose here is different from previous proposals in that we detect accumulated effect remaining in target medium rather than measuring individual events themselves. If the accumulated magnetization is above the sensitivity level of detectors, for instance, a high quality SQUID, then the method is found to be very sensitive to Majorana/Dirac distinction, as shown below. Our approach does not assume the nature of neutrino masses, but lanthanoid experiments can determine whether neutrinos are of Majorana or of Dirac type by measuring angular distribution of magnetization caused by interference terms intrinsic to Majorana neutrino, but absent in the Dirac neutrino case [@my-07]. Measurement of smallest neutrino mass at several meV level in three flavor scheme requires high statistics data, or use of smaller Stark level transitions in crystals. Since there exist a rich variety of J-manifold levels in some lanthanoid ions, both measurement of Majorana/Dirac distinction and smallest neutrino mass seems possible.
Process without photon emission $|e \rangle \rightarrow
| g \rangle + \nu\bar{\nu}$ might be considered for magnetization measurement, but estimate gives rate much smaller than detection level. Radiative neutrino pair emission $|e \rangle \rightarrow
| g \rangle + \gamma+ \nu\bar{\nu}$ [@renp; @overview] gives 10 orders of magnitude larger rates than the process without photon emission. Raman stimulated radiative emission $\gamma + |e \rangle \rightarrow
| g \rangle + \gamma+ \nu\bar{\nu}$ [@ranp] is also conceivable [@ect], but the present scheme is simpler and has a better sensitivity to neutrino properties.
We use the natural unit of $\hbar = c = 1$ throughout the present work unless otherwise stated.
**Magnetization generated at RENP (Radiative Emission of Neutrino Pair)**
=========================================================================
RENP de-excitation path of ion states is defined as $| e \rangle \rightarrow | p \rangle \rightarrow | g \rangle $ with energies $\epsilon_e > \epsilon_p > \epsilon_g$. This is a cascade process in which neutrino pair is emitted at $| e \rangle \rightarrow | p \rangle$ followed by stimulated photon emission at $ | p \rangle \rightarrow | g \rangle $: $| e \rangle \rightarrow | g \rangle + \gamma_0 + \nu{\bar \nu} $.
Probability amplitude of RENP including parity violating terms is given by $$\begin{aligned}
&&
{\cal M} = \frac{G_F}{\sqrt{2}} \frac{\vec{\mu}_{pg}\cdot \vec{B}_0 }
{\omega_0 - \epsilon_{pg} + i \gamma_p/2 }
(\vec{S}_{ep} \cdot \vec{{\cal N}}_b+ \frac{\vec{p}_{ep}}{m_e}\cdot
\vec{{\cal N}}_c )
\,,\end{aligned}$$ where $\vec{{\cal N}}_{b,c}$ is the neutrino pair emission current defined by eq.(\[weak cc\]). Magnetic dipole transition given by $\vec{\mu}_{pg}\cdot \vec{B}_0 $ in this formula is usually dominant for 4f$^n$ lanthanoid optical transitions among ion lower levels, where $\vec{B}_0$ is RENP trigger field strength with its frequency given by $\omega_0$. There may be a forced electric dipole transition of the form, $\vec{e}_{pg}\cdot \vec{E}_0 $, as well in lanthanoid ion transitions. To maximize RENP rates, we irradiate the trigger beam at resonant frequency, $\omega_0 = \epsilon_{pg}$, equated to a level energy spacing. Convolution integral of squared resonance function $\propto 1/\left( (\omega_0 - \epsilon_{pg})^2 + \gamma_p^2/4 \right)$ with the Lorentzian spectrum gives $4/ (\Delta \omega_0 \gamma_p)$, hence we use, in rate formulas below, the laser peak power $P_0 = \vec{B}_0^2 = \vec{E}_0^2$ and its (angular) frequency width $\Delta \omega_0 = 2 \pi \Delta \nu_0$. The width factor $ \gamma_p$ in the resonance formula is equal to $ {\rm Max\,} ( \Delta \omega_0, $ inverse lifetime of state $|p \rangle$ or $|e \rangle )$.
We fully exploit macro-coherent amplification mechanism [@renp; @overview] to enhance rates of weak processes. The macro-coherence amplification has been experimentally verified in weak QED processes of two-photon emission of para-hydrogen vibrational transitions [@psr; @exp]. Compared with spontaneous emission rate, the amplification factor was $\sim 10^{18}$ in these experiments. Rates of macro-coherent amplification are in proportion to $n^2 V = n N$ where $n$ is the number density of target atoms and $V$ is the volume of target region with $N$ the number of total target atoms. Macro-coherent rates are thus enhanced by $n$ in contrast to spontaneous emission rate $\propto N$. When macro-coherence works, both the energy and the momentum conservation holds (neglecting very small atomic recoil), to give rates given by a phase-space integrated quantity of $$\begin{aligned}
&&
|{\cal M}|^2 n^2 V (2\pi)^4 \delta^{(4)} ( p_e - p_g - k_0)
\,.\end{aligned}$$
The phase-space integrated RENP rates are given by $$\begin{aligned}
&&
\Gamma_{{\rm RENP 1}} =
\frac{G_F^2}{ 24 \pi} \frac{P_0}{\Delta \nu_0 }
\frac{\gamma_{pg} }{\epsilon_{pg}^3 \gamma_p } |\rho_{eg}|^2 |\vec{S}_{ep} |^2%S(S+1)
n^2 V {\cal F}
\,, \hspace{0.5cm}
{\cal F} = \sum_{ij} {\cal F}_{ij} \Theta \left( {\cal M}^2 - (m_i + m_j)^2
\right)\,,
\label {renp rate}
\\ &&
4 \pi {\cal F}_{ij}
=
\left\{
\left(1 - \frac{ (m_i + m_j)^2}{ {\cal M}^2} \right)
\left(1 - \frac{ (m_i - m_j)^2}{ {\cal M}^2} \right)
\right\}^{1/2}
\left[
| b_{ij} |^2 \left({\cal M}^2 - m_i^2 - m_j^2 \right)
- 2 \delta_M \Re( b_{ij}^2 )\,m_i m_j \right]
\,,
\nonumber \\ &&
\label {2nu integral pc}
\\ &&
{\cal M}^2 = (\epsilon_{eg} - \omega_0 )^2 - (\vec{p}_{eg} - \vec{k}_0)^2
= (1-r^2 )\epsilon_{eg}^2 - 2 \epsilon_{eg} \epsilon_{pg} \left( 1 - r \cos \theta
\right) \equiv {\cal M}^2(\theta)
\,,
\label {mass sq f}\end{aligned}$$ for parity conserving part, and $$\begin{aligned}
&&
\Gamma_{{\rm RENP 2}} =
\frac{G_F^2}{ 12 \pi} \frac{P_0}{\Delta \nu_0 }
\frac{\gamma_{pg} }{\epsilon_{pg}^3 \gamma_p } |\rho_{eg}|^2
\vec{S}_{ep}\cdot \frac{\vec{p}_{ep}}{m_e} %S(S+1)
n^2 V {\cal G}
\,, \hspace{0.5cm}
{\cal G} = \sum_{ij} {\cal G}_{ij} \Theta \left( {\cal M}^2 - (m_i + m_j)^2
\right)\,,
\label {renp rate 2}
\\ &&
\hspace*{-1cm}
4 \pi {\cal G}_{ij}
=
\left\{
\left(1 - \frac{ (m_i + m_j)^2}{ {\cal M}^2} \right)
\left(1 - \frac{ (m_i - m_j)^2}{ {\cal M}^2} \right)
\right\}^{1/2}
\left[
\Re(c_{ij}^* b_{ij} ) \left({\cal M}^2 - m_i^2 - m_j^2 \right)
- 2 \delta_M \Re(c_{ij} b_{ij} )\,m_i m_j \right]
\,,
\nonumber \\ &&
\label {2nu integral pv}\end{aligned}$$ for parity violating part. Majorana/Dirac difference appears in terms $\propto \delta_M$: $\delta_M = 1$ for Majorana neutrino due to the effect of anti-symmetrized wave functions of two identical fermions [@my-07], and = 0 for Dirac neutrino. We neglected small contribution in $\Gamma_{{\rm RENP 1}} $ from squared vector current $\propto (\vec{p}_{pe}/m_e )^2$.
The mass squared function $ {\cal M}^2(\theta) $ is invariant mass squared given to neutrino pairs and a function of measurable quantity $\theta$. The same function determines six neutrino-pair emission thresholds at different directions $\theta$ given by $ {\cal M}^2(\theta) = (m_i + m_j)^2\,, i,j = 1,2,3$. It also plays important roles in QED background rejection, as discussed below. The initial spatial phase vector $\vec{p}_{eg} $ given at excitation was parametrized by $|\vec{p}_{eg}| = r \epsilon_{eg}$ with its direction $\theta $ away from the RENP trigger beam.
We adopt for excitation scheme two-photon cascade process using two pulsed lasers: $\gamma_1 + | g\rangle^{\pm} \rightarrow | q \rangle^{\mp} $ and $\gamma_2 + | q\rangle^{\mp} \rightarrow | e \rangle^{\pm} $. This way two relevant states, $|g\rangle $ and $|e \rangle $, have the same time reversal quantum numbers, T $= \pm$, since single photon process is governed by time reversal odd operator whether it is of magnetic dipole or electric dipole. Counter-propagating two-photon cascade excitation requires frequencies of two lasers to satisfy $\omega_1 + \omega_2 = \epsilon_{eg}\,,
\omega_1 - \omega_2 = r \epsilon_{eg}$. A maximal excitation rate is provided when two frequencies $\omega_i$ are matched to level spacings, namely resonant excitation. The neutrino-pair emission operator is also T-odd. T-even two-photon excitation here helps to reject QED background processes, as discussed later.
The quantity $\rho_{eg}$ (called coherence in the optics literature) that appears in rate formulas, eq.(\[renp rate\]) and eq.(\[renp rate 2\]), is generated at excitation to $| e\rangle $, its maximum value being 1/2. It is in general time dependent, too. Estimate of this quantity requires detailed simulations of Maxwell-Bloch equations, a set of non-linear partial differential equations that deal with ion state wave functions and propagating electromagnetic fields in a simplified spacetime of one space and one time dimensions [@renp; @overview]. In a off-resonance Raman excitation simulations suggest that this quantity is of order $10^{-3}$ or less. We expect a larger coherence of order $10^{-1} \sim 10^{-2}$ in the on-resonance cascade excitation adopted here.
Magnetization is a macroscopic quantity given by $$\begin{aligned}
&&
2 \mu_{{\rm eff}} n \approx 6.6\, {\rm G} \frac{n}{ 10^{18}{\rm cm}^{-3}}
\; ({\rm for \; Er}^{3+})
\,,\end{aligned}$$ using $\mu_{{\rm eff}} = g \sqrt{J(J+1)} \approx 9.5 \mu_B $ for Er$^{3+}$ in its ground state as an illustration. The generated magnetization by RENP is this value times interference rate, $\Gamma_{{\rm RENP 2}} $ of eq.(\[renp rate 2\]).
Both rates $\propto {\cal F}$ and magnetization $\propto {\cal G}$ are functions of the mass squared function $ {\cal M}^2(\theta) $. In measuring the angular distribution of signals one may use a single trigger beam at different directions of irradiation to determine neutrino-pair thresholds. Six thresholds have different weights, which are calculable from neutrino oscillation data [@pdg] and listed in the following table, assuming CP conservation case described by real number $b_{ij}, c_{ij}$.
(11) (12) (22) (13) (23) (33)
------------------- -------- ------- -------- -------- -------- -------
[rate]{} 0.0311 0.405 0.0401 0.0325 0.0144 0.227
[magnetization]{} 0.508 0.405 0.517 0.0325 0.0144 0.704
Spin factors are calculated as follows. We consider unpolarized targets, hence average over initial $|e \rangle$ magnetic quantum numbers and sum over final $|p \rangle$ magnetic quantum numbers. Using Wigner-Eckart theorem applied to manifolds of the same $J$ value (in actual application $J=11/2$), one may relate $\vec{S}_{ep}^2$ and $\vec{S}_{ep}\cdot \vec{p}_{ep}/m_e$ to electric and magnetic dipole transition rates. Summation and average over magnetic $J$ quantum numbers of two state gives $( \vec{S}_{ep}\cdot \vec{p}_{ep}/m_e)^2 = \vec{S}_{ep}^2 ( \vec{p}_{ep}/m_e)^2$. Relation of spin and velocity matrix elements to transition rates is $\gamma^E_{ep} = e^2 ( \vec{p}_{ep}/m_e)^2 \Delta_{ep}/3\pi$ and $\gamma^M_{ep} = \vec{S}_{ep}^2 \mu_{ep}^2\Delta_{ep}^3/3\pi$ where $\Delta_{ep}$ is the energy difference of two states. Hence $ ( \vec{p}_{ep}/m_e)^2/\vec{S}_{ep}^2
= e^2/(\mu_{ep}^2\Delta_{ep}^2 ) = (2m_e/g \Delta_{ep})^2 $ with the Lande g-factor $g=2$ of electron spin. This gives $$\begin{aligned}
&&
\frac{|\vec{v}_{ep}| }{ | \vec{S}_{ep} |}
= \frac{ \Delta_{ep} } { m_e} \sqrt{ \frac{ \gamma^{ E}_{ep} }{ \gamma^{M}_{ep} }} \equiv (\frac{v}{S})_{ep}
\,.\end{aligned}$$ We illustrate an example of $v/S$ calculations for Er$^{3+}$ in Appendix. For unpolarized targets $\vec{S}_{ep}^2$ is of order unity, close to $(2 J_p +1)/(2J_e+1) $.
To obtain a large PV/PC ratio it is desirable to use a large electric/magnetic transition ratio, since the ratio $ \frac{\Delta_{pe}} {m_e} \approx 10^{-6} $ is small. In lanthanoid ions doped in crystals transitions among low lying levels are predominantly magnetic, and it is desirable to have a large forced electric dipole transition. For trivalent Kramers ions such as Er$^{3+}$ placed at a less symmetric cite (not inversion center for instance) are necessary. Even in cubic crystals trivalent lanthanoid ions often substitute at sites of less symmetry, C$_2$ site instead of C$_{3i}$ IC (Inversion Center). In this respect trivalent Er ion doped in host crystal such as YSO may give larger magnetization than the example here.
**Dominant QED background**
===========================
It is easy to understand that both of macro-coherently amplified QED events, $| e\rangle^{\pm} \rightarrow | g\rangle^{\pm} + \gamma \gamma $ and $| e\rangle^{\pm} \rightarrow | g\rangle^{\mp} + \gamma $ do not exist by kinematic choice of a positive mass squared function $ {\cal M}^2(\theta)$, which constrains trigger directions $\theta$ for a specified chosen parameter $r$. Major backgrounds appear to be macro-coherently amplified $| e\rangle^{\pm} \rightarrow | g\rangle^{\mp} + \gamma \gamma \gamma$ (called McQ3) and spontaneous emission of a single photon. McQ3 events are however characterized by a time reversal odd transition, which is distinguished from time reversal even RENP process, $| e\rangle^{\pm} \rightarrow | g\rangle^{\pm} + \gamma + \nu\bar{\nu} $. This way one can kill time reversal odd transition McQ3 and spontaneous single photon emission, leaving McQ4 $| e\rangle^{\pm} \rightarrow | g\rangle^{\pm} +
\gamma \gamma\gamma \gamma$ the major QED background.
McQ4 do not contribute to magnetization. The only thing one has to worry about is depletion of prepared state $| e\rangle^{\pm} $. A crude estimate of McQ4 event rate shows that it is a few to several orders of magnitudes larger than RENP rate. With extra emitted photon detection, this remaining background is presumably controllable even in rate measurements.
**Example of Er$^{3+}$ RENP scheme**
====================================
Trivalent Er ion has a rich J-manifold structure, as illustrated in Appendix. The ion is at inversion center of this crystal, but J-manifold structure is insensitive to crystal environments. We consider the following de-excitation path of time reversal degeneracy: $$\begin{aligned}
&&
|e \rangle =^4{\rm H}_{11/2}^{\pm}(2.3946) \rightarrow
^4{\rm I}_{11/2}^{\mp}
|g \rangle =^4{\rm I}_{15/2}^{\pm} (0)
\,,
\label {higher level 2}\end{aligned}$$ and cascade excitation of frequencies, $$\begin{aligned}
&&
\omega_1 =1.5894 {\rm eV}(= \epsilon(^4{\rm H}_{11/2} - ^4{\rm I}_{13/2})\,)
\,, \hspace{0.5cm}
\omega_2 =0.805 {\rm eV} (= \epsilon(^4{\rm I}_{13/2} - ^4{\rm I}_{15/2})\,)
\,, \hspace{0.5cm}
r = 0.3276
\,.\end{aligned}$$ J-manifolds are split by crystal field, giving Stark levels having two-fold Kramers degeneracy. We ignore effects of different Stark levels for simplicity. In order to maintain the doubled rate of time reversal degeneracy, it is necessary to shield earth magnetic field.
Using Er$^{3+}$ data given in Appendix, we estimate RENP rate $\Gamma$ and generated magnetization $M$: $$\begin{aligned}
&&
\Gamma = 1.3 \times 10^{-3} {\rm sec}^{-1} \frac{{\cal F} }{{\rm eV}^2 } |\rho_{eg}|^2
|\vec{S}_{ep} |^2
( \frac{ n}{ 10^{18}{\rm cm}^{-3}})^2 \frac{V }{10^{-2}{\rm cm}^3 }
\frac{P_0 }{{\rm GW cm}^{-2} } \frac{100 {\rm MHz} }{ \Delta \nu_0}
\,,
\label {rate number}
\\ &&
M = 7.3 \times 10^{-9} {\rm G\, sec}^{-1}
\frac{{\cal G} }{{\rm eV}^2 } |\rho_{eg}|^2 |\vec{S}_{ep} |^2
( \frac{ n}{ 10^{18}{\rm cm}^{-3}})^3 \frac{V }{10^{-2}{\rm cm}^3 }
\frac{P_0 }{{\rm GW cm}^{-2} } \frac{100 {\rm MHz} }{ \Delta \nu_0}
\,.
\label {mag number}\end{aligned}$$ For this estimate we used numbers appropriate for pulsed trigger laser. For (continuous wave) CW operation $P_0 = 1 {\rm kW cm}^{-2}\,,
\Delta \nu_0 = 1 {\rm kHz} $, hence 1/10 reductions of rate and magnetization are more appropriate for CW.
Magnetization calculated this way is measurable, being comparable to, or above, the SQUID sensitivity level used in fundamental physics experiments such as in axion force experiments, [@quax-gpgs], [@axion; @force], [@moody-wilczek] and axion haloscope experiment [@axion; @haloscope], [@sikivie].
We show magnetization curves, its angular $\theta$ distributions at a specific excitation parameter $r$, in Fig(\[angular dist 3\]) and Fig(\[angular dist 4\]). For comparison rate angular distribution is shown in Fig(\[rate angular dist\]). Magnetization depends on $b_{ij} \times c_{ij}$ above $(ij)$ pair thresholds, while rate depends on $b_{ij}^2$. This difference, as numerically shown in the table, explains a high sensitivity of magnetization to the Majorana/Dirac distinction. On the other hand, it is difficult to distinguish mass types from rate, as seen in Fig(\[rate angular dist\]). Absolute mass determination is harder, and one needs a high statistics data near the end point of angular distribution, as shown in Fig(\[angular dist 4\]).
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{width="7cm"}
**Towards Er$^{3+}$ experimental design**
=========================================
There are many works to be done before we prepare technical design report of actual experiments.
First, results in the present work are incomplete in several places:
\(1) Lack of theoretical calculations or experimental results of crystal field effect in a variety of host crystals, namely, Stark levels and forced electric dipole transition rates along with magnetic transition among low lying levels for trivalent Er ions.
\(2) Simulation estimate of the coherence $\rho_{eg}(t)$ assuming excitation scheme and laser specification.
\(3) Simulation of McQ4 QED backgrounds in order to identify and isolate background events.
\(4) Design of SQUID detection system.
Experimental R and D studies are even more important. One may list some items:
\(5) Optical measurements of 4f$^{11}$ manifolds in candidate host crystals.
\(6) Study of relaxation processes, in particular, phonon related relaxation at low temperatures.
**Appendix: Er$^{3+}$ data and estimated parity violating component**
=====================================================================
We consider a special host crystal, Er$^{3+}$:Cs$_2$NaYF$_6$ or :Y$_2$O$_3$ for concreteness. Trivalent Er ion substitutes inversion center of Y. According to [@er3+], calculated radiative decay rates of 10% doped Er$^{3+}$ in host Cs$_2$NaYF$_6$ are
$$\begin{aligned}
\begin{array}{cccccc}
{\rm initial } & {\rm final } & {\rm energy/eV } & {\rm rate/sec}^{-1} & {\rm radiative\; life/msec} & v/S (10^{-6} )\\
^4{\rm I}_{13/2} \rightarrow & ^4{\rm I}_{15/2} & 0.805 & 24.82^{{\rm MD}} + 2.46^{{\rm ED}} & 36.7 & 0.25 \\
^4{\rm I}_{11/2} \rightarrow & ^4{\rm I}_{15/2} & 1.286 & 4.13^{{\rm ED}} & 113.4 &\\
& ^4{\rm I}_{13/2} & 0. 481 & 4.26^{{\rm MD}} + 0.42^{{\rm ED}}& & 0.15 \\
^4{\rm I}_{9/2} \rightarrow & ^4{\rm I}_{15/2} & 1.563 & 9.46^{{\rm ED}} & 73.3 &\\
& ^4{\rm I}_{13/2} & 0.7575 & 3.48^{{\rm ED}} & & \\
& ^4{\rm I}_{11/2} & 0.277 & 0.67^{{\rm MD}} + 0.02^{{\rm ED}} & & 0.047 \\
^4{\rm F}_{9/2} \rightarrow & ^4{\rm I}_{15/2} & 1.9014 &58.91^{{\rm ED}} &14.6 & \\
& ^4{\rm I}_{13/2} & 1.0964 & 3.5^{{\rm ED}} & & \\
& ^4{\rm I}_{11/2} & 0.619 & 3.49^{{\rm MD}} +1.06^{{\rm ED}} & & 0.33\\
& ^4{\rm I}_{9/2} & 0.339 & 1.34^{{\rm MD}} +0.18^{{\rm ED}} & & 0.12\\
^4{\rm S}_{3/2} \rightarrow & ^4{\rm I}_{15/2} & 2.2645 {\rm eV} & 4.93^{{\rm ED}} & 131.6 & \\
& ^4{\rm I}_{13/2} & 1.4593{\rm eV} & 2.02^{{\rm ED}} & & \\
& ^4{\rm I}_{11/2} & 0.9785{\rm eV} & 0.18^{{\rm ED}} & &\\
&^4{\rm I}_{9/2} & 0.7017{\rm eV} & 0.48^{{\rm ED}} & &\\
^4{\rm H}_{11/2} \rightarrow & ^4{\rm I}_{15/2} & 2.3946 {\rm eV} &518.02^{{\rm ED}} & 1.7 & \\
& ^4{\rm I}_{13/2} & 1.5894{\rm eV} & 49.43^{{\rm MD}}+7.91^{{\rm ED}} & & 0.62 \\
& ^4{\rm I}_{11/2} & 1.1091{\rm eV} & 5.58^{{\rm MD}} +4.73^{{\rm ED}} & & 1.0\\
&^4{\rm I}_{9/2} & 0.8318{\rm eV} & 0.49^{{\rm MD}} + 5.52^{{\rm ED}}& & 2.7
\end{array}\end{aligned}$$
Low lying levels of Er$^{3+}$ have configuration 4f$^{11}$ and its ground state $^4{\rm I}_{15/2}$ has the effective magnetic moment related to the Lande factor $\mu_{{\rm eff}} = g \sqrt{J(J+1)} = 9.5$ (experimental value). Ratio of parity violating to conserving amplitude given by $v/S$ was estimated by using the formula in the text. Low values of $v/S$ here are presumably related to Er$^{3+}$ site at inversion center (IC) of host crystals. Without IC $v/S$’s are expected to be larger.
This research was partially supported by Grant-in-Aid 17H02895 from the Ministry of Education, Culture, Sports, Science, and Technology.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We study the symmetry properties of the spectra of normalized Laplacians on signed graphs. We find a new machinery that generates symmetric spectra for signed graphs, which includes bipartiteness of unsigned graphs as a special case. Moreover, we prove a fundamental connection between the symmetry of the spectrum and the existence of damped two-periodic solutions for the discrete-time heat equation on the graph.\
**MSC2010:** 05C50, 05C22, 39A12
address:
- 'Max Planck Institute for Mathematics in the Sciences, Leipzig 04103, Germany'
- 'School of Mathematical Sciences, LMNS, Fudan University, Shanghai 200433, China; Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, China'
author:
- 'Fatihcan M. Atay'
- Bobo Hua
title: On the symmetry of the Laplacian spectra of signed graphs
---
Introduction
============
A signed graph refers to a graph together with a labeling of edges by a sign $\pm 1$, so as to represent two different types of relationships between adjacent vertices. Such graphs originate from studies of social networks where one distinguishes between friend and foe type relations [@Harary53]. They are also appropriate models of electrical circuits with negative resistance or neuronal networks with inhibitory or excitatory connections.
Let $G(V,E,\mu)$ be a finite, undirected, weighted graph without isolated vertices, having the vertex set $V$, edge set $E$, and edge weight $\mu:E\to \mathbb{R}_+$. Two vertices $x,y$ are called neighbours, denoted $x\sim y$, if $(x,y)\in E$. The weighted degree measure $m: V\to (0,\infty)$ on the vertices is defined by $m(x)=\sum_{y:y\sim x}\mu_{xy}$. It reduces to the combinatorial degree for the unweighted case, i.e. when $\mu=1_E,$ the constant function $1$ on $E.$ A function $\eta: E \to \{1,-1\}$ is called a sign function on $E.$ We call $\Gamma=(G,\eta)$ a *signed graph* based on $G$ with the sign function $\eta.$ Let $\ell^2(V,m)$ denote the space of real functions on $V$ equipped with the $\ell^2$ norm with respect to the measure $m.$ The *normalized signed Laplacian* $\Delta: \ell^2(V,m) \to \ell^2(V,m)$ for a signed graph $\Gamma$ is defined by $$\Delta f(x)=\frac{1}{m(x)}\sum_{y\in V:y\sim x}\mu_{xy}(f(x)-\eta_{xy}f(y)), \quad \forall f\in \ell^2(V,m).$$ One can show that $\Delta$ is a self-adjoint linear operator on $\ell^2(V,m).$ The eigenvalues of $\Delta$ will be referred to as the spectrum of the signed graph $\Gamma$ and denoted by $\sigma(\Gamma)$. The spectrum is always contained in the interval $[0,2]$. The spectral properties of signed graphs have been studied by several authors, e.g. [@HouLi03; @Hou05; @LiLi08; @LiHonghaiLi09; @GerminaHameedZaslavsky11; @AtayTuncel; @Belardo14].
We say that the spectrum of a signed graph $\Gamma$ is *symmetric* (w.r.t. the point $\{1\}$) if $$\sigma(\Gamma)=2-\sigma(\Gamma),$$ that is, $\lambda$ is an eigenvalue of the signed Laplacian $\Delta$ of $\Gamma$ if and only if $2-\lambda$ is. For unsigned graphs, i.e. for $\eta=1_E$, it is well known that the spectrum is symmetric if and only if the graph is bipartite [@Chung97]. For signed graphs, however, the symmetry of the spectrum and bipartiteness are not equivalent: While bipartite signed graphs do have symmetric spectra (see Lemma \[l:bipartite spectrum\]), we will see that the reverse implication does not hold, and there exist non-bipartite signed graphs which nevertheless have symmetric spectra. In fact, we will present a general machinery to produce symmetric spectra for signed graphs that has no counterpart in the unsigned case. This can be seen as one of the major differences between spectral theories of unsigned and signed graphs.
To state our result, we use the concept of a *switching function*, i.e., a function $\theta: V\to \{1,-1\}$: Given $\Gamma=(G,\eta)$, a switching function $\theta$ can be used to define a new graph $\Gamma^{\theta}=(G,\eta^\theta)$ having the sign function $\eta^{\theta}(xy)=\theta(x)\eta_{xy}\theta(y)$. For a signed graph $\Gamma=(G,\eta),$ we denote by $-\Gamma=(G,-\eta)$ the signed graph with the opposite sign function. We say that two signed weighted graphs $\Gamma=(V,E,\mu,\eta)$ and $\Gamma'=(V',E',\mu',\eta')$ are *isomorphic*, denoted $\Gamma\simeq\Gamma'$, if there is a bijective map $S:V\to V'$ such that $Sx\sim Sy$ iff $x\sim y,$ and $\mu_{SxSy}'=\mu_{xy}$ and $\eta_{SxSy}'=\eta_{xy}$ for any $x\sim y$. That is, two signed graphs are isomorphic if they only differ up to a relabeling of vertices. Based on these notions we prove the following in Theorem \[t:symmetry of the spectrum\]:
*Given $\Gamma=(G,\eta)$, if there is a switching function $\theta: V\to \{1,-1\}$ such that $\Gamma^{\theta}\simeq -\Gamma$, then the spectrum of $\Gamma$ is symmetric.*
Motivated by this new machinery, we provide a nontrivial example (Example \[ex:symmetric spectrum\]) of a signed graph that possesses a symmetric spectrum but is not bipartite. Although the symmetry of the spectrum of a signed graph is a purely algebraic property, we find that it plays an important role in the analysis and dynamics on the graph. We say that a function $f:({{\mathbb N}}\cup\{0\})\times V\to {{\mathbb R}}$ solves the discrete-time heat equation on $(G,\eta)$ with the initial data $g:V\to {{\mathbb R}}$, if for any $n\in \mathbb{N}\cup\{0\}$ and $x\in V$, $$\label{e:discrete time heat}
\left\{\begin{array}{ll}f(n+1,x)-f(n,x)=-\Delta f(n,x), & \\
f(0,x)=g(x). &
\end{array}
\right.$$ This definition mimics continuous-time heat equations on Euclidean domains or Riemannian manifolds. Among all solutions to the discrete-time heat equation, a special class, namely damped $2$-periodic solutions (see Definition \[d:periodic solution\]), will be of particular interest. These solutions are motivated from the well-known oscillatory solutions on unsigned graphs: For a bipartite unsigned graph $G(V,E,\mu)$ with bipartition $V=V_1\cup V_2,$ the solution to the discrete-time heat equation with initial data $f(0,\cdot)=1_{V_1}$ (i.e. the characteristic function on $V_1$) is given by (see e.g. [@Grigoryan09 pp. 43]) $$\label{e:periodic_solution}
f(n,\cdot)=\left\{\begin{array}{ll}1_{V_1}, & n \text{ even}, \\
1_{V_2}, & n \text{ odd}.
\end{array}
\right.$$ That is, the solution oscillates between two phases, $1_{V_1}$ and $1_{V_2},$ which justifies the name of $2$-periodic solution. We generalize such solutions to signed graphs and also allow temporal damping with decay rate $\lambda\in[0,1]$. (In the example of , $\lambda=1$.) We show that this analytic property of the solutions, i.e. the periodicity of order two, is deeply connected to the symmetry of the spectrum of the signed graph. Precisely, we prove the following in Theorem \[t:symmetry and periodic\]:
*Let $(G,\eta)$ be a signed graph. Then $u$ is a damped $2$-periodic solution with decay rate $\lambda$ ($\lambda\neq0$) if and only if $$u=f+g,$$ where $f$ and $g$ are eigenfunctions corresponding to the eigenvalues $1-\lambda$ and $1+\lambda$, respectively, of the normalized Laplacian.*
In the last section, we study the spectral properties of the motif replication of a signed graph. A subset of vertices, say $\Omega$, of a signed graph $\Gamma=(G,\eta)$ is sometimes referred to as a motif. By motif replication we refer to the enlarged graph $\Gamma^{\Omega}$ that contains a replica of the subset $\Omega$ with all its connections and weights; see [@AtayTuncel] or Section \[s:motif\]. Let $\Delta_{\Omega}$ be the signed Laplacian on $\Omega$ with Dirichlet boundary condition whose spectrum is denoted by $\sigma(\Delta_{\Omega})$; see [@BHJ12] for the unsigned case or Section \[s:motif\]. We prove in Theorem \[t:motif replication\] that $\sigma(\Delta_{\Omega})\subset \sigma(\Gamma^{\Omega}).$ This follows from a discussion with Bauer-Keller [@BauerKeller12] which obviously generalizes [@AtayTuncel Theorem 13]. As a consequence, if the subgraph $\Omega$ admits damped 2-periodic solutions, then so does the larger graph $\Gamma^\Omega$ after replication.
The paper is organized as follows. In the next section we introduce the concepts of signed graphs and normalized signed Laplacians, and study their spectral properties. In Section \[s:symmetry\], we explore a general machinery to create symmetry in the spectrum. Section \[s:Periodic solutions\] is devoted to damped $2$-periodic solutions of the discrete-time heat equation and their connection to the symmetry of the spectrum. The last section contains the spectral properties of motif replication.
Basic properties of signed graphs {#s:basic properties signed graphs}
=================================
In this section, we study the basic properties of signed graphs and the spectral properties of the normalized signed Laplacian. Let $G(V,E)$ be a finite (combinatorial) graph with the set of vertices $V$ and the set of edges $E$ where $E$ is a symmetric subset of $V\times V$. A graph is called connected if for any $x,y\in V$ there is a finite sequence of vertices, $\{x_i\}_{i=0}^n,$ such that $$x=x_0\sim x_1\sim \cdots\sim x_n=y.$$ In this paper, we consider finite, connected, undirected graphs without isolated vertices.
We assign symmetric weights on edges, $$\mu:E\to (0,\infty),\quad E\ni (x,y)\mapsto \mu_{xy}$$ which satisfies $\mu_{xy}=\mu_{yx}$ for any $x\sim y,$ and call the triple $G(V,E,\mu)$ a *weighted graph*. The special case of $\mu=1_E$ is also referred to as an unweighted graph. For any $x\in V$, the weighted degree of $x$ is defined as $$m(x)=\sum_{y\sim x} \mu_{xy}.$$ The weighted degree function $m: V\to (0,\infty),$ $x\mapsto m(x)$ can be understood as a measure on $V.$ We denote by $\ell^2(V,m)$ the space of real functions on $V$ equipped with an inner product with respect to the measure $m$, defined by $\langle u,v\rangle=\sum_{x\in
V}u(x)v(x)m(x)$ for $u,v\in \ell^2(V,m).$
\[d:signed graphs\] Let $G(V,E,\mu)$ be a weighted graph. A symmetric function $\eta: E\to \{1,-1\},$ $E\ni(x,y)\mapsto\eta_{xy}$, is called a *sign function* on $G$. We refer to the quadruple $(V,E,\mu,\eta)=(G,\eta)$ as a *weighted signed graph*.
In the special case $\eta=1_E$, $(G,\eta)$ is called an unsigned graph. In the following, by signed graphs we always mean weighted signed graphs. For convenience, we define the *signed weight* of a signed graph $(V,E,\mu,\eta)$ by ${{\kappa}}=\eta \mu: E\to {{\mathbb R}},$ i.e. ${{\kappa}}_{xy}=\eta_{xy}\mu_{xy}$ for any $x\sim y.$
Let $\Gamma=(V,E,\mu,\eta)$ be a signed graph. The *normalized signed Laplacian* of $\Gamma$, denoted by $\Delta_{\Gamma},$ is defined as $$\Delta_{\Gamma} f(x)=\frac{1}{m(x)}\sum_{y\sim x}\mu_{xy}(f(x)-\eta_{xy}f(y)),\ \ \ \forall\ f:V\to{{\mathbb R}}.$$
The adjacency matrix of a signed graph $\Gamma$ is defined by $$A_{\Gamma}(x,y)=\left\{\begin{array}{lr} {{\kappa}}_{xy}=\eta_{xy}\mu_{xy},& \mathrm{if}\ x\sim y \\
0,&\mathrm{otherwise}\end{array}\right.$$ The degree matrix is defined as $D_{\Gamma}(x,y):=m(x)\delta_{xy}$, where $\delta_{xy}=1$ if $y=x$, and $0$ otherwise. Hence, the normalized Laplacian of the signed graph $\Gamma=(G,\eta)$ can be expressed as the matrix $$\Delta_{\Gamma}=I-D_{\Gamma}^{-1}A_{\Gamma},$$ or as an operator on $\ell^2(V,m)$, $$\Delta_{\Gamma} f(x)=f(x)-\frac{1}{m(x)}\sum_{y\sim x}f(y){{\kappa}}_{xy}.$$ for any $f:V\to{{\mathbb R}}$. We call $P_{\Gamma}=D_{\Gamma}^{-1}A_{\Gamma}$ the *generalized transition matrix* in analogy to the transition matrix of the simple random walk on an unsigned graph. One notices that the row sum of $P_{\Gamma}$ is not necessarily equal to $1$ for a generic signed graph, which is an obvious difference between signed and unsigned graphs. We will often omit the subscripts and simply write $\Delta$ and $P$ for $\Delta_{\Gamma}$ and $P_\Gamma$, respectively unless we want to emphasize the underlying signed graph $\Gamma$.
One can show that $\Delta:\ell^2(V,m)\to
\ell^2(V,m)$ is a bounded self-adjoint linear operator on a finite dimensional Hilbert space, and hence the spectrum is real and discrete. We denote by $\sigma(T)$ the spectrum of a linear operator $T$ on $\ell^2(V,m)$. By the spectrum of a signed graph $\Gamma$, denoted by $\sigma(\Gamma),$ we mean the spectrum of the normalized signed Laplacian of $\Gamma,$ $\sigma(\Delta_{\Gamma}).$ Since $\Delta=I-P,$ we have $\sigma(\Delta)=1-\sigma(P)$, where the right hand side is understood as $\{1-\lambda:\lambda\in
\sigma(P)\}.$ By the Cauchy-Schwarz inequality, one can show that the operator norm of $P$ on $\ell^2(V,m)$ is bounded by 1. Hence, $\sigma(P)\subset [-1,1]$, and consequently $\sigma(\Gamma)\subset [0,2]$ for any signed graph $\Gamma.$
Given a weighted graph $G(V,E,\mu)$, we let $$\mathcal{G}=\{(G,\eta)\, | \, \eta \ {\rm is\ a\ sign\ function}\}$$ denote the set of all signed graphs with a common underlying weighted graph $G$. We distinguish two special cases where the edges have all positive or all negative signs, namely $(G,+):=(G,1_E)$ and $(G,-):=(G,-1_E)$, respectively.
Given a signed graph $\Gamma=(G,\eta)$, a function $\theta: V\to \{1,-1\}$ is called a *switching function* (on $V$). Using the switching function $\theta,$ one can define a new signed graph as $$\label{d:switched}\Gamma^{\theta}=(G,\eta^\theta),\quad\quad \mathrm{where}\quad
\eta^{\theta}(xy)=\theta(x)\eta_{xy}\theta(y),\quad x\sim y.$$ For a switching function $\theta,$ let $S^{\theta}$ denote the diagonal matrix defined as $S^{\theta}(x,y):=\theta(x)\delta_{xy}$. Then the adjacency matrix of $\Gamma^{\theta}$ can be written as $$A_{\Gamma^{\theta}}=S^{\theta}A_{\Gamma} S^{\theta},$$ Note that the degree matrix is invariant under the switching operation, i.e. $D_{\Gamma^{\theta}}=D_{\Gamma},$ and $(S^{\theta})^{-1}=S^{\theta}$. Clearly, $S^{\theta}D=DS^{\theta}$, and hence $\Delta_{\Gamma^{\theta}}=(S^{\theta})^{-1}\Delta_{\Gamma}
S^{\theta}$, which implies that the spectrum is invariant under the switching operation $\theta.$ This observation yields the following lemma.
\[l:invariant by switching\] Let $(G,\eta)$ be a signed graph and $\theta:V\to\{1,-1\}$ a switching function. Then the switched signed graph $\Gamma^{\theta}$ defined in has the same spectrum as $\Gamma$, i.e. $$\sigma(\Gamma^{\theta})=\sigma(\Gamma).$$ Moreover, if $f:V\to{{\mathbb R}}$ is an eigenfunction of $\Delta_\Gamma$ corresponding to the eigenvalue $\lambda$, then the function $f^{\theta}:V\to{{\mathbb R}}$, defined by $f^{\theta}(x)=\theta(x)f(x)$ for $x\in V$, is an eigenfunction of $\Delta_{\Gamma^{\theta}}$ corresponding to the eigenvalue $\lambda$.
Given a weighted graph $G(V,E,\mu)$ with a fixed labeling of vertices, we introduce an equivalence relation on the set of all signed graphs $\mathcal{G}$ based on $G$: Two signed graphs $\Gamma_1,\Gamma_2\in \mathcal{G}$ are called equivalent, denoted $\Gamma_1\sim\Gamma_2,$ if there exists a switching function $\theta: V\to \{1,-1\}$ such that $\Gamma_1^{\theta}=\Gamma_2$. For $\Gamma\in \mathcal{G}$, we denote by $\overline{\Gamma}$ the equivalence class of $\Gamma$ and by $\overline{\mathcal{G}}:=\{\overline{\Gamma}:\Gamma\in\mathcal{G}\}$ the set of all equivalence classes. For a finite signed graph $\Gamma=(G,\eta)$, we order the eigenvalues of the normalized Laplacian in a nondecreasing way: $$0\leq \lambda_1\leq \lambda_2\leq\cdots\leq \lambda_N\leq 2,$$ where $N=|V|.$ We always denote the smallest and largest eigenvalues by $\lambda_1(\Gamma)$ and $\lambda_N(\Gamma)$, respectively. The following result is well known for unsigned graphs (e.g. [@Chung97]).
\[l:spectrum unsigned\] Let $G(V,E,\mu)$ be a finite connected unsigned graph. Then the following are equivalent:
(i) $G$ is bipartite.
(ii) $\lambda_N(G)=2.$
(iii) The spectrum of $G$ is symmetric with respect to 1, i.e. $\sigma(G)=2-\sigma(G)$.
Clearly, statement $(iii)$ is equivalent to saying that the spectrum of the transition matrix $P=D^{-1}A$ is symmetric with respect to 0, i.e. $\sigma(P)=-\sigma(P).$
In this section, we characterize property $(ii)$ of Lemma \[l:spectrum unsigned\] for signed graphs. Property $(iii),$ i.e. the symmetry of the spectrum, will be postponed to the next section.
Let $C$ be a cycle, i.e. $C=\{x_i\}_{i=0}^k$ such that $x_0\sim x_1\sim\cdots\sim x_k\sim x_0,$ in a signed graph $\Gamma=(G,\eta)$. The sign of $C$ is defined as $${\rm sign}(C)=\prod_{e\in C}\eta_e$$ where the product is taken over all edges $e$ in the cycle. A signed graph $\Gamma=(G,\eta)$ is called *balanced* if every cycle in $\Gamma$ has positive sign. The following characterization of balanced signed graphs is well known [@Harary53 Theorem 1], [@Zaslavsky82], [@Hou05 Corollary 2.4] and [@LiHonghaiLi09 Theorem 1].
Let $\Gamma=(G,\eta)$ be a signed graph. Then the following statements are equivalent:
(a) $\Gamma$ is balanced.
(b) $\Gamma\in \overline{(G,+)}.$
(c) There exists a partition of $V,$ $V=V_1\cup V_2$ and $V_1\cap
V_2=\emptyset$, such that every edge connecting $V_1$ and $V_2$ has negative sign and every edge within $V_1$ or $V_2$ has positive sign.
(d) $\lambda_1(\Gamma)=0.$
For a signed graph $\Gamma=(G,\eta)$, the *reverse signed graph* is defined as $-\Gamma:=(G,-\eta)$. Clearly $(G,-)=-(G,+).$ A signed graph $\Gamma=(G,\eta)$ is called *antibalanced* if $-\Gamma$ is balanced. Furthermore, since $P_{-\Gamma}=-P_{\Gamma},$ we have $$\label{e:minusoperation}
\sigma(-\Gamma)=2-\sigma(\Gamma).$$ Based on these observations, one obtains the following lemma; see also [@LiHonghaiLi09 Theorem 1].
Let $\Gamma=(G,\eta)$ be a signed graph. Then the following are equivalent:
(a) $\Gamma$ is antibalanced.
(b) $\Gamma\in \overline{(G,-)}.$
(c) There exists a partition of $V,$ $V=V_1\cup V_2$ ($V_1\cap
V_2=\emptyset$), such that every edge connecting $V_1$ and $V_2$ has positive sign and each edge within $V_1$ or $V_2$ has negative sign.
(d) $\lambda_N(\Gamma)=2.$
For any weighted graph, $$\lambda_1(G,\eta)\geq 0=\lambda_1(G,+),$$ $$\lambda_N(G,\eta)\leq 2=\lambda_N(G,-),$$ where the equalities hold only for balanced or antibalanced graphs, respectively.
In the remainder of this section, we discuss the first eigenvalue and eigenvectors of signed graphs. As usual, the smallest eigenvalue of a finite signed graph $\Gamma$ is characterized by the Rayleigh quotient $$\lambda_1(\Gamma)=\inf_{f\neq 0}\frac12\frac{\sum_{x,y\in V}\mu_{xy}(f(x)-\eta_{xy}f(y))^2}{\sum_{x\in V}f^2(x)\mu(x)}.$$ Is there any special property for the first eigenvalue and eigenvector? It is well known that the first eigenvalue of an unsigned weighted graph $G$ is simple if the graph is connected. However, this is not the case for signed graphs:
Let $(K_N,+)$ be an unsigned complete graph of $N$ vertices and $\Gamma=(K_N,-).$ Then $$\sigma(\Gamma)=\left\{\frac{N-2}{N-1},\dots,\frac{N-2}{N-1},2\right\},$$ where the multiplicity of the first eigenvalue is $N-1$.
Note that the multiplicity of the first eigenvalue of a signed graph can be quite large. The example above concerns antibalanced graphs. We next give an example of a generic signed graph, neither balanced nor antibalanced, whose first eigenvalue has multiplicity larger than 1.
Let $\Gamma=(C_4,\eta)$ be a cycle graph of order $4$ with edge signs $\{1,1,1,-1\}$. Clearly, it is neither balanced nor antibalanced. An explicit calculation shows that $$\sigma(\Gamma)=\left\{1-\frac{\sqrt2}{2},1-\frac{\sqrt2}{2},1+\frac{\sqrt2}{2},1+\frac{\sqrt2}{2}\right\},$$ where the multiplicity of the first eigenvalue is $2$. Thus, the first eigenvalue of a signed graph may have high multiplicity.
A well-known fact is that the first eigenfunction of an unsigned graph can be chosen to be positive everywhere (in fact, it is a constant function). One then has the following natural question: If $(G,\eta)$ is a signed graph with a positive first eigenfunction, should this graph be the unsigned weighted graph $(G,\eta)=(G,+)$? We provide a negative answer.
For any $\overline{\Gamma}\in \overline{\mathcal{G}},$ there exists a signed graph $\Gamma'\in\overline{\Gamma}$ such that the first eigenvector of $\Delta_{\Gamma'}$ is nonnegative everywhere. Moreover, if one of the first eigenvectors of $\Gamma$ vanishes nowhere, then there exists a signed graph $\Gamma'\in\overline{\Gamma}$ such that the first eigenvector of $\Delta_{\Gamma'}$ is strictly positive everywhere.
Let $f$ be a first eigenvector of $\Delta(\Gamma).$ We define a switching function $\theta:V\to{{\mathbb R}}$ by $$\theta(x)=\left\{\begin{array}{rl}-1, & f(x)<0,\\
1, & \rm{otherwise}.
\end{array}
\right.$$ Clearly, $f^{\theta}:=\theta\cdot f$ is nonnegative everywhere. Then by Lemma \[l:invariant by switching\], $f^{\theta}$ is the first eigenvector to $\Delta_{\Gamma^{\theta}}$. The signed graph $\Gamma ' = \Gamma^{\theta}$ satisfies the assertions of the theorem.
Symmetry of the spectrum {#s:symmetry}
========================
In this section, we study the symmetry of the spectra of signed graphs. Recall that for unsigned graphs bipartiteness is the only reason for the symmetry of the spectra. However, for signed graphs some new phenomena emerge.
A signed graph is called *bipartite* if its underlying graph is bipartite, i.e. there is a partition of $V,$ $V=V_1\cup V_2$, such that any edge in $E$ connects a vertex in $V_1$ to a vertex in $V_2.$ One notices that the sign function plays no role in the definition of bipartiteness. By the same techniques as in the unsigned case, one can prove that the spectrum of a signed graph is symmetric if it is bipartite; see [@AtayTuncel Lemma 4].
\[l:bipartite spectrum\] If $\Gamma=(G,\eta)$ is a bipartite signed graph, then the spectrum of $\Gamma$ is symmetric.
The next proposition gives a characterization of bipartite signed graphs.
\[p:bipartite\] A signed graph $\Gamma=(G,\eta)$ is bipartite if and only if $\overline{\Gamma}=\overline{-\Gamma}.$
$\Longrightarrow$: Let $\Gamma=(G,\eta)$ be a bipartite signed graph with bipartition $V_1,V_2$, i.e. $V=V_1\cup V_2,$ $V_1\cap V_2=\emptyset,$ $V_1,V_2\neq\emptyset$ and there is no edge in the induced subgraphs $V_i,$ $i=1,2$. Set $$\theta(x)=\left\{\begin{array}{rl}1, & \text{if }x\in V_1,\\
-1, & \text{if }x\in V_2.\end{array} \right.$$ Then we have $\Gamma^{\theta}=-\Gamma.$
$\Longleftarrow$: By $\overline{\Gamma}=\overline{-\Gamma},$ there exists a switching function $\theta:V\to\{1,-1\}$ such that $$\label{e:eq1}\Gamma^{\theta}=-\Gamma.$$Let $V_1=\{x\in V \, | \, \theta(x)=1\}$ and $V_2=\{x\in V \, | \, \theta(x)=-1\}.$ Then there are no edges within the induced subgraphs $V_1$ and $V_2$. Indeed, if there were an edge in the subgraph $V_1$ or $V_2$, this would contradict . Hence we obtain a bipartition, $V_1\cup V_2$, of $G.$
Recall that for unsigned weighted graphs the symmetry of the spectrum is completely equivalent to the bipartiteness of the graph (see in Lemma \[l:spectrum unsigned\]$(iii)$). A main difference in signed graphs is that there are more structural conditions which may create symmetric spectra. In the following, we present a general machinery to produce symmetric spectrum for signed graphs that has no counterpart in the unsigned case.
As defined in the introduction, two signed graphs $(G,\eta),(G',\eta')$ are called isomorphic if they have same combinatorial, weighted, and signed graph structure. Hence $$\sigma(G,\eta)=\sigma(G',\eta').$$ Now we are ready to prove one of the main results of this paper.
\[t:symmetry of the spectrum\] Let $\Gamma=(G,\eta)$ be a signed weighted graph. If there is a switching function $\theta: V\to \{1,-1\}$ such that $\Gamma^{\theta}\simeq -\Gamma$, then the spectrum of $\Gamma$ is symmetric.
Combining Lemma \[l:invariant by switching\], and the invariance of the spectrum under the isomorphism, we have $$\sigma(\Gamma)=\sigma(\Gamma^{\theta})=\sigma(-\Gamma)=2-\sigma(\Gamma).$$ This proves the theorem.
Note that for bipartite signed graphs $\overline{\Gamma}=\overline{-\Gamma}$ by Proposition \[p:bipartite\]. This indicates that Lemma \[l:bipartite spectrum\] is a special case of Theorem \[t:symmetry of the spectrum\]. In the following, inspired by Theorem \[t:symmetry of the spectrum\], we provide an example of a signed graph with symmetric spectrum, although it is non-bipartite.
\[ex:symmetric spectrum\] Let $\Gamma$ be the signed graph shown in Figure 1, with $+1$ and $-1$ edge weights as indicated. Consider the switching function $\theta$ given by $$\theta(i)=\left\{\begin{array}{rl}-1, & i=2,4,\\
1, & \rm{otherwise}. \end{array}
\right.$$ Now the permutation $T=(1,3)\in S_5$, which is an isomorphism of signed graphs, transforms $\Gamma^{\theta}$ to $-\Gamma.$ By Theorem \[t:symmetry of the spectrum\], the spectrum of $\Gamma$ is symmetric with respect to $1$. Indeed, an explicit calculation gives $$\sigma(\Gamma)=\left\{1\pm \frac{\sqrt{4-\sqrt3}}{3},1\pm
\frac{\sqrt{4+\sqrt3}}{3},1\right\}.$$ Thus, $\Gamma$ is a nontrivial (i.e. non-bipartite) example for the symmetry of the spectrum.
![A non-bipartite signed graph with symmetric spectrum. The eigenvalues are $1\pm \frac{\sqrt{4-\sqrt3}}{3},1\pm
\frac{\sqrt{4+\sqrt3}}{3}$, and 1.[]{data-label="fig1"}](Fig_symmetry){width="80.00000%"}
Damped two-periodic solutions {#s:Periodic solutions}
=============================
In this section, we connect the symmetry of the spectrum of signed graphs to the existence of period-two oscillatory solutions for the discrete-time heat equation on the graph.
Let $(G,\eta)$ be a finite signed graph. We say that a function $f:({{\mathbb N}}\cup{0})\times V\to {{\mathbb R}}$ satisfies the discrete-time heat equation on $(G,\eta)$ if $$\left\{\begin{array}{ll}f(n+1,x)-f(n,x)=-\Delta f(n,x),
\quad n\in \mathbb{N}\cup \{0\},& \\
f(0,x)=g(x). &
\end{array}
\right.$$ For simplicity, we denote the function $f_n:V\to{{\mathbb R}},$ $n\geq 0$, by $f_n(x):=f(n,x)$. Then the heat equation can be written as $$\label{heateq}
\left\{\begin{array}{ll}f_{n+1}=P f_n, & \\
f_0=g, &
\end{array}
\right.$$ where $P=D^{-1}A$ is the generalized transition matrix. As usual, the notation $\|\cdot\|$ and $\langle \cdot,\cdot\rangle$ will denote the $\ell^2$ norm and inner product, respectively, of the Hilbert space $\ell^2(V,m).$
\[d:periodic solution\] A function $u\neq 0$ is called a generic damped periodic solution of order two to the discrete-time heat equation , or a *damped $2$-periodic solution* for short, if $u$ is an eigenfunction of $P^2$ but not of $P$; that is, if there exists a constant $\lambda\geq0$ such that $P^2u=\lambda^2 u$ and $Pu\neq \pm\frac{\|Pu\|}{\|u\|}u.$ We call $\lambda$ the *decay rate*.
Let $u$ be a damped $2$-periodic solution of with decay rate $\lambda$. Then the following hold:
(a) $ku$ is a damped $2$-periodic solution for any $k\neq0.$
(b) $\lambda=\sqrt{\frac{\|P^2u\|}{\|u\|}}=\frac{\|Pu\|}{\|u\|}$.
(c) $0<\lambda\le 1.$
Statement $(a)$ follows by definition. By $P^2u=\lambda^2u$ and $u\neq 0,$ we have $\lambda=\sqrt{\frac{\|P^2u\|}{\|u\|}}.$ Moveover, noting that $P$ is a self-adjoint operator with respect to the inner product on $\ell^2(V,m)$, we obtain $$\|Pu\|^2=\langle Pu,Pu\rangle=\langle
P^2u,u\rangle=\lambda^2\|u\|^2.$$ This proves statement $(b)$. For $(c)$, to prove that $\lambda\neq0$, we argue by contradiction. Suppose $\lambda=0$. Then by statement $(b)$ we have $\|Pu\|=0$, and thus $Pu=0$, which implies that $u$ is an eigenvector of $P$ pertaining to the eigenvalue $0$. This, however, contradicts the definition of $2$-periodic solutions. Finally, $\lambda\le 1$ follows from the fact that all eigenvalues of $P$ belong to the interval $[-1,1]$ and $P^2 u = \lambda^2 u$ by definition.
For a $2$-periodic oscillatory solution $u$ with decay rate $\lambda$, we set $v=\frac1{\lambda}Pu$. (Notice that $v$ is not parallel to $u$ by definition). Hence we have the system of equations $$\label{e:system of equations to periodic}\left\{\begin{array}{ll}Pu=\lambda v, & \\
Pv=\lambda u. &
\end{array}
\right.$$ If we set $u$ as the initial data of the discrete-time heat equation, then the solution to reads $$\label{e:2 periodic solution}f_n=P^nu=\left\{\begin{array}{ll}\lambda^n u, & n \text{ even}, \\
\lambda^n v, & n \text{ odd}.
\end{array}
\right.$$ Since the vectors $u$ and $v$ are linearly independent, the solution $f_n$ oscillates between two phases, $u$ and $v,$ with an amplitude that decays exponentially at a rate $\lambda$, since $\lambda \le 1$. This motivates the meaning of damped $2$-periodic solution given in Definition \[d:periodic solution\].
We study the machinery to produce a $2$-periodic solution. We will see that the existence of $2$-periodic solutions is equivalent to certain symmetry of the spectrum of the normalized signed Laplacian operator. The next theorem states that all damped $2$-periodic solutions are in fact encoded by the symmetry of the spectrum.
\[t:symmetry and periodic\] Let $(G,\eta)$ be a signed graph. Then $u$ is a damped $2$-periodic solution with decay rate $\lambda$ if and only if $$u=f+g,$$ where $f$ and $g$ are eigenfunctions corresponding to the eigenvalues $1-\lambda$ and $1+\lambda$, respectively, of the normalized Laplacian $\Delta$, with $\lambda\neq0$.
$\Leftarrow:$ By the assumption and the fact that $\Delta=I-P$, $f$ and $g$ are eigenfunctions pertaining to the eigenvalues $\lambda$ and $-\lambda$, respectively, of $P$. Hence $Pu=\lambda(f-g)$ and $P^2u=\lambda^2u.$ Since $f,g$ are eigenvectors pertaining to different eigenvalues, $\langle f,g\rangle=0.$ It is easy to check that $u=f+g$ and $f-g$ are linearly independent. This yields that $u$ is not an eigenvector to $P$, and proves the assertion.
$\Rightarrow:$ Setting $v=\frac1{\lambda}Pu,$ we have the system for $u$ and $v$. Define $f:=\frac12(u+v)$ and $g:=\frac{1}{2}(u-v)$. It is easy to see that $f$ and $g$ are nonzero since $u$ and $v$ are linearly independent. By definition $u=f+g$. Direct calculation shows that $Pf=\lambda f$ and $Pg=-\lambda g$. This completes the proof.
(a) We have the following corollary of Theorem \[t:symmetry and periodic\]: If there is a damped $2$-periodic solution with decay rate $\lambda$ on a signed graph $\Gamma$, then both $1-\lambda$ and $1+\lambda$ belong to the spectrum of $\Gamma$.
(b) For signed weighted graphs whose spectrum is symmetric with respect to 1, one can construct many $2$-periodic solutions by virtue of Theorem \[t:symmetry and periodic\].
Finally, as a consequence of Theorem \[t:symmetry and periodic\], we have the following formulation. Let $\Gamma=(G,\eta)$ be a signed weighted graph. We set $$\Lambda(\Gamma):=\{\lambda: \lambda\neq 1, \lambda\in
\sigma(\Gamma), 2-\lambda\in\sigma(\Gamma)\}.$$ We denote by $E_{\lambda}$ the eigenspace pertaining to the eigenvalue $\lambda$ of $\Delta_\Gamma.$ Then the set of damped $2$-periodic solutions, denoted by $\mathcal{P}$, has the representation $$\mathcal{P}=\bigcup_{\lambda\in\Lambda(\Gamma)}E_{\lambda}\oplus E_{2-\lambda}\setminus(E_{\lambda}\oplus 0\cup0\oplus E_{2-\lambda}).$$
Motif replication {#s:motif}
=================
In this section, we use the normalized Laplace operator with Dirichlet boundary conditions to study the spectral changes under motif replication.
The term *motif* refers to a subgraph $\Omega$ of a signed graph $\Gamma$. By motif replication we refer to the operation of appending an additional copy of $\Omega$ together with all its connections and corresponding weights, yielding the enlarged graph denoted by $\Gamma^{\Omega}$. More precisely, let $\Omega$ be a subgraph on vertices $\{x_1,\dots,x_n\}$ and let $\Omega'=\{x_1',\dots,x_n'\}$ be an exact replica of $\Omega$. Then the enlarged graph $\Gamma^\Omega$, obtained by replicating $\Omega$, is defined on the vertex set $V(\Gamma^{\Omega})=V(\Gamma)\cup \Omega'$ with the signed edge weights given by $${{\kappa}}=\begin{cases} {{\kappa}}(x,y), & x,y\in V(\Gamma), \\
{{\kappa}}(x_i,x_j), & x_i',x_j'\in \Omega',\\
{{\kappa}}(x_i,y), & x_i'\in \Omega', y\in V(\Gamma)\setminus \Omega,\\
\end{cases}
$$ using the edge weights $\kappa(x,y)$ from the original graph $\Gamma$ [@AtayTuncel]. In particular, there are no edges between $\Omega$ and $\Omega'$ in the replicated graph $\Gamma^\Omega$.
Let $\Omega$ be a finite subset of $V$ and $\ell^2(\Omega,m)$ be the space of real-valued functions on $\Omega$ equipped with the $\ell^2$ inner product. Note that every function $f\in\ell^2(\Omega,m)$ can be extended to a function $\tilde{f}\in\ell^2(V,m)$ by setting $\tilde{f}(x)=0$ for all $x\in V\setminus \Omega$. The Laplace operator with Dirichlet boundary conditions $\Delta_\Omega:
\ell^2(\Omega,m) \to \ell^2(\Omega,m)$ is defined as $$\Delta_\Omega f = (\Delta \tilde{f})_{|\Omega}.$$ Thus, for $x\in
\Omega$ the Dirichlet Laplace operator is given pointwise by $$\begin{aligned}
\Delta_\Omega f(x) = f(x) - \frac{1}{m(x)}\sum_{y\in\Omega: y\sim x} {{\kappa}}_{xy}f(y) \\
=
\tilde{f}(x) - \frac{1}{m(x)}\sum_{y\sim x} {{\kappa}}_{xy}\tilde{f}(y).\end{aligned}$$ A simple calculation shows that $\Delta_\Omega$ is a positive self-adjoint operator. We arrange the eigenvalues of the Dirichlet Laplace operator $\Delta_\Omega$ in nondecreasing order, $\lambda_1(\Omega) \leq \lambda_2(\Omega) \leq \dots \leq
\lambda_N(\Omega)$, where $N=|\Omega|$ denotes the cardinality of the set $\Omega$.
We next prove that all eigenvalues of the Dirichlet Laplacian on $\Omega$ are preserved in the motif replication.
\[t:motif replication\] Let $\Omega$ be a motif in a signed weighted graph $\Gamma$ and $\Gamma^{\Omega}$ be the new graph obtained after replicating $\Omega$. Then $\sigma(\Delta_{\Omega})\subset \sigma(\Gamma^{\Omega}).$
Let $f$ be an eigenvector of the Dirichlet Laplacian $\Delta_{\Omega}$ corresponding to eigenvalue $\lambda$. Then by direct calculation it can be seen that the following function is an eigenvector of the Laplacian on $\Gamma^{\Omega}$: $$\label{e:motif mod}f'(x)=\left\{\begin{array}{rl}f(x), & x\in \Omega, \\
-f(x), & x\in \Omega',\\
0, & \rm{otherwise},
\end{array}
\right.$$ where $\Omega'$ is the copy of $\Omega.$ This proves the theorem.
By Theorem \[t:motif replication\], if the motif $\Omega$ supports a damped 2-periodic solution $u$ with respect to the Dirichlet boundary condition, then $u',$ defined as in , is a damped 2-periodic solution for the replicated graph $\Gamma^{\Omega}.$ In this way, symmetric eigenvalues $\lambda$ and 2-periodic solutions with decay rate $\lambda$ are carried over to the larger graph after motif replication.
**Acknowledgements.** The authors thank the ZiF (Center for Interdisciplinary Research) of Bielefeld University, where part of this research was conducted under the program *Discrete and Continuous Models in the Theory of Networks*. FMA acknowledges the support of the European Union’s 7th Framework Programme under grant \#318723 (MatheMACS). BH is supported by NSFC, grant no. 11401106.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
The advent of multicast and the growth and complexity of the Internet has complicated network protocol design and evaluation. Evaluation of Internet protocols usually uses random scenarios or scenarios based on designers’ intuition. Such approach may be useful for average-case analysis but does not cover [*boundary-point*]{} (worst or best-case) scenarios. To synthesize boundary-point scenarios a more systematic approach is needed.
In this paper, we present a method for automatic synthesis of worst and best case scenarios for protocol boundary-point evaluation. Our method uses a fault-oriented test generation (FOTG) algorithm for searching the protocol and system state space to synthesize these scenarios. The algorithm is based on a global finite state machine (FSM) model. We extend the algorithm with timing semantics to handle end-to-end delays and address performance criteria. We introduce the notion of a [*virtual LAN*]{} to represent delays of the underlying multicast distribution tree. The algorithms used in our method utilize implicit backward search using branch and bound techniques and start from given [*target events*]{}. This aims to reduce the search complexity drastically.
As a case study, we use our method to evaluate variants of the timer suppression mechanism, used in various multicast protocols, with respect to two performance criteria: overhead of response messages and response time. Simulation results for reliable multicast protocols show that our method provides a scalable way for synthesizing worst-case scenarios automatically. Results obtained using stress scenarios differ dramatically from those obtained through average-case analyses. We hope for our method to serve as a model for applying systematic scenario generation to other multicast protocols.
bibliography:
- 'dissertation-2.bib'
---
[**The STRESS Method for Boundary-point Performance Analysis of End-to-end Multicast Timer-Suppression Mechanisms**]{} Ahmed Helmy+, Sandeep Gupta+, Deborah Estrin\*
+ University of Southern California (USC), \* University of California at Los Angeles (UCLA)
Introduction
============
The longevity and power of Internet technologies derives from its ability to operate under a wide range of operating conditions (underlying topologies and transmission characteristics, as well as heterogeneous applications generating varied traffic inputs). Perhaps more than any other technology, the range of operating conditions is enormous (it is the cross product of the top and bottom of the IP protocol stack).
Perhaps it is this enormous set of conditions that has inhibited the development of systematic approaches to analyzing Internet protocol designs. How can we test correctness or characterize performance of a protocol when the set of inputs is intractable? Nevertheless, networking infrastructure is increasingly critical and there is enormous need to increase the understanding and robustness of network protocols. Most current approaches for protocol evaluation use average-case analysis and are based on random or intuitive scenarios. Such approach does not address protocol robustness or [*boundary-point*]{} analysis, in which the protocol exhibits worst or best-case behavior. We believe that such protocol breaking points should be identified and studied in depth to understand and hopefully increase protocol robustness. It is time to develop techniques for systematic testing of protocol behavior, even in the face of the above challenges and obstacles. At the same time we do not expect that complex adaptive protocols will be automatically verifiable under their full range of conditions. Rather, we are proposing a framework in which a protocol designer can follow a set of systematic steps, assisted by automation where possible, to cover a specific part of the design and operating space. Our goal is to complement average case studies and enrich the evaluation test-suites for multicast protocols.
In our proposed framework, a protocol designer will still need to create the initial mechanisms, describe it in the form of a finite state machine, and identify the performance criteria or correctness conditions that need to be investigated. Our automated method will pick up at that point, providing algorithms that generate scenarios or test suites that stress the protocol with respect to the identified criteria. The algorithms used in our method utilize implicit backward search using branch and bound techniques and start from given [*target events*]{}. This aims to reduce the search complexity drastically, as we shall discuss based on our case studies.
This paper demonstrates our progress in realizing this vision as we present our method and apply it to [*boundary-point*]{} (worst and best-case) performance evaluation of the timer suppression mechanism used in numerous multicast protocols.
Motivation
----------
The recent growth of the Internet and its increased heterogeneity has introduced new failure modes and added complexity to protocol design and testing. In addition, the advent of multicast applications has introduced new challenges of qualitatively different nature than the traditional point-to-point protocols. Multicast applications involve a group of receivers and one or more senders. As more complex multicast applications and protocols are coming to life, the need for systematic and automatic methods to study and evaluate such protocols is becoming more apparent. Such methods aim to expedite the protocol development cycle and improve resulting protocol robustness and performance.
Through our proposed methodology for test synthesis, we hope to address the following key issues of protocol design and evaluation.
- Scenario dependent evaluation, and the use of validation test suites: Protocols may be evaluated for correctness and performance. In many evaluation studies of multicast protocols, the results are dependent upon several factors, such as membership distribution and network topology, among others. Hence, conclusions drawn from these studies depend heavily upon the evaluation scenarios.
Protocol development usually passes through iterative cycles of refinement, which requires revisiting the evaluation scenarios to ensure that no erroneous behavior has been introduced. This brings about the need for validation test suites. Constructing these test suites can be an onerous and error-prone task if performed manually. Unfortunately, little work has been done to automate the generation of such tests for multicast network protocols. In this paper, we propose a method for synthesizing test scenarios automatically for boundary-point analysis of timer-suppression mechanisms employed by several classes of Internet multicast protocols.
- Worst-case analysis of protocols: It is difficult to design a protocol that would perform well in all environments. However, identifying breaking points that violate correctness or exhibit worst-case performance behaviors of a protocol may give insight to protocol designers and help in evaluating design trade-offs. In general, it is desirable to identify, early on in the protocol development cycle, scenarios under which the protocol exhibits worst or best case behavior. The method presented in this paper automates the generation of scenarios in which multicast protocols exhibit worst and best case behaviors.
- Performance benchmarking: New protocols may propose to refine a mechanism with respect to a particular performance metric, using for evaluation those scenarios that show performance improvement. However, without systematic evaluation, these refinement studies often (though unintentionally) overlook other scenarios that may be relevant. To alleviate such a problem we propose to integrate stress test scenarios that provide an objective benchmark for performance evaluation.
Using our scenario synthesis methodology we hope to contribute to the understanding of better performance benchmarking and the design of more robust protocols.
Background {#building_blocks}
----------
The design of multicast protocols has introduced new challenges and problems. Some of the problems are common to a wide range of protocols and applications. One such problem is the [*multi-responder*]{} problem, where multiple members of a group may respond (almost) simultaneously to an event, which may cause a flood of messages throughout the network, and in turn may lead to synchronized responses, and may cause additional overhead (e.g., the well-known [*Ack implosion*]{} problem), leading to performance degradation.
One common technique to alleviate the above problem is the [*multicast damping*]{} technique, which employs a [*timer suppression*]{} mechanism (TSM). TSM is employed in several multicast protocols, including the following:
- IP-multicast protocols, e.g., PIM [@PIM-SMv2-Spec] [@PIM-DM-SPEC] and IGMP [@igmp], use TSM on LANs to reduce Join/Prune control overhead.
- Reliable multicast schemes, e.g., SRM [@SRM] and MFTP [@mftp], use this mechanism to alleviate [*Ack implosion*]{}. Variants of the SRM timers are used in registry replication (e.g., RRM [@rrm] [@rrm2]) and adaptive web caching [@awc].
- Multicast address allocation schemes, e.g., AAP and SDr [@sdr], use TSM to avoid an implosion of responses during the collision detection phase.
- Active services [@elan] use multicast damping to launch one service agent ‘servent’ from a pool of servers.
TSM is also used in self-organizing hierarchies (SCAN [@scan]), and transport protocols (e.g., XTP [@xtp] and RTP [@RTP]).
We believe TSM is a good building block to analyze as our first end-to-end case study, since it is rich in multicast and timing semantics, and can be evaluated using standard performance criteria. As a case study, we examine its worst and best case behaviors in a systematic, automatic fashion[^1].
In TSM, a member of a multicast group that has detected loss of a data packet multicasts a request for recovery. Other members of the group, that receive this request and that have previously received the data packet, schedule transmission of a response. In general, randomized timers are used in scheduling the response. While a response timer is running at one node, if a response is received from another node then the response timer is suppressed to reduce the number of responses triggered. Consequently, the response time may be delayed to allow for more suppression.
Two main performance evaluation criteria used in this case are overhead of response messages and time to recover from packet loss. Depending on the relative delays between group members and the timer settings, the mechanism may exhibit different performance. In this study, our method attempts to obtain scenarios of best case and worst case performance according to the above criteria.
We are not aware of any related work that attempts to achieve this goal systematically. However, we borrow from previous work on protocol verification and test generation. The rest of the paper is organized as follows. Section \[model\] introduces the protocol and topology models. Section \[apply\] outlines the main algorithm, and Section \[timer\] presents the model for TSM. Sections \[overhead\] and \[response\] present performance analyses for protocol overhead and response time, and Section \[simulation\] presents simulation results. Related work is described in Section \[related\]. Issues and future work are discussed in Section \[issues\]. We present concluding remarks in Section \[conclusion\]. Algorithmic details, mathematical models and example case studies are given in the appendices.
The Model {#model}
=========
The model is a processable representation of the system under study that enables automation of our method. Our overall model consists of: A) the protocol model, B) the topology model, and C) the fault model.
The Protocol Model {#fsm}
------------------
We represent the protocol by a finite state machine (FSM) and the overall system by a global FSM (GFSM).
[*I. FSM model:*]{} Every instance of the protocol, running on a single end-system, is modeled by a deterministic FSM consisting of: (i) a set of states, (ii) a set of stimuli causing state transitions, and (iii) a state transition function (or table) describing the state transition rules. A protocol running on an end-system $i$ is represented by the machine ${\mathcal{M}}_{i} = ({\mathcal{S}}_{i},\tau_{i},\delta_{i})$, where ${\mathcal{S}}_{i}$ is a finite set of state symbols, $\tau_{i}$ is the set of stimuli, and $\delta_{i}$ is the state transition function ${\mathcal{S}}_{i} \times \tau_{i} \rightarrow {\mathcal{S}}_{i}$.
[*II. Global FSM model:*]{} The global state is defined as the composition of individual end-system states. The behavior of a system with $n$ end-systems may be described by $\mathcal{M}_{\mathcal{G}} =
(\mathcal{S}_{\mathcal{G}},\tau_{\mathcal{G}},\delta_{\mathcal{G}})$, where $\mathcal{S}_{\mathcal{G}}$: ${\mathcal{S}}_{1} \times
{\mathcal{S}}_{2}
\times \dots \times {\mathcal{S}}_{n}$ is the global state space, $\tau_{\mathcal{G}}$: $\overset{n}{\underset{i=1}{\bigcup}}
\tau_i$ is the set of stimuli, and $\delta_{\mathcal{G}}$ is the global state transition function $\mathcal{S}_{\mathcal{G}} \times \tau_{\mathcal{G}}
\rightarrow \mathcal{S}_{\mathcal{G}}$.
The Topology Model
------------------
The topology cannot be captured simply by one metric. Indeed, its dynamics may be complex to model and sometimes intractable. We model the topology at the network layer and we abstract the network using end-to-end delays. We model the delays using the delay matrix and loss patterns using the fault model. We use a [*virtual LAN*]{} (VLAN) model to represent the underlying network topology and multicast distribution tree. The VLAN captures delay semantics using a delay matrix $D$ (see Figure \[vlan\_figure\]), where $d_{i,j}$ is the delay from system $i$ to system $j$[^2]. The VLAN model may seem as an over-simplification of the topology as it abstracts the internal network connectivity and queues. This, however, renders our model tractable and is quite useful in obtaining characteristics of boundary-point scenarios. We shall further investigate the utility and accuracy of our model in Section \[simulation\] through detailed packet level simulations of sophisticated timer mechanisms over complex topologies.
The Fault Model
---------------
A [*fault*]{} is a low level (e.g., physical layer) anomalous behavior that may affect the protocol under test. Faults may include packet loss, system crashes, or routing loops. For brevity, we only consider selective packet loss in this study. Selective packet loss occurs when a multicast message is received by some group members but not others. The selective loss of a message prevents the transition that this message triggers at the intended recipient.
Algorithm and Objectives {#apply}
========================
To apply our method, the designer specifies the protocol as a global FSM model. In addition, the evaluation criteria, be it related to performance or correctness, are given as input to the method. In this paper we address performance criteria, correctness has been addressed in previous studies [@stress; @fotg]. The algorithm operates on the specified model and synthesizes a set of ‘test scenarios’; protocol events and relations between topology delays and timer values, that stress the protocol according to the evaluation criteria (e.g., exhibit maximum overhead or delay). In this section, we outline the algorithmic details of our method. The algorithm is further discussed in Section \[overhead\] and illustrated by a case study. Algorithmic complexity issues are discussed in Section \[issues\].
Algorithm Outline {#algorithm}
-----------------
Our algorithm is a variant of the fault-oriented test generation (FOTG) algorithm presented in [@fotg]. It includes the topology synthesis, the backward search and the forward search stages. Here we describe those aspects of our algorithm that deal with timing and performance semantics. The basic algorithm passes through three main steps (1) the target event identification, (2) the search, and (3) the task specific solution. The algorithm is outlined in Figure \[block\_diag\].
1. [**The target event:**]{} The algorithm starts from a given event, called the ‘target event’. The target event (e.g., sending a message) is identified by the designer based on the protocol evaluation criteria, e.g., overhead.
2. [**The search:**]{} Three steps are taken in the search: (a) identifying conditions, (b) obtaining sequences, and (c) formulating inequalities.
1. [*Identifying conditions:*]{} The algorithm uses the protocol transition rules to identify transitions necessary to trigger the target event and those that prevent it, these transitions are called [*wanted transitions*]{} and [*unwanted transitions*]{}, respectively.
2. [*Obtaining sequences:*]{} Once the above transitions are identified, the algorithm uses backward and forward search to build event sequences leading to these transitions and calculates the times of these events as follows.
1. [**Backward search**]{} is used to identify events preceding the wanted and unwanted transitions, and uses implication rules that operate on the protocol’s transition table. Section \[implication\_rules\] describes the implication rules.
2. [**Forward search**]{} is used to verify the backward search. Every backward step must correspond to valid forward step(s). Branches leading to contradictions between forward and backward search are rejected. Forward search is also used to complete event sequences necessary to maintain system consistency[^3].
3. [*Formulating inequalities:*]{} Based on the transitions and timed sequences obtained in the previous steps, the algorithm formulates relations between timer values and network delays that trigger the wanted transitions and avoid the unwanted transitions.
3. [**Task specific solution:**]{} The output of the search is a set of event sequences and inequalities that satisfy the evaluation criteria. These inequalities are solved mathematically to find a topology or timer configuration, depending on the task definition.
Task Definition
---------------
We apply our method to two kinds of tasks:
1. [**Topology synthesis**]{} is performed to identify the delays, $d_{i,j}$, in the dealy matrix $D$ that produce the best or worst case behavior, given the timer values[^4].
2. [**Timer configuration**]{} is performed to obtain the timer values that cause the best and worst case behavior, given the topology delay matrix $D$.
The Timer Suppression Mechanism (TSM) {#timer}
=====================================
In this section, we present a simple description of TSM, then present its model, used thereafter in the analysis. TSM involves a request, $q$, and one or more responses, $p$. When a system, $Q$, detects the loss of a data packet it sets a request timer and multicasts a request $q$. When a system $i$ receives $q$ it sets a response timer (e.g., randomly), the expiration of which, after duration $Exp_i$, triggers a response $p$. If the system $i$ receives a response $p$ from another system $j$ while its timer is running, it suppresses its own response.
Performance Evaluation Criteria
-------------------------------
We use two performance criteria to evaluate TSM:
1. Overhead of response messages, where the worst case produces the maximum number of responses per data packet loss. As an extreme case, this occurs when all potential responders respond and no suppression takes place.
2. The response delay, where worst case scenario produces maximum loss recovery time.
Timer Suppression Model
-----------------------
Following is the TSM model used in the analysis.
### Protocol states ($\mathcal{S}$)
Following is the state symbol table for the TSM model.
State Meaning
------- ---------------------------------------
$R$ original state of the requester $Q$
$R_T$ requester with the request timer set
$D$ potential responder
$D_T$ responder with the response timer set
### Stimuli or events
1. Sending/receiving messages: transmitting response ($p_t$) and request ($q_t$), receiving response ($p_r$) and request ($q_r$).
2. Timer events and other events: the events of firing the request timer, $Req$, and response timer, $Res$, and the event of detecting packet loss, $L$.
### Notation
Following are the notations used in the transition table and the analysis thereafter.
- An event subscript denotes the system initiating the event, e.g., $p_{t_i}$ is response sent by system $i$, while the subscript $m$ denotes multicast reception, e.g., $p_{r_m}$ denotes scheduled reception of a response by all members of the group if no loss occurs. When system $i$ receives a message sent by system $j$, this is denoted by the subscript $i,j$, e.g., $p_{r_{i,j}}$ denotes system $i$ receiving response from system $j$.
- The state subscript $T$ denotes the existence of a timer, and is used by the algorithm to apply the [*timer implication*]{} to fire the timer event after the expiration period $Exp$.
- A state transition has a $start$ state and an $end$ state and is expressed in the form $startState \rightarrow endState$ (e.g. $D
\rightarrow D_T$). It implies the existence of a system in the $startState$ (i.e., $D$) as a condition for the transition to the $endState$ (i.e., $D_T$).
- [*Effect*]{} in the transition table may contain transition and stimulus in the form ($startState \rightarrow endState).stimulus$, which indicates that the condition for triggering $stimulus$ is the state transition. An effect may contain several transitions (e.g., $Trans1$, $Trans2$), which means that out of these transitions only those with satisfied conditions will take effect.
- To describe event sequences in the backward search we denote $a
\Leftarrow b$, where $a$ and $b$ are global states, which means that $a$ succeeds $b$ in the event sequence, and that $b$ can be implied from $a$. Also, for forward search we use $a \Rightarrow b$, which means that $a$ precedes $b$ in the event sequence and that $b$ can be implied from $a$.
### Transition table {#transition_table}
Following is the transition table for TSM.
Symbol Event Effect Meaning
------------ --------- ------------------------------------------ --------------------------------------------------------------------------------
$loss$ $L$ $(R \rightarrow R_T).q_t$ Loss detection causes $q_t$ and setting of request timer
$tx\_req$ $q_{t}$ $q_{r_m}$ Transmission of $q$ causes multicast reception of $q$ after network delay
$rcv\_req$ $q_{r}$ $D \rightarrow D_T$ Reception of $q$ causes a system in $D$ state to set response timer
$res\_tmr$ $Res$ $(D_T \rightarrow D).p_{t}$ Response timer expiration causes transmission of $p$ and a change to $D$ state
$tx\_res$ $p_{t}$ $p_{r_m}$ Transmission of $p$ causes multicast reception of $p$ after network delay
$rcv\_res$ $p_{r}$ $R_T \rightarrow R$, $D_T \rightarrow D$ Reception of $p$ by a system with the timer set causes suppression
$req\_tmr$ $Req$ $q_t$ Expiration of request timer causes re-transmission of $q$
The model contains a requester, $Q$, and several potential responders (e.g., $i$ and $j$).[^5] Initially, the requester, $Q$, exists in state $R$ and all potential responders exist in state $D$. Let $t_0$ be the time at which $Q$ sends the request, $q$. The request sent by $Q$ is received by $i$ and $j$ at times $d_{Q,i}$ and $d_{Q,j}$, respectively. When the request, $q$, is sent, the requester transitions into state $R_T$ by setting the request timer. Upon receiving a request, a potential responder in state $D$ transitions into state $D_T$, by setting the response timer. The time at which an event occurs is given by $t(event)$, e.g., $q_{r_j}$ occurs at $t(q_{r_j})$.[^6]
### Implication rules {#implication_rules}
The backward search uses the following cause-effect implication rules:
1. Transmission/Reception ([**Tx\_Rcv**]{}): By the reception of a message, the algorithm implies the transmission of that message –without loss– sometime in the past (after applying the network delays). An example of this implication is $p_{r_{i,j}} \Leftarrow
p_{t_j}$, where $t(p_{r_{i,j}}) = t(p_{t_j}) + d_{j,i}$.
2. Timer Expiration ([**Tmr\_Exp**]{}): When a timer expires, the algorithm infers that it was set $Exp$ time units in the past, and that no event occurred during that period to reset the timer. An example of this implication is $Res_i.(D_i \leftarrow
D_{T_i}) \Leftarrow D_{T_i}$, where $t(Res_i) = t(D_{T_i}) + Exp_i$, and $Exp_i$ is the duration of the response timer $Res_i$.[^7]
3. State Creation ([**St\_Cr**]{}): To build a history of events leading to a certain state, we reverse the transition rules and get to the $startState$ of the transitions leading to the creation of the state in question. For example, $D_{T_i} \Leftarrow (D_{T_i}
\leftarrow D_i)$ means that for the system to be in state $D_{T_i}$ the system must have existed in state $D_i$ and this is implied from the transition $(D_{T_i}
\leftarrow D_i)$.
In the following sections we use the above model to synthesize worst and best case scenarios according to protocol overhead and response time.
Protocol Overhead Analysis {#overhead}
==========================
In this section, we conduct worst and best case performance analyses for TSM with respect to the number of responses triggered per packet loss. Initially, we assume no loss of request or response messages until recovery, and that the request timer is high enough that the recovery will occur within one request round. The case of multiple request rounds is discussed in Appendix C.
Worst-Case Analysis {#worst_case}
-------------------
Worst-case analysis aims to obtain scenarios with maximum number of responses per data loss. In this section we present the algorithm to obtain inequalities that lead to worst-case scenarios. These inequalities are a function of network delays and timer expiration values.
### Target event
Since the overhead in this case is measured as the number of response messages, the designer identifies the event of triggering a response, $p_t$, as the target event, and the goal is to maximize the number of response messages.
### The search
As previously described in Section \[algorithm\], the main steps of the search algorithm are to: (1) identify the wanted and unwanted transitions, (2) obtain sequences leading to the above transitions, and calculating the times for these sequences, and (3) formulate the inequalities that achieve the time constraints required to invoke wanted transitions and avoid unwanted transitions.
- [**Identifying conditions:**]{} The algorithm searches for the transitions necessary to trigger the target event, and their conditions, recursively. These are called [*wanted transitions*]{} and [*wanted conditions*]{}, respectively. The algorithm also searches for transitions that nullify the target event or invalidate any of its conditions. These are called [*unwanted transitions*]{}.
In our case the target event is the transmission of a response (i.e., $p_t$). From the transition table described in Section \[transition\_table\], the algorithm identifies transition [*res\_tmr*]{}, or $Res.(D_T \rightarrow
D).p_{t}$, as a [*wanted transition*]{} and its condition $D_T$ as a [*wanted condition*]{}. Transition [*rcv\_req*]{}, or $q_{r}.(D
\rightarrow D_T)$, is also identified as a [*wanted transition*]{} since it is necessary to create $D_T$. The [*unwanted transition*]{} is identified as transition [*rcv\_res*]{}, or $p_{r}.(D_T \rightarrow D)$, since it alters the $D_T$ state without invoking $p_t$.
- [**Obtaining sequences:**]{} Using backward search, the algorithm obtains sequences and calculates time values for the following transitions: (1) wanted transition, [ *res\_tmr*]{}, (2) wanted transition [*rcv\_req*]{}, and (3) unwanted transition [*rcv\_res*]{}, as follows:
1. To obtain the sequence of events for transition [*res\_tmr*]{}, the algorithm applies implication rules (see Section \[implication\_rules\]) Tmr\_Exp, St\_Cr, Tx\_Rcv in that order, and we get $res\_tmr_i \Leftarrow rcv\_req_i \Leftarrow tx\_req$, or
$Res_i.(D_i \leftarrow D_{T_i}).p_{t_i}
\Leftarrow q_{r_i}.(D_{T_i} \leftarrow D_i)
\Leftarrow q_{t}$.
Hence the calculated time for $t(p_{t_i})$ becomes
$$t(p_{t_i}) = t_0 + d_{Q,i} + Exp_i,$$
where $t_0$ is the time at which $q_{t}$ occurs.
2. To obtain the sequence of events for transition [*rcv\_req*]{} the algorithm applies implication rule Tx\_Rcv, and we get $rcv\_req_i \Leftarrow
tx\_req$, or
$q_{r_i}.(D_{T_i} \leftarrow D_i) \Leftarrow
q_{t}$.
Hence the calculated time for $t(q_{r_i})$ becomes
$$t(q_{r_i}) = t_0 + d_{Q,i}.$$
3. To obtain sequence of events for transition [*rcv\_res*]{} for systems $i$ and $j$ the algorithm applies implication rules Tx\_Rcv,Tmr\_Exp, St\_Cr, Tx\_Rcv in that order, and we get $rcv\_res_i \Leftarrow
res\_tmr_j \Leftarrow rcv\_req_j \Leftarrow tx\_req$, or
$p_{r_{i,j}}.(D_i \leftarrow D_{T_i}) \Leftarrow
Res_j.(D_j \leftarrow D_{T_j}).p_{t_j} \Leftarrow
q_{r_j}.(D_{T_j} \leftarrow D_j) \Leftarrow
q_{t}$.
Hence the calculated time for $t(p_{r_{i,j}})$ becomes
$$t(p_{r_{i,j}}) = t_0 + d_{Q,j} + Exp_j + d_{j,i}.$$
- [**Formulating Inequalities:**]{} Based on the above wanted and unwanted transitions the algorithm forms constraints and conditions to aoivd the unwanted transition, [*rcv\_res*]{}, while invoking the wanted transition, [*res\_tmr*]{}, to transit out of $D_T$. To achieve this, the algorithm automatically derives the following inequality (see Appendix A for more details):
$$t(p_{t_i}) < t(p_{r_{i,j}}).$$
Substituting expressions for $t(p_{t_i})$ and $t(p_{r_{i,j}})$ previously derived, we get:
$d_{Q,i} + Exp_i < d_{Q,j} + Exp_j + d_{j,i}$.
Alternatively, we can avoid the unwanted transition [*rcv\_res*]{} if the system did not exist in $D_T$ when the response is received. Hence, the algorithm automatically derives the following inequality (see Appendix A for more details):
$$t(p_{r_{i,j}}) < t(q_{r_i}).$$
Again, substituting expressions derived above, we get:
$d_{Q,i} > d_{Q,j} + Exp_j + d_{j,i}$.
Note that equations (1) and (2) are general for any number of responders, where $i$ and $j$ are any two responders in the system. Figure \[time\_fig\] (a) and (b) show equations (1) and (2), respectively.
### Task specific solutions
- [**Topology synthesis:**]{} Given the timer expiration values or ranges we want to find a feasible solution for the worst-case delays. A feasible solution in this context means assigning positive values to the delays $d_{i,j} \forall i, j$.
In equation (1) above, if we take $d_{Q,i} = d_{Q,j}$[^8], we get:
$Exp_i - Exp_j < d_{j,i}$.
These inequalities put a lower limit on the delays $d_{j,i}$, hence, we can always find a positive $d_{j,i}$ to satisfy the inequalities. Note that, the delays used in the delay matrix reflect delays over the multicast distribution tree. In general, these delays are affected by several factors including the multicast and unicast routing protocols, tree type and dynamics, propagation, transmission and queuing delays. One simple topology that reflects the delays of the delay matrix is a completely connected network where the underlying multicast distribution tree coincides with the unicast routing. There may also exist many other complex topologies that satisfy the delay matrix $D$[^9].
- [**Timer configuration:**]{} Given the delay values, ranges or bounds, we want to obtain timer expiration values that produce worst-case behavior. We obtain a range for the relative timer settings (i.e., $Exp_i - Exp_j$) using equation (1) above.
The solution for the system of inequalities given by (1) and (2) above can be solved in the general case using linear programming (LP) techniques (see Appendix B for more details). Section \[simulation\] uses the above solutions to synthesize simulation scenarios.
Note, however, that it may not be feasible to satisfy all these constraints, due to upper bounds on the delays for example. In this case the problem becomes one of maximization, where the worst-case scenario is one that triggers maximum number of responses per packet loss. This problem is discussed in Appendix B.
Best-Case Analysis {#best_case}
------------------
Best case overhead analysis constructs constraints that lead to maximum suppression, i.e., minimum number of responses. The following conditions are formulated using steps similar to those given in the worst-case analysis:
$$t(p_{t_i}) > t(p_{r_{i,j}}),$$
and
$$t(p_{r_{i,j}}) > t(q_{r_i}).$$
These are complementary conditions to those given in the worst case analysis. Figure \[time\_fig\] (c) shows equations (3) and (4). Refer to the Appendix A for more details on the inequality derivation. This concludes our description of the algorithmic details to construct worst and best-case delay-timer relations for overhead of response messages. Solutions to these relations represent delay and timer settings for stress scenarios that are used later on for simulations.
Response Time Analysis {#response}
======================
In this section, we conduct the performance analysis with respect to response time, i.e., the time for the requester to recover from the packet loss. The algorithm obtains possible sequences leading to the target event and calculates the response time for each sequence. To synthesize the worst case scenario that maximizes the response time, for example, the sequence with maximum time is chosen.
To systematically approach this problem we consider the following three cases: (1) The case of [*no*]{} loss to the response message. This case leads to single round of request-response messages. Without loss of response messages this problem becomes one of maximizing the round trip delay between the requester and the first responder. (2) The case of single selective[^10] loss of the response message. This case may lead to two rounds of request-response messages. We analyze this case in the first part of this section. (3) The case of multiple selective losses of the response messages. This case may lead to more than two rounds of request-response messages, and is discussed at the end of this section.
We now consider the case of [**single selective loss**]{} of the response message during the recovery phase. For selective losses, transition rules are applied to only those systems that receive the message.
Target Event
------------
The response time is the time taken by the mechanism to recover from the packet loss, i.e., until the requester receives the response $p$ and resets its request timer by transitioning out of the $R_T$ state. In other words, the response interval is $t(p_{r_Q}) - t(q_{t}) = t(p_{r_Q}) - t_0$. The designer identifies $t(p_{r_Q})$ as the target time, hence, $p_{r_Q}$ is the target event.
The Search
----------
We present in detail the case of single responder, then discuss the multiple responders case.
- [**Backward search:**]{} As shown in Figure \[response\_diag\] (a), the backward search starts from $p_{r_Q}$ and is performed over the transition table (see Section \[transition\_table\]) using the implication rules in Section \[implication\_rules\], yielding $rcv\_res_Q \Leftarrow res\_tmr_j \Leftarrow rcv\_req_j$, or [^11]:
$D_j.p_{r_Q}.(R \leftarrow R_T) \Leftarrow p_{t_j}.(D_j
\leftarrow D_{T_j}).Res_j.R_T \Leftarrow q_{r_j}.(D_{T_j}
\leftarrow D_j).R_T$
At which point the algorithm reaches a branching point, where two possible preceding states could cause $q_{r_j}$:
- The first is transition [*loss*]{}, or $D_j.q_{t}.(R_T \leftarrow
R)$, and since the initial state $R$ is reached, the backward search ends for this branch.
- The second is transition [*req\_tmr*]{}, or $D_j.Req.q_{t}.R_T$. Note that $Req$ indicates the need for a transition to $R_T$, i.e., ($R_T \leftarrow R$), and the search for this last state yields the intial (data packet loss) state $loss$: $D_j.q_{t}.(R_T \leftarrow R)$. However, $q_t$ is message transmission, which implies that the message must be received (or lost). Hence, there are gaps in the event sequence (indicated by the $dots$ in Figure \[response\_diag\] (a)) that are filled through forward search (in Figure \[response\_diag\] (b)).
- [**Forward search:**]{} The algorithm performs a forward search and checks for consistency of the GFSM. The forward search step may lead to contradiction with the original backward search, causing rejection of that branch as a feasible sequence. For example, as shown in Figure \[response\_diag\] (b), one possible forward sequence from the initial state gives $loss \Rightarrow tx\_req \Rightarrow rcv\_req_j \Rightarrow
res\_tmr_j \Rightarrow tx\_res_j$, or:
$D_j.q_{t}.(R \rightarrow R_T) \Rightarrow
q_{r_j}.(D_j \rightarrow D_{T_j}).R_T \Rightarrow
p_{t_j}.(D_{T_j} \rightarrow D_j).Res_j.R_T$
The algorithm then searches two possible next states:
- If $p_{t_j}$ is not lost, and hence causes $p_{r_Q}$, then the next state is $D_j.R$. But the original backward search started from $D_j.q_{t}.Req.R_T$ which cannot be reached from $D_j.R$. Hence, we get contradiction and the algorithm rejects this sequence.
- If the response $p$ is lost by $Q$, we get $D_j.R_T$ that leads to $D_j.Req.q_{t}.R_T$. The algorithm identifies this as a feasible sequence.
Calculating the time for each feasible sequence, the algorithm identifies the latter sequence as one of maximum response time.
For [**multiple responders**]{}, the algorithm automatically explores the different possible selective loss patterns of the response message. The search identifies the sequence with maximum response as one in which only one responder triggers a response that is selectively lost by the requester. To construct such a sequence, the algorithm creates conditions and inequalities similar to those formulated for the best-case overhead analysis with respect to number of responses (see Section \[best\_case\]).
Effectively, the sequence obtained above occurs when the response is lost by the requester, which triggers another request. Intuitively, the response delay is increased with multiple request rounds. The case of [**multiple selective loss**]{} of the response messages may trigger multiple (more than two) request rounds. Practically, the number of request rounds is bounded by the protocol implementation, which imposes an upper bound on the number of requests sent per packet loss. This, in turn, imposes an upper bound on the worst-case response time. This bound can be easily integrated into the search to end the search when the maximum number of allowed request rounds is reached[^12].
After conducting the above analyses, we have applied our method to generate worst-case overhead scenarios for topology synthesis and timer configuration tasks using determinsitic and adaptive timers (see Appendix D). We also applied it to response time analysis and to best-case analyses. In the next section we show network simulations using our generated worst-case overhead scenarios.
Simulations Using Systematic Scenarios {#simulation}
======================================
To evaluate the utility and accuracy of our method, we have conducted a set of detailed simulations for the Scalable Reliable Multicast (SRM) [@SRM] based on our worst-case scenario synthesis results for the timer-suppression mechanism. We tied our method to the network simulator (NS) [@ns]. The output of our method, in the form of inequalities (see Section \[overhead\]), is solved using a mathematical package (LINDO). The solution, in terms of a delay matrix, is then used to generate the simulation topologies for NS automatically.
For our simulations we measured the number of responses triggered for each data packet loss. We have conducted two sets of simulations, each using two sets of topologies. The simulated topologies included topologies with up to 200 receivers. The first set of topologies was generated according to the overhead analysis presented in this paper. We call this set of topologies the [*stress*]{} topologies. Example [ *stress*]{} topology is shown in Figure \[stress\_topo\] (a), and its corresponding fully-connected topology is shown in Figure \[stress\_topo\] (b). Both topologies satisfy the delay matrix, $D$, produced by stress[^13]. The second set of topologies was generated by the GT-ITM topology generator [@gt_itm], generating random and transit stub topologies[^14]. We call this set of topologies the [*random*]{} topologies[^15].
The first set of simulations was conducted for the SRM deterministic timers[^16].The results of the simulation are shown in Figure \[sim\_res1\] (a). The number of responses triggered for all the $stress$ topologies was $n-1$, where $n$ is the number of receivers (i.e., no suppression occurred). For the $random$ topologies, with up to 200 receivers, the number of responses triggered was less than 20 responses in the worst case.
Using the same two sets of topologies, the second set of simulations was conducted for the SRM adaptive timers[^17]. The results are given in Figure \[sim\_res1\] (b). For the $stress$ topologies almost 50% of the receivers triggered responses. Whereas $random$ topologies simulation generated almost 10 responses in the worst case, for topologies with 100-200 receivers.
These simulations illustrate how our method may be used to generate consistent worst-case scenarios in a scalable fashion. It is interesting to notice that worst-case topologies generated for simple deterministic timers also experienced substantial overhead (perhaps not the worst, though) for more complicated timers (such as the adaptive timers). It is also obvious from the simulations that [*stress*]{} scenarios are more consistent than the other scenarios when used to compare different mechanisms, in this case deterministic and adaptive timers; the performance gain for adaptive timers is very clear under [*stress*]{} scenarios.
So, in addition to experiencing the worst-case behavior of a mechanism, our stress methodology may be used to compare protocols in the above fashion and to aid in investigating design trade-offs. It is a useful tool for generating meaningful simulation scenarios that we believe should be considered in performance evaluation of protocols in addition to the average case performance and random simulations. We plan to apply our method to test a wider range of protocols through simulation[^18].
Related Work {#related}
============
Related work falls mainly in the areas of protocol verification, VLSI test generation and network simulation.
There is a large body of literature dealing with verification of protocols. Verification systems typically address well-defined properties –such as [*safety*]{}, [*liveness*]{}, and [ *responsiveness*]{} [@proto_design]– and aim to detect violations of these properties. In general, the two main approaches for protocol verification are theorem proving and reachability analysis [@formal_survey1]. Theorem proving systems define a set of axioms and relations to prove properties, and include [*model-based*]{} and [ *logic-based*]{} formalisms [@nqthm; @z]. These systems are useful in many applications. However, these systems tend to abstract out some network dynamics that we study (e.g., selective packet loss). Moreover, they do not synthesize network topologies and do not address performance issues per se.
Reachability analysis algorithms [@reachability], on the other hand, try to inspect reachable protocol states, and suffer from the ‘state space explosion’ problem. To circumvent this problem, state reduction techniques could be used [@partial_reachability2]. These algorithms, however, do not synthesize network topologies. Reduced reachability analysis has been used in the verification of cache coherence protocols [@cache_coherence], using a global FSM model. We adopt a similar FSM model and extend it for our approach in this study. However, our approach differs in that we address end-to-end protocols, that encompass rich timing, delay, and loss semantics, and we address performance issues (such as overhead or response delays).
There is a good number of publications dealing with conformance testing [@Yannakakis] [@conformance] [@conformance1] [@conformance2]. However, conformance testing verifies that an implementation (as a black box) adheres to a given specification of the protocol by constructing input/output sequences. Conformance testing is useful during the implementation testing phase –which we do not address in this paper– but does not address performance issues nor topology synthesis for design testing. By contrast, our method synthesizes test scenarios for protocol design, according to evaluation criteria.
Automatic test generation techniques have been used in several fields. VLSI chip testing [@testability] uses test vector generation to detect target faults. Test vectors may be generated based on circuit and fault models, using the fault-oriented technique, that utilizes [ *implication*]{} techniques. These techniques were adopted in [@fotg] to develop fault-oriented test generation (FOTG) for multicast routing. In [@fotg], FOTG was used to study correctness of a multicast routing protocol on a LAN. We extend FOTG to study performance of end-to-end multicast mechanisms. We introduce the concept of a virtual LAN to represent the underlying network, integrate timing and delay semantics into our model and use performance criteria to drive our synthesis algorithm.
In [@stress], a simulation-based stress testing framework based on heuristics was proposed. However, that method does not provide automatic topology generation, nor does it address performance issues. The VINT [@vint] tools provide a framework for Internet protocols simulation. Based on the network simulator (NS) [@ns] and the network animator (NAM) [@nam], VINT provides a library of protocols and a set of validation test suites. However, it does not provide a generic tool for generating these tests automatically. Work in this paper is complementary to such studies, and may be integrated with network simulation tools similar to our work in Section \[simulation\].
Issues and Future Work {#issues}
======================
In this paper we have presented our first endeavor to automate the test synthesis as applies to boundary-point performance evaluation of multicast timer suppression protocols. Our case studies were by no means exhaustive. However, they gave us insights into the research issues involved. Particularly, in this section we shall discuss issues of algorithmic complexity. In addition, we present our future plans to explore several potential extensions and applications of our method.
- [**Algorithmic complexity**]{}
One goal of our case studies is to understand and evaluate the computational complexity of our method and algorithms. Our main algorithm uses a mix of backward and forward search techniques. The algorithm starts from [*target events*]{} and uses implicit backward search and branch and bound techniques to synthesize the required scenario sequences. Complexity of such algorithm depends on the finite state machine (FSM), the state transition rules, and the target events from which the algorithm starts. Hence, it is hard to quantify, in general terms, the complexity of our algorithm. Nonetheless, we shall comment on the nature of the method and the algorithm qualitatively based on our case studies. We note the following: (a) Our algorithms use branch and bound techniques and utilize implicit backward search starting from a [*target*]{} event (vs. explicit forward reachability analysis starting from initial states). Branch and bound techniques are, generally, hard to quantify in terms of (worst-case or average case) complexity in abstract terms. Although the worst-case for branch and bound could be exponential, through our experiments we found that, on average, the target-based approach has far less complexity than forward search. In many cases the branch bounds immediately (e.g., due to contradiction if the sequence is not feasible). For all our STRESS case studies, we have found our search algorithms to be quite manageable. (b) Scenarios synthesized using the STRESS method usually are simple and include relatively small topologies. Thus, they often experience low computational complexity. It is our observation, in all our case studies thus far, that erroneous and worst-case protocol behaviors may be invoked using relatively simple (yet carefully synthesized) scenarios. Also, it was often the case that these simple scenarios were extensible to larger and more complex scenarios using simple heuristics. In Section \[simulation\], we have demonstrated how the simple scenarios generated by STRESS, with only a few receivers, could be scaled up to include hundreds of receivers. Accuracy of such extrapolation was validated through detailed simulations.
- [**Automated generation of simulation test suites**]{}
Simulation is a valuable tool for designing and evaluating network protocols. Researchers usually use their insight and expertise to develop simulation inputs and test suites. Our method may be used to assist in automating the process of choosing simulation inputs and scenarios. The inputs to the simulation may include the topology, host events (such as traffic models), network dynamics (such as link failures or packet loss) and membership distribution and dynamics. Our future work includes implementing a more complete tool to automate our method (including search algorithms and modeling semantics) and tie it to a network simulator to be applied to a wider range of multicast protocols.
- [**Validating protocol building blocks**]{}
The design of new protocols and applications often borrows from existing protocols or mechanisms. Hence, there is a good chance of re-using established mechanisms, as appropriate, in the protocol design process. Identifying, verifying and understanding building blocks for such mechanisms is necessary to increase their re-usability. Our method may be used as a tool to improve that understanding in a systematic and automatic manner. Ultimately, one may envision that a library of these building blocks will be available, from which protocols (or parts thereof) will be readily composable and verifiable using CAD tools; similar to the way circuit and chip design is carried out today using VLSI design tools. In this work and earlier works [@fotg] [@stress], some mechanistic building blocks for multicast protocols were identified, namely, the timer-suppression mechanism and the Join/Prune mechanism (for multicast routing). More work is needed to identify more building blocks to cover a wider range of protocols and mechanisms.
- [**Generalization to performance bound analysis**]{}
An approach similar to the one we have taken in this paper may be based on performance bounds, instead of worst or best case analyses. We call such approach ‘condition-oriented test generation’.
For example, a target event may be defined as ‘the response time exceeding certain delay bounds’ (either absolute or parametrized bounds). If such a scenario is not feasible, that indicates that the protocol gives absolute guarantees (under the assumptions of the study). This may be used to design and analyze quality-of-service or real-time protocols, for example.
- [**Applicability to other problem domains**]{}
So far, our method has been applied mainly to case studies on multicast protocols in the context of the Internet.
Other problem and application domains may introduce new mechanistic semantics or assumptions about the system or environment. One example of such domains includes sensor networks. These networks, similar to ad-hoc networks, assume dynamic topologies, lossy channels, and deal with stringent power constraints, which differentiates their protocols from Internet protocols [@sensornets].
Possible research directions in this respect include:
- Extending the topology representation or model to capture dynamics, where delays vary with time.
- Defining new evaluation criteria that apply to the specific problem domain, such as power usage.
- Investigating the algorithms and search techniques that best fit the new model or evaluation criteria.
Conclusion
==========
We have presented a methodology for scenario synthesis for boundary-point performance evaluation of multicast protocols. In this paper we applied our method to worst and best-case evaluation of the timer suppression mechanism; a common building block for various multicast protocols. We introduced a virtual LAN model to represent the underlying network topology and an extended global FSM model to represent the protocol mechanism. We adopted the fault-oriented test generation algorithm for search, and extended it to capture timing/delay semantics and performance issues for end-to-end multicast protocols.
Two performance criteria were used for evaluation of the worst and best case scenarios; the number of responses per packet loss, and the response delay. Simulation results illustrate how our method can be used in a scalable fashion to test and compare reliable multicast protocols.
We do not claim to have a generalized algorithm that applies to any arbitrary protocol. However, we hope that similar approaches may be used to identify and analyze other protocol building blocks. We believe that such systematic analysis tools will be essential in designing and testing protocols of the future.
\
In this appendix we present details of inequality formulation for the end-to-end performance evaluation. In addition, we present the mathematical model to solve these inequalities. We also discuss the case of multiple request rounds for the timer suppression mechanism, and present several example case studies.
Deriving Stress Inequalities {#myapp}
============================
Given the target event, transitions are identified as either wanted or unwanted transitions, according to the maximization or minimization objective. For maximization, wanted transitions are those that establish conditions to trigger the target event, while unwanted transitions are those that nullify these conditions.
Let $W$ be the wanted transition, and let $t(W)$ be the time of its occurrence. Let $C$ be the condition for the wanted transition, and let $t(C)$ be the time at which it is satisfied. Let $U$ be the unwanted transition occurring at $t(U)$.
We want to establish and maintain $C$ until $W$ occurs, i.e., in the duration \[$t(C)$, $t(W)$\]. Hence, $U$ may only occur outside (before or after) that interval. In Figure \[timeline\], this means that $U$ can only occur in $Region (1)$ or $Region (3)$.
Hence, the inequalities must satisfy the following
1. the condition for the wanted transition, $C$, must be established before the event for the wanted transition, $W$, triggers, i.e., $t(C) < t(W)$, and
2. one of the following two conditions must be satisfied:
1. the unwanted transition, $U$, must occur before $C$, i.e., $t(U) < t(C)$, or
2. the unwanted transition, $U$, must occur after the wanted transition, $W$, i.e., $t(W) < t(U)$.
These conditions must be satisfied for all systems. In addition, the algorithm needs to verify, using backward search and implication rules, that no contradiction exists between the above conditions and the nature of the events of the given protocol.
Worst-case Overhead Analysis
----------------------------
The target event for the overhead analysis is $p_t$. The objective for the worst case analysis is to maximize the number of responses $p_t$. The wanted transition is transition [*res\_tmr*]{}, or $Res.(D_T \rightarrow D).p_t$ (see Section \[timer\]). Hence $t(W) = t(p_t)$. The condition for the wanted transition is $D_T$ and its time is $t(C) =
t(q_r)$, from transition [*tx\_req*]{}, or $q_r.(D \rightarrow D_T)$.
The unwanted transition is one that nullifies the condition $D_T$. Transition [*rcv\_res*]{}, or $p_r.(D_T \rightarrow D)$, is identified by the algorithm as the unwanted transition, hence $t(U) =
t(p_r)$.
For a given system $i$, the inequalities become:
$$t(q_{r_i}) < t(p_{t_i}),$$ and either $$t(p_{r_{i,j}}) < t(q_{r_i})$$ or $$t(p_{t_i}) < t(p_{r_{i,j}}).$$
The above automated process is shown in Figure \[table\_algo\]. From the timer expiration implication rule, however, we get that the response time must have been set earlier by the request reception, i.e., $Res_i.(D_i \leftarrow D_{T_i}).p_{t_i}
\Leftarrow q_{r_i}.(D_{T_i} \leftarrow D_i)$ and $t(p_{t_i}) = t(q_{r_i}) + Exp_i$. Hence, $t(q_{r_i}) < t(p_{t_i})$ is readily satisfied and we need not add any constraints on the expiration timers or delays to satisfy this condition. Thus, the inequalities formulated by the algorithm to produce worst-case behavior are:
$$t(p_{r_{i,j}}) < t(q_{r_i}),$$ or $$t(p_{t_i}) < t(p_{r_{i,j}}).$$
Best-case Analysis {#best-case-analysis}
------------------
Using a similar approach to the above analysis, the algorithm identifies transition [*rcv\_res*]{}, or $p_r.(D_T \rightarrow D)$, as the wanted transition. Hence $t(W) = t(p_r)$, and $t(C) = t(q_r)$. The unwanted transition is transition [*res\_tmr*]{}, and $t(U) = t(p_t)$.
For system $i$ the inequalities become:
$$t(q_{r_i}) < t(p_{r_{i,j}}),$$ and either $$t(p_{t_i}) < t(q_{r_i})$$ or $$t(p_{r_{i,j}}) < t(p_{t_i}).$$
But from the backward implication we have $t(q_{r_i}) < t(p_{t_i})$. Hence, the algorithm encounters contradiction and the inequality $t(p_{t_i}) < t(q_{r_i})$ cannot be satisfied.
Thus, the inequalities formulated by the algorithm to produce best-case behavior are:
$$t(q_{r_i}) < t(p_{r_{i,j}}),$$ and $$t(p_{r_{i,j}}) < t(p_{t_i}).$$
Solving the System of Inequalities {#math_model}
==================================
In this section we present the general model of the constraints (or inequalities) generated by our method. As a first step, we form a linear programming problem and attempt to find a solution. If a solution is not found, then we form a mixed non-linear programming problem to get the maximum number of feasible constraints.
In general, the system of inequalities generated by our method to obtain worst or best case scenarios, can be formulated as a linear programming problem. In our case, satisfying all the constraints, regardless of the objective function, leads to obtaining the absolute worst/best case. For example, in the case of worst case overhead analysis, this means obtaining the scenario leading to no-suppression. The formulated inequalities by our method as given in Section \[overhead\] are as follows.
- for the worst case behavior: $$d_{Q,i} + Exp_i < d_{Q,j} + Exp_j + d_{j,i},$$ or $$d_{Q,i} > d_{Q,j} + Exp_j + d_{j,i}.$$
- for the best case behavior: $$d_{Q,i} + Exp_i > d_{Q,j} + Exp_j + d_{j,i},$$ and $$d_{Q,i} < d_{Q,j} + Exp_j + d_{j,i}.$$
The above systems of inequalities can be represented by a linear programming model. The general form of a linear programming (LP) problem is: $$Maximize Z = C^TX = \sum_{0\le i\le n} c_i \cdot x_i$$ subject to: $$AX \le B$$ $$X \geq 0$$
where $Z$ is the objective function (in our case it is a dummy objective function such as $Z = const$), $C$ is a vector of $n$ constants $c_i$, $X$ is a vector of $n$ variables $x_i$, $A$ is $m \times n$ matrix, and $B$ is a vector of $m$ elements. This problem can be solved practically in polynomial time using Karmarkar [@karmarkar] or simplex method [@simplex], if a feasible solution exists.
In some cases, however, the absolute worst/best case may not be attainable, and it may not be possible to find a feasible solution to the above problem. In such cases we want to obtain the maximum feasible set of constraints in order to get the worst/best case scenario. To achieve this, we define the problem as follows:
$$Maximize \sum_{0\le i\le m} y_i$$ subject to: $$y_i \cdot f_i(x) \le 0, \forall i$$ $$y_i \in \{0,1\}$$ or $$y_i \cdot (1 - y_i) = 0$$
where $f_i(x)$ is the original constraint from the previous problem.
This problem is a mixed integer non-linear programming (MINLP) problem, that can be solved using branch and bound methods [@minlp].
[**Obtaining**]{} [***Link***]{} [**Delays:**]{}
In the previous discussion we assumed that the model deals only with end-to-end delays ($d_{i,j}$ of the delay matrix $D$). In some cases, however, it may be the case that the connectivity of the network topology is given and the task is to find the [*link*]{} delays (instead of end-to-end delays). We present a very simple extension to the model to accommodate such situation, as follows. Let $l_x$ be any link in the topology and let $d_{l_x}$ be its delay. Take any two end systems $i$ and $j$ and let the path from $i$ to $j$ pass through links $l_a, l_b, \dots \l_n$. Hence, we get $d_{i,j} = \sum_{x\in L} d_{l_x}$, where $L = \{l_a, l_b, \dots \l_n\}$. Substituting these relations in the above inequalities we can formulate the problem in terms of link delays.
Multiple request rounds {#multi_request}
=======================
In Section \[overhead\] we conducted the protocol overhead analysis with the assumption that recovery will occur in one round of request. In general, however, loss recovery may require multiple rounds of request, and we need to consider the request timer as well as the response timers. Considering multiple timers or stimuli adds to the branching factor of the search. Some of these branches may not satisfy the timing and delay constraints. It would be more efficient then to incorporate timing semantics into the search technique to prune off infeasible branches.
Let us consider forward search first. For example, consider the state $q_{t_i}.R_{T_i}$ having a transmitted request message and a request timer running. Depending on the timer expiration value $Exp_i$ and the delay experienced by the message $d_{i,j}$, we may get different successor states. If $d_{i,j} > Exp_i$ then the request timer fires first triggering the event $Req_i$ and we get $q_{t_i}.Req_i$ as the successor state. Otherwise, the request message will be received first, and the successor state will be $q_{r_j}.R_{T_i}$. Note that in this case the timer value must be decremented by $d_{i,j}$. This is illustrated in figure \[multi\_fwd\]. The condition for branching is given on the arrow of the branch, and the timer value of $i$ is given by $T_i$.
For backward search, instead of decreasing timer values (as is done with forward search), timer values are increased, and the starting point of the search is arbitrary in time, as opposed to time ‘0’ for forward search.
To illustrate, consider the state having $(D_i \leftarrow D_{T_i}).R_{T_j}$, with the request timer running at $j$ and the response timer firing at $i$.
Figure \[multi\_bkwd\], shows the backward branching search, with the timer values at each step and the condition for each branch. In the first state, the timer $T_Q$ starts at an arbitrary point in time $x$, and the timer $T_i$ is set to ‘0’ (i.e. the timer expired triggering a response $p_{t_i}$). One step backward, either the timer at $i$ must have been started ‘$Exp_Q - x$’ units in the past, or the response timer must have been started ‘$Exp_i$’ units in the past. Depending on the relative values of these times some branch(es) become valid. The timer values at each step are updated accordingly. Note that if a timer expires while a message is in flight (i.e. transmitted but not yet received), we use the $m$ subscript to denote it is still multicast, as in $q_{r_m}$ in the figure.
Sometimes, the values of the timers and the delays are given as ranges or intervals. Following we present how branching decision are made when comparing intervals.
[**Branching decision for intervals**]{}
In order to conduct the search for multiple stimuli, we need to check the constraints for each branch. To decide on the branches valid for search, we compare values of timers and delays. These values are often given as intervals, e.g. $[a,b]$.
Comparison of two intervals $Int_1 = [a_1,b_1]$ and $Int_2 = [a_2,b_2]$ is done according to the following rules.
Branch $Int_1 > Int_2$ becomes valid if there exists a value in $[a_1,b_1]$ that is greater than a value in $[a_2,b_2]$, i.e. if there is overlap of more than one number between the intervals. We define the ‘$<$’ and ‘$=$’ relations similarly, i.e., if there are any numbers in the interval that satisfy the relation then the branch becomes valid.
For example, if we have the following branch conditions: (i) $Exp_i < Exp_j$, (ii) $Exp_i = Exp_j$, and (iii) $Exp_i > Exp_j$. If $Exp_i = [3,5]$ and $Exp_j = [4,6]$, then, according to our above definitions, all the branch conditions are valid. However, if $Exp_i = [3,5]$ and $Exp_j = [5,7]$, then only branches (i) and (ii) are valid.
The above definitions are sufficient to cover the forward search branching. However, for backward search branching, we may have an arbitrary value $x$ as noted above.
For example, take the state $(D_i \leftarrow D_{T_i}). R_{T_Q}$. Consider the timer at $Q$, the expiration duration of which is $Exp_Q$ and the value of which is $x$, and the timer at $i$, the expiration duration of which is $Exp_i$ and the value of which is ‘0’, as given in figure \[multi\_bkwd\]. Depending on the relevant values of $Exp_i$ and $Exp_Q - x$ the search follows some branch(es). If $Exp_Q = [a_1,b_1]$, then $x = [0,b_1]$ and $Exp_Q - x = [0,b_1]$. Hence, we can apply the forward branching rules described earlier by taking $Exp_Q - x = [0,b_1]$, as follows. Since $Exp_i = [a_2,b_2]$, where $a_2 >0$ and $b_2 > 0$, hence, the branch condition $Exp_i > Exp_Q - x$ is always true. The condition $Exp_i = Exp_Q - x$ is valid when: (i) $Exp_i =
Exp_Q$, or (ii) $Exp_i < Exp_Q$. The last condition, $Exp_i <
Exp_Q - x$, is valid only if $Exp_i < Exp_Q$.
These rules are integrated into the search algorithm for our method to deal with multiple stimuli and timers simultaneously.
Example Case Studies {#example}
====================
In this section, we present several case studies that show how to apply the previous analysis results to examples in reliable multicast and related protocol design problems.
Topology Synthesis
------------------
In this subsection we apply the test synthesis method to the task where the timer values are known and the topology (i.e., $D$ matrix) is to be synthesized according to the worst-case behavior. We explore various timer settings. We investigate two examples of topology synthesis, one uses timers with fixed randomization intervals and the other uses timers that are a function of distance.
Let $Q$ be the requester and $1$, $2$ and $3$ be potential responders. Let $V_{t_i}$ be the time required for system $i$ to trigger a response transmission from the time a request was sent, i.e., $V_{t_i} = d_{Q,i} + Exp_i$. From Section \[overhead\], we get $V_{t_i} < V_{t_j} + d_{j,i}$ for worst-case overhead.
At time $t_0$ $Q$ sends the request. For simplicity we assume, without loss of generality, that the systems are ordered such that $V_{t_i} < V_{t_j}$ for $i < j$ (e.g., system $1$ has the least $d_{Q,1} + Exp_1$, then 2, and then 3). Thus the inequalities $V_{t_i} < V_{t_j} + d_{j,i}$ are readily satisfied for $i < j$ and we need only satisfy it for $i > j$.
From equation (1) for the worst-case (see Section \[overhead\]) we get:
$$\begin{aligned}
V_{t_2} < V_{t_1} + d_{1,2}, \nonumber\\
V_{t_3} < V_{t_1} + d_{1,3}, \nonumber\\
V_{t_3} < V_{t_2} + d_{2,3}.\end{aligned}$$
By satisfying these inequalities we obtain the delay settings of the worst case topology, as will be shown in the rest of this section.
### Timers with fixed randomization intervals
Some multicast applications and protocols (such as wb [@SRM], IGMP [@igmp] or PIM [@PIM-ARCHv2]) employ fixed randomization intervals to set the suppression timers. For instance, for the shared white board (wb) [@SRM], the response timer is assigned a random value from the (uniformly distributed) interval \[t,2\*t\] where t = 100 msec for the source $src$, and 200 msec for other responders.
Assume $Q$ is a receiver with a lost packet. Using wb parameters we get $Exp_{src} = [100,200]$ msec, and $Exp_i = [200,400]$ msec for all other nodes.
To derive worst-case topologies from inequalities (A.1) we may use a standard mathematical tool for linear or non-linear programming, for more details see Appendix B. However, in the following we illustrate general techniques that may be used to obtain the solution.
From inequalities (A.1) we get:
$d_{Q,2} + Exp_2 = V_{t_2} < V_{t_1} + d_{1,2} = d_{Q,1}
+ Exp_1 + d_{1,2}$.
This can be rewritten as
$$d_{Q,2} - (d_{Q,1} + d_{1,2}) < Exp_1 - Exp_2 =
diff_{1,2},$$
where
$$diff_{1,2} =
\begin{cases}
$[100,200] - [200,400] = [-300,0]$ & \text{if 1 is src},\\
$[200,400] - [100,200] = [0,300]$ & \text{if 2 is src},\\
$[200,400] - [200,400] = [-200,200]$ &
\text{Otherwise}.
\end{cases} \notag$$
Similarly, we derive the following from inequalities for $V_{t_3}$:
$d_{Q,3} - (d_{Q,1} + d_{1,3}) < diff_{1,3}$, and
$d_{Q,3} - (d_{Q,2} + d_{2,3}) < diff_{2,3}$.
If we assume system 1 to be the source, and for a conservative solution we choose the minimum value of $diff$, we get:
$min(diff_{1,2}) = min(diff_{1,3}) = -300$,
$min(diff_{2,3}) = -200$.
We then substitute these values in the above inequalities, and assign the values of some of the delays to compute the others.
[*Example:*]{} if we assign $d_{Q,1} = d_{Q,2} = d_{Q,3} = 100$msec, we get: $d_{1,2} > 300$, $d_{1,3} > 300$ and $d_{2,3} > 200$.
These delays exhibit worst-case behavior for the [*timer suppression mechanism*]{}.
### Timers as function of distance
In contrast to fixed timers, this section uses timers that are function of an estimated distance. The expiration timer may be set as a function of the distance to the requester. For example, system $i$ may set its timer to repond to a request from system $Q$ in the interval: $[C_1 * E_{i,Q} , (C_1+C_2) * E_{i,Q}]$, where $E_{i,Q}$ is the estimated distance/delay from $i$ to $Q$, which is calculated using message exchange (e.g. SRM session messages) and is equal to $(d_{i,Q} + d_{Q,i})/2$. (Note that this estimate assumes symmetry which sometimes is not valid.)
[@SRM] suggests values for $C_1$ and $C_2$ as 1 or $log_{10} G$, where $G$ is the number of members in the group.
We take $C_1 = C_2 = 1$ to synthesize the worst-case topology. We get the expression
$Exp_1 - Exp_2 = [(d_{1,Q}+d_{Q,1})/2,d_{1,Q}+d_{Q,1}] -
[(d_{2,Q}+d_{Q,2})/2,d_{2,Q}+d_{Q,2}]$.
[*Example:*]{} If we assume that $d_{1,Q}=d_{Q,1}=d_{2,Q}=d_{Q,2}=100msec$, we can rewrite the above relation as $Exp_1 - Exp_2 = [-100, 100]$ msec.
Substituting in equation (A.2) above, we get $d_{1,2} > 100$msec. Under similar assumptions, we can obtain $d_{2,3} > 100$msec, and $d_{1,3} > 100$msec.
Topologies with the above delay settings will experience the worst case overhead behavior (as defined above) for the [*timer suppression*]{} mechanism.
As was shown, the inequalities formulated automatically by our method in section \[overhead\], can be used with various timer strategies (e.g., fixed timers or timers as function of distance). Although the topologies we have presented are limited, a mathematical tool (such as LINDO) can be used to obtain solutions for larger topologies.
Timer configuration
-------------------
In this subsection we give simple examples of the timer configuration task solution, where the delay bounds (i.e., D matrix) are given and the timer values are adjusted to achieve the required behavior.
In these examples the delay is given as an interval \[x,y\] msec. We show an example for worst-case analysis.
### Worst-case analysis {#worst-case-analysis}
If the given ranges for the delays are \[2,200\] msec for all delays, then the term $d_{Q,j} - d_{Q,i} + d_{j,i}$ evaluates to \[-196,398\]. From equation (A.2) above, we get
$Exp_i < Exp_j - 196$, to guarantee that a response is triggered.
If the delays are \[5,50\] msec, we get:
$$Exp_i < Exp_j - 45,$$
i.e., $i$’s expiration timer must be less than $j$’s by at least 45 msecs. Note that we have an implied inequality that $Exp_i > 0$ for all $i$.
These timer expiration settings would exhibit worst-case behavior for the given delay bounds.
[^1]: Such behavior is not protocol specific, and if a protocol is composed of previously checked building blocks, these parts of the protocol need not be revalidated in full. However, interaction between the building blocks still needs to be validated.
[^2]: Throughout this documents, we use the term [*topology synthesis*]{} to denote the assignment of delay values which constitute the entries of the $D$ matrix.
[^3]: The role of forward search will be further illustrated in the response time analysis in Section \[response\].
[^4]: If the topology connectivity is also given, the task may also include obtaining [*link*]{} delays, not only end-to-end delays as in the $D$ matrix. For our discussion in this document we will assume that identifying the entries of the $D$ matrix is the task. Appendix B discusses the problem formulation to accommodate [*link*]{} delays.
[^5]: Since there is only one requester, we simply use $q_t$ instead of $q_{t_Q}$, $q_{r_i}$ instead of $q_{r_{i,Q}}$, $Req$ instead of $Req_Q$, $R$ instead of $R_Q$ and $R_T$ instead of $R_{T_Q}$.
[^6]: The time of a state is when the state was first created, so $t(D_{T_i})$ is the time at which $i$ transited into state $D_T$.
[^7]: We use the notation $Event.Effect$ to represent a transition.
[^8]: The number of inequalities ($n^2$, where $n$ is the number of responders) is less then the number of the unknowns $d_{i,j}$ ($n^2 - n$), hence there are multiple solutions. We can obtain a solution by assigning values to $n$ unknowns (e.g., $d_{Q,i}$) and solving for the others.
[^9]: Mapping from the delay matrix $D$ into complex topologies is not covered in this document.
[^10]: In selective loss the response may be received by some systems but not others.
[^11]: The GFSM may be represented by composition of individual states (e.g., $State_1.State_2$ or $transition_1.State_2$).
[^12]: The theoretical, trivial, worst-case response time is an infinite number of request rounds. The goal of this analysis, however, is to provide a scenario in which response time is maximized. It was a finding of our algorithm that if multiple rounds are forced then the response time increases. It was also part our algorithm to formulate conditions under which multiple response rounds are forced.
[^13]: One may perceive the fully-connected graph as an abstraction of more complex topologies that satisfy the same delay matrix, $D$. Mapping of delay matrix into complex topologies is out of scope of this document.
[^14]: This topology generator is probably representative of a standard tool for topology generation used in networking research. Using GT-ITM we have covered most topologies used in several SRM studies [@kannan] [@poly].
[^15]: We faced difficulties when choosing the lossy link for the [*random*]{} topologies in order to maximize the number of responses. This is an example of the difficulties networking researchers face when trying to stress networking protocols in an ad-hoc way.
[^16]: SRM response timer values are selected randomly from the interval \[$D_1.d_r$,$(D_1 + D_2).d_r$\], where $d_r$ is the estimated distance to the requester, and $D_1$, $D_2$ depend on the timer type. For deterministic timers $D_2=0$ and $D_1=1$.
[^17]: Adaptive timers adjust their interval based on the number of duplicate responses received and the estimated distance to the requester.
[^18]: We have conducted other case studies using our STRESS method on multicast routing (PIM-DM [@fotg] [@stress_ic3n], PIM-SM [@stress]), MARS, Mobile-IP [@mars], and multicast-congestion control [@stress_pgmcc] [@sim_pgmcc]. We are currently investigating ad hoc network protocols (e.g., MAC layer and ad hoc routing).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In order to make a unified treatment for estimation problems of a very small noise or a very weak signal in a quantum process, we introduce the notion of a low-noise quantum channel with one noise parameter. It is known in several examples that prior entanglement together with nonlocal output measurement improves the performance of the channel estimation. In this paper, we study this “ancilla-assisted enhancement” for estimation of the noise parameter in a general low-noise channel. For channels on two level systems we prove that the enhancement factor, the ratio of the Fisher information of the ancilla-assisted estimation to that of the original one, is always upper bounded by $3/2$. Some conditions for the attainability are also given with illustrative examples.'
author:
- Masahiro Hotta
- Tokishiro Karasawa
- Masanao Ozawa
date: 'Revised August 29, 2005'
title: 'Ancilla-Assisted Enhancement of Channel Estimation for Low-Noise Parameters'
---
Introduction
============
One of the formidable obstacles for the realization of quantum computers is decoherence caused by the coupling between computational qubits and the environment. Recent study of quantum error correction has shown that fault-tolerant quantum computing is in principle possible, but it requires that the noise caused by the decoherence should be lower than the very stringent threshold. Obviously, such a statement has a physical meaning only if we have an efficient method for quantitatively estimating very small noise in quantum devices in real experiments. However, if the noise is very small, so is our success probability of observing the disturbance caused by that noise. This difficulty makes evident the demand for the study of optimal quantum estimation of very small nose in general quantum channels based on well-established quantum estimation theory.
Quantum estimation theory was instituted by Helstrom in the late 1960’s and has been developed with various applications until recently; for standard reviews we refer to Helstrom [@Hel76] and Holevo [@Hol82], and see also Hayashi [@Hay05] for recent progress. A typical problem of quantum estimation is to ask what is the best observable, possibly in an extended system with ancilla, to measure in order to estimate the true value of $\theta$ provided that the system is known to be in one of the state in a given family $\{\rho_{\theta}\}$. A well-established solution for this problem is given as follows. We call an observable $A$ a (locally) unbiased estimator at $\theta=\theta_0$ if the expectation value $E_{\theta}[A]$ of $A$ in the state $\rho_{\theta}$ satisfies $$\begin{aligned}
E_{\theta_0}[A]&=&\theta_0,\\
\partial_{\theta}E_{\theta}[A]|_{\theta=\theta_0}&=&1.\end{aligned}$$ In general there are many unbiased estimators. In order to select a good one, we consider the variance $V_\theta [A]$ of an arbitrary unbiased estimator $A$ in the state $\rho_{\theta}$. Then, the quantum Cramér-Rao inequality $$\label{CR}
V_\theta [A]\ge \frac{1}{J(\rho_\theta)}$$ holds for any unbiased estimator $A$ at $\theta$, where $$\label{FI}
J(\rho_\theta)={\rm Tr}[\rho_{\theta}L^{2}_{\theta}]$$ is the (quantum) Fisher information defined through the symmetric logarithmic derivative (SLD) $L_{\theta}$ that is characterized by the relations $$\begin{aligned}
\partial_\theta \rho_\theta
&=& \frac{1}{2}(L_\theta \rho_\theta +\rho_\theta L_\theta),
\label{SLD1}\\
L_{\theta}^{\dagger}&=&L_{\theta}.
\label{SLD2}\end{aligned}$$ The SLD is determined uniquely on the range of $\rho_{\theta}$, i.e., $L_{\theta}\rho_{\theta}=L'_{\theta}\rho_{\theta}$ holds for any two SLDs $L_{\theta}$ and $L'_{\theta}$. The Cramér-Rao inequality (\[CR\]) follows from a simple application of the Schwarz inequality for the Hilbert-Schmidt inner product. From the equality condition for that the lower bound $J_\theta^{-1}$ in Eq. (\[CR\]) is always achieved by any observable $A$ satisfying $$\label{Optimality}
A\rho_{\theta}=(J_{\theta}^{-1}L_{\theta}+\theta)\rho_{\theta},$$ see Refs. [@Hel76; @Hol82] and for a straightforward derivation see Appendix of Ref. [@04QEL]. In general, to find an optimal estimator for the true value $\theta$ needs prior information on the value $\theta$, which might be collected by prior estimations assuming prior probabilities on the unknown parameter, so that the optimal estimator is considered as an ultimate limit allowed by physics. However, there are some cases in which the optimal estimator can be chosen uniformly over unknown values of $\theta$ [@FN95]. In these cases the ultimate limit can be certainly achieved without prior information.
From the quantum estimation theory for state parameters mentioned above, we can construct an estimation theory for unknown parameters of physical processes, such as coupling constants of the interaction. Suppose that we prepare a quantum system in an initial state $\rho_{in}$ and leave it in an evolution process characterized by an unknown parameter $\theta$. Then, the final state $\rho_{out}(\theta)$ of this process depends on the parameter $\theta$. The problem of finding the optimal estimation of the parameter $\theta$ is solved by maximizing the Fisher information $J_{\theta}$ over all the possible initial states $\rho_{in}$ and all the possible observable $A$ in the final state [@CN97; @PCZ97]. The above physical process can be represented by a mapping $\Gamma_\theta$ that transform the initial state $\rho_{in}$ to the final state $\rho_{out}$ as $$\rho_{out} =\Gamma_\theta [\rho_{in}].$$ It is now fairly well-known that every general state change, called a quantum operation or a quantum channel, such as $\Gamma_\theta$, physically realizable with probability one should be a trace-preserving completely positive (TPCP) mapping, and conversely that every TPCP map can be realized as a unitary process of the system augmented by an ancilla prepared in a fixed state as shown by Kraus [@Kra71; @Kra83]; see also Ref. [@83CR; @84QC] for the generalization of the above statement to generalized measurements and see Ref. [@04URN] for the latest elaboration.
As pointed out in Ref. [@Fuj01], one can improve the parameter estimation if a correlation, or in particular an entanglement, is allowed between the input system $S$ and an ancilla $A$. It should be stressed that in doing so one needs no physical process to occur on the ancilla system $A$ while the system $S$ passes through the channel $\Gamma_{\theta}$. In this case, the extended channel is represented as $\Gamma_\theta \otimes id_{A}$, where $id_A$ stands for the identity channel for $A$. Then, the improvement can be achieved by the initial preparation of the composite system in an entangled state together with the measurement of the composite system after the process.
Recent progress has been reported on problems for special families of quantum channels, in particular, SU(2) channel [@Fuj02], a generalized Pauli channel [@FI03], a generalized amplitude-damping channel [@Fuj04], U(N) channel and its Abelian subgroup channel [@Bal04a; @Bal04b]. A review by Fujiwara [@Fuj04b] is also available. For earlier contributions see also [@CPR00; @Aci01; @DPP01; @FMCF01; @BFF01; @SBB02]. In this paper, we are devoted to the ancilla-assisted enhancement of Fisher information derived by the quantum Cramér-Rao bound, whereas ancilla-assisted enhancements have been recently investigated within the Bayesian approach [@CDS05; @Sac05; @Sac05a] and the minimax approach [@DSK05].
This enhancement effect not only projects a theoretical profundity of quantum mechanics, but also suggests many physical applications including the low-noise estimation in quantum computing, where the enhanced noise estimation is expected to contribute to developing the quantum error correction and quantum noise reduction technology [@NC00].
We can find another application of the low-noise estimation in elementary particle physics. Today, because of technological difficulties of high-energy experiments, direct researches of new physics far beyond the TeV energy scale are almost impossible. This is one of reasons why the low-energy rare processes predicted by the new physics recently attract much attention. (The CPT symmetry violation in the $K-\bar{K}$ oscillation is one of the typical processes [@Ebe72; @CCE76; @EHN84; @HP95].) Clearly, the number of signals for the new-physics evidence is predicted very small, even if the process really exists in nature. The new-physics data should be separated from an enormous number of ordinary data explained by the standard model. This means that the new-physics data can be regarded as a sort of background low noise in the standard data. Hence, we can treat the rare process as a low-noise channel. It is very significant to estimate the intensity of the low noise because indirect information about physics beyond the standard model is obtained. In the estimation, the above ancilla-assisted enhancement may effectively reduce the trial number of the experiment.
In this paper, we study the estimation theory of the parameter characterizing a small noise in a general quantum channel on a system with finite dimensional state space. We can always decompose the quantum channel into two channels so that the input state of the original channel passes through the first noiseless channel and consecutively passes through the second noisy quantum channel called the noise channel. Thus, we can concentrate our attention on the noise channel. We are interested in the case where the noise is so small that the noise channel deviates only a little from the identity channel. In such a case, the channel is called a low-noise channel, and the parameter representing the noise is called the low-noise parameter denoted by $\epsilon$. Let $\Gamma_\epsilon$ be a low-noise channel with low-noise parameter $\epsilon$. We assume that the low-noise parameter is scaled so that $\Gamma_0$ is the identity channel. We can formulate natural mathematical requirements for the behavior of the low-noise parameter in a neighborhood of $\epsilon=0$. It is an interesting problem to figure out how much ancilla-assisted enhancement can be achievable in the estimation of the low-noise parameter $\epsilon$. In this paper we shall discuss this problem and obtain several upper bounds for this ancilla-assisted enhancement factor in the low-noise parameter estimation.
In Section 2, we explain a theorem [@Fuj01] states that the Fisher information is attained in a pure initial state, so that we can always assume that the input of the channel is a pure state. In Section 3, we discuss parameter estimation for unitary channels, which do not couple with the environment, and show that we have no ancilla-assisted enhancement. Thus, the ancilla-assisted enhancement is possible only for channels coupled with the environment. In Section 4, we introduce the notion of low-noise channels mentioned above with rigorous mathematical requirements, and we obtain a general formula for the upper bound for the ancilla-assisted enhancement factor. In Section 5, we introduce two physical examples of low-noise channel. In Section 6, we give a concrete evaluation of the enhancement factor in two level systems. Let us consider a low-noise channel $\Gamma_\epsilon$ with low-noise parameter $\epsilon$ in a two level system $S_2$. We obtain a universal upper bound for the enhancement factor $\eta$ defined by $$\eta =\frac{{\cal L}[\max[J_{S_2 +A}]_{\rho_{S_2 +A}}]}
{{\cal L}[\max[J_{S_2}]_{\rho_{S_2}}]} \label{ed}$$ for any finite level ancilla $A$. Here, $\rho_{S_2}$ is the input in the system $S$, $J_{S_2}$ is the Fisher information of $\Gamma_\epsilon [\rho_{S_2}]$, $\rho_{S_2 +A}$ is the channel input in the composite system $S_2 +A$, $J_{S_2 +A}$ is the Fisher information of the output states $(\Gamma_\epsilon \otimes id_A) [\rho_{S_2 +A}]$, and $\max[ \cdot ]_\rho$ stands for the maximum over all the state $\rho$. As shown later, $J_{S_2}$ and $J_{S_2 +A}$ shows a singular behavior $\propto 1/\epsilon$ in the $\epsilon$ expansion, and ${\cal L}[J]$ is coefficient of $\propto 1/\epsilon$, i.e., $$J(\epsilon) =\frac{1}{\epsilon}{\cal L}[J] +O(\epsilon^0).$$
The universal upper bound of the enhancement factor $\eta$ for all the two level systems is given by $$\eta \leq \frac{3}{2} .$$ This upper bound is attainable by various channels $\Gamma_\epsilon$, and the corresponding optimal input state is a maximal entangled state, and holds for any low-noise channels on two level systems.
The Maximum Is Attained by a Pure Input State: The Fujiwara theorem
===================================================================
In this section we briefly review an important theorem due to Fujiwara [@Fuj01]: [*the maximum of the Fisher information of output states $\rho_\theta(=\Gamma_\theta [\rho])$ over all possible input states $\rho$ is attained by a pure input state for an arbitrary fixed channel $\Gamma_\theta $.* ]{}
To show this following Fujiwara, let $L_{\theta}$ be the SLD defined by Eqs. (\[SLD1\]), (\[SLD2\]) for the output state $\rho_\theta$. Then the Fisher information $J(\rho_{\theta})$ is given by Eq. (\[FI\]). Fujiwara [@Fuj01] showed that the Fisher information has a convexity property, i.e., $$J(\lambda \sigma_\theta +(1-\lambda) \tau_\theta)
\leq
\lambda J(\sigma_\theta)
+(1-\lambda)J(\tau_\theta).\label{convex}$$ for any $0<\lambda<1$, where $\sigma_\theta$ and $\tau_\theta$ are states with parameter $\theta$.
To see the above relation, let Hermitian operators $L^{\sigma}_\theta$ and $L^{\tau}_\theta$ be the SLDs of $\sigma_\theta$ and $\tau_\theta$, respectively, i.e., $$\begin{aligned}
\partial_\theta \sigma_\theta
& =& \frac{1}{2}(L^\sigma_\theta \sigma_\theta +\sigma_\theta L^\sigma_\theta),\\
\partial_\theta \tau_\theta
& =& \frac{1}{2}(L^\tau_\theta \tau_\theta +\tau_\theta L^\tau_\theta).\end{aligned}$$ Let us consider the tensor product Hilbert space ${\cal K}={\cal H}\otimes {\bf C}^{2}$, where ${\cal H}$ is the state space of $S$ and ${\bf C}^{2}$ is a 2-dimensional state space. With fixed basis $\{{|0\rangle},{|1\rangle}\}$ of ${\bf C}^{2}$, let $\tilde{\rho}_{\theta}$ be a density operator on ${\cal K}$ such that $$\tilde{\rho}_{\theta}
=\lambda\sigma_{\theta}\otimes{|0\rangle}{\langle0|}+(1-\lambda)\tau_{\theta}\otimes{|1\rangle}{\langle1|}.$$ Then, it is easy to see that the SLD of $\tilde{\rho}_{\theta}$ is $L^{\sigma}_{\theta}\otimes{|0\rangle}{\langle0|}+L^{\tau}\otimes{|1\rangle}{\langle1|}$, so that the Fisher information of $\tilde{\rho}_{\theta}$ is given by $$\begin{aligned}
J(\tilde{\rho}_{\theta})
&=&
{\rm Tr}[\tilde{\rho}_\theta
(L^{\sigma}_{\theta}\otimes{|0\rangle}{\langle0|}+L^{\tau}\otimes{|1\rangle}{\langle1|})^{2}]\nonumber\\
&=&
\lambda{\rm Tr}[{\sigma}_\theta (L^{\sigma}_{\theta})^{2}]+
(1-\lambda){\rm Tr}[{\tau}_\theta (L^{\tau}_{\theta})^{2}]\nonumber\\
&=&
\lambda J({\sigma}_\theta)+(1-\lambda)J({\tau}_\theta).
\label{Fujiwara}\end{aligned}$$ On the other hand, the partial trace of $\tilde{\rho}_{\theta}$ over ${\bf C}^{2}$ is given by $$\begin{aligned}
{\rm Tr}_{{\bf C}^{2}}[\tilde{\rho}_{\theta}]
=
\lambda\sigma_{\theta}+(1-\lambda)\tau_{\theta}.\end{aligned}$$ Since the partial trace is a trace-preserving completely positive map, the monotonicity of the Fisher information under trace-preserving completely positive maps [@MC90; @Pet96; @PS96] concludes $$J(\lambda \sigma_\theta +(1-\lambda) \tau_\theta)\leq
J(\tilde{\rho}_{\theta}).
\label{Petz}$$ Therefore, from Eq. (\[Fujiwara\]) and Eq. (\[Petz\]) the convexity relation (\[convex\]) follows.
Now suppose that an input state $\bar\rho$ maximizes the Fisher information, i.e., $$J(\Gamma_\theta[\bar\rho])=\max[J(\Gamma_\theta[\rho])]_{\rho}.$$ Let $$\bar\rho =\sum_n p_n |n\rangle\langle n|$$ be the spectral decomposition, where $0< p_n \leq 1$ and $\sum p_n=1$. The output state $\bar\rho_\theta$ is given by $$\bar\rho_\theta =\Gamma_\theta [\bar\rho]=
\sum_n p_n \Gamma_\theta [ |n\rangle\langle n|].$$ By using relation (\[convex\]) repeatedly, we have $$J(\bar\rho_\theta)
\leq
\sum_n p_n J(\Gamma_\theta[|n\rangle\langle n|]).$$ Since $\bar\rho$ maximizes the Fisher information, we also have $$\sum_n p_n J(\Gamma_\theta[|n\rangle\langle n|])\leq J(\bar\rho_\theta),$$ and this concludes the relation $J(\bar\rho_\theta)=J(\Gamma_\theta[|n\rangle\langle n|])$ for all $n$. Thus, the maximum of the Fisher information is also attained by a pure input state.
From now on, we assume without any loss of generality that the input state of the channel is always a pure state by virtue of this theorem.
One-Parameter Unitary Channels Have No Enhancement
==================================================
Before we go to general analysis of low-noise channels, let us consider the case where the channel is unitary, or the channel does not interact with the environment. Interestingly, the maximization of the output Fisher information $J[\rho_\theta]$ with respect to the input $\rho$ can be explicitly accomplished. After the calculation of the maximum, one can notice that the ancilla-assisted enhancement does not take place at all. The result makes it clear that, in order to gain the ancilla-assisted enhancement for channel parameter estimations, the channels must have the effective interaction between the system and the environment.
Let $U(\theta)$ be a unitary operator with an unknown parameter $\theta$. Then the output state of the unitary channel determined by $U(\theta)$ for an input state $|\Psi\rangle$ is given by $$\rho(\theta) =|\Psi (\theta)\rangle\langle \Psi (\theta)| ,$$ where the output state $|\Psi(\theta)\rangle$ is defined by $$|\Psi(\theta)\rangle= U(\theta)|\Psi\rangle.$$ By introducing the (logarithmic) Hamiltonian operator $H(\theta)$ such that $$\begin{aligned}
H(\theta)&=&i(\partial_\theta U(\theta) )U(\theta)^\dagger,\\
H(\theta)^{\dagger}&=&H(\theta),\end{aligned}$$ and using the result in Ref. [@FN95], the Fisher information of the output state is evaluated as $$\label{FisherInformation}
J_S [\rho_\theta]=4V_{\Psi(\theta)}[H(\theta)],$$ where $V_{\Psi(\theta)}[H(\theta)]$ is the variance of $H(\theta)$ in the state $\Psi(\theta)$, i.e., $$V_{\Psi(\theta)}[H(\theta)]=
\langle \Psi(\theta) |H(\theta)^2|\Psi(\theta)\rangle
-\langle \Psi(\theta)|H(\theta)|\Psi(\theta )\rangle^2.$$ To obtain the maximum of $J_S$, let us consider the maximum and minimum of the eigenvalues $E_n$ of $H(\theta)$: $$E_{\max}(\theta) = \max[E_n ]_n,$$ $$E_{\min}(\theta) = \min[E_n]_n.$$ Let $|\max(\theta)\rangle$ and $|\min(\theta)\rangle$ be eigenstates corresponding to $E_{\max}(\theta)$ and $E_{\min}(\theta)$, respectively. By a straightforward manipulation, it is easy to see that the maximum of $J_S$ is taken by a pure input state $|\Phi\rangle =U(\theta)^\dagger |\Phi (\theta)\rangle $, where $|\Phi (\theta)\rangle $ is given by $$|\Phi(\theta)\rangle =
\frac{1}{\sqrt{2}}
\left[
|\max(\theta)\rangle +|\min(\theta)\rangle
\right].$$ For a fixed value of $\theta$, the maximum is given by $$\begin{aligned}
\max[ J_S ]_{|\Psi\rangle} &=&
J_S[ |\Phi(\theta)\rangle\langle\Phi(\theta)|]\nonumber\\
&=&(E_{\max} (\theta) -E_{\min} (\theta))^2.\label{JU}\end{aligned}$$
To obtain the corresponding result for ancilla-assisted estimations, let us introduce an ancilla system $A$ and the extended channel defined by $$|\tilde{\Psi} (\theta)\rangle
=(U(\theta)\otimes {\bf 1}_A ) |\tilde{\Psi}\rangle,$$ where $|\tilde{\Psi}\rangle$ is a state of the composite system $S+A$ to be put in the extended channel. For the output state $\tilde{\rho}(\theta) =|\tilde{\Psi}(\theta)\rangle\langle\tilde{\Psi}
(\theta) |$, the Fisher information is given by $$J_{S+A}[\tilde{\rho}(\theta)]
=
4V_{\tilde{\Psi}(\theta)}[H(\theta) \otimes {\bf 1}_A],$$ where $V_{\tilde{\Psi}(\theta)}[H(\theta) \otimes {\bf 1}_A]$ is the variance of $H(\theta) \otimes {\bf 1}_A$ in the state $\tilde{\Psi}(\theta)$. Note that the maximum and minimum of the eigenvalues of $H\otimes {\bf 1}_A$ are taken in the states $|\max (\theta)\rangle |a\rangle$ $|\min (\theta)\rangle |a\rangle$, respectively, with an arbitrary ancilla state $|a\rangle$, i.e., $$\begin{aligned}
H(\theta)\otimes {\bf 1}_A |\max (\theta)\rangle |a\rangle
&=&E_{\max} (\theta) |\max (\theta)\rangle |a\rangle, \\
H(\theta)\otimes {\bf 1}_A |\min (\theta)\rangle |a\rangle
&=&E_{\min} (\theta)|\min (\theta)\rangle |a\rangle.\end{aligned}$$ Hence, the input state given by $$\begin{aligned}
|\tilde{\Phi}\rangle
&=&
\frac{1}{\sqrt{2}}
(U(\theta)^\dagger \otimes {\bf 1}_A)
\left[
|\max(\theta)\rangle |a\rangle +|\min(\theta)\rangle |a\rangle
\right]\nonumber\\
&=&
|\Phi\rangle |a\rangle\end{aligned}$$ takes the maximum value of $J_{S+A}$, which turns out to be the same as that given in (\[JU\]), i.e., $$\begin{aligned}
\max[ J_{S+A} ]_{|\tilde{\Psi}\rangle}
&=& J_{S+A}
[ |\tilde{\Phi}(\theta)\rangle\langle\tilde{\Phi}(\theta)|]
\nonumber\\
&=&(E_{\max} (\theta) -E_{\min} (\theta))^2.\end{aligned}$$ Consequently, no enhancement by the ancilla extension is observed in this unitary case, i.e., $$\frac{\max[ J_{S+A} ]_{\rho_{S+A}}}{\max[ J_S ]_{\rho_S}}
=
\frac{\max[ J_{S+A} ]_{|\Psi_{S+A} \rangle}}{\max[ J_S ]_{|\Psi_S \rangle}}
=1.$$
It should be noted here that the above argument applies only to one-parameter unitary channels, for which Eq. (\[FisherInformation\]) can be applied, whereas a generalization to Abelian group parameters may follow. For multiple phase parameter estimation of unitary channels, Ballester [@Bal04a; @Bal04b] showed, ancilla-assisted enhancement actually takes place, whereas for commuting phase parameter estimation no enhancement occurs.
Within Bayesian approach, Chiribella, D’Ariano, and Sacchi [@CDS05] showed that unitary channels with non-Abelian group parameter can have ancilla-assisted improvement of a large class of cost functions. In this connection, Sacchi [@Sac05; @Sac05a] gave extensive analysis on the condition for ancilla-assisted improvement of the error probability for discrimination of Pauli channels.
Low-Noise Channels
==================
In this section, we introduce the notion of a low-noise channel $\Gamma_\epsilon$ with unknown parameter $\epsilon$, which takes only small values $\epsilon\sim 0$, by requiring a physically natural assumption of the channel $\Gamma_\epsilon$ for the parameter values near $\epsilon =0$. The small parameter $\epsilon$ is assumed to control the low noise well enough and is called the low-noise parameter.
As mentioned in the introduction, we will focus on the ancilla extension of the low-noise channel defined by $\Gamma_\epsilon \otimes id_{A}$. The ancilla-assisted enhancement factor $\eta$ is also defined as the ratio of the Fisher information of the ancilla-assisted estimation to that of the original one and is analyzed in detail.
The concept of the noise in a quantum process to implement a target unitary process can be understood under the following consideration. Suppose that we would like to implement a unitary channel $\Lambda^{(U)}$ for a system $S$, so that the output state corresponding to an input state $\rho_{in}$ of $S$ is designed to be $$\rho_{out} =\Lambda^{(U)} [\rho_{in}]= U\rho_{in} U^\dagger.$$ Without any noise, the unitary operator can be normally implemented as $$\label{eq:unitary}
U=\exp[-it H_S/\hbar ],$$ where $t$ is the time interval from input to output, and $H_S$ is the Hamiltonian of $S$ under control (Fig. 1).
(75,10)(0,0)
(0,5)[(5,0)\[l\][$\rho_{in}$]{}]{}
(8,5)[(1,0)[10]{}]{} (49,5)[(1,0)[10]{}]{}
(21,0)[(26,10)\[i\]]{} (62,5.5)[(5,0)\[l\][$\rho_{out} = \Lambda^{(U)}[ \rho_{in}]$]{}]{}
In real life, the system $S$ is coupled weakly with the environment $E$ and causes the decoherence that cannot be corrected by controlling the Hamiltonian $H_S$ of the system $S$, so that the noise is brought from the environment. Assume that the noise is controlled by one unknown positive parameter $\nu$. The estimation of the noise parameter $\nu$ often becomes critical in development of quantum devices such as quantum computers.
The total Hamiltonian reads $$H_{tot} =H_S +H_{SE} +H_E,$$ where $H_E$ is the Hamiltonian of $E$ and $H_{SE}$ is the interaction Hamiltonian between $S$ and $E$. Because of the noise, the actual output state $\rho_{out}'$ deviates from the intended output state $\rho_{out}$ (Fig. 2).
(80,70)(0,0)
(25.5,68.5)[(5,0)\[l\][environ-]{}]{} (25.7,65)[(5,0)\[l\][ ment]{}]{} (31.5,60)(29,60.5)(31.5,61) (31.5,59)(34,59.5)(31.5,60) (31.5,58)(29,58.5)(31.5,59) (31.5,57)(34,57.5)(31.5,58) (31.5,56)(29,56.5)(31.5,57) (31.5,55.1)(34,55.5)(31.5,56) (31.5,67)[[(15.2,12)]{}]{}
(0,51)[(5,0)\[l\][$\rho_{in}$]{}]{} (8,51)[(1,0)[13]{}]{} (24,46)[(15,9)\[c\]]{} (42,51)[(1,0)[13]{}]{} (58,51.5)[(5,0)\[l\][$\rho_{out}' ={\mathrm{Tr}}_{E} \left[
e^{-itH_{tot}} (\rho_{in}\otimes \rho_{\mathrm{E}} )\hspace{1mm}e^{itH_{tot}}\right] $]{}]{}
(32,39)(33.5,41.5)(31.4,42) (32.6,36)(30.5,36.5)(32,39) (29.5,42)[(0,-1)[6]{}]{} (0,28)[(5,0)\[l\][$\rho_{in}$]{}]{} (8,28)[(1,0)[13]{}]{} (24,23)[(15,9)\[c\]]{} (42,28)[(1,0)[13]{}]{} (58,28.5)[(5,0)\[l\][$\rho_{out}'
= \Lambda_{\nu}\left[ \rho_{in} \right] $]{}]{}
(32,16)(33.5,18.5)(31.4,19) (32.6,13)(30.5,13.5)(32,16) (29.5,19)[(0,-1)[6]{}]{} (0,5)[(5,0)\[l\][$\rho_{in}$]{}]{} (8,5)[(1,0)[5]{}]{} (15,0)[(12,9)\[c\]]{} (29,5)[(1,0)[5]{}]{} (36,0)[(12,9)\[c\]]{} (50,5)[(1,0)[5]{}]{} (58,5.5)[(5,0)\[l\][$\rho_{out}'
= \Gamma_{\nu}\left[ \Lambda^{(U)}[\rho_{in}] \right] $]{}]{}
By using $H_{tot}$, the output state $\rho_{out}'$ is determined in principle by $$\begin{aligned}
\rho_{out}'
=
{\rm Tr}_E \left[
e^{-itH_{tot}} \left(\rho_{in}\otimes \rho_E\right) e^{itH_{tot}}
\right],
\label{1ds}\end{aligned}$$ where ${\rm Tr}_E$ is the partial trace over $E$ and $\rho_E$ is the initial state of $E$. Theoretically, it is preferable that we determine the value of the noise parameter $\nu$ via [Eq. (\[1ds\])]{}; however, the explicit calculation of [Eq. (\[1ds\])]{} is too complicated to perform in many cases. Hence, adopting a reasonable theoretical model of the noise effect, the actual value of its noise parameter of the model should be experimentally estimated.
Without assuming any detailed knowledge about $H_E$ and $H_{SE}$, it is natural to represent the noisy process by a TPCP map $\Lambda_\nu$ such that $$\rho_{out}' = \Lambda_\nu [\rho_{in} ],$$ where the relation $\Lambda_0 =\Lambda^{(U)}$ holds as the noiseless case. In quantum theory, the channel $\Lambda_\nu$ can be equivalently described by a sequence of two channels (the third line of Fig. 2). The first one is the target unitary channel $\Lambda^{(U)}$ and the second represents the genuine noise part. This means that the general noisy process is equivalent to the noiseless unitary process followed by an instantaneous noise process. The second channel is called the noise channel $\Gamma_\nu$ and defined by $$\Gamma_\nu [\rho]:= \Lambda_\nu [U^\dagger \rho U]
=\Lambda_\nu [(\Lambda^{(U)})^{-1}[\rho] ].$$ Using the definition and the ideal output state $\rho_{out}$, it is possible to write the actual output state $\rho_{out}'$ such that $$\rho_{out}' = \Gamma_\nu [ U \rho_{in} U^\dagger ]
=\Gamma_\nu [\rho_{out}]. \label{noise}$$ When the noise vanishes, the channel reduces to the identity channel: $$\Gamma_0 =id_S.\label{ic}$$
(82,35)(0,0)
(0,28)[(5,0)\[l\][$\rho_{in}$]{}]{} (8,28)[(1,0)[17]{}]{} (27,23)[(15,9)\[c\]]{} (44,28)[(1,0)[17]{}]{} (64,28.5)[(5,0)\[l\][$\rho_{\nu\hspace{0.5mm} out}'
= \Gamma_{\nu}\left[ \rho_{in} \right] $]{}]{}
(32.7,19)[(0,-1)[6]{}]{} (35.2,16)(36.7,18.5)(34.6,19) (35.9,13)(33.7,13.5)(35.2,16) (-28,5)[(5,0)\[l\][$( \Lambda^{(U)})^{-1} [\rho_{in} ]=\rho_{in}'$]{}]{} (8,5)[(1,0)[17]{}]{} (27,0)[(15,9)\[c\]]{} (44,5)[(1,0)[17]{}]{} (64,5.5)[(5,0)\[l\][$\rho_{\nu, \hspace{0.5mm} out}'
= \Lambda_\nu \left[(\Lambda^{(U)})^{-1} [\rho_{in}] \right] $]{}]{}
It is stressed that despite that the noise channel $\Gamma_\nu$ is conceptual constituent, it can be simulated in a real experiment by use of the actual channel $\Lambda_\nu$ (Fig. 3). In fact, the output state of the channel $\Gamma_\nu$ defined by $$\rho_{\nu, out}=\Gamma_\nu [\rho_{in}]$$ is exactly reproduced by $$\rho_{\nu, out}= \Lambda_\nu [(\Lambda^{(U)})^{-1}[\rho_{in}]],$$ for an arbitrary input state $\rho_{in}$. Therefore, by adopting a known state $\rho'_{in}=(\Lambda^{(U)})^{-1}[\rho_{in}] $, which is independent of $\nu$, as the input state of the actual channel $\Lambda_\nu$, we experimentally obtain the output state $\rho_{\nu, out}$ of the noise channel $\Gamma_\nu$. This aspect sounds very significant. Actually, we can replace, not only theoretically but also experimentally, the estimation problem for a given real channel $\Lambda_\nu$ into the equivalent estimation problem for the noise channel $\Gamma_\nu$. Hence, we later concentrate on estimation of the noise parameters for $\Gamma_\nu$ which satisfies relation (\[ic\]).
Next let us define mathematically the low-noise channel $\Gamma_\epsilon$. This is a kind of the noise channel and its noise parameter $\nu$ takes small positive values, which is denoted by $\epsilon$. We call $\epsilon$ the low-noise parameter. Physically, $\Gamma_\epsilon$ is expected to have an analytic $\epsilon$ dependence near $\epsilon =0$. A rigorous mathematical formulation of this requirement is given as follows.
Since the low-noise channel $\Gamma_{\epsilon}$ is a TPCP map, it has a Kraus representations determined by a family of Kraus operators. We shall define low-noise channels in terms of their Kraus operators. A family of TPCP maps $\Gamma_{\epsilon}$ with one parameter $\epsilon>0$ is called a [*low-noise channel with low-noise parameter $\epsilon$*]{} if each $\Gamma_{\epsilon}$ has a Kraus representation $$\Gamma_\epsilon [\rho]
=\sum_a B_a (\epsilon) \rho\, B^\dagger_{a} (\epsilon)
+\epsilon
\sum_\alpha C_\alpha (\epsilon) \rho\, C^\dagger_\alpha (\epsilon)
\label{k1}$$ with two classes of Kraus operators $\{B_a (\epsilon)\}$ and $\{\sqrt{\epsilon}C_\alpha (\epsilon)\}$ satisfying the following conditions:
\(i) $B_a (\epsilon)$ is analytic at $\epsilon=0$, so that we have the power series expansion $$B_a (\epsilon) =\kappa_a {\bf 1}_S
-\sum^\infty_{n=1} N^{(n)}_a \epsilon^n,\label{1}$$ in a neighborhood of $\epsilon=0$, where $\kappa_a$ and $N^{(n)}_a$ are constant coefficients and operators, respectively, independent of $\epsilon$. The noise channel condition in [Eq. (\[ic\])]{} requires $$\sum |\kappa_a|^2 =1.$$
\(ii) $C_\alpha (\epsilon)$ is analytic at $\epsilon=0$, so that we have the power series expansion $$C_\alpha (\epsilon) =
M_\alpha +\sum^\infty_{n=1} M^{(n)}_\alpha \epsilon^n ,
\label{2}$$ in a neighborhood of $\epsilon=0$, where $M_\alpha$ and $M^{(n)}_\alpha$ are constant operators independent of $\epsilon$.
Needless to say, the Kraus operators satisfies the trace-preserving condition $${\bf 1}_S =
\sum_a B^\dagger_a (\epsilon) B_a (\epsilon)
+\epsilon\sum_\alpha C^\dagger_\alpha (\epsilon) C_\alpha (\epsilon),
\label{cp}$$ where ${\bf 1}_S$ is the identity operator. By definition, the relation $$\lim_{\epsilon\to +0} \Gamma_\epsilon =id_S \label{icln}$$ is automatically satisfied.
It should be emphasized that our definition of the low-noise channel is general from the physical point of view. Except that $\Gamma_\epsilon$ satisfies [Eq. (\[icln\])]{} and has analytic dependence of $\epsilon $ near the origin, the channel $\Gamma_\epsilon$ can be said to be a general quantum operation acting on the input state. Therefore, the low-noise channel should be always found in the weak-interaction limit of $H_{SE}$ for rather general physical processes.
A useful comment is given here. Expanding [Eq. (\[cp\])]{} in terms of $\epsilon$ generates a lot of recursion relations between $\kappa_a$, $N^{(n)}_a$ and $M_\alpha^{(n)}$. The higher components of the operators and the coefficients are determined recursively and systematically by solving the equations using their lower components. The first-order relation in the $\epsilon$ expansion of [Eq. (\[cp\])]{} is given by $$\sum_\alpha M^\dagger_\alpha M_\alpha
=\sum_a (\kappa_a N^{(1)\dagger}_a
+\kappa_a^\ast N^{(1)}_a ).
\label{nic}$$
One of our fundamental interests is to ask a question: which input state for the low-noise channel does maximize the Fisher information of its output state $\rho_{\epsilon}$? By virtue of the theorem reviewed in Section 2, the optimal input state is a pure state. Denote the input state by $|\phi\rangle\langle\phi |$. Then, from [Eq. (\[1\])]{} and [Eq. (\[2\])]{}, $\rho_\epsilon$ can be expanded as $$\rho_\epsilon:=
\Gamma_{\epsilon} [|\phi\rangle\langle\phi |]
=|\phi\rangle\langle\phi | -\epsilon \rho_1
+O(\epsilon^2).$$ Here $\rho_1$ is given by $$\begin{aligned}
\rho_1 &=&
\sum_a [
\kappa_a |\phi\rangle\langle\phi|N_a^{(1)\dagger}
+
N^{(1)}_a |\phi\rangle\langle\phi|\kappa_a^\ast]
\nonumber\\
&&
-
\sum_{\alpha} M_\alpha |\phi\rangle\langle\phi |M^\dagger_\alpha.
\label{3}\end{aligned}$$ For this output state $\rho_\epsilon$, we perturbatively solve the equation, $$\partial_\epsilon \rho_\epsilon = \frac{1}{2}(L_\epsilon\rho_\epsilon +\rho_\epsilon L_\epsilon),
\label{sld}$$ in order to get the SLD operator $L_\epsilon$. It is possible to check that the following solution actually satisfies [Eq. (\[sld\])]{} by substitution. $$L_\epsilon=\frac{1}{\epsilon}
\left[
{\bf 1} -|\phi\rangle \langle \phi|
\right]
-\rho_1 +O(\epsilon).\label{4}$$ By substituting [Eq. (\[4\])]{} into the definition of the Fisher information, we get the value of the information such that $$J_S [ \rho_\epsilon] ={\rm Tr}[\rho_\epsilon L_\epsilon^2]
=\frac{1}{\epsilon}\langle \phi|\rho_1 |\phi\rangle +O(\epsilon^0).$$ By using [Eq. (\[nic\])]{} and [Eq. (\[3\])]{}, the Fisher information is evaluated in the leading order of $\epsilon$ as $$J_S \left[ \rho_\epsilon \right]
=\frac{1}{\epsilon}\sum_\alpha
\left[
\langle\phi|M^\dagger_\alpha M_\alpha|\phi\rangle
-\left|
\langle \phi|M_\alpha |\phi\rangle
\right|^2
\right]+O(\epsilon^0) .$$
From Eq. (\[Optimality\]) the optimal output-measurement observable $A_{opt}$ for any input state ${|\psi\rangle}$ is given by $A_{opt}\rho_{\epsilon}
=(J_S \left[ \rho_\epsilon \right]^{-1}L_\epsilon+\epsilon)\rho_{\epsilon}$. The optimal input state $|\phi_{opt}\rangle$ can be determined by maximizing $J_S [\rho_\epsilon]$ with respect to the state $|\phi\rangle$.
Let us next discuss the low-noise channel in the ancilla-extended system $S+A$. Its extended channel is now given by $\Gamma_\epsilon \otimes id_A$. The input pure state $|\Psi\rangle$ can be decomposed into $$|\Psi\rangle = \sum_{n=1} C_n |n\rangle\otimes |A_n\rangle,$$ where $\{|n\rangle\}$ is an arbitrary orthonormal basis of $S$ and $|A_n\rangle$’s are normalized pure states of $A$, which are not necessarily orthogonal to each other. The constants $C_n$ should satisfy the normalization condition: $$\sum_n |C_n|^2 =1.$$ The SLD for the extended state is given by $$\begin{aligned}
\tilde{L}_\epsilon =\frac{1}{\epsilon} (1-|\Psi\rangle\langle\Psi|)
+O(\epsilon^0).\end{aligned}$$ The Fisher information $J_{S+A}[\tilde{\rho}_{\epsilon}]$ for the output state $\tilde{\rho}_{\epsilon}=\Gamma_{\epsilon}\otimes id_{A}[{|\Psi\rangle}{\langle\Psi|}]$ is also evaluated in a similar manner. We have $$J_{S+A}[\tilde{\rho}_{\epsilon}]=\frac{1}{\epsilon}
\sum_\alpha
\left[
{\rm Tr}[\tilde{\rho}M^\dagger_\alpha M_\alpha]
-\left|
{\rm Tr}\left[\tilde{\rho} M_\alpha \right]
\right|^2
\right]+O(\epsilon^0),$$ where $\tilde{\rho}$ is a state of $S$ defined by $$\begin{aligned}
\tilde{\rho} ={\rm Tr}_{A} [|\Psi\rangle\langle\Psi|]
=\sum_{n\bar{n}}
C^\ast_{\bar{n}} \langle A_{\bar{n}}|A_n\rangle C_n |n\rangle\langle \bar{n}|.\end{aligned}$$
The optimal output-measurement observable $\tilde{A}_{opt}$ for any input state ${|\Psi\rangle}$ is given by $\tilde{A}_{opt}\tilde{\rho}_{\epsilon}
=(J_{S+A}[\tilde{\rho}_{\epsilon}]^{-1}\tilde{L}_\epsilon
+\epsilon {\bf 1})\tilde{\rho}_{\epsilon}$. The optimal input state $|\Psi_{opt}\rangle$ for the extended system is determined by maximizing $J_{S+A}[\tilde{\rho}_{\epsilon}]$ with respect to $\tilde{\rho}={\rm Tr}_A [|\Psi\rangle\langle \Psi|]$.
If the dimension of $A$ is not less than that of $S$, we are able to make $|A_n\rangle$’s orthogonal to each other: $$\langle A_{\bar{n}}|A_{n}\rangle =\delta_{\bar{n} n}.$$ Then the state $\tilde{\rho}$ is reduced into a form such that $$\tilde{\rho} =\sum_n |C_n|^2 |n\rangle \langle n|.\label{5}$$ Note that the orthonormal basis $\{|n\rangle \}$ of $S$ and the coefficients $C_n$ can be arbitrarily chosen except that $\sum |C_n|^2 =1$. Hence, $\tilde{\rho}$ in [Eq. (\[5\])]{} is able to describe any possible state of $S$. Therefore, the dimension of the ancilla Hilbert space suffices to be at most the same as the system Hilbert space.
By combining both results of $J_S$ and $J_{S+A}$, we have the ancilla-assisted enhancement factor $\eta$ such that $$\eta
=\frac{\max\left[\sum_\alpha\left[
{\rm Tr}\left[\rho_S M^\dagger_\alpha M_\alpha \right]
-
\left|
{\rm Tr}\left[\rho_S M_\alpha \right]
\right|^2\right]
\right]_{\rho_S }
}
{\max\left[\sum_{\alpha}\left[
\langle\phi_S |M^\dagger_\alpha M_\alpha|\phi_S \rangle
-
\left|
\langle \phi_S |M_\alpha |\phi_S \rangle
\right|^2\right]
\right]_{|\phi_S \rangle}
}.\label{eta}$$ Here $\max[\ ]_{\rho_S}$ means the maximum value over all possible states of $S$ and $\max[\ ]_{|\phi_S\rangle}$ the maximum value over all possible pure states of $S$.
Because the set of pure states of $S$ is a subset of the set of states of $S$, the following inequality trivially holds: $$\eta \geq 1.$$
Examples of Low-Noise Channels
==============================
Low-noise channels introduced in the previous section are found in a lot of applications. Checking that low-noise channels really appear in some physical phenomena may lead to a deeper understanding. Thus we give two critical examples in this section. The details of the channels we introduce below can be seen in Ref. [@NC00].
Isotropic Depolarizing Channels
-------------------------------
An isotropic depolarizing channel is given by $$\Gamma_\epsilon [\rho]
=
\left( 1-\frac{3}{4}\epsilon\right) \rho
+\frac{1}{4} \epsilon \sum_{a=1}^3 \sigma_a \rho \sigma_a.
\label{idc}$$ This is a well known example induced by quantum noise. The parameter $\epsilon(\geq 0)$ is just a probability that the qubit system becomes depolarized. The Kraus operators in [Eq. (\[1\])]{} and [Eq. (\[2\])]{} are given by $$\begin{aligned}
B_0 (\epsilon) &=&\left(1-\frac{3}{4}\epsilon\right)^{1/2}{\bf 1}_S,\\
C_a (\epsilon) &=&\frac{1}{2}\sigma_a,\end{aligned}$$ where $a=1,2,3$. Hence the expansion coefficients in [Eq. (\[1\])]{} and [Eq. (\[2\])]{} are given by $$\begin{aligned}
\kappa_0& =&1,\\
N^{(1)}_0 &=&\frac{3}{8}{\bf 1}_S,\\
N^{(n)}_0
&=&\frac{(2n-3)!!}{n!}\left( \frac{3}{8} \right)^n {\bf 1}_S,\\
M_a &=&\frac{1}{2}\sigma_a ,\\
M_a^{(n)} &=&{\bf 0}.\end{aligned}$$ In this case, the Fisher informations have been already calculated [@Fuj01]. For the isolated original system $S$, the information is independent of the input state and given by $$J_S = \frac{1}{\epsilon(2-\epsilon)}.$$ For the extended channel $\Gamma_\epsilon \otimes id_A$, the optimal input state is the maximally entangled state and the information is given by $$\tilde{J}_{S+A} =\frac{3}{\epsilon(4-3\epsilon)},$$ as long as the parameter $\epsilon$ is small.
Generalized Amplitude-Damping Channels
--------------------------------------
A generalized amplitude-damping channel is given by $$\Gamma[\rho]
=\sum^2_{a=1} B_a (\epsilon) \rho B^\dagger_a (\epsilon)
+\epsilon\sum^2_{\alpha =1} C_\alpha (\epsilon) \rho
C^\dagger_\alpha (\epsilon),\label{gadc}$$ where $B_\nu$ and $C_\alpha$ are given by $$\begin{aligned}
&&
B_1 (\epsilon) =
\sqrt{\frac{1}{1+e^{-\beta E}}}
\left[
\begin{array}{cc}
1 & 0 \\
0 & \sqrt{1-\epsilon}
\end{array}
\right],\\
&&
B_2 (\epsilon) =
\sqrt{\frac{e^{-\beta E}}{1+e^{-\beta E}}}
\left[
\begin{array}{cc}
\sqrt{1-\epsilon} & 0 \\
0 & 1
\end{array}
\right],\\
&&
C_1 (\epsilon) =
\sqrt{\frac{1}{1+e^{-\beta E}}}
\left[
\begin{array}{cc}
0 & 1 \\
0 & 0
\end{array}
\right],\\
&&C_2 (\epsilon) =
\sqrt{\frac{e^{-\beta E}}{1+e^{-\beta E}}}
\left[
\begin{array}{cc}
0 & 0 \\
1 & 0
\end{array}
\right].\end{aligned}$$ The channel describes a relaxation process of the two-level system driven by a finite-temperature thermal bath. The temperature is $(k_B \beta)^{-1}$ where $k_B$ is the Boltzmann constant. Here the small noise parameter $\epsilon$ is related with the survival rate $s$ of the initial state under the relaxation such that $\epsilon =1-s$. The rate $s$ is given by $s=e^{-\gamma t}$, where t is time and $\gamma$ the relaxation rate constant. The corresponding coefficients in the $\epsilon$ expansion are given by $$\begin{aligned}
\kappa_1 &=&
\sqrt{\frac{1}{1+e^{-\beta E}}},\\
\kappa_2 &=&\sqrt{\frac{e^{-\beta E}}{1+e^{-\beta E}}},\\
N^{(n)}_1
&=&
\sqrt{\frac{1}{1+e^{-\beta E}}}\frac{(2n-3)!!}{2^n n!}
\left[
\begin{array}{cc}
0 & 0 \\
0 & 1
\end{array}
\right],\\
N^{(n)}_2
&=&
\sqrt{\frac{e^{-\beta E}}{1+e^{-\beta E}}}\frac{(2n-3)!!}{2^n n!}
\left[
\begin{array}{cc}
1 & 0 \\
0 & 0
\end{array}
\right],\\
M_1
&=&
\sqrt{\frac{1}{1+e^{-\beta E}}}
\left[
\begin{array}{cc}
0 & 1 \\
0 & 0
\end{array}
\right],\\
M_2
&=&
\sqrt{\frac{e^{-\beta E}}
{1+e^{-\beta E}}}
\left[
\begin{array}{cc}
0 & 0 \\
1 & 0
\end{array}
\right],\\
M_{1,2}^{(n)}
&=&{\bf 0}.\end{aligned}$$ The calculation of the Fisher information has been performed in Ref. [@Fuj04].
The two examples in this section will be discussed again in Section 6.
Channels on Two-Level Systems
=============================
In this section we concentrate on a two-level system $S_2$ and an arbitrary ancilla system $A$. The dimension of $A$ is not necessarily two, but assumed finite. Let us derive a universal bound on the ancilla-assisted enhancement factor $\eta$ such that $$\eta \leq \frac{3}{2}.$$ The bound must hold for all low-noise channels of $S_2$.
As well known, any state $\rho$ of the two-dimensional system $S_2$ can be written by $$\rho= \frac{1}{2}{\bf 1}_S
+\frac{1}{2} \vec{x} \cdot \vec{\sigma}$$ where $\vec{\sigma}$ is the Pauli matrix vector and the three-dimensional real parameter vector $\vec{x}$ takes values which satisfies $$0 \leq |\vec{x}|^2 \leq 1.$$ For pure states, the vector is normal: $$|\vec{x}|^2 =1.$$
Similarly, the matrix $M_{\alpha}$ in eqn(\[2\]) is uniquely expanded as $$M_\alpha =m_{a 0}{\bf 1}_S +\sum^3_{a=1} m_{a\alpha} \sigma^a. \label{9}$$ Now let us define complex vectors $\vec{\mu}_a$$(a=0\sim 3)$ by using the coefficients $m_{a\alpha}$ in [Eq. (\[9\])]{} as $$\vec{\mu}_a =(m_{a\alpha}).$$ In the vector space, there exists a natural inner product defined by $$(\vec{u},\vec{v})=\sum_{\alpha} u^\ast_\alpha v_\alpha.$$ A metric is also induced naturally from the inner product such that $$g_{ab}:=(\vec{\mu}_a,\vec{\mu}_b )=g_{ba}^\ast ,\label{gab}$$ where $a,b=1\sim 3$. For later convenience, define a real non-negative symmetric matrix $H$ by $$\begin{aligned}
H=[h_{ab}]=[\mbox{\rm Re } g_{ab}]\geq 0,\end{aligned}$$ and a real three-dimensional vector $\vec{J}$ by $$\begin{aligned}
\vec{J} =[J_a]=[\mbox{\rm Im } g_{23},\mbox{\rm Im } g_{31}, \mbox{\rm Im } g_{12}].\end{aligned}$$ Here denote by $h_1 ,h_2 ,h_3$ the eigenvalues of $H$. Without loss of generality, we can assume that $$0 \leq h_1 \leq h_2 \leq h_3. \label{h123}$$
Assume later that $\vec{\mu}_a (a=1,2,3)$ are linearly independent. Even if it is not so, because of the continuity of $\eta$, we can take three linearly-independent vectors $\vec{\mu}_a (t)$ parametrized by a real parameter $t$ such that $$\lim_{t\rightarrow 0} \vec{\mu}_a ( t)=\vec{\mu}_a.$$ In order to get $\eta$, we first calculate the factor $\eta (t)$ for $\{\vec{\mu}_a (t)\}$ and just take a limit as $$\lim_{t\rightarrow 0} \eta(t) =\eta.$$ Note that the linearly independence of $\{\vec{\mu}_a\}$ also means $$H >0.$$ This allows us to assume the existence of $H^{-1}$.
By a simple manipulation, we have $$\begin{aligned}
\eta(\vec{J})
=
\frac
{
{\rm Tr} H + \vec{J} H^{-1} \vec{J}
-\min
\left[
(\vec{x} +H^{-1}\vec{J}) H (\vec{x} +H^{-1} \vec{J})
\right]_{|\vec{x}| \leq 1}
}
{
{\rm Tr} H + \vec{J} H^{-1} \vec{J}
-\min
\left[
(\vec{x} +H^{-1}\vec{J}) H (\vec{x} +H^{-1} \vec{J})
\right]_{|\vec{x}| =1}
}.\label{eta2}\end{aligned}$$
For the original system $S$, the optimal input state is given by $$\begin{aligned}
|\phi\rangle\langle \phi|= \frac{1}{2}{\bf 1}_S
+\frac{1}{2}\vec{x}_{opt}\cdot
\vec{\sigma}\end{aligned}$$ where $\vec{x}_{opt}$ is the vector which minimizes $(\vec{x} +H^{-1} \vec{J})H(\vec{x} +H^{-1}\vec{J})$ among whole the unit vectors. The optimal input state $|\Psi\rangle$ for the extended system is also given as follows. Find a vector $\vec{X}$ which minimizes $(\vec{x} +H^{-1} \vec{J})H(\vec{x} +H^{-1}\vec{J})$ among whole the vectors with $|\vec{x}| \leq 1$. Then the optimal state $|\Psi\rangle$ is determined by solving the equation $$\begin{aligned}
Tr_A [|\Psi\rangle\langle \Psi|]
=
\frac{1}{2}{\bf 1}_S +\frac{1}{2}\vec{X}\cdot
\vec{\sigma}.\end{aligned}$$
Let us consider the case where $\vec{J} =\vec{0}$. The factor $\eta$ is given by $$\begin{aligned}
\eta
&=&
\frac
{
{\rm Tr} H
-\min\left[
\sum^3_{a,b=1} x^a x^b h_{ab}
\right]_{|\vec{x}| \leq 1}
}
{
{\rm Tr} H
-\min\left[
\sum^3_{a,b=1} x^a x^b h_{ab}
\right]_{|\vec{x}| = 1}
}\nonumber
\\
&=&
\frac
{
h_{1} +h_{2}+h_{3}
}
{
h_{1} +h_{2}+h_{3}
-\min\left[
\sum^3_{a=1} h_a ({x'}^a )^2
\right]_{|\vec{x}'| = 1}
}.\nonumber\end{aligned}$$ Here we have made $H$ diagonalized in the last equality. Consequently we obtain an expression of $\eta$ such that $$\eta
=\frac{h_{1} +h_{2} +h_{3}}
{h_{1} +h_{2} +h_{3}-
\min\left[h_{1} ,h_{2} ,h_{3}
\right]} .$$ Taking account of $h_1 \leq h_2 \leq h_3$, we can easily prove $\eta \leq 3/2$ as follows. $$\begin{aligned}
\eta
&=&
\frac{h_{1} +h_{2} +h_{3}}
{h_{2} +h_{3}}
\nonumber\\
&\leq&
\frac{2h_{2} +h_{3}}
{h_{2} +h_{3}}
\nonumber\\
&\leq&
\frac{3h_{3}}
{2h_{3}}
=\frac{3}{2}.\end{aligned}$$
Next let us discuss the case where $\vec{J}\neq \vec{0}$. Suppose that $|H^{-1} \vec{J}|\leq 1$. Then we have $$\begin{aligned}
\label{eq:100}
\min
\left[
(\vec{x} +H^{-1}\vec{J}) H (\vec{x} +H^{-1} \vec{J})
\right]_{|\vec{x}| \leq 1} = 0,\label{mh}\end{aligned}$$ because we can always take a vector $\vec{x}$ such that $\vec{x} =-H^{-1}\vec{J} $. For later convenience, let us introduce a function $G(\vec{J})$ as $$\begin{aligned}
G(\vec{J}):= \min
\left[
(\vec{x} +H^{-1}\vec{J}) H (\vec{x} +H^{-1} \vec{J})
\right]_{|\vec{x}| = 1}. \label{Gd}\end{aligned}$$ Then we can prove that the function $G$ satisfies $$\begin{aligned}
G(\vec{0}) \geq G(\vec{J}).\nonumber\end{aligned}$$ To show this, we transform $G(\vec{J})$ as $$\begin{aligned}
G(\vec{J})= \min
\left[
\vec{X} H \vec{X}
\right]_{|\vec{X} -H^{-1} \vec{J}| = 1}.\end{aligned}$$ By denoting $\vec{K}=H^{-1}\vec{J}$, the function $G$ is given in the diagonal basis of $H$ by $$G= \min
\left[
\sum_a h_a (X_a')^2
\right]_{|\vec{X}' -\vec{K}'| = 1}.$$ Note that the relation $|\vec{K}'|\leq 1$ trivially holds. Also notice from definition (\[Gd\]) that if $\vec{J}=\vec{K}'=\vec{0}$, $G$ takes the minimum value of the eigenvalues of $H$, that is, $h_1$: $$G(\vec{0})=h_1.$$ To compare $G(\vec{J})$ with this value $h_1$, suppose a point $\vec{X}_o'$ on a trajectory defined by $|\vec{X}' -\vec{K}'| = 1$ such that $$\vec{X}_o' =\left(K_1' \pm \sqrt{1-(K_2')^2 -(K_3')^2}, 0,0\right).$$ Then we have $$\vec{X}_o' H\vec{X}_o'
=h_1 \left(K_1' \pm \sqrt{1-(K_2')^2 -(K_3')^2}\right)^2.$$ Here we fix the double sign in the above equation so as to satisfy the relation: $$\vec{X}_o' H\vec{X}_o'
=h_1 \left(|K_1'| - \sqrt{1-(K_2')^2 -(K_3')^2}\right)^2.$$ Since the relation $|\vec{K}'|\leq 1$ holds, it is guaranteed that $$\begin{aligned}
1\geq \left(|K_1'| - \sqrt{1-(K_2')^2 -(K_3')^2}\right)^2.\end{aligned}$$ Therefore the important inequality $$G(\vec{0}) \geq G(\vec{J}) \label{G}$$ really arises as follows. $$\begin{aligned}
G(\vec{0}) &=&
h_1
\nonumber\\
&\geq&
h_1 \left(|K_1'| - \sqrt{1-(K_2')^2 -(K_3')^2}\right)^2
=\vec{X}_o' H\vec{X}_o'
\nonumber\\
&\geq& \min\left[
\vec{X'} H \vec{X'}
\right]_{|\vec{X'} -\vec{K'}| = 1} =G(\vec{J}).\end{aligned}$$ Note that $$\begin{aligned}
\vec{J}H^{-1}\vec{J} \geq 0 \label{JHJ}\end{aligned}$$ and $${\rm Tr} H -G(\vec{0})=h_2+h_3 >0. \label{h-g}$$ Keeping [Eq. (\[G\])]{}, [Eq. (\[JHJ\])]{} and [Eq. (\[h-g\])]{} in mind, let us go back to the proof of $\eta(\vec{J} ) \leq \eta (\vec{0})$. By using [Eq. (\[mh\])]{}, we have $$\begin{aligned}
\label{eq:113}
\eta(\vec{J})
=
\frac
{
{\rm Tr} H + \vec{J} H^{-1} \vec{J}
}
{
{\rm Tr} H + \vec{J} H^{-1} \vec{J}
-G(\vec{J})}.\end{aligned}$$ By replacing $G(\vec{J})$ by $G(\vec{0})$ in the above equality, from Eq. (\[G\]) we obtain $$\begin{aligned}
\eta(\vec{J})
&\leq&
\frac
{
{\rm Tr} H + \vec{J} H^{-1} \vec{J}
}
{
[{\rm Tr} H-G(\vec{0}) ]+ \vec{J} H^{-1} \vec{J}
}.\end{aligned}$$ By using an inequality such that $$\begin{aligned}
\frac{b+\epsilon}{a+\epsilon} \leq \frac{b}{a}\end{aligned}$$ for $a \leq b$ and $\epsilon \geq 0$ with $\epsilon =\vec{J}H^{-1}\vec{J}$, we have $$\begin{aligned}
\eta(\vec{J})
&\leq&
\frac
{
{\rm Tr} H
}
{
{\rm Tr} H-G(\vec{0})
}=\eta (\vec{0}).\end{aligned}$$ Consequently, we have obtained the bound $$\eta(\vec{J} ) \leq \eta (\vec{0})\leq 3/2.$$
For the remaining case where $|H^{-1}\vec{J}| >1$, the problem becomes much trivial. This is because $$\begin{aligned}
\lefteqn{
\min
\left[
(\vec{x} +H^{-1}\vec{J}) H (\vec{x} +H^{-1} \vec{J})
\right]_{|\vec{x}| \leq 1}}\quad\nonumber\\
& =&
\min
\left[
(\vec{x} +H^{-1}\vec{J}) H (\vec{x} +H^{-1} \vec{J})
\right]_{|\vec{x}| = 1}\end{aligned}$$ holds in this case. Therefore the relation $\eta =1$ is satisfied in [Eq. (\[eta2\])]{}.
Therefore, for all the possible low-noise channels, the bound $\eta \leq 3/2$ has been proven. The equality $\eta =3/2$ can be attained by the channels satisfying $$g_{ab}\propto \delta_{ab}$$ with the maximally-entangled input pure states of $S+A$.
The optimal input state depends on the vector $\vec{J}$ of the channel. When $\vec{J}=\vec{0}$, the optimal input state is the maximally entangled state. If $|H^{-1} \vec{J}| \geq 1$, a factorized input state takes the maximum and gives $\eta=1$. When $1> |H^{-1}\vec{J}| >0$, the optimal input state is neither the maximally entangled state nor the factorized state. From the argument below [Eq. (\[eq:100\])]{} the output state ${|\psi\rangle}_{S+A}$ satisfies $$\mbox{\rm Tr}_A[{|\psi\rangle}_{S+A}{\langle\psi|}_{S+A}]=\frac{1}{2}{\bf 1}_S
-\frac{1}{2}\vec{J}H^{-1}\vec{\sigma}.$$ The value of $\eta$ given by [Eq. (\[eq:113\])]{} also changes continuously between $1\leq \eta <3/2$ depending on $|H^{-1}\vec{J}|$.
The channel dependence of the optimal input state has been already noticed in a generalized amplitude-damping channel [@Fuj04] by changing the temperature of the thermal bath. Because of the simplicity of the model, it is possible to estimate the unknown parameter even in a finite parameter region. On the other hand, in this paper, the parameter region of the low-noise channel is constrained to a neighborhood of a fixed value ($\epsilon =0$). However, we would like to stress that our channel includes an enormous number of degrees of freedom corresponding to $\kappa_a$, $N^{(n)}_a$ and $M^{(n)}_\alpha$, compared with the generalized amplitude-damping channel.
Note that the isotropic depolarizing channel ([Eq. (\[idc\])]{} in Section 5) is one of the channels attaining the bound ($\eta =3/2$). The vectors $\vec{\mu}_a$ are calculated as $$\begin{aligned}
&&
\vec{\mu}_1
=\frac{1}{2}
\left[
\begin{array}{c}
1 \\
0 \\
0
\end{array}
\right],\\
&&
\vec{\mu}_2
=\frac{1}{2}
\left[
\begin{array}{c}
0 \\
1 \\
0
\end{array}
\right],\\
&&
\vec{\mu}_3
=\frac{1}{2}
\left[
\begin{array}{c}
0 \\
0 \\
1
\end{array}
\right].\end{aligned}$$ The corresponding matrix $g_{ab}$ is just evaluated as $$g_{ab} =\frac{1}{4}\delta_{ab}.$$ Thus the channel can achieve $\eta =3/2$.
On the other hand, the generalized amplitude-damping channels [Eq. (\[gadc\])]{} in Section 5) cannot achieve the bound. The vectors $\vec{\mu}_a$ are now described by $$\begin{aligned}
&&
\vec{\mu}_1
=\frac{1}{2}
\left[
\begin{array}{c}
\sqrt{\frac{1}{1+e^{-\beta E}}} \\
\sqrt{\frac{e^{-\beta E}}{1+e^{-\beta E}}} \\
0
\end{array}
\right],\\
&&
\vec{\mu}_2
=\frac{i}{2}
\left[
\begin{array}{c}
\sqrt{\frac{1}{1+e^{-\beta E}}} \\
-\sqrt{\frac{e^{-\beta E}}{1+e^{-\beta E}}} \\
0
\end{array}
\right],\\
&&
\vec{\mu}_3
=
\left[
\begin{array}{c}
0 \\
0 \\
0
\end{array}
\right].\end{aligned}$$ The corresponding $g_{ab}$ is now given by $$\begin{aligned}
[g_{ab}]
=
\frac{1}{4}
\left[
\begin{array}{ccc}
1 & i\frac{1-e^{-\beta E}}{1+e^{-\beta E}} &0 \\
- i\frac{1-e^{-\beta E}}{1+e^{-\beta E}} & 1 & 0\\
0 &0&0
\end{array}
\right].\end{aligned}$$ Because $g_{ab} \propto \delta_{ab}$ does hold, the channel cannot satisfy $\eta =3/2$ for any parameter value. In spite of the ancilla extension, the ancilla-assisted enhancement does not appear at all ($\eta =1$), as long as the low-noise parameter $\epsilon$ is small enough. This is because the value of $|H^{-1}\vec{J}|$ diverges and the relation $|H^{-1}\vec{J}| >1$ always holds.
The authors thank Akio Fujiwara and Gen Kimura for useful comments and discussions. This work was supported by the SCOPE project of the MPHPT of Japan and by the Grant-in-Aid for Scientific Research of the JSPS.
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'V. Renvoizé, I. Baraffe, U. Kolb'
- 'H. Ritter'
date: 'Received /Accepted'
title: 'Distortion of secondaries in semi-detached binaries and the cataclysmic variable period minimum'
---
Introduction
============
The description of close binary systems is usually based on the Roche model which defines the shape of a binary component distorted by tidal and rotational forces. In the framework of the Roche model one assumes that the binary components (the primary and the secondary) either are point masses, or are corotating and have a spherically symmetric mass distribution irrespective of their proximity or mass ratio (Kopal 1959, 1978). In a semi–detached system, one of the components fills its critical equipotential lobe defined by the potential of the inner Lagrangian point, and which determines the maximum extent of a star in a close binary. This is the so–called Roche lobe within the Roche model. Cataclysmic variables (hereafter CVs), composed of a white dwarf as the primary and a low–mass star or a brown dwarf as the secondary, belong to this type of systems: the secondary fills its critical lobe and transfers mass towards the primary. When applying the Roche model to problems of binary evolution one makes implicitly the following assumptions (among others). First, the Roche potential is a good approximation of the true potential that one would obtain by solving the Poisson equation. Second, the effects of tidal and rotational forces on the internal structure of the star are negligible, i.e. that they result in only small corrections compared to stellar models assuming spherical geometry. Third, that for the purpose of evolutionary computations involving one–dimensional stellar models the lobe–filling star may be replaced by a spherical star of the same volume. This is tantamount to assuming that though tidal and rotational forces change the shape of a star they leave its volume invariant. The radius of the lobe–filling star then only depends on the geometry of the system, and can be calculated by means of simple analytical fits (Paczyński 1971, Eggleton 1983). The main purpose of the present paper is to examine in some detail the third and to some extent also the second of the above assumptions, both of which have so far not been tested.
Recent 3D simulations (Rezzolla et al. 2001; Motl & Frank, priv. comm.) confirm that, at least in the case of a semi-detached system, the Roche potential is a good approximation if the lobe-filling star is sufficiently centrally condensed, i.e. if the effective polytropic index is $N \simgr$ 3/2. The analysis of Rezzolla et al. (2001) is based on numerical models of semi-detached binaries that account for the finite size of the secondary star, thus relaxing the first assumption inherent in the Roche model. With the validity of this approximation for the determination of the potential, they also show that such effects hardly affect gravitational quadrupole radiation. Moreover, a comparison between the angular momentum loss and mass-transfer timescales predicted by the Roche model and their numerical models shows small differences. They thus conclude that finite size effects cannot account for the mismatch between the observed minimum period $\pmin$ at 80 min of CV systems and the theoretical value $\pturn$. The latter is indeed $\sim$ 15% shorter than the observed value, according to recent calculations based on improved stellar physics (see Kolb & Baraffe 1999). Since Rezzolla et al. (2001) do not consider thermal relaxation effects in their calculations, they can only determine a differential correction to $\pturn$ when going from Roche model to self-consistent potential. However, in doing so, the second and third assumptions mentioned above remain untested. The main purpose of our paper is thus to explore the consequences of making these two assumptions. Our main goal is to determine quantitatively the departure from spherical symmetry of the secondary in semi-detached binaries and to analyse the consequences on the mass transfer rates and orbital period in CV systems. We use smoothed particle hydrodynamics (SPH) techniques to study equilibrium configurations of semi-detached binaries and estimate for different mass ratios the geometrical deformation of the secondary as it fills its critical lobe. The numerical models and results are described in §2. In §3, we analyse some of the consequences of the tidal and rotational forces on the secular evolution of the low mass donor on grounds of models constructed by Kolb & Baraffe (1999) and Baraffe & Kolb (2000). We focus on the problem of the minimum period and the discrepancy between observations and models (see e.g King 1988 and Kolb 2001 for a review on the properties of CV systems). A discussion and conclusions follow in §4.
Numerical models of semi-detached binary systems
================================================
The method
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We used a SPH code, originally developed and kindly provided by Willy Benz (see details in Benz et al. 1990) to perform numerical simulations of a close binary system composed of a point mass (primary) and a polytropic star (secondary). The SPH method has been described extensively in the literature (see Monaghan 1992 and references therein) and is often applied to the study of close binary systems (e.g Benz et al. 1990; Lai et al. 1994; Rasio & Shapiro 1995; Segrétain et al. 1997) In all our simulations, the primary is a 1 $\msol$ point-like mass. The secondary is described by a polytropic equation of state $p = K \rho ^{1+1/N}$, where $p$ is the pressure and $\rho$ the density. The polytropic constant $K$ is fixed for a given index $N$ by the mass $M$ and radius $R$ of the spherical secondary. We adopt two polytropic indices, i.e. $N$ = 3/2, which provides a good description of fully convective objects such as low mass stars, and $N$ = 3 characteristic of solar type stars with $M \sim 1 \msol$. For the particular case of CVs, systems below the period gap are well described by $N$ = 3/2 polytropes, whereas $N$ = 3 applies to systems with periods $>$ 6 h and typical masses around $\sim$ 1 $\msol$. The two values of $N$ thus represent limiting cases for the description of CV secondaries. Assuming that $K$ remains constant in space applies well to fully convective objects with a fully adiabatic structure, and implies a chemically homogeneous structure (constant molecular weight) for the standard models with $N$ = 3. This is a reasonable approximation for the present study.
The simulations use $\sim$ 15000 particles. In order to check the accuracy of our results, we ran a limited number of simulations with 57000 particles. We find that 15000 particles is a good compromise between computational demand and accuracy. The particles are initially uniformly distributed on a hexagonal close-packed lattice. The initial number density of particles is constant throughout the volume of the sphere describing the initial configuration of the secondary. The particle masses are proportional to the local mass density. This provides a good spatial resolution near the stellar surface, which is crucial for our problem of critical lobe determination where surface effects are predominant. The simulations are performed in a corotating reference frame with the origin at the center of mass of the system. The initial separation $A_{\rm init}$ of the two components is arbitrarily fixed at four times the separation required for the secondary to fill its Roche-lobe $A_{\rm Roche}$, estimated from the Eggleton (1983) fit. For such a separation, tidal and rotational effects on the secondary are negligible. The orbital separation is decreased with the arbitrary constant rate $(A_{\rm init} - A_{\rm Roche})/\tau_{\rm simu}$, so that the total timescale of the simulation $\tau_{\rm simu}$ is $\sim$ 1000 times the typical hydrodynamical relaxation time $\tau_{\rm relax} \simeq \left(\frac{R^3}{GM}\right)^{1/2}$ of the secondary. The simulation is stopped when the secondary fills its critical lobe [*i.e.*]{} when the first particles from the secondary reach the inner saddle point of the potential [^1]. This marks the onset of mass transfer. Once the critical separation is reached, we check that the model has reached an equilibrium configuration, starting from such critical separation and letting it relax in a non rotating reference frame.
Our goal is to estimate the deformation effects on the secondary due to tidal and rotational forces as it fills its critical lobe. The deformation can be measured in terms of the ratio of the final to initial stellar radius $D = R_{\rm f}/R_{\rm i}$. $R_{\rm i}$ is the radius of the unperturbed spherical polytrope. $R_{\rm f}$ is an effective radius defined as the radius of the sphere with the volume $V_{\rm f}$ of the secondary filling its critical lobe. $V_{\rm f}$ is provided by our SPH simulation at the onset of mass transfer. The method to estimate $V_{\rm f}$ is described in Appendix A.
Results
-------
We ran a grid of simulations for various mass ratios $q=M_2/M_1$ between the secondary and the primary. Typically, CV systems with periods from $\sim$ 10 h down to the minimum period cover a range of $q$ between 1 and 0.06. Fig. \[fig1\] displays the final configuration of a $N$ = 3 polytrope with mass ratio $q$ = 0.8. This illustrates the case of CV systems with periods $\simgr$ 6 h (see Baraffe & Kolb 2000). Fig. \[fig2\] shows the results for the case $N$ = 3/2 and $q$ = 0.07, characteristic of secondaries approaching the period bounce $\pturn$ (Kolb & Baraffe 1999). We note that in the case $N$ = 3/2 (Fig. \[fig2\], lower panel), the surface value of $\phi$ is not constant. We did not find any satisfactory explanation for such behavior. This feature has already been noted in some cases by Rasio and Shapiro (1995) and interpreted in terms of number density of SPH particles being not exactly constant around the surface of a star with large tidal deformation. Increasing the number of particles from 15000 to 57000 and double-checking that the models have reached an equilibrium configuration do not solve the problem. We do not expect that this affects the accuracy of our final results, since our deformation calculations are in excellent agreement with similar calculations by other authors (see below).
The resulting deformations $D$ as a function of $q$ are summarized in Table 1 for $N$ = 3 and $N$ = 3/2. As expected, tidal and rotational distortion yields an expansion of the secondary’s volume with respect to the unperturbed spherical configuration. In terms of effective radius, the expansion is typically 11% for $N$ = 3 and 5% for $N$ = 3/2. The dependence of $D$ on $N$ can be understood in terms of the compressibility $\chi \, = \,
\partial \log \rho/\partial p$ = $N/(1 + N)/p$, which is larger for $N$ = 3 than for $N$ = 3/2. The larger the compressibility, the larger the deformation.
$q$ $D_3$ $D_{3/2}$
------ ------- -----------
0.06 - 1.06
0.07 - 1.06
0.1 - 1.05
0.2 1.12 1.05
0.3 1.12 1.05
0.4 1.12 1.04
0.5 1.12 1.04
0.6 1.11 1.04
0.7 1.11 1.04
0.8 1.11 1.04
0.9 1.11 1.04
1.0 1.10 1.04
: Deformation $D = R_{\rm f}/R_{\rm i}$ as a function of mass ratio $q \, = \, M_{\rm donor}/M_{\rm accretor}$ of the secondary in a semi-detached binary for polytropic indices $N=3$ ($D_3$) and $N=3/2$ ($D_{3/2}$)[]{data-label="tab1"}
In order to visualize the deformation of the secondary compared to the spherical case, Fig. \[fig3\] displays lines of constant density for $N$ = 3 and $N$ = 3/2. An inspection of Fig. \[fig3\] shows that the largest departure from spherical symmetry is observed in the outermost layers of the polytropic star, whereas the central regions are only slightly affected. The results displayed in Table 1 are in excellent agreement with the work of Uryu & Eriguchi (1999), based on a different numerical method. Indeed, for $N$ = 3/2 they found distortion effects of $\sim$ 4% for 0.1 $\le \, q \, \le $ 1. They however did not analyse the case $N$ = 3. A comparison of the numerical ratio $R_{\rm f}/A$, where $A$ is the orbital separation, and the ratio given within the Roche model according to Eggleton (1983) shows small differences (less than 2%), in agreement with the results of Rezzolla et al. (2001) and confirming indeed that the Roche potential is a good approximation in the present case (the so-called first assumption, see §1).
Application to cataclysmic variable evolutionary models
=======================================================
In order to analyse the consequences of the distortion effects found in the previous section on period and mass transfer rate in CV systems, we follow the secular evolution of the secondary using the same models and input physics as described in Kolb & Baraffe (1999) and Baraffe & Kolb (2000). We focus on systems below the 2-3 h period gap and specifically on the minimum period discrepancy between observations and models. Although distortion effects seem to be more important for systems above the period gap ($P > 3$ h) (see Tab. \[tab1\]), their consequences are difficult to quantify given the large uncertainties of evolutionary models describing such systems, such as the magnetic braking law and the resulting mass transfer rate, the evolutionary stage of the secondary at onset of mass transfer or the mixing length parameter. Below the period gap, such uncertainties are fortunately considerably reduced (see Baraffe & Kolb 2000; Kolb et al. 2001 for details).
Geometrical effects
-------------------
In the following, we only consider the effects of distortion on the geometry of the system. The orbital properties, e.g. the orbital period and separation, and the mass transfer rate will be indeed affected by the larger effective radius of the donor, estimated in §2, compared to the undistorted case. However, for the moment, we ignore the rotational and tidal effects on the thermal structure of the star, assuming that its inner structure is unaffected and determined by the unperturbed stellar structure equations in spherical symmetry. A rough estimate of the thermal effects on the secondary’s properties resulting from its expansion is derived in the next section (§3.2).
We analyse an evolutionary sequence with an initial donor mass of 0.21 $\msol$, a primary mass of 0.6 $\msol$, and gravitational radiation (GR) as angular momentum loss mechanism (see Kolb & Baraffe 1999). From the radius $R_2$ obtained from integration of the standard stellar structure equations, and the mass ratio $q$, which varies along the sequence of evolution, the effective radius is derived according to Tab. \[tab1\]. The mass transfer rate is then calculated as a function of the difference between effective donor radius and Roche radius, following Ritter (1988).
The comparison between sequences without distortion (solid line) and with distortion (dashed line) is shown in the orbital period - effective temperature diagram (Fig. \[fig4\]). Although reducing the discrepancy with the observed $P_{\rm min}$, distortion effects provide an increase of the minimum period $P_{\rm turn}$ of only $\sim$ 6% (or $\sim$ 4-5 min), compared to the undistorted case. This is slightly less than what is naively expected from the period - radius relation $P \propto (R_2^3/M_2)^{1/2}$. An increase of the radius by $\sim$ 6%, as expected from distortion effects near the minimum period (see Tab. \[tab1\]), should indeed yield $\sim$ 9% increase of $P$. The smaller effect found on $P$ stems from the dependence of angular momentum loss driven by GR on the secondary radius $\dot J_{\rm GR}/J\, \propto \, P^{-8/3} \, \propto \, R_2^{-4}$. Consequently, the larger radius in the distorted sequence implies a decrease of $\dot J_{\rm GR}$, and thus a smaller mass transfer rate $-\dot M_2$. As shown below $P_{\rm turn}$ depends on the ratio $\tau=t_{\rm KH}/t_M$ of the secondary’s Kelvin–Helmholtz time $t_{\rm KH}$ and the mass transfer time $t_M =
M_2/(-\dot M_2)$. The decrease of $-\dot M_2$ thus yields a decrease of $\tau$, implying less departure from thermal equilibrium and thus a smaller $P_{\rm turn}$. Because $\dot J_{\rm GR}$ depends explicitly on the mass of the primary, $P_{\rm turn}$ does also depend on it, but only weakly, as shown by Paczyński & Sienkiewicz (1983), who found that $\partial \ln P_{\rm turn}/\partial \ln M_1 \approx
0.09$. In fact our computations show that $P_{\rm turn}$ varies from 71 min for $M_1$ = 0.6 $\msol$ to 74 min for $M_1$ = 1.2 $\msol$, when distortion effects are included.
Thermal relaxation
------------------
In the numerical computations discussed in the previous section, we have not taken into account changes in the thermal reaction of the secondary which must result from its inflation due to tidal and rotational forces. A fully consistent treatment of the distortion effects would imply solving the multi-dimensional stellar structure equations. Rather than doing this we shall in the following derive a rough estimate of the thermal relaxation effects and explore their consequences for the minimum period of CV systems. We indeed expect that the changed surface area of the more distended secondary, as a result of the distorsion effects, will affect its surface luminosity, and thus its thermal properties.
We denote by $R_{2,0}, L_{2,0}, T_{\rm eff, 0}, \dots$ the quantities of the donor star which result from assuming a pure $1/r$ potential, and by $R_2, L_2, T_{\rm eff}, \dots$ the corresponding quantities of the spherical equivalent of the critical lobe–filling star. Obviously we have $$\label{tr1}
R_2 = R_{2,0} \, D_{\rm N} \,,$$ where $D_{\rm N}$ is the deformation factor for a polytropic index $N$ determined in §2.2 (see Table 1). We thus have for the orbital separation $A$ and the orbital period $P$ $$\label{tr2}
A = A_0 \, D_{\rm N} \,,$$ and $$\label{tr3}
P =P_0 \, {D_{\rm N}}^{3/2} \,.$$ Since mass transfer below the period gap is assumed to be driven by loss of angular momentum via gravitational radiation alone, the mass transfer rate can be written as (see e.g. Ritter 1996) $$\label{tr4}
-\dot M_2 = \frac{M_2}{\zeta_{\rm eff} - \zeta_{\rm CL}} \, \left(-2\,
\frac{\dot J_{\rm GR}}{J}\right) \,.$$ Here $\zeta_{\rm eff} = d\ln R_2/d \ln M_2$ is the effective mass radius exponent of the donor star, and $\zeta_{\rm CL}$ the mass radius exponent of the volume–equivalent critical lobe radius[^2]. Because $$\label{tr5}
\frac{\dot J_{\rm GR}}{J} \propto a^{-4}$$ we have $$\label{tr6}
-\dot M_2 = -\dot M_{2,0} \, {D_{\rm N}}^{-4}\,,$$ assuming that $\zeta_{\rm eff}$ is the same.
Because of the larger radius of the critical lobe–filling star, its surface is larger, thereby affecting its luminosity, effective temperature and Kelvin–Helmholtz time $$\label{tr7}
t_{\rm KH} = \frac{G {M_2}^2}{R_2 L_2}
= \frac{G {M_2}^2}{4 \pi \sigma {R_2}^3 {T_{\rm eff}}^4}.$$ Consequently, its thermal relaxation, i.e. its gravo–thermal luminosity $L_{\rm g}$ will also be different.
In order to estimate the change of $t_{\rm KH}$, we need to determine the change in $T_{\rm eff}$ with radius $R_2$. Since donor stars below the period gap are fully convective, we can apply the theory of the Hayashi–line, as described by Kippenhahn and Weigert (1990). Accordingly we get $$\begin{aligned}
\label{tr8}
\log T_{\rm eff} & = & \frac{3a-1}{5a+2b+5}\, \log R_2 \nonumber \\
& & + \frac{a+3}{5a+2b+5}\, \log M_2 + const. \end{aligned}$$ Here $a=(\partial \log \kappa/\partial \log p)_T$ and $b=(\partial \log \kappa/\partial \log T)_p$, where $\kappa$ is the photospheric opacity. Typically, for very low–mass stars with atmospheric opacities dominated by molecular absorption, we have $a \approx 1$ and $b \approx 0$. Therefore $$\label{tr9}
\left( \frac{ \partial \log T_{\rm eff}}{\partial \log R_2}\right)_{\rm M}
\approx \frac{1}{5}\, .$$
We can now use (\[tr1\]) together with (\[tr9\]) in (\[tr7\]) and obtain $$\label{tr10}
t_{\rm KH} = t_{\rm KH,0} \, {D_{\rm N}}^{-2.8} \,.$$ We note that in deriving (\[tr10\]) we have applied in (\[tr7\]) the factor $D_{\rm N}$ only to the $R$–dependence coming from the luminosity $L_2$. Indeed, the $R_2$ in the denominator of (\[tr7\]) comes from the gravitational binding energy of the star which is related via the Virial theorem to its thermal energy, i.e. essentially to its central temperature. Since the central regions of the star are expected to be hardly affected by the distortion of the outer layers (cf. Fig. 3), it is not appropriate to also propagate the factor $D_{\rm N}$ to the remaining factor $R_2$.
Let us now examine the conditions at the period $P_{\rm turn}$. For a polytope of index $N$ losing mass the effective mass radius exponent can be written as (e.g. Ritter 1996) $$\label{tr11}
\zeta_{\rm eff} = \zeta_{\rm ad} + \frac{5-N}{3-N}\,
\frac{t_{\rm M_2}}{t_{\rm KH}}\,
\frac{L_{\rm g}}{L}\, ,$$ where $$\label{tr12}
\zeta_{\rm ad}= \left( \frac{\partial \log R}{\partial \log M}\right)_{\rm K}
= \frac{1 - N}{3 - N}$$ is the adiabatic mass radius exponent and $$\label{tr13}
t_{\rm M_2} = -\frac{M_2}{\dot M_2}$$ is the mass loss time scale. Because at $P = P_{\rm turn}$ the donor star is characterized by $N=3/2$ and $\zeta_{\rm eff}=+1/3$ [^3] we obtain from (\[tr11\]) and (\[tr12\]) $$\label{tr14}
\frac{t_{\rm M_2}}{t_{\rm KH}} \, \frac{L_{\rm g}}{L} =
\frac{2}{7}$$ at $P=P_{\rm turn}$, or $$\label{tr15}
\frac{L_{\rm g}}{L}=\frac{2}{7} \,\frac{t_{\rm KH}}{t_{\rm M_2}}
\approx \left(\frac{2}{7} \,\frac{t_{\rm KH}}
{t_{\rm M_2}}\right)_0 \,
{D_{3/2}}^{-6.8}\,.$$ With $D_{3/2} \approx 1.06$ (cf. Table 1) (\[tr15\]) yields $$\label{tr16}
\frac{L_{\rm g}}{L} \approx 0.67 \,
\left(\frac{L_{\rm g}}{L}\right)_0\, .$$ Eq. (\[tr16\]) means that the 3D–effects, i.e. the reduced mass loss rate and the increased surface luminosity of the more distended star, result in a smaller deviation from thermal equilibrium at $P=P_{\rm turn}$. Hence the star is systematically less inflated by the effects of thermal disequilibrium, and this, in turn, compensates at least partially for the systematic increase of the orbital period due to the factor $D_{3/2} > 1$.
A quantitative estimate of the decrease of $\pturn$ suggested from (\[tr16\]) can be derived by recomputing the evolutionary sequences including the effect of distortion, as done in §3.1, and by artificially increasing the radiating surface of the donor in the Stefan–Boltzmann law by a factor ${D_{3/2}}^{2.8}$, as suggested from (10). Note that this is equivalent to increasing the radiating surface by a factor ${D_{3/2}}^2$, and to reducing the surface gravity in the integration of the stellar atmosphere by the same factor. The result of such a numerical experiment is displayed in Fig. 4 (dash-dotted line) and shows a slight decrease of $P_{\rm turn}$ by $\sim$ 2-3% compared to the case with pure geometrical effects (dashed line). These results fully confirm the expectation derived from (16), namely that the value of $P_{\rm turn}$ is reduced by taking into account the effects of the changed thermal relaxation.
Discussion and conclusions
==========================
Although reducing the discrepancy between observed and predicted minimum period, distortion effects seem insufficient to provide a satisfactory solution of the mismatch between calculated and observed minimum period. A combination of distortion effects as estimated in §2 and an angular momentum loss rate of 2–2.5 $\times \dot J_{\rm GR}$ can reconcile $P_{\rm turn}$ with the observed 80 min value (see dotted line in Fig. \[fig4\]). Note that without distortion effects, one would need 4 $\times \dot J_{\rm GR}$ to reach $\sim 80$ min, as estimated in Kolb & Baraffe (1999). The more modest increase of $\dot J$ required according to our calculations is also in better agreement with Patterson’s (1998) estimate based on space density considerations. Additional physical processes can also result in an inflation of the secondary, e.g. irradiation from the primary (Ritter et al. 2000) or star spots (Spruit & Ritter 1983). A rough estimate of irradiation effects or star spots can be derived by following Ritter et al. (2000), i.e. by reducing the effective radiating surface of the star by a factor $(1-s_{\rm eff}$). Adopting in our secular evolution calculation a factor $s_{\rm eff} = 1/2$ and $\dot J =\dot J_{\rm GR}$ yields a sequence very similar to the one obtained with deformation (dashed line in Fig. \[fig4\]). If $s_{\rm eff} = 2/3$ the result resembles the sequence with deformation and $\dot J = 2.5 \times \dot J_{\rm GR}$ (dotted line in Fig. \[fig4\]). In order to know whether such values of $s_{\rm eff}$, yielding effects comparable to the distortion effects, are reasonable requires a sophisticated treatment of star spots or irradiation. An investigation of irradiation effects on non-gray stellar atmospheres is in progress (Barman 2001). We stress however that even if distortion, irradiation, star spots or additional sources of $\dot J$ are possible solutions for removing the mismatch between observed and predicted minimum period, the so-called period spike problem still remains. A period spike which is a consequence of the accumulation of systems near $P_{\rm turn}$ (where $\dot P = 0$) is indeed predicted by all models for which $\dot J$ or $s_{\rm
eff}$ are assumed to be the same for all systems. Even if they are not, it is very difficult to “smear out” the period spike in a population of systems with different individual bounce periods (Barker & Kolb, in preparation). Such a period spike is, however, not observed (see Kolb & Baraffe 1999).
Finally, we note that Kolb & Baraffe (1999) obtained negligible effects on the secondary structure and evolution when applying tidal and rotational corrections to the 1D stellar structure equations, on the basis of the scheme by Chan & Chau (1979). The effect on the total radius in Kolb & Baraffe (1999) is much smaller ($< 2$%) than that found from the present SPH simulations. Since two different numerical methods, on the one hand the work by Uryu & Eriguchi (1999) and on the other hand the present work, predict the same quantitative deformation effects, we are confident that the results of our SPH simulation are accurate. Although in the SPH simulations we do not take into account the thermal reaction of the star to its inflation and deal with polytropes, on the basis of our simple estimate given in §3.2 we do not expect the thermal effects to significantly reduce the radius of the deformed star. A possible reason for the discrepancy between the calculations by Chan & Chau (1979) and the present results could be the limitation of the former 1D scheme to describe multi-dimensional effects. Figure 3 shows strong effects in the outermost layers which may be difficult to account for with such a scheme. In any case, both approaches have their shortcomings, but they both provide the same conclusion regarding the mismatch of the observed and predicted minimum period. To conclude, our SPH simulations suggest that tidal and rotational distortion effects on the secondary in semi-detached binaries may not be negligible, and may reach observable levels of $\sim$ 10% on the radius for specific cases of polytropic index and mass ratio. Although this effect yields an increase of the predicted minimum period for CV systems, it remains too small to explain the observed value of 80 min. Additional effects such as irradiation, star spots or extra sources of angular momentum loss still seem to be required, leaving the problem of the minimum period of CV systems unsettled.
We thank W. Benz and H-C. Thomas for valuable discussions. I.B thanks the Max-Planck Institut for Astrophysik in Garching for hospitality during elaboration of this work. The calculations were performed using facilities at Centre d’Etudes Nucléaires de Grenoble.
Baraffe, I., Kolb, U. 2000, , 318, 354 Barman, T.S. 2001, in The Physics of Cataclysmic Variables and Related Objects, eds. B. Gänsicke et al., ASP Conf. Series, in press Benz, W., Bowers R. L., Cameron A. G. W., Press W. H. 1990, , 348, 647 Chan, K.L., Chau, W.Y. 1979, , 233, 950 Eggleton, P.P. 1983, , 268, 368 King A.R. 1988, QJRAS, 29, 1 Kippenhahn, R., Weigert, A. 1990, [*Stellar Structure and Evolution*]{}, ed. M. Harvit, R. Kippenhahn, V. Trimble and J.P Zahn Kolb, U. 2001, in The Physics of Cataclysmic Variables and Related Objects, eds. B. Gänsicke et al., ASP Conf. Series, in press (astro-ph/0110253) Kolb, U., Baraffe, I. 1999, , 309, 1034 Kolb, U., King, A.R., Baraffe, I. 2001, MNRAS, 321, 544 Kopal, Z. 1959, in Close Binary Systems (New York: Wiley) Kopal, Z. 1978, in Dynamics of Close Binary Systems (Dordrecht: Reidel) Lai, D., Rasio, F.A., Shapiro, S.L. 1994, , 423, 244 Monaghan, J.J. 1992, , 30, 543 Paczyński, B. 1971, , 9, 183 Paczyński, B., Sienkiewicz, R. 1983, , 268, 825 Patterson, J. 1998, PASP, 110, 1132 Press, W. H., Teukolsky S.A., Vetterling W.T., Flannery B.P., 1992, Numerical Recipes in fortran, Cambridge Univ. Press Rasio, F.A., Shapiro, S.L. 1995, , 438, 887 Rezzolla, L., Uryu, K., Yosida, S. 2001, , in press (gr-qc/0107019) Ritter, H. 1988, , 202, 93 Ritter, H. 1996, in Evolutionary Processes in Binary Stars, Proceedings of the NATO Advanced Study Institute, Cambridge, Eds. R. Wijers and M. Davies, Vol. 477, 223 Ritter, H., Zhang, Z.Y., Kolb, U. 2000, , 360, 969 Spruit, H.C., Ritter, H., 1983, , 124, 267 Uryu, K., Eriguchi, Y. 1999, , 303, 329 Segrétain, L., Chabrier, G., Mochkovitch, R. 1997, , 481, 355
Determination of the volume in the SPH calculation
==================================================
To estimate the final volume of the secondary at the onset of mass transfer in our SPH simulations, we proceed as follows. We first determine the smallest rectangular box containing the secondary star. This box with volume $V_{\rm box}$ is then filled with $N_{\rm tot}$ points following a Sobol sequence of pseudo-random numbers. Such a sequence is self-avoiding, [*i.e.*]{} the points are spread out randomly but in a uniform way (see Press et al. 1992), allowing a more uniform filling of a volume than a standard random method. Each point within twice the smoothing length of a particle is counted, otherwise it is discarded. The number of points $N_{\rm effective}$ fulfilling such condition determines the volume $V_{\rm star}$ of the star:
$$V_{\rm star} =\frac{N_{\rm effective}}{N_{\rm tot}} \times V_{\rm box}.$$
The radius of the star $R_{\rm star}$ is then defined as the radius of the sphere of same volume:
$$V_{\rm star}=\frac{4}{3} \pi R_{\rm star}^3$$
This method has been tested on spherical and elliptical structures of known volumes. According to these tests, it provides the radius within the size of one particle $R_{\rm part}$. Typically, for $N_{part}=15000$, we have
$$R_{\rm part} \simeq \frac{R_{\rm star}}{N_{\rm part}^{1/3}} \simeq
\frac{R_{\rm star}}{24}.$$
Our method thus determines the radius of a star within a systematic error of 4%. Although rough for a precise determination of a stellar radius, the uncertainty is much smaller on the [*ratio*]{} of the radius of the same star at two different times of the simulation. This is the case for the deformation $D$ which is the main quantity of interest in our analysis. If $\epsilon$ is the absolute error on the radius, $R_1$ and $R_2$ the stellar radii at respectively time $t_1$ and $t_2$, one can write:
$$\frac{R_2 + \epsilon}{R_1 + \epsilon} \simeq
\frac{R_2}{R_1}\left(1+\epsilon\left(\frac{1}{R_2}-\frac{1}{R_1}
\right)+o(\epsilon)\right).$$
For the specific case of distortion calculation, $R_1$ is the radius of the unperturbed polytrope and $R_2$ the radius of the polytrope filling its critical lobe. We find typical values of $R_2/R_1 \leq$ 12% (see Tables \[tab1\]) with $\epsilon \sim$ 4%. Thus the first-order term in our last equation provides a correction of at most 0.5%. Consequently, our method used for estimating the volume is accurate enough for the present study. Note that the test simulations done with 57000 particles give the same distortion factor (by less than 1%) than calculations with 15000 particles.
[^1]: defined as $L_1$ in the Roche potential
[^2]: $\zeta_{\rm CL}$ is equivalent to the mass radius exponent of the Roche radius $\zeta_{\rm R}$ in the Roche model (see e.g. Ritter 1996)
[^3]: this follows directly from Kepler’s 3rd law, the fact that $(1 +1/q)V_{\rm f}/A^3 \approx {\rm const.}$ for $q \simle 0.8$, and $\dot P = 0$
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'An attempt is made to describe the general-relativistic equations of motion for the Schwarzschild geometry in terms of the classical concepts of energy and angular momentum. Using the customary terms the geodesic equations can be viewed in a way that is very helpful in providing the physical meaning of the mathematical development.'
author:
- Yong Gwan Yi
title: 'General-Relativistic Equations of Motion in terms of Energy and Angular Momentum'
---
=14.5cm
The general theory of relativity has led to a completely new picture of gravitational phenomena in geometrical terms. The gravitational field is represented by metric tensor, and the equations of free fall are geodesics. Although the geodesic equation gives a constant of motion corresponding to energy, accordingly, most textbooks introduce approaches that exclude serious use of energy concept \[1\]. However, it would seem desirable to use the relativistic energy in describing the central force problem. For many applications, the equation of motion containing the energy and angular momentum is the natural one. In order to discuss the comparison with Newton’s theory or the transition to quantum theory, it is important that the description of the motion be in terms of its energy and angular momentum. In a certain sense, the use of relativistic energy is considered necessary and important.
Let us see what can be learned from Einstein’s theory of gravitation. In this paper, we define $a_{\mu}b^{\mu}$ as $a_0b^0 - {\bf a} \cdot {\bf b}$. We begin by pointing out that the metric of our space-time is $ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}$. The Schwarzschild expression for the metric around a mass $M$ is $$c^2d\tau^2=c^2g_{00}dt^2
-g_{rr}dr^2-r^2d\theta^2-r^2\sin^2\theta d\psi^2$$ with $g_{00}=1/g_{rr}=1-2GM/c^2r$. We consider the central force motion in the plane $\psi=\pi/2$. The square of velocities will then be $$c^2\rightarrow c^2g_{00} \quad\mbox{and}\quad v^2\rightarrow g_{rr}
\biggl(\frac{dr}{dt}\biggr)^2+r^2\biggl(\frac{d\theta}{dt}\biggr)^2.$$ In consequence of this relation the Schwarzschild metric can be written $$d\tau=g_{00}^{1/2}dt\biggl(1-\frac{v^2}{c^2}\biggr)^{1/2},$$ to a first approximation. This form of equation reduces to the familiar equation leading to the Lorentz time dilation in the limit as $g_{00}$ approaches to unity. In this sense one may see the relation in (3) as the Schwarzschild time dilation.
The equations of motion in the Schwarzschild field yield two constants of motion. One of them is given by $$g_{00}\frac{dt}{d\tau}=\mbox{constant},$$ which corresponds to the energy of the system. The other constant is obtained from $r^2(d\theta/d\tau)=$ constant, and is absorbed immediately into the definition of the angular momentum $l$. It would seem at first sight that the constant in (4) is of no importance in the geometrical approach. However, the constant has an important physical significance, for it can lead to the formulation of the resulting relativistic mechanics in terms of the energy of a particle as in the case of special relativity. The relativistic equations of motion must be such that in the nonrelativistic limit they go over into the customary forms given by Newton’s theory. Thus the task of identifying the constant is greatly facilitated by seeking the form which it would have in the nonrelativistic limit. In the nonrelativistic limit, Eq. (4) can be expanded as $$g_{00}\frac{dt}{d\tau}\simeq \frac{1}{mc^2}\biggl(mc^2+
\frac 12mv^2-\frac{GMm}{r}\biggr),$$ where $m$ is the mass of a particle. By comparison with Newton’s theory, we can identify the constant with $$g_{00}\frac{dt}{d\tau}=\frac{1}{mc^2}\bigl(mc^2+E\bigr).$$ Consequently it yields the expression $$E=mc^2g_{00}\frac{dt}{d\tau}-mc^2$$ for the energy of a particle in the static isotropic gravitational field.
The geodesic equations teach us a four-velocity of the form $g_{\mu\mu}dx^{\mu}/d\tau$. In the Schwarzschild metric the scalar product of two four-vectors is defined as $g_{\mu\nu}a^{\mu}b^{\nu}$ or $g^{\mu\nu}a_{\mu}b_{\nu}$. With this definition the square of the magnitude of the velocity four-vector is a constant, $c^2$. From the covariant form of velocity we can write the relativistic expression for momentum as $$p_r=mg_{rr}\frac{dr}{d\tau} \quad\mbox{and}\quad
p_{\theta}=mr^2\frac{d\theta}{d\tau}\equiv l.$$ This is a definition of momentum which is deduced from the constants of motion in the Schwarzschild metric field. Equations (7) and (8) are the necessary relativistic generalizations for the energy and momentum of a particle, consistent with the conservation laws and the postulates of general relativity.
As in the special theory of relativity, it is natural to attempt to identify the four equations of energy and momentum conservation as relations among the energy-momentum four-vectors. In special relativity, the connection between the kinetic energy $T$ and the momentum is expressed in the statement that the magnitude of the momentum four-vector is constant: $$p_{\mu}p^{\mu}=\frac{T^2}{c^2}-p^2=m^2c^2.$$ This must be generalized to provide an expression satisfying the general-relativity formulation. We observe that the momentum in (8) is proportional to the space components of the four-vector velocity. The time component of the four-vector velocity is $cg_{00}dt/d\tau$. Comparison with (7) shows that the energy of a particle differs from its time component by the rest energy $mc^2$. We are thus led to $$E=mc^2g_{00}\frac{dt}{d\tau}$$ as the covariant form of the total energy, for then $p_r$, $l$, and $E/c$ form a four-vector momentum. The desired generalization of energy-momentum equation must be $$g^{\mu\mu}p_{\mu}p_{\mu}=\frac{E^2}{c^2g_{00}}
-\frac{p_r^2}{g_{rr}}-\frac{l^2}{r^2}=m^2c^2.$$ It should be noted that the gravitational potential lends itself to incorporation in the metric of space-time geometrization, so the potential energy is absorbed automatically into the path length of a particle and its motion therein. In general relativity, therefore, kinetic energy and potential energy individually become meaningless; only the total energy of a particle is significant.
We can now proceed to the relativistic equation for the orbit of a planet. We can still talk in terms of the system energy and the system angular momentum. For comparison with Newton’s theory, it is preferable to define the energy $E$ as in (7), which would bring $E$ in line with the nonrelativistic value. The Schwarzschild metric in (1) can now be expressed in terms of two constants of motion $E$ and $l$ as $$\frac{(mc^2+E)^2}{c^2g_{00}}-m^2g_{rr}
\biggl(\frac{dr}{d\tau}\biggr)^2-\frac{l^2}{r^2}=m^2c^2.$$ This form of the equation of motion can also be obtained from a combination of the differential equations of geodesics \[2\]. Most often we are more interested in the shape of orbits, that is, in $r$ as a function of $\theta$, than in their time history. The angular momentum relation can then be used directly to convert (12) into the differential equation for the orbit; this gives $$\frac{(mc^2+E)^2}{c^2l^2g_{00}}-\frac{g_{rr}}{r^4}
\biggl(\frac{dr}{d\theta}\biggr)^2-\frac{1}{r^2}=
\frac{m^2c^2}{l^2}.$$ The solution may thus be determined by a quadrature: $$\triangle\theta=\int\biggl[\frac{(mc^2+E)^2}{c^2l^2g_{00}}-
\frac{m^2c^2}{l^2}-\frac{1}{r^2}\biggr]^{-1/2}
\frac{g_{rr}^{1/2}dr}{r^2}.$$
At perihelia and aphelia, $r$ reaches its minimum and maximum values $r_-$ and $r_+$, and at both points $dr/d\theta$ vanishes, so (13) gives $$\frac{(mc^2+E)^2}{c^2l^2g_{00}(r_{\pm})}-\frac{1}{r_{\pm}^2}=
\frac{m^2c^2}{l^2},$$ where $g_{00}(r_{\pm})=1-2GM/c^2r_{\pm}$. From these two equations we can derive values for the two constants of the motion: $$\biggl(1+\frac{E}{mc^2}\biggr)^2=
\frac{r_+^2-r_-^2}{r_+^2g_{00}^{-1}(r_+)-r_-^2g_{00}^{-1}(r_-)},
\quad \frac{m^2c^2}{l^2}=
\frac{r_+^{-2}g_{00}(r_+)-r_-^{-2}g_{00}(r_-)}{g_{00}(r_-)-g_{00}(r_+)}.$$ The expressions for the energy and angular momentum appear here in somewhat different forms involving the metric tensors $g_{00}(r_{\pm})$, but their equivalence in the limit as $g_{00}\to 1$ with the respective nonrelativistic Newtonian relations are shown by expanding the equations to a first approximation: $$E\simeq-\frac{GMm}{r_++r_-}, \quad
l^2\simeq\frac{2GMm^2}{r_+^{-1}+r_-^{-1}}.$$ Using the exact values of the constants given by (16) in (14) yield the formula for $\triangle\theta$ as $$\triangle\theta=\int\biggl[\frac{r_+^{-2}(g_{00}^{-1}(r)-
g_{00}^{-1}(r_-)) - r_-^{-2}(g_{00}^{-1}(r)-g_{00}^{-1}(r_+))}
{g_{00}^{-1}(r_+)-g_{00}^{-1}(r_-)}-\frac{1}{r^2}\biggr]^{-1/2}
\frac{g_{rr}^{1/2}(r)dr}{r^2}.$$ We can make the argument of the first square root in the integrand a quadratic function of $1/r$ which vanishes at $r=r_{\pm}$, so $$\triangle\theta\simeq\int\biggl[C\biggl(\frac{1}{r_-}-
\frac 1r\biggr)\biggl(\frac 1r -\frac{1}{r_+}\biggr)\biggr]^{-1/2}
\biggl(1+\frac{GM}{c^2r}\biggr)\frac{dr}{r^2},$$ where $C\simeq 1-(2GM/c^2)(r_+^{-1}+r_-^{-1})$. The constant $C$ could be determined by letting $r\to\infty$.
We can obtain the same result much more simply. It is both easier and more instructive to expand $g_{00}$ in the formal solution (14). It preserves the advantage that the orbit equation is evaluated in terms of the energy and the angular momentum of the system. Note that we have to expand to second order in $GM/c^2r$. The angle swept out by the position vector is then given by (14) as $$\triangle\theta\simeq\int\biggl[
\frac{2mE}{l^2}\biggl(1+\frac{E}{2mc^2}\biggr)+
\frac{2GMm^2}{l^2r}\biggl(1+\frac{2E}{mc^2}\biggr)-
\frac{1}{r^2}\biggl(1-\frac{4G^2M^2m^2}{c^2l^2}\biggr)
\biggr]^{-1/2}\biggl(1+\frac{GM}{c^2r}\biggr)\frac{dr}{r^2}.$$ As it stands, this integral is of the standard form. The integrand differs from the corresponding nonrelativistic expression in that the second term in each pair of parenthesis represents the relativistic correction. In form it is the general-relativity analogue of Sommerfeld’s treatment of the hydrogen atom in special relativity. It has been said that we need $g_{00}$ to second order in $GM/c^2r$ to calculate $\triangle\theta$ to first order. To put this another way, the high accuracy of the orbit precession serves as a touchstone for the possible forms of $g_{00}$ by requiring the degree of agreement to second order. The procedure described here is particularly simple and is sufficient to enable one to confirm the fact.
On carrying out the integration, the equation of the orbit is found to be $$\frac 1r\simeq A\biggl[1+\epsilon\cos\biggl(
(\theta-\theta_0)\biggl(1-\frac{3G^2M^2m^2}{c^2l^2}\biggr)
-\frac{GM}{c^2r^2}\biggl(\frac{dr}{d\theta}\biggr)
\biggr)\biggr],$$ where $$A\simeq \frac{GMm^2}{l^2}
\biggl(1+\frac{2E}{mc^2}+\frac{4G^2M^2m^2}{c^2l^2}\biggr),
\quad \epsilon\simeq\biggl[1+\frac{2El^2}{G^2M^2m^3}
\biggl(1-\frac{7E}{2mc^2}-\frac{4G^2M^2m^2}{c^2l^2}\biggr)
\biggr]^{1/2},$$ and $\theta_0$ is a constant of integration. In addition to the motion of a planet’s perihelion of $2\pi(3G^2M^2m^2/c^2l^2)$ per revolution, the relativity effect produces the term $(GM/c^2r^2)(dr/d\theta)$ in the angle swept out by the radius vector of the planet. This term is not a new result but merely a result of rewriting the square root in the integrand which the integration of (20) actually yields, using (14) to a first approximation. It is evident therefore that the relativity effect in planetary motion obeying (19) or (20) is to cause not only the precession of the perihelion of the orbit of a planet but also the change in the angular displacement of the planet due to its radial velocity. The additional change appearing in the angular displacement of the planet, which does not appear in a circular orbit, might be an effect due to the finite velocity of propagation of the solar gravitational field.
The formulation presented in this paper is mathematically equivalent to the familiar formulations. There are therefore no fundamentally new results. However, the point of view which has been taken here regarding the central force problem differs from the usual point of view. There is a certain lack of energy concept in the geometrical approach to the subject. A prominent feature of the present formulation is that the customary concepts of classical mechanics are emphasized throughout within the mathematical framework required by general relativity. This is very helpful for grasping the physical meaning behind the mathematical development. The present point of view offers a distinct advantage.
[2]{}
P. G. Bergmann, [*Introduction to the Theory of Relativity*]{} (Prentice-Hall, New Delhi, 1942), Sec. 6.4; C. Möller, [*The Theory of Relativity*]{} (Oxford, New York, 1972), 2nd ed., Sec. 12.2; L. D. Landau and E. M. Lifshitz, [*The Classical Theory of Fields*]{} (Pergamon, New York, 1975), 4th ed., Sec. 101; S. Weinberg, [*Gravitation and Cosmology*]{} (John Wiley & Sons, New York, 1972), Sec.8.6.
In his classic book [*Gravitation and Cosmology*]{}, Weinberg assumed a new constant of motion from a combination of geodesic equations. But the constant in his equation (8.4.13) is devoid of physical significance. I should mention one more point. His equation (8.5.6) for the deflection of light is in formal agreement with the equation for rays in geometrical optics! For the development of an optical analogy, see [*e-print*]{} physics/0006006.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In the present work, the temperature dependence of the scalar mesons parameters is investigated in the framework of thermal QCD sum rules. We calculate $\sigma$ -pole and the non-resonant two-pion continuum contributions to the spectral density. Taking into account additional operators appearing at finite temperature, the thermal QCD sum rules are derived. The temperature dependence of the shifts in the mass and leptonic decay constant of scalar $\sigma (600)$ meson is calculated.'
author:
- |
Elşen Veli Veliev \*, Takhmassib M. Aliev \*\*\
\*\* Physics Department, Kocaeli University, Umuttepe Yerleşkesi\
41380 Izmit, Turkey\
e-mail: [email protected]\
\*\*\* Physics Department, Middle East Technical University,\
06531 Ankara, Turkey\
e-mail: [email protected]
title: ' **Thermal QCD Sum Rules for $\sigma (600)$ Meson**'
---
Introduction
============
The QCD sum rules method [@1], proposed about three decades ago, is one of the powerful methods for investigating the properties of hadrons. This method has been extensively used as an efficient tool to study the masses, decay form factors and so on [@2]. In recent years there has been increasing interest in the modification of hadronic properties at finite temperature in order to understand the results of the heavy ion collision experiments.
The QCD sum rules method is extended to the finite temperature in [@3] and finite temperature sum rules have several new features. One of them is the interaction of the current with the particles of the medium. This effect requires modifying hadron spectral function. The other novel feature is the breakdown of Lorentz invariance by the choice of reference frame [@4]-[@6]. Due to the residual O(3) symmetry more operators with the same dimension appear in the operator product expansion (OPE) at finite temperature compared to those at zero temperature. Taking into account both complications, the investigation of OPE for thermal correlator of the two vector currents, and the thermal QCD sum rules for vector mesons have been realized in [@7] and [@8], respectively. Also, nuclear medium modifications of meson parameters are widely discussed in the literature [@9]-[@10].
In the present work, we investigate the properties of the scalar $\sigma$ meson in the framework of thermal QCD sum rules. The basic idea of thermal QCD sum rules is to get information about temperature dependence of hadron parameters, studying the same correlator, both at high temperature where the quark-gluon plasma is expected and at low temperature, where the hadronic phase is dominated. Note that the nature of light scalar mesons is still an open problem and is the subject of intensive and continuous theoretical [@11] and experimental investigations [@12]. Can we get any new information about the nature of the scalar mesons from the thermal QCD analysis? Present work is addressed to the investigation of this problem.
The paper is organized as follows. In section 2 we derive the thermal QCD sum rules for scalar $\sigma (600)$ meson. In section 3 we present our numerical calculations. This section also contains discussion and our conclusion.
Thermal QCD sum rules for scalar sigma mesons
=============================================
In this section we construct the thermal sum rules for scalar $\sigma (600)$ meson. For this purpose we consider the thermal average of correlation function $$\label{eqn1}
T(q)=i \int d^{4}x e^{iq\cdot x} \langle T(J(x)J(0))\rangle, \\$$ where $J(x)=\frac{1}{\sqrt{2}}(\overline{u}u+\overline{d}d)$ is the interpolating current with the $\sigma$ meson quantum numbers. The thermal average of any operator is determined by following expression $$\label{eqn2}
\langle O\rangle=Tr e^{-\beta H}O/Tr e^{-\beta H}, \\$$ where $H$ is the QCD Hamiltonian, and $\beta=1/T$ stands for the inverse of the temperature $T$ and traces are carried out over any complete set of states.
The fundamental assumption of Wilson expansion is that the product of operators at different points can be expanded as the sum of local operators with momentum dependent coefficients in the form: $$\label{eqn3}
T(q)=\sum_n C_{n}(q^2)\langle O_{n}\rangle , \\$$ where $C_{n}(q^2)$ are called Wilson coefficients, and $O_{n}$ are a set of local operators. In this expansion, the operators are ordered according to their dimension $d$ . The lowest dimension operator with $d=0$ is the unit operator associated with the perturbative contribution. In the vacuum sum rules operators with dimensions $d=3$ and $d=4$ composed of quark and gluon fields are the quark condensate $\langle \overline{\psi}\psi \rangle$ and the gluon condensate $\langle G^{a}_{\mu\nu}G^{a\mu\nu}\rangle$, respectively. At finite temperature Lorentz invariance is broken by the choice of a preferred frame of reference, and therefore new operators appear in the Wilson expansion. In order to restore Lorentz invariance in thermal field theory, four-vector velocity of the medium $u^{\mu}$ is introduced. Using four-vector velocity and quark/gluon fields, we can construct a new set of low dimension operators $\langle
u\Theta^{f}u\rangle$ and $\langle u\Theta^{g}u\rangle$ with dimension $d=4$ , where $\Theta^{f}_{\mu\nu}$ and $\Theta^{g}_{\mu\nu}$ are fermionic and gluonic parts of energy momentum tensor $\Theta_{\mu\nu}$, respectively. So, we can write thermal correlation function in terms of operators up to dimension four: $$\label{eqn4}
T(q)=C_{1} I + C_{2}\langle \overline{\psi}\psi\rangle +
C_{3}\langle G^{a}_{\mu\nu}
G^{a\mu\nu}\rangle + C_{4}\langle u\Theta^{f}u\rangle + C_{5}\langle
u\Theta^{g}u\rangle . \\$$ The Wilson coefficients in Eq.(\[eqn4\]) are calculated in [@13], and renormalization group improved expression of OPE leads to the following result $$\begin{aligned}
\label{eqn5}
&&T(Q)=\frac{3}{
8\pi^2}Q^2\left(\gamma-\ln\frac{4\pi}{Q^2}\right)+\frac{3}{Q^2}m
\langle \overline{\psi}\psi\rangle+\frac{g^2}{32 \pi^2 Q^2}\langle
G^{a}_{\mu\nu}G^{a\mu\nu}\rangle
\nonumber \\
&&+\frac{4}{16+3n_{f}}\left(\frac{4(u\cdot
Q)^2}{Q^4}+\frac{1}{Q^2}\right) \left[\langle u \Theta u \rangle +
\lambda\left(Q^{2}\right)\left(\frac{16}{3}\langle
u\Theta^{f}u\rangle-\langle u\Theta^{g}u\rangle\right)\right],\end{aligned}$$ where $Q$ is the Euclidean momentum, $\lambda(Q^2)=\left[\alpha_{s}\left(\mu^{2}\right)
/\alpha_{s}\left(Q^{2}\right)\right]^{-\delta/b}$ and $\Theta_{\mu
\nu}=n_f \Theta^{f}_{\mu\nu}+\Theta^{g}_{\mu\nu}$. At one loop level the constants $\delta$ and $b$ are given by $$\label{eqn6}
\delta=\frac{2}{3}\left(\frac{16}{3}+ n_{f}\right)\,\,\,\,\,\, and
\,\,\,\,\,\, b=11-\frac{2}{3}n_{f} \,\,, \\$$ where $n_{f}$ is quark flavours number. The spectral representation for the correlation function in $q_{0}$ at fixed $|\textbf{q}|$ can be written as [@8] $$\label{eqn7}
T\left(q_{0}^{2},|\textbf{q}|\right)= \int_{0}^{\infty}d{q'}_{0}^{2}
\frac{N\left(q_{0}^{'},|\textbf{q}|\right)}{{q'}_0^{2}+Q_0^2}+
subtraction \,\,\,\,\,\,terms \,\, , \\$$ where $$\label{eqn8}
N\left(q_{0},|\textbf{q}|\right)= \frac{1}{\pi}
ImT\left(q_{0},|\textbf{q}|\right)\tanh\left(\beta
q_{0}/2\right)\,\,\,\,\,\, and \,\,\,\,\,\, Q_{0}^{2}=-q_{0}^{2} . \\$$ Note that, the subtraction terms are removed by the Borel transformation. For this reason in further discussion we omit these terms. Equating spectral representation and OPE, and performing Borel transformations with respect to $Q_{0}^{2}$ from both sides Eq.(\[eqn7\]), we obtain the QCD sum rules $$\label{eqn9}
\int_{0}^{\infty}dq_{0}^{2}e^{-q_{0}^{2}/M^2}
N\left(q_{0},|\textbf{q}|\right)=e^{-|\textbf{q}|^{2}/M^2}
\left[\frac{3 M^4}{8 \pi^2}+\langle O_1 \rangle+ \left(1-\frac{4\textbf{q}^2}{3 M^2}\right)\langle O_2 \rangle\right]\,\, , \\$$ where $M$ is Borel parameter, $\langle O_1\rangle$ and $\langle O_2\rangle$ are the non-perturbative contributions of higher dimensional operators, $$\label{eqn10}
\langle O_1\rangle=3m \langle \overline{\psi}\psi\rangle +
\frac{g^2}{32 \pi^2} \langle G^{a}_{\mu\nu}G^{a\mu\nu}\rangle \,\, ,\\$$ $$\label{eqn11}
\langle O_2\rangle=-\frac{12}{16+3n_{f}}\left[\langle u\Theta u
\rangle + \lambda (M^2)\left(\frac{16}{3}\langle u\Theta^{f} u
\rangle-\langle
u\Theta^{g} u \rangle\right)\right]\,\, . \\$$ Now we consider the phenomenological part of the correlation function. We shall work below the critical temperature, where the physical spectrum is saturated by hadrons . In this case, similar to the vacuum QCD sum rules, the dominant contribution to the spectral density comes from $\sigma$ mesons. We also calculate the contribution of the non-resonant two-pion continuum.
Let us calculate $\sigma$-pole contribution to the correlator. The leptonic decay constant $\lambda_{\sigma}$ of the $\sigma$-meson is given by, $\langle 0|J(0)|\sigma\rangle =
m_{\sigma}\lambda_{\sigma}$, where $m_{\sigma}$ is $\sigma$-meson mass. In thermal field theory, the parameters $m_{\sigma}$ and $\lambda_{\sigma}$ must be replaced by their temperature dependent values. The vacuum value of the leptonic decay constant is obtained from two point QCD sum rules and $\lambda_{\sigma}=200MeV$ [@14]. The absorptive part of the correlation function is calculated by using the following field-current identity $$\label{eqn12}
J(x)=m_{\sigma}\lambda_{\sigma}\sigma(x)\,\, , \\$$ and $\sigma$-meson contribution to thermal correlator can be written as $$\label{eqn13}
T(q)=i m_{\sigma}^{2}\lambda_{\sigma}^{2}D_{11}^{\sigma}(q)\,\, . \\$$ Here $D_{11}^{\sigma}(q)= \int d^{4}x e^{iq\cdot x} \langle
T(\sigma(x)\sigma(0))\rangle$ is the time ordered product of two $\sigma$-meson fields (11-component of the finite temperature scalar field propagator with mass $m_{\sigma}$ in the real time formalism) and has the following form [@15]-[@16] $$\label{eqn14}
D_{11}^{\sigma}(q)=\frac{i}{q^2- m_{\sigma}^{2}+i\varepsilon}+2\pi n
(\omega_{q})\delta(q^2-m_{\sigma}^{2})\,\, , \\$$ where $n(\omega_{q})$ is the Bose distribution function, $n(\omega_{q})=\left[exp(\beta \, \omega_{q})-1\right]^{-1}$ and $\omega_{q}=\sqrt{\textbf{q}^2+m_{\sigma}^{2}}$. The imaginary part of correlation function can be simply evaluated using the formula $\frac{i}{x+i \varepsilon}=\pi \delta(x)+i
P\left(\frac{1}{x}\right)$, which leads to $$\label{eqn15}
Im T(q)=\pi m_{\sigma}^{2}\lambda_{\sigma}^{2}(2 n(\omega_q)+1)
\delta\left(q^2-m_{\sigma}^{2}\right)\,\, . \\$$ With the help of $\delta$-function we obtain the following result for $\sigma$-pole contribution to the spectral function $$\label{eqn16}
N(q)=m_{\sigma}^{2}\lambda_{\sigma}^{2}
\delta\left(q^2-m_{\sigma}^{2}\right). \\$$ In order to calculate the dependence of $m_{\sigma}$ and $\lambda_{\sigma}$ on temperature, we consider appropriate loop diagrams. Let us calculate the $\pi\pi$-contribution to the amplitudes, which describes the interaction of the current with the particles in the medium. This contribution to the correlation function can be written as $$\label{eqn17}
T(q)=i g_{\sigma}^{2} \int \frac{d^4 k}{(2 \pi)^4}
D_{11}^{\pi}(k)D_{11}^{\pi}(k-q) \,\, , \\$$ where $D_{11}^{\pi}(k)$ is the 11-component of the finite temperature propagator for pions and $g_{\sigma}=2,0 \,GeV$ [@17]-[@18]. The integration over $k_0$ in Eq.(\[eqn17\]) can be evaluated using the residue theorem. After integration and some simplifications for the imaginary part of the correlation function we obtain $$\begin{aligned}
\label{eqn18}
Im T(q)&=&\pi g_{\sigma}^{2} \int
\frac{d\textbf{k}}{(2\pi)^3}\frac{1}{4
\omega_{1}\omega_{2}}\left(\left(1+n_1\right)\left(1+n_2\right)+n_1
n_2\right)\left(\delta\left(q_0-\omega_{1}-\omega_{2}\right)+
\delta\left(q_0+\omega_{1}+\omega_{2}\right)\right)
\nonumber \\
&+&\left(\left(1+n_1\right)n_2+\left(1+n_2\right)n_1\right)
\left(\delta\left(q_0-\omega_{1}+\omega_{2}\right)+
\delta\left(q_0+\omega_{1}-\omega_{2}\right)\right) \,\, ,\end{aligned}$$ where, $$\label{eqn19}
n_1=n(\omega_1)\,\, , \,\,\,\,\, n_2=n(\omega_2)\,\, , \,\,\,\,\,
\omega_1=\sqrt{\textbf{k}^{2}+m_{\pi}^{2}}\,\, , \,\,\,\,\,
\omega_2=\sqrt{(\textbf{k}-\textbf{q})^{2}+m_{\pi}^{2}} \,\,\, . \\$$ At values $q_0=\omega_1+\omega_2$ and $q_0=\omega_1-\omega_2$ the terms involving the density distributions can be written as $$\label{eqn20}
\left[\left(1+n_1\right)\left(1+n_2\right)+n_1 n_2 \right] \tanh \left( \frac{\beta q_0}{2}\right )= \left( n_1+n_2+1\right) \,\, , \\$$ $$\label{eqn21}
\left[\left(1+n_1\right)n_2+\left(1+n_2\right)n_1 \right] \tanh
\left( \frac{\beta q_0}{2}\right )= \left(n_2-n_1 \right) \,\, ,$$ respectively. As can be seen, delta function $\delta(q_0-\omega_1-\omega_2)$ in Eq.(\[eqn18\]) gives the first branch cut, $q^2\geq 4m_{\pi}^{2}$, which coincides with zero temperature cut that describes the standard threshold for particle decays. On the other hand, delta function $\delta(q_0-\omega_1+\omega_2)$ in Eq.(\[eqn18\]) shows that an additional branch cut arises at finite temperature, $q^2\leq 0$ , which corresponds to particle absorption from the medium. Therefore, delta functions $\delta(q_0-\omega_1-\omega_2)$ and $\delta(q_0-\omega_1+\omega_2)$ in Eq.(\[eqn18\]) contribute in regions $q^2\geq 4m_{\pi}^{2}$ and $q^2\leq 0$, respectively. Taking into account both contributions, the spectral function can be written as $$\begin{aligned}
\label{eqn22}
N(q)&=&g_{\sigma}^{2} \int\frac{k^2 \sin\theta dk d\theta}{(2 \pi)^2
\,\,2\omega_1}\left[\left(n_1+n_2+1\right)\theta\left(q^2-4m_{\pi}^{2}\right)
+\left(n_2-n_1\right)\theta\left(-q^2\right)\right]
\nonumber \\
& &\times \delta \left( q^2-2 q_0
\omega_1+2|\textbf{k}||\textbf{q}|\cos \theta \right)\,\, .\end{aligned}$$ The integration over angle $\theta$ in Eq.(\[eqn22\]) can be evaluated using the constraint $|\cos
\theta_{\textbf{q},\textbf{k}}|\leq1$ , which leads to following inequality $$\label{eqn23}
\frac{|q^2-2q_0 \omega_1|}{2|\textbf{k}|
|\textbf{q}|}\leq1 \,\, .\\$$ The solution of this inequality at values $q^2\geq 4m_{\pi}^{2}$ give us the integration range of $\omega_1$ as $\omega_{-} \leq
\omega\leq \omega_{+}$ , where $$\label{eqn24}
\omega_{\pm}=\frac{1}{2}(q_0\pm |\textbf{q}|v) \,\, , \\$$ $$\label{eqn25}
v(q^2)=\sqrt{1-4m_{\pi}^{2}/q^2} \,\, . \\$$ At $q^2\leq 0$ the region of variation of $\omega_{1}$ must be $\omega_{+} \leq \omega_{1}<\infty$. Finally, the thermal spectral function can be written as $$\label{eqn26}
N(q)=\frac{g_{\sigma}^{2}}{2
|\textbf{q}|}\int^{\omega_{+}}_{\omega_{-}} \frac{d
\omega_1}{(2\pi)^2}\left(n_1+n_2+1\right)\theta
\left(q^2-4 m_{\pi}^{2}\right)+\frac{g_{\sigma}^{2}}{2 |\textbf{q}|}\int^{\infty }_{\omega_{+}}
\frac{d \omega_1}{(2\pi)^2}\left(n_2-n_1\right)\theta
\left(-q^2\right)\,\, . \\$$ Changing the variable $\omega_1$ to $x$ given by $\omega_1=\frac{1}{2}\left(q_0+|\textbf{q}|x\right) $ , we finally get the two pion contribution to the spectral function as $$\label{eqn27}
N(q)\equiv{g_{\sigma}^{2}}\frac{v(q^2)}{8\pi^2}+N_1(q)=\frac{g_{\sigma}^{2}\,v(q^2)}{8\pi^2}+
\frac{g_{\sigma}^{2}}{8\pi^2}\int^{v}_{-v}dx\,\,
n\left(\frac{1}{2}\left(q_0+|\textbf{q}|x\right)\right)\,\, ,
\,\,\,\,\,\, q^2\geq 4m^2$$ $$\label{eqn28}
N(q)\equiv
N_2(q)=\frac{g_{\sigma}^{2}}{16\pi^2}\int^{\infty}_{v}dx\,\, \left [
n\left(\frac{1}{2}\left(|\textbf{q}|x-q_0\right)\right)-
n\left(\frac{1}{2}\left(|\textbf{q}|x+q_0\right)\right)\right]\,\, ,
\,\,\,\, q^2\leq 0$$ The QCD sum rules are obtained by equating theoretical and phenomenological parts of correlation function. Taking into account of expressions Eq.(\[eqn27\]) and Eq.(28) in Eq.(\[eqn9\]) we get $$\begin{aligned}
\label{eqn29}
&&m_{\sigma}^2(T)\lambda_{\sigma}^2(T)e^{-m_{\sigma}^2(T)/M^2}+
\frac{g_{\sigma}^{2}}{8 \pi^2}e^{|\textbf{q}|^2/M^2}
\int_{4m_{\pi}^2+|\textbf{q}|^2}^{\infty}dq_0^2 e^{-q_0^2/M^2} v
(q^2)
\nonumber \\
&+&e^{|\textbf{q}|^2/M^2}\left(\int_{4m_{\pi}^2+
\textbf{q}|^2}^{\infty}dq_0^2
e^{-q_0^2/M^2}N_1\left(q_0,|\textbf{q}|\right)+
\int_{0}^{|\textbf{q}|^2}dq_0^2
e^{-q_0^2/M^2}N_2\left(q_0,|\textbf{q}|\right)\right)
\nonumber \\
&=&\frac{3 M^4}{8 \pi^2}+\langle O_1
\rangle+\left(1-\frac{4\textbf{q}^2}{3 M^2}\right)\langle O_2
\rangle \, .\end{aligned}$$ As the temperature approaches to zero, the two terms in bracket go to zero and the thermal average of the operators on the right become the expectation values, recovering the vacuum sum rules [@14]. In the limit $|\textbf{q}|\rightarrow 0$, the sum rule (29) simplifies considerably. Finally we obtain that $$\label{eqn30}
m_{\sigma}^2(T)\lambda_{\sigma}^2(T)exp(-m_{\sigma}^2(T)/M^2)+I_0(M^2)
+I_1(M^2)=\frac{3 M^4}{8 \pi^2}+\langle O \rangle \,\, ,\\$$ where $$\label{eqn30}
I_0(M^2)=\frac{g_{\sigma}^{2}}{8 \pi^2}\int^{\infty}_{4m_{\pi}^2}ds
\,\,
v(s)exp(-s/M^2) \,\, , \\$$ $$\label{eqn31}
I_1(M^2)=\frac{g_{\sigma}^{2}}{4 \pi^2}\int^{\infty}_{4m_{\pi}^2}ds
\,\,
v(s)\,\,n(\sqrt{s}/2)exp(-s/M^2) \,\, ,\\$$ $$\label{eqn311}
\langle O \rangle=\langle O_1 \rangle+\langle O_2 \rangle \,\, ,\\$$
Numerical analysis of the shifts in mass and leptonic decay constant
====================================================================
In this section we present our results for the temperature dependence of the shifts in $\sigma$ meson mass and leptonic decay constant. By derivativing with respect to $1/M^2$ from both sides of the sum rules (30), and making some transformations we obtain $$\label{eqn32}
m_{\sigma}^2(T)=\frac{m_{\sigma}^4 \lambda_{\sigma}^2
exp(-m_{\sigma}^2/M^2)-J_1(M^2)+\eta \overline{\langle O_3
\rangle}}{m_{\sigma}^2 \lambda_{\sigma}^2
exp(-m_{\sigma}^2/M^2)-I_1(M^2)+ \overline{\langle O\rangle} } \,\, , \\$$ $$\label{eqn33}
\lambda_{\sigma}^2(T)=\lambda_{\sigma}^2 \frac{m_{\sigma}^2
\lambda_{ \sigma}^2+\left(\overline{\langle O
\rangle}-I_1(M^2)\right) exp(m_{\sigma}^2/M^2)}{m_{\sigma}^2
\lambda_{
\sigma}^2+\left(\frac{1}{M^2}-\frac{1}{m_{\sigma}^2}\right)\left[
J_1(M^2)-\eta\overline{\langle O_3
\rangle}+m_{\sigma}^2\left(\overline{\langle O
\rangle}-I_1(M^2)\right)\right]exp(m_{\sigma}^2/M^2)} \,\, ,
\\$$ where the bar on the operators means subtractions of their vacuum expectation values and $$\label{eqn34}
\eta(M^2)=\frac{\delta M^2}{ b \ln(M^2/\Lambda^2)}\,\, , \\$$ $$\label{eqn35}
J_1(M^2)=\frac{g_{\sigma}^{2}}{4 \pi^2} \int^{\infty}_{4 m_{\pi}^2}
ds\,\, s\,\,
\upsilon(s)\,\,n \left(\sqrt{s}/2\right)exp(-s/M^2) \,\, , \\$$ $$\label{eqn36}
\overline{\langle O_3
\rangle}=-\frac{12}{16+3n_{f}}\lambda(M^2)\left(\frac{16}{3}\langle
u\Theta^{f}u \rangle-\langle u\Theta^{g}u \rangle \right)\,\, . \\$$ For the numerical analysis, let us list thermal average of operators contributing to the QCD sum rules. The temperature dependence of quark condensate is known from chiral perturbation theory [@19]-[@20] $$\label{eqn37}
\langle \overline{\psi}\psi \rangle=\langle 0|\overline{\psi}\psi |0
\rangle \left[1-\frac{n_f^2-1}
{n_f}\frac{T^2}{12 F^2}+ O(T^4) \right] , \\$$ where $n_f$ is number of quark flavors and $F=0.088\,\, GeV$. The low temperature expansion of the gluon condensate has been studied in article [@21] $$\label{eqn38}
\frac{g^2}{4 \pi^2} \overline{\langle G^{a}_{\mu\nu} G^{a \mu \nu }
\rangle}=-\frac{8}{9} \left( \langle \Theta ^{\mu}_{\mu}\rangle +
\sum_f m_f \overline{\langle \overline{\psi}\psi
\rangle}\right) , \\$$ where the trace of the total energy momentum tensor $\Theta
^{\mu}_{\mu}$ is given by $\langle\Theta
^{\mu}_{\mu}\rangle=\langle\Theta \rangle-3p$, and for two massless quarks in the low temperature chiral perturbation limit the trace has following form [@19] $$\label{eqn39}
\langle\Theta
^{\mu}_{\mu}\rangle=\frac{\pi^2}{270}\frac{T^8}{F_{\pi}^{4}}\ln
\frac{\Lambda_{p}}{T}+ O(T^{10})\,\, . \\$$ Here $\langle\Theta \rangle$ is the total energy density and $p$ is the pressure, whose expressions are known in the low temperature region [@20]. The pion decay constant has the value of $F_{\pi}=0.093GeV$ and the logarithmic scale factor is $\Lambda_{p}=0.275GeV$. We also use the fact that, the quark and gluon energy densities at finite temperature can be expressed as $n_f\langle\Theta^{f}\rangle=\langle\Theta^{g}\rangle=\frac{1}{2}\langle\Theta\rangle$, which agrees both with the naive counting of the degrees of freedom and empirical studies of the pion structure functions [@4], [@8].
For the numerical evolution of the above sum rule, we use the values $m\langle\overline{\psi}\psi\rangle=-0.82\times10^{-4}\,\,GeV^{4} $, $\Lambda=0.230\,\,GeV$ and $m_{\sigma}=0.6\,\,GeV$. We study the dependence of $\sigma$ meson mass and leptonic decay constant on $M^2$, when $M^2$ changes between $0.9\,\,GeV^{2}$ and $1.4\,\,GeV^{2}$. This region of $M^2$ is obtained from the mass sum rule analysis of the $\sigma$ meson [@14].
The shifts in $\sigma$ meson mass and leptonic decay constant as a function of temperature for different values of $M^2$ is shown in Fig.1 and Fig.2, respectively.
As seen, the results for $\Delta m_{\sigma}$ are stable for temperatures up to $120\,\,MeV$. At high temperatures the results for $\Delta m_{\sigma}$ becomes unstable and the contributions of higher dimensional operators become important here, whose inclusions might restore the stability in $M^2$ to higher temperatures. The results for $\Delta \lambda_{\sigma}$ are stable and leptonic decay constant decreases with increasing temperature and vanishes approximately at temperature $T=160\,\,MeV$ . This situation may be interpreted as a signal for deconfinement and agrees with heavy-light mesons investigations [@23]. Numerical analysis shows that the temperature dependence of $\Delta \lambda_{\sigma}$ is the same, when $M^2$ changes between $0.9\,\,GeV^{2}$ and $1.4\,\,GeV^{2}$.
Obtained results can be used for interpretation heavy ion collision experiments. It is also essential to compare these results with other model calculations. We believe these studies to be of great importance for understanding phenomenological and theoretical aspects of thermal QCD.
Acknowledgement
===============
This work is supported by the Scientific and Technological Research Council of Turkey (TUBITAK), research project no.105T131, and the Research Fund of Kocaeli University under grant no. 2004/4.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A supersymmetric way of imposing the constraint of no double occupancy in models with strong on-site Coulomb repulsion is presented in this paper. In this formulation the physical operators in the constrainted Hilbert space are invariant under local unitary transformations mixing boson and fermion representations. As an illustration the formulation is applied to the $t-J$ model. The model is studied in the mean-field level in the $J=0$ limit where we show how both the slave-boson and slave-fermion formulations are included naturally in the present approach and how further results beyond both approaches are obtained.'
address: 'Dept. of Physics, HKUST, Kowloon, Hong Kong'
author:
- 'T. K. Ng and C. H. Cheng'
date:
title: ' Supersymmetry in models with strong on-site Coulomb repulsion - application to t-J model '
---
The $t-J$ model has became a focus in the study of strongly correlated metals and High-$T_c$ superconductors since it was proposed in late eighties[@and]. Because of lack of small parameters for expansion, analytical understandings of the model were largely depending on mean-field theories which treat the constraint of no double occupancy only on average. So far, the most successful mean-field approaches to the $t-J$ model seems to be based on either the slave-fermion mean-field theory(SFMFT)[@sf] which is successful at very small doping when antiferromagnetic correlation is important, and the slave-boson mean-field theory(SBMFT)[@sb] which is successful at larger value of doping when the system becomes superconducting. The only difference between the two approaches is that two different representations of spin and electron operators are used to impose the constraint of no double occupancy. More recently, the focus in the study of High-$T_c$ superconductors has turned to the underdoped and spin-glass regimes where it is believed that the subtle interplay between antiferromagnetism and superconductivity determines the properties of this crossover region. In particular, the importance of $SU(2)$ symmetry in the underdoped regime of the $t-J$ model has been pointed out[@wl; @lw2]. Alternatively, it was also suggested that an $SO(5)$ symmetry may play an important role in determining the competition between antiferromagnetism and superconductivity in the high-$T_c$ cuprates[@zhang].
To understand the complicated behaviour in this regime of the $t-J$ model, it seems that a unified approach which incorporate both the advantages of the slave-fermion mean-field theory and the slave-boson mean-field theory is essential. In this paper, we shall show that it is possible in general to formulate models with constraint of no double occupancy in a way which incorporates the advantages of both slave-fermion and slave-boson representations. In this new formulation the physical operators are [*supersymmetric*]{} and are invariant under unitary transformations mixing fermion and boson representations. The formulation suggests that supersymmetry exists naturally in strongly-correlated systems where on-site Coulomb repulsions are strong. In the following, we shall use the $t-J$ model as an example to illustrate our approach. To begin with, we first consider the Hilbert space of a lattice model with constraint of no double occupancy imposed.
The constraint of no double occupancy implies that there are three possible states on any single lattice site in the model. The site can be either empty (hole state), or can be occupied by either an up- or down- spin electron. In the slave-boson approach, the hole state is represented as a boson, whereas spins are represented as fermions[@sb]. It is also equally valid to represent spins as bosons, and holes as slave fermions, as in the slave fermion treatment[@sf]. In our formulation we shall consider an enlarged Hilbert space where both possibility of representing hole and spins coexist as different states of the system, i.e. there are now six possible states per site in the Hilbert space, represented by,
\[hilbert\] $$\label{hil1}
|\sigma_i^{(f)}>=c^+_{i\sigma}|0>\;, \;\;\;
|h_i^{(b)}>=b^+_i|0>,$$ and $$\label{hil2}
|\sigma_i^{(b)}>=\bar{Z}_{i\sigma}|0>\;, \;\;\;
|h_i^{(f)}>=f^+_{i}|0>,$$
where $\sigma=\uparrow,\downarrow$, $c^+_{i\sigma}$ and $\bar{Z}_{i\sigma}$ are fermionic and bosonic spin creation operators, respectively and $|0>$ is the vacuum state. Similarly, $b^+_i$ and $f^+_i$ are bosonic and fermionic hole creation operators, respectively. Notice that we have seperated the states into “slave-boson” (1a) and “slave-fermion” (1b) groups in Eq. (\[hilbert\]). For a system of $N$-lattice sites, both groups of states are allowed at all sites in our formulation and the total Hilbert space is thus $2^N$ times larger than the Hilbert space of the original model. The essence of our approach is to construct a Hamiltonian which is equivalent to the original model in all these $2^N$ groups of states, and consequently our system with enlarged Hilbert space is equivalent to $2^N$ replicas of the original model. To see how this Hamiltonian can be constructed for the $t-J$ model we first consider spin operators.
We consider the spin operator $\vec{s}_i$ at site $i$,
\[spino\] $$\label{sp1}
\vec{s}_i=\vec{s}^{(f)}_i+\vec{s}^{(b)}_i,$$ where $$\label{sp2}
\vec{s}^{(f(b))}_i=\left(c^+(\bar{Z})_{i\uparrow},
c^+(\bar{Z})_{i\downarrow}\right)
\vec{\sigma}\left(\begin{array}{r}
c(Z)_{i\uparrow} \\
c(Z)_{i\downarrow}
\end{array}\right),$$
and $\sigma$ is the usual Pauli matrix. Notice that $\vec{s}^{(f)}$ and $\vec{s}^{(b)}$ are the spin operators in usual slave-boson and slave-fermion representations, respectively. It is obvious that $\vec{s}_i$ is itself a spin operator since it is the sum of two spin operators. The matrix elements $<\sigma_i^{(\alpha)}|\vec{s}_i|\sigma{'}_i^{(\beta)}>$ where $\alpha,\beta=f,b$ and $\sigma,\sigma{'}=\uparrow,\downarrow$ can be computed easily where it is easy to see that $<\sigma_i^{(f)}|\vec{s}_i|\sigma{'}_i^{(f)}>=
<\sigma_i^{(b)}|\vec{s}_i|\sigma{'}_i^{(b)}>$ and gives the usual spin operator matrix elements between spin-$1/2$ states whereas all other matrix elements with $\alpha\neq\beta$ are equal to zero. Thus the states $|\sigma_i^{(f)}>$ and $|\sigma_i^{(b)}>$ together form two identical replicas of spin-$1/2$ states on site $i$ with our definition of spin operator (\[spino\]). It can then be shown by direct evaluation that the Hamiltonian $$H_J=J\sum_{<i,j>}\vec{s}_i.\vec{s}_j,$$ represents $2^N$ identical copies of Heisenberg interaction in our system of enlarged Hilbert space.
The electron annihilation and creation operators in our system can be defined as $\psi_{i\sigma}=
h^+_i\xi_{i\sigma}$, and $\psi^+_{i\sigma}=\xi^+_{i\sigma}h_i$, respectively, where $$\label{hxi}
h_i=\left(\begin{array}{r}
f_i \\
b_i
\end{array}\right)\; , \;\;\;
\xi_{i\sigma}=\left(\begin{array}{r}
Z_{i\sigma} \\
c_{i\sigma}
\end{array}\right),$$ are doublets of hole and spin operators carrying fermion and boson statistics. It is straightforward to show that the electron operators defined this way satisfies the usual electron commutation relations in the Hilbert space spanned by Eq. (\[hilbert\]). The kinetic energy term $H_t$ of the $t-J$ model can be constructed by requiring that the hopping matrix elements $<\sigma_j^{\alpha'},h_i^{\beta'}|H_t|h_j^{\alpha},\sigma_i^{\beta}>
=-t_{ij}\delta_{\alpha\alpha'}\delta_{\beta\beta'}$, where $\alpha,
\alpha',\beta,\beta'=b,f$ and $|h_j^{\alpha},\sigma^{\beta}>$ represents a state with a hole belonging to group $\alpha$ on site $j$ and a spin $\sigma$ belonging to group $\beta$ on site $i$. Notice that the group indices $\alpha, \beta$’s are “conserved” in constructing $H_t$. It is straightforward to show that $$\label{t-term}
H_t=-t\sum_{<i,j>,\sigma}\left(c^+_{j\sigma}b_jb^+_ic_{i\sigma}+
c^+_{j\sigma}b_jf^+_iZ_{i\sigma}
+\bar{Z}_{j\sigma}f_jb^+_ic_{i\sigma}
-\bar{Z}_{j\sigma}f_jf_i^+Z_{i\sigma}+c.c.\right),$$ where we have considered a Hamiltonian with nearest neighbor hopping only. The four different terms in $H_t$ give the matrix elements of the hopping term between the four possible combination of groups of states at sites $<i,j>$. With this it is easy to verify that our system with Hamiltonian $H=H_J+H_t$ is equivalent to $2^N$ copies of the usual $t-J$ model. Notice that the hopping term cannot be simply represented as $H_t=-t\sum(\psi^+_{i\sigma}\psi_{j\sigma}+c.c.)$ because of the sign difference in the “slave-fermion” term $\bar{Z}_{j\sigma}f_jf_i^+Z_{i\sigma}$. This sign difference is well known in studies of slave-fermion mean-field theory[@sf].
The invariance of our system under change of group of states defined in Eq. (\[hilbert\]) at any lattice site can be expressed in the language of supersymmetry. To see that first we note that the spin operator (\[spino\]) can be written using the $\xi_{\sigma}$ fields as $$\label{spinxi}
s_i^z ={1\over2}(\xi^+_{i\uparrow}\xi_{i\uparrow}-\xi^+_{i\downarrow}
\xi_{i\downarrow})\;, \;\;\;
s^+_i = \xi^+_{i\uparrow}\xi_{i\downarrow}\; , \;\;\;
s^-_i = \xi^+_{i\downarrow}\xi_{i\uparrow},$$ where $\xi^+_{i\sigma}\xi_{i\sigma'}=
\bar{Z}_{i\sigma}Z_{i\sigma'}+c^+_{i\sigma}c_{i\sigma}$. It is now easy to see that the spin and electron operators are invariant under the super-unitary transformation $$\label{un}
\xi_{i\sigma}\rightarrow{U}_i\xi_{i\sigma} \;, \;\;\;
h_i\rightarrow{U}_ih_i$$ where $U_i$’s are local $2\times2$ unitary super-matrices mixing the fermion and boson representations of spins and holes. The generators of the super-unitary transformations are superspin operators $$\begin{aligned}
\label{gen}
S^z_i & = & {1\over2}\left((\sum_{\sigma}c^+_{i\sigma}c_{i\sigma}+b^+_ib_i)
-(\sum_{\sigma}\bar{Z}_{i\sigma}Z_{i\sigma}+f^+_if_i)\right),
\\ \nonumber
S^+_i & = & \sum_{\sigma}c^+_{i\sigma}Z_{i\sigma}-b^+_if_i,
\\ \nonumber
S^-_i & = & \sum_{\sigma}\bar{Z}_{i\sigma}c_{i\sigma}-f^+_ib_i,\end{aligned}$$ where we also define the (magnitude$)^2$ of the superspin as $S^2=S^zS^z+{1\over2}(S^+S^-+S^-S^+)$. Using the fact that the allowed states in our Hilbert space can only be singly ocupied by either spin or hole, it is easy to show that $$\label{cas}
S_i^2={3\over4}\left(\sum_{\sigma}\xi^+_{i\sigma}\xi_{i\sigma}
+h^+_ih_i\right)={3\over4},$$ where we have used the requirement that $\sum_{\sigma}(\bar{Z}_{i\sigma}
Z_{i\sigma}+c^+_{i\sigma}c_{i\sigma})+f^+_if_i+b^+_ib_i=1$ in our Hilbert space (\[hilbert\]) in writing down the last equality. Notice that the constraint of no double occupancy is equivalent to the condition that the [*magnitudes*]{} of the superspins are fixed ($=1/2$) on all lattice sites! In terms of the superspin operators, the slave-boson and slave-fermion representations are equivalent to fixing the direction of superspins to be pointing up and down, respectively and supersymmetry expresses the fact that the physical observables in the system are in fact, invariant under local (but time-independent) rotations in the superspin space. Notice that because of the “minus” sign in the “slave-fermion” hopping term $\bar{Z}_{j\sigma}f_jf^+_iZ_{i\sigma}$, supersymmetry is straightly speaking, broken by the $t-$ term in the $t-J$ model. However, our formulation of $t-J$ model in the enlarged Hilbert space (\[hilbert\]) does not require supersymmetry in the Hamiltonian and is still valid.
The Lagrangian of the $t-J$ model in the enlarged Hilbert space is $$\begin{aligned}
\label{lag}
L & = & i\hbar\sum_{i,\sigma}\xi_{i\sigma}^+{\partial\over\partial{t}}
\xi_{i\sigma}+i\hbar\sum_ih^+_i{\partial\over\partial{t}}h_i-H_t-H_J
\\ \nonumber
& & +\sum_i\lambda_i\left(\sum_{\sigma}\xi^+_{i\sigma}\xi_{i\sigma}+
h^+_ih_i-1\right)-\mu\sum_ih^+_ih_i,\end{aligned}$$ and a Path-integral formulation of the problem can be written down as usual. In particular, the partition function of the present model is $2^N$ times the partition function of the original $t-J$ model. Notice that the constraint of no double occupancy in our enlarged Hilbert space is imposed by an Lagrange multiplier term as usual.
It is obvious that the same approach can be applied to other models with constraint of no double occupancy, as long as a proper Hamiltonian can be constructed in the enlarged Hilbert space. For example, we find that the Hamiltonian for the Infinite-$U$ Anderson model in the supersymmetric representation is, $$\label{and}
H_{and}=\sum_{\vec{k},\sigma}\epsilon_{\vec{k}}a^+_{\vec{k}\sigma}
a_{\vec{k}\sigma}+\epsilon_o\sum_{i,\sigma}\xi^+_{i\sigma}\xi_{i\sigma}
+V\sum_{i,\sigma}(a^+_{i\sigma}\psi_{i\sigma}+\psi^+_{i\sigma}a_{i\sigma}),$$ where $a(a^+)_{\vec{k}\sigma}$ is the annihilation(creation) operator for conduction electrons with momentum $\vec{k}$ and spin $\sigma$ and $\xi(\xi^+)_{i\sigma}$ are annihilation(creation) operator for localized spins on site $i$ where constraint of no double occupancy is imposed. $\epsilon_{\vec{k}}$, $\epsilon_o$ and $V$ have their usual meaning in Anderson model. The spin and electron operators in the localized orbitals are represented by $\xi_{i\sigma}$ and $\psi_{i\sigma}$ operators which are supersymmetric in our formulation and the constraint of no double occupancy can be imposed by Langrange multiplier fields as in the $t-J$ model. Notice that the Infinite-$U$ Anderson model is, in fact, completely supersymmetric, in constrast to the $t-J$ model where supersymmetry is broken by $t-$ term.
To see the advantage of present formulation we shall consider in the following the $t-J$ model in the $J=0$ limit, and shall study the model at two dimension in the mean-field level. The model is equivalent to the $U=\infty$ Hubbard model which is a model of interests in itself[@u1; @u2].
A mean-field theory in the supersymmetric $t-(J=0)$-model can be obtained by making the following decouplings, $$\begin{aligned}
\label{ct}
H & \rightarrow & -t\sum_{<i,j>,\sigma}\left(<c^+_{j\sigma}c_{i\sigma}>
b_jb^+_i+<b_jb^+_i>c^+_{j\sigma}c_{i\sigma}-<c^+_{j\sigma}c_{i\sigma}>
<b_jb^+_i>+c.c.\right. \\ \nonumber
& & +<c^+_{j\sigma}Z_{i\sigma}>b_jf^+_i+c^+_{j\sigma}Z_{i\sigma}<b_jf^+_i>
-<c^+_{j\sigma}Z_{i\sigma}><b_jf^+_i>+c.c. \\ \nonumber
& & +f_jb^+_i<\bar{Z}_{j\sigma}c_{i\sigma}>+<f_jb^+_i>\bar{Z}_{j\sigma}
c_{i\sigma}-<f_jb^+_i><\bar{Z}_{j\sigma}c_{i\sigma}>+c.c. \\ \nonumber
& & \left.-<\bar{Z}_{j\sigma}Z_{i\sigma}>f_jf^+_i-\bar{Z}_{j\sigma}Z_{i\sigma}
<f_jf^+_i>+<\bar{Z}_{j\sigma}Z_{i\sigma}><f_jf^+_i>+c.c.\right) \\ \nonumber
& & +\lambda\sum_{i,\sigma}(\bar{Z}_{i\sigma}Z_{i\sigma}+
c^+_{i\sigma}c_{i\sigma})+(\lambda-\mu)\sum_i(f^+_if_i+b^+_ib_i)\end{aligned}$$ where $<A>$ is the expectation value of operator $A$ evaluated with the mean-field Hamiltonian. The mean-field parameters $\lambda$ and $\mu$ are chosen so that $<f^+_if_i+b^+_ib_i>=\delta$ (hole concentration) and $\sum_{\sigma}
<\bar{Z}_{i\sigma}Z_{i\sigma}+c^+_{i\sigma}c_{i\sigma}>=1-\delta$. We shall be interested at translationally invariant solutions where the mean-field parameters are independent of $<i,j>$ in the following.
Using the fact that $<C>=0$ for Grassman variables $C$’s, we find that the second and third lines in the mean-field Hamiltonian (\[ct\]) are equal to zero. However, both slave-boson and slave-fermion type terms are still present in the mean-field Hamiltonian (first and fourth lines in (\[ct\])). The relative weight of the two kinds of mean-field terms are determined by the occupation numbers of spins (and holes) with fermion (boson) and boson (fermion) statistics. These numbers are in term, determined by minimizing the free energy of the system with the constraint that the [*total*]{} number of spins and holes are fixed to be $1-\delta$ and $\delta$, respectively.
Solving the mean-field equations we find that at low temperatures there are two solutions corresponds to local minima in the mean-field free energy for any given hole concentration $\delta$. The two solutions have either $<c^+c>,<b^+b>\neq0$ and $<\bar{Z}Z>,<f^+f>=0$ or the other way around and corresponds to the usual slave-boson and slave-fermion solutions. New solutions where both slave-fermion and slave-boson mean-field parameters are nonzero are also present. However, these solutions correspond to saddle points in free energy landscape and are unstable. Comparing the free energies of the two stable solutions we find that at zero temperature and at low doping $\delta\leq0.33$ the slave-fermion-like solution has lower energy whereas for $\delta\geq0.33$ slave-boson-like solution has lower energy. The slave-fermion solution where the spinons condensed with $<Z_{i\uparrow}>=<Z_{i
\downarrow}>$ corresponds to [*ferromagnetic*]{} state with spin pointing in $x-$ direction whereas the slave-boson solutions correspond to paramagnetic state. The mean-field theory predicts a first-order transition at zero temperature from ferromagnetic to paramagnetic state as concentration of hole increases across $\sim0.33$. Notice that numerical[@u1] and analytical[@u2] works have established that the Nagaoka (ferromagnetic) state is unstable in the infinite-U Hubbard model for any finite concentration of doping $\delta$ and our mean-field phase diagram is incorrect. Nevertheless, our mean-field theory does produce the qualitative features that the spins (holes) excitations are bosonic (fermionic) like in the ferromagnetic state of the model, and are fermionic (bosonic) like in the paramagnetic state[@u3].
It has to be emphasized that although the mean-field solutions are either slave-boson or slave-fermion like, spins and holes excitations with both statistics are present in the supersymmetric mean-field theory. For example, in the slave-boson-like solution where $<\bar{Z}Z>,<f^+f>=0$, bosonic spin and fermionic hole excitations still appear as dispersionless high energy excitations with energies $\lambda$ and $\lambda-\mu$, respectively in the mean-field theory. Similar situation occurs also in the slave-fermion-like solution. The simultaneous appearance of spin and hole excitations with both statistics in the supersymmetric mean-field theory implies that at finite temperature, the physical properties of the theory is quite different from usual slave-boson or slave-fermion mean-field theories. For example, extra incoherent parts in the one-electron Green’s function which does not exist in usual slave-boson or slave-fermion theories will appear in the supersymmetric theory at nonzero temperature.
At high temperature $T>>\delta(1-\delta){t}$ the only stable mean-field solution which exists is with mean-field parameters $<\bar{Z}Z>,<f^+f>,<c^+c>,<b^+b>$ all equal to zero, indicating that the high-temperature phase is completely incoherent. The transition from coherent low temperature states to incoherent high temperature state is a first-order phase transition in mean-field theory. However, similar results were also obtained in conventional slave-fermion or slave-boson mean-field theories where it is believed that the prediction of first-order phase transition from low-temperature to high-temperature phases is a defect of mean-field theory and should be replaced by a smooth crossover when temperature increases. The mean-field phase diagram is shown in figure 1, where the three stable phases are seperated by lines indicating first-order phase transitions in mean-field theory.
Summarizing, by extending the size of Hilbert space, we present in this paper a new way of formulating $t-J$ model where supersymmetry is inherent in the physical observables. The consequence of this hidden supersymmetry is exploited in the $t-J$ model where a new, supersymmetric mean-field theory of the model is introduced and is studied in the $J\rightarrow0$ limit. The new mean-field theory unifies the slave-boson and slave-fermion mean-field treatments of $t-J$ model by including them as subsets of possible solutions of the new mean-field theory. More generally, new phases where both slave-boson and slave-fermion type mean-field parameters are nonzero may also exist.
Perhaps the more important message in this paper is the demonstration of existence of supersymmetry in general strongly correlated systems where the low energy physics is described by effective Hamiltonians with constraint of no double occupancy like the $t-J$ or infinite-$U$ Anderson model. The consequence of this supersymmetry is never exploited in studies of strongly-correlated systems. Note that supersymmetry is broken by the $t-$ term in the $t-J$ model and it’s consequences are not fully exploited in our paper presenting only the mean-field treatment of $t-(J=0)$ model. Further studies of the $t-J$ model is underway and the results will be presented in future papers.
We acknowledges the support of HKRGC through grant no. HKUST6143/97P.
P.W. Anderson, Science [**235**]{}, 1196 (1987). G. Baskaran, Z.Zou and P.W. Anderson, Solid State Commun. [**63**]{}, 973 (1987); C. Gros, R. Joynt and T.M. Rice, , 8190 (1987); G. Kotliar and J. Liu, , 5142 (1988). C. Jayaprakash, H.R. Krishnamurthy, and S. Sarker, , 2610 (1989); C.L. Kane, P.A. Lee, T.K. Ng, B. Chakraborty and N. Read, , 2653 (1990). X.-G. Wen and P.A. Lee, , 503 (1996). P.A. Lee, N. Nagaosa, T.K. Ng and X.-G. Wen, to appear in S.C. Zhang, Science [**275**]{}, 1089 (1997). W.O. Putikka, M.U. Luchini and M. Ogata, , 2288 (1992). B.S. Shastry, H.R. Krishnamurthy and P.W. Anderson, , 2375 (1990); S.-Q. Shen, Z.-M. Qiu and G.-S. Tian, Phys. Lett. A[**178**]{}, 426 (1993). W. Long and X. Zotos, , 317 (1993).
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Let $z=(z_1, \cdots, z_n)$ and $\Delta=\sum_{i=1}^n {\frac}{{\partial}^2}{{\partial}z^2_i}$ the Laplace operator. A formal power series $P(z)$ is said to be [*Hessian Nilpotent*]{}(HN) if its Hessian matrix ${ \text{Hes\,} }P(z)=({\frac}{{\partial}^2 P}{{\partial}z_i{\partial}z_j})$ is nilpotent. In recent developments in [@BE1], [@M] and [@HNP], the Jacobian conjecture has been reduced to the following so-called [*vanishing conjecture*]{}(VC) of HN polynomials: [*for any homogeneous HN polynomial $P(z)$ $($of degree $d=4$$)$, we have $\Delta^m P^{m+1}(z)=0$ for any $m>>0$.*]{} In this paper, we first show that, the VC holds for any homogeneous HN polynomial $P(z)$ provided that the projective subvarieties ${\mathcal Z}_P$ and ${\mathcal Z}_{\sigma_2}$ of ${{\mathbb C}}P^{n-1}$ determined by the principal ideals generated by $P(z)$ and $\sigma_2(z)\!:=\sum_{i=1}^n z_i^2$, respectively, intersect only at regular points of ${\mathcal Z}_P$. Consequently, the Jacobian conjecture holds for the symmetric polynomial maps $F=z-\nabla P$ with $P(z)$ HN if $F$ has no non-zero fixed point $w\in {{\mathbb C}}^n$ with $\sum_{i=1}^n w_i^2=0$. Secondly, we show that the VC holds for a HN formal power series $P(z)$ if and only if, for any polynomial $f(z)$, $\Delta^m (f(z)P(z)^m)=0$ when $m>>0$.'
author:
- 'Arno van den Essen$^{*}$ and Wenhua Zhao$^{**}$'
title: Two Results on Homogeneous Hessian Nilpotent Polynomials
---
**Introduction and Main Results**
=================================
Let $z=(z_1, z_2, \cdots, z_n)$ be commutative free variables. Recall that the well-known Jacobian conjecture claims that: [*any polynomial map $F(z): {{\mathbb C}}^n \to {{\mathbb C}}^n$ with the Jacobian $j(F)(z)\equiv 1$ is an autompophism of ${{\mathbb C}}^n$ and its inverse map must also be a polynomial map*]{}. Despite intense study from mathematicians in more than sixty years, the conjecture is still open even for the case $n=2$. In 1998, S. Smale [@S] included the Jacobian conjecture in his list of $18$ important mathematical problems for $21$st century. For more history and known results on the Jacobian conjecture, see [@BCW], [@E] and references there.
Recently, M. de Bondt and the first author [@BE1] and G. Meng [@M] independently made the following remarkable breakthrough on the Jacobian conjecture. Namely, they reduced the Jacobian conjecture to the so-called [*symmetric*]{} polynomial maps, i.e the polynomial maps of the form $F=z-\nabla P$, where $\nabla P\! :=({\frac}{{\partial}P}{{\partial}z_1}, {\frac}{{\partial}P}{{\partial}z_2},
\cdots ,{\frac}{{\partial}P}{{\partial}z_n})$, i.e. $\nabla P(z)$ is the [*gradient*]{} of $P(z) \in {{\mathbb C}}[z]$.
For more recent developments on the Jacobian conjecture for symmetric polynomial maps, see [@BE1]–[@BE4].
Based on the symmetric reduction above and also the classical homogeneous reduction in [@BCW] and [@Y], the second author in [@HNP] further reduced the Jacobian conjecture to the following so-called vanishing conjecture.
Let $\Delta\! :=\sum_{i=1}^n {\frac}{{\partial}^2}{{\partial}z^2_i}$ the Laplace operator and call a formal power series $P(z)$ [*Hessian nilpotent*]{}(HN) if its Hessian matrix ${ \text{Hes\,} }P(z)\! :=({\frac}{{\partial}^2 P}{{\partial}z_i{\partial}z_j})$ is nilpotent. It has been shown in [@HNP] that the Jacobian conjecture is equivalent to
[**(Vanishing Conjecture of HN Polynomials)**]{}\
[*For any homogeneous HN polynomial $P(z)$ $($of degree $d=4$$)$, we have $\Delta^m P^{m+1}=0$ when $m>>0$.*]{}
Note that, it has also been shown in [@HNP] that $P(z)$ is HN if and only if $\Delta^m P^{m}=0$ for $m\geq 1$.
In this paper, we will prove the following two results on HN polynomials.
Let $P(z)$ be a homogeneous HN polynomial of degree $d\geq 3$ and $\sigma_2(z)\!\! :=\!\sum_{i=1}^n z_i^2$. We denote by ${\mathcal Z}_P$ and ${\mathcal Z}_{\sigma_2}$ the projective subvarieties of ${{\mathbb C}}P^{n-1}$ determined by the principal ideals generated by $P(z)$ and $\sigma_2(z)$, respectively. The first main result of this paper is the following theorem.
\[MainResult-1\] Let $P(z)$ be a homogeneous HN polynomial of degree $d \geq 4$. Assume that ${\mathcal Z}_P$ intersects with ${\mathcal Z}_{\sigma_2}$ only at regular points of ${\mathcal Z}_P$, then the vanishing conjecture holds for $P(z)$. In particular, the vanishing conjecture holds if the projective variety ${\mathcal Z}_P$ is regular.
Note that, when $\deg P(z)=d=2$ or $3$, the Jacobian conjecture holds for the symmetric polynomial map $F=z-\nabla P$. This is because, when $d=2$, $F$ is a linear map with $j(F)\equiv 1$. Hence $F$ is an automorphism of ${{\mathbb C}}^n$; while when $d=3$, we have $\deg F=2$. By Wang’s theorem [@Wa], the Jacobian conjecture holds for $F$ again. Then, by the equivalence of the vanishing conjecture for the homogeneous HN polynomial $P(z)$ and the Jacobian conjecture for the symmetric map $F=z-\nabla P$ established in [@HNP], we see that, when $\deg P(z)=d=2$ or $3$, Theorem \[MainResult-1\] actually also holds even without the condition on the projective variety ${\mathcal Z}_P$.
For any non-zero $z\in {{\mathbb C}}^n$, denote by $[z]$ its image in the projective space ${{\mathbb C}}P^{n-1}$. Set $$\begin{aligned}
\widetilde {\mathcal Z}_{\sigma_2}\! :=\{z \in {{\mathbb C}}^n \,|\, z\neq 0;\,
[z] \in {\mathcal Z}_{\sigma_2}\}. \end{aligned}$$ In other words, $\widetilde {\mathcal Z}_{\sigma_2}$ is the set of non-zero $z\in {{\mathbb C}}^n$ such that $\sum_{i=1}^n z_i^2=0$.
Note that, for any homogeneous polynomial $P(z)$ of degree $d$, it follows from the Euler’s formula $dP=\sum_{i=1}^n z_i\frac{d P}{d z_i}$, that any non-zero $w\in {{\mathbb C}}^n$, $[w]\in {{\mathbb C}}P^{n-1}$ is a singular point of ${\mathcal Z}_P$ if and only if $w$ is a fixed point of the symmetric map $F=z-\nabla P$. Furthermore, it is also well-known that, $j(F)\equiv 1$ if and only if $P(z)$ is HN.
By the observations above and Theorem \[MainResult-1\], it is easy to see that we have the following corollary on symmetric polynomial maps.
\[MR1-corol-1\] Let $F=z-\nabla P$ with $P$ homogeneous and $j(F)\equiv 1$ $($or equivalently, $P$ is HN$)$. Assume that $F$ does not fix any $w\in \widetilde {\mathcal Z}_{\sigma_2}$. Then the Jacobian holds for $F(z)$. In particular, if $F$ has no non-zero fixed point, the Jacobian conjecture holds for $F$.
Our second main result is following theorem which says that the vanishing conjecture is actually equivalent to a formally much stronger statement.
\[MainResult-2\] For any HN polynomial $P(z)$, the vanishing conjecture holds for $P(z)$ if and only if, for any polynomial $f(z)\in {{\mathbb C}}[z]$, $\Delta^m (f(z)P(z)^m)=0$ when $m>>0$.
**Proof of the Main Results**
=============================
Let us first fix the following notation. Let $z=(z_1, z_2,\cdots, z_n)$ be free complex variables and ${{\mathbb C}}[z]$ (resp.${{\mathbb C}}[[z]]$) the algebra of polynomials (resp.formal power series) in $z$. For any $d \geq 0$, we denote by $V_d$ the vector space of homogeneous polynomials in $z$ of degree $d$.
For any $1\leq i\leq n$, we set $D_i={\frac}{{\partial}}{{\partial}z_i}$ and $D=(D_1, D_2,\cdots, D_n)$. We define a ${{\mathbb C}}$-bilinear map $\{\cdot, \cdot\}: {{\mathbb C}}[z] \times {{\mathbb C}}[z] \to {{\mathbb C}}[z]$ by setting $$\begin{aligned}
\{f, \, g \}\! := f(D)g(z)\end{aligned}$$ for any $f(z), g(z)\in {{\mathbb C}}[z]$.
Note that, for any $m \geq 0$, the restriction of $\{\cdot, \cdot\}$ on $V_m \times V_m $ gives a ${{\mathbb C}}$-bilinear form of the vector subspace $V_m$, which we will denote by $B_m(\cdot, \cdot)$. It is easy to check that, for any $m\geq 1$, $B_m(\cdot, \cdot)$ is symmetric and non-singular.
The following lemma will play a crucial role in our proof of the first main result.
\[MainLemma\] For any homogeneous polynomials $g_i(z)$ $(1\leq i\leq k)$ of degree $d_i\geq 1$, let $S$ be the vector space of polynomial solutions of the following system of PDEs: $$\begin{aligned}
\label{Sys-1}
\begin{cases}
g_1 (D)\, u(z)=0, \\
g_2 (D)\, u(z)=0, \\
\quad ..... \quad \\
g_k (D)\, u(z)=0.
\end{cases}\end{aligned}$$ Then, $\dim S<+\infty$ if and only if $g_i(z)$ $(1\leq i\leq k)$ have no non-zero common zeroes.
[[*Proof:*]{}]{}Let $I$ the homogeneous ideal of ${{\mathbb C}}[z]$ generated by $\{ g_i(z) | 1\leq i\leq k\}$. Since all $g_i(z)$’s are homogeneous, $S$ is a homogeneous vector subspace $S$ of ${{\mathbb C}}[z]$.
Write $$\begin{aligned}
S& =\bigoplus_{m=0}^\infty S_m, \\
I& =\bigoplus_{m=0}^\infty I_m.\end{aligned}$$ where $I_m\! :=I\cap V_m$ and $S_m\! :=I\cap V_m$ for any $m\geq 0$.
First, by the definitions of $I$ and $S$, we have $\{I_m, S_m\}=0$ for any $m\geq 1$, hence $S_m\subseteq I_m^\perp$. Therefore, we need only show that, for any $u(z)\in I_m^\perp\subset V_m$, $g_i(D)u(z)=0$ for any $1\leq i\leq n$.
We first fix any $1\leq i\leq n$. If $m < d_i$, there is nothing to prove. If $m=d_i$, then $g_i(z) \in I_m$, hence $\{g_i, u\}=g_i(D)u=0$. Now suppose $m>d_i$. Note that, for any $v(z)\in V_{m-d_i}$, $v(z) g_i(z)\in I_m$. Hence we have $$\begin{aligned}
0&= \{ v(z) g_i(z), u(z) \}\\
&=v(D)g_i(D) u(z)\\
&=v(D)\left (g_i(D) u \right )(z)\\
&=\{v(z), \left (g_i(D)u\right )(z) \}.\end{aligned}$$
Therefore, we have $$B_{m-d_i}\left( (g_i(D)u) (z),\, V_{m-d_i} \right )=0.$$ Since $B_{m-d_i}(\cdot, \cdot)$ is a non-singular ${{\mathbb C}}$-bilinear form of $V_{m-d_i}$, we have $g_i(D)u=0$. Hence, the Claim holds. [$\Box$]{}
By a well-known fact in Algebraic Geometry (see Exercise $2.2$ in [@H], for example), we know that the homogeneous polynomials $g_i(z)$ $(1\leq i\leq k)$ have no non-zero common zeroes if and only if $I_m=V_m$ when $m>>0$. While, by the Claim above, we know that, $I_m=V_m$ when $m>>0$ if and only if $S_m=0$ when $m>>0$, and if and only if the solution space $S$ of the system (\[Sys-1\]) is finite dimensional. Hence, the lemma follows. [$\Box$]{}
Now we are ready to prove our first main result, Theorem \[MainResult-1\].
: Let $P(z)$ be a homogeneous HN polynomial of degree $d\geq 4$ and $S$ the vector space of polynomial solutions of the following system of PDEs: $$\begin{aligned}
\label{Sys-2}
\begin{cases}
\frac{{\partial}P}{{\partial}z_1} (D)\, u(z)=0, \\
\frac{{\partial}P}{{\partial}z_2} (D)\, u(z)=0, \\
\quad ..... \quad \\
\frac{{\partial}P}{{\partial}z_n} (D)\, u(z)=0, \\
\Delta \, u(z)=0.
\end{cases}\end{aligned}$$
First, note that the projective subvariety ${\mathcal Z}_P$ intersects with ${\mathcal Z}_{\sigma_2}$ only at regular points of ${\mathcal Z}_P$ if and only if $\frac{{\partial}P}{{\partial}z_i}(z)$ $(1\leq i\leq n)$ and $\sigma_2=\sum_{i=1}^n z_i^2$ have no non-zero common zeros (agian use Euler’s formula). Then, by Lemma \[MainLemma\], we have $\dim S < +\infty$.
On the other hand, by Theorem $6.3$ in [@HNP], we know that $\Delta^m P^{m+1}\in S$ for any $m\geq 0$. Note that $\deg \Delta^m P^{m+1}=(d-2)m+d$ for any $m\geq 0$. So $\deg \Delta^m P^{m+1}>\deg \Delta^k P^{k+1}$ for any $m>k$. Since $\dim S < +\infty$ (from above), we have $\Delta^m P^{m+1}=0$ when $m>>0$, i.e. the vanishing conjecture holds for $P(z)$. [$\Box$]{}
Next, we give a proof for our second main result, Theorem \[MainResult-2\].
: The $(\Leftarrow)$ part follows directly by choosing $f(z)$ to be $P(z)$ itself.
To show $(\Rightarrow)$ part, let $d=\deg f(z)$. If $d=0$, $f$ is a constant. Then, $\Delta^m (f(z)P(z)^m)=f(z)\Delta^m P^m=0$ for any $m\geq 1$.
So we assume $d\geq 1$. By Theorem $6.2$ in [@HNP], we know that, if the vanishing conjecture holds for $P(z)$, then, for any fixed $a\geq 1$, $\Delta^m P^{m+a}=0$ when $m>>0$. Therefore there exists $N>0$ such that, for any $0\leq b\leq d$ and any $m>N$, we have $\Delta^m P^{m+b}=0$.
By Lemma $6.5$ in [@HNP], for any $m\geq 1$, we have $$\begin{aligned}
\label{MainResult-2-pe1}
&\Delta^m (f(z) P(z)^m)= \\
&\, \sum_{\substack{k_1+ k_2+ k_3= m\\
k_1, k_2, k_3 \geq 0}} 2^{k_2}\binom {m}{k_1, k_2, k_3}
\sum_{\substack{ {\bf s} \in {{\mathbb N}}^n \\ |{\bf s}|=k_2}}
\binom {k_2}{\bf s} \frac {{\partial}^{k_2} \Delta^{k_1} f(z)}{{\partial}z^{\bf s}}
\frac {{\partial}^{k_2}\Delta^{k_3} P^m(z) }{{\partial}z^{\bf s}},{\nonumber}\end{aligned}$$ where $\binom {m}{k_1, k_2, k_3}$ and $\binom{k_2}{\bf s}$ denote the usual binomials.
Note first that, the general term in the sum above is non-zero only if $2k_1+k_2 \leq d$. But on the other hand, since $$\begin{aligned}
\label{MainResult-2-pe2}
0\leq k_1+k_2\leq 2k_1+k_2\leq d,\end{aligned}$$ by the choice of $N\geq 1$, we have $\Delta^{k_3} P^m(z)=\Delta^{k_3} P^{k_3+(k_1+k_2)}(z)$ is non-zero only if $$\begin{aligned}
\label{MainResult-2-pe3}
k_3\leq N.\end{aligned}$$
From the observations above and Eqs.(\[MainResult-2-pe1\]), (\[MainResult-2-pe2\]), (\[MainResult-2-pe3\]) it is easy to see that, $\Delta^m (f(z) P(z)^m)\neq 0$ only if $m=k_1+k_2+k_3 \leq d+N$. In other words, $\Delta^m (f(z) P(z)^m)=0$ for any $m>d+N$. Hence Theorem \[MainResult-2\] holds. [$\Box$]{}
Note that all results used in the proof above for the $(\Leftarrow)$ part of the theorem also hold for all HN formal power series. Therefore we have the following corollary.
Let $P(z)$ be a HN formal power series such that the vanishing conjecture holds for $P(z)$. Then, for any polynomial $f(z)$, we have $\Delta^m (f(z) P(z)^m)=0$ when $m>>0$.
[FLM2]{}
H. Bass, E. Connell, D. Wright, *The Jacobian conjecture, reduction of degree and formal expansion of the inverse*. Bull. Amer. Math. Soc. **7**, (1982), 287–330. \[MR83k:14028\], \[Zbl.539.13012\].
M. de Bondt and A. van den Essen, [*A Reduction of the Jacobian Conjecture to the Symmetric Case*]{}, Proc. Amer. Math. Soc. [**133**]{} (2005), no. 8, 2201–2205. \[MR2138860\].
M. de Bondt and A. van den Essen, [*Nilpotent Symmetric Jacobian Matrices and the Jacobian Conjecture*]{}, J. Pure Appl. Algebra [**193**]{} (2004), no. 1-3, 61–70. \[MR2076378\].
M. de Bondt and A. van den Essen, [*Nilpotent symmetric Jacobian matrices and the Jacobian conjecture II*]{}, J. Pure Appl. Algebra [**196**]{} (2005), no. 2-3, 135–148. \[MR2110519\].
M. de Bondt and A. van den Essen, [*Singular Hessians*]{}, J. Algebra [**282**]{} (2004), no. 1, 195–204. \[MR2095579\].
A. van den Essen, [*P*olynomial automorphisms and the Jacobian conjecture]{}. Progress in Mathematics, 190. Birkhäuser Verlag, Basel, 2000. \[MR1790619\].
R. Hartshorne, [*Algebraic Geometry*]{}, Springer-Verlag, New York-Heidelberg-Berlin, 1977.
O. H. Keller, [*Ganze Gremona-Transformation*]{}, Monats. Math. Physik [**47**]{} (1939), 299-306.
G. Meng, [*Legendre Transform, Hessian Conjecture and Tree Formula*]{}, Appl. Math. Lett. 19 (2006), no. 6, 503–510. \[MR2170971\]. See also math-ph/0308035.
S. Smale, [*Mathematical Problems for the Next Century*]{}, Math. Intelligencer 20, No. 2, 7-15, 1998. \[MR1631413 (99h:01033)\].
S. Wang, [*A Jacobian criterion for Separability*]{}, J. Algebra [**65**]{} (1980), 453-494. \[MR 83e:14010\].
A. V. Jagžev, [*On a problem of O.-H. Keller.*]{} (Russian) Sibirsk. Mat. Zh. 21 (1980), no. 5, 141–150, 191. \[MR0592226\].
W. Zhao, [*Hessian Nilpotent Polynomials and the Jacobian Conjecture*]{}. Trans. Amer. Math. Soc. 359 (2007), 249-274. \[MR2247890\]. See also math.CV/0409534.
[${}^{*}$ Department of Mathematics, Radboud University Nijmegen, Postbus 9010, 6500 GL Nijmegen, The Netherlands.]{}
[*E-mail*]{}: [email protected]
[${}^{**}$ Department of Mathematics, Illinois State University, Normal, IL 61790-4520.]{}
[*E-mail*]{}: [email protected].
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{
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}
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---
author:
- 'Yassir Jedra and Alexandre Proutiere[^1] [^2]'
bibliography:
- 'references.bib'
- 'references2.bib'
title: |
Finite-time Identification of Stable Linear Systems\
Optimality of the Least-Squares Estimator
---
[^1]: This work was supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation.
[^2]: Y. Jedra and A. Proutiere are with the Division of Decision and Control Systems, School of Electrical Engineering and Computer Science, Royal institute of Technology (KTH), Stockholm, Sweden. Emails: {[*[email protected], [email protected]*]{}}.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We consider the Schrödinger operator on graphs and study the spectral statistics of a unitary operator which represents the quantum evolution, or a quantum map on the graph. This operator is the quantum analogue of the classical evolution operator of the corresponding classical dynamics on the same graph. We derive a trace formula, which expresses the spectral density of the quantum operator in terms of periodic orbits on the graph, and show that one can reduce the computation of the two-point spectral correlation function to a well defined combinatorial problem. We illustrate this approach by considering an ensemble of simple graphs. We prove by a direct computation that the two-point correlation function coincides with the CUE expression for $2\times2$ matrices. We derive the same result using the periodic orbit approach in its combinatorial guise. This involves the use of advanced combinatorial techniques which we explain.'
address:
- |
$^\dag$Max-Planck-Institut für Strömungsforschung,\
37073 Göttingen, Germany\
- |
$^\ddag$[Department of Physics of Complex Systems,]{}\
[The Weizmann Institute of Science, Rehovot 76100, Israel]{}\
author:
- 'Holger Schanz$^\dag$ and Uzy Smilansky$^\ddag$\'
date: 'April 25, 1999'
title: |
Spectral Statistics for Quantum Graphs:\
Periodic Orbits and Combinatorics
---
[(]{}
[)]{}
[\[]{}
[\]]{}
[i]{}
[[N]{}]{}
\
To be published in the\
[*Proceedings of the Australian Summer School on Quantum Chaos and Mesoscopics*]{}\
Canberra, Australia, January 1999
[**Introduction**]{} \[introduction\]
=====================================
We have recently shown [@KS97; @KS99] that the Schrödinger operator on graphs provides a useful paradigm for the study of spectral statistics and their relations to periodic orbit theory. In particular, the universal features which are observed in quantum systems whose classical counterpart is chaotic, appear also in the spectra of quantum graphs. This observation was substantiated by several numerical studies. The relevance to quantum chaology was established by identifying the underlying [*mixing*]{} classical evolution on the graphs, which provides the stability coefficients and actions of periodic orbits in whose terms an exact trace formula can be written [@R83; @KS97; @KS99].
In spite of the large amount of effort invested in the past fifteen years [@berry; @keatbog], we have only a limited understanding of the reasons for the universality of spectral statistics in systems whose classical dynamics is chaotic. The main stumbling block is the lack of understanding of the intricate and delicate interference between the contributions of (exponentially many) periodic orbits. This genuinely quantum quantity, (also known as the “off-diagonal" contribution), is the subject of several researches, which address it from various points of view [@keatbog; @ADDKKSS93; @CPS98; @Miller97; @Agam95]. The present contribution attempts to illuminate this issue from yet another angle, and we harness for this purpose quantum graphs and combinatorics.
Our material is presented in the following way. We shall start by defining the quantum dynamics on the graph in terms of a quantum map. This map will be represented by a unitary matrix, which is the quantum analogue of the classical Frobenius-Perron operator of the properly defined classical dynamics on the graph. The spectrum of the quantum operator is on the unit circle, and its statistics is the main object of the present work. After defining the two-point correlation function of interest, we shall write it down in terms of periodic orbits and discuss the combinatorial problem which should be addressed in order to obtain a complete expression which includes the “off-diagonal" contribution. Since the RMT is known to reproduce the two-point correlation function for generic graphs, we propose that the RMT expression could be obtained from a combinatorial theory, perhaps as the leading term in an asymptotic expansion. For one particular example we show that this is indeed the case in the last section. There we construct an ensemble of simple graphs with non-trivial spectral statistics, which can be solved in two independent ways. The direct way yields the statistics of RMT for the $2\times 2$ circular unitary ensemble (CUE). The corresponding periodic orbit calculation is converted into a combinatorial problem, which is solved by proving a previously unknown combinatorial identity.
[**The Quantum Scattering Map and its Classical Analogue**]{} \[maps\]
======================================================================
General Definitions for Quantum Graphs
--------------------------------------
We shall start with a few general definitions. Graphs consist of $V$ [*vertices*]{} connected by $B$ [*bonds*]{} (or [*edges*]{}). The [*valency*]{} $v_{i}$ of a vertex $i$ is the number of bonds meeting at that vertex. Associated to every graph is its [*connectivity (adjacency) matrix*]{} $C_{i,j}$. It is a square matrix of size $V$ whose matrix elements $C_{i,j}$ are given in the following way $$\begin{aligned}
C_{i,j}=C_{j,i}=\left\{
\begin{array}{l}
1\qquad\text{if}\ i,j\ \text{ are connected} \\
0\qquad\text{otherwise}
\end{array}\right\}
\qquad(i,j=1,\dots,V)\,.
\label{cmat}\end{aligned}$$ The valency of a vertex is given in terms of the connectivity matrix, by $v_i=
\sum_{j=1}^V C_{i,j}$ and the total number of bonds is $B= {1\over
2}\sum_{i,j=1}^VC_{i,j}$.
When the vertices $i$ and $j$ are connected, we shall assume that the connection is achieved by a single bond, such that multiple bonds are excluded. We denote the connecting bond by $b=[i,j]$. Note that the notation $[i,j]$ will be used whenever we do not need to specify the [*direction*]{} on the bond. Hence $[i,j]=[j,i]$. [*Directed bonds*]{} will be denoted by $(i,j)$, and we shall always use the convention that the bond is directed from the first index to the second one. To each bond $[i,j]$ we assign a length $L_{[i,j]} = L_{(i,j)} = L_{(j,i)}$. In most applications we would avoid non-generic degeneracies by assuming that the $L_{[i,j]}$ are [*rationally independent*]{}. The mean length is defined by $\left \langle L \right \rangle
\equiv {1\over B}\sum_{b=1}^B L_b$.
For the quantum description we assign to each bond $b=[i,j]$ a coordinate $x_b$ which measures distances along the bond. We may use $x_{(i,j)}$ which is defined to take the value $0$ at the vertex $i$ and the value $L_{(i,j)}
\equiv L_{(j,i)}$ at the vertex $j$. We can also use $x_{(j,i)}$ which vanishes at $j$ and takes the value $L_{(i,j)}$ at $i$.
The wave function $\Psi$ is a $B-$component vector and will be written as $(\Psi_{b_1}(x_{b_1})$, $\Psi_{b_2}(x_{b_2}),\dots$, $\Psi_{b_B}(x_{b_B}))^T$ where the set $\{b_i\}_{i=1}^B$ consists of all the $B$ distinct bonds on the graph. We will call $\Psi_b(x_b)$ the component of $\Psi$ on the bond $b$. The bond coordinates $x_b$ were defined above. When there is no danger of confusion, we shall use the shorthand notation $\Psi_b(x)$ for $\Psi_b(x_b)$ and it is understood that $x$ is the coordinate on the bond $b$ to which the component $\Psi_b$ refers.
The Schrödinger equation is defined on the graph in the following way [@A83; @A94] (see also [@KS99] for an extensive list of references on the subject): On each bond $b$, the component $\Psi_b$ of the total wave function $\Psi$ is a solution of the one-dimensional equation $$\begin{aligned}
\left(-\i\;{\text{d}/ \text{d}x_{(i,j)}}
-A_{(i,j)}\right)^2\Psi_b(x_{(i,j)})=k^2\Psi_b(x_{(i,j)})
\qquad(b=[i,j])\,.
\label{schrodinger}\end{aligned}$$ We included a “magnetic vector potential" $A_{(i,j)}$, with $A_{(i,j)}=
-A_{(j,i)}$ which breaks time-reversal symmetry.
On each of the bonds, the general solution of (\[schrodinger\]) is a superposition of two counter-propagating waves $$\begin{aligned}
\psi_{(i,j)}(x_{(i,j)}) = \exp
\left (\i\[ kx_{(i,j)} +
A_{(i,j)}x_{(i,j)}\]\right ) \nonumber
\\
\psi_{(j,i)}(x_{(j,i)}) = \exp \left (\i\[ kx_{(j,i)}
+
A_{(j,i)}x_{(j,i)}\]\right )\,.
\label{counterprop}\end{aligned}$$ Note that the above functions are normalised to have an amplitude $1$ at the points from which they “emerge", namely, $\psi_{(i,j)}=1$ at the vertex $i$ and $\psi_{(j,i)}=1$ at the vertex $j$. The Hilbert space of the solutions of (\[schrodinger\]) is spanned by the set of functions defined above, such that for all $b=[i,j]$ $$\Psi_b = a_{(i,j)} \psi_{(i,j)}(x_{(i,j)}) +
a_{(j,i)} \psi_{(j,i)}(x_{(j,i)})\,.
\label {aijdef}$$ Thus, the yet undetermined coefficients $a_{(i,j)}$ form a $2B$-dimensional vector of complex numbers, which uniquely determines an element in the Hilbert space of solutions. This space corresponds to “free wave" solutions since we did not yet impose any conditions which the solutions of (\[schrodinger\]) have to satisfy at the vertices.
The Quantum Scattering Map
--------------------------
The [*quantum scattering map*]{} is a unitary transformation acting in the space of free waves, and it is defined as follows.
In a first step, we prescribe at each vertex $i=1,\dots,V$ a [*vertex scattering matrix*]{} which is a unitary matrix of dimension $v_i$. The vertex scattering matrices may be $k$ dependent and they are denoted by $\sigma_{l,m}^{(i)} (k)$, where the indices $l,m$ take the values of the vertices which are connected to $i$, that is, $C_{i,l}=C_{i,m}=1$. The vertex scattering matrix is a property which is attributed to the vertex under consideration. It can either be derived from appropriate boundary conditions as in [@KS97; @KS99], or, it can be constructed to model other physical situations. The important property of $\sigma_{l,m}^{(i)} (k)$ in the present context is, that any wave which is [*incoming*]{} to the vertex $i$ from the bonds $(l,i)$, and which has an amplitude $1$ at the vertex, is scattered and forms [*outgoing*]{} waves in the bonds $(i,m)$ with amplitudes $\sigma_{l,m}^{(i)} (k)$.
Now, the quantum scattering map is represented by its effect on the $2B$-dimensional vector of coefficients ${\bf a} = \left \{a_{(i,j)}\right \}
$, namely, $\bf a$ is mapped to $\bf a'$ with components $$a'_{ b'} = \sum _{b=1}^{2B}a_{b}
S_{B_{{b,b'}}}\,,
\label{QSM}$$ where $b$ and $b'$ run over all directed bonds, and if we denote $b=(i,j)$ and $b'= (l,m)$ $${S_B}_{(i,j),(l,m)}(k) = \delta _{j,l}
{\rm
e}^{\i L_{(i,j)}(k+A_{(i,j)})}\sigma_{i,m}^{(j)} (k)\,.
\label{S_Bdef}$$ The effect of $S_B$ on a wave function can be intuitively understood as follows. The coefficient $a_{(i,j)}$ is the (complex) amplitude of the wave which emerges from the vertex $i$ and propagates to the vertex $j$. Once it reaches the vertex $j$, it has accumulated a phase ${\rm e}^{i
L_{(i,j)}(k+A_{(i,j)})}$ and it scatters into the bonds which emanate from $j$ with an amplitude given by the appropriate vertex scattering matrix. The new amplitude $a'_{(l=j,m)}$ consists of the superposition of all the amplitudes contributed by waves which impinge on the vertex $l=j$ and then scatter. The name “quantum scattering" map is justified by this intuitive picture.
The resulting matrix $S_B$ is a $2B \times 2B$ unitary matrix. The unitarity follows simply from the unitarity of the vertex scattering matrices, and from the fact that $S_B$ has non-vanishing entries between connected directed bonds: the incoming bond aims at the vertex from which the outgoing bond emerges. The unitarity of $S_B$ implies that its spectrum is restricted to the unit circle. In this paper we shall mainly be concerned with the spectral statistics of the eigenphases, and their relation to the underlying classical dynamics on the graph. The spectral statistics will be discussed in the next chapter. We shall use the remaining part of the present chapter to clarify two important issues. We shall first show how one can use the quantum scattering map to construct the space of solutions of the Schrödinger operator on the graph with boundary conditions. Then, we shall introduce the classical dynamics which corresponds to the scattering map.
To define the space of “bound states" on the graph, one has to restrict the space of wave functions by imposing appropriate boundary conditions on the vertices. The boundary conditions guarantee that the resulting Schrödinger operator is self-adjoint. In [@KS97; @KS99], we described and used one particular set of boundary conditions, which ensure continuity (uniqueness) and current conservation. Here we shall use a slight generalisation, which matches well with the spirit of the present article. We shall impose the boundary conditions in terms of a consistency requirement that the coefficients $a_{(i,j)}$ have to obey. Namely, we require that the wave function (\[aijdef\]) is [*stationary*]{} under the action of the quantum scattering map. In other words, the vector [**a**]{} must be an eigenvector of $S_B(k)$ with a unit eigenvalue. (see also [@Rochus]). This requirement can be fulfilled when $$\det (I-S_B(k)) =0\,.
\label {secular}$$ In [@KS97; @KS99] we have actually derived (\[secular\]), for the particular case in which the vertex scattering matrices where computed form a particular set of vertex boundary conditions which impose continuity and current conservation on the vertices. The resulting vertex scattering matrices read $$\sigma _{j,j^{\prime }}^{(i)}=\left( -\delta _{j,j^{\prime }}+{\frac{
(1+\e^{-\i\omega _i})}{v_i}}\right) C_{i,j}C_{i,j^{\prime
}},\medskip\
\,\,\,\omega _i=2\arctan \frac{\lambda _i}{v_ik}\,.
\label{smatrix}$$ Here, $0\le \lambda_i \le \infty$ are arbitrary constants. The “Dirichlet" (“Neumann") boundary conditions correspond to $ \lambda_i =\infty \ \ (0)$, respectively. The Dirichlet case implies total reflection at the vertex, $\sigma_{j,j^{\prime }} ^{(i)}= -\delta _{j,j^{\prime }}$. For the Neumann boundary condition we have $\sigma _{j,j^{\prime }}^{(i)}=-\delta _{j,j^{\prime
}}+2/{v_i}$ which is independent of $k$. For any intermediate boundary condition, the scattering matrix approaches the Neumann expression as $k
\rightarrow \infty$. Note that in all non-trivial cases ($v_i> 2$), back-scattering ($j =j^{\prime}$) is singled out both in sign and in magnitude: $\sigma _{j,j }^{(i)}$ has always a negative real part, and the reflection probability $|\sigma _{j,j }^{(i)}|^2 $ approaches $1$ as the valency $v_i$ increases. One can easily check that $\sigma ^{(i)}$ is a symmetric unitary matrix, ensuring flux conservation and time reversal symmetry at the vertex. For Neumann boundary conditions $\sigma ^{(i)}$ is a real orthogonal matrix.
The spectral theory of the Schrödinger operators on graphs can be developed using (\[secular\]) as the starting point. In particular, the corresponding trace formula [@R83] can naturally be derived, and related to the underlying classical dynamics [@KS97; @KS99]. Here, we shall study the quantum scattering map on its own right, without a particular reference to its rôle in the construction of the spectrum. We shall consider the ensemble of unitary, $2B\times 2B$ matrices $S_B(k)$, where $k$ is allowed to vary in a certain interval to be specified later. Our main concern will be the statistical properties of the eigenvalues of $S_B$. This will be explained in the next chapter.
The Classical Scattering Map
----------------------------
The last point to be introduced and discussed in the present chapter is the classical dynamics on the graph and the corresponding scattering map.
We consider a classical particle which moves freely as long as it is on a bond. The vertices are singular points, and it is not possible to write down the analogue of Newton’s equations at the vertices. Instead, one can employ a Liouvillian approach based on the study of the evolution of phase-space densities. This phase-space description will be constructed on a Poincaré section which is defined in the following way. Crossing of the section is registered as the particle encounters a vertex, thus the “coordinate" on the section is the vertex label. The corresponding “momentum" is the direction in which the particle moves when it emerges from the vertex. This is completely specified by the label of the next vertex to be encountered. In other words, $$\left\{
\begin{array}{c}
{\rm position} \\
{\rm
momentum}
\end{array}
\right\} \Longleftrightarrow
\left\{
\begin{array}{c}
{\rm vertex}\text{ }{\rm index} \\
{\rm
next}\text{ }{\rm index}
\end{array}
\right\}\,.$$ The set of all possible vertices and directions is equivalent to the set of $2B$ directed bonds. The evolution on this Poincaré section is well defined once we postulate the transition probabilities $P_{j\to
j^{\prime}}^{(i)}$ between the directed bonds $b=\{j,i\}$ and $b^{\prime
}=\{i,j^{\prime }\}$. To make the connection with the quantum description, we adopt the quantum transition probabilities, expressed as the absolute squares of the $S_B$ matrix elements $$P_{j\to j^{\prime }}^{(i)}=\left|
\sigma_{j,j^{\prime
}}^{(i)}(k)\right|
^2\,.
\label{cl1}$$ When the vertex scattering matrices are constructed from the standard matching conditions on the vertices (\[smatrix\]), we get the explicit expression $$P_{j\to
j^{\prime }}^{(i)} =
\left| -\delta _{j,j^{\prime }}+{\frac{(1+\e^{-\i\omega_i})}{v_i}}\right|^2 \,.
\label{cl11}$$ For the two extreme cases corresponding to Neumann and Dirichlet boundary conditions this results in $$\begin{aligned}
P_{j\to j^{\prime }}^{(i)} &=&
\left\{
\begin{array}{ll}
\left( -\delta _{j,j^{\prime }}+{2/{v_i}}\right) ^2 & \text{Neumann} \cr
\delta _{j,j^{\prime}} & \text{Dirichlet}
\end{array}
\right\}\,.\end{aligned}$$ The transition probability $P_{j\to j^{\prime }}^{(i)}$ for the Dirichlet case admits the following physical interpretation. The particle is confined to the bond where it started and thus the phase space is divided into non-overlapping ergodic components ($\approx$ “tori”). For all other boundary conditions the graph is dynamically connected.
The classical Frobenius-Perron evolution operator is a $2B\times 2B$ matrix whose elements $U_{b,b^{\prime }}$ are the classical transition probabilities between the bonds $b,b^{\prime }$ $$U_{ij,nm}=\delta_{j,n} P^{(j)}_{i\to m}\,.
\label{cl3}$$ $U$ does not involve any metric information on the graph, and for Dirichlet or Neumann boundary conditions $U$ is independent of $k$. This operator is the classical analogue of the quantum scattering matrix $S_B$. Usually, one “quantises" the classical operator to generate the quantum analogue. For graphs the process is reversed, and the classical evolution is derived from the more fundamental quantum dynamics.
Let $\rho _b(t), \ b=1,\dots, 2B$ denote the distribution of probabilities to occupy the directed bonds at the (topological) time $t$. This distribution will evolve after the first return to the Poincaré section according to $$\rho _b(t+1)=\sum_{b^{\prime }}U_{b,b^{\prime }}\rho _{b^{\prime }}(t)\,.
\label{master}$$ This is a Markovian master equation which governs the evolution of the classical probability distribution. The unitarity of the graph scattering matrix $S_B$ guarantees $\sum_{b=1}^{2B}U_{b,b^{\prime }}=1$ and $0\leq
U_{b,b^{\prime }}\leq 1$, such that the probability that the particle is on any of the bonds is conserved during the evolution. The spectrum of $U$ is restricted to the unit circle and its interior, and $\nu_1 = 1$ is always an eigenvalue with the corresponding eigenvector $|1\rangle = \frac 1{2B} \left(
1,1,...,1\right) ^T$. In most cases, the eigenvalue $1$ is the only eigenvalue on the unit circle. Then, the evolution is ergodic since any initial density will evolve to the eigenvector $|1\rangle $ which corresponds to a uniform distribution (equilibrium). $$\rho (t)\ \overrightarrow{\scriptstyle t{\rightarrow \infty }}\ |1\rangle\,.
\label{epart}$$ The mixing rate $-\ln \left| \nu _2\right|$ at which equilibrium is approached is determined by the gap between the next largest eigenvalue $\nu_2$ and $1$. This is characteristic of a classically mixing system.
However, there are some non-generic cases such as, e.g., bipartite graphs when $-1$ belongs to the spectrum. In this case the asymptotic distribution is not stationary. Nevertheless an equivalent description is possible for bipartite graphs when $U$ is replaced by $U^2$ which has then two uncoupled blocks of dimension $B$. The example that we are going to discuss in the last section will be of this type.
Periodic orbits on the graph will play an important rôle in the sequel and we define them in the following way. An [*orbit*]{} on the graph is an itinerary (finite or infinite) of successively connected [*directed*]{} bonds $\{i_1,i_2\}, \{i_2,i_3\},\dots $ For graphs without loops or multiple bonds this is uniquely defined by the sequence of vertices $i_1,i_2, \dots$ with $i_m \in [1,V]$ and $C_{i_m,i_{m+1}} =1$ for all $m$. An orbit is [*periodic*]{} with period $n$ if for all $k$, $ (i_{n+k},i_{n+k+1})
=(i_k,i_{k+1})$. The [*code*]{} of a periodic orbit of period $n$ is the sequence of $n$ vertices $i_1,\dots,i_n$ and the orbit consists of the bonds $(i_m,i_{m+1})$ (with the identification $i_{m+n} \equiv i_{m}$). In this way, any cyclic permutation of the code defines the same periodic orbit.
The periodic orbits (PO’s) can be classified in the following way:
- [*Irreducible periodic orbits*]{} - PO’s which do not intersect themselves such that any vertex label in the code can appear at most once. Since the graphs are finite, the maximum period of irreducible PO’s is $V$. To each irreducible PO corresponds its time reversed partner whose code is read in the reverse order. The only PO’s which are both irreducible and conjugate to itself under time reversal are the PO’s of period 2.
- [*Reducible periodic orbits*]{} - PO’s whose code is constructed by inserting the code of any number of irreducible PO’s at any position which is consistent with the connectivity matrix. All the PO’s of period $n >V$ are reducible.
- [*Primitive periodic orbits*]{} - PO’s whose code cannot be written down as a repetition of a shorter code.
We introduced above the concept of orbits on the graph as strings of vertex labels whose ordering obeys the required connectivity. This is a finite coding which is governed by a Markovian grammar provided by the connectivity matrix. In this sense, the symbolic dynamics on the graph is Bernoulli. This property adds another piece of evidence to the assertion that the dynamics on the graph is chaotic. In particular, one can obtain the topological entropy $\Gamma$ from the symbolic code. Using the relation $$\Gamma = \lim_{n\to \infty} {1\over n} \log{\rm
tr} (C^n)$$ one gets $\Gamma = \log \bar v$, where $\bar v$ is the mean valency.
Of prime importance in the discussion of the relation between the classical and the quantum dynamics are the traces $u_n={\rm tr}(U^n)$ which are interpreted as the mean classical probability to perform $n$-periodic motion. Using the definition (\[cl3\]) one can write the expression for $u_n$ as a sum over contributions of $n$-periodic orbits $$u_n=\sum_{p\in {\cal P}_n} n_p \exp (-r \gamma_p
n_p ) \,,
\label{classicalsum1}$$ where the sum is over the set ${\cal P}_n$ of primitive PO’s whose period $n_p$ is a divisor of $n$, with $r=n/n_p$. To each primitive orbit one can assign a [*stability factor*]{} $\exp (-\gamma_p n_p ) $ which is accumulated as a product of the transition probabilities as the trajectory traverses its successive vertices: $$\exp (-\gamma_p n_p) \equiv\prod _{j=1}^{n_p}
P^{(i_j)}_{i_{j-1}\to i_{j+1}}\,.
\label{lyapunov}$$ The stability exponents $\gamma_p$ correspond to the Lyapunov exponents in periodic orbit theory.
When only one eigenvalue of the classical evolution operator $U$ is on the unit circle, one has, $u_n\overrightarrow {\scriptstyle {n\rightarrow \infty
}}\ 1$. This leads to a classical sum-rule $$u_n=\sum_{p\in P_n}n_p\exp (-r \gamma_p n_p ) \ \
\overrightarrow{
\scriptstyle {n\rightarrow \infty }}\ 1\,.
\label{classicalsum}$$ This last relation shows again that the number of periodic orbits must increase exponentially with $n$ to balance the exponentially decreasing stability factors of the individual periodic orbits. The topological entropy can be related to the mean stability exponent through this relation.
Using the expression (\[classicalsum1\]) for $u_n$ one can easily write down the complete thermodynamic formalism for the graph. Here, we shall only quote the periodic orbit expression for the Ruelle $\zeta $ function $$\begin{aligned}
\zeta _R(z) &\equiv &\left( \det (I-zU)\right) ^{-1}={\rm \exp }\left[ -{\rm
tr}\left( \ln (I-zU)\right) \right] \label{cl4} \\
&=&\exp \left[ \sum_n\frac{z^n}nu_n\right] =\prod_p\frac 1{\left(
1-z^{n_p}\exp (-n_p\gamma _p)\right)}\,, \nonumber\end{aligned}$$ where the product extends over all primitive periodic orbits.
The above discussion of the classical dynamics on the graph shows that it bears a striking similarity to the dynamics induced by area preserving hyperbolic maps. The reason underlying this similarity is that even though the graph is a genuinely one-dimensional system, it is not simply connected, and the complex connectivity is the origin and reason for the classically chaotic dynamics.
The Spectral Statistics of the Quantum Scattering Map {#statistics}
=====================================================
We consider the matrices $S_B$ defined in (\[S\_Bdef\]). Their spectrum consist of $2B$ points confined to the unit circle (eigenphases). Unitary matrices of this type are frequently studied since they are the quantum analogues of classical, area preserving maps. Their spectral fluctuations depend on the nature of the underlying classical dynamics [@S89]. The quantum analogues of classically integrable maps display Poissonian statistics while in the opposite case of classically chaotic maps, the statistics of eigenphases conform quite accurately with the results of Dyson’s random matrix theory (RMT) for the [*circular*]{} ensembles. The ensemble of unitary matrices which will be used for the statistical study will be the set of matrices $S_B(k)$ with $k$ in the range $|k-k_0| \le \Delta_k/2$. The interval size $\Delta_k$ must be sufficiently small such that the vertex matrices do not vary appreciably when $k$ scans this range of values. Then the $k$ averaging can be performed with the vertex scattering matrices replaced by their value at $k_0$. When the vertex scattering matrices are derived from Neumann or Dirichlet boundary conditions, the averaging interval is unrestricted because the dimension of $S_B$ is independent of $k$. In any case $\Delta_k$ must be much larger than the correlation length between the matrices $S_B(k)$, which was estimated in [@KS99] to be inversely proportional to the width of the distribution of the bond lengths. The ensemble average with respect to $k$ will be denoted by $$\label{ensaveS}
\left \langle \ \cdot \ \right \rangle _k \equiv \frac {1}{\Delta_k}
\int_{k_0-\Delta_k/2}^{k_0+\Delta_k/2}\cdot
\,\, dk \,.$$ Another way to generate an ensemble of matrices $S_{B}$ is to randomise the length matrix $L$ or the magnetic vector potentials $A_{(i,j)}$, while the connectivity (topology of the graph) is kept constant. In most cases, the ensembles generated in this way will be equivalent. In the last section we will also consider an [*additional*]{} average over the vertex scattering matrices.
In the following subsections we compare statistical properties of the eigenphases $\left\{ \theta _l(k)\right\} $ of $S_B$ with the predictions of RMT [@M90] and with the results of periodic orbit theory for the spectral fluctuations of quantised maps [@BS88]. The statistical measure which we shall investigate is the spectral form factor. Explicit expressions for this quantity are given by RMT [@HKSSZ96], and a semiclassical discussion can be found in [@keatbog; @UScorr; @camb].
**The Form Factor** {#the_form_factor}
-------------------
The matrix $S_B$ for a fixed value of $k$ is a unitary matrix with eigenvalues $\e^{\i\theta_l(k)}$. The spectral density of the eigenphases reads $$d(\theta;k )\equiv \sum_{l=1}^{2B}\delta (\theta -\theta
_l(k))=\frac{2B}{2\pi }+
\frac 1{2\pi }\sum_{n=1}^\infty\e^{-\i\theta n}{\rm tr}S^n_B(k) +{\rm c.c.}\,,
\label{sms1}$$ where the first term on the r.h.s. is the smooth density $\overline{d}=\frac{2B}{2\pi }$. The oscillatory part is a Fourier series with the coefficients ${\rm tr}S^n_B(k) $. This set of coefficients will play an important rôle in the following. Using the definitions (\[S\_Bdef\]) one can expand ${\rm tr}S^n_B(k) $ directly as a sum over $n-$periodic orbits on the graph $${\rm tr} S^n_B(k)
=\sum_{p\in {\cal P}_n}n_p{\cal A}_p^r{\rm e}^{i(kl_p+
\Phi_p)r}{\rm e}
^{i
\mu _p r }\,,
\label{posum}$$ where the sum is over the set ${\cal P}_n$ of primitive PO’s whose period $n_p$ is a divisor of $n$, with $r=n/n_p$. $l_p = \sum_{b \in p} L_{b}$ is the length of the periodic orbit. $\Phi_p = \sum_{b \in p}L_b A_b$ is the “magnetic flux" through the orbit. If all the parameters $A_b$ have the same absolute size $A$ we can write $\Phi_p = A b_p$, where $b_p$ is the directed length of the orbit. $\mu _p$ is the phase accumulated from the vertex matrix elements along the orbit, and it is the analogue of the Maslov index. For the standard vertex matrices (\[smatrix\]) $\mu_p/\pi$ gives the number of [*backscatterings*]{} along $p$. The amplitudes ${\cal A}_p$ are given by $${\cal A}_p=\prod _{j=1}^{n_p} \left
|\sigma^{(i_j)}_{i_{j-1},i{j+1}}\right| \equiv
{\rm e}^{-{\frac{ \gamma
_p}2}n_p}\,, \label{amplitude}$$ where $i_{j}$ runs over the vertex indices of the periodic orbit, and $j$ is understood ${\rm mod}\,n_{p}$. The Lyapunov exponent $\gamma_p$ was defined in (\[lyapunov\]). It should be mentioned that (\[posum\]) is the building block of the periodic orbit expression for the spectral density of the graph, which can be obtained starting from the secular equation (\[secular\]). In the quantisation of classical area preserving maps similar expressions appear as the leading semiclassical approximations. In the present context (\[posum\]) is an identity.
The two-point correlations are expressed in terms of the excess probability density $R_2(r)$ of finding two phases at a distance $r$, where $r$ is measured in units of the mean spacing ${ 2\pi \over 2B}$ $$R_2(r;k_0)={2\over 2\pi}\sum_{n=1}^\infty \cos
\left(
\frac{2\pi rn}{2B}\right) \frac 1{2B}\left\langle\left|
{\rm
tr}S_B^n\right|^2\right\rangle_k\,\, .
\label{sms3}$$ The form factor $$\label{ff}
K(n/2B)={\frac 1{2B}}<|{\rm
tr}S_B^n|^2>_k$$ is the Fourier transform of $R_2(r,k_0)$. For a Poisson spectrum, $K(n/2B)=1$ for all $n$. RMT predicts that $K(n/2B)$, depends on the scaled time ${n/2B}$ only [@S89], and explicit expressions for the orthogonal and the unitary circular ensembles are known [@HKSSZ96].
As was indicated above, if the vertex scattering matrices are chosen by imposing Dirichlet boundary conditions on the vertices, the classical dynamics is “integrable". One expects therefore the spectral statistics to be Poissonian, $$K(n/2B)= 1\qquad
{\rm for\ all}\ n\ge 1
\,.$$ For Dirichlet boundary conditions the vertex scattering matrices (\[smatrix\]) couple only time reversed bonds. $S_B$ is reduced to a block diagonal form where each bond and its time reversed partner are coupled by a $2\times 2$ matrix of the form $$\begin{aligned}
S^{(b)}(k,A)
=\left (
{\begin{array}{ll}
0 & \e^{\i(k+A)L_b} \\
\e^{\i(k-A)L_b } & 0
\end{array}}\right ) \,.\end{aligned}$$ The spectrum of each block is the pair $\pm \e^{\i kL_b}$, with the corresponding symmetric and antisymmetric eigenvectors $ {1\over \sqrt {2}}(1,
\pm1)$. As a result, we get $$K(n/2B)=1+(-1)^n \ \
\
{\rm for \ \ all} \ \ \ n\geq 1 \,.
\label{poissonform}$$ This deviation from the expected Poissonian result is due to the fact that the extra symmetry reduces the matrix $S_B$ further into the symmetric and antisymmetric subspaces. The spectrum in each of them is Poissonian, but when combined together, the fact that the eigenvalues in the two spectra differ only by a sign leads to the anomaly (\[poissonform\]).
Having successfully disposed of the integrable case, we address now the more general situation. In Fig. \[v20\] we show typical examples of form factors, computed numerically for a fully connected graph with $V= 20$. The data for Neumann boundary conditions and $A=0$ (Fig. 1(a)) or $A\ne 0$ (Fig. 1(b)) are reproduced quite well by the predictions of RMT, which are shown by the smooth lines. For this purpose, one has to scale the topological time $n$ by the corresponding “Heisenberg time" which is the dimension of the matrix, i.e., $2B$. The deviations from the smooth curves are not statistical, and cannot be ironed out by further averaging. Rather, they are due to the fact that the graph is a dynamical system which cannot be described by RMT in all detail. To study this point in depth we shall express the form factor in terms of the PO expression (\[posum\]). $$\begin{aligned}
K(n/2B)&=&\frac 1{2B}\left\langle \left|\sum_{p\in {\cal P}_n}n_p{\cal
A}_p^r\e^{\i(kl_p+A
b_p+ \pi \mu_p)r}\right|^2 \right\rangle_k
\label{sms5} \\
&=&\left . \frac 1{2B} \sum_{p,p'\in {\cal P}_n}
n_pn_{p\prime} {\cal A}_p^r
{\cal A}_{p\prime}^{r^{\prime}}
\exp \left
\{\i A(r b_p-r'b_{p\prime}) +i\pi (r\mu_p-r'\mu_{p\prime})\right\}
\right
|_{rl_p = r^{\prime}l_{p^{\prime}}}\,. \nonumber\end{aligned}$$ The $k$ averaging is carried out on such a large interval that the double sum above is restricted to pairs of periodic orbits which have exactly the same length. The fact that we choose the lengths of the bonds to be rationally independent will enter the considerations which follow in a crucial way.
The largest deviations between the numerical data and the predictions of RMT occur for $n=1,2$. For $n=1$ one gets $0$ instead of the COE (CUE) values $1/B$ ($1/2B$), simply because the graph has no periodic orbits of period $1$. This could be modified by allowing loops, which were excluded here from the outset. The $2$-periodic orbits are self-retracing (i.e. invariant under time reversal), and each has a distinct length. Their contribution is enhanced because back scattering is favoured when the valency is large. Self-retracing implies also that their contribution is insensitive to the value of $A$. The form factor for $n=2$ calculated for a fully connected graph with $v=V-1$ is $$\label{tsampi}
K(n/2B)= 2\(\[1-\frac 2v\]\)^4\,,$$ independent of the value of $A$. This is different from the value expected from RMT. The repetitions of the 2-periodic orbits are also the reason for the odd-even staggering which is seen for low values of $\tau\equiv n/2B$. They contribute a term which is $\approx 2\exp(-2V\tau)$ and thus decays faster with the scaled time $\tau$ when the graph increases.
The deviations between the predictions of RMT and periodic orbit theory for low values of $\tau$ are typical and express the fact that for deterministic systems in general, the short time dynamics is not fully chaotic. The short time domain becomes less prominent as $B$ becomes larger because the time $n$ has to be scaled by $2B$. This limit is the analogue of the limit $\hbar
\rightarrow 0$ in a general system.
Consider now the domain $2 <n \ll 2B$. The PO’s are mostly of the irreducible type, and the length restriction limits the sum to pairs of orbits which are conjugate under time reversal. Neglecting the contributions from repetitions and from self-retracing orbits we get $$\label{transuzy1}
K(n/2B)\approx \frac 1{2B} \sum_{p\in {\cal P}_n} n^2 {\cal A}_p^2 \ \
4\cos ^2A b_p = {2n\over 2B} u_n \left \langle \cos ^2 A b_p\right
\rangle _n \,.$$ The classical return probability $u_n$ approaches $1$ as $n$ increases (see (\[classicalsum\])). Neglecting the short time deviations, we can replace $u_n$ by $1$, and we see that the remaining expression is the classical expectation of $\cos ^2 A b_p$ over PO’s of length $n$. For $A=0$ this factor is identically $1$ and one obtains the leading term of the COE expression for $n\ll 2B$. If $A$ is sufficiently large $ \left \langle \cos ^2 A b_p \right
\rangle_n \approx 1/2 $, one obtains the short-time limit of the CUE result. The transition between the two extreme situations is well described by $$\label{transuzy2}
\left \langle \cos ^2 A b_p\right \rangle _n \approx {1\over 2} \left (
\e^{- A^2\left \langle L_b^2\right \rangle {n\over 2}} +1 \right ) \,.$$ This formula is derived by assuming that the total directed length $b_p$ of a periodic orbit is a sum of elementary lengths with random signs.
The basic approximation so far was to neglect the interference between contributions of periodic orbits with different codes (up to time reversal). This can be justified as long as periodic orbits with different codes have different lengths. This is the case for low values of $n$. As $n$ approaches $B$ the degeneracy of the length spectrum increases, and for $n>2B$ all the orbits are degenerate. In other words, the restriction $rl_p =
r^{\prime}l_{p^{\prime}}$ in (\[sms5\]) does not pick up a unique orbit and its time reversed partner, but rather a group of [*isometric*]{} but distinct orbits. Therefore, the interference of the contributions from these orbits must be calculated. The relative sign of the terms is determined by the “Maslov" index. The computation of the interfering contributions from different periodic orbits with neighbouring actions is an endemic problem in the semiclassical theory of spectral statistics. These contributions are referred to as the [*non-diagonal*]{} terms, and they are treated by invoking the concept of periodic orbit correlations [@ADDKKSS93; @CPS98]. The dynamical origin of these correlations is not known. In the case of graphs, they appear as correlations of the “Maslov" signs within a class of isometric $n$-periodic orbits.
To compute $K(n/2B)$ from (\[sms5\]) one has to sum the contributions of all the $n$-periodic orbits after grouping together those which have exactly the same lengths. We shall discuss the case $A=0$, so a further restriction on the orbits to have the same directed length is not required here. Since the lengths of the individual bonds are assumed to be rationally independent, a group of isometric $n$-periodic orbits is identified by the non-negative integers $q_i, i=1,\dots,B$ such that $$l_{\bf q} \equiv
\sum_{i=1} ^B q_i l_i
\qquad{\rm
with}\qquad\sum_{i=1}^Bq_i=n\,,
\label{qdef}$$ i.e., each bond $i$ is traversed $q_i$ times. The orbits in the group differ only in the [*order*]{} by which the bonds are traversed. We shall denote the number of isometric periodic orbits by $D_{n}(\bf q)$. Note that not all the integer vectors ${\bf q}$ which satisfy (\[qdef\]) correspond to periodic orbits. Rather, the connectivity required by the concept of an orbit imposes restrictions, which render the problem of computing $D_n({\bf q})$ a very hard combinatorial problem [@Urigavish]. Writing (\[sms5\]) explicitly for the case of a fully connected graph with Neumann vertex scattering matrices, we get $$K(n/2B)={1\over 2B}\left({2\over v}\right)^{2n}
\sum_{\bf q} \left|\sum_{\alpha=1}^{D_n(\bf q)} {n \over r_{\alpha}}
(- \xi )^{\mu_{\alpha}}\right| ^2 \ , \ \
{\rm with} \ \ \ \xi \equiv \left({v-2\over 2}\right)\,,
\label {tracecomp}$$ and the $\alpha$ summation extends over the $n$-periodic orbits in the class ${\bf q}$. $\mu_{\alpha}$ is the number of back scattering along the orbit, and $r_{\alpha}$ is different from unity if the orbit is a repetition of a shorter primitive orbit of period $n/r_{\alpha}$.
Equation (\[tracecomp\]) is the starting point of the new approach to spectral statistics, which we would like to develop in the present paper. The actual computation of (\[tracecomp\]) can be considered as a [*combinatorial*]{} problem, since it involves counting of loops on a graph, and adding them with appropriate (signed) weights. For Neumann boundary conditions, the weights are entirely determined by the connectivity of the graph. Our numerical data convincingly show that in the limit of large $B$ the form factors for sufficiently connected graphs reproduce the results of RMT. The question is, if this relation can be derived using asymptotic combinatorial theory. The answer is not yet known, but we would like to show in the next section that for a very simple graph one can use combinatorics to evaluate the periodic orbit sums, and recover in this way the exact values of the form factor.
**The $2$-star Model**
======================
In this section we will investigate the classical and quantum dynamics in a very simple graph using two different methods. We shall use periodic orbit theory to reduce the computation of the trace of the classical evolution operator $u_{n}$ and the spectral form factor $K(n/2B)$ to combinatorial problems, namely sums over products of binomial coefficients. The result will be compared to a straight forward computation starting from the eigenvalues of the classical and quantum scattering maps.
An $n$-star graph consists of a “central" vertex (with vertex index $o$) out of which emerge $n$ bonds, all terminating at vertices (with indices $j=1,\dots, n$) with valencies $v_j=1$. The bond lengths are $L_{oj}\equiv
L_j$. This simple model (sometimes called a [*hydra*]{}) was studied at some length in [@KS99]. The star with $n=2$ is not completely trivial if the central vertex scattering matrix is chosen as $$\sigma^{(o)}(\eta ) = \left (
{\begin{array} {ll}
\cos \eta & {\rm i}\sin \eta \\
{\rm i}\sin \eta & \cos \eta
\end{array}} \right )\,,
\label {2-starsigma}$$ where the value $0\le \eta \le \pi /2 $ is still to be fixed. The scattering matrices at the two other vertices are taken to be $1$ and correspond to Neumann boundary conditions. The dimension of $U$ and $S_B$ is $4$, but it can be immediately reduced to $2$: due to the trivial scattering at the reflecting tips, $a_{jo}=a_{oj}\equiv a_j$ for $j=1,2$. In this representation the space is labelled by the indices of the two loops (of lengths $2L_1$ and $2L_2$ respectively) which start and end at the central vertex. After this simplification the matrix $S_B$ reads $$S_B(k;\eta) =
\left (
{\begin{array} {ll}
{\rm e}^{2ikL_1} & 0 \\
0 & {\rm e}^{2ikL_2}
\end{array}} \right )
\left (
{\begin{array} {ll}
\cos \eta & {\rm i}\sin \eta \\
{\rm i}\sin \eta & \cos \eta
\end{array}} \right )\,.
\label {2-starSB}$$ We shall compute the form-factor for two ensembles. The first is defined by a fixed value of $\eta = \pi/4$, and the average is over an infinitely large $k$ range. The second ensemble includes an additional averaging over the parameter $\eta$. We will show that the measure for the integration over $\eta$ can be chosen such that the model yields the CUE form factor. This is surprising at first sight, since the model defined above is clearly time-reversal invariant. However, if we replace $kL_1$ and $kL_2$ in (\[2-starSB\]) by $L(k\pm A)$, (\[2-starSB\]) will allow for an interpretation as the quantum scattering map of a graph with a single loop of length $L$ and a vector potential $A$, i.e., of a system with broken time-reversal invariance (see Fig. \[tst\]). In particular, the form factors of the two systems will coincide exactly, when an ensemble average over $L$ is performed. Clearly, this is a very special feature of the model considered, and we will not discuss it here in more detail.
**Periodic Orbit Representation of $u_n$**
------------------------------------------
The classical evolution operator corresponding to (\[2-starSB\]) is $$U(\eta) = \left (
{\begin{array} {ll}
\cos ^2\eta & \sin ^2 \eta \\
\sin ^2\eta & \cos ^2 \eta
\end{array}} \right )\,.
\label {2-starclass}$$ The spectrum of $U$ consists of $\{1,\cos 2\eta \}$, such that $$\label{un}
u_n (\eta )=1+\cos ^n 2\eta\,.$$ We will now show how this result can be obtained from a sum over the periodic orbits of the system, grouped into classes of isometric orbits. This grouping is not really necessary for a classical calculation, but we would like to stress the analogy to the quantum case considered below.
The periodic orbits are uniquely encoded by the loop indices, such that each $n$-tuple of two symbols $1$ and $2$ corresponds (up to a cyclic permutation) to a single periodic orbit. When $n$ is prime, the number of different periodic orbits is $N_2(n)=2+(2^n-2)/n$, otherwise there are small corrections due to the repetitions of shorter orbits. These corrections are the reason why it is more convenient to represent a sum over periodic orbits of length $n$ as a sum over all possible code words, though some of these code words are related by a cyclic permutation and consequently denote the same orbit. If we do so and moreover replace the stability factor of each orbit by (\[lyapunov\]), the periodic orbit expansion of the classical return probability becomes $$\begin{aligned}
\label{un_po}
u_{n}&=&
\sum_{i_{1}=1,2}
\dots
\sum_{i_{n}=1,2}\prod_{j=1}^{n}
P_{i_{j}\rightarrow i_{j+1}}\,,\end{aligned}$$ where $j$ is a cyclic variable such that $i_{n+1}\equiv i_{1}$. In fact (\[un\_po\]) can be obtained without any reference to periodic orbits if one expands the intermediate matrix products contained in $u_{n}=\Tr U^{n}$ and uses $P_{i_{j}\rightarrow i_{j+1}}=U_{i_{j},i_{j+1}}(\eta)$.
We will now order the terms in the multiple sum above according to the classes of isometric orbits. In the present case a class is completely specified by the integer $q\equiv q_1$ which counts the traversals of the loop $1$, i.e., the number of symbols $1$ in the code word. Each of the $q$ symbols $1$ in the code is followed by an uninterrupted sequence of $t_{j}\ge 0$ symbols $2$ with the restriction that the total number of symbols $2$ is given by $$\sum_{j=1}^{q}t_{j}=n-q\,.$$ We conclude that each code word in a class $0<q<n$ which starts with a symbol $i_{1}=1$ corresponds to an ordered partition of the number $n-q$ into $q$ non-negative integers, while the words starting with $i_{1}=2$ can be viewed as partition of $q$ into $n-q$ summands.
To make this step very clear, consider the following example: All code words of length $n=5$ in the class $q=2$ are $11222$, $12122$, $12212$, $12221$ and $22211$, $22121$, $21221$, $22112$, $21212$, $21122$. The first four words correspond to the partitions $0+3=1+2=2+1=3+0$ of $n-q=3$ into $q=2$ terms, while the remaining $5$ words correspond to $2=0+0+2=0+1+1=1+0+1=0+2+0=1+1+0=2+0+0$.
In the multiple products in (\[un\_po\]), a forward scattering along the orbit is expressed by two different consecutive symbols $i_{j}\ne i_{j+1}$ in the code and leads to a factor $\sin^2\eta$, while a back scattering contributes a factor $\cos^2\eta$ . Since the sum is over periodic orbits, the number of forward scatterings is always even and we denote it with $2\nu$. It is then easy to see that $\nu$ corresponds to the number of positive terms in the partitions introduced above, since each such term corresponds to an uninterrupted sequence of symbols $2$ enclosed between two symbols $1$ or vice versa and thus contributes two forward scatterings. For the codes starting with a symbol $1$ there are ${q\choose \nu}$ ways to choose the $\nu$ positive terms in the sum of $q$ terms, and there are ${n-q-1\choose \nu-1}$ ways to decompose $n-q$ into $\nu$ [*positive*]{} summands. After similar reasoning for the codes starting with the symbol $2$ we find for the periodic orbit expansion of the classical return probability $$\begin{aligned}
\label{un_ex}
u_{n}(\eta)&=&2\cos^{2n}\eta+\sum_{q=1}^{n-1}\sum_{\nu}
\[{q\choose \nu}{n-q-1\choose \nu-1}+{n-q\choose \nu}{q-1\choose \nu-1}\]
\sin^{4\nu}\!\eta\,\cos^{2n-4\nu}\!\eta
\nonumber \\
&=&2\cos^{2n}\eta+\sum_{q=1}^{n-1}\sum_{\nu}{n\/\nu}
{q-1\choose \nu-1}{n-q-1\choose \nu-1}\sin^{4\nu}\!\eta\,\cos^{2n-4\nu}\!\eta\,
\nonumber \\
&=&2\sum_{\nu}{n\choose 2\nu}\sin^{4\nu}\!\eta\,\cos^{2n-4\nu}\!\eta
\nonumber \\
&=&(\cos^{2}\!\eta+\sin^{2}\!\eta)^{n}+(\cos^{2}\!\eta-\sin^{2}\!\eta)^{n}\,,\end{aligned}$$ which is obviously equivalent to (\[un\]). The summation limits for the variable $\nu$ are implicit since all terms outside vanish due to the properties of the binomial coefficients. In order to get to the third line we have used the identity $$\label{ci1}
\sum_{q=1}^{n-1}{q-1\choose \nu-1}{n-q-1\choose \nu-1}=
{n-1\choose 2\nu-1}
={2\nu\over n}{n\choose 2\nu}\,.$$ It can be derived by some straightforward variable substitutions from $$\sum_{k=l}^{n-m}{k\choose l}{n-k\choose m}={n+1\choose l+m+1}\,.$$ which, in turn, is found in the literature [@prudnikov].
**Quantum Mechanics: Spacing Distribution and Form Factor** {#qm}
-----------------------------------------------------------
Starting from (\[2-starSB\]), and writing the eigenvalues as ${\rm
e}^{ik(L_1+L_2)} {\rm e}^{\pm i\lambda/2}$, we get for $\lambda$, the difference between the eigenphases, $$\lambda = 2\,{\rm arcos}\[\cos\eta\,\cos k(L_1-L_2)\]\,.
\label {2-starlambda}$$ For fixed $\eta$, the $k$ averaged spacing distribution (which is essentially equivalent to $R_{2}(r)$ for the considered model) is given by $$\begin{aligned}
\label{2-starspacing}
P(\theta ;\eta) &=& {1\over
\Delta_k} \int _{k_0-\Delta_k/2}^{k_0+\Delta_k/2} {\rm d}k
\
\delta
\left (\theta - 2 {\rm arcos} \left [\cos \eta
\cos k(L_1-L_2) \right ]
\right ) \nonumber \\ \nonumber \\
&=& \left \{
{\begin {array} {cl}
0 &
\qquad\cos(\theta/2)> |\cos \eta\,| \\ \\
{\displaystyle\sin(\theta/2)
\over \sqrt{\displaystyle\cos^2\eta-\cos^2(\theta/2)}}
&
\qquad\cos(\theta/2) < |\cos\eta\,|
\end {array}} \right.\end{aligned}$$ We have assumed that $\theta$ is the smaller of the intervals between the two eigenphases, i.e. $0\le \theta \le \pi$.
The spacings are excluded from a domain centered about $0$ $(\pi)$, i.e., they show very strong level repulsion. The distribution is square-root singular at the limits of the allowed domain.
$P(\theta ;\eta)$ can be written as $$P(\theta;\eta ) =
{1\over 2\pi} + {1\over \pi}\sum_{n=1}^\infty
\cos(n\theta)\,\(
{1\over 2}\left\langle\left|{\rm tr}S_B(\eta) ^n\right|^2\right\rangle_k -1\)\,,
\label{2-starPassum}$$ and, by a Fourier transformation, we can compute the form factor $$K_{2}(n;\eta)={1\over2} \left\langle\left|{\rm tr}S_B(\eta) ^n\right|^2\right\rangle\,.$$ In particular, for $\eta =\pi/4$ one finds $$\begin{aligned}
\label{K2PI4_UZY}
K_{2}(n;\pi/4)&=&1+{(-1)^{m+n}\over 2^{2m+1}}{2m\choose m} \\
&\approx& 1 + {(-1)^{m+n}\over 2\sqrt{\pi n }}\,.
\label{k2pi4_uzy_app}\end{aligned}$$ Where $ m =[n/2]$ and $[\cdot]$ stands for the integer part. The slow convergence of $K_{2}(n;\pi/4)$ to the asymptotic value $1$ is a consequence of the singularity of $P(\theta;\pi /4 )$.
We now consider the ensemble for which the parameter $\eta$ is distributed with the measure ${\rm d}\mu(\eta) = |\cos\eta \sin\eta |{\rm d}\eta$. The [*only*]{} reason for the choice of this measure is that upon integrating (\[2-starPassum\]) one gets $$P(\theta) = 2\sin^2(\theta/2)\,,
\label {2-starCUE}$$ which coincides with the CUE result for $2\times 2$ matrices. A Fourier transformation results in $$K_2(n) = \left \{ {\begin {array} {ll} {1\over 2} & {\rm for} \ \ n=1 \\
1 & {\rm for} \ \ n\ge 2 \end {array} } \right.\,.
\label {2-starK(n)CUE}$$ The form factors (\[K2PI4\_UZY\]), (\[k2pi4\_uzy\_app\]) and (\[2-starK(n)CUE\]) are displayed in Fig. \[tst\] below.
Periodic Orbit Expansion of the Form Factor {#po}
-------------------------------------------
As pointed out at the end of section \[the\_form\_factor\], the $k$-averaged form factor can be expressed as a sum over classes of isometric periodic orbits. The analogue of (\[tracecomp\]) for the 2-star is $$K_2(n;\eta)={1\over 2}\sum_{q=0}^{n}
\left|\sum_{\alpha=1}^{D_{n}(q)}{n\over r_{\alpha}}
\i^{2\nu_{\alpha}}\sin^{2\nu_{\alpha}}\!\eta\cos^{n-2\nu_{\alpha}}\!\eta
\right|^{2}\,,$$ where the number of forward and backward scatterings along the orbits are $2\nu_\alpha$ and $\mu_{\alpha}=n-2\nu_{\alpha}$, respectively. Again, it is very inconvenient to work with the repetition number $r_{\alpha}$, and consequently we replace—as in the derivation of (\[un\_ex\])—the sum over orbits by a sum over all code words and use the analogy with the compositions of integer numbers to obtain $$\begin{aligned}
\label{K2eta}
K_2(n;\eta)&=&
\cos^{2n}\!\eta+{n^2\over 2}\sum_{q=1}^{n-1}
\[\sum_{\nu}{(-1)^{\nu}\/\nu}{q-1\choose \nu-1}{n-q-1\choose \nu-1}
\sin^{2\nu}\!\eta\,\cos^{n-2\nu}\!\eta\]^{2}\,.\end{aligned}$$ The inner sum over $\nu$ can be written in terms of Krawtchouk polynomials [@kp1; @kp2] as $$\begin{aligned}
\label{K2eta_K}
K_2(n;\eta)&=& \cos^{2n}\!\eta+{1\/2}\sum_{q=1}^{n-1}
{n-1\choose n-q}\cos^{2q}\!\eta\sin^{2(n-q)}\!\eta
\[{n\over q}P_{n-1,n-q}^{(\cos^{2}\!\eta,\sin^2\!\eta)}(q)\]^2\,,\end{aligned}$$ and the Krawtchouk polynomials are defined as in [@kp1; @kp2] by $$\begin{aligned}
\label{krawtchouk}
P_{N,k}^{(u,v)}(x)=\[{N\choose k}(uv)^{k}\]^{-1/2}\sum_{\nu=0}^{k}
(-1)^{k-\nu}{x\choose \nu}{N-x\choose k-\nu}u^{k-\nu}v^{\nu}\qquad
\(\begin{array}{l}0\le k \le N\cr u+v=1\end{array}\)\,.\end{aligned}$$ These functions form a complete system of orthogonal polynomials of integer $x$ with $0\le x\le N$. They have quite diverse applications ranging from the theory of covering codes [@cohen] to the statistical mechanics of polymers [@schulten], and are studied extensively in the mathematical literature [@kp1; @kp2]. The same functions appear also as a building block in our periodic orbit theory of Anderson localisation on graphs [@HSUS]. Unfortunately, we were not able to reduce the above expression any further by using the known sum-rules and asymptotic representations for Krawtchouk polynomials. The main obstacle stems from the fact that in our case the three numbers $N,k,x$ in the definition (\[krawtchouk\]) are constrained by $N=k+x-1$.
We will now consider the special case $\eta=\pi/4$ for which we obtained in the previous subsection the solution (\[K2PI4\_UZY\]). The result can be expressed in terms of Krawtchouk polynomials with $u=v=1/2$ which is also the most important case for the applications mentioned above. We adopt the common practice to omit the superscript $(u,v)$ in this special case and find $$\begin{aligned}
\label{k2pi4}
K_2(n;\pi/4)&=&
{1\over 2^{n}}+{1\/2^{n+1}}\sum_{q=1}^{n-1}
{n-1\choose n-q}\[{n\over q}P_{n-1,n-q}(q)\right]^2\,.\end{aligned}$$ It is convenient to introduce $$\begin{aligned}
\label{}
\nq(s,t)&=&(-1)^{s+t}{s+t-1\choose s}^{1/2}P_{s+t-1,s}(t)
\nonumber\\
&=&\sum_{\nu}(-1)^{t-\nu}{t\choose \nu}{s-1\choose \nu-1}\end{aligned}$$ and to rewrite (\[k2pi4\]) with the help of some standard transformations of binomial coefficients as $$\begin{aligned}
\label{K2PI4N}
K_2(n;\pi/4)&=&{1\over 2^{n}}+{1\/2^{n+1}}\sum_{q=1}^{n-1}
\[{n\over q}\nq(q,n-q-1)\right]^2
\nonumber\\
&=&{1\over 2^{n}}+{1\over 2^{n+1}}\sum_{q=1}^{n-1}\[\nq(q,n-q)+(-1)^{n}\nq(n-q,q)\]^{2}\end{aligned}$$ This expression is displayed in Fig. \[tst\] together with (\[K2PI4\_UZY\]) in order to illustrate the equivalence of the two results. An independent proof for this equivalence can be given by comparing the generating functions of $K_2(n;\pi/4)$ in the two representations [@gregory]. We defer this to appendix \[proof\].
Please note, that in this way we have found a proof for two identities involving Krawtchouk polynomials $$\label{ci2a}
\sum_{q=1}^{2m-1}{2m-1\choose 2m-q}\[{2m\over q}P_{2m-1,2m-q}(q)\]^{2}
=2^{2m+1}+(-1)^{m}{2m\choose m}-2$$ and $$\label{ci2b}
\sum_{q=1}^{2m}{2m\choose 2m+1-q}\[{2m+1\over q}P_{2m,2m+1-q}(q)\]^{2}
=2^{2m+2}-2\,(-1)^{m}{2m\choose m}-2\,,$$ which were obtained by separating even and odd powers of $n$ in (\[K2PI4\_UZY\]) and (\[k2pi4\]). To the best of our knowledge, (\[ci2a\]) and (\[ci2b\]) were derived here for the first time.
Finally we will derive the CUE result (\[2-starK(n)CUE\]) for the ensemble of graphs defined in the previous subsection starting from the periodic orbit expansion (\[K2eta\]). We find $$\begin{aligned}
K_{2}(n)&=&\int_{0}^{\pi/2}{\rm d}\mu(\eta) K_2(n;\eta)\,.\end{aligned}$$ Inserting (\[K2eta\]), expanding into a double sum and using $$\int_{0}^{\pi/2}{\rm d}\eta
\sin^{2(\nu+\nu')+1}\!\eta\cos^{2(n-\nu-\nu')+1}\!\eta=
{1\over 2(n+1)}{n\choose \nu+\nu'}^{-1}$$ we get $$\begin{aligned}
\label{intermediate}
K_{2}(n)&=&{1\/n+1}+
\\\nonumber &&+
{n^2\over 4(n+1)}\sum_{q=1}^{n-1}
\sum_{\nu,\nu'}
{(-1)^{\nu+\nu'}\/\nu\nu'}{n\choose \nu+\nu'}^{-1}
{q-1\choose \nu-1}{n-q-1\choose \nu-1}{q-1\choose \nu'-1}{n-q-1\choose
\nu'-1}\,.\end{aligned}$$ Comparing this to the equivalent result (\[2-starK(n)CUE\]) we were again led to a previously unknown identity involving a multiple sum over binomial coefficients. It can be expressed as $$\label{ci3}
S(n,q)=\sum_{\nu,\nu'}F_{\nu,\nu'}(n,q)=1\qquad (1\le q < n)$$ with $$\begin{aligned}
F_{\nu,\nu'}(n,q)&=&{(n-1)n\/2}{(-1)^{\nu+\nu'} \over \nu\nu'}
{n\choose \nu+\nu'}^{-1}
{q-1\choose \nu-1}{q-1\choose \nu'-1}{n-q-1\choose \nu-1}{n-q-1\choose
\nu'-1}\,.\end{aligned}$$ In this case, an independent computer-generated proof was found [@akalu], which is based on the recursion relation $$\label{recursion}
q^2F_{\nu,\nu'}(n,q)-(n-q-1)^{2}F_{\nu,\nu'}(n,q+1)+(n-1)(n-2q-1)F_{\nu,\nu'}(n+
1,q+1)=0\,.$$ This recursion relation was obtained with the help of a Mathematica routine [@multisum], but it can be checked manually in a straight forward calculation. By summing (\[recursion\]) over the indices $\nu,\nu'$, the same recursion relation is shown to be valid for $S(n,q)$ [@multisum; @A=B] and the proof is completed by demonstrating the validity of (\[ci3\]) for a few initial values. Having proven (\[ci3\]) we can use it to perform the summation over $\nu,\nu'$ in (\[intermediate\]) and find $$\begin{aligned}
K_{2}(n)={1\/n+1}+\sum_{q=1}^{n-1}{n\/n^2-1}={1\/n+1}+{n\over
n+1}(1-\delta_{n,1})\,,\end{aligned}$$ which is now obviously equivalent to the random matrix form factor (\[2-starK(n)CUE\]). To the best of our knowledge, this is the first instance in which a combinatorial approach to random matrix theory is employed.
**Conclusions**
===============
We have shown how within periodic orbit theory the problem of finding the form factor (the spectral two-point correlation function) for a quantum graph can be exactly reduced to a well-defined combinatorial problem. For this purpose it was necessary to go beyond the diagonal approximation and to take into account the correlations between the periodic orbits.
In our model, these correlations are restricted to groups of isometric periodic orbits. This fits very well with the results of [@CPS98], where for a completely different system (the Sinai billiard), the classical correlations between PO’s were analysed and found to be restricted to relatively small groups of orbits. The code words of the orbits belonging to one group were conjectured to be related by a permutation and a symmetry operation, which is in complete analogy to the isometric orbits on graphs.
Even for the very small and simple graph model that we considered in the last section the combinatorial problems involved were highly non-trivial. In fact we encountered previously unknown identities which we could not have obtained if it were not for the second independent method of computing the form factor. However, since the pioneering work documented in [@A=B] the investigation of sums of the type we encountered in this paper is a rapidly developing subject, and it can be expected that finding identities like (\[ci2a\]), (\[ci2b\]) and (\[ci3\]) will shortly be a matter of computer power.
The universality of the correlations between periodic orbits in all chaotic systems poses the problem to identify the common dynamical reasons for their occurrence and to find a common mathematical structure which is capable to describe them. A very interesting question in this respect is, if the correlations between PO’s in a general chaotic system can be related to combinatorial problems.
**Acknowledgements**
====================
This research was supported by the Minerva Center for Physics of Nonlinear Systems, and by a grant from the Israel Science Foundation. We thank Tsampikos Kottos for preparing the data for Fig. \[v20\]. We were introduced to the [*El Dorado*]{} of combinatorial theory by Uri Gavish and we thank him as well as Brendan McKay and Herbert Wilf for their interest and support. We are indebted to Gregory Berkolaiko for his idea concerning the proof of (\[ci2a\]) and (\[ci2b\]), and to Akalu Tefera for his kind help in obtaining a computer-aided proof of (\[ci3\]). HS wishes to thank the Weizmann Institute of Science for the kind hospitality during the visit where this work was initiated.
Proof of equivalence for Eqs. (\[K2PI4\_UZY\]) and (\[K2PI4N\]) {#proof}
===============================================================
In this appendix we give an independent proof for the equivalence between the two results (\[K2PI4\_UZY\]) and (\[K2PI4N\]) obtained in sections \[qm\] and \[po\], respectively, for the form factor of the 2-star with $\eta=\pi/4$. We define the generating function $$\begin{aligned}
\label{Gfun}
G(x)&=&\sum_{x=1}^{\infty}K_2(n;\pi/4)\,(2x)^{n}\qquad(|x|<1/2)\end{aligned}$$ and find from (\[K2PI4\_UZY\]) $$\begin{aligned}
\label{Gfun_uzy}
G(x)&=&{2x\/1-2x}-{1\over 2}+\sum_{m=0}^{\infty}{(-1)^{m}\/2}{2m\choose m}x^{2m}
(1-2x)
\nonumber\\
&=&{1\over 2}{1-2x\over \sqrt{1+4x^2}}-{1\over 2}{1-6x\/1-2x}\,.\end{aligned}$$ On the other hand we have from (\[K2PI4N\]) $$\begin{aligned}
G(x)={x\over 1-x}+G_{1}(x)+G_{2}(-x)\end{aligned}$$ with $$\begin{aligned}
\label{G1}
G_{1}(x)=\sum_{s,t=1}^{\infty}\nq^{2}(s,t)\,x^{s+t}\end{aligned}$$ and $$\begin{aligned}
\label{G2}
G_{2}(x)=\sum_{s,t=1}^{\infty}\nq(s,t)\,\nq(t,s)\,x^{s+t}\,.\end{aligned}$$ A convenient starting point to obtain $G_{1}$ and $G_{2}$ is the integral representation $$\begin{aligned}
\label{irep}
\nq(s,t)=-{(-1)^{t}\over 2\pi\i}\oint{\rm d}z\,(1+z^{-1})^{t}(1-z)^{s-1}\,,\end{aligned}$$ where the contour encircles the origin. With the help of (\[irep\]) we find $$\begin{aligned}
\label{g}
g(x,y)&=&\sum_{s,t=1}^{\infty}\nq(s,t)\,x^{s}\,y^{t}
\nonumber\\
&=&
-{1\over 2\pi\i}\sum_{s,t=1}^{\infty}
\oint{\rm d}z\,\sum_{s,t=1}^{\infty}(1+z^{-1})^{t}(1-z)^{s-1}\,x^{s}\,(-y)^{t}
\nonumber\\
&=&
{xy\over 2\pi\i}\sum_{s,t=0}^{\infty}\oint{\rm d}z\,
{1\over 1-x(1-z)}\,{1+z\over z+y(1+z)}
\nonumber\\
&=&
{xy\over (1+y)(1-x+y-2xy)}\qquad(|x|,|y|<1/\sqrt{2})\,.\end{aligned}$$ The contour $|1+z^{-1}|=|1-z|=\sqrt{2}$ has been chosen such that both geometric series converge everywhere on it. Now we have [$$\begin{aligned}
G_{1}(x^2)&=&{1\over (2\pi\i)^{2}}\oint{{\rm d}z\,{\rm d}z'\/zz'}
\sum_{s,t=1}^{\infty}\sum_{s',t'=1}^{\infty}\nq(s,t)\,\nq(s',t')\,
(x\,z)^{s}(x/z)^{s'}(x\,z')^{t}(x/z')^{t'}
\nonumber\\
&=&
{x^{4}\over (2\pi\i)^{2}}\oint{{\rm d}z\,{\rm d}z'}\,
{1\over (1+xz')(1+x[z'-z]-2x^2zz')}{z'\over (z'+x)(zz'+x[z-z']-2x^2)}\,,\end{aligned}$$ ]{}where $|x|<1/\sqrt{2}$ and the contour for $z,z'$ is the unit circle. We perform the double integral using the residua inside the contour and obtain $$\begin{aligned}
G_{1}(x)={x\over 2x-1}\({1\over \sqrt{4x^2+1}}-{1\/1-x}\)\,.\end{aligned}$$ In complete analogy we find $$\begin{aligned}
G_{2}(x)={1\over 2}{4x^2+2x+1\/(2x+1)\sqrt{4x^2+1}}-{1\over 2}\end{aligned}$$ such that $$\label{GfunPO}
G(x)={x\/1-x}+{x\over 2x-1}\({1\over \sqrt{4x^2+1}}-{1\/1-x}\)+
{1\over 2}{4x^2-2x+1\/(1-2x)\sqrt{4x^2+1}}-{1\over 2}\,.$$ The proof is completed by a straightforward verification of the equivalence between the rational functions (\[Gfun\_uzy\]) and (\[GfunPO\]).
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We generalize a recent construction of Exel and Pardo, from discrete groups acting on finite directed graphs to locally compact groups acting on topological graphs. To each cocycle for such an action, we construct a $C^*$-correspondence whose associated Cuntz-Pimsner algebra is the analog of the Exel-Pardo $C^*$-algebra.'
address:
- 'Institute of Mathematics, University of Oslo, PB 1053 Blindern, 0316 Oslo, Norway'
- 'School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287'
- 'School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287'
author:
- Erik Bédos
- 'S. Kaliszewski'
- John Quigg
date: 'October 27, 2017'
title: 'On Exel-Pardo algebras'
---
Introduction {#intro}
============
Let $E$ denote a directed graph with vertex set $E^0$ and edge set $E^1$, as in [@Rae Section 5]. When $E$ is finite with no sources, Exel and Pardo have shown in [@EP] how to attach a Cuntz-Krieger-like $C^*$-algebra ${\mathcal O}_{G,E}$ to an action of a countable discrete group $G$ on $E$, equipped with a cocycle $\varphi$ from $ G \times E^1$ into $G$ that is compatible with the action of $G$ on $E^0$. This set-up is powerful as it encompasses the $C^*$-algebras ${\mathcal O}_{(G,X)}$ of self-similar groups introduced by Nekrashevych [@Nek09] and the Katsura’s algebras ${\mathcal O}_{A,B}$ associated with two $N\times N$ integer matrices $A$ and $B$ in [@Kat08]. Our aim with this paper is to generalize Exel and Pardo’s construction, allowing $G$ to be uncountable and $E$ to be infinite, possibly with sources. In fact, we develop the basic construction of the $C^*$-algebra at a significantly greater level of generality: we start with a locally compact group $G$ acting on a topological graph $E$ and a cocycle $\varphi$ for this action.
The main idea is that there is a natural way to associate to the given data $(E, G, \varphi)$ a $C^*$-correspondence ${Y^\varphi}$ over the crossed product $C_0(E^0) \rtimes G$, and the Cuntz-Pimsner algebra associated with this correspondence provides the desired algebra. In [@EP Section 10], Exel and Pardo also give a description of ${\mathcal O}_{G,E}$ (in the case they consider) as a Cuntz-Pimsner algebra, but our approach has an interesting conceptual feature, besides that it works without any restriction on $G$ and $E$. Considering first the case where $\varphi$ is the “trivial” cocycle, that sends $(g, e)$ to $g$ for every $g\in G, e \in E^1$, our correspondence ${Y^\varphi}$ reduces to the crossed product $X\rtimes G$ of the graph correspondence $X=X_E$ by the action of $G$ naturally associated with the action of $G$ on $E$. For a general $\varphi$, the correspondence ${Y^\varphi}$ is equal to $X\rtimes G$ as a right Hilbert $(C_0(E_0)\rtimes G)$-module, but the cocycle $\varphi$ is used to deform the left action of $C_0(E_0)\rtimes G$ on $X\rtimes G$.
The paper is organized as follows. In Section 2 we review some facts about cocycles for actions of locally compact groups on locally compact Hausdorff spaces. In Section 3 we consider an action of a locally compact group $G$ on a topological graph $E$ as in [@dkq Section 3], introduce the concept of a cocycle $\varphi$ for such an action (in a slightly more general way than Exel and Pardo) and show how to construct the desired $C^*$-correspondence $Y^\varphi$. As a result, we can form the Toeplitz algebra ${\mathcal T}_{{Y^\varphi}}$ and the Cuntz-Pimsner algebra ${\mathcal O}_{{Y^\varphi}}$ associated to ${Y^\varphi}$. In Section 4 we show that if two systems $(E, G, \varphi)$ and $(E',G, \varphi')$ are cohomology conjugate in a natural sense, then $Y^\varphi$ is isomorphic to $Y^{\varphi'}$, hence the resulting algebras are isomorphic. In Section 5 we restrict our attention to the case where $G$ is discrete and $E$ is a directed graph and give a description of ${\mathcal T}_{Y^\varphi}$ in terms of generators and relations. When $E$ is row-finite, we also give a similar description of ${\mathcal O}_{{Y^\varphi}}$, which in particular shows that ${\mathcal O}_{{Y^\varphi}}$ is isomorphic to the Exel-Pardo algebra ${\mathcal O}_{G,E}$ when $E$ is finite and sourceless. For completeness we also show in Section 6 that the Exel-Pardo correspondence obtained in [@EP Section 10] is isomorphic to our $Y^\varphi$. Finally, in Section 7 we present several examples of triples $(E, G, \varphi)$ that illustrate the flexibility of our setting and indicate the diversity of $C^*$-algebras that arise from this construction.
Preliminaries {#prelims}
=============
We recall some of the well-known theory of cocycles for group actions (see, e.g., [@Zimmer Section 4.2]). Let $G$ be a locally compact group acting continuously by homeomorphisms on a nonempty locally compact Hausdorff space $S$, and let $T$ be a locally compact group. We sometimes write the action as a map $\sigma{\colon}G\times S\to S$, and we also write $$gx=g\cdot x=\sigma(g,x){\quad\text{for }}g\in G,x\in S.$$ A *cocycle for the action $G{\curvearrowright}S$ with values in $T$* is a continuous map $\varphi{\colon}G\times S\to T$ satisfying the *cocycle identity* $$\varphi(gh,x)=\varphi(g,hx)\varphi(h,x){\quad\text{for all }}g,h\in G,x\in S.$$ We will primarily be concerned with the case $T=G$.
When $S$ is discrete, the action $G{\curvearrowright}S$ is a disjoint union of transitive actions on the orbits $Gx$, and the restriction $\varphi|_{G\times Gx}$ is a cocycle $\varphi_x$ for this transitive action. In fact, the cocycle $\varphi$ can be reconstructed from these restricted cocycles $\varphi_x$; indeed the cocycles for the actions on the orbits may be chosen willy-nilly.
If $S'$ is another $G$-space that is conjugate to $S$ via a homeomorphism $\theta{\colon}S'\to S$, then $\theta$ transports (in the reverse direction) the cocycle $\varphi$ for the action on $S$ to a cocycle $\varphi'$ for the action on $S'$ via $$\varphi'(g,x)=\varphi(g,\theta(x)){\quad\text{for }}g\in G,x\in S'.$$
If $\pi{\colon}G\to T$ is any continuous homomorphism, then the map $\varphi$ defined by $$\varphi(g,x)=\pi(g)$$ is a $T$-valued cocycle for the action $G{\curvearrowright}S$, and these are precisely the cocycles that are constant in the second coordinate. In particular, the map $\varphi(g,x)=1$ is a cocycle, where 1 denotes the identity element of $T$.
If $G={\mathbb Z}$ then there is a bijection between the set of $T$-valued cocycles for $G{\curvearrowright}S$ and the set of continuous maps $\xi{\colon}S\to T$, given by $$\xi(x)=\varphi(1,x),$$ where 1 denotes the identity element of $G$. We will need to use this, and for convenience we will call $\xi$ the *generating function* of $\varphi$.
If $\varphi{\colon}G\times S\to T$ is a cocycle and $\pi{\colon}T\to R$ is a continuous homomorphism to another locally compact group, then $\pi\circ\varphi$ is an $R$-valued cocycle.
The cocycle identity is precisely what is needed so that the equation $$\label{induced action}
g\cdot (x,t)=(gx,\varphi(g,x)t){\quad\text{for }}g\in G,x\in S,t\in T$$ defines an action of $G$ on $S\times T$. Two $T$-valued cocycles $\varphi$ and $\varphi'$ for the action $G{\curvearrowright}S$ are *cohomologous* if there is a continuous map $\psi{\colon}S\to T$ such that $$\label{cohomologous}
\varphi'(g,x)=\psi(gx)\varphi(g,x)\psi(x){^{-1}}{\quad\text{for all }}g\in G,x\in S.$$ Conversely, starting with a $T$-valued cocycle $\varphi$ for the action $G{\curvearrowright}S$ and a continuous map $\psi{\colon}S\to T$, the map $\varphi'$ defined by is also a cocycle for the action $G{\curvearrowright}S$ (which is then cohomologous to $\varphi$ by construction). Moreover, the respective actions $\cdot$ and $\cdot'$ of $G$ on $S\times T$ are *conjugate*: the homeomorphism $\theta$ on $S\times T$ defined by $$\theta(x,t)=\bigl(x,\psi(x)t\bigr)$$ satisfies $$g\cdot'\theta(x,t)=\theta\bigl(g\cdot(x,t)\bigr){\quad\text{for all }}g\in G,x\in S,t\in T.$$ A cocycle is a *coboundary* if it is cohomologous to the trivial cocycle $\varphi(g,x)=1$. If the group $T$ is abelian, then the set of $T$-valued cocycles for the action $G{\curvearrowright}S$ is an abelian group, the coboundaries form a subgroup, and the set of cohomology classes of cocycles is the quotient group.
Let $\varphi,\varphi'$ be $T$-valued cocycles for an action ${\mathbb Z}{\curvearrowright}S$, with respective generating functions $\xi,\xi'$. Let the ${\mathbb Z}$-action be generated by the homeomorphism $\tau$ on $S$. Then $\varphi$ and $\varphi'$ are cohomologous if and only if there is a continuous map $\psi{\colon}S\to T$ such that $$\xi'(x)=\psi(\tau(x))\xi(x)\psi(x){^{-1}}{\quad\text{for all }}x\in S,$$ in which case $\psi$ also satisfies .
The following elementary result is presumably folklore, but we could not find it in the literature, so we include the short proof:
\[g cbdy\] Let $G{\curvearrowright}S$, and let $\varphi{\colon}G\times S\to G$ be a cocycle. Then the following are equivalent:
1. The cocycle $(g,e)\mapsto g$ is a coboundary.
2. There is a continuous map $\psi{\colon}S\to G$ such that $$\begin{aligned}
\psi(gx)&=g\psi(x){\quad\text{for all }}g\in G,x\in S.\end{aligned}$$
3. $S$ is $G$-equivariantly homeomorphic to a space of the form $G\times R$, where $G$ acts by left translation in the first factor.
(1)${\ensuremath{\Leftrightarrow}}$(2) follows immediately from the definitions, and (3)${\ensuremath{\Rightarrow}}$(2) is trivial. Assuming (2), put $R=\psi{^{-1}}(\{1\})$. It is an elementary exercise to show that the map $\theta{\colon}S\to G\times R$ defined by $$\theta(x)=\bigl(\psi(x),\psi(x){^{-1}}x\bigr)$$ is a $G$-equivariant homeomorphism, giving (3).
We have not seen the following terminology in the literature, but it surely expresses a standard relationship. Since we will need it, we record it formally.
\[cohomology conjugate\] Suppose that we have two actions of $G$ on respective spaces $S$ and $S'$, with respective $T$-valued cocycles $\varphi$ and $\varphi'$. We say that the systems $(G,S,\varphi)$ and $(G,S',\varphi')$ are *cohomology conjugate* if there is a homeomorphism $\theta{\colon}S'\to S$ that intertwines the actions and transports $\varphi$ to a cocycle that is cohomologous to $\varphi'$.
Suppose that $G$ acts on a finite set $S$. Let the group $T$ be abelian, and write it additively. Let $\varphi{\colon}G\times S\to T$ be a cocycle. Then the function $$g\mapsto \sum_{x\in S}\varphi(g,x)$$ is a cohomology invariant, because for any map $\psi{\colon}S\to T$ we have $$\begin{aligned}
\sum_{x\in S}\bigl(\varphi(g,x)+\psi(gx)-\psi(x)\bigr)
&=\sum_{x\in S}\varphi(g,x)+\sum_{x\in S}\psi(gx)-\sum_{x\in S}\psi(x)
\\&=\sum_{x\in S}\varphi(g,x),\end{aligned}$$ since $x\mapsto gx$ is a permutation of $S$. In particular, if $G={\mathbb Z}$ and $\varphi$ has generating function $\xi{\colon}S\to T$, then the number $$\sum_{x\in S}\xi(x)$$ is a cohomology invariant. We call this number the *signature* of the cocycle $\varphi$. We will find it useful to record the following consequence, which is surely folklore:
\[signature\] Let ${\mathbb Z}{\curvearrowright}S$ and ${\mathbb Z}{\curvearrowright}S'$, and let $\varphi$ and $\varphi'$ be $T$-valued cocycles for the respective actions. If $S$ and $S'$ are finite, $T$ is abelian, and the cocycles $\varphi$ and $\varphi'$ have different signatures, then the systems $({\mathbb Z},S,\varphi)$ and $({\mathbb Z},S',\varphi')$ are not cohomology conjugate in the sense of [Definition ]{}.
[Lemma ]{} below is [@Zimmer 4.2.13]. Zimmer proved the result in greater generality, involving Borel actions and cocycles, but we restrict ourselves to the discrete case. We briefly summarize the proof for convenient reference.
\[Zimmer\] Let $G$ and $T$ be discrete groups, let $H$ be a subgroup of $G$, and let $\varphi{\colon}G\times G/H\to T$ be a $T$-valued cocycle for the canonical action by left translation. Define $\pi_\varphi{\colon}H\to T$ by $$\pi_\varphi(h)=\varphi(h,H).$$ Then $\pi_\varphi$ is a homomorphism, and moreover the map $\varphi\mapsto \pi_\varphi$ gives a bijection from the set of cohomology classes of $T$-valued cocycles for the action $G\curvearrowright G/H$ to the set of conjugacy classes of homomorphisms from $H$ to $T$.
It follows immediately from the cocycle identity that $\pi_\varphi$ defines a homomorphism.
Now let $\pi{\colon}H\to T$ be a homomorphism. Choose a cross-section $\eta{\colon}G/H\to G$ such that $\eta(H)=1$, and define $\varphi_0{\colon}G\times G/H\to H$ by $$\varphi_0(g,x)=\eta(gx){^{-1}}g\eta(x).$$ It is an easy exercise in the definitions to check that $\varphi_0$ is a cocycle with values in $H$, and that $$\pi_{\varphi_0}={\text{\textup{id}}}_H.$$ Then $\varphi:=\pi\circ\varphi_0{\colon}G\times G/H\to T$ is a cocycle, and one readily checks that with $\pi_\varphi=\pi$. Thus the map $\varphi\mapsto \pi_\varphi$ is onto the set of homomorphisms from $H$ to $T$.
Let $\varphi,\varphi'{\colon}G\times G/H\to T$ be cocycles, with associated homomorphisms $\pi,\pi'$. Suppose that $\varphi'$ is cohomologous to $\varphi$, and choose a map $\psi{\colon}G/H\to T$ such that $$\varphi'(g,x)=\psi(gx)\varphi(g,x)\psi(x){^{-1}}{\quad\text{for all }}g\in G,x\in G/H.$$ Then for all $h\in H$ we have $$\begin{aligned}
\pi'(h)
=\psi(hH)\pi(h)\psi(H){^{-1}}=\operatorname{Ad}\psi(H)\circ \pi(h),\end{aligned}$$ so the element $\psi(H)\in T$ conjugates $\pi$ to $\pi'$.
Conversely, let $t\in T$, and suppose that $\pi'=\operatorname{Ad}t\circ\pi$. Note that $$\begin{aligned}
\pi\bigl(\eta(gx){^{-1}}g\eta(x)\bigr)
&=\varphi\bigl(\eta(gx){^{-1}}g\eta(x),H\bigr)
\\&=\varphi\bigl(\eta(gx){^{-1}},g\eta(x)H\bigr)
\varphi\bigl(g\eta(x),H\bigr)
\\&=\varphi\bigl(\eta(gx),\eta(gx){^{-1}}g\eta(x)H\bigr){^{-1}}\varphi\bigl(g,\eta(x)H\bigr)
\varphi\bigl(\eta(x),H\bigr)
\\&=\varphi\bigl(\eta(gx),H\bigr){^{-1}}\varphi(g,x)\varphi\bigl(\eta(x),H\bigr),\end{aligned}$$ because $\eta(gx){^{-1}}g\eta(x)\in H$ and $\eta(x)H=x$, and similarly for $\pi'$ and $\varphi'$. Thus $$\begin{aligned}
&\varphi'(g,x)
\\&\quad=\varphi'\bigl(\eta(gx),H\bigr)t{^{-1}}\varphi\bigl(\eta(gx),H\bigr){^{-1}}\varphi(g,x)\varphi\bigl(\eta(x),H\bigr)t\varphi'\bigl(\eta(x),H\bigr){^{-1}}\\&\quad=\psi(gx)\varphi(g,x)\psi(x){^{-1}},\end{aligned}$$ where $\psi{\colon}G/H\to T$ is defined by $$\psi(x)=\varphi'\bigl(\eta(x),H\bigr)t{^{-1}}\varphi\bigl(\eta(x),H\bigr){^{-1}},$$ and hence $\varphi'$ is cohomologous to $\varphi$.
Note that, in the notation of the above proof, if we are given a cocycle $\varphi$, we can explicitly compute how the cocycle $\pi_\varphi\circ\varphi_0$ is cohomologous to $\varphi$: $$\begin{aligned}
\pi_\varphi\circ\varphi_0(g,x)
&=\varphi\bigl(\eta(gx){^{-1}}g\eta(x),H\bigr)
\\&=\varphi\bigl(\eta(gx),H\bigr){^{-1}}\varphi(g,x)\varphi\bigl(\eta(x),H\bigr)
\\&=\tau(gx)\varphi(g,x)\tau(x){^{-1}},\end{aligned}$$ where $\tau{\colon}G/H\to T$ is defined by $$\tau(x)=\varphi(\eta(x),H){^{-1}}.$$
\[Z hom\] With the hypotheses of [Lemma ]{}, if $T$ is abelian then the group of cohomology classes of cocycles of $T$-valued cocycles for $G{\curvearrowright}G/H$ is isomorphic to the group of homomorphisms from $H$ to $T$.
The following result is surely standard, but since we could not find it in the literature and we need to refer to it later, we give the elementary proof.
\[H1Z\] Let $a$ be a positive integer, and let $${\mathbb Z}_a={\mathbb Z}/a{\mathbb Z}=\{0,1,\dots,a-1\}$$ be the quotient group. Let ${\mathbb Z}$ act on ${\mathbb Z}_a$ in the canonical manner, by translation modulo $a$. For any $c\in{\mathbb Z}$ define $\xi_c{\colon}{\mathbb Z}_a\to{\mathbb Z}$ by $$\xi_c(x)=\begin{cases}0{& \text{if }}x<a-1\\c{& \text{if }}x=a-1,\end{cases}$$ and let $\varphi_c$ be the cocycle with generating function $\xi_c$. Then $\{\varphi_c:c\in{\mathbb Z}\}$ is a complete set of representatives for the set of cohomology classes of ${\mathbb Z}$-valued cocycles for the canonical action ${\mathbb Z}{\curvearrowright}{\mathbb Z}_a$.
The action of ${\mathbb Z}$ on ${\mathbb Z}_a$ is generated by the permutation $\tau$ of ${\mathbb Z}_a$ given by $$\tau(x)=x+1.$$ In the notation of the proof of [Lemma ]{}, choose the cross section $\eta{\colon}{\mathbb Z}_a\to{\mathbb Z}$ to be given by $$\eta(k+a{\mathbb Z})=k{\quad\text{for }}k=0,1,\dots,a-1.$$ The special $a{\mathbb Z}$-valued cocycle $\varphi_0$ as in the proof of [Lemma ]{} has generating function $$\varphi_0(1,x)=1+\eta(x)-\eta(\tau(x))
=\begin{cases}0{& \text{if }}x<a-1\\a{& \text{if }}x=a-1.\end{cases}$$ As in the proof of [Lemma ]{} (see also [Remark ]{}), every ${\mathbb Z}$-valued cocycle is cohomologous to a unique cocycle of the form $\pi\circ\varphi_0$ for a homomorphism $\pi{\colon}a{\mathbb Z}\to{\mathbb Z}$. The homomorphism $\pi$ is uniquely determined by the number $c=\pi(a)\in{\mathbb Z}$, and a routine calculation shows that the generating function of the cocycle $\pi\circ\varphi_0$ is given by $\xi_c$.
With the notation of [Lemma ]{}, we of course see immediately that for distinct $c$ the cocycles $\varphi_c$ are noncohomologous, since $\varphi_c$ has signature $c$. But in fact the following corollary (which again is surely folklore) shows that much more is true:
\[transitive\] For systems $({\mathbb Z},S,\varphi)$, where ${\mathbb Z}{\curvearrowright}S$ transitively, $S$ is finite, and $\varphi{\colon}{\mathbb Z}\times S\to{\mathbb Z}$ is a cocycle, the signature of $\varphi$ is a complete invariant for cohomology conjugacy.
Let $({\mathbb Z},S,\varphi)$ be such a system, and let $S$ have cardinality $a$. By transitivity this system is cohomology conjugate to a system of the form $({\mathbb Z},{\mathbb Z}_a,\varphi')$, where ${\mathbb Z}$ acts on ${\mathbb Z}_a$ by the usual translation modulo $a$. It follows from [Lemma ]{} that the cocycle $\varphi'$ is determined up to cohomology by its signature, and moreover every integer can occur as the signature of some cocycle for this action ${\mathbb Z}{\curvearrowright}{\mathbb Z}_a$.
Cocycles for graph actions {#withphi}
==========================
Let $E=(E^0,E^1,r,s)$ be a topological graph in the sense of Katsura [@Ka1], that is, $E^0$ and $E^1$ are locally compact Hausdorff spaces, $r{\colon}E^1\to E^0$ is continuous, and $s{\colon}E^1\to E^0$ is a local homeomorphism.
In [@Ka1 Section 2], Katsura constructs a correspondence $X=X_E$ over the commutative $C^*$-algebra $A:=C_0(E^0)$ as the completion of the pre-correspondence $C_c(E^1)$, with operations defined for $a\in A$ and $x,y\in C_c(E^1)$ by $$\begin{aligned}
(a\cdot x)(e)&=a(r(e))x(e)
\\
(x\cdot a)(e)&=x(e)a(s(e))
\\
{\langle}x,y{\rangle}_A(v)&=\sum_{s(e)=v}{\overline}{x(e)}y(e).\end{aligned}$$ We will call $X$ the *graph correspondence* of $E$.
Katsura defines $C^*(E)$ as the Cuntz-Pimsner algebra ${\mathcal O}_X$ [@Ka1 Definition 2.10]. Here we use the conventions of [@Kat04 Definition 3.5] for Cuntz-Pimsner algebras.
Let $G$ be a locally compact group acting continuously on $E$ in the sense of [@dkq Section 3], that is, $G$ acts in the usual way by homeomorphisms on the spaces $E^0$ and $E^1$, and for each $g\in G$ the maps $e\mapsto ge$ on edges and $v\mapsto gv$ on vertices constitute an automorphism of the topological graph $E$.
Let $\alpha$ denote the associated action of $G$ on $A$: $$\alpha_g(a)(v)=a(g{^{-1}}v){\quad\text{for }}g\in G,a\in A,v\in E^0.$$ By [@dkq Proposition 5.4], we can define an $\alpha$-compatible action $\gamma$ of $G$ on the graph correspondence $X$ via $$\gamma_g(x)(e)=x(g{^{-1}}e){\quad\text{for }}g\in G,x\in C_c(E^1),e\in E^1.$$ Here we use the conventions of [@taco Definition 3.1] for actions on correspondences. By [@taco Proposition 3.5], there is a $C^*$-correspondence $Y:=X\rtimes_\gamma G$ over the crossed product $B:=A\rtimes_\alpha G$, which contains $C_c(G,X)$ as a dense subspace, and which satisfies $$\begin{aligned}
b\cdot \xi(g)&=\int_G b(h)\cdot \gamma_h\bigl(\xi(h{^{-1}}g)\bigr)\,dh
\\
\xi\cdot b(g)&=\int_G \xi(h)\cdot \alpha_h\bigl(b(h{^{-1}}g)\bigr)\,dh
\\
{\langle}\xi,\eta{\rangle}_{A\rtimes_\alpha G}(g)&=\int_G \alpha_{h{^{-1}}}\bigl({\langle}\xi(h),\eta(hg){\rangle}_A\bigr)\,dh\end{aligned}$$ for $b\in B$, $\xi,\eta\in C_c(G,X)$, and $g\in G$. We call $Y$ the *crossed product* of the action $(X,G)$. This correspondence is both *full* in the sense that $\operatorname*{\overline{\operatorname*{span}}}{\langle}Y,Y{\rangle}_B=B$, and *nondegenerate* in the sense that $BY=Y$. The left $B$-module multiplication is given by a homomorphism $\phi_Y{\colon}B\to {\mathcal L}(Y)=M({\mathcal K}(Y))$, which is the integrated form of a covariant pair $(\pi,U)$, where $\pi{\colon}A\to {\mathcal L}(Y)$ is the nondegenerate representation determined by $$\label{left A}
\bigl(\pi(a)\xi\bigr)(h)=a\cdot \xi(h){\quad\text{for }}a\in A,\xi\in C_c(G,X),h\in G,$$ and $U{\colon}G\to {\mathcal L}(Y)$ is the strongly continuous unitary representation determined by $$\label{U}
(U_g\xi)(h)=\gamma_g(\xi(g{^{-1}}h)){\quad\text{for }}g\in G,\xi\in C_c(G,X),h\in G$$ (see the proof of [@taco Proposition 3.5]).
We will want to compute with the $B$-correspondence $Y$ using two-variable functions. Since $A=C_0(E^0)$, we can identify the crossed product $A\rtimes_\alpha G$ as a completion of the convolution $*$-algebra $C_c(E^0\times G)$ with operations $$\begin{aligned}
(b*c)(v,g)&=\int_G b(v,h)c(h{^{-1}}v,h{^{-1}}g)\,dh
\label{convolution two}
\\
b^*(v,g)&=\Delta(g{^{-1}}){\overline}{b(g{^{-1}}v,g{^{-1}})}
\label{star two}\end{aligned}$$ for $b,c\in C_c(E^0\times G)$ and $(v,g)\in E^0\times G$. (See, for example, [@danacrossed page 53].) Since we will play a similar game with $X\rtimes_\gamma G$, we pause to provide a little detail on how – are derived from the usual operations on the convolution algebra $C_c(G,A)$, given by $$\begin{aligned}
(b*c)(g)&=\int_G b(h)\alpha_h(c(h{^{-1}}g))\,dh
\label{convolution one}
\\
b^*(g)&=\Delta(g{^{-1}})\alpha_g(b(g{^{-1}}))^*.
\label{star one}\end{aligned}$$ We do it for ; it is much easier for . Using the embeddings $$C_c(E^0\times G)\subset C_c(G,C_c(E^0))\subset C_c(G,A),$$ for $b,c\in C_c(E^0\times G)$ and $(v,g)\in E^0\times G$, we have $$\begin{aligned}
(b*c)(v,g)
&=(b*c)(g)(v)
\\&=\int_G b(h)\alpha_h(c(h{^{-1}}g))\,dh(v)
\\&\overset{(*)}=\int_G \bigl(b(h)\alpha_h(c(h{^{-1}}g))\bigr)(v)\,dh
\\&=\int_G b(h)(v)\alpha_h(c(h{^{-1}}g))(v)\,dh
\\&=\int_G b(v,h)c(h{^{-1}}g)(h{^{-1}}v)\,dh
\\&=\int_G b(v,h)c(h{^{-1}}v,h{^{-1}}g)\,dh.\end{aligned}$$ The point is that at the equality $(*)$ we are using that in the line above we have a norm-convergent integral of a continuous $A$-valued function with compact support, and evaluation at $v$ is a bounded linear functional.
Now we argue similarly for $$C_c(E^1\times G)\subset C_c(G,C_c(E^1))\subset C_c(G,X),$$ where in a couple of computations we will have a norm-convergent integral of an $X$-valued function with compact support, and we use the property that evaluation at an edge $e$ is a bounded linear functional on $X$, since on $C_c(E^1)$ the uniform norm is less than the norm from the Hilbert $A$-module $X$. For $b\in C_c(E^0\times G)$, $\xi,\eta\in C_c(E^1\times G)$, $(e,g)\in E^1\times G$, and $v\in E^0$, we have $$\begin{split}
(b\cdot \xi)(e,g)&=\int_G b(r(e),h)\xi(h{^{-1}}e,h{^{-1}}g)\,dh,\\
(\xi\cdot b)(e,g)&=\int_G \xi(e,h)b(h{^{-1}}s(e),h{^{-1}}g)\,dh,\ \text{ and}\\
{\langle}\xi,\eta{\rangle}_{A\rtimes_\alpha G}(v,g)&=\int_G \sum_{s(e)=hv} {\overline}{\xi(e,h)}\eta(e,hg)\,dh.
\end{split}$$ Indeed, $$\begin{split}
(b\cdot \xi)(e,g)
&=(b\cdot \xi)(g)(e)
\\&=\int_G b(h)\cdot \gamma_h(\xi(h{^{-1}}g))\,dh(e)
\\&=\int_G \bigl(b(h)\cdot \gamma_h(\xi(h{^{-1}}g))\bigr)(e)\,dh
\\&=\int_G b(h)(r(e))\gamma_h(\xi(h{^{-1}}g))(e)\,dh
\\&=\int_G b(r(e),h)\xi(h{^{-1}}g)(h{^{-1}}e)\,dh
\\&=\int_G b(r(e),h)\xi(h{^{-1}}e,h{^{-1}}g)\,dh,
\end{split}$$ $$\begin{split}
(\xi\cdot b)(e,g)
&=(\xi\cdot b)(g)(e)
\\&=\int_G \xi(h)\cdot \alpha_h(b(h{^{-1}}g))\,dh(e)
\\&=\int_G \bigl(\xi(h)\cdot \alpha_h(b(h{^{-1}}g))\bigr)(e)\,dh
\\&=\int_G \xi(h)(e)\alpha_h(b(h{^{-1}}g))(s(e))\,dh
\\&=\int_G \xi(e,h)b(h{^{-1}}g)(h{^{-1}}s(e))\,dh
\\&=\int_G \xi(e,h)b(h{^{-1}}s(e),h{^{-1}}g)\,dh,
\end{split}$$ and $$\begin{split}
{\langle}\xi,\eta{\rangle}_{A\rtimes_\alpha G}(v,g)
&={\langle}\xi,\eta{\rangle}_{A\rtimes_\alpha G}(g)(v)
\\&=\int_G \alpha_{h{^{-1}}}\bigl({\langle}\xi(h),\eta(hg){\rangle}_A\bigr)\,dh(v)
\\&=\int_G \bigl(\alpha_{h{^{-1}}}\bigl({\langle}\xi(h),\eta(hg){\rangle}_A\bigr)(v)\bigr)\,dh
\\&=\int_G {\langle}\xi(h),\eta(hg){\rangle}_A(hv)\,dh
\\&\overset{(*)}=\int_G \sum_{s(e)=hv} {\overline}{\xi(h)(e)}\eta(hg)(e)\,dh
\\&=\int_G \sum_{s(e)=hv} {\overline}{\xi(e,h)}\eta(e,hg)\,dh,
\end{split}$$ where in the equality at $(*)$ the sum is finite by [@Ka1 Lemma 1.4], since $\xi(h)\in C_c(E^1)$.
We can also compute with the covariant pair $(\pi,U)$ of and using two-variable functions: for $a\in A=C_0(E^0)$, $\xi\in C_c(E^1\times G)$, $g,h\in G$, and $e\in E^1$ we have $$\begin{aligned}
\bigl(\pi(a)\xi\bigr)(e,h)&=a(r(e))\xi(e,h)
\label{left A two}
\\
(U_g\xi)(e,h)&=\xi(g{^{-1}}e,g{^{-1}}h).
\label{U two}\end{aligned}$$
\[ind lim\] The inductive limit topology on $C_c(E^1\times G)$ is stronger than the norm topology from $Y$.
It suffices to show that if $\{\xi_i\}$ is a net in $C_c(E^1\times G)$ converging uniformly to 0 and such the supports of the $\xi_i$’s are all contained in some fixed compact set $K$, then $$\|{\langle}\xi_i,\xi_i{\rangle}_B\|\to 0.$$ Choose compact sets $K_1\subset E^1$ and $K_2\subset G$ such that $K\subset K_1\times K_2$. By the elementary [Lemma ]{} below, we can choose $n\in{\mathbb N}$ such that for all $v\in E^0$ the set $s{^{-1}}(v)\cap K_1$ has at most $n$ elements. Let ${\varepsilon}>0$, and choose $i_0$ such that for all $i\ge i_0$ we have $$|\xi_i(e,g)|<{\varepsilon}{\quad\text{for all }}(e,g)\in E^1\times G.$$ For all $i\ge i_0$ and $(v,g)\in E^0\times G$, $$\begin{aligned}
{\langle}\xi_i,\xi_i{\rangle}_B(v,g)
&=\int_G \sum_{s(e)=hv} {\overline}{\xi_i(e,h)}\xi_i(e,hg)\,dh
\\&=\int_{K_2} \sum_{s(e)=hv} {\overline}{\xi_i(e,h)}\xi_i(e,hg)\,dh,\end{aligned}$$ which has absolute value bounded above by $n{\varepsilon}^2$ times the measure of $K_2$, and this suffices to show that $\|{\langle}\xi_i,\xi_i{\rangle}_B\|\to 0$ uniformly.
In the above proof we used the following elementary lemma, which we could not find in the literature (although it is similar to [@dkq Corollary 3.9 (2)]):
\[finite\] For any compact set $K\subset E^1$ there is a positive integer $n$ such that for all $v\in E^0$ the set $s{^{-1}}(v)\cap K$ has at most $n$ elements.
Let $L=s(K)$, a compact subset of $E^0$. By [@Ka1 Lemma 1.4], each $v\in E^0$ has a neighborhood $U_v$ such that for some positive integer $n_v$ the set $K\cap s{^{-1}}(U_v)$ has at most $n_v$ elements. Covering $L$ by finitely many $U_v$’s gives the lemma.
We will soon modify the $B$-correspondence $Y$ using a cocycle. The following definition of cocycle generalizes that of [@EP Section 2], where the authors consider discrete groups acting on finite graphs.
\[cocycle\] A *cocycle* for the action of $G$ on the topological graph $E$ is a cocycle $\varphi$ for the action of $G$ on the edge space $E^1$ that also satisfies the *vertex condition* $$\label{vertex condition}
\varphi(g,e)s(e)=gs(e){\quad\text{for all }}g\in G,e\in E^1.$$
In [@EP (2.3.1)] (for finite graphs), Exel and Pardo impose a stronger version of , namely $\varphi(g,e)v=gv$ for all $g\in G$, $e\in E^1$, and $v\in E^0$; our weakened version above is all that is needed, and allows for greater flexibility. For example, the elementary theory of cohomology for cocycles (see [Section ]{}) would be significantly hampered with the Exel-Pardo version.
Note that implies that for all $(g,e)\in G\times E^1$ the product $g{^{-1}}\varphi(g,e)$ lies in the isotropy subgroup $G_{s(e)}$ of $G$ at the vertex $s(e)$. Thus, the existence of nontrivial cocycles, i.e., other than the map $(g,e)\mapsto g$, depends upon having nontrivial isotropy of the action on vertices.
Let $\varphi$ be a cocycle for the action of $G$ on $E$. We use $\varphi$ to modify the $B$-correspondence $Y$ as follows: we keep the same structure as a Hilbert $B$-module, as well as the same left $A$-module action determined by . In the following we use the technique of to define an action of $G$ on the Hilbert $B$-module $Y$: for $g\in G$ and $\xi\in C_c(E^1\times G)$ define the function $V_g\xi\in C_c(E^1\times G)$ by $$\label{V}
(V_g\xi)(e,h)=\xi(g{^{-1}}e,\varphi(g{^{-1}},e)h).$$
\[G varphi\]
1. The map $g\mapsto V_g$ given by is a strongly continuous unitary representation of $G$ on the Hilbert $B$-module $Y$.
2. With $\pi$ as in and $V$ as above, the pair $(\pi,V)$ is a covariant representation of the system $(A,G,\alpha)$ on the Hilbert $B$-module $Y$.
\(1) First note that the map $g\mapsto V_g$ is multiplicative from $G$ into the set of linear operators on $C_c(E^1\times G)$: $$\begin{aligned}
(V_gV_h\xi)(e,k)
&=(V_h\xi)\bigl(g{^{-1}}e,\varphi(g{^{-1}},e)k\bigr)
\\&=\xi\bigl(h{^{-1}}g{^{-1}}e,\varphi(h{^{-1}},g{^{-1}}e)\varphi(g{^{-1}},e)k\bigr)
\\&=\xi\bigl(h{^{-1}}g{^{-1}}e,\varphi(h{^{-1}}g{^{-1}},e)k\bigr)
\\&=(V_{gh}\xi)(e,k).\end{aligned}$$ Since $V_{1_G}\xi=\xi$ for all $\xi\in C_c(E^1\times G)$ (where $1_G$ here denotes the identity element of $G$), we deduce that $V$ is a homomorphism from $G$ to the group of invertible linear operators on $C_c(E^1\times G)$.
Now we show that the inner products on $C_c(E^1\times G)$ are preserved by each $V_g$: $$\begin{aligned}
&{\langle}V_g\xi,V_g\eta{\rangle}_B(v,h)
\\&\quad=\int_G \sum_{s(e)=kv}{\overline}{(V_g\xi)(e,k)}(V_g\eta)(e,kh)\,dk
\\&\quad=\int_G \sum_{s(e)=kv}
{\overline}{\xi(g{^{-1}}e,\varphi(g{^{-1}},e)k)}\eta(g{^{-1}}e,\varphi(g{^{-1}},e)kh)\,dk
\\&\quad=\int_G \sum_{s(e)=kv}
{\overline}{\xi(e,k)}\eta(e,kh)\,dk
\intertext{\big[after $e\mapsto ge$ and $k\mapsto \varphi(g{^{-1}},e){^{-1}}k$, since a short computation using the cocycle identity and \eqref{vertex condition} shows that $s(e)=kv$ if and only if $s(g{^{-1}}e)=\varphi(g{^{-1}},e){^{-1}}kv$\big]
}
&\quad={\langle}\xi,\eta{\rangle}_B(v,h).\end{aligned}$$ In particular, $V_g$ is isometric on $C_c(E^1\times G)$, and hence extends uniquely to an isometry on the completion $Y$; moreover, since $V_g$ maps $C_c(E^1\times G)$ onto itself, this extension, which we continue to denote by $V_g$, is in fact an isometric linear map of $Y$ onto itself. Upon taking limits we see that these extensions still satisfy $V_gV_h=V_{gh}$ for all $g,h\in G$.
For $b\in C_c(E^0\times G)$ we have $$\begin{aligned}
&\bigl(V_g(\xi\cdot b)\bigr)(e,h)
\\&\quad=(\xi\cdot b)(g{^{-1}}e,\varphi(g{^{-1}},e)h)
\\&\quad=\int_G \xi(g{^{-1}}e,k)b\bigl(k{^{-1}}s(g{^{-1}}e),k{^{-1}}\varphi(g{^{-1}},e)h\bigr)\,dk
\\&\quad=\int_G \xi(g{^{-1}}e,\varphi(g{^{-1}},e)k)b\bigl(k{^{-1}}\varphi(g{^{-1}},e){^{-1}}s(g{^{-1}}e),k{^{-1}}h\bigr)\,dk
\\&\hspace{1in}\text{(after $k\mapsto \varphi(g{^{-1}},e)k$)}
\\&\quad=\int_G \xi(g{^{-1}}e,\varphi(g{^{-1}},e)k)b\bigl(k{^{-1}}\varphi(g,g{^{-1}}e)s(g{^{-1}}e),k{^{-1}}h\bigr)\,dk
\\&\quad=\int_G \xi(g{^{-1}}e,\varphi(g{^{-1}},e)k)b\bigl(k{^{-1}}gs(g{^{-1}}e),k{^{-1}}h\bigr)\,dk
{\quad\text{(by \eqref{vertex condition}) }}
\\&\quad=\int_G (V_g\xi)(e,k)b\bigl(k{^{-1}}s(e),k{^{-1}}h\bigr)\,dk
\\&\quad=\bigl((V_g\xi)\cdot b\bigr)(e,h).\end{aligned}$$ Thus by continuity the map $V_g$ on $Y$ is right $B$-linear, and this combined with its other properties makes it a unitary operator on the Hilbert $B$-module $Y$ [@lance Theorem 3.5].
For the strong continuity, by uniform boundedness it suffices to show that if $\xi\in C_c(E^1\times G)$ and $g_i\to 1$ in $G$ then $\|V_{g_i}\xi-\xi\|\to 0$. Arguing by contradiction, we can replace $\{g_i\}$ by a subnet and relabel so that no subnet of $\{\|V_{g_i}\xi-\xi\|\}$ converges to 0. Again replacing $\{g_i\}$ by a subnet, we can suppose that the $g_i$’s are all contained in some compact neighborhood $U$ of 1. It then follows from continuity of the operations, and of the function $\varphi$, that the supports of the functions $V_{v_i}\xi$’s are all contained in some fixed compact set $K\subset E^1\times G$. Then by [Lemma ]{} it suffices to show that $V_{g_i}\xi\to \xi$ uniformly. Arguing by contradiction, we can replace by a subnet so that no subnet of $V_{g_i}\xi$ converges uniformly to $\xi$. Then we can find ${\varepsilon}>0$ such that, after again replacing by a subnet, for all $i$ there exists $(e_i,h_i)\in E^1\times G$ such that $$|V_{g_i}\xi(e_i,h_i)-\xi(e_i,h_i)|\ge {\varepsilon}.$$ In particular, we must have $(e_i,h_i)\in K$ for all $i$, so that after replacing by a subnet again we have $(e_i,h_i)\to (e,h)$ for some $(e,h)\in E^1\times G$. But then by continuity we have $${\varepsilon}\le |V_{g_i}\xi(e_i,h_i)-\xi(e_i,h_i)|\to |\xi(e,h)-\xi(e,h)|=0,$$ which is a contradiction.
\(2) It suffices to show that for all $g\in G$, $a\in A$, and $\xi\in C_c(E^1\times G)$ we have $V_g\pi(a)\xi=\pi(\alpha_g(a))V_g\xi$, and we check this by evaluating at an arbitrary pair $(e,h)\in E^1\times G$: $$\begin{aligned}
(V_g\pi(a)\xi)(e,h)
&=(\pi(a)\xi)\bigl(g{^{-1}}e,\varphi(g{^{-1}},e)h\bigr)
\\&=a\bigl(r(g{^{-1}}e)\bigr)\xi\bigl(g{^{-1}}e,\varphi(g{^{-1}},e)h\bigr)
\\&=a\bigl(g{^{-1}}r(e)\bigr)(V_g\xi)(e,h)
\\&=\alpha_g(a)(r(e))(V_g\xi)(e,h)
\\&=\bigl(\pi(\alpha_g(a))V_g\xi\bigr)(e,h).
\qedhere\end{aligned}$$
The integrated form of the covariant representation $(\pi,V)$ of [Proposition ]{} is a nondegenerate representation of $B$ in ${\mathcal L}(Y)$, giving $Y$ the structure of a $B$-correspondence that we denote by ${Y^\varphi}$.
For the special cocycle $(g,e)\mapsto g$ the correspondence ${Y^\varphi}$ reduces to the crossed product $Y=X\rtimes_\gamma G$.
It will be useful to handle the left module action of $B$ on ${Y^\varphi}$ in terms of two-variable functions: for $b\in C_c(E^0\times G)\subset B$ and $\xi\in C_c(E^1\times G)\subset {Y^\varphi}$, the function $b\cdot \xi\in C_c(E^1\times G)$ is given by $$\begin{aligned}
(b\cdot \xi)(e,g)
&=\bigl(\int_G \bigl(i_A(b(h))i_G(h)\cdot \xi\bigr)\,dh\bigr)(e,g)
\\&=\int_G \bigl(i_A(b(h))i_G(h)\cdot \xi\bigr)(e,g)\,dh
{\quad\text{by {Lemma~\textup{\ref{ind lim}}} }}
\\&=\int_G b(h)(r(e))\bigl(i_G(h)\cdot \xi\bigr)(e,g)\,dh
{\quad\text{by \eqref{left A two} }}
\\&=\int_G b(r(e),h)(V_h\xi)(e,g)\,dh
\\&=\int_G b(r(e),h)\xi(h{^{-1}}e,\varphi(h{^{-1}},e)g)\,dh.\end{aligned}$$
Following [@Kat04] we can associate two $C^*$-algebras to the correspondence ${Y^\varphi}$: the Toeplitz algebra ${{\mathcal T}_{{Y^\varphi}}}$ and the Cuntz-Pimsner algebra ${{\mathcal O}_{{Y^\varphi}}}$. It will follow from Corollary \[universal O\] (or Corollary \[alt EP alg\]) that ${{\mathcal O}_{{Y^\varphi}}}$ is isomorphic to the $C^*$-algebra ${\mathcal O}_{G, E}$ associated to $(E, G, \varphi)$ by Exel and Pardo in [@EP] in the case where $E$ is finite and sourceless, $G$ is discrete and $\varphi$ satisfies $\varphi(g, e) v = v$ for all $(g, e)\in G\times E^1$ and $v\in E^0$.
We will call ${{\mathcal O}_{{Y^\varphi}}}$ the [*Exel-Pardo algebra*]{} associated to the system $(E, G, \varphi)$.
Exel and Pardo show that in the case they consider, ${\mathcal O}_{G, E}$ is nuclear whenever $G$ is amenable (cf. [@EP Corollary 10.12]). In our more general context, assume that the action of the locally compact group $G$ on the locally compact Hausdorff space $E^0$ is amenable in the sense of Anantharaman-Delaroche (see [@AD02 Section 2]). Then $B=C_0(E^0)\rtimes_\alpha G$ is nuclear [@AD02 Theorem 5.3], and it then follows from [@Kat04 Theorem 7.2 and Corollary 7.4] that ${{\mathcal T}_{{Y^\varphi}}}$ and ${{\mathcal O}_{{Y^\varphi}}}$ are nuclear. If $G$ is discrete, then $B$ is nuclear if and only if the action of $G$ on $E^0$ is amenable, cf. [@AD87 Th[é]{}or[è]{}me 4.5]. Hence, when $G$ is discrete, [@Kat04 Theorem 7.2] gives that ${{\mathcal T}_{{Y^\varphi}}}$ is nuclear if and only if $G$ acts amenably on $E^0$.
By [@enchilada Proposition 3.2], there is also a $C^*$-correspondence $Y_r:=X\rtimes_{\gamma,r} G$ over the reduced crossed product $B_r:=C_0(E^0)\rtimes_{\alpha, r} G$, which contains $C_c(G,X)$ as a dense subspace and is constructed in a similar way as $Y$. To be able to talk about the reduced $C^*$-correspondence $Y_r^\varphi$, i.e., to define a left action of $B_r$ on $Y_r$ involving $\varphi$, one will have to find out if $ \pi \times V$ factors through $B_r$ in general. If $G$ acts amenably on $E^0$, then $B=B_r$ (cf. [@AD02 Theorem 5.3]), so the problem does not show up in this case.
\[category\] In [@EP Section 2], Exel and Pardo show how to extend the action and cocycle to the set $E^*$ of finite paths, and it is clear that their proof works whenever $E$ is a directed graph and $G$ is discrete. We see a way to carry this further, to form a sort of Zappa-Sz[é]{}p product of $E^*$ by $G$ with respect to $\varphi$, and thereby obtain a new category of paths $E^*\rtimes^{\varphi} G$ in the sense of Spielberg [@Spi11], except that right cancellativity will not hold in general, and a little bit of work is necessary to force the category to have no inverses. Several natural questions arise: is the algebra $C^*(E^*\rtimes^{\varphi} G)$ that Spielberg’s theory associates to this category of paths isomorphic (or related) to the Toeplitz algebra ${{\mathcal T}_{{Y^\varphi}}}$? And then is a suitable quotient of $C^*(E^*\rtimes^\varphi G)$ isomorphic to the Cuntz-Pimsner algebra ${{\mathcal O}_{{Y^\varphi}}}$? We plan to pursue this in subsequent work.
Cohomology for graph cocycles {#cohom sec}
=============================
Throughout this section $G$ will be a locally compact group acting on a topological graph $E$.
Let $\varphi,\varphi'$ be cocycles for the action $G{\curvearrowright}E$. We say $\varphi$ and $\varphi'$ are *cohomologous* if there is a continuous function $\psi{\colon}E^1\to G$ such that for all $g\in E$ and $e\in E^1$ we have $$\begin{aligned}
\varphi'(g,e)&=\psi(ge)\varphi(g,e)\psi(e){^{-1}}\label{cohomologous graph}
\\
\psi(e)s(e)&=s(e).\label{cochain}\end{aligned}$$
Note that just says that $\varphi$ and $\varphi'$ are cohomologous as cocycles for the action $G{\curvearrowright}E^1$. The extra condition is necessary to make the theory work for actions on topological graphs.
\[is a cocycle\] If $\varphi$ is a cocycle for the action of $G$ on the topological graph $E$ and $\psi{\colon}E^1\to G$ is a continuous map satisfying , then the map $\varphi'{\colon}E^1\times G\to G$ defined by is also a cocycle for the action of $G$ on $E$.
As we mentioned above, the cocycle identity holds for $\varphi'$ by the standard theory of actions on spaces (and is a routine computation). We verify : $$\begin{aligned}
\varphi'(g,e)s(e)
&=\psi(ge)\varphi(g,e)\psi(e){^{-1}}s(e)
\\&=\psi(ge)\varphi(g,e)s(e)
\\&=\psi(ge)gs(e)
\\&=\psi(ge)s(ge)
\\&=s(ge)
\\&=gs(e).
\qedhere\end{aligned}$$
In the general theory of cocycles for actions on spaces, the constant function $(g,e)\mapsto 1$ is a cocycle (where 1 here denotes the identity element of $G$). But not necessarily for the action of $G$ on the topological graph $E$:
\[fix\] For an action of $G$ on a topological graph $E$, the following are equivalent:
1. The constant function $(g,e)\mapsto 1$ is a cocycle for the action $G\curvearrowright E$.
2. $gs(e)=s(e)$ for all $(g,e)\in G\times E^1$.
3. In [Definition ]{}, the axiom is redundant.
\(2) trivially implies (3), which in turn trivially implies (1). Assume (1). Then for all $g\in E$ and $e\in E^1$ we have $$gs(e)=1_Gs(e)=s(e),$$ giving (2).
We say that an action of $G$ on a topological graph $E$ *fixes sources* if it satisfies the equivalent conditions (1)–(3) in Lemma \[fix\].
\[trivial\] If the action $G\curvearrowright E$ fixes sources, and if $\psi{\colon}E^1\to G$ is a continuous map satisfying , then the map $\varphi{\colon}G\times E^1\to G$ defined by $$\label{coboundary}
\varphi(g,e)=\psi(ge)\psi(e){^{-1}}$$ is a cocycle for the action $G\curvearrowright E$.
If the action $G\curvearrowright E$ fixes sources, a cocycle $\varphi$ as in is a *coboundary* for the action $G\curvearrowright E$.
Thus, when the action $G\curvearrowright E$ fixes sources, coboundaries are precisely the cocycles that are cohomologous to the cocycle taking the constant value $1$.
Cohomologous cocycles give isomorphic correspondences:
\[coho cocy\] If $\varphi$ and $\varphi'$ are cohomologous cocycles for the action $G\curvearrowright E$, then the $B$-correspondences ${Y^\varphi}$ and $Y^{\varphi'}$ are isomorphic.
Let $\varphi'(e,g)=\psi(eg)\varphi(e,g)\psi(e){^{-1}}$ for a continuous map $\psi:E^1\to G$ satisfying . We will construct an isomorphism $\Phi{\colon}Y^{\varphi}\to Y^{\varphi'}$. To begin, we define $\Phi$ as a linear map on $C_c(E^1\times G)$ by $$(\Phi\xi)(e,g)=\xi(e,\psi(e){^{-1}}g){\quad\text{for }}\xi\in C_c(E^1\times G).$$ We show that $\Phi$ preserves inner products: $$\begin{aligned}
{\langle}\Phi\xi,\Phi\eta{\rangle}(v,g)
&=\int_G \sum_{s(e)=hv} {\overline}{(\Phi\xi)(e,h)}(\Phi\eta)(e,hg)\,dh
\\&=\int_G \sum_{s(e)=hv} {\overline}{\xi(e,\psi(e){^{-1}}h)}(\eta(e,\psi(e){^{-1}}hg)\,dh
\\&=\int_G \sum_{s(e)=hv} {\overline}{\xi(e,h)}(\eta(e,hg)\,dh,
\intertext{after $h\mapsto \psi(e)h$, since by \eqref{cochain} $s(e)=hv$ if and only if $s(e)=\psi(e)hv$,}
&={\langle}\xi,\eta{\rangle}(v,g).\end{aligned}$$ Thus $\Phi$ extends uniquely to an isometric linear operator on the Hilbert $B$-module $Y$. The following computation implies that $\Phi$ is right $B$-linear: for $\xi\in C_c(E^1\times G)$ and $b\in C_c(E^0\times G)$ we have $$\begin{aligned}
\bigl(\Phi(\xi\cdot b)\bigr)(e,g)
&=(\xi\cdot b)(e,\psi(e){^{-1}}g)
\\&=\int_G \xi(e,h)b(h{^{-1}}s(e),h{^{-1}}\psi(e){^{-1}}g)\,dh
\\&=\int_G \xi(e,\psi(e){^{-1}}h)b(h{^{-1}}\psi(e)s(e),h{^{-1}}g)\,dh,
\\&\hspace{1in}\text{after $h\mapsto \psi(e){^{-1}}h$}
\\&=\int_G \xi(e,\psi(e){^{-1}}h)b(h{^{-1}}s(e),h{^{-1}}g)\,dh,
{\quad\text{(by \eqref{cochain}) }}
\\&=\int_G (\Phi\xi)(e,h)b(h{^{-1}}s(e),h{^{-1}}g)\,dh
\\&=\bigl((\Phi\xi)\cdot b\bigr)(e,g).\end{aligned}$$ This combined with the other properties of $\Phi$ makes it a unitary map from the Hilbert $B$-module ${Y^\varphi}$ to the Hilbert $B$-module $Y^{\varphi'}$ [@lance Theorem 3.5].
Then the following computation implies that $\Phi$ is left $B$-linear, from which the theorem will follow: $$\begin{aligned}
\bigl(\Phi(b\cdot \xi)\bigr)(e,g)
&=(b\cdot \xi)(e,\psi(e){^{-1}}g)
\\&=\int_G b(r(e),h)\xi(h{^{-1}}e,\varphi(h{^{-1}},e) \psi(e){^{-1}}g)\,dh
\\&=\int_G b(r(e),h)\xi(h{^{-1}}e,\psi(h{^{-1}}e){^{-1}}\varphi'(h{^{-1}},e)g)\,dh
\\&=\int_G b(r(e),h)(\Phi\xi)(h{^{-1}}e,\varphi'(h{^{-1}},e)g)\,dh
\\&=\bigl(b\cdot (\Phi\xi)\bigr)(e,g).
\qedhere\end{aligned}$$
As an immediate consequence of [Theorem ]{}, we get:
\[cohomology conjugate 2\] Assume that $G$ also acts on another topological graph $F=(F^0, F^1, r', s')$ and that $\varphi$ and $\varphi'$ are cocycles for $G{\curvearrowright}E$ and $G{\curvearrowright}F$, respectively. If $(E, G, \varphi)$ and $(F, G, \varphi')$ are *cohomology conjugate* in the sense that there exist $G$-equivariant homeomorphisms $\theta_j{\colon}F^j\to E^j$ for $j=0,1$ such that $r\circ \theta_1= \theta_0\circ r'$, $s\circ \theta_1= \theta_0\circ s'$, and the map $(g,f) \to \varphi(g, \theta_1(f))$ is a cocycle for $G{\curvearrowright}F$ that is cohomologous to $\varphi'$, then ${Y^\varphi}$ is isomorphic to $Y^{\varphi'}$, and it follows that ${\mathcal T}_{{Y^\varphi}}$ $resp.\ ${\mathcal O}_{{Y^\varphi}}$$ is isomorphic to ${\mathcal T}_{Y^{\varphi'}}$ $resp.\ ${\mathcal O}_{Y^{\varphi'}}$$.
In the following proposition we consider the questions of whether the cocycle $(g,e)\mapsto g$ can be a coboundary for a graph action.
\[g cby\] Suppose the action $G\curvearrowright E$ fixes sources. Then the following are equivalent:
1. The cocycle $(g,e)\mapsto g$ is a coboundary.
2. There is a continuous map $\psi{\colon}E^1\to G$ such that for all $g\in G$ and $e\in E^1$ we have $$\begin{aligned}
\psi(ge)&=g\psi(e).\end{aligned}$$
3. $E^1$ is $G$-equivariantly homeomorphic to $G\times \Omega$ for some space $\Omega$, where $G$ acts by left translation in the first factor.
Since the action fixes sources, we have $gs(e)=s(e)$ for all $g\in G$ and $e\in E^1$, and consequently it is easy to see that $(g,e)\mapsto g$ is a coboundary for the action of $G$ on the topological graph $E$ if and only if it is a coboundary for the action of $G$ on the space $E^1$, so the result follows immediately from [Proposition ]{}.
Suppose that the action $G\curvearrowright E$ fixes sources and that the map $(g,e)\mapsto g$ is a coboundary. In view of Proposition \[g cby\], we may assume that $E^1=G\times\Omega$ and $g(h,x)=(gh,x)$ for all $g,h\in G$ and $x\in\Omega$. It is interesting to examine the range and source maps of $E$. Define continuous maps $\sigma,\rho{\colon}\Omega\to E^0$ by $$\begin{aligned}
\sigma(x)&=s(1,x)
\\
\rho(x)&=r(1,x).\end{aligned}$$ Then for all $(g,x)\in G\times \Omega$ we have $$s(g,x)
=gs(1,x)
=s(1,x)
=\sigma(x).$$ On the other hand, for the range map we have $$r(g,x)=gr(1,x)=g\rho(x).$$ It is tempting to conjecture that much more can be said about this situation. As a kind of converse, let $\rho, \sigma{\colon}\Omega\to E^0$ be continuous maps between some locally compact Hausdorff spaces $\Omega$ and $E^0$. Assume that $\sigma$ is a local homeomorphism and that $G$ is a discrete group acting by homeomorphisms on $E^0$ in such a way that $$g\sigma(x) = \sigma(x)$$ for all $g\in G, x \in \Omega$. Set $E^1:=G\times \Omega$ and define $r, s {\colon}E^1\to E^0$ by $$r(h,x) = h \rho(x), \quad s(h,x) = \sigma(x)$$ for all $(h,x) \in E^1$. Then one checks readily that $E=(E^1, E^0, r, s)$ is a topological graph. Moreover, letting $G$ act on $E^1$ by $g(h,x) = (gh,x)$ for all $g\in G$ and $(h,x)\in E^1$, we obtain an action of $G$ on $E$ that is easily seen to satisfy $gs(e) = s(e) $ for all $e\in E^1$.
If the action of $G$ on $E$ fixes sources, what can be said about the correspondence ${Y^\varphi}$ for the cocycle $\varphi(g,e)=1$? In the case where $E$ is finite with no sources, Exel and Pardo [@EP Example 3.6] show that ${\mathcal O}_{G,E}\simeq C^*(E)$. But we would like to understand this (admittedly rather trivial) situation better. We discuss a special case toward the end of [Example ]{}.
Generators and relations {#gen-rel}
========================
Throughout this section, $E=(E^0,E^1,r,s)$ will be a directed graph, $G$ will be a discrete group acting on $E$, and $\varphi{\colon}G\times E^1\to G$ will be a cocycle for this action. We will describe the Toeplitz algebra ${\mathcal T}_{{Y^\varphi}}$ in terms of generators and relations, and give a similar description of the Cuntz-Pimsner algebra ${\mathcal O}_{{Y^\varphi}}$ when $E$ is assumed to be row-finite. In the case where $E$ is finite and sourceless, we thereby recover Exel and Pardo’s initial definition of ${\mathcal O}_{G,E}$ in [@EP Section 3].
We use the notation introduced in Section 3 and refer the reader to [@Kat04] for undefined terminology and notation on $C^*$-correspondences. Let $(t_{{Y^\varphi}},t_B)$ denote the universal Toeplitz representation of $({Y^\varphi},B)$ in ${{\mathcal T}_{{Y^\varphi}}}$, $(k_{{Y^\varphi}},k_B)$ the universal Cuntz-Pimsner covariant representation of $({Y^\varphi},B)$ in ${{\mathcal O}_{{Y^\varphi}}}$, and $(i_A,i_G)$ the universal covariant homomorphism of $(A,G)$ in $M(B)$. (Of course, since $G$ is discrete we have $i_A{\colon}A\to B$.)
We work with the crossed product $B=A\rtimes_\alpha G$ and the $B$-correspondence ${Y^\varphi}$ in terms of the generators:
- ${\raisebox{2pt}{\ensuremath{\chi}}}_{e,g}$ denotes the element of $C_c(E^1\times G)\subset {Y^\varphi}$ given by the characteristic function of $\{(e,g)\}$.
- $\delta_{v,g}$ denotes the element of $C_c(E^0\times G)\subset B$ given by the characteristic function of $\{(v,g)\}$.
- Similarly for ${\raisebox{2pt}{\ensuremath{\chi}}}_e\in C_c(E^1)\subset X$ and $\delta_v\in C_c(E^0)\subset A$.
Thus
- $C_c(E^1\times G)=\operatorname*{span}\{{\raisebox{2pt}{\ensuremath{\chi}}}_{e,g}:e\in E^1,g\in G\}$.
- $C_c(E^0\times G)=\operatorname*{span}\{\delta_{v,g}:v\in E^0,g\in G\}$.
- $A$ is the $c_0$-direct sum of the 1-dimensional ideals generated by the projections $\delta_v$ for $v\in E^0$.
Let $D$ be a $C^*$-algebra. A *representation of $(E,G,\varphi)$ in $D$* is a family $\{P_v,S_e,U_g:v\in E^0,e\in E^1,g\in G\}$ such that:
1. \[ep 1\] $\{P_v,S_e:v\in E^0,e\in E^1\}$ is a Toeplitz $E$-family in $D$,
2. \[ep 2\] the map $U:g \mapsto U_g$ is a unitary representation of $G$ in $M(D)$,
3. \[ep 3\] for all $g\in G$, $v\in E^0$, and $e\in E^1$ we have $$U_gP_v=P_{gv}U_g{\quad\text{and}\quad}U_gS_e=S_{ge}U_{\varphi(g,e)},$$ and
4. \[ep 4\] $D$ is generated as a $C^*$-algebra by $$\{P_vU_g:v\in E^0,g\in G\}\cup \{S_eU_g:e\in E^1,g\in G\}.$$
We frequently shorten the notation for the family to $\{P_v,S_e,U_g\}$. If the above condition is replaced by
1. \[epprime 1\] $\{P_v,S_e:v\in E^0,e\in E^1\}$ is a Cuntz-Krieger $E$-family in $D$,
then we say $\{P_v,S_e,U_g\}$ is a *CK-representation of $(E,G,\varphi)$ in $D$*.
Since the left $B$-module ${Y^\varphi}$ is nondegenerate, the canonical homomorphisms $t_B{\colon}B\to {{\mathcal T}_{{Y^\varphi}}}$ and $k_B{\colon}B\to {{\mathcal O}_{{Y^\varphi}}}$ are nondegenerate; we denote by ${\overline}{t_B}$ and ${\overline}{k_B}$ their extensions to the multiplier algebra $M(B)$. We define a representation $\{p_v,s_e,u_g\}$ of $(E,G,\varphi)$ in ${{\mathcal T}_{{Y^\varphi}}}$ by $$\begin{aligned}
p_v&=t_B(i_A(\delta_v))
\\
s_e&=t_{{Y^\varphi}}({\raisebox{2pt}{\ensuremath{\chi}}}_{e,1})
\\
u_g&={\overline}{t_B}(i_G(g)),\end{aligned}$$ and a CK-representation $\{p'_v,s'_e,u'_g\}$ in ${{\mathcal O}_{{Y^\varphi}}}$ by $$\begin{aligned}
p'_v&=k_B(i_A(\delta_v))
\\
s'_e&=k_{{Y^\varphi}}({\raisebox{2pt}{\ensuremath{\chi}}}_{e,1})
\\
u'_g&={\overline}{k_B}(i_G(g)).\end{aligned}$$
Note that $$\begin{aligned}
i_A(\delta_v)&=\delta_{v,1}
\\
i_G(g)&=\sum_{v\in E^0}\delta_{v,g},\end{aligned}$$ where $1$ denotes the identity element of $G$ and the sum converges in the strict topology of $M(B)$. Also, the technology of discrete crossed products is set up so that $$i_A(\delta_v)i_G(g)=\delta_{v,g},$$ and it follows that $$\delta_{v',g}\delta_{v,h}
=\begin{cases}
\delta_{gv,gh}{& \text{if }}v'=gv\\
0{& \text{otherwise}}.
\end{cases}$$
We have $$\begin{aligned}
i_A(\delta_v)\cdot {\raisebox{2pt}{\ensuremath{\chi}}}_{e,h}&=
\begin{cases}
{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h}{& \text{if }}v=r(e)\\
0{& \text{otherwise}}\end{cases}
\\
{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h}\cdot i_A(\delta_v)&=
\begin{cases}
{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h}{& \text{if }}s(e)=hv\\
0{& \text{otherwise}}\end{cases}
\\
i_G(g)\cdot {\raisebox{2pt}{\ensuremath{\chi}}}_{e,h}&={\raisebox{2pt}{\ensuremath{\chi}}}_{ge,\varphi(g,e)h}
\\
{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h}\cdot i_G(g)&={\raisebox{2pt}{\ensuremath{\chi}}}_{e,hg}.\end{aligned}$$
Consequently $$\begin{aligned}
\delta_{v,g}\cdot {\raisebox{2pt}{\ensuremath{\chi}}}_{e,h}&=
\begin{cases}
{\raisebox{2pt}{\ensuremath{\chi}}}_{ge,\varphi(g,e)h}{& \text{if }}v=r(ge)\\
0{& \text{otherwise}}\end{cases}
\\
{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h}\cdot \delta_{v,g}&=
\begin{cases}
{\raisebox{2pt}{\ensuremath{\chi}}}_{e,hg}{& \text{if }}s(e)=hv\\
0{& \text{otherwise}}.
\end{cases}\end{aligned}$$
The inner product on basis elements satisfies $${\langle}{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1},{\raisebox{2pt}{\ensuremath{\chi}}}_{e',1}{\rangle}=
\begin{cases}
\delta_{s(e),1}{& \text{if }}e=e'\\
0{& \text{otherwise}},
\end{cases}$$ and so $$\begin{aligned}
{\langle}{\raisebox{2pt}{\ensuremath{\chi}}}_{e,g},{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h}{\rangle}&=\bigl{\langle}{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1}\cdot i_G(g),{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1}\cdot i_G(h)\bigr{\rangle}\\&=i_G(g{^{-1}}){\langle}{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1},{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1}{\rangle}i_G(h)
\\&=i_G(g{^{-1}})\delta_{s(e),1}i_G(h)
\\&=\delta_{s(g{^{-1}}e),g{^{-1}}h},\end{aligned}$$ while ${\langle}{\raisebox{2pt}{\ensuremath{\chi}}}_{e,g},{\raisebox{2pt}{\ensuremath{\chi}}}_{e',h}{\rangle}=0$ if $e\ne e'$.
Also, $$\sum_{v\in E^0}\delta_v=1$$ in $M(A)$, where the series converges strictly, and similarly $$\sum_{v\in E^0}\delta_{v,1}=1$$ in $M(B)$. Consequently (as has been mentioned in the literature, probably many times), $$\sum_{v\in E^0}p_v=1$$ strictly in $M({\mathcal T}_{{Y^\varphi}})$, and similarly for the projections $p_v'$ in $M({\mathcal O}_{{Y^\varphi}})$.
The following shows that ${{\mathcal T}_{{Y^\varphi}}}$ is the universal $C^*$-algebra for representations of $(E,G,\varphi)$, which gives a presentation of ${{\mathcal T}_{{Y^\varphi}}}$ in terms of generators and relations.
\[universal T\] Let $\{P_v,S_e,U_g\}$ be a representation of $(E,G,\varphi)$ in a $C^*$-algebra $D$. Then there is a unique surjective homomorphism $\Phi$ from ${{\mathcal T}_{{Y^\varphi}}}$ onto $D$ such that $$\Phi(p_v)=P_v,\quad
\Phi(s_e)=S_e, {\quad\text{and}\quad}
\overline{\Phi}(u_g)=U_g$$ for all $v\in E^0$, $e\in E^1$, and $g\in G$.
We will construct a Toeplitz representation $(\psi,\zeta)$ of the correspondence $({Y^\varphi},B)$ in $D$, and we work primarily with the generators. First of all, the family $\{P_v\}$ of orthogonal projections uniquely determines a homomorphism ${\widetilde}P{\colon}A\to D$. Then the relation immediately implies that the pair $({\widetilde}P,U)$ is a covariant homomorphism of the system $(A,G,\alpha)$ in $D$, and the integrated form is a homomorphism $\zeta{\colon}B\to D$, given on generators by $$\zeta(\delta_{v,g})=P_vU_g{\quad\text{for }}v\in E^0,g\in G.$$ Next, we define a linear map $\psi{\colon}C_c(E^1\times G)\to D$ as the unique linear extension of the map given on generators by $$\psi({\raisebox{2pt}{\ensuremath{\chi}}}_{e,g})=S_eU_g{\quad\text{for }}e\in E^1,g\in G.$$ The computation $$\begin{aligned}
\psi({\raisebox{2pt}{\ensuremath{\chi}}}_{e,g})^*\psi({\raisebox{2pt}{\ensuremath{\chi}}}_{e',h})
&=(S_eU_g)^*(S_{e'}U_h)
\\&=U_g^*S_e^*S_{e'}U_h,
\intertext{which is 0 unless $e=e'$, in which case we can continue as}
&=U_{g{^{-1}}}P_{s(e)}U_h
\\&=P_{gs(e)}U_{g{^{-1}}h}
\\&=\delta_{gs(e),g{^{-1}}h}
\\&=\zeta\bigl({\langle}{\raisebox{2pt}{\ensuremath{\chi}}}_{e,g},{\raisebox{2pt}{\ensuremath{\chi}}}_{e',h}{\rangle}\bigr)
{\quad\text{(which is also 0 if $e\ne e'$) }}\end{aligned}$$ implies that $\psi$ is bounded, and hence extends uniquely to a bounded linear map $\psi{\colon}{Y^\varphi}\to D$. Then combining the above with the computation $$\begin{aligned}
\zeta(\delta_{v,g})\psi({\raisebox{2pt}{\ensuremath{\chi}}}_{e,h})
&=P_vU_gS_eU_h
\\&=P_vS_{ge}U_{\varphi(g,e)h},
\intertext{which is 0 unless $v=r(ge)$, in which case we can continue as}
&=S_{ge}U_{\varphi(g,e)h}
\\&=\psi({\raisebox{2pt}{\ensuremath{\chi}}}_{ge,\varphi(g,e)h})
\\&=\psi\bigl(\delta_{v,g}\cdot {\raisebox{2pt}{\ensuremath{\chi}}}_{e,h}\bigr)
{\quad\text{(which is also 0 if $v\ne r(ge)$) }}\end{aligned}$$ shows that $(\psi,\zeta)$ is a Toeplitz representation of $(Y^\varphi,B)$ in $D$.
The associated homomorphism $\Phi=\psi\times_{\mathcal T}\zeta$ from ${\mathcal T}_{{Y^\varphi}}$ to $D$ is surjective, by the properties of representations. Moreover, by construction this homomorphism is the unique one satisfying $$\Phi(p_v)=P_v,\quad
\Phi(s_e)=S_e,{\quad\text{and}\quad}
\overline{\Phi}(u_g)=U_g$$ for all $v\in E^0$, $e\in E^1$, and $g\in G$.
In the following lemma we will gather some information about the Katsura ideal of the $B$-correspondence ${Y^\varphi}$, under a mild assumption on $E$.
\[JY\] Assume that $E$ is row-finite, i.e., $|r{^{-1}}(v)| < \infty $ for all $v\in E^0$. Then the Katsura ideal $J_X$ for the $A$-correspondence $X$ is $G$-invariant, the image of the left-module map $\phi{\colon}B\to{\mathcal L}({Y^\varphi})$ is contained in ${\mathcal K}({Y^\varphi})$, and $$\begin{aligned}
J_{{Y^\varphi}}
&\subset J_X\rtimes_\alpha G
=\operatorname*{\overline{\operatorname*{span}}}\{\delta_{v,g}:(v,g)\in E^0\times G,0<|r{^{-1}}(v)|\}.\end{aligned}$$
Let $$\begin{aligned}
E^0_{{\textup{rg}}}&=r(E^1)
\\
E^0_{{\textup{so}}}&=E^0{\setminus}E^0_{{\textup{rg}}}.\end{aligned}$$ Thus $E^0_{{\textup{so}}}$ is the set of sources and $E^0_{{\textup{rg}}}$ is the set of regular vertices. Also, $E^0$ is the disjoint union of these two $G$-invariant subsets. Put $J_0=c_0(E^0_{{\textup{so}}})$. It is well-known that $$\begin{aligned}
J_X&=c_0(E^0_{{\textup{rg}}})
\\
J_0&=\ker\pi.\end{aligned}$$ Moreover, we have a direct-sum decomposition $A=J_X\oplus J_0$ into complementary $G$-invariant ideals. The crossed product is thus a direct sum $$B=(J_X\rtimes_\alpha G)\oplus (J_0\rtimes_\alpha G)$$ of complementary ideals. Since the left-module map $\phi{\colon}B\to{\mathcal L}({Y^\varphi})$ coincides with $\pi \times V$, we get $$J_0\rtimes_\alpha G\subset \ker\phi.$$ Thus $$(\ker\phi){^\perp}\subset (J_0\rtimes_\alpha G){^\perp}=J_X\rtimes_\alpha G.$$ As $J_{{Y^\varphi}}= \phi^{-1}({\mathcal K}({Y^\varphi})) \cap (\ker\phi){^\perp}$, we can finish by showing that $\phi(B)\subset {\mathcal K}({Y^\varphi})$, and by the above it suffices to show that if $v\in E^0_{{\textup{rg}}}$ and $g\in G$ then $$\label{finite rank}
\phi(\delta_{v,g})
=\sum_{r(e)=v}\theta_{{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1},{\raisebox{2pt}{\ensuremath{\chi}}}_{g{^{-1}}e,\varphi(g{^{-1}},e)}},$$ which is finite rank. Since $C_c(E^1\times G)$ is dense in ${Y^\varphi}$, it suffices to check the equality of the above two operators on a basis vector ${\raisebox{2pt}{\ensuremath{\chi}}}_{e',h}$. Recall that $\delta_{v,g}\cdot {\raisebox{2pt}{\ensuremath{\chi}}}_{e',h}={\raisebox{2pt}{\ensuremath{\chi}}}_{ge,\varphi(g,e')h}$ if $v=r(ge')$ and 0 otherwise.
We first show that $$\label{v,1}
\phi(\delta_{v,1})=\sum_{r(e)=v}\theta_{{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1},{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1}}.$$ For any $e\in r{^{-1}}(v)$ we have $$\theta_{{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1},{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1}}{\raisebox{2pt}{\ensuremath{\chi}}}_{e',h}
={\raisebox{2pt}{\ensuremath{\chi}}}_{e,1}\cdot {\langle}{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1},{\raisebox{2pt}{\ensuremath{\chi}}}_{e',h}{\rangle},$$ which is 0 if $e\ne e'$. Thus if $r(e')\ne v$ we have $$\sum_{r(e)=v}\theta_{{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1},{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1}}{\raisebox{2pt}{\ensuremath{\chi}}}_{e',h}
=0=\phi(\delta_{v,1}){\raisebox{2pt}{\ensuremath{\chi}}}_{e',h}.$$ So now suppose that $r(e')=v$. Then $$\begin{aligned}
\sum_{r(e)=v}\theta_{{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1},{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1}}{\raisebox{2pt}{\ensuremath{\chi}}}_{e',h}
&=\theta_{{\raisebox{2pt}{\ensuremath{\chi}}}_{e',1},{\raisebox{2pt}{\ensuremath{\chi}}}_{e',1}}{\raisebox{2pt}{\ensuremath{\chi}}}_{e',h}
\\&={\raisebox{2pt}{\ensuremath{\chi}}}_{e',1}\cdot {\langle}{\raisebox{2pt}{\ensuremath{\chi}}}_{e',1},{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h}{\rangle}\\&={\raisebox{2pt}{\ensuremath{\chi}}}_{e',1}\cdot \delta_{s(e'),h}
\\&={\raisebox{2pt}{\ensuremath{\chi}}}_{e',h}=\phi(\delta_{v,1}){\raisebox{2pt}{\ensuremath{\chi}}}_{e',h},\end{aligned}$$ verifying .
Now we can prove : $$\begin{aligned}
\phi(\delta_{v,g})
&=\phi(\delta_{v,1}i_G(g))
=\phi(\delta_{v,1})V_g
\\&=\sum_{r(e)=v}\theta_{{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1},{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1}}V_g
\\&=\sum_{r(e)=v}\theta_{{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1},V_g^*{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1}}
\\&=\sum_{r(e)=v}\theta_{{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1},V_{g{^{-1}}}{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1}}
\\&=\sum_{r(e)=v}\theta_{{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1},{\raisebox{2pt}{\ensuremath{\chi}}}_{g{^{-1}}e,\varphi(g{^{-1}},e)}}.
\qedhere\end{aligned}$$
We can now deduce that, keeping the row-finiteness assumption on the graph $E$, the Cuntz-Pimsner algebra ${{\mathcal O}_{{Y^\varphi}}}$ is the universal $C^*$-algebra for CK-representations of $(E,G,\varphi)$, which gives a presentation of ${{\mathcal O}_{{Y^\varphi}}}$ in terms of generators and relations.
\[universal O\] Let $\{P_v,S_e,U_g\}$ be a CK-representation of $(E,G,\varphi)$ in a $C^*$-algebra $D$. If $E$ is row-finite, then there is a unique surjective homomorphism $\Phi{\colon}{{\mathcal O}_{{Y^\varphi}}}\to D$ such that $$\Phi(p'_v)=P_v,\quad
\Phi(s'_e)=S_e, {\quad\text{and}\quad}
\overline{\Phi}(u'_g)=U_g$$ for all $v\in E^0$, $e\in E^1$, and $g\in G$.
With the notation from the proof of [Theorem ]{}, we must show that the Toeplitz representation $(\psi,\zeta)$ is Cuntz-Pimsner covariant. That is, we must show that for all $b$ in the Katsura ideal $J_{{Y^\varphi}}$ of ${Y^\varphi}$ we have $\psi^{(1)}\circ \phi(b)=\zeta(b)$, where $\phi{\colon}B\to {\mathcal L}({Y^\varphi})$ is the left module homomorphism and $\psi^{(1)}{\colon}{\mathcal K}({Y^\varphi})\to D$ is the homomorphism associated to the Toeplitz representation $(\psi,\zeta)$. By [Lemma ]{}, and by linearity, density, and continuity, it suffices to compute that for all $(v,g)\in E^0_{{\textup{rg}}}\times G$ $$\begin{aligned}
\psi^{(1)}\circ \phi(\delta_{v,g})
&=\psi^{(1)}\left(\sum_{r(e)=v}\theta_{{\raisebox{2pt}{\ensuremath{\chi}}}_{e,1},{\raisebox{2pt}{\ensuremath{\chi}}}_{g{^{-1}}e,\varphi(g{^{-1}},e)}}\right)
\\&=\sum_{r(e)=v}\psi({\raisebox{2pt}{\ensuremath{\chi}}}_{e,1})\psi({\raisebox{2pt}{\ensuremath{\chi}}}_{g{^{-1}}e,\varphi(g{^{-1}},e)})^*
\\&=\sum_{r(e)=v}S_e\bigl(S_{g{^{-1}}e}U_{\varphi(g{^{-1}},e)}\bigr)^*
\\&=\sum_{r(e)=v}S_e\bigl(U_{g{^{-1}}}S_e\bigr)^*
\\&=\sum_{r(e)=v}S_eS_e^*U_g
\\&=P_vU_g
\\&=\zeta(\delta_{v,g}).
\qedhere\end{aligned}$$
If $E$ is finite, then ${\mathcal O}_{{Y^\varphi}}$ is unital, and [Corollary ]{} shows that it has exactly the same universal properties as the Exel-Pardo algebra ${\mathcal O}_{G,E}$ (cf. [@EP Definition 3.2]). Hence, if $E$ is finite with no sources, then ${{\mathcal O}_{{Y^\varphi}}}$ is isomorphic to ${\mathcal O}_{G,E}$. We will give another proof of this fact in [Corollary ]{}.
In the proof of [Lemma ]{} we showed that when $E$ is row-finite we have $\phi(B)\subset {\mathcal K}({Y^\varphi})$, where $\phi{\colon}B\to {\mathcal L}({Y^\varphi})$ is the left-module homomorphism. In fact, assuming a bit more about $E$, we can identify the Katsura ideal:
\[ideal\] If $E$ is row-finite and has no sources, then the Katsura ideal $J_{{Y^\varphi}}$ of the $B$-correspondence ${Y^\varphi}$ coincides with $B$.
By the preceding, we only need to show that the left-module homomorphism $\phi{\colon}B\to {\mathcal K}({Y^\varphi})$ is injective. Our new hypotheses imply that, in the notation of the proof of [Lemma ]{}, $B=J_X\rtimes_\alpha G$. Recall the CK-representation $$\begin{aligned}
p_v'&=k_B(i_A(\delta_v)),
&
s_e'&=k_{{Y^\varphi}}({\raisebox{2pt}{\ensuremath{\chi}}}_{e,1}),
&
u_g'&={\overline}{k_B}(i_G(g))\end{aligned}$$ of $(E,G,\varphi)$ in ${{\mathcal O}_{{Y^\varphi}}}$, and let $(\psi,\zeta)$ be the associated Toeplitz representation of the $B$-correspondence ${Y^\varphi}$ in ${\mathcal O}_{{Y^\varphi}}$, so that in particular $$\begin{aligned}
\zeta=\pi_A\times u',\end{aligned}$$ where $\pi_A{\colon}A\to {\mathcal O}_{{Y^\varphi}}$ is determined by $$\pi_A(a)=\sum_{v\in E^0}a(v)p_v'{\quad\text{for }}a\in C_c(E^0).$$ Then clearly $\zeta=k_B$, the canonical homomorphism from $B$ to ${{\mathcal O}_{{Y^\varphi}}}$. Thus $\zeta$ is injective by [@Kat04 Proposition 4.11]. Since we have shown above that $\psi^{(1)}\circ \phi=\zeta$, it follows that $\phi$ is injective.
\[varphi trivial\] We imposed the row-finite hypothesis on the graph in [Corollary ]{} because otherwise it would be problematic to get our hands on the Katsura ideal $J_{{Y^\varphi}}$ of the correspondence ${Y^\varphi}$. Even when $\varphi$ is the cocycle $(g,e)\mapsto g$, so that ${Y^\varphi}=X\rtimes_\gamma G$, the relationship between the two ideals $J_{X\rtimes_\gamma G}$ and $J_X\rtimes_\alpha G$ of $B=A\rtimes_\alpha G$ is murky. There are partial results: the two ideals coincide when $G$ is amenable [@HaoNg Proposition 2.7], or is discrete and has Exel’s Approximation Property [@bkqr Theorem 5.5], but it is unknown whether the two ideals coincide for arbitrary $G$. However, when $G$ is discrete, $E$ is row-finite with no sources, and $\varphi$ is the cocycle $(g,e)\mapsto g$, Corollary \[ideal\] gives that $J_{X\rtimes G} = B = A\rtimes G = J_X\rtimes G$, and we can then conclude from [@bkqr Theorem 4.1] that ${\mathcal O}_{X\rtimes G}\simeq {\mathcal O}_X \rtimes G$, i.e., ${\mathcal O}_{{Y^\varphi}} \simeq C^*(E)\rtimes G$. In the case where $E$ is finite and sourceless, this was pointed out by Exel and Pardo in [@EP Example 3.5].
The Exel-Pardo correspondence
=============================
When $G$ is discrete and the graph $E$ is finite, Exel and Pardo [@EP Section 10] define a correspondence, that they denote by $M$, over the crossed product $B=A\rtimes_\alpha G$. (Warning: they call this crossed product $A$, whereas we write $A$ for $C_0(E^0)$.) Exel and Pardo also require $E$ to have no sources, but they remark in [@EP Section 2] that this assumption, as well as finiteness of $E$, are probably only necessary in Section 3 of their paper, which it so happens does not concern us in our paper.
Throughout this section we assume that $G$ is a discrete group acting on a directed graph $E$, and that $\varphi$ is a cocycle for this action.
Our construction of the $B$-correspondence ${Y^\varphi}$ in [Section ]{} is different from that of Exel and Pardo [@EP Section 10], so it behooves us to compare them.
\[M\] Let $M$ be the $B$-correspondence constructed in [@EP Section 10]. Then ${Y^\varphi}\simeq M$ as $B$-correspondences.
We review the construction of $M$, but using slightly different notation and adapting it to our more general context. It should be clear that we produce the same structure as in [@EP]. For $v\in E^0$ let $\delta_v\in C_c(E^0)\subset A$ be the characteristic function of $\{v\}$. For each $E\in E^1$ let $$B^e=i_A(\delta_{s(e)})B,$$ which is a closed right ideal of $B$, and hence a Hilbert $B$-module in the obvious way. Then form a new Hilbert $B$-module as the direct sum $$M=\bigoplus_{e\in E^1}B^e.$$ An element $m\in M$ is an $E^1$-tuple $$m=(m_e)_{e\in E^1},$$ and the coordinates have the form $$m_e=i_A(\delta_{s(e)})b_e,{\quad\text{with }}b_e\in B.$$ The left $B$-module structure on $M$ is the integrated form of a covariant pair of left module multiplications of $A$ and $G$, defined on the generators by $$\begin{aligned}
(\delta_v\cdot m)_e&=\begin{cases}m_e{& \text{if }}v=r(e)\\0{& \text{otherwise}}\end{cases}
\\
(g\cdot m)_e&=i_A(\delta_{s(e)})i_G(\varphi(g,g{^{-1}}e))m_{g{^{-1}}e}.\end{aligned}$$
We will define an isomorphism $\Psi{\colon}{Y^\varphi}\to M$ of $B$-correspondences. We begin by defining $\Psi$ on the dense subspace $C_c(E^1\times G)$, and by linear independence it suffices to define $$\bigl(\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,g}\bigr)_{e'}=\begin{cases}
i_A(\delta_{s(e)})i_G(g){& \text{if }}e'=e\\
0{& \text{otherwise}}.
\end{cases}$$ The following computation implies that $\Psi$ preserves inner products on $C_c(E^1\times G)$: for $e,f\in E^1$ and $g,h\in G$ we have $$\begin{aligned}
{\langle}\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,g},\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{f,h}{\rangle}&=\sum_{e'\in E^1}(\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,g})_{e'}^*(\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{f,h})_{e'},\end{aligned}$$ which is 0 unless $e=f=e'$, and when $e=f$ we have $$\begin{aligned}
{\langle}\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,g},\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h}{\rangle}&=(\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,g})_e^*(\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h})_e
\\&=\bigl(i_A(\delta_{s(e)})i_G(g)\bigr)^*\bigl(i_A(\delta_{s(e)})i_G(h)\bigr)
\\&=i_G(g{^{-1}})i_A(\delta_{s(e)})i_G(h)
\\&=i_A(\delta_{g{^{-1}}s(e)}i_G(g{^{-1}}h)
\\&=\delta_{g{^{-1}}s(e),g{^{-1}}h}
\\&={\langle}{\raisebox{2pt}{\ensuremath{\chi}}}_{e,g},{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h}{\rangle}.\end{aligned}$$ Thus $\Psi$ extends uniquely to an isometric linear map from ${Y^\varphi}$ to $M$, which we continue to denote by $\Psi$.
As pointed out in [@EP Section 10], $$B^e=\operatorname*{\overline{\operatorname*{span}}}\{i_A(\delta_{s(e)})i_G(g):g\in G\},$$ and it follows that $\Psi$ has dense range, and hence is surjective.
The following computations imply that $\Psi$ is right $B$-linear: $$\begin{aligned}
\bigl(\Psi({\raisebox{2pt}{\ensuremath{\chi}}}_{e,h}\cdot \delta_v)\bigr)_{e'}
&=\begin{cases}
(\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h})_{e'}{& \text{if }}s(e)=v\\
0{& \text{otherwise}}\end{cases}
\\&=\begin{cases}
i_A(\delta_{s(e)})i_G(h){& \text{if }}s(e)=v,e'=e\\
0{& \text{otherwise}},
\end{cases}\end{aligned}$$ while $$\begin{aligned}
\bigl((\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h})\cdot \delta_v\bigr)_{e'}
&=\begin{cases}
(\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h})_{e'}{& \text{if }}s(e')=v\\
0{& \text{otherwise}}\end{cases}
\\&=\begin{cases}
i_A(\delta_{s(e)})i_G(h){& \text{if }}s(e')=v,e'=e\\
0{& \text{otherwise}},
\end{cases}\end{aligned}$$ so $\Psi({\raisebox{2pt}{\ensuremath{\chi}}}_{e,h}\cdot \delta_v)=(\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h})\cdot \delta_v$, and $$\begin{aligned}
\bigl(\Psi({\raisebox{2pt}{\ensuremath{\chi}}}_{e,h}\cdot g)\bigr)_{e'}
&=\bigl(\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,hg}\bigr)_{e'}
\\&=\begin{cases}i_A(\delta_{s(e)})i_G(hg){& \text{if }}e'=e\\0{& \text{otherwise}},\end{cases}\end{aligned}$$ while $$\begin{aligned}
\bigl((\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h})\cdot g\bigr)_{e'}
&=(\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h})_{e'}\cdot g
\\&=\begin{cases}i_A(\delta_{s(e)})i_G(h)\cdot g{& \text{if }}e'=e\\0{& \text{otherwise}}\end{cases}
\\&=\begin{cases}i_A(\delta_{s(e)})i_G(hg){& \text{if }}e'=e\\0{& \text{otherwise}}\end{cases}\end{aligned}$$ so $\Psi({\raisebox{2pt}{\ensuremath{\chi}}}_{e,h}\cdot g)=(\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h})\cdot g$. This combined with the other properties of $\Psi$ makes it a unitary map from the Hilbert $B$-module ${Y^\varphi}$ to the Hilbert $B$-module $M$ [@lance Theorem 3.5].
The following computations imply that $\Psi$ is left $B$-linear: $$\begin{aligned}
\bigl(\Psi(\delta_v\cdot {\raisebox{2pt}{\ensuremath{\chi}}}_{e,h})\bigr)_{e'}
&=\begin{cases}
\bigl(\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h}\bigr)_{e'}{& \text{if }}v=r(e)\\
0{& \text{otherwise}}\end{cases}
\\&=\begin{cases}
i_A(\delta_{s(e)})i_G(h){& \text{if }}v=r(e),e'=e\\
0{& \text{otherwise}},
\end{cases}\end{aligned}$$ while $$\begin{aligned}
\bigl(\delta_v\cdot (\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h})\bigr)_{e'}
&=\begin{cases}
\bigl(\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h}\bigr)_{e'}{& \text{if }}v=r(e')\\
0{& \text{otherwise}}\end{cases}
\\&=\begin{cases}
i_A(\delta_{s(e)})i_G(h){& \text{if }}v=r(e'),e'=e\\
0{& \text{otherwise}},
\end{cases}\end{aligned}$$ so $\Psi(\delta_v\cdot {\raisebox{2pt}{\ensuremath{\chi}}}_{e,h})=\delta_v\cdot (\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h})$, and $$\begin{aligned}
\bigl(\Psi(g\cdot {\raisebox{2pt}{\ensuremath{\chi}}}_{e,h})\bigr)_{e'}
&=\bigl(\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{ge,\varphi(g,e)h}\bigr)_{e'}
\\&=\begin{cases}
i_A(\delta_{s(ge)})i_G(\varphi(g,e)h){& \text{if }}e'=ge\\
0{& \text{otherwise}}\end{cases}\end{aligned}$$ while $$\begin{aligned}
&\bigl(g\cdot (\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h})\bigr)_{e'}
\\&\quad=\begin{cases}
i_A(\delta_{s(e')})i_G(\varphi(g,g{^{-1}}e'))
i_A(\delta_{s(e)})i_G(h){& \text{if }}g{^{-1}}e'=e\\
0{& \text{otherwise}},
\end{cases}
\intertext{since $(\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h})_{g{^{-1}}e'}
=i_A(\delta_{s(e)})i_G(h)$ if $g{^{-1}}e'=e$ and 0 if not,}
&=\begin{cases}
i_A(\delta_{s(ge)})i_A(\delta_{\varphi(g,e)s(e)})i_G(\varphi(g,e)h){& \text{if }}e'=ge\\
0{& \text{otherwise}}\end{cases}
\\&=\begin{cases}
i_A(\delta_{s(ge)})i_G(\varphi(g,e)h){& \text{if }}e'=ge\\
0{& \text{otherwise}},
\end{cases}\end{aligned}$$ since $\varphi(g,e)s(e)=gs(e)=s(ge)$, so $\Psi(g\cdot {\raisebox{2pt}{\ensuremath{\chi}}}_{e,h})=g\cdot (\Psi{\raisebox{2pt}{\ensuremath{\chi}}}_{e,h})$. Therefore $\Psi$ is an isomorphism of $B$-correspondences.
\[alt EP alg\] Assume that $E$ is finite with no sources. Then the Cuntz-Pimsner algebra ${{\mathcal O}_{{Y^\varphi}}}$ is isomorphic to the Exel-Pardo algebra ${\mathcal O}_{G,E}$.
It follows immediately from [Theorem ]{} that the Cuntz-Pimsner algebras ${{\mathcal O}_{{Y^\varphi}}}$ and ${\mathcal O}_M$ are isomorphic. As ${\mathcal O}_M$ is isomorphic to ${\mathcal O}_{G,E}$ (cf. [@EP Theorem 10.15]), the result follows.
Examples
========
{#71}
Assume that a discrete group $G$ acts on a nonempty set $S$ and that $\varphi$ is a $G$-valued cocycle for $G{\curvearrowright}S$, so we have $$\label{S-cocy-eq}
\varphi(gh, x) = \varphi(g, h\cdot x) \varphi(h, x) {\quad\text{for all }} g, h \in G,x \in S.$$ As in [@EP Example 3.3], we may regard $G$ as acting on the graph $E_S$ that has one single vertex and $S$ as its edge set (so $E_S$ is bouquet of loops). The cocycle $\varphi$ for $G{\curvearrowright}S$ is then automatically a cocycle for $G {\curvearrowright}E_S$, which we also denote by $\varphi$. We may then form the Toeplitz algebra ${{\mathcal T}_{{Y^\varphi}}}$ and the Cuntz-Pimsner algebra ${{\mathcal O}_{{Y^\varphi}}}$. Since $E_S$ is sourceless, it follows from [Corollary ]{} that ${{\mathcal O}_{{Y^\varphi}}}$ is isomorphic to the Exel-Pardo algebra ${\mathcal O}_{G, E_S}$ whenever $S$ is finite. Moreover, an important motivation in [@EP] is that if $(G,S)$ is a self-similar group, then ${\mathcal O}_{G, E_S}$ is isomorphic to the $C^*$-algebra ${\mathcal O}(G,S)$ introduced in [@Nek09]. Similarly, the $C^*$-algebra $\mathcal{T}(G,S)$ studied in [@lrrw] is easily seen to be isomorphic to ${\mathcal T}_{{Y^\varphi}}$ in this case.
For completeness, we include some comments on self-similar groups (sometimes called self-similar actions) in the terminology of this paper. Given an action $G{\curvearrowright}S$ as above and a $G$-valued cocycle $\varphi$ for $G{\curvearrowright}S$, let $S^*$ denote the set of all finite words in the alphabet $S$ and let $\varnothing \in S^*$ denote the empty word. One may then inductively extend the action of $G$ on $S$ to an action of $G$ on $S^*$ and $\varphi$ to a cocycle for $G{\curvearrowright}S^*$, also denoted by $\varphi$, such that $g\cdot \varnothing = \varnothing$, $\varphi(g,\varnothing) = g$ and $$\label{SS1}
g\cdot(vw)=(g\cdot v)\bigl(\varphi(g,v)\cdot w\bigr),$$ for all $g\in G$ and $v,w\in S^*$. We refer to [@Law08 Lemma 5.1] for a proof. Alternatively, we note that this is just a special case of [@EP Proposition 2.4] if one identifies $S^*$ with the set of finite paths on $E_S$.
If $S$ is finite and the action $G {\curvearrowright}S^*$ is faithful, then equation (\[SS1\]) says that the pair $(G,S)$ is a self-similar group in the sense of [@Nek05; @Nek09] (see also [@lrrw]). (Note that $\varphi(g,v)$ is denoted by $g_{\mid v}$ in these references.) Conversely, assume that $(G,S)$ is a self-similar group, that is, $S$ is a nonempty finite set, a faithful action of $G$ on $S^*$ fixing the empty word is given and $\varphi{\colon}G\times S^*\to G$ is a map such that $\varphi(g,\varnothing) = g$ and (\[SS1\]) holds. Then it can be shown (see [@Nek05 Section 1.3]) that $\varphi$ is a cocycle for $G{\curvearrowright}S^*$ and that $G{\curvearrowright}S^*$ restricts to an action of $G$ on $S$. In particular, the restriction of $\varphi$ to $G\times S$ is a $G$-valued cocycle for $G{\curvearrowright}S$.
As pointed out in [@Law08], see also [@LW14], it appears that self-similar (actions of) groups in a generalized sense were already considered in the 1972 thesis of Perrot, without assuming finiteness of $S$ or faithfulness of $G {\curvearrowright}S^*$. Considering $S^*$ as the free monoid on a given set $S$, the key issue in Perrot’s work is the existence of a left action $(g,w) \to g\cdot w$ of $G$ on $S^*$ such that $\varnothing$ is fixed, and of a right action $(g,w) \to \varphi(g,w)$ of $S^*$ on $G$ such that (\[SS1\]) holds. It follows from [@Law08] (see in particular subsection 5.1) that this happens if and only if there exist an action of $G$ on $S$ and a $G$-valued cocycle for this action. This setting is precisely the one that is generalized in the work of Exel and Pardo.
{#section}
A natural class of examples of ${\mathbb Z}$-valued cocycles for actions of ${\mathbb Z}$ on finite sets, related to the work of Katsura in [@Kat08] (see also [@EP Example 3.4]), is as follows. Let $a\in{\mathbb N}$ and $b\in{\mathbb Z}$. For any $m\in{\mathbb N}$ and $k\in {\mathbb Z}_a$, let $\varphi_{a,b}(m,k)\in{\mathbb Z}$ and $\sigma_{a,b}(m,k)\in{\mathbb Z}_a$ be the unique numbers satisfying $$bm+k=\varphi_{a,b}(m,k)a+\sigma_{a,b}(m,k).$$ It is well-known that $\sigma_{a,b}{\colon}{\mathbb Z}\times{\mathbb Z}_a\to{\mathbb Z}_a$ is the action of ${\mathbb Z}$ on ${\mathbb Z}_a$ given by $$\sigma_{a,b}(m,k)=bm+k \mod a$$ and that $\varphi_{a,b}{\colon}{\mathbb Z}\times {\mathbb Z}_a\to {\mathbb Z}$ is a cocycle for $\sigma_{a,b}$. We call $\varphi_{a,b}$ an *EPK cocycle* (for “Exel-Pardo-Katsura”) and the triple $({\mathbb Z}_a,\sigma_{a,b},\varphi_{a,b})$ an *EPK system*. Clearly, we have $$\sigma_{a,b+\ell a}=\sigma_{a,b}{\quad\text{for all }}\ell\in{\mathbb Z},$$ so when $a$ is fixed we really only have $a$ distinct actions $\sigma_{a,b}$ with $ b=0,1,\dots,a-1$. Moreover, for $b,b' \in{\mathbb Z}_a$, the actions $\sigma_{a,b}$ and $\sigma_{a,b'}$ are conjugate if and only if $\gcd(a,b)=\gcd(a, b')$, so $$\{\sigma_{a,d}:\text{$d\in {\mathbb Z}_a$ is either 0 or a positive divisor of $a$}\}$$ forms a complete set of representatives of conjugacy classes for the actions $\sigma_{a,b}$. However, as we will see below, something interesting happens with the cocycles.
For $b\in {\mathbb Z}$, writing $b = q a + r$ where $q\in {\mathbb Z}$ and $0 \leq r \leq a-1$, we get $$\sigma_{a,b}(1, k) = r + k \mod a.$$ Set $c = a-r$. The generating function of $\varphi_{a,b} $ is then given by $$\varphi_{a,b}(1,k)=\begin{cases}q{& \text{if }}k<c\\q+1{& \text{if }}k\ge c.\end{cases}$$ Thus the signature of the cocycle $\varphi_{a,b}$ is $$\sum_{k=0}^{a-1}\varphi_{a,b}(1,k)=qc+(q+1)r=qa+r=b.$$ Hence, it follows from Lemma \[signature\] that if $b,b' \in {\mathbb Z}$, then the two EPK-systems $({\mathbb Z}_a,\sigma_{a,b},\varphi_{a,b})$ and $({\mathbb Z}_a,\sigma_{a,b'},\varphi_{a,b'})$ are not cohomology conjugate whenever $b\neq b'$.
If $b$ is relatively prime to $a$, then the action $\sigma_{a,b}$ of ${\mathbb Z}$ on ${\mathbb Z}_a$ is obviously transitive. Otherwise, the EPK-system $({\mathbb Z}_a,\sigma_{a,b},\varphi_{a,b})$ may be decomposed as follows. Setting $d= \gcd(a,b) = \gcd(a,r)$ and $a'=a/d$, one finds that there are $d$ orbits $$\big\{i+d{\mathbb Z}_a\big\}_{i=0}^{d-1},$$ each having $a'$ elements. Set $b'=b/d$ and $r' = r/d $, so that $b'$ is relatively prime to $a'$ and $b'=qa'+r'$ with $0\le r'<a'$. For each $i=0,\dots,d-1$, the restriction of the cocycle $\varphi_{a,b}$ to the orbit $i+d{\mathbb Z}_a$ has generating function given by $$k\mapsto\begin{cases}q{& \text{if }}i+kd<(a'-r')d\\q+1{& \text{if }}i+kd\ge (a'-r')d.\end{cases}$$ A quick computation shows that the inequality $i+kd<(a'-r')d$ is equivalent to $
k<(a'-r')
$ for each $i=0,\dots,d-1$. Thus this restricted cocycle has signature $$(a'-r')q+r'(q+1)=a'q+r'=b'.$$ Since the cocycle $\varphi_{a',b'}$ for the transitive action $\sigma_{a',b'}$ of ${\mathbb Z}$ on ${\mathbb Z}_{a'}$ also has signature $b'$, we conclude from Corollary \[transitive\] that the restriction of the action $\sigma_{a,b}$ and the cocycle $\varphi_{a,b}$ to the orbit $i+d{\mathbb Z}_a$ is cohomology conjugate to the EPK system $({\mathbb Z}_{a'},\sigma_{a',b'},\varphi_{a',b'})$. (In fact, a routine computation shows that the map $k\mapsto i+kd$ transports the system $({\mathbb Z}_{a'},\sigma_{a',b'},\varphi_{a',b'})$ to the restriction of the action $\sigma_{a,b}$ and the cocycle $\varphi_{a,b}$ to $i+d{\mathbb Z}_a$.) In this way, we see that the EPK system $({\mathbb Z}_a, \sigma_{a,b},\varphi_{a,b})$ is cohomology conjugate with the system obtained from pasting $d$ disjoint copies of the transitive system $({\mathbb Z}_{a'},\sigma_{a',b'},\varphi_{a',b'})$.
More generally, let us now consider a bijection $\sigma$ of ${\mathbb Z}_a$ and let $\xi{\colon}{\mathbb Z}_a \to {\mathbb Z}$. We get an action of ${\mathbb Z}$ on ${\mathbb Z}_a$ by setting $m\cdot k = \sigma^m(k)$ and we may then form the ${\mathbb Z}$-valued cocycle $\varphi$ determined by $\xi$ with respect to this action. Letting $E_{{\mathbb Z}_a}$ be the graph having one vertex and ${\mathbb Z}_a$ as its edge set, we get an action of ${\mathbb Z}$ on $E_{{\mathbb Z}_a}$ and we may regard $\varphi$ as a cocycle for ${\mathbb Z}{\curvearrowright}E_{{\mathbb Z}_a}$. The Cuntz-Pimsner algebra ${\mathcal O}_{Y^\varphi}$ is then the universal unital $C^*$-algebra generated by Cuntz isometries $s_0, \ldots, s_{a-1}$ and a unitary $u$ satisfying the relations $$us_k = s_{ \sigma(k)}u^{\xi(k)},\quad k=0, 1, \ldots, a-1.$$ Indeed, using these relations, one computes readily that$$u^ms_k= s_{\sigma^m(k)} u^{\varphi(m,k)}$$ for $m\in {\mathbb Z}$ and $k\in {\mathbb Z}_a$, and these are precisely the relations for the associated Exel-Pardo algebra.
To ease notation, when $m \in {\mathbb Z}$ and $m = qa + r$ for $q\in {\mathbb Z}$ and $0\leq r \leq a-1$, we will write $ q = m|a$ and $[m]_a = r $. If $b\in {\mathbb Z}$ is given, and we let $\sigma{\colon}{\mathbb Z}_a\to {\mathbb Z}_a$ be defined by $
\sigma(k) = [b+k]_a
$ and $\xi{\colon}{\mathbb Z}_a\to {\mathbb Z}$ be given by $
\xi(k) = (b+k)|a,
$ then the associated action of ${\mathbb Z}$ on ${\mathbb Z}_a$ is $\sigma_{a,b}$, while $\varphi = \varphi_{a,b}$. Hence ${\mathcal O}^{a,b}:={\mathcal O}_{Y^{\varphi_{a,b}}}$ is the universal unital $C^*$-algebra generated by Cuntz isometries $s_0, \ldots, s_{a-1}$ and a unitary $u$ satisfying the relations $$us_k = s_{[b+k]_a}u^{(b+k)|a},\quad k=0, 1, \ldots, a-1.$$ This gives, for example, ${\mathcal O}^{a, 0} \simeq {\mathcal O}_a = C^*(E_{{\mathbb Z}_a})$ (in accordance with the fact that $\varphi_{a,0}(m,k) = 0 $ for all $m\in {\mathbb Z}$ and $k\in {\mathbb Z}_a$), and ${\mathcal O}^{a,a}= {\mathcal O}_a\otimes C(\mathbb{T})\simeq C^*(E_{{\mathbb Z}_a})\rtimes_{\rm id}{\mathbb Z}$ (in accordance with the fact that $\varphi_{a,a}(m,k) = m $ for all $m\in {\mathbb Z}$ and $k\in {\mathbb Z}_a$). More interestingly, ${\mathcal O}^{2,1}$ is the universal unital $C^*$-algebra generated by two Cuntz isometries $s_0, s_1$ and a unitary $u$ satisfying the relations $$us_0 = s_1,\quad us_1 = s_0u.$$ It is then not difficult to see that ${\mathcal O}^{2,1}$ is the universal unital $C^*$-algebra generated by an isometry $s_0$ and a unitary $u$ satisfying the relations $$u^2s_0 = s_0u,\quad s_0s_0^* + us_0s_0^*u^* = 1,$$ that is, ${\mathcal O}^{2,1} \simeq \mathcal{Q}_2$, where $\mathcal{Q}_2$ is the $C^*$-algebra studied in [@LarLi2adic] (see also references therein). As mentioned in [@LarLi2adic] (right after Remark 3.2), $\mathcal{Q}_2$ is isomorphic to the $C^*$-algebra ${\mathcal O}(E_{2,1})$ considered in [@Ka4 Example A.6]. In fact, we have ${\mathcal O}^{a,b}\simeq {\mathcal O}(E_{a,b})$ in general, where $E_{a,b}$ denotes the topological graph defined in [@Ka4 Example A.6]; this follows readily from the description of ${\mathcal O}(E_{a,b})$ given on page 1182 of [@Ka4].
{#section-1}
The class of EPK-systems may be put in a general framework. Let us first remark that if a discrete group $G$ acts on a set $S \neq \varnothing$, $\varphi$ is a $G$-valued cocycle for $G{\curvearrowright}S$, and $\tau$ is an endomorphism of $G$, then we may define another action $\cdot'$ of $G$ on $S$ by setting $$g\cdot'x = \tau(g) \cdot x$$ and a $G$-valued cocycle $\varphi_\tau$ for this action by setting $$\varphi_\tau(g,x)= \varphi(\tau(g), x),$$ as is easily verified.
Next, let $\rho$ be an injective endomorphism of a discrete group $G$ and set $H=\rho(G)$. To be interesting for what follows, $G$ should be infinite and $\rho$ should not be surjective. Choose a set $S_\rho$ of coset representatives for $G/H$ containing $e$. For each $g\in G$, let $s(g)$ denote the unique element of $S_\rho$ satisfying $s(g)H = gH$. For $g\in G$ and $x\in S_\rho$, set $$\begin{gathered}
g\cdot x = s(gx),
\\
\varphi(g,x) = \rho^{-1}(s(gx)^{-1} gx).\end{gathered}$$ Since $s(gx)H = gxH$, we have $s(gx)^{-1} gx \in H=\rho(G)$, so $\varphi(g,x)$ is well-defined and lies in $G$. It is then not difficult to check that this gives an action of $G$ on $S_\rho$, that $\varphi$ is a cocycle for this action and that this construction does not depend on the choice of coset representatives for $G/H$, up to cohomology conjugacy. Such a construction appears in [@lrrw Example 2.2] in the case where $G={\mathbb Z}^n$ for some $n\in {\mathbb N}$ and $\rho{\colon}{\mathbb Z}^n \to {\mathbb Z}^n$ is of the form $\rho(m) = Am$ for some $A\in M_n({\mathbb Z})$ with $|\det A\, | > 1$, in which case $S_\rho$ is finite with $|S_\rho| = |\det A \,| $.
Now, let $\tau$ be another endomorphism of $G$. We then get an action $\cdot'$ of $G$ on $S_\rho$ and a $G$-valued cocycle $\varphi_\tau$ for this action, given by $$\begin{gathered}
g\cdot'x = s\big(\tau(g)x\big),
\\
\varphi_\tau(g,x) = \rho^{-1}\Big(s\big(\tau(g)x\big)^{-1} \tau(g)x\Big)\end{gathered}$$ for $g\in G$ and $x\in S_\rho$.
For example, let $G={\mathbb Z}$, $a\in {\mathbb N}$ ($a\geq 2$) and $b\in {\mathbb Z}$, set $\rho(m) = am$ and $ \tau(m)=bm$ for $m\in {\mathbb Z}$, and choose $S_\rho={\mathbb Z}_a$. Then the action $\cdot'$ of ${\mathbb Z}$ on ${\mathbb Z}_a$ is equal to $\sigma_{a,b}$ and $\varphi_\tau$ is equal to $\varphi_{a,b}$, so we recover the EPK-system associated with $a$ and $b$. When $G={\mathbb Z}^n$, one may similarly consider $\rho$ associated with some $A \in M_n({\mathbb Z})$ ($|\det A\,| > 1$) and $\tau$ associated with some $B\in M_n({\mathbb Z})$.
{#section-2}
Triples $(E, G, \varphi)$ where $G$ is a discrete group acting on a directed graph $E$, in the trivial way on $E^0$, might be produced as follows:
- Pick a directed graph $E$ and a discrete group $G$.
- Let $G$ act trivially on $E^0$.
- For each $v,w\in E^0$, set ${}_{v}E^1_w=\{ e\in E^1: r(e) = v, s(e) = w\}$. Note that $E^1$ is the disjoint union of all these sets.
- Set $R_E= \{ (v,w) \in E^0\times E^0: {}_{v}E^1_w \neq \varnothing\}$.
- For each $(v,w) \in R_E$, pick an action of $G$ on ${}_{v}E^1_w$ and a cocycle ${}_{v}\varphi_w$ for this action.
- Paste these actions and these cocycles together to obtain an action of $G$ on $E^1$ and a cocycle $\varphi$ for it.
Since $G$ acts trivially on $E^0$, it is clear that we get an action of $G$ on the graph $E$ and that $\varphi$ is a cocycle for this action. Moreover, it is easy to see that if ${}_{v}\varphi'_w$ is also a cocycle for the chosen action of $G$ on ${}_{v}E^1_w$ for each $(v,w) \in R_E$, then the resulting cocycle $\varphi'$ will be cohomologous to $\varphi$ if and only if ${}_{v}\varphi'_w$ is cohomologous to ${}_{v}\varphi_w$ for each $(v,w) \in R_E$.
To illustrate this procedure, set $G={\mathbb Z}$ and let $E$ be a directed graph such that the number $A(v, w)$ of edges in ${}_{v}E^1_w$ is finite for all $v,w \in E^0$. Note that this hypothesis is much weaker than requiring that $E$ be *locally finite* in the sense that each vertex only receives and emits finitely many edges. Let $B{\colon}E^0\times E^0\to {\mathbb Z}$ be a map. For each $(v,w) \in E^0\times E^0$ such that $A(v,w) \geq 1$, i.e., for each $(v,w)\in R_E$, we may choose a bijection from ${\mathbb Z}_{A(v,w)}$ onto ${}_{v}E^1_w$ and use it to transfer the EPK-system associated with the pair $A(v,w), B(v,w)$ into an action of ${\mathbb Z}$ on ${}_{v}E^1_w$ and a cocycle for this action. Using these choices in the construction outlined above, we obtain an action of ${\mathbb Z}$ on $E$ fixing all vertices and a cocycle $\varphi_B$ for this action. Let $B_E{\colon}R_E\to{\mathbb Z}$ denote the restriction of $B$ to $R_E$. Note that if $C{\colon}E^0 \times E^0\to {\mathbb Z}$ is any other map such that $B_E \neq C_E$, then it follows from our previous analysis of EPK-systems that the systems $(E,{\mathbb Z},\varphi_B)$ and $(E,{\mathbb Z},\varphi_{C})$ are not cohomology conjugate. Note also that if $E$ is a countable row-finite graph with no sources, then we just get the class of $C^*$-algebras $\mathcal{O}_{A,B}$ introduced by Katsura in [@Kat08], as presented in [@EP Example 3.4] when $E$ is finite with no sources.
As a concrete example, let $a \in {\mathbb N}$ and consider the graph $E$ given by $E^0={\mathbb Z}$, $E^1 = {\mathbb Z}_a \times {\mathbb Z}$, $r(t,j) = j-1$, and $ s(t,j) = j $ for $(t,j) \in E^1$, so that $A(i,j) = a$ when $i= j-1$ and is zero otherwise. Only the coefficients $B(j-1,j)$ along the first subdiagonal of $B$ will then matter. In this example, $E$ is row-finite with no sources, so it will give one of Katsura’s $\mathcal{O}_{A,B}$. But it can easily be changed so that $E$ is not row-finite with no sources (for example by adding one edge $e_j$ (or more) going from 0 to $j$ for each $j\in {\mathbb Z}$), but still satisfies the requirement that $|A(i,j)| < \infty$ for all $i, j\in {\mathbb Z}=E^0$).
{#strings}
Consider again a triple $(S, G, \varphi)$ where a discrete group $G$ acts on a set $S$ and $\varphi$ is a cocycle for this action. Pick any symbol $\omega \not\in S$. Let then $F=F_S$ be the directed graph where $F^0 = S \cup \{\omega\}$, $F^1= S$, and $ r, s{\colon}F^1\to F^0$ are given by $$r(x) = x,\quad s(x) = \omega{\quad\text{for }} x \in F^1=S.$$ Obviously, $F$ has exactly one source, namely $\omega$. (If $S$ is finite, $F$ may be thought of as a bouquet of $|S|$ disjoint strings (that are not loops) emanating from $\omega$.) The action of $G$ on $S$ induces a natural action of $G$ on $F$ in an obvious way: we just set $g\omega = \omega$ for all $g\in G$, and let $G$ act on $F^0\setminus \{\omega\} = S$ and on $F^1=S$ via its given action on $S$. The cocycle $\varphi$ is then a cocycle for the action of $G$ on $F$: the first condition is automatically satisfied (since $F^1=S$); because $$\varphi(g,x) s(x) = \varphi(g,x) \omega = \omega =g\omega = g s(x)$$ for all $g\in G$ and $x\in F^1=S$, the second condition is trivially satisfied.
### Special case {#special-case .unnumbered}
Set $S=G$ and let $G$ act on itself by left translation. As the map ${\text{\textup{id}}}{\colon}F^1=G \to G$ trivially satisfies condition (2) in [Proposition ]{} (and the assumption in this proposition is fulfilled), we get that the cocycle $(g,e) \to g$ is a coboundary, i.e., it is cohomologous to the cocycle $(g,e)\to 1$. Hence we conclude that the correspondences associated to these cocycles are isomorphic. For the first of these cocycles, it follows from [Remark ]{} that we have $${{\mathcal O}_{{Y^\varphi}}}= {\mathcal O}_{X_F\rtimes G}\simeq C^*(F)\rtimes G,$$ which is frequently not isomorphic to $C^*(F)$. As an explicit example, consider the cocycle $\varphi(g,d)=g$ for the action ${\mathbb Z}_2{\curvearrowright}{\mathbb Z}_2$ by translation. Since any action of ${\mathbb Z}_2$ on $C^*(F)=M_3$ is inner, we get $$\begin{aligned}
{\mathcal O}_{{Y^\varphi}} &
\simeq C^*(F)\rtimes {\mathbb Z}_2
\simeq M_3\rtimes {\mathbb Z}_2\\
&\simeq M_3\otimes {\mathbb C}^2
\simeq M_3\oplus M_3 \\
&\not\simeq M_3,\end{aligned}$$ and we obtain the same $C^*$-algebra for the cocycle $\varphi=1$.
This is in contrast to the situation in [@EP Example 3.6], where the graph $E$ is finite and has no sources, and the action fixes the vertices; Exel and Pardo then show that for the cocycle $\varphi=1$ we have ${{\mathcal O}_{{Y^\varphi}}}\simeq C^*(E)$, because the unitaries $u_g$ for $g\in G$ can be expressed in terms of the partial isometries $s_e$ for $e\in E^1$. Note that Exel and Pardo’s observation does not apply to the graph $F$ above simply because $F$ has a source, namely $\omega$.
{#section-3}
A more general construction in the same vein as the one in \[strings\] is as follows. Let $(S, G, \varphi)$ be as in \[strings\]. Assume that we are also given an action of $G$ on a nonempty set $I$ and a $G$-equivariant map $\rho{\colon}S\to I$. Pick a symbol $\omega \not\in I$ and let $F$ be the directed graph where $F^0 = I \cup \{ \omega\}$, $F^1=S$ and $ r, s{\colon}F^1\to F^0$ are given by $$r(x) = \rho(x),\quad s(x) = \omega{\quad\text{for }} x \in F^1=S.$$ The two actions of $G$ induce a natural action of $G$ on $F$ by setting $g\omega = \omega$ for all $g\in G$ and letting $G$ act on $F^0\setminus \{\omega\} = I$ and on $F^1=S$ via the given actions of $G$ on $I$ and $S$, respectively. The cocycle $\varphi$ is then again a cocycle for the action of $G$ on $F$.
In this example, all edges of $F$ have the same source $\omega$, which is a source for $F$, and all vertices different from $\omega$ are sinks for $F$, undoubtedly a somewhat special situation. Next we define a similar class of examples, but without sinks.
Assume that $G$ also acts on a nonempty set $T$ and pick a symbol $\omega \not\in S\cup T$. Let then $K = (K^0, K^1, r, s)$ be the directed graph where $$K^0 = S \cup \{\omega\},\quad K^1= S\times (T \cup\{\omega\})$$ and $ r, s{\colon}K^1\to K^0$ are given by $$\begin{gathered}
r(x, \omega) = x,\quad s(x,\omega) = \omega,
\\
r(x, y) = x = s(x,y)
\end{gathered}$$ for $x \in S$ and $ y \in T.$ Define an action of $G$ on $K$ as follows:
- $G$ acts on $K^0\setminus \{\omega\} = S$ via the given action of $G$ on $S$,
- $g\omega = \omega,$
- $ g(x, \omega) = (gx, \omega)$
- $ g(x,y) = (gx, gy)$
for $g \in G$, $x\in S$, and $y \in T$. Moreover, define $\widetilde{\varphi}{\colon}G\times K^1\to G$ by $$\begin{aligned}
\widetilde{\varphi}\big(g, (x,\omega)\big) &= \varphi(g, x),
\\
\widetilde{\varphi}\big(g, (x,y)\big)&= g\end{aligned}$$ for $g \in G$, $x\in S$, and $ y \in T$. Then $\widetilde{\varphi}$ is a cocycle for $G {\curvearrowright}K$ that is not cohomologous to the trivial cocycle if $\varphi$ is not cohomologous to the trivial cocycle for $G {\curvearrowright}S$.
This graph still has one source, namely $\omega$. To obtain a system with a graph having no sources, one can for example add one loop (or more) at $\omega$, let $G$ act on this loop (or these loops) by fixing it (or them), and set $\varphi(g, e) = g$ for all $g$ when $e$ is this loop (or any of these loops).
{#section-4}
In [@Ka2 Example 2], Katsura constructs a topological graph from a locally compact Hausdorff space $S$ and a homeomorphism $\sigma{\colon}S\to S$. The associated topological graph $E_\sigma$ has $E_\sigma^0=E_\sigma^1=S$, $s={\text{\textup{id}}}_S$, and $r=\sigma$. The main point of this class of examples of topological graphs is the natural isomorphism $$C^*(E_\sigma)\simeq C_0(S)\rtimes_\alpha {\mathbb Z},$$ where $\alpha$ is the associated action of ${\mathbb Z}$ on $C_0(S)$.
Actions of ${\mathbb Z}$ on the topological graph $E_\sigma$ are in 1-1 correspondence with homeomorphisms $\tau{\colon}S\to S$ that commute with $\sigma$, via $n\cdot x=\tau^n(x)$ for $n\in{\mathbb Z},x\in S$. We can regard a cocycle $\varphi$ for such an action as a continuous map $\varphi{\colon}{\mathbb Z}\times S\to {\mathbb Z}$, and the generating function of $\varphi$ as a continuous map $\xi{\colon}S\to {\mathbb Z}$ satisfying $$\xi(x)-1\in S_x:=\{k\in{\mathbb Z}:\tau^k(x)=x\},$$ so that $\xi(x)$ is congruent to 1 modulo the period of the orbit ${\mathbb Z}\cdot x$ (where by convention the period is defined to be 0 if the orbit is free, in which case $\varphi(n,x)=n$ for all $n\in{\mathbb Z}$).
{#section-5}
Assume that $H$ is a discrete group acting by homeomorphisms on a locally compact space Hausdorff space $E^0$. Set $E^1= H\times E^0$ and define $r,s{\colon}E^1\to E^0$ by $$r(h,x) = h\cdot x, \quad s(h,x) = x$$ for all $(h,x) \in E^1$. This gives a topological graph $E$. Note that $C^*(E)$ is in general not isomorphic to $C_0(E^0) \rtimes H$. (For example, if $H$ is finite and abelian, $E^0$ is finite and the action of $H$ on $E^0$ is trivial, then $C_0(E^0) \rtimes H\simeq C(E^0) \otimes C^*(H)$ is abelian, while $C^*(E)$ is the direct sum of $|E^0| $ copies of the Cuntz algebra ${\mathcal O}_{|H|}$).
Now, assume that a discrete group $G$ also acts on $E^0$ by homeomorphisms and that this action commutes with the action of $H$. We may then define an action of $G$ on $E^1$ by $$g\cdot(h,x) = (h, g\cdot x)$$ One easily verifies that this gives an action of $G$ on $E$.
Let $\phi{\colon}G\times E^0 \to G$ be a cocycle for $G\curvearrowright E^0$ satisfying $$\phi(g,x)\cdot x = g\cdot x$$ for all $(g,x)\in G\times E^0$. Then the map $\varphi{\colon}G\times E^1\to G$ defined by $$\varphi\big(g,(h,x)\big) = \phi(g, h\cdot x)$$ is a cocycle for the action of $G$ on $E$. Indeed, since the actions of $G$ and $H$ on $E^0$ commute, we have $$\begin{aligned}
\varphi\big(g_1g_2, (h,x)\big) &= \phi(g_1g_2, h\cdot x) \\
&= \phi\big(g_1, g_2\cdot(h\cdot x)\big) \phi(g_2, h\cdot x)\\
&= \phi\big(g_1, h\cdot(g_2\cdot x)\big) \phi(g_2, h\cdot x)\\
&= \varphi\big(g_1, (h, g_2\cdot x)\big) \varphi\big(g_2, (h,x)\big)\\
&=\varphi\big(g_1, g_2\cdot(h,x)\big) \varphi\big(g_2, (h,x)\big)\end{aligned}$$ for all $g_1, g_2 \in G, (h,x)\in E^1$, and $$\begin{aligned}
\varphi\big(g, (h,x)\big)\cdot s(h,x)&= \phi(g, h\cdot x)\cdot x \\
&= h^{-1}\cdot\big(\phi(g, h\cdot x)\cdot (h\cdot x)\big)\\
&= h^{-1}\cdot\big(g\cdot(h\cdot x)\big) = g\cdot x\\
&= g\cdot s(h,x)\end{aligned}$$ for all $g\in G, (h,x)\in E^1$.
Note that if the action of $G$ on $E^0$ is free, then $\phi$ has to be the trivial cocycle $(g, x) \to g$ for the action $G{\curvearrowright}E^0$, so $\varphi$ can only be the trivial cocycle for $G{\curvearrowright}E$. A simple example where the action $G{\curvearrowright}E^0$ is not free is as follows. Set $E^0 = {\mathbb T}$, $G=H={\mathbb Z}$, pick $\lambda, \mu \in {\mathbb T}$ such that $\lambda$ has period $p$, and $k\in {\mathbb Z}$. Let $G{\curvearrowright}E^0$ (resp. $H {\curvearrowright}E^0$) be given by $(m,z) \to \lambda^m z$ (resp. $(n,z) \to \mu^n z$) and define $\phi: G\times E^0\to G$ by $\phi(m, z) = (1 + kp)m$. Then $G{\curvearrowright}E^0$ is not free and all the required conditions are easily verified. Note that the cocycle $\varphi$ we get for the action of $G$ on $E$ is simply given by $\varphi(m, (n,z)) = (1 + kp)m$. It would be interesting to know whether more exotic examples can be produced.
{#section-6}
Assume that a discrete group $G$ acts on a nonempty set $S$ and that $\varphi$ is a $G$-valued cocycle for $G{\curvearrowright}S$. We recall from [Subsection ]{} that $G {\curvearrowright}S$ extends to an action of $G$ on $S^*$, where $S^*$ denotes the set of words on the alphabet $S$, and that $\varphi$ extends to a cocycle for $G{\curvearrowright}S^*$, also denoted by $\varphi$.
As Nekrashevych [@Nek09] points out in the case of a self-similar group, see also [@lrrw Section 2], $S^*$ may be used to build a directed rooted tree $T$ (sometimes called an arborescence), with the empty word $\varnothing$ as the root, and with vertex set $T^0=S^*$ and edge set $$T^1=\{(w,wx):w\in S^*,x\in S\}.$$ In view of our conventions (which in this respect conform to those of [@EP]), namely that paths in a directed graph should go from right to left, we dictate that an edge $(w,wx)$ has source $wx$ and range $w$. Since $G$ acts on $S^*=T^0$, we clearly get an action of $G$ on $T$ when we define $G{\curvearrowright}T^1$ by setting $$g\cdot(w,wx)= \big(g\cdot w, g\cdot(wx)\big)$$ for $g\in G, w\in S^*$ and $x\in S$. We can also define a map $\varphi{\colon}G\times T^1 \to G$ by $$\varphi\big(g, (w,wx)\big) = \varphi(g, wx)$$ for $g\in G, w\in S^*$ and $x\in S$. It is then straightforward to check that $\varphi$ is a cocycle for $G{\curvearrowright}T^1$. To become a graph cocycle for $G{\curvearrowright}T$, $\varphi$ must satisfy $$\label{CT1}
\varphi\big(g, (w, wx)\big)\cdot (wx) = g\cdot (wx) \quad \text{ for all } g\in G, w\in S^*,
x\in S,$$ that is, $$\label{CT2}
\varphi(g,wx)\cdot (wx) = g\cdot (wx) \quad \text{ for all } g\in G, w\in S^*,
x\in S.$$ In particular, $\varphi$ must then satisfy $$\label{CT3}
\varphi(g,x)\cdot x = g\cdot x \quad \text{ for all } g\in G \text{ and } x \in S.$$ If $G{\curvearrowright}S$ is free, then only holds when $\varphi$ is the trivial cocycle $(g,x) \mapsto g$ for $G{\curvearrowright}S$, hence the trivial cocycle is the only possible one for $G{\curvearrowright}T$. Interestingly, this is also the case if we assume that the action $G{\curvearrowright}S^*$ is faithful and, instead of , we impose the stronger Exel-Pardo vertex condition $\varphi(g,e)\cdot v=g\cdot v$ for all $g\in G, e\in T^1$, and $v\in T^0$. It is not difficult to construct examples where $G{\curvearrowright}S$ is not free and there exist cocycles $\varphi$ for $G {\curvearrowright}S$ that satisfy and are different from the trivial cocycle. Conceivably, there might exist cases where such cocycles satisfy , that is, give cocycles for $G{\curvearrowright}T$, but we don’t know of any concrete example.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'For smooth manifolds equipped with various geometric structures, we construct complexes that replace the de Rham complex in providing an alternative fine resolution of the sheaf of locally constant functions. In case that the geometric structure is that of a parabolic geometry, our complexes coincide with the Bernstein-Gelfand-Gelfand complex associated with the trivial representation. However, at least in the cases we discuss, our constructions are relatively simple and avoid most of the machinery of parabolic geometry. Moreover, our method extends to contact and symplectic geometries (beyond the parabolic realm).'
address:
- |
Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley,\
CA 94720-5070, USA, and\
Department of Mathematics, University of California, Berkeley,\
CA 94720-3840, USA
- |
Mathematical Sciences Institute, Australian National University,\
ACT 0200, Australia
- |
Department of Mathematics, University of Auckland, Private Bag 92019,\
Auckland 1142, New Zealand, and\
Mathematical Sciences Institute, Australian National University,\
ACT 0200, Australia
- |
Mathematical Sciences Institute, Australian National University,\
ACT 0200, Australia
author:
- 'Robert L. Bryant'
- 'Michael G. Eastwood'
- 'A. Rod. Gover'
- Katharina Neusser
title: Some differential complexes within and beyond parabolic geometry
---
Introduction
============
In [@CSS], Čap, Slovák, and Souček construct sequences of invariant differential operators on [*parabolic geometries*]{} of any type $G/P$, one for each finite-dimensional representation ${\mathbb{V}}$ of $G$. (Here, $G$ is a semisimple Lie group and $P\subset G$ a parabolic subgroup.) These sequences are known as [*Bernstein-Gelfand-Gelfand*]{} (BGG) sequences since, for the homogeneous model $G/P$ of such a geometry, these sequences are complexes, which are dual to a parallel construction due to these authors [@BGG] on the level of Verma modules. In [@CD] Calderbank and Diemer simplify the construction of BGG sequences in [@CSS]. In addition they provide [@CD p. 87], for regular parabolic geometries, alternative BGG sequences, which only coincide with the ones in [@CSS] if the geometry is [*torsion-free*]{}. The latter sequences not only appear to be more natural, they also have the advantage that if ${\mathbb{V}}$ is taken to be the trivial representation, then they form complexes, providing fine resolutions of the locally constant sheaf ${\mathbb{R}}$ (as one sees by suitably modifying [@CD Proposition 5.5(iv)]). For the sequences of [@CSS] this is only true if the geometry is torsion-free and, in this case, the two sequences are anyway the same. In combination with the construction of canonical Cartan connections given in [@CSc], this shows that one can find alternatives to the de Rham resolution for any parabolic geometry defined in terms of a regular infinitesimal flag structure [@CSl §3.1.6]. A hallmark of these resolutions is that the ranks of the bundles involved are diminished as compared to the de Rham complex. The price one pays is that the operators may be higher than first order. The construction of these resolutions in [@CD; @CSS], entails firstly constructing the Cartan connection as described in [@CSc] and this is not at all straightforward.
In this article we present some examples constructed by a more elementary route. As we show, our method extends to certain non-parabolic geometries, namely arbitrary contact and symplectic geometries. We shall use the spectral sequence of a filtered complex [@Ch] without comment and merely as a replacement for tedious diagram chasing.
The Rumin complex {#rumin}
=================
For our first example we shall construct the Rumin complex [@R]. It is defined on an arbitrary contact manifold but, for simplicity, we shall present the $5$-dimensional case, which is typical. So let $M$ be a $5$-dimensional smooth manifold with $H\subset TM$ a contact distribution. Equivalently, the contact structure may be defined by $L\equiv H^\perp$, a line sub-bundle of the bundle of $1$-forms $\Lambda^1$. If we define a rank $4$ vector bundle $\Lambda_H^1$ as the quotient $\Lambda^1/L$, then there are induced short exact sequences $$0\to \Lambda_H^{p-1}\otimes L\to\Lambda^p\to\Lambda_H^p\to 0,
\quad\mbox{for }1\leq p\leq 5$$ and the spectral sequence of the de Rham complex filtered in this way reads, at the $E_0$-level, $$\begin{picture}(300,45)
\put(160,25){\makebox(0,0){$\begin{array}{ccccccccccccc}
\Lambda^0&&\Lambda_H^1&&\Lambda_H^2 &&\Lambda_H^3
&&\Lambda_H^4&&0\\
&&&&\makebox[0pt][r]{${\mathcal{L}}$}\uparrow&&\uparrow&&\uparrow\\
&&0&&L&&\Lambda_H^1\otimes L&&\Lambda_H^2\otimes L&
&\Lambda_H^3\otimes L&&\Lambda_H^4\otimes L
\end{array}$}}
\put(0,0){\vector(1,0){40}}
\put(0,0){\vector(0,1){45}}
\put(30,5){$p$}
\put(3,35){$q$}
\end{picture}$$where ${\mathcal{L}}$ is the composition $L\to\Lambda^1\xrightarrow{\,d\,}\Lambda^2\to\Lambda_H^2$. The Leibniz rule shows that ${\mathcal{L}}$ is linear over the functions and is, therefore, a homomorphism of vector bundles. It is called the [*Levi form*]{}. By definition of contact manifold, the range of ${\mathcal{L}}$ is non-degenerate as a skew form on $H$, defined up to scale. Equivalently, we can choose local co-framings $(\omega_1,\omega_2,\omega_3,\omega_4,\omega_5)$ with $\omega_1$ a section of $L$ such that $$d\omega_1\equiv \omega_2\wedge\omega_3+\omega_4\wedge\omega_5\bmod\omega_1.$$ Notice that $$\begin{array}{cc}
d(\omega_1\wedge\omega_2)
\equiv\omega_2\wedge\omega_4\wedge\omega_5\bmod\omega_1&\quad
d(\omega_1\wedge\omega_3)
\equiv\omega_3\wedge\omega_4\wedge\omega_5\bmod\omega_1\\
d(\omega_1\wedge\omega_4)
\equiv\omega_2\wedge\omega_3\wedge\omega_4\bmod\omega_1&\quad
d(\omega_1\wedge\omega_5)
\equiv\omega_2\wedge\omega_3\wedge\omega_5\bmod\omega_1
\end{array}$$whence the $E_0$-differential $\Lambda_H^1\otimes L\to \Lambda_H^3$ is an isomorphism of vector bundles. Similar reasoning shows that $\Lambda_H^2\otimes L\to\Lambda_H^4$ is surjective. Hence, at the $E_1$-level we obtain $$\begin{picture}(300,45)
\put(163,25){\makebox(0,0){$\begin{array}{ccccccccccccc}
\Lambda^0&\!\!\!\stackrel{d_\perp}{\longrightarrow}\!\!\!&\Lambda_H^1&
\!\!\!\stackrel{d_\perp}{\longrightarrow}\!\!\!&
\Lambda_{H\perp}^2&&\!\!0\!\!&&0&&0\\[4pt]
&&0&&0&&\!\!0\!\!&&\Lambda_{H\perp}^2\otimes L&
\!\!\!\stackrel{d_\perp}{\longrightarrow}\!\!\!
&\Lambda_H^3\otimes L&
\!\!\!\stackrel{d_\perp}{\longrightarrow}\!\!\!&\Lambda_H^4\otimes L
\end{array}$}}
\put(0,0){\vector(1,0){40}}
\put(0,0){\vector(0,1){45}}
\put(30,5){$p$}
\put(3,35){$q$}
\end{picture}$$and deduce that there is a complex $$\label{rumincomplex}
0\to{\mathbb{R}}\to\Lambda^0\stackrel{d_\perp}{\longrightarrow}\Lambda_H^1
\stackrel{d_\perp}{\longrightarrow}\Lambda_{H\perp}^2
\stackrel{d_\perp^{(2)}}{\longrightarrow}\Lambda_{H\perp}^2\otimes L
\stackrel{d_\perp}{\longrightarrow}\Lambda_H^3\otimes L
\stackrel{d_\perp}{\longrightarrow}\Lambda_H^4\otimes L\to 0$$where $\Lambda_{H\perp}^p$ denotes the sub-bundle of $\Lambda_H^p$, trace-free with respect to the Levi-form. The operator $d_\perp^{(2)}$ is second order and, because the spectral sequence converges to the local cohomology of the de Rham complex, it follows that this complex is exact on the level of sheaves. Already, the Rumin complex goes beyond parabolic geometry. Notice that, although a convenient co-framing was chosen to perform some calculation, the construction itself and the resulting complex are independent of any such choice. This is a repeated theme in this article.
The Engel complex {#engel}
=================
In this section we shall be concerned with a smooth $4$-manifold $M$ equipped with a generic distribution $H\subset TM$ of rank $2$. Genericity entails that $[H,H]$ has rank $3$ and that $[H,[H,H]]=TM$. Dually, if we let $K\equiv
H^\perp$ and $L\equiv [H,H]^\perp$ then the $1$-forms are filtered $L\subset
K\subset\Lambda^1$ by the line-bundle $L$ and rank $2$ bundle $K$. In fact, there is a canonically defined finer filtration constructed as follows. One easily checks that the Levi form $K\to\Lambda_H^2$, defined as the composition $K\to\Lambda^1\xrightarrow{\,d\,}\Lambda^2\to\Lambda_H^2$, is a surjective homomorphism of vector bundles with $L$ as kernel. It follows that the other Levi form, defined as the composition $$L\to\Lambda^1\xrightarrow{\,d\,}\Lambda^2\to\Lambda^2(\Lambda^1/L)$$ has range in the kernel of $\Lambda^2(\Lambda^1/L)\to\Lambda_H^2$. However, the short exact sequence $$0\to K/L\to\Lambda^1/L\to\Lambda_H^1\to0$$ identifies this kernel as $\Lambda_H^1\otimes K/L$. In other words, we have a canonically defined inclusion $L\otimes(K/L)^*\hookrightarrow\Lambda_H^1$ the range of which defines a line sub-bundle $\xi$ of $\Lambda_H^1$. The result is that we can write $$\Lambda^1=\Lambda_H^1/\xi+\xi+K/L+L,$$ meaning that $\Lambda^1$ is filtered with composition factors being line bundles as indicated (ordered so that $\Lambda_H^1/\xi$ is a canonical quotient and $L$ is a canonical sub-bundle). All in all, if we write $\lambda$ for $\Lambda_H^1/\xi$ and untangle the identifications found above, then we conclude that $$\label{filtration}\Lambda^1=
\lambda+\xi+\lambda\xi+\lambda\xi^2.$$ Equivalently, we can work locally with $(\omega^1,\omega^2,\omega^3,\omega^4)$, an [*adapted*]{} co-framing such that $$\label{structure}
d\omega^1\equiv\omega^2\wedge\omega^3\bmod\omega^1\quad\mbox{and}\quad
d\omega^2\equiv\omega^3\wedge\omega^4\bmod \omega^1,\omega^2,$$ noting that the freedom in such a co-framing comprises exactly the triangular endomorphisms of the filtration (\[filtration\]), where $$\begin{array}{c}
L=\lambda\xi^2={\mathrm{span}}\{\omega^1\},\enskip
K=\lambda\xi+\lambda\xi^2={\mathrm{span}}\{\omega^1,\omega^2\},\\[4pt]
\xi+K={\mathrm{span}}\{\omega^1,\omega^2\,\omega^3\}.\end{array}$$ So far, this is the structure of an [*Engel manifold*]{}. As with the Rumin complex, it is clear that the first order operator $d_H:\Lambda^0\to\Lambda_H^1$ defined as the composition $\Lambda^0\xrightarrow{\,d\,}\Lambda^1\to\Lambda_H^1$ has the locally constant functions as its kernel. We now seek differential conditions on a section of $\Lambda_H^1$ in order that it be in the range of the operator $d_H$. Starting with any $1$-form $\omega$, $$\label{primary_recipe}
\begin{array}{l}
\bullet\enskip\mbox{define }f\mbox{ by }
d\omega \equiv f\,\omega^3\wedge\omega^4 \bmod \omega^1,\omega^2\\
\bullet\enskip\mbox{define }p\mbox{ by }
d(\omega-f\omega^2)\equiv p\,\omega^2\wedge\omega^4+
g\,\omega^2\wedge\omega^3\bmod \omega^1.
\end{array}$$ The structure equations (\[structure\]) show that $p$ is well-defined and one easily checks that the equivalence class $$[p\,\omega^2\wedge\omega^4]\in
\frac{{\mathrm{span}}\{\omega^2\wedge\omega^4,\omega^2\wedge\omega^3,
\omega^1\wedge\omega^4,\omega^1\wedge\omega^3,\omega^1\wedge\omega^2\}}
{{\mathrm{span}}\{\omega^2\wedge\omega^3,
\omega^1\wedge\omega^4,\omega^1\wedge\omega^3,\omega^1\wedge\omega^2\}}
\cong\lambda^2\xi$$ depends only on the equivalence class $[\omega]\in\Lambda_H^1$ and is independent of choice of co-framing. We have a well-defined second order differential operator $$\Lambda_H^1\ni[\omega]\stackrel{{\mathcal{P}}}{\longmapsto}
[p\,\omega^2\wedge\omega^4]\in\lambda^2\xi,$$ giving what we shall call the [*primary*]{} obstruction to $[\omega]$ being in the range of $d_H$. In a chosen co-frame, one can easily proceed to find a [*secondary*]{} obstruction $s$ as follows. Define $f,p,g$ by (\[primary\_recipe\]) and then $$\bullet\enskip\mbox{define }s\mbox{ by }
d(\omega-f\omega^2-g\omega^1)=p\,\omega^2\wedge\omega^4
+r\,\omega^1\wedge\omega^4+s\,\omega^1\wedge\omega^3
+t\,\omega^1\wedge\omega^2.$$ If $p$ vanishes, then $$0=d^2(\omega-f\omega^2-g\omega^1)=r\,\omega^2\wedge\omega^3\wedge\omega^4+
\cdots$$ so $r$ vanishes. If, in addition $s$ vanishes, then $$0=d^2(\omega-f\omega^2-g\omega^1)=d(t\,\omega^1\wedge\omega^2)=
-t\,\omega^1\wedge\omega^3\wedge\omega^4$$ so $t$ vanishes. Hence, if both $p$ and $s$ vanish, then $d(\omega-f\omega^2-g\omega^1)=0$. By the Poincaré Lemma, it follows that $[\omega]$ is locally in the range of $d_H$, as required. If the primary obstruction vanishes, then the equivalence class $$[s\,\omega^1\wedge\omega^3]\in
\frac{{\mathrm{span}}\{\omega^1\wedge\omega^3,\omega^1\wedge\omega^2\}}
{{\mathrm{span}}\{\omega^1\wedge\omega^2\}}
\cong\lambda\xi^3$$ is independent of choice of co-framing. Otherwise, the change $$\label{badchange}
\omega_4\mapsto\omega_4+h\omega_3$$ induces severe complications with $s$ changing by $r$ and its derivatives. If one wants to avoid these complications, it suffices to prohibit (\[badchange\]) to arrive at an invariantly defined differential operator $$({\mathcal{P}},{\mathcal{S}}):\Lambda_H^1\to
\lambda^2\xi\oplus\lambda\xi^3,$$ whose kernel is locally the range of $d_H$. More precisely, we may eliminate (\[badchange\]) by choosing a complement to the line sub-bundle $\xi\hookrightarrow\Lambda_H^1$. In other words, we choose a splitting $\Lambda_H^1=\lambda\oplus\xi$. An adapted co-framing yields such a splitting and, conversely, a fixed choice of splitting restricts the choice of adapted co-framings precisely by preventing the addition of any multiple of $\omega^3$ to $\omega^4$. The forms on an Engel manifold endowed with this extra structure are filtered as follows. $$\begin{array}{rcl}
\Lambda^1&=&(\lambda\oplus\xi)+\lambda\xi+\lambda\xi^2\\[2pt]
\Lambda^2&=&\lambda\xi+(\lambda^2\xi\oplus\lambda\xi^2)+
(\lambda^2\xi^2\oplus\lambda\xi^3)+\lambda^2\xi^3\\[2pt]
\Lambda^3&=&\lambda^2\xi^2+\lambda^2\xi^3+(\lambda^3\xi^3\oplus\lambda^2\xi^4)
\end{array}$$ and the spectral sequence of the de Rham complex filtered in this way reads, at the $E_0$-level, $$\begin{picture}(300,85)(0,5)
\put(170,50){\makebox(0,0){$\begin{array}{cccccccc}
\Lambda^0&\lambda\oplus\xi&\lambda\xi &0&0&0&0\\
&&\uparrow\\
&0&\lambda\xi&\lambda^2\xi\oplus\lambda\xi^2&\lambda^2\xi^2&0&0\\
&&&\uparrow&\uparrow\\
&0&0&\lambda\xi^2&\lambda^2\xi^2\oplus\lambda\xi^3&
\lambda^2\xi^3&0\\
&&&&&\uparrow\\
&0&0&0&0&\lambda^2\xi^3&\lambda^3\xi^3\oplus\lambda^2\xi^4&\Lambda^4\,.
\end{array}$}}
\put(0,0){\vector(1,0){40}}
\put(0,0){\vector(0,1){45}}
\put(30,5){$p$}
\put(3,35){$q$}
\end{picture}$$The $E_0$-differentials are easily computed in our adapted co-frame. For example $$\begin{array}{rcl}
f\,\omega^1\wedge\omega^2&\!\stackrel{d}{\longrightarrow}\!&
f\,\omega^1\wedge\omega^3\wedge\omega^4
\scriptstyle\bmod\omega^1\wedge\omega^2\\
h\,\omega^1\wedge\omega^4+g\,\omega^1\wedge\omega^3+f\,\omega^1\wedge\omega^2
&\!\stackrel{d}{\longrightarrow}\!&
h\,\omega^2\wedge\omega^3\wedge\omega^4
\scriptstyle\bmod\omega^1\wedge\omega^2,\,\omega^1\wedge\omega^3\wedge\omega^4
\end{array}$$deals with the two rightmost differentials. Consequently, at the $E_1$-level we obtain $$\begin{picture}(300,75)(0,5)
\put(165,45){\makebox(0,0){$\begin{array}{ccccccccccccccc}
\Lambda^0&\!&\lambda\oplus\xi&&0&&0&&0&&0&&0\\[4pt]
&\!&0&&0&&\lambda^2\xi&&0&&0&&0\\[4pt]
&\!&0&&0&&0&&\lambda\xi^3&&0&&0\\[4pt]
&\!&0&&0&&0&&0&&0&&\lambda^3\xi^3\oplus\lambda^2\xi^4&\!&\Lambda^4\,.
\end{array}$}}
\put(0,0){\vector(1,0){40}}
\put(0,0){\vector(0,1){45}}
\put(30,5){$p$}
\put(3,35){$q$}
\end{picture}$$ The bundles $\lambda\oplus\xi$ and $\lambda^3\xi^3\oplus\lambda^2\xi^4$ may be identified with $\Lambda_H^1$ and $\Lambda_H^1\otimes\Lambda^2\!K$, respectively. The line bundles $\lambda^2\xi$ and $\lambda\xi^3$ combine to give a rank $2$ vector bundle $\lambda^2\xi+\lambda\xi^3$ but, in fact, this bundle canonically splits as can readily be seen in our adapted co-frame: $$\textstyle\lambda^2\xi+\lambda\xi^3=
\frac{{\mathrm{span}}\{\omega^2\wedge\omega^4,\omega^1\wedge\omega^2,
\omega^1\wedge\omega^4\}}
{{\mathrm{span}}\{\omega^1\wedge\omega^2,\omega^1\wedge\omega^4\}}\oplus
\frac{{\mathrm{span}}\{\omega^1\wedge\omega^3,\omega^1\wedge\omega^2\}}
{{\mathrm{span}}\{\omega^1\wedge\omega^2\}},$$ independent of choice of co-frame. We conclude that there is a complex of differential operators (cf. [@P]) $$\label{fullEngel}
\Lambda^0\xrightarrow{\,d_H\,}\Lambda_H^1\to \lambda^2\xi\oplus\lambda\xi^3
\to\Lambda_H^1\otimes\Lambda^2\!K\to\Lambda^4\to 0$$ resolving the locally constant sheaf ${\mathbb{R}}$. Following through the spectral sequence more explicitly as a diagram chase shows that $\Lambda_H^1\to\lambda^2\xi\oplus\lambda\xi^3$ is given by our previous recipe. We shall see later in §\[engelrevisited\], that (\[fullEngel\]) is a BGG complex for an appropriate parabolic geometry.
The Rumin complex revisited
===========================
Since a contact manifold with no extra structure is not a parabolic geometry, the Rumin complex lies outside the realm of parabolic geometry. Nevertheless, there is a parabolic geometry in which the Rumin complex finds its genesis. Let us denote by ${\mathrm{Sp}}(2n,{\mathbb{R}})$ the simple Lie group of linear automorphisms of ${\mathbb{R}}^{2n}$ preserving a fixed non-degenerate symplectic form. Viewing the $(2n+1)$-sphere $S^{2n+1}$ as $$\{x\in{\mathbb{R}}^{2n+2}\mbox{ s.t. }x\not=0\}/\{x\sim\lambda x
\mbox{ for }\lambda>0\}$$ (i.e. the space of rays emanating from the origin in ${\mathbb{R}}^{2n+2}$), the group $G={\mathrm{Sp}}(2n+2,{\mathbb{R}})$ acts smoothly and transitively on $S^{2n+1}$. The stabiliser subgroup $P$ of this action is parabolic. Parabolic geometries modelled on this particular homogeneous space $S^{2n+1}=G/P$ are known as [*contact projective*]{} [@CSl §4.2.6]. In any case, when viewed in this way, the sphere $S^{2n+1}$ inherits a $G$-invariant contact structure from the symplectic form on ${\mathbb{R}}^{2n+2}$. As in §\[rumin\], let us now consider the case $n=2$. Adopting the notation from [@BE], this homogeneous space is written as
(42,5) (20,.1)[(1,0)[16]{}]{} (20,3.3)[(1,0)[16]{}]{} (20,1.5)[(0,0)[$\bullet$]{}]{} (36,1.5)[(0,0)[$\bullet$]{}]{} (28,1.5)[(0,0)[$\langle$]{}]{} (4,1.5)[(0,0)[$\times$]{}]{} (4,1.5)[(1,0)[15]{}]{}
and the Bernstein-Gelfand-Gelfand complex corresponding to the trivial representation of ${\mathrm{Sp}}(6,{\mathbb{R}})$ is $$\begin{array}{l}0\to
\begin{picture}(42,5)
\put(20,.1){\line(1,0){16}}
\put(20,3.3){\line(1,0){16}}
\put(20,1.5){\makebox(0,0){$\bullet$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){15}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\to
\begin{picture}(42,5)
\put(20,.1){\line(1,0){16}}
\put(20,3.3){\line(1,0){16}}
\put(20,1.5){\makebox(0,0){$\bullet$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){15}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\to
\begin{picture}(42,5)
\put(20,.1){\line(1,0){16}}
\put(20,3.3){\line(1,0){16}}
\put(20,1.5){\makebox(0,0){$\bullet$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){15}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\end{picture}\to
\begin{picture}(42,5)
\put(20,.1){\line(1,0){16}}
\put(20,3.3){\line(1,0){16}}
\put(20,1.5){\makebox(0,0){$\bullet$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){15}}
\put(4,8){\makebox(0,0){$\scriptstyle -3$}}
\end{picture}\\[8pt]
\mbox{ }\hspace{100pt}\to
\begin{picture}(42,5)
\put(20,.1){\line(1,0){16}}
\put(20,3.3){\line(1,0){16}}
\put(20,1.5){\makebox(0,0){$\bullet$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){15}}
\put(4,8){\makebox(0,0){$\scriptstyle -5$}}
\end{picture}\to
\begin{picture}(42,5)
\put(20,.1){\line(1,0){16}}
\put(20,3.3){\line(1,0){16}}
\put(20,1.5){\makebox(0,0){$\bullet$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){15}}
\put(4,8){\makebox(0,0){$\scriptstyle -6$}}
\end{picture}\to
\begin{picture}(42,5)
\put(20,.1){\line(1,0){16}}
\put(20,3.3){\line(1,0){16}}
\put(20,1.5){\makebox(0,0){$\bullet$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){15}}
\put(4,8){\makebox(0,0){$\scriptstyle -6$}}
\end{picture}\to 0.\end{array}$$ This coincides with the Rumin complex (\[rumincomplex\]). The reason for the notation is fully explained in [@BE]. Here, suffice it to say that $$\begin{picture}(24,5)
\put(4,.1){\line(1,0){16}}
\put(4,3.3){\line(1,0){16}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(20,1.5){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle q$}}
\put(20,8){\makebox(0,0){$\scriptstyle r$}}
\end{picture}(\Lambda_H^1)\otimes L^s=
\begin{picture}(52,5)(-20,0)
\put(30,.1){\line(1,0){16}}
\put(30,3.3){\line(1,0){16}}
\put(30,1.5){\makebox(0,0){$\bullet$}}
\put(46,1.5){\makebox(0,0){$\bullet$}}
\put(38,1.5){\makebox(0,0){$\langle$}}
\put(30,8){\makebox(0,0){$\scriptstyle q$}}
\put(46,8){\makebox(0,0){$\scriptstyle r$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){25}}
\put(0,8){\makebox(0,0){$\scriptstyle -2s-2q-3r$}}
\end{picture}$$ where $\begin{picture}(24,5)
\put(4,.1){\line(1,0){16}}
\put(4,3.3){\line(1,0){16}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(20,1.5){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle q$}}
\put(20,8){\makebox(0,0){$\scriptstyle r$}}
\end{picture}(\Lambda_H^1)$ denotes the bundle induced by the irreducible representation $\begin{picture}(24,6)
\put(4,.1){\line(1,0){16}}
\put(4,3.3){\line(1,0){16}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(20,1.5){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,7){\makebox(0,0){$\scriptstyle q$}}
\put(20,7){\makebox(0,0){$\scriptstyle r$}}
\end{picture}$ of ${\mathrm{Sp}}(4,{\mathbb{R}})$ (meaning that its highest weight is $[q,r]$ with respect to the standard Bourbaki-ordered basis of fundamental weights).
In summary, there is a homogeneous contact geometry $G/P$, with $G$ simple and $P$ parabolic, for which the BGG complex coincides with the Rumin complex.
Pfaffian systems of rank three in five variables {#five}
================================================
Let $M$ be a $5$-manifold equipped with $H\subset TM$, a generic distribution of rank $2$. Equivalently, let $I\subset\Lambda^1$ be a Pfaffian system of rank $3$ that is generic in Cartan’s sense, i.e. the first derived system $I^\prime$ has rank $2$ and the second derived system $I^{\prime\prime}$ is zero. We have a filtration of the tangent bundle $$H\subset [H,H]\subset TM\quad\mbox{by vector bundles of ranks}\quad 2,3,5$$ and a dual filtration of the cotangent bundle, which we shall write as $$\label{filter}\Lambda^1=\Lambda_H^1+L+I^\prime,$$ where $L$ is the line-bundle $I/I^\prime$. There are locally defined co-framings $(\omega^1,\omega^2,\omega^3,\omega^4,\omega^5)$ so that the following congruences hold $$\label{G2structure}
\begin{array}{c}
d\omega^1\equiv\omega^3\wedge\omega^4\bmod\omega^1,\omega^2\qquad
d\omega^2\equiv\omega^3\wedge\omega^5\bmod\omega^1,\omega^2\\[6pt]
d\omega^3\equiv\omega^4\wedge\omega^5\bmod\omega^1,\omega^2,\omega^3
\end{array}$$ with $I^\prime={\mathrm{span}}\{\omega^1,\omega^2\}$ and $I={\mathrm{span}}\{\omega^1,\omega^2,\omega^3\}$. We shall refer to such co-framings as [*adapted*]{}. The Levi form for $I$, defined as the composition $I\to\Lambda^1\xrightarrow{\,d\,}\Lambda^2\to\Lambda_H^2$, has $I^\prime$ as its kernel (by definition of $I^\prime$ or by viewing this form in an adapted co-frame). Hence, the line-bundle $L$ may be canonically identified with $\Lambda_H^2$. Similarly, the Levi form for $I^\prime$, namely $I^\prime\to\Lambda^1\xrightarrow{\,d\,}\Lambda^2\to
\Lambda_H^2+\Lambda_H^1\otimes L$, canonically identifies $I^\prime$ with $\Lambda_H^1\otimes L$. Therefore, we may rewrite (\[filter\]) as $$\label{betterfilter}
\Lambda^1=\Lambda_H^1+\Lambda_H^2+\Lambda_H^1\otimes\Lambda_H^2.$$ To proceed, it is useful to have a more compact notation for the bundles induced by $\Lambda_H^1$. Following a common convention for the irreducible representations of ${\mathrm{GL}}(2,{\mathbb{C}})$, let us write $$\label{Clebsch}
(a,b)\in{\mathbb{Z}}^2\enskip\mbox{with $a\leq b$}\enskip
\mbox{for the bundle}
\enskip\textstyle\bigodot^{b-a}\!\Lambda_H^1\otimes(\Lambda_H^2)^a,$$ where $\bigodot$ means symmetric tensor product. Then (\[betterfilter\]) becomes $$\Lambda^1=(0,1)+(1,1)+(1,2)$$ and the induced filtration on $2$-forms is $$\Lambda^2=(1,1)+(1,2)+\begin{array}c(1,3)\\[-2pt] \oplus\\[-2pt] (2,2)
\end{array}+(2,3)+(3,3).$$ Without further ado, we may now consider the spectral sequence of the de Rham complex filtered in this way. At the $E_0$-level we obtain $$\begin{picture}(300,125)(0,5)
\put(151,70){\makebox(0,0){$\begin{array}{ccccccccccc}
\Lambda^0&\!(0,1)\!&\!(1,1)\!&0&0&0\\
&&\uparrow\\
&0&\!(1,1)\!&\!(1,2)\!&(2,2)&0&0\\
&&&\uparrow&\uparrow\\
&&0&\!(1,2)\!&\!(1,3)\!\oplus\!(2,2)\!&\!(2,3)\!&0&0\\
&&&&&\uparrow\\
&&&0&0&\!(2,3)\!&\!(2,4)\!\oplus\!(3,3)\!&\!(3,4)\!&0\\
&&&&&&\uparrow&\uparrow\\
&&&&0&0&(3,3)&\!(3,4)\!&\!(4,4)\!&0\\
&&&&&&&&\uparrow\\
&&&&&0&0&0&\!(4,4)\!&\!(4,5)\!&\!\Lambda^5\,.
\end{array}$}}
\put(0,0){\vector(1,0){40}}
\put(0,0){\vector(0,1){45}}
\put(30,5){$p$}
\put(3,35){$q$}
\end{picture}$$The $E_0$-level differentials are easily computed from the structure equations (\[G2structure\]), the $E_1$-level is $$\begin{picture}(300,80)(0,5)
\put(150,50){\makebox(0,0){$\begin{array}{ccccccccccccc}
\Lambda^0&\!\to\!&\!(0,1)\!&\;0\;&\;0\;&\;0\;&\;0\\[2pt]
&&0&0&0&0&0&0\\[2pt]
&&&0&0&\!(1,3)\!&0&0&0\\[2pt]
&&&&0&0&0&\!(2,4)\!&0&0\\[2pt]
&&&&&0&0&0&0&0&0\\[2pt]
&&&&&&\;0\;&\;0\;&\;0\;&\;0\;&\!(4,5)\!&\!\to\!&\!(5,5)\,,
\end{array}$}}
\put(0,0){\vector(1,0){40}}
\put(0,0){\vector(0,1){45}}
\put(30,5){$p$}
\put(3,35){$q$}
\end{picture}$$and we have shown that there is a differential complex $$\Lambda^0\xrightarrow{\,\nabla\,}\Lambda_H^1
\xrightarrow{\,\nabla^{3}\,}(1,3)\xrightarrow{\,\nabla^{2}\,}(2,4)
\xrightarrow{\,\nabla^{3}\,}(4,5)\xrightarrow{\,\nabla\,}(5,5)$$ resolving the constant sheaf ${\mathbb{R}}$, where $\nabla^k$ simply indicates a differential operator of order $k$. If necessary, the structure equations (\[G2structure\]) can be used to compute the operators precisely. To compare with the usual BGG complex, we follow Cartan [@Ca] in realising the flat model for this geometry as a homogeneous space $G/P$ where $G$ is the exceptional non-compact Lie group $G_2$ and $P$ is a parabolic subgroup. Specifically, following the notation of [@BE], the homogeneous space is $\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\end{picture}$. The Levi factor of the parabolic subgroup is ${\mathrm{GL}}(2,{\mathbb{R}})$ but it is useful to identify its Lie algebra with ${\mathfrak{g}}_0$ where we have graded the Lie algebra of $G_2$ $${\mathfrak{g}}={\mathfrak{g}}_{-3}\oplus
{\mathfrak{g}}_{-2}\oplus{\mathfrak{g}}_{-1}\oplus
\underbrace{{\mathfrak{g}}_0\oplus{\mathfrak{g}}_1
\oplus{\mathfrak{g}}_2\oplus{\mathfrak{g}}_3}_{\mathfrak{p}}$$ in accordance with the parabolic subalgebra ${\mathfrak{p}}$ (see [@CSl]). With these conventions, the cotangent bundle is $$\label{cotangent}\Lambda^1=
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}+
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -1$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}+
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -3$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}$$ and the BGG complex is $$\label{G2BGG}\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\xrightarrow{\,\nabla\,}
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\xrightarrow{\,\nabla^3\,}
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -5$}}
\put(20,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}\xrightarrow{\,\nabla^2\,}
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -6$}}
\put(20,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}\xrightarrow{\,\nabla^3\,}
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -6$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\xrightarrow{\,\nabla\,}
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -5$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}.$$ More generally, in Dynkin diagram notation the bundle $(a,b)$ is written as $\begin{picture}(34,12)
\put(5.6,0){\line(1,0){24.4}}
\put(4,1.6){\line(1,0){26}}
\put(5.6,3.2){\line(1,0){24.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(30,1.4){\makebox(0,0){$\bullet$}}
\put(17,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle a-2b$}}
\put(30,8){\makebox(0,0){$\scriptstyle b-a$}}
\end{picture}$.
In fact, there are several other complexes that can be created from the de Rham complex by choosing to carry out only some of the diagram chasing involved in creating the BGG complex. We now explain two of these complexes and their motivation. Keeping the Dynkin diagram notation, the filtration of the $2$-forms induced from (\[cotangent\]) is $$\Lambda^2=
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -1$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}+
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -3$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}+
\begin{array}c\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -5$}}
\put(20,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}\\[0pt] \oplus\\[0pt]
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\end{array}+
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -4$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}+
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -3$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture},$$ which suggests that one might cancel $\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -1$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}+
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -3$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}$ from the exterior derivative $\Lambda^1\xrightarrow{\,d\,}\Lambda^2$. But, as a sub-bundle of $\Lambda^1$, this is precisely the original Pfaffian system $I$. So, we are trying to cancel from $\Lambda^1\to\Lambda^2$, the homomorphism defined as the composition $$I\hookrightarrow\Lambda^1\xrightarrow{\,d\,}\Lambda^2\to
\frac{\Lambda^2}{\Lambda^1\wedge I^\prime}.$$ We can accomplish this explicitly by means of an adapted co-frame. Specifically, we define a differential operator $$\label{euler}{\mathcal{E}}:\Lambda_H^1=\Lambda^1/I\to
\Lambda^1\wedge I^\prime\subset\Lambda^2$$ by the following steps. Starting with any $1$-form $\omega$,
- define $f$ by $d\omega \equiv f\,\omega^4\wedge\omega^5\bmod \omega^1,\omega^2,\omega^3$,
- define $g,h$ by $d(\omega-f\omega^3)\equiv g\,\omega^3\wedge\omega^4+h\,\omega^3\wedge\omega^5
\bmod \omega^1,\omega^2$.
This is possible according to the structure equations (\[G2structure\]), which also imply that $$d(\omega-f\omega^3-g\omega^1-h\omega^2)\equiv 0\bmod\omega^1,\omega^2,$$ in other words that $${\mathcal{E}}\omega\equiv d(\omega-f\omega^3-g\omega^1-h\omega^2)
\in\Lambda^1\wedge I^\prime\subset\Lambda^2.$$ One checks easily that this definition of ${\mathcal{E}}\omega$ is independent of choice of adapted co-framing. Moreover, if $\omega$ is actually a section of $I$, say $\omega=F\omega^3+G\omega^1+H\omega^2$, then $f=F$, $g=G$, and $h=H$, whence ${\mathcal{E}}\omega=0$. In other words, the differential operator ${\mathcal{E}}$ descends to $\Lambda_H^1$, as claimed in (\[euler\]).
\[thm1\] The sequence $$0\to{\mathbb{R}}\to\Lambda^0\xrightarrow{\,d_H\,}\Lambda_H^1
\xrightarrow{\,{\mathcal{E}}\,}\Lambda^1\wedge I^\prime\xrightarrow{\,d\,}
\Lambda^3\xrightarrow{\,d\,}\Lambda^4\xrightarrow{\,d\,}\Lambda^5\to 0$$ is a locally exact complex.
This is just a matter of unravelling definitions, bearing in mind that the de Rham complex is itself locally exact. Suppose, for example, that $\omega$ is a $1$-form representing a section of $\Lambda_H^1$ that is annihilated by ${\mathcal{E}}$. Locally, we need to find a smooth function $\phi$ such that $\omega-d\phi$ is a section of $I$. By construction of ${\mathcal{E}}$ we know $d(\omega-f\omega^3-g\omega^1-h\omega^2)=0$ for some smooth functions $f,g,h$. Thus, by exactness of the de Rham complex, locally we can write $\omega-f\omega^3-g\omega^1-h\omega^2 =d\phi$ and then $\omega-d\phi=f\omega^3+g\omega^1+h\omega^2$ is a section of $I$, as required. The remaining verifications are similarly straightforward.
The operator ${\mathcal{E}}:\Lambda_H^1\to\Lambda^1\wedge I^\prime$ has a geometric meaning: a section $\phi$ of $\Lambda_H^1$ can be regarded as a Lagrangian for an integral curve of $I$. [From]{} this point of view ${\mathcal{E}}\phi$ are the Euler-Lagrange equations associated to this Lagrangian. [From]{} its construction, one can easily verify that ${\mathcal{E}}$ is third order. More specifically, by construction, its symbol $$\textstyle\bigodot^3\!\Lambda^1\otimes\Lambda_H^1\to
\Lambda^1\wedge I^\prime
=\begin{array}c\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -5$}}
\put(20,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}\\[0pt] \oplus\\[0pt]
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\end{array}+
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -4$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}+
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -3$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}$$ composes with the projection to $\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -5$}}
\put(20,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}$ as the homomorphism $$\begin{array}{l}\bigodot^3\!\Lambda^1\otimes\Lambda_H^1=
\textstyle\bigodot^3\!\big(\,\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}+\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -1$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}+\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -3$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\big)\otimes
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\\[5pt]
\qquad\to\bigodot^3\!\big(\,\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\big)\otimes
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\to
\bigodot^2\!\big(\,\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\big)\otimes
\Lambda^2\big(\,\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\big)\\[5pt]
\mbox{ }\hspace{145pt}{}=\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -4$}}
\put(20,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}\otimes
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -1$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}=
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -5$}}
\put(20,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}.\end{array}$$ Furthermore, not only does the symbol have no component in $\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}$ but, in fact, the range of the operator ${\mathcal{E}}$ is entirely contained in the sub-bundle where this component vanishes. This is easily seen in an adapted co-frame: since $$d(\omega^1\wedge\omega^5-\omega^2\wedge\omega^4)=
2\,\omega^3\wedge\omega^4\wedge\omega^5\bmod\omega^1,\omega^2$$ and since $\omega^1\wedge\omega^5-\omega^2\wedge\omega^4$ spans $\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,7){\makebox(0,0){$\scriptstyle -2$}}
\put(20,7){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}$, any exact $2$-form in $\Lambda^1\wedge I^\prime$ has vanishing component in $\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,7){\makebox(0,0){$\scriptstyle -2$}}
\put(20,7){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}$. If we denote by $B^2$ the rank $6$ sub-bundle of $\Lambda^1\wedge I^\prime$ defined as the kernel of the natural projection $\Lambda^1\wedge I^\prime\to
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,7){\makebox(0,0){$\scriptstyle -2$}}
\put(20,7){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}$, and by $B^3$ the rank $9$ sub-bundle of $\Lambda^3$ generated by $\omega^1,\omega^2$, then we have cancelled $\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,7){\makebox(0,0){$\scriptstyle -2$}}
\put(20,7){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}$ from the complex of Theorem \[thm1\] and demonstrated the following improvement.
The sequence $$\label{basiccomplex}
0\to{\mathbb{R}}\to\Lambda^0\xrightarrow{\,d_H\,}\Lambda_H^1
\xrightarrow{\,{\mathcal{E}}\,}B^2\xrightarrow{\,d\,}
B^3\xrightarrow{\,d\,}\Lambda^4\xrightarrow{\,d\,}\Lambda^5\to 0$$ is a locally exact complex.
The ranks of the bundles and the orders of the differential operators in (\[basiccomplex\]) are $$1\xrightarrow{\,\nabla\,}2\xrightarrow{\,\nabla^3\,}6
\xrightarrow{\,\nabla\,}9\xrightarrow{\,\nabla\,}5\xrightarrow{\,\nabla\,}1$$ but if we consider $d:B^2\to B^3$ in more detail $$B^2=\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -5$}}
\put(20,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}+\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -4$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}+\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -3$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\rightarrow
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -4$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}+\!\!
\begin{array}c\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -6$}}
\put(20,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}\\[0pt] \oplus\\[0pt]
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -3$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\end{array}
\!\!+\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -5$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}+
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -4$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}=B^3,$$ then it suggests that we should be able to eliminate $\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -4$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}+\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -3$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}$ from both bundles. This is, indeed, the case as can be seen in an adapted co-frame: writing a general section of $B^2$ as $$\omega=\mu\wedge\omega^1+\nu\wedge\omega^2\mbox{ s.t.\ }
d\omega\equiv 0\bmod\omega^1,\omega^2$$ we may define a differential operator ${\mathcal{F}}:B^2\to B^3$ by the following familiar steps.
--------------- -------------------------------------------------------------------------------------
$\bullet\!\!$ Define $f,g$ by $d\omega\equiv\begin{array}[t]{l}
f\,\omega^1\wedge\omega^4\wedge\omega^5+
g\,\omega^2\wedge\omega^4\wedge\omega^5\\
\quad{}\bmod\omega^1\wedge\omega^2,\omega^1\wedge\omega^3,\omega^2\wedge\omega^3,
\end{array}$
$\bullet\!\!$ Define $h$ by $d(\omega-f\,\omega^1\wedge\omega^3-g\,\omega^2\wedge\omega^3)\equiv$
$\begin{array}[t]{l}
h\,
(\omega^2\wedge\omega^3\wedge\omega^4-\omega^1\wedge\omega^3\wedge\omega^5)\\
{}+p\,\omega^1\wedge\omega^3\wedge\omega^4
+q\,\omega^2\wedge\omega^3\wedge\omega^5\\
\enskip{}+r\,
(\omega^2\wedge\omega^3\wedge\omega^4+\omega^1\wedge\omega^3\wedge\omega^5)\\
\quad{}\bmod\omega^1\wedge\omega^2.\end{array}$
--------------- -------------------------------------------------------------------------------------
This is possible according to the structure equations (\[G2structure\]), which also imply that $${\mathcal{F}}\omega\equiv
d(\omega-f\,\omega^1\wedge\omega^3-g\,\omega^2\wedge\omega^3
-h\,\omega^1\wedge\omega^2)$$ lies in the sub-bundle $$C^3\equiv\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -6$}}
\put(20,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}+\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -5$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}+\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -4$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}$$ of $B^3\subset\Lambda^3$ and that it descends to the quotient $$B^2=\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -5$}}
\put(20,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}+\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -4$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}+\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -3$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\twoheadrightarrow
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
\put(5.6,3.2){\line(1,0){14.4}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -5$}}
\put(20,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}\equiv C^2$$ of $B^2$. It is easily verified that this definition of ${\mathcal{F}}$ is independent of choice of co-framing and that, if we denote by $\bar{\mathcal{E}}$ the composition $$\Lambda_H^1\xrightarrow{\,{\mathcal{E}}\,}B^2\to C^2,$$ then the expected theorem follows:
The sequence $$\label{secondcomplex}
0\to{\mathbb{R}}\to\Lambda^0\xrightarrow{\,d_H\,}\Lambda_H^1
\xrightarrow{\,\bar{\mathcal{E}}\,}C^2\xrightarrow{\,{\mathcal{F}}\,}
C^3\xrightarrow{\,d\,}\Lambda^4\xrightarrow{\,d\,}\Lambda^5\to 0$$ is a locally exact complex.
The ranks of the bundles and the orders of the differential operators in (\[secondcomplex\]) are $$1\xrightarrow{\,\nabla\,}2\xrightarrow{\,\nabla^3\,}3
\xrightarrow{\,\nabla^3\,}6\xrightarrow{\,\nabla\,}5
\xrightarrow{\,\nabla\,}1.$$ Writing (\[secondcomplex\]) as $$\begin{array}{l}\begin{picture}(24,5)
\put(5.6,0){\line(1,0){14.4}}
\put(4,1.6){\line(1,0){16}}
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\end{picture}\to\\[5pt]
\mbox{ }\hspace{152pt}\begin{picture}(24,5)
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\put(4,8){\makebox(0,0){$\scriptstyle -5$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture},\end{array}$$ suggests one final cancellation, specifically of $\begin{picture}(24,5)
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\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}$ from $C^3$ and $\Lambda^4$. The reader can readily verify that this gives the BGG complex (\[G2BGG\]). It is interesting to note that the ranks of the bundles and orders of differential operators in the BGG complex are $$1\xrightarrow{\,\nabla\,}2\xrightarrow{\,\nabla^3\,}3
\xrightarrow{\,\nabla^2\,}3\xrightarrow{\,\nabla^3\,}2
\xrightarrow{\,\nabla\,}1.$$ In particular, the order of the differential operator in the middle has gone down from $3$ to $2$. Since our filtering on the de Rham complex is, by construction, compatible with the tautological Hodge isomorphisms $\Lambda^p=\Lambda^5\otimes(\Lambda^{5-p})^*$, and since we have run the spectral sequence to its end, it follows that the BGG complex is formally self-adjoint.
Pfaffian systems of rank three in six variables {#threeinsix}
===============================================
Let $M$ be a $6$-manifold equipped with $H\subset TM$, a generic distribution of rank $3$. Equivalently, let $I\subset\Lambda^1$ be a Pfaffian system of rank $3$ that is generic in Cartan’s sense, i.e. the first derived system $I^\prime$ is zero. Locally there are co-framings $(\omega^1,\omega^2,\omega^3,\omega^4,\omega^5,\omega^6)$ so that $\omega^1,\omega^2,\omega^3$ span $I$ and the following congruences hold. $$\label{Spin7structure}
\begin{array}{c}
d\omega^1\equiv\omega^5\wedge\omega^6\bmod\omega^1,\omega^2,\omega^3\quad
d\omega^2\equiv\omega^6\wedge\omega^4\bmod\omega^1,\omega^2,\omega^3\\[3pt]
d\omega^3\equiv\omega^4\wedge\omega^5\bmod\omega^1,\omega^2,\omega^3.
\end{array}$$
In the terminology of [@B], these co-framings are [*$1$-adapted*]{}. As usual, let us write $\Lambda_H^1$ for $\Lambda^1/I$. Then the Levi form ${\mathcal{L}}:I\to\Lambda_H^2$ defined as the composition $I\hookrightarrow\Lambda^1\xrightarrow{\,d\,}
\Lambda^2\twoheadrightarrow\Lambda_H^2$ is an isomorphism and we can canonically identify $I$ with $\Lambda_H^2$ as vector bundles. Indeed, this isomorphism is apparent in our $1$-adapted co-framing (\[Spin7structure\]). We may mimic (\[Clebsch\]) to write, up to isomorphism, the general Schur-irreducible bundle induced by $\Lambda_H^1$ as $(a,b,c)\in{\mathbb{Z}}^3$ with $a\leq b\leq c$ for the bundle $$\textstyle
(\bigodot^{c-b}\!\Lambda_H^1\otimes\bigodot^{b-a}(\Lambda_H^1)^*)_\circ
\otimes(\Lambda_H^3)^b,$$ where $\circ$ as a subscript means to take the trace-free part. These observations mean that we may write the filtration $$\Lambda^1=\Lambda_H^1+I\quad\mbox{as}\quad\Lambda^1=(0,0,1)+(0,1,1)$$ and decompose the induced filtrations on the higher forms as $$\label{higherforms}\begin{array}{l}
\Lambda^2=
(0,1,1)+\begin{array}c(0,1,2)\\[-2pt] \oplus\\[-2pt] (1,1,1)\end{array}
+(1,1,2)\\[20pt]
\Lambda^3=
(1,1,1)+\begin{array}c(0,2,2)\\[-2pt] \oplus\\[-2pt] (1,1,2)\end{array}+
\begin{array}c(1,1,3)\\[-2pt] \oplus\\[-2pt] (1,2,2)\end{array}+(2,2,2)\\[20pt]
\Lambda^4=
(1,2,2)+\begin{array}c(1,2,3)\\[-2pt] \oplus\\[-2pt] (2,2,2)\end{array}+
(2,2,3)\enskip\quad
\Lambda^5=(2,2,3)+(2,3,3)\\[12pt]
\mbox{ }\hspace{140pt}\Lambda^6=(3,3,3).
\end{array}$$ [From]{} the structure equations (\[Spin7structure\]) for a $1$-adapted co-framing it is easily verified that all expected cancellations at the $E_0$-level of the associated spectral sequence actually take place and we have proved the following result.
\[preBGG\] There is a canonically defined locally exact differential complex $$\begin{array}{r}
0\to{\mathbb{R}}\to(0,0,0)\to(0,0,1)\to(0,1,2)\to
\big[(0,2,2)+(1,1,3)\big]\to\quad\\
(1,2,3)\to(2,3,3)\to(3,3,3)\to 0
\end{array}$$ on any smooth $6$-manifold equipped with a generic $3$-distribution.
For the moment, the bundle $\big[(0,2,2)+(1,1,3)\big]$ is a canonically defined sub-quotient of $\Lambda^3$ but, in fact, one can improve matters as the following theorem shows (the analogous step was not necessary in §\[five\]).
\[preferred\] A splitting of the short exact sequence $$0\to(0,1,1)\to\Lambda^1\to(0,0,1)\to 0$$ gives rise to a homomorphism of vector bundles defined as the composition $$\label{hom}
(1,1,1)\to\Lambda^2\xrightarrow{\,d\,}\Lambda^3\to(0,2,2)$$ and there is a preferred class of splittings characterised by requiring that this induced homomorphism vanish. This preference canonically splits the bundle $\big[(0,2,2)+(1,1,3)\big]$.
Certainly, a splitting of the $1$-forms splits all the other forms and so, from (\[higherforms\]), one may consider the composition (\[hom\]) obtained by splitting the $2$-forms and $3$-forms. To see that it is a homomorphism, rather than the differential operator it might appear to be, notice that if $\Omega$ is a $2$-form in $(1,1,1)$, then $$\label{Leibniz}d(f\Omega)=fd\Omega+df\wedge\Omega$$ and it is clear that $df\wedge\Omega$ has components only in $$\Lambda^1\otimes (1,1,1)=\big((0,0,1)\oplus(0,1,1)\big)\otimes(1,1,1)
=(1,1,2)\oplus(1,2,2)$$ inside $$\Lambda^3=(1,1,1)\oplus
\begin{array}c(0,2,2)\\[-2pt] \oplus\\[-2pt] (1,1,2)\end{array}
\oplus\begin{array}c(1,1,3)\\[-2pt] \oplus\\[-2pt] (1,2,2)\end{array}
\oplus(2,2,2).$$ In particular, if we project to $(0,2,2)$ as in (\[hom\]), then $df\wedge\Omega$ does not contribute and, from (\[Leibniz\]), the result is linear over the functions. Now suppose we change the splitting of the $1$-forms. The freedom in doing so lies in $$\label{basicfreedom}
{\mathrm{Hom}}(\big(0,0,1),(0,1,1)\big)=(-1,1,1)\oplus(0,0,1).$$ This same freedom shows up in splitting the first part of $\Lambda^3$: $$\Lambda^3=(1,1,1)+
\begin{array}c(0,2,2)\\[-2pt] \oplus\\[-2pt] (1,1,2)\end{array}+\cdots{}$$ and $${\mathrm{Hom}}\!\left(\!(1,1,1),
\begin{array}c(0,2,2)\\[-2pt] \oplus\\[-2pt] (1,1,2)\end{array}\right)=
\begin{array}c(-1,1,1)\\[-2pt] \oplus\\[-2pt] (0,0,1)\end{array}.$$ Bearing in mind that the composition $(1,1,1)\to\Lambda^2\to\Lambda^3\to(1,1,1)$ is an isomorphism, independent of choice of splitting (it is responsible for one of the cancellations occurring at the $E_0$-level of the spectral sequence), we conclude that we can spend the $(-1,1,1)$-freedom in splitting precisely in setting the homomorphism (\[hom\]) to zero. Now let us consider how this impacts on the sub-quotient $\big[(0,2,2)+(1,1,3)\big]$ of $\Lambda^3$. The freedom in splitting this sub-quotient lies in $$\label{morefreedom}
{\mathrm{Hom}}\big((0,2,2),(1,1,3)\big)=(-1,-1,3)\oplus(-1,0,2)\oplus(-1,1,1)$$ and one sees that the only way that (\[basicfreedom\]) can enter is through $(-1,1,1)$. Having eliminated this freedom by a preferred choice of splittings for $\Lambda^1$, it is thereby eliminated from (\[morefreedom\]) and we have obtained our canonical splitting.
The preferred splittings of $\Lambda^1$ afforded by Theorem \[preferred\] can be conveniently expressed in terms of our $1$-adapted co-framings satisfying (\[Spin7structure\]). If such a co-framing is used to split $\Lambda^1$, then the resulting sub-bundle $(1,1,1)$ of $\Lambda^2$ is spanned by $\Omega\equiv\omega^1\wedge\omega^4+\omega^2\wedge\omega^5
+\omega^3\wedge\omega^6$ and, following through its proof, the preferred splittings of Theorem \[preferred\] are characterised by requiring that $$d\Omega\equiv3\,\omega^4\wedge\omega^5\wedge\omega^6+
\omega\wedge\Omega{\scriptstyle\bmod\omega^1\wedge\omega^2,
\omega^2\wedge\omega^3,\omega^3\wedge\omega^1},
\enskip\mbox{ for some $1$-form }\omega.$$ In the terminology of [@B], co-framings satisfying this extra congruence are called [*$2$-adapted*]{}.
The Lie algebra ${\mathfrak{so}}(4,3)$ admits a grading of the form $$\label{grading}
\begin{array}{l}{\mathfrak{g}}_{-2}\oplus{\mathfrak{g}}_{-1}\oplus
\underbrace{{\mathfrak{g}}_0
\oplus{\mathfrak{g}}_1\oplus{\mathfrak{g}}_2}_{\mathfrak{p}}=\\
\enskip\begin{picture}(42,5)
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\end{picture}\oplus\!
\begin{array}{c}
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\end{picture}\\[0pt]
\oplus\\[2pt]
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\end{picture}
\end{array}\!\oplus
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\put(20,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\bullet$}}
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\end{picture}\oplus
\begin{picture}(42,5)
\put(20,.1){\line(1,0){14.3}}
\put(20,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\bullet$}}
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\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle -2$}}
\end{picture}\end{array}$$ and one can see from this grading that the corresponding $6$-dimensional homogeneous space $G/P$ is equipped with a canonical $3$-dimensional distribution. The corresponding infinitesimal flag structure [@CSl §3.1.6] is exactly the geometry of such $3$-distributions and the irreducible bundles are related in our two notations by $$(a,b,c)=\enskip
\begin{picture}(54,5)
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\put(26,3.2){\line(1,0){20.3}}
\put(26,1.5){\makebox(0,0){$\bullet$}}
\put(48,1.6){\makebox(0,0){$\times$}}
\put(37,1.5){\makebox(0,0){$\rangle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){21}}
\put(4,8){\makebox(0,0){$\scriptstyle b-a$}}
\put(26,8){\makebox(0,0){$\scriptstyle c-b$}}
\put(48,8){\makebox(0,0){$\scriptstyle -2c$}}
\end{picture}.$$ Recall in the proof of Theorem \[preferred\] that we reduced the freedom in splitting $\Lambda^1$ to $(0,0,1)$ in (\[basicfreedom\]). In the Dynkin diagram notation this remaining freedom lies in
(42,5) (20,.1)[(1,0)[14.3]{}]{} (20,3.2)[(1,0)[14.3]{}]{} (20,1.5)[(0,0)[$\bullet$]{}]{} (36,1.6)[(0,0)[$\times$]{}]{} (28,1.5)[(0,0)[$\rangle$]{}]{} (4,1.5)[(0,0)[$\bullet$]{}]{} (4,1.5)[(1,0)[15]{}]{} (4,8)[(0,0)[$\scriptstyle 0$]{}]{} (20,8)[(0,0)[$\scriptstyle 1$]{}]{} (36,8)[(0,0)[$\scriptstyle -2$]{}]{}
, which is exactly the action of ${\mathfrak{g}}_1$ on $${\mathfrak{g}}/{\mathfrak{p}}={\mathfrak{g}}_{-2}\oplus{\mathfrak{g}}_{-1}=
\begin{picture}(42,5)
\put(20,.1){\line(1,0){14.3}}
\put(20,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\bullet$}}
\put(36,1.6){\makebox(0,0){$\times$}}
\put(28,1.5){\makebox(0,0){$\rangle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){15}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\oplus
\begin{picture}(42,5)
\put(20,.1){\line(1,0){14.3}}
\put(20,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\bullet$}}
\put(36,1.6){\makebox(0,0){$\times$}}
\put(28,1.5){\makebox(0,0){$\rangle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){15}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture},$$ as can be seen in (\[grading\]). The geometric import of this observation is that Theorem \[preferred\] reduces the structure group of the tangent bundle from general $H$-preserving and Levi-form-preserving automorphisms to the subgroup of ${\mathrm{Aut}}({\mathfrak{g}}/{\mathfrak{p}})$ defined by the Adjoint action of $P$, namely the group $P/\exp({\mathfrak{g}}_2)$ with Lie algebra ${\mathfrak{g}}_0\oplus{\mathfrak{g}}_1$. Dually, the $2$-adapted co-framings are preserved by exactly this group.
Finally, we can take the complex of Theorem \[preBGG\], use the splitting of $\big[(0,2,2)+(1,1,3)\big]$ afforded by Theorem \[preferred\], and write the result in Dynkin diagram notation to obtain the following.
On any smooth $6$-manifold equipped with a generic $3$-distribution, there is a canonically defined locally exact differential complex $$\begin{array}{r}0\to{\mathbb{R}}\to\!
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\!\to\!
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\!\to\!
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\begin{array}{c}\nearrow\\ \searrow\end{array}\!\!
\begin{array}{c}
\begin{picture}(42,5)
\put(20,.1){\line(1,0){14.3}}
\put(20,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\bullet$}}
\put(36,1.6){\makebox(0,0){$\times$}}
\put(28,1.5){\makebox(0,0){$\rangle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){15}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle -4$}}
\end{picture}\\[2pt]
\oplus\\[4pt]
\begin{picture}(42,5)
\put(20,.1){\line(1,0){14.3}}
\put(20,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\bullet$}}
\put(36,1.6){\makebox(0,0){$\times$}}
\put(28,1.5){\makebox(0,0){$\rangle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){15}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle 2$}}
\put(36,8){\makebox(0,0){$\scriptstyle -6$}}
\end{picture}
\end{array}
\!\!\begin{array}{c}\searrow\\ \nearrow\end{array}\qquad\\[20pt]
\begin{picture}(42,5)
\put(20,.1){\line(1,0){14.3}}
\put(20,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\bullet$}}
\put(36,1.6){\makebox(0,0){$\times$}}
\put(28,1.5){\makebox(0,0){$\rangle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){15}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\put(36,8){\makebox(0,0){$\scriptstyle -6$}}
\end{picture}
\!\to\!
\begin{picture}(42,5)
\put(20,.1){\line(1,0){14.3}}
\put(20,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\bullet$}}
\put(36,1.6){\makebox(0,0){$\times$}}
\put(28,1.5){\makebox(0,0){$\rangle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){15}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle -6$}}
\end{picture}
\!\to\!
\begin{picture}(42,5)
\put(20,.1){\line(1,0){14.3}}
\put(20,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\bullet$}}
\put(36,1.6){\makebox(0,0){$\times$}}
\put(28,1.5){\makebox(0,0){$\rangle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){15}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle -6$}}
\end{picture}
\!\to 0.\end{array}$$
This is the BGG complex in standard notation.
The Engel complex revisited {#engelrevisited}
===========================
Although the complex $\Lambda^0\xrightarrow{\,d_H\,}\Lambda_H^1
\xrightarrow{\,{\mathcal{P}}\,}\lambda\xi^2$ constructed in §\[engel\] used nothing beyond an Engel structure, for the full-blown resolution (\[fullEngel\]) it was necessary to choose some extra structure, namely a complement to $\xi\subset\Lambda_H^1$ (equivalently, a complement to $(\xi+K)^\perp\subset H$, the [*Engel line field*]{} [@M]). As pointed out to us by Boris Doubrov, there is a unique homogeneous space of the form $G/P$, for $G$ semisimple and $P$ parabolic, that carries a $G$-invariant Engel structure. Specifically, if $G={\mathrm{Sp}}(4,{\mathbb{R}})$ and $P$ is its Borel subgroup, then $${\mathfrak{g}}={\mathfrak{sp}}(4,{\mathbb{R}})={\mathfrak{g}}_{-3}\oplus
{\mathfrak{g}}_{-2}\oplus{\mathfrak{g}}_{-1}\oplus
\underbrace{{\mathfrak{g}}_0\oplus{\mathfrak{g}}_1
\oplus{\mathfrak{g}}_2\oplus{\mathfrak{g}}_3}_{\mathfrak{p}},$$ which, in Dynkin diagram notation, reads $$\begin{picture}(24,5)
\put(4,0){\line(1,0){16}}
\put(4,3.2){\line(1,0){16}}
\put(4,1.3){\makebox(0,0){$\bullet$}}
\put(20,1.4){\makebox(0,0){$\bullet$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}=
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\oplus
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\oplus\!\!
\begin{array}c\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -1$}}
\end{picture}\\ \oplus\\
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}\end{array}\!\!\oplus\!\!
\begin{array}c\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\\ \oplus\\
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\end{array}\!\!\oplus\!\!
\begin{array}c\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\end{picture}\\ \oplus\\
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\end{array}\!\!\oplus
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -1$}}
\end{picture}\oplus
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}.$$ The $1$-forms on this homogeneous space $G/P$ are filtered $$\Lambda^1=\begin{array}c\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\end{picture}\\ \oplus\\
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\end{array}\!\!+
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -1$}}
\end{picture}+
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}=
\begin{array}c\lambda\\ \oplus\\ \xi\end{array}\!\!+
\lambda\xi+\lambda\xi^2$$ and the corresponding regular infinitesimal flag structure is exactly that of an Engel manifold equipped with a choice of splitting $\Lambda_H^1=\lambda\oplus\xi$ as discussed in §\[engel\]. The BGG complex (\[fullEngel\]) in Dynkin diagram notation reads $$0\to{\mathbb{R}}\to\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\begin{array}{c}\nearrow\\ \searrow\end{array}\!\!
\begin{array}c\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\end{picture}\\[2pt] \oplus\\[4pt]
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\end{array}\!\!
\begin{array}c\longrightarrow\\ \mbox{\Large\begin{picture}(0,0)
\put(0,0){\makebox(0,0){$\nearrow$}}
\put(0,0){\makebox(0,0){$\searrow$}}
\end{picture}}\\[4pt] \longrightarrow\end{array}\!\!
\begin{array}c\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\end{picture}\\[2pt] \oplus\\[4pt]
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -4$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\end{array}\!\!
\begin{array}c\longrightarrow\\ \mbox{\Large\begin{picture}(0,0)
\put(0,0){\makebox(0,0){$\nearrow$}}
\put(0,0){\makebox(0,0){$\searrow$}}
\end{picture}}\\[4pt] \longrightarrow\end{array}\!\!
\begin{array}c\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\end{picture}\\[2pt] \oplus\\[4pt]
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -4$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\end{array}\!\!
\begin{array}{c}\searrow\\ \nearrow\end{array}
\begin{picture}(24,5)
\put(5.6,0){\line(1,0){12.8}}
\put(5.6,3.2){\line(1,0){12.8}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(12,1.5){\makebox(0,0){$\langle$}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\end{picture}\to 0.$$
Another geometry in five variables
==================================
Recall that an Engel manifold is a $4$-dimensional manifold equipped with a generic $2$-dimensional distribution. The geometry considered in §\[engel\] and §\[engelrevisited\] was defined on an Engel manifold by a choice of splitting of $\Lambda_H^1$, the bundle of $1$-forms along $H$ (rather than the filtration that is canonically present). The geometry to be considered in this section will very much resemble this case.
Let us consider a Pfaffian system $I\subset\Lambda^1$ of rank $2$ on a smooth $5$-manifold. As usual, we define the [*Levi form*]{} ${\mathcal{L}}$ as the composition $$I\to\Lambda^1\xrightarrow{\,d\,}\Lambda^2\to\Lambda_H^2,$$ where $H\equiv I^\perp$. Notice that $\Lambda_H^2$ has rank $3$ and we shall suppose that ${\mathcal{L}}$ is injective, as is generically the case. Under the canonical identification $\Lambda_H^2=\Lambda_H^3\otimes H$ we see that the rank $2$ sub-bundle ${\mathcal{L}}(I)\subset\Lambda_H^2$ gives rise to a rank $2$ sub-distribution $D\subset H$. In [@M2] it is observed that $D$ is the unique rank $2$ sub-bundle of $H$ such that $[D,D]\subseteq H$. It is not necessarily the case, however, that $[D,D]=H$ (in which case we would be back in the five variables geometry of §\[five\]). To proceed further, let us write $L$ for the line sub-bundle $D^\perp\subset\Lambda_H^1$ and choose a complementary rank $2$ sub-bundle $Q$ so that we now have a splitting $\Lambda_H^1=Q\oplus L$. This completes the definition of the structure to be considered in this section. Equivalently, we are considering a $5$-manifold $M$ equipped with a pair of transverse distributions $D$ and $\ell$ of ranks $2$ and $1$, respectively, such that $$\label{anothergeometry}
[D,D]\subseteq \ell\oplus D\qquad\mbox{and}\qquad[\ell\oplus D,\ell\oplus D]
=TM.$$ This is precisely the regular infinitesimal flag structure associated with the grading $$\begin{array}{l}
{\mathfrak{g}}_{-2}\oplus{\mathfrak{g}}_{-1}\oplus
\underbrace{{\mathfrak{g}}_0
\oplus{\mathfrak{g}}_1\oplus{\mathfrak{g}}_2}_{\mathfrak{p}}=\\
\quad\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\oplus
\begin{array}{c}
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\\[-2pt]
\oplus\\[1pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -1$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}
\end{array}\oplus
\begin{array}{c}
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\\[-2pt]
\oplus\\[1pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}\\[-2pt]
\oplus\\[1pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}
\end{array}\oplus
\begin{array}{c}
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\\[-2pt]
\oplus\\[1pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}
\end{array}\oplus
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\end{array}$$ of ${\mathfrak{sl}}(4,{\mathbb{R}})$. The bundles $$\textstyle\bigodot^c\!Q\otimes(\Lambda^2\!Q)^b\otimes L^a
\enskip\mbox{for}\enskip
c\in{\mathbb{Z}}_{\geq 0}\enskip\mbox{become}\quad
\begin{picture}(78,5)
\put(50,1.5){\makebox(0,0){$\times$}}
\put(74,1.5){\makebox(0,0){$\bullet$}}
\put(14,1.5){\makebox(0,0){$\times$}}
\put(14,1.5){\line(1,0){60}}
\put(14,8){\makebox(0,0){$\scriptstyle c+2b-2a$}}
\put(50,8){\makebox(0,0){$\scriptstyle a-3b-2c$}}
\put(74,8){\makebox(0,0){$\scriptstyle c$}}
\end{picture}$$ and the $1$-forms are filtered $$\label{oneforms}\Lambda^1=
\raisebox{2pt}{$\begin{array}{c}
Q\\[-2pt] \oplus\\[-2pt] L\end{array}$}+\,I=
\begin{array}{c}
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\\[-2pt]
\oplus\\[1pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}
\end{array}+
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture},$$ as expected. This induces filtrations on the higher forms as follows. $$\Lambda^2=
\begin{array}{c}
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\\[-2pt]
\oplus\\[1pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}
\end{array}+
\begin{array}{c}
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}\\[-2pt]
\oplus\\[1pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\\[-2pt]
\oplus\\[1pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -3$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}
\end{array}+
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}$$ and $$\begin{array}{r}
\Lambda^3=
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}+
\begin{array}{c}
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -4$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\\[-2pt]
\oplus\\[1pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\\[-2pt]
\oplus\\[1pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}
\end{array}+
\begin{array}{c}
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\\[-2pt] \oplus\\[1pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -4$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}
\end{array}\\
\Lambda^4=
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}+
\begin{array}{c}
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -4$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\\[-2pt]
\oplus\\[1pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -3$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}
\end{array}\qquad\qquad\qquad
\Lambda^5=\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}
\end{array}$$ One can readily verify using an adapted co-frame $\{\omega^1,\omega^2,\omega^3,\omega^4,\omega^5\}$ with $$\begin{array}{c}I={\mathrm{span}}\{\omega^1,\omega^2\},\quad
L+I={\mathrm{span}}\{\omega^1,\omega^2,\omega^3\},\\[4pt]
Q+I={\mathrm{span}}\{\omega^1,\omega^2,\omega^4,\omega^5\}
\end{array}$$ and such that $$\label{yetanothercoframe}
d\omega^1\equiv\omega^3\wedge\omega^4\bmod\omega^1,\omega^2
\quad\mbox{and}\quad
d\omega^2\equiv\omega^3\wedge\omega^5\bmod\omega^1,\omega^2,$$ that the expected cancellations in the $E_0$-level of the associated spectral sequence actually take place and we have found a differential complex as follows.
\[protoBGGcomplex\] On any $5$-dimensional manifold equipped with a geometric structure defined by transverse distributions $D$ and $\ell$ of ranks $2$ and $1$, respectively, and satisfying [(\[anothergeometry\])]{}, there is a canonically defined locally exact differential complex $$\begin{array}{r}0\to{\mathbb{R}}\to
\begin{array}{c}
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\\
\oplus\\[3pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}
\end{array}\!\to\!
\begin{array}{c}
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\\
+\\[3pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}\oplus
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -3$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}
\end{array}\!\!\to\hspace{77pt}\\[30pt]
\begin{array}{c}
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}\oplus
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -4$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\\
+\\[3pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -4$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}
\end{array}\!\!\to\!
\begin{array}{c}
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -4$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\\
\oplus\\[3pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -3$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}
\end{array}\!\to
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\to 0.\end{array}$$
As in §\[threeinsix\], one can make a further normalisation in order to split the two bundles that have arisen from the spectral sequence, or from the equivalent diagram chasing, only as filtered bundles. For the first of these we note that the freedom in its splitting lies in $${\mathrm{Hom}}\big(\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture},\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}\oplus\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -3$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\big)=
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}\oplus\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -1$}}
\put(20,8){\makebox(0,0){$\scriptstyle 3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}$$ whereas, from (\[oneforms\]), the freedom in splitting $\Lambda^1$ lies in $$\begin{array}{l}
{\mathrm{Hom}}\big(\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\oplus\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture},\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\big)\\[6pt]
\qquad{}=\big(\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -1$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\oplus\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\big)\otimes\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\\[6pt]
\qquad\qquad{}=\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}\oplus\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\oplus\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}.\end{array}$$ We see that only $\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}$ is common to both. Therefore, it is only this freedom that need be eliminated from the freedom to split $\Lambda^1$. In fact, once this freedom is eliminated, then Theorem \[protoBGGcomplex\] is improved as follows.
\[fullBGGcomplex\] On any $5$-dimensional manifold equipped with a regular infinitesimal flag structure defined by
(42,5) (20,1.5)[(0,0)[$\times$]{}]{} (36,1.5)[(0,0)[$\bullet$]{}]{} (4,1.5)[(0,0)[$\times$]{}]{} (4,1.5)[(1,0)[32]{}]{}
, there is a canonically defined locally exact differential complex $$\begin{array}{r}
0\to{\mathbb{R}}\to\!
\begin{array}{c}
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\\
\oplus\\[3pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}
\end{array}\!\to\!
\begin{array}{c}
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\\
\oplus\\[3pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}\\
\oplus\\[3pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -3$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}
\end{array}\!\to\!
\begin{array}{c}
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -4$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\\
\oplus\\[3pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}\\
\oplus\\[3pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -4$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}
\end{array}\!\to\!
\begin{array}{c}
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -4$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\\
\oplus\\[3pt]
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -3$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}
\end{array}\enskip\quad\\
{}\to\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\to 0.\end{array}$$
As already remarked, to complete the proof we should find a preferred class of splittings of the $1$-forms (\[oneforms\]) so that the $\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}$-freedom present in the general splitting is eliminated. As in §\[threeinsix\], this can be achieved by restricting a particular component of the exterior derivative $d:\Lambda^2\to\Lambda^3$ defined via an arbitrary splitting. In this case, we may consider the composition $$\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\to\Lambda^2\xrightarrow{\,d\,}\Lambda^3\to
\begin{picture}(42,5)
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\makebox(0,0){$\times$}}
\put(4,1.5){\line(1,0){32}}
\put(4,8){\makebox(0,0){$\scriptstyle -2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}.$$ Using an adapted co-framing (\[yetanothercoframe\]), one may readily verify that
- this is actually a homomorphism of vector bundles,
- insisting that it vanish reduces the freedom in splitting $\Lambda^1$\
exactly as desired,
- this also eliminates the freedom in splitting the filtered\
occurring bundles in Theorem \[protoBGGcomplex\],
which completes the proof.
The differential complex in Theorem \[fullBGGcomplex\] is our BGG complex for this parabolic geometry. Furthermore, one can easily check that in case $[D,D]=\ell\oplus D$, equivalently if the
(42,5) (20,1.5)[(0,0)[$\times$]{}]{} (36,1.5)[(0,0)[$\bullet$]{}]{} (4,1.5)[(0,0)[$\times$]{}]{} (4,1.5)[(1,0)[32]{}]{} (4,8)[(0,0)[$\scriptstyle 4$]{}]{} (20,8)[(0,0)[$\scriptstyle -4$]{}]{} (36,8)[(0,0)[$\scriptstyle 0$]{}]{}
-component of curvature does not vanish, then further cancellations may be effected and one reduces to the BGG complex for the five variables geometry previously discussed in §\[five\].
Pfaffian systems of rank three in seven variables {#threeinseven}
=================================================
Let $M$ be a $7$-manifold endowed with a generic distribution $H\subset TM$ of rank $4$. Equivalently, let $I\subset\Lambda^1$ be a Pfaffian system of rank $3$ that is generic in Cartan’s sense, meaning that its first derived system $I'$ is zero. We write the corresponding filtration of the cotangent bundle as $$\Lambda^1=\Lambda^1_H+I.$$ Genericity says that the Levi form, defined as the composition $$I\rightarrow\Lambda^1\rightarrow\Lambda^2\rightarrow\Lambda^2_H,$$ is injective.
It turns out that there exactly two types of generic rank $4$ distributions in dimension $7$, corresponding to the two open orbits of the action of ${\mathrm{GL}}(4,\mathbb R)\times{\mathrm{GL}}(3,\mathbb R)$ on the space of linear maps ${\mathrm{Hom}}(\Lambda^2\mathbb R^4,\mathbb R^3)$ called elliptic, respectively hyperbolic; see [@M2]. We shall treat these two cases simultaneously.
The Lie algebra $\mathfrak s\mathfrak p(6,\mathbb C)$ admits a grading of the form $$\begin{array}{l}{\mathfrak{g}}_{-2}\oplus{\mathfrak{g}}_{-1}\oplus
\underbrace{{\mathfrak{g}}_0
\oplus{\mathfrak{g}}_1\oplus{\mathfrak{g}}_2}_{\mathfrak{p}}=\\[-5pt]
\qquad\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\oplus
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\oplus
\begin{array}{c}
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -1$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\\[-1pt]
\oplus\\[2pt]
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}\\[-1pt]
\oplus\\[2pt]
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}
\end{array}\oplus
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\oplus
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}.\end{array}$$ There are two real forms of this grading, namely $\mathfrak s\mathfrak p(2,1)$ and the split real form $\mathfrak s\mathfrak p(6,\mathbb R)$. One can see that these gradings give rise to an elliptic generic rank $4$ distribution on the corresponding $7$-dimensional homogeneous space $G/P$ in the first case and to a hyperbolic generic rank $4$ distribution on the corresponding homogeneous space $G/P$ in the second case. The parabolic geometries based on these particular $G/P$ are known as [*quaternionic contact*]{} [@CSl §4.3.3] and [*split quaternionic contact*]{} [@CSl §4.3.4], respectively. Regular infinitesimal flag structures of these types correspond exactly to generic rank $4$ distributions on $7$-manifolds and the irreducible bundles of these geometries can be written as $$\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle a$}}
\put(20,8){\makebox(0,0){$\scriptstyle b$}}
\put(36,8){\makebox(0,0){$\scriptstyle c$}}
\end{picture},$$ where $a,c$ are non-negative integers. Accordingly, we can write the filtration of the cotangent bundle of a generic rank $4$ distribution as $$\Lambda^1=
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}+
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}$$ and the filtration of the higher forms as $$\Lambda^2=
\begin{array}{c}
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\\[-1pt]
\oplus\\[2pt]
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}
\end{array}
+
\begin{array}{c}
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 3$}}
\put(20,8){\makebox(0,0){$\scriptstyle -4$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\\[-1pt]
\oplus\\[2pt]
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}
\end{array}
+
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}$$ $$\Lambda^3 =
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}
+
\begin{array}{c}
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 4$}}
\put(20,8){\makebox(0,0){$\scriptstyle -4$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\\[-1pt]
\oplus\\[2pt]
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}
\\[-1pt]
\oplus\\[2pt]
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}
\\[-1pt]
\oplus\\[2pt]
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
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\oplus\\[2pt]
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\oplus\\[2pt]
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\oplus\\[2pt]
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\oplus\\[2pt]
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\oplus\\[2pt]
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+
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\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}$$ $$\Lambda^5 =
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\end{picture}
+
\begin{array}{c}
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\end{picture}\\[-1pt]
\oplus\\[2pt]
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\end{array}
+
\begin{array}{c}
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\end{picture}\\[-1pt]
\oplus\\[2pt]
\begin{picture}(42,5)
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\end{picture}
\end{array}$$ $$\Lambda^6=
\begin{picture}(42,5)
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\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
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\put(28,1.5){\makebox(0,0){$\langle$}}
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\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}
+
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
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\end{picture}
\quad\quad\quad\quad\quad
\Lambda^7=
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
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\put(4,1.5){\line(1,0){16}}
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\end{picture}$$ Choosing an adapted co-framing of the Pfaffian system, one can explicitly verify that all the expected cancellations at the $E_0$-level of the associated spectral sequence take place and, therefore, one obtains the following.
There is a canonically defined locally exact differential complex $$\begin{array}{l}
0\rightarrow{\mathbb{R}}\rightarrow
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\rightarrow\!\!
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\end{picture}\\
+\\[3pt]
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\end{array}
\!\!\rightarrow\!\!
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\end{picture}\\
\oplus\\[3pt]
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\!\!\rightarrow\!\!
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\end{picture}\\
\oplus\\[3pt]
\begin{picture}(42,5)
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\end{array}
\!\!\rightarrow\\
\qquad\begin{array}{c}
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\end{picture}\\
+\\[3pt]
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
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\end{array}
\!\!\rightarrow
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\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
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\put(28,1.5){\makebox(0,0){$\langle$}}
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\rightarrow
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\put(20,1.5){\makebox(0,0){$\times$}}
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\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
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\rightarrow 0\end{array}$$ on any smooth $7$-manifold equipped with a generic distribution of rank $4$.
The bundles $\begin{picture}(42,5)
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\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}
+
\begin{picture}(42,5)
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\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
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\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 3$}}
\put(20,8){\makebox(0,0){$\scriptstyle -4$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}$ and $\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
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\put(20,8){\makebox(0,0){$\scriptstyle -6$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}
+
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -6$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}$ are sub-quotients of $\Lambda^2$, respectively $\Lambda^5$. However, there is a preferred class of splittings of the filtration of $\Lambda^1$ that canonically splits these bundles as the following result shows.
\[anotherpreferredsplitting\] The splittings of the short exact sequence $$0\rightarrow \begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\rightarrow
\Lambda^1
\rightarrow
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}
\rightarrow 0$$ are acted freely upon by $${\mathrm{Hom}}\big( \begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture},\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\big)=
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 3$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\oplus \begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}.$$ There is a preferred class of splittings in which the
(42,5) (21.4,.1)[(1,0)[14.3]{}]{} (21.4,3.2)[(1,0)[14.3]{}]{} (20,1.5)[(0,0)[$\times$]{}]{} (36,1.6)[(0,0)[$\bullet$]{}]{} (28,1.5)[(0,0)[$\langle$]{}]{} (4,1.5)[(0,0)[$\bullet$]{}]{} (4,1.5)[(1,0)[16]{}]{} (4,8)[(0,0)[$\scriptstyle 3$]{}]{} (20,8)[(0,0)[$\scriptstyle -3$]{}]{} (36,8)[(0,0)[$\scriptstyle 1$]{}]{}
-freedom is eliminated. This restricted choice of splittings canonically splits the bundles
------------------------------------------------------------------------
$\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}
+
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 3$}}
\put(20,8){\makebox(0,0){$\scriptstyle -4$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}$ and $\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 3$}}
\put(20,8){\makebox(0,0){$\scriptstyle -6$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}
+
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -6$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}$.
The only difficulty is in restricting the class of splittings and, as usual, one looks to the exterior derivative $d:\Lambda^2\to\Lambda^3$ in the presence of a chosen splitting. More specifically, one checks (e.g., in an adapted co-frame) that, having chosen a splitting of $\Lambda^1$, the resulting component of $d:\Lambda^2\to\Lambda^3$ mapping $\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}$ to $\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 4$}}
\put(20,8){\makebox(0,0){$\scriptstyle -4$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}$ is actually a homomorphism. However, $$\label{torsionishere}
\begin{array}{rcc}{\mathrm{Hom}}\big(\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture},\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 4$}}
\put(20,8){\makebox(0,0){$\scriptstyle -4$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\big)&=&\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle 0$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\otimes\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 4$}}
\put(20,8){\makebox(0,0){$\scriptstyle -4$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\\[5pt]
&&\|\\[5pt]
&&\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 5$}}
\put(20,8){\makebox(0,0){$\scriptstyle -4$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\oplus\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 3$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\end{array}$$ and one checks (again using an adapted co-frame or by arguing with irreducible bundles and Schur’s lemma) that the original freedom in splitting $\Lambda^1$ can be used to eliminate the $\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 3$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}$-component. The remaining freedom in splitting $\Lambda^1$ is thereby restricted in the appropriate manner (with the stated knock-on effect on the previously identified subquotients of $\Lambda^2$ and $\Lambda^5$).
Of course, one could rephrase Theorem \[anotherpreferredsplitting\] as defining the notion of a $2$-adapted co-framing and observe that the effect is to reduce the structure bundle of $\Lambda^1$ to $P/\exp({\mathfrak{g}}_2)$. In any case, combining the two theorems above we immediately obtain the following improved complex.
On any smooth $7$-manifold endowed with a generic $4$-distribution, there is a canonically defined locally exact differential complex $$\begin{array}{l}
0\rightarrow{\mathbb{R}}\rightarrow
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -2$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}
\rightarrow\!\!
\begin{array}{c}
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -3$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}\\
\oplus\\[3pt]
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 3$}}
\put(20,8){\makebox(0,0){$\scriptstyle -4$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}
\end{array}
\!\!\rightarrow\!\!
\begin{array}{c}
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -5$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}\\
\oplus\\[3pt]
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 4$}}
\put(20,8){\makebox(0,0){$\scriptstyle -4$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}
\end{array}
\!\!\rightarrow\!\!
\begin{array}{c}
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 4$}}
\put(20,8){\makebox(0,0){$\scriptstyle -5$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}\\
\oplus\\[3pt]
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 2$}}
\put(20,8){\makebox(0,0){$\scriptstyle -6$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}
\end{array}
\!\!\rightarrow\\
\qquad\begin{array}{c}
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 3$}}
\put(20,8){\makebox(0,0){$\scriptstyle -6$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}\\
\oplus\\[3pt]
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -6$}}
\put(36,8){\makebox(0,0){$\scriptstyle 2$}}
\end{picture}
\end{array}
\!\!\rightarrow
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 1$}}
\put(20,8){\makebox(0,0){$\scriptstyle -6$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}
\rightarrow
\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 0$}}
\put(20,8){\makebox(0,0){$\scriptstyle -5$}}
\put(36,8){\makebox(0,0){$\scriptstyle 0$}}
\end{picture}
\rightarrow 0.\end{array}$$
Of course, on the homogeneous model $G/P$ this is the standard BGG complex. Finally, as mentioned already in the introduction, let us reiterate that our construction does not see the torsion of this parabolic geometry. In fact, the torsion lies in $\begin{picture}(42,5)
\put(21.4,.1){\line(1,0){14.3}}
\put(21.4,3.2){\line(1,0){14.3}}
\put(20,1.5){\makebox(0,0){$\times$}}
\put(36,1.6){\makebox(0,0){$\bullet$}}
\put(28,1.5){\makebox(0,0){$\langle$}}
\put(4,1.5){\makebox(0,0){$\bullet$}}
\put(4,1.5){\line(1,0){16}}
\put(4,8){\makebox(0,0){$\scriptstyle 5$}}
\put(20,8){\makebox(0,0){$\scriptstyle -4$}}
\put(36,8){\makebox(0,0){$\scriptstyle 1$}}
\end{picture}$, which is exactly the component of (\[torsionishere\]) that we have not eliminated.
The Rumin-Seshadri complex
==========================
Although not a replacement for the de Rham complex in resolving the constants, we take the opportunity here to describe another natural differential complex, the Rumin-Seshadri complex [@S], the construction of which follows the same general technique. This complex is defined on any symplectic $2n$-manifold $M$ as follows. Denoting by $J$ the symplectic $2$-form, let us consider the filtered differential complex $E^\bullet$ defined by $$E^p=\Lambda^p\oplus\Lambda^{p-1}\enskip\mbox{for }
p=0,1,\ldots,{2n+1}$$ with differentials $$(\omega,\mu)\mapsto(d\omega+(-1)^pJ\wedge\mu,d\mu).$$ Notice that this complex has local cohomology at both $p=0$ and $p=1$. Specifically, the kernel of $E^0\to E^1$ is $\{(f,0)\mbox{ s.t.\ }f\mbox{ is constant}\}$ and the cohomology at $p=1$ is generated by $(\alpha,-1)$, where $\alpha$ is any local potential for the symplectic form $J$, meaning that $d\alpha=J$. Evidently, this is a filtered complex: the de Rham complex is a sub-complex. The associated spectral sequence immediately gives rise to the Rumin-Seshadri differential complex on $M$. It has the form $$\label{RScomplex}\begin{array}{ccccccccccc}
\Lambda^0&\stackrel{d}{\longrightarrow}&\Lambda^1
&\stackrel{d_\perp}{\longrightarrow}&\Lambda_\perp^2
&\stackrel{d_\perp}{\longrightarrow}&\Lambda_\perp^3
&\stackrel{d_\perp}{\longrightarrow}&\cdots
&\stackrel{d_\perp}{\longrightarrow}&\Lambda_\perp^{n}\\[2pt]
&&&&&&&&&&\big\downarrow\makebox[0pt][l]{\scriptsize$d_\perp^{(2)}$}\\
\Lambda^0&\stackrel{d_\perp}{\longleftarrow}&\Lambda^1
&\stackrel{d_\perp}{\longleftarrow}&\Lambda_\perp^2
&\stackrel{d_\perp}{\longleftarrow}&\Lambda_\perp^3
&\stackrel{d_\perp}{\longleftarrow}&\cdots
&\stackrel{d_\perp}{\longleftarrow}&\Lambda_\perp^{n}
\end{array}$$ where $\Lambda_\perp^p$ denotes the $p$-forms that are trace-free with respect to $J$. The conclusion is as follows.
On any symplectic manifold, there is a differential complex [(\[RScomplex\])]{} with local cohomology in degrees $0$ and $\,1$. On the level of sheaves, in both these degrees the cohomology is the locally constant sheaf $\,{\mathbb{R}}$. In all other degrees it is locally exact. On a compact symplectic manifold of dimension $\geq 4$, $$\label{firstcohomology}
\frac{{\mathrm{ker}}\,d_\perp:\Gamma(M,\Lambda^1)\to
\Gamma(M,\Lambda_\perp^2)}
{{\mathrm{im}}\,d:\Gamma(M,\Lambda^0)\to
\Gamma(M,\Lambda^1)}\cong H^1(M,{\mathbb{R}}).$$
The construction of the complex and the identification of its local cohomology are immediate form the spectral sequence. To see (\[firstcohomology\]), note that for a $1$-form $\omega$ to be in the kernel of $d_\perp$ is to say that $d\omega=fJ$ for some smooth function $f$ but then $$\begin{array}{ccccl}
0=d^2\omega=df\wedge J&\implies&df=0&\implies&
f\mbox{ is constant}\\
&&&\implies&f=0
\enskip\mbox{or}\enskip J=d(\omega/f).\end{array}$$ However, the symplectic form cannot be exact for $M$ compact so $f=0$ and thus $d\omega=0$.
In four dimensions, the complex (\[RScomplex\]) is due to R.T. Smith [@Sm]. In higher dimensions, it was also found by L.-S. Tseng and S.-T. Yau [@SY] who show that it is elliptic and go on to study its cohomology on compact manifolds. The complex of first order operators after the second-order operator in the middle, was introduced by T. Bouche [@Bo] and who dubbed it the [*coeffective*]{} complex (he regarded it as a subcomplex of the second half of the de Rham complex $\Lambda^{n}\to\cdots\Lambda^{2n}$). The coeffective cohomology was further studied by M. Fernández, R. Ibáñez, and M. de León (see, for example, [@fil]).
Acknowledgements {#acknowledgements .unnumbered}
================
RLB gratefully acknowledges NSF support from grants DMS-8352009 and DMS-8905207 (from the 1980s when some of this work was done), and current NSF support from DMS-1105868. MGE, ARG, and KN would like to thank the Erwin Schrödinger Institute for hospitality in July 2011 during which this work was crucially advanced. MGE gratefully acknowledges support from the Australian Research Council. ARG gratefully acknowledges support from the Royal Society of New Zealand via Marsden Grant 10-UOA-113.
We thank Boris Doubrov for pointing out to us the homogeneous Engel manifold of §\[engelrevisited\], Jean-Pierre Demailly for drawing our attention to [@fil], and Li-Sheng Tseng for drawing our attention to [@Sm].
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{
"pile_set_name": "ArXiv"
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---
abstract: 'In this paper we develop a statistical mechanical interpretation of the noiseless source coding scheme based on an absolutely optimal instantaneous code. The notions in statistical mechanics such as statistical mechanical entropy, temperature, and thermal equilibrium are translated into the context of noiseless source coding. Especially, it is discovered that the temperature $1$ corresponds to the average codeword length of an instantaneous code in this statistical mechanical interpretation of noiseless source coding scheme. This correspondence is also verified by the investigation using box-counting dimension. Using the notion of temperature and statistical mechanical arguments, some information-theoretic relations can be derived in the manner which appeals to intuition.'
author:
-
title: A Statistical Mechanical Interpretation of Instantaneous Codes
---
Introduction
============
We introduce a statistical mechanical interpretation to the noiseless source coding scheme based on an absolutely optimal instantaneous code. The notions in statistical mechanics such as statistical mechanical entropy, temperature, and thermal equilibrium are translated into the context of noiseless source coding.
We identify a coded message by an instantaneous code with an energy eigenstate of a quantum system treated in statistical mechanics, and the length of the coded message with the energy of the eigenstate. The discreteness of the length of coded message naturally corresponds to statistical mechanics based on quantum mechanics and not on classical mechanics. This is because the energy of a quantum system takes discrete value while an energy takes continuous value in classical physics in general. Especially, in this statistical mechanical interpretation of noiseless source coding, the energy of the corresponding quantum system is bounded to the above, and therefore the system has negative temperature. We discover that the temperature $1$ corresponds to the average codeword length of an instantaneous code in the interpretation. This correspondence is also verified by the investigation based on box-counting dimension.
Note that, we do not stick to the mathematical strictness of the argument in this paper. We respect the statistical mechanical intuition in order to shed light on a hidden statistical mechanical aspect of information theory, and therefore make an argument on the same level of mathematical strictness as statistical mechanics.
Instantaneous codes
===================
We start with some notation on instantaneous codes from information theory [@S48; @A90; @CT91].
For any set $S$, $\#S$ denotes the number of elements in $S$. We denote the set of all finite binary strings by ${\{0,1\}^*}$. For any $s\in{\{0,1\}^*}$, ${\left\lverts\right\rvert}$ is the *length* of $s$. We define an *alphabet* to be any nonempty finite set.
Let $X$ be an arbitrary random variable with an alphabet ${\mathcal{H}}$ and a probability mass function $p_X(x)=\operatorname{Pr}\{X=x\}$, $x\in{\mathcal{H}}$. Then the *entropy* $H(X)$ of $X$ is defined by $$H(X)\equiv-\sum_{x\in{\mathcal{H}}}p_X(x)\log p_X(x),$$ where the $\log$ is to the base $2$. We will introduce the notion of a statistical mechanical entropy later. Thus, in order to distinguish $H(X)$ from it, we particularly call $H(X)$ the *Shannon entropy* of $X$. A subset $S$ of ${\{0,1\}^*}$ is called a *prefix-free set* if no string in $S$ is a prefix of any other string in $S$. An *instantaneous code* $C$ for the random variable $X$ is an injective mapping from ${\mathcal{H}}$ to ${\{0,1\}^*}$ such that $C({\mathcal{H}})\equiv\{C(x)|x\in{\mathcal{H}}\}$ is a prefix-free set. For each $x\in{\mathcal{H}}$, $C(x)$ is called the *codeword* corresponding to $x$ and ${\left\lvertC(x)\right\rvert}$ is denoted by $l(x)$. A sequence $x_1,x_2,\dots,x_N$ with $x_i\in{\mathcal{H}}$ is called a $message$. On the other hand, the finite binary string $C(x_1)C(x_2)\dotsm C(x_N)$ is called the *coded message* for a message $x_1,x_2,\dots,x_N$.
An instantaneous code play an important role in the noiseless source coding problem described as follows. Let $X_1,X_2,\dots,X_N$ be independent identically distributed random variables drawn from the probability mass function $p_X(x)$. The objective of the noiseless source coding problem is to minimize the length of the binary string $C(x_1)C(x_2)\dotsm C(x_N)$ for a message $x_1,x_2,\dots,x_N$ generated by the random variables $\{X_i\}$ as $N\to\infty$. For that purpose, it is sufficient to consider the *average codeword length* $L_X(C)$ of an instantaneous code $C$ for the random variable $X$, which is defined by $$L_X(C)\equiv\sum_{x\in{\mathcal{H}}}p_X(x)l(x)$$ independently on the value of $N$. We can then show that $L_X(C)\ge H(X)$ for any instantaneous code $C$ for the random variable $X$. Hence, the Shannon entropy gives the data compression limit for the noiseless source coding problem based on instantaneous codes. Thus, it is important to consider the notion of absolutely optimality of an instantaneous code, where we say that an instantaneous code $C$ for the random variable $X$ is *absolutely optimal* if $L_X(C)=H(X)$. We can see that an instantaneous code $C$ is absolutely optimal if and only if $p_X(x)=2^{-l(x)}$ for all $x\in{\mathcal{H}}$.
Finally, for each $x^N=(x_1,x_2,\dots,x_N)\in{\mathcal{H}}^N$, we define $p_X(x^N)$ as $p_X(x_1)p_X(x_2)\dotsm p_X(x_N)$.
Statistical Mechanical Interpretation
=====================================
In this section, we develop a statistical mechanical interpretation of the noiseless source coding by an instantaneous code. In what follows, we assume that an instantaneous code $C$ for a random variable $X$ is absolutely optimal.
In statistical mechanics [@Re65; @TKS92; @Ru99], we consider a quantum system $\mathcal{S}_{\text{total}}$ which consists in a large number of identical quantum subsystems. Let $N$ be a number of such subsystems. For example, $N\sim 10^{22}$ for $1\,\mathrm{cm^3}$ of a gas at room temperature. We assume here that each quantum subsystem can be distinguishable from others. Thus, we deal with quantum particles which obey Maxwell-Boltzmann statistics and not Bose-Einstein statistics or Fermi-Dirac statistics. Under this assumption, we can identify the $i$th quantum subsystem $\mathcal{S}_i$ for each $i=1,\dots,N$. In quantum mechanics, any quantum system is described by a quantum state completely. In statistical mechanics, among all quantum states, energy eigenstates are of particular importance. Any energy eigenstate of each subsystem $\mathcal{S}_i$ can be specified by a number $n=1,2,3,\dotsc$, called a *quantum number*, where the subsystem in the energy eigenstate specified by $n$ has the energy $E_n$. Then, any energy eigenstate of the system $\mathcal{S}_{\text{total}}$ can be specified by an $N$-tuple $(n_1,n_2,\dots,n_N)$ of quantum numbers. If the state of the system $\mathcal{S}_{\text{total}}$ is the energy eigenstate specified by $(n_1,n_2,\dots,n_N)$, then the state of each subsystem $\mathcal{S}_i$ is the energy eigenstate specified by $n_i$ and the system $\mathcal{S}_{\text{total}}$ has the energy $E_{n_1}+E_{n_2}+ \dots +E_{n_N}$. Then, the fundamental postulate of statistical mechanics is stated as follows.
**Fundamental Postulate:** If the energy of the system $\mathcal{S}_{\text{total}}$ is known to have a constant value in the range between $E$ and $E+\delta E$, where $\delta E$ is the indeterminacy in measurement of the energy of the system $\mathcal{S}_{\text{total}}$, then the system $\mathcal{S}_{\text{total}}$ is equally likely to be in any energy eigenstate specified by $(n_1,n_2,\dots,n_N)$ such that $E\le E_{n_1}+E_{n_2}+ \dots +E_{n_N}\le E+\delta E$.
Let $\Omega(E,N)$ be the total number of energy eigenstates of $\mathcal{S}_{\text{total}}$ specified by $(n_1,n_2,\dots,n_N)$ such that $E\le E_{n_1}+E_{n_2}+ \dots +E_{n_N}\le E+\delta E$. The above postulate states that any energy eigenstate of $\mathcal{S}_{\text{total}}$ whose energy lies between $E$ and $E+\delta E$ occurs with the probability $1/\Omega(E,N)$. This uniform distribution of energy eigenstates whose energy lies between $E$ and $E+\delta E$ is called a *microcanonical ensemble*. In statistical mechanics, the *entropy* $S(E,N)$ of the system $\mathcal{S}_{\text{total}}$ is then defined by $$S(E,N)\equiv k\ln \Omega(E,N),$$ where $k$ is a positive constant, called the *Boltzmann Constant*, and the $\ln$ denotes the natural logarithm. Note that, in statistical mechanics, the entropy $S(E,N)$ is normally estimated to first order in $N$ and $E$. Thus the magnitude of the indeterminacy $\delta E$ of the energy does not matter unless it is too small. The *temperature* $T(E,N)$ of the system $\mathcal{S}_{\text{total}}$ is defined by $$\frac{1}{T(E,N)}\equiv \frac{\partial S}{\partial E}(E,N).$$ Thus the temperature is a function of $E$ and $N$. The average energy $\varepsilon$ per one subsystem is given by $E/N$.
Now we give a statistical mechanical interpretation to the noiseless source coding scheme based on an instantaneous code. Let $X$ be an arbitrary random variable with an alphabet ${\mathcal{H}}$, and let $C$ be an absolutely optimal instantaneous code for the random variable $X$. Let $X_1,X_2,\dots,X_N$ be independent identically distributed random variables drawn from the probability mass function $p_X(x)$ for a large $N$, say $N\sim 10^{22}$. We relate the noiseless source coding based on $C$ to the above statistical mechanics as follows. The sequence $X_1,X_2,\dots,X_N$ corresponds to the quantum system $\mathcal{S}_{\text{total}}$, where each $X_i$ corresponds to the $i$th quantum subsystem $\mathcal{S}_i$. We relate $x\in{\mathcal{H}}$, or equivalently, $C(x)$ to an energy eigenstate of a subsystem, and we relate $l(x)={\left\lvertC(x)\right\rvert}$ to an energy $E_n$ of the energy eigenstate of the subsystem. Then a sequence $(x_1,\dots,x_N)\in{\mathcal{H}}^N$, or equivalently, a finite binary string $C(x_1)\dotsm C(x_N)$ corresponds to an energy eigenstate of $\mathcal{S}_{\text{total}}$ specified by $(n_1,\dots,n_N)$. Thus, $l(x_1)+\dots+l(x_N)={\left\lvertC(x_1)\dotsm C(x_N)\right\rvert}$ corresponds to the energy $E_{n_1}+ \dots +E_{n_N}$ of the energy eigenstate of $\mathcal{S}_{\text{total}}$.
We define a subset $C(L,N)$ of ${\{0,1\}^*}$ as the set of all coded messages $C(x_1)\dotsm C(x_N)$ whose length lies between $L$ and $L+\delta L$. Then $\Omega(L,N)$ is defined as $\#C(L,N)$. Therefore $\Omega(L,N)$ is the total number of coded messages whose length lies between $L$ and $L+\delta L$. We can see that if $C(x_1)\dotsm C(x_N)\in C(L,N)$, then $2^{-L}\le p(x^N)\le 2^{-(L+\delta L)}$. This is because $C$ is an absolutely optimal instantaneous code. Thus all coded messages $C(x_1)\dotsm C(x_N)\in C(L,N)$ occur with the probability $2^{-L}$. Note here that we care nothing about the magnitude of $\delta L$, as in the case of statistical mechanics. Thus, given that the length of coded message is $L$, all coded messages occur with the same probability $1/\Omega(L,N)$. We introduce a micro-canonical ensemble on the noiseless source coding in this manner. Thus we can develop a certain sort of statistical mechanics on the noiseless source coding scheme.
The *statistical mechanical entropy* $S(L,N)$ of the instantaneous code $C$ is defined by $$S(L,N)\equiv \log \Omega(L,N).$$ The *temperature* $T(L,N)$ of $C$ is then defined by $$\frac{1}{T(L,N)}\equiv \frac{\partial S}{\partial L}(L,N).$$ Thus the temperature is a function of $L$ and $N$. The average length $\lambda$ of coded message per one codeword is given by $L/N$. The average length $\lambda$ corresponds to the average energy $\varepsilon$ in the statistical mechanics above.
Properties of Statistical Mechanical Entropy {#psme}
============================================
In statistical mechanics, it is important to know the values of the energy $E_n$ of subsystem $\mathcal{S}_i$ for all quantum numbers $n$, since the values determine the entropy $S(E,N)$ of the quantum system $\mathcal{S}_{\text{total}}$. Corresponding to this fact, the knowledge of $l(x)$ for all $x\in{\mathcal{H}}$ is important to calculate $S(L,N)$. We investigate some properties of $S(L,N)$ and $T(L,N)$ based on $l(x)$ in the following.
As is well known in statistical mechanics, if the energy of a quantum system $\mathcal{S}_{\text{total}}$ is bounded to the above, then the system can have negative temperature. The same situation happens in our statistical mechanics developed on an instantaneous code $C$, since there are only finite codewords of $C$. We define $l_{\text{min}}$ and $l_{\text{max}}$ as $\min\{l(x)\mid x\in{\mathcal{H}}\}$ and $\max\{l(x)\mid x\in{\mathcal{H}}\}$, respectively. Given $N$, the statistical mechanical entropy $S(L,N)$ is a unimodal function of $L$ and takes nonzero value only between $Nl_{\text{min}}$ and $Nl_{\text{max}}$. Let $L_0$ be the value $L$ which maximizes $S(L,N)$. If $L<L_0$ then $T(L,N)>0$. On the other hand, if $L>L_0$ then $T(L,N)<0$. The temperature $T(L,N)$ takes $\pm\infty$ at $L=L_0$.
According to the method of Boltzmann and Planck (see e.g. [@TKS92]), we can show that $$\label{BP}
S(L,N)=NH(G(C,T(L,N))),$$ where $G(C,T)$ is the random variable with the alphabet ${\mathcal{H}}$ and the probability mass function $p_{G(C,T)}(x)=\operatorname{Pr}\{G(C,T)=x\}$ defined by $$p_{G(C,T)}(x)\equiv
\frac{2^{-l(x)/T}}{\sum_{a\in{\mathcal{H}}}2^{-l(a)/T}}.$$ The temperature $T(L,N)$ is implicitly determined through the equation $$\label{al}
\frac{L}{N}=\sum_{x\in{\mathcal{H}}} l(x)p_{G(C,T(L,N))}(x)$$ as a function of $L$ and $N$. These properties of $S(L,N)$ and $T(L,N)$ are derived only based on a combinatorial aspect of $S(L,N)$.
Now, let us take into account the probabilistic issue given by the random variables $X_1,X_2,\dots,X_N$. Since the instantaneous code $C$ is absolutely optimal, a particular coded message of length $L$ occurs with probability $2^{-L}$. Thus the probability that some coded message of length $L$ occurs is given by $2^{-L}\Omega(L,N)$. Hence, by differentiating $2^{-L}\Omega(L,N)$ on $L$ and setting the result to $0$, we can determine the most probable length $L^*$ of coded message, given $N$. Thus we have the relation $$\frac{\partial}{\partial L}\{-L+ S(L,N)\}\Big|_{(L,N)=(L^*,N)}=0,$$ which is satisfied by $L^*$. It follows that $T(L^*,N)=1$, Thus, the temperature $1$ corresponds to the most probable length $L^*$. On the other hand, $p_{G(C,1)}(x)=2^{-l(x)}$ at $T(L^*,N)=1$, and therefore, by , we have $L^*/N=H(X)=L_X(C)$. Since $C$ is absolutely optimal, this result is consistent with the law of large numbers. Thus, the temperature $1$ corresponds to the average codeword length $L_X(C)$, which is equal to the average length $\lambda$ of coded message per one codeword at the temperature $1$.
Thermal Equilibrium between Two Instantaneous Codes {#te}
===================================================
Let $X^{\text{I}}$ be an arbitrary random variable with an alphabet ${\mathcal{H}}^{\text{I}}$, and let $C^{\text{I}}$ be an absolutely optimal instantaneous code for the random variable $X^{\text{I}}$. Let $X^{\text{I}}_1,X^{\text{I}}_2,\dots,X^{\text{I}}_{N^{\text{I}}}$ be independent identically distributed random variables drawn from the probability mass function $p_{X^{\text{I}}}(x)$ for a large $N^{\text{I}}$. On the other hand, let $X^{\text{II}}$ be an arbitrary random variable with an alphabet ${\mathcal{H}}^{\text{II}}$, and let $C^{\text{II}}$ be an absolutely optimal instantaneous code for the random variable $X^{\text{II}}$. Let $X^{\text{II}}_1,X^{\text{II}}_2,\dots,X^{\text{II}}_{N^{\text{II}}}$ be independent identically distributed random variables drawn from the probability mass function $p_{X^{\text{II}}}(x)$ for a large $N^{\text{II}}$.
Consider the following problem: Find the most probable values $L^{\text{I}}$ and $L^{\text{II}}$, given that the sum $L^{\text{I}}+L^{\text{II}}$ of the length $L^{\text{I}}$ of coded message by $C^{\text{I}}$ for the random variables $\{X^{\text{I}}_i\}$ and the length $L^{\text{II}}$ of coded message by $C^{\text{II}}$ for the random variables $\{X^{\text{II}}_j\}$ is equal to $L$.
In order to solve this problem, the statistical mechanical notion of “thermal equilibrium” can be used. We first note that a particular coded message by $C^{\text{I}}$ of length $L_{\text{I}}$ and a particular coded message by $C^{\text{II}}$ of length $L_{\text{II}}$ occur with probability $2^{-L_{\text{I}}}2^{-L_{\text{II}}}=2^{-L}$, since the instantaneous codes $C^{\text{I}}$ and $C^{\text{II}}$ are absolutely optimal. Thus, any particular pair of coded messages by $C^{\text{I}}$ and $C^{\text{II}}$ occurs with an equal probability, given that the total length of coded messages for $\{X^{\text{I}}_i\}$ and $\{X^{\text{II}}_j\}$ is $L$. Therefore, the most probable allocation ${L_{\text{I}}}^*$ and ${L_{\text{II}}}^*$ of $L=L_{\text{I}}+L_{\text{II}}$ maximizes the product $\Omega_{\text{I}}(L_{\text{I}},N_{\text{I}})
\Omega_{\text{II}}(L_{\text{II}},N_{\text{II}})$. We see that this condition is equivalent to the equality: $$T_{\text{I}}(L_{\text{I}}^*,N_{\text{I}})=
T_{\text{II}}(L_{\text{II}}^*,N_{\text{II}}),$$ where the functions $T_{\text{I}}$ and $T_{\text{II}}$ are the temperature of $C^{\text{I}}$ and $C^{\text{II}}$, respectively. This equality corresponds to the condition on the thermal equilibrium between two systems, given a total energy, in statistical mechanics. Using , the value of $T_{\text{I}}(L_{\text{I}}^*,N_{\text{I}})=
T_{\text{II}}(L_{\text{II}}^*,N_{\text{II}})$ is obtained by solving the equation on $T$: $$\begin{aligned}
&&\frac{N^{\text{I}}}{L}
\sum_{x\in{\mathcal{H}}^{\text{I}}}
{\left\lvertC^{\text{I}}(x)\right\rvert}p_{G(C^{\text{I}},T)}(x)+\\
&&\frac{N^{\text{II}}}{L}
\sum_{x\in{\mathcal{H}}^{\text{II}}}
{\left\lvertC^{\text{II}}(x)\right\rvert}p_{G(C^{\text{II}},T)}(x)\\
&&=1.\end{aligned}$$ Then, again by , the most probable values $L_{\text{I}}^*$ and $L_{\text{II}}^*$ are determined.
Dimension of Coded Messages
===========================
The notion of dimension plays an important role in fractal geometry [@F90]. In this section, we investigate our statistical mechanical interpretation of the noiseless source coding from the point of view of dimension. Let $F$ be a bounded subset of ${\mathbb{R}}$, and let $N_n(F)$ be the number of $2^{-n}$-mesh cubes that intersect $F$, where $2^{-n}$-mesh cube is a subset of ${\mathbb{R}}$ in the form of $[m2^{-n},(m+1)2^{-n}]$ for some integer $m$. The *box-counting dimension* $\dim_B F$ of $F$ is then defined by $$\dim_B F \equiv \lim_{n\to\infty} \frac{\log N_n(F)}{n}.$$ Let ${\{0,1\}^\infty}=\{b_1b_2b_3\dotsm\mid b_i=0,1\text{ for all }i=1,2,3,\dotsc\}$ be the set of all infinite binary strings. In [@T02] we investigate the dimension of sets of coded messages of infinite length, where the number of distinct codewords is finite or infinite. In a similar manner, we investigate the set of coded messages of infinite length by an absolutely optimal instantaneous code $C$.
By , the ratio $L/N$ is uniquely determined by temperature $T$. Thus, by letting $L,N\to\infty$ while keeping the ratio $L/N$ constant, we can regard the set $C(L,N)$ as a subset of ${\{0,1\}^\infty}$. This kind of limit is called the *thermodynamic limit* in statistical mechanics. Taking the thermodynamic limit, we denote $C(L,N)$ by $F(T)$, where $T$ is related to the limit value of $L/N$ through . Although $F(T)$ is a subset of ${\{0,1\}^\infty}$, we can regard $F(T)$ as a subset of $[0,1]$ by identifying $\alpha\in{\{0,1\}^\infty}$ with the real number $0.\alpha$. In this manner, we can consider the box-counting dimension $\dim_B F(T)$ of $F(T)$.
We investigate the dependency of $\dim_B F(T)$ on temperature $T$ with $-\infty\le T\le \infty$. First it can be shown that $$\begin{aligned}
\dim_B F(T)
&=&\lim_{L,N\to\infty} \frac{\log \Omega(L,N)}{L}\\
&=&\lim_{L,N\to\infty} \frac{S(L,N)}{L},\end{aligned}$$ where the limits are taken while satisfying for each $T$. Thus the statistical mechanical entropy $S(L,N)$ and the box-counting dimension $\dim_B F(T)$ of $F(T)$ are closely related. By and , we can obtain, as an explicit formula of $T$, $$\label{fd}
\dim_B F(T)=
\frac{1}{T}+\frac{1}{\lambda(T)}\log \sum_{x\in{\mathcal{H}}} 2^{-l(x)/T},$$ where $\lambda(T)$ is defined by $$\lambda(T)\equiv\sum_{x\in{\mathcal{H}}} l(x)p_{G(C,T)}(x).$$ We define the “degeneracy factors” $d_{\text{min}}$ and $d_{\text{max}}$ of the lowest and highest “energies” by $d_{\text{min}}\equiv\#\{x\in{\mathcal{H}}\mid\l(x)=l_{\text{min}}\}$ and $d_{\text{max}}\equiv\#\{x\in{\mathcal{H}}\mid\l(x)=l_{\text{max}}\}$, respectively. Note here that since $C$ is assumed to be absolutely optimal, $\sum_{x\in{\mathcal{H}}}2^{-l(x)}=1$ and therefore $d_{\text{max}}$ can be shown to be an even number. In the increasing order of the ratio $L/N$ (i.e. $\lambda(T)$), we see from that $$\begin{aligned}
\lim_{T\to +0} \dim_B F(T)&=&\frac{\log d_{\text{min}}}{l_{\text{min}}},\\
\dim_B F(1)&=&1,\\
\lim_{T\to \pm\infty} \dim_B F(T)&=&\frac{n\log n}{\sum_{x\in{\mathcal{H}}}l(x)},\\
\lim_{T\to -0} \dim_B F(T)&=&\frac{\log d_{\text{max}}}{l_{\text{max}}}.\end{aligned}$$ We can show that $n\log n<\sum_{x\in{\mathcal{H}}}l(x)$ unless all codewords have the same length, and obviously $\log d_{\text{min}}/l_{\text{min}}<1$ and $\log d_{\text{max}}/l_{\text{max}}<1$ except for such a trivial case. Thus, in general, the dimension $\dim_B F(T)$ is maximized at the temperature $T=1$. This can be checked using based on the differentiation of $\dim_B F(T)$. That is, we can show that, if all codewords do not have the same length, then the following hold:
1. $\displaystyle\frac{d}{dT}\dim_B F(T)\Big|_{T=T_0}=0$ if and only if $T_0=1$,
2. $\displaystyle\frac{d^2}{dT^2}\dim_B F(T)\Big|_{T=1}<0$.
Note that all coded messages $C(x_1)C(x_2)\dotsm$ of infinite length form the set ${\{0,1\}^\infty}$ and therefore the interval $[0,1]$, since $C$ is an absolutely optimal instantaneous code. Thus, since $\dim_B F(1)$ is equal to $\dim_B [0,1]$, the set $F(1)$ is as rich as the set $[0,1]$ in a certain sense. This can be explained as follows. Since $L/N=L_X(C)$ at the temperature $T=1$, as seen in Section \[psme\], by the law of large numbers, the length of coded message for a message of length $N$ is likely to equal $NL_X(C)$, for a sufficiently large $N$. Thus $F(1)$ contains almost all finite binary string of length $L$. In other words, $F(1)$ consists in coded messages for all messages which form the typical set in a sense.
Conclusion
==========
In this paper we have developed a statistical mechanical interpretation of the noiseless source coding scheme based on an absolutely optimal instantaneous code. The notions in statistical mechanics such as statistical mechanical entropy, temperature, and thermal equilibrium are translated into the context of information theory. Especially, it is discovered that the temperature $1$ corresponds to the average codeword length $L_X(C)$ in this statistical mechanical interpretation of information theory. This correspondence is also verified by the investigation using box-counting dimension. The argument is not necessarily mathematically rigorous. However, using the notion of temperature and statistical mechanical arguments, several information-theoretic relations can be derived in the manner which appeals to intuition.
A statistical mechanical interpretation of the general case where the underlying instantaneous code is not necessarily absolutely optimal is reported in another work.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author is grateful to the 21st Century COE Program and the Research and Development Initiative of Chuo University for the financial supports.
[99]{}
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Let $X$ be a Banach space. Then $X$ is complemented in the bidual $X^{**}$ if and only if there exists an invariant mean $\ell_\infty(G, X)\to X$ with respect to a free Abelian group $G$ of rank equal to the cardinality of $X^{**}$, and this happens if and only if there exists an invariant mean with respect to the additive group of $X^{**}$. This improves upon previous results due to Bustos Domecq [@felix] and the second-named author [@kania], where certain idempotent semigroups of cardinality equal to the cardinality of $X^{**}$ were considered, and answers a question of J.M.F. Castillo (private communication) that was also considered in [@kania]. *En route* to the proof of the main result, we endow the family of all finite-dimensional subspaces of an infinite-dimensional vector space with a structure of a free commutative monoid with the property that the product of two subspaces contains the respective subspaces, which is possibly of interest in itself.'
address:
- 'Department of Pure Mathematics and Mathematical Statistics, University of Cambridge'
- 'Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Prague 1, Czech Republic and Institute of Mathematics, Jagiellonian University, [Ł]{}ojasiewicza 6, 30-348 Kraków, Poland'
author:
- 'Adam P. Goucher'
- Tomasz Kania
date:
-
-
title: |
Invariant means on Abelian groups capture complementability of Banach spaces\
in their second duals
---
[^1]
Introduction and the main result
================================
Every Banach space $X$ embeds canonically (linearly and isometrically) into its second dual $X^{**}$ via the map $\kappa_X\colon X\to X^{**}$ given by $\langle \kappa_X x, f\rangle = \langle f,x\rangle$ ($x\in X, f\in X^*$), so that it is customary to identify $X$ with a closed subspace of $X^{**}$. The question of when is $X$ complemented in $X^{**}$ (that is, when does there exist a bounded linear projection from $X^{**}$ onto the image of $\kappa_X$) is a frequently recurring problem in Banach space theory as Banach spaces complemented in the bidual enjoy various, otherwise often unavailable, averaging properties such as the existence of vector valued invariant means with respect to amenable semigroups. Banach spaces complemented in their biduals include, besides reflexive spaces, dual spaces and weakly sequentially complete Banach lattices; $c_0$, the space of sequences convergent to 0, is the most prominent example of a space not complemented in its bidual.
Let $X$ be a Banach space, $(S, \cdot)$ a semigroup, and $\lambda\geqslant 1$. A bounded linear operator $M\colon \ell_\infty(S,X)\to X$ of norm at most $\lambda$ such that for every $x\in X$, $s\in S$, and for all $f\in \ell_\infty(S,X)$ one has
$M(x\mathds{1}_S) = x$;
$M(f) = M({}_sf) = M(f_s)$,
is called an $X$*-valued invariant $\lambda$-mean* on $S$. Here, $\ell_\infty(S,X)$ denotes the Banach space of all bounded $X$-valued functions on $S$, $x\mathds{1}_S$ stands for the function constantly equal to $x$ on $S$, and ${}_sf(t) = f(s\cdot t)$ and $f_s(t) = (t\cdot s)$ ($s,t\in S$). When there is no need to emphasise the constant $\lambda$, $M$ is simply called an $X$*-valued invariant mean on* $S$.
A semigroup is *amenable*, whenever it admits an $X$-valued 1-mean with respect to the one-dimensional space $X$ (namely, the scalar field). By the Markov–Kakutani fixed-point theorem, all commuative semigroups are amenable. In the literature, means invariant with respect to the additive monoid of natural numbers (with zero) are called *Banach limits* (*vector-valued Banach limits*, when $\dim X \geqslant 2$).
Invariant means with respect to more complicated groups were employed by Pe[ł]{}czyński, who proved that for a Banach space $X$, if $Y\subseteq X$ is a closed subspace, which is complemented in $Y^{**}$ and there exists a Lipschitz map $r\colon X\to Y$ such that $r(y)=y$ for $y\in Y$, then $Y$ is (linearly) complemented in $X$ ([@pelczynski pp. 61–62], see also [@benyamini Theorem 3.3]).
Suppose that $S$ is an amenable semigroup and $X$ is a Banach space complemented in $X^{**}$. Then one can construct an $X$-valued invariant $\lambda$-mean on $S$ as follows. Let $m$ be a (scalar-valued) invariant mean on $S$. We define a map $\widetilde{M}\colon \ell_\infty(S, X)\to X^{**}$ by $$\langle \widetilde{M}f, \varphi\rangle = \big\langle m, \langle \varphi, f(\boldsymbol{\cdot} )\rangle \big\rangle \quad (\varphi \in X^*, f\in \ell_\infty(S, X)).$$ Since $\|m\|=1$, $\widetilde{M}$ is a norm-one bounded linear operator. If $P$ is a projection from $X^{**}$ onto the canonical copy of $X$, then $$M = \kappa_X^{-1}P\widetilde{M}$$ is an $X$-valued invariant $\|P\|$-mean on $S$.
In order to prove that the converse holds true as well, in [@felix; @kania] the existence of an invariant mean on a certain idempotent semigroup $S$ was assumed; the semigroup $S$ comprises pairs $(F,\varepsilon)$, where $F\subset X^{**}$ is a finite-dimensional subspace and $\varepsilon > 0$; the semigroup operation was defined as $(F, \varepsilon) \cdot (G, \delta) = (F+G, \min\{\varepsilon, \delta\})$; related questions were also considered in [@radek].
In this note we consider the following question communicated privately to the second-named author by J.M.F. Castillo:
> *Suppose that a Banach space $X$ admits an invariant mean with respect to every/some Abelian group. Must $X$ be complemented in $X^{**}$?*
The question arises from the observation that the semigroup $S$ used to prove the existence of a projection from $X^{**}$ onto $X$ is idempotent, so in a sense, it is as far from being a (subset of a) group as possible. In the present paper we demonstrate that free Abelian groups of big enough rank suffice to prove the existence of a projection from the bidual.
\[main\]Let $X$ be a Banach space and $\lambda\geqslant 1$. Then the following assertions are equivalent.
\[complement\] $X$ is complemented in $X^{**}$ by a projection of norm at most $\lambda$;
\[allamenable\] for every amenable semigroup $S$ there exists an $X$-valued invariant $\lambda$-mean on $S$;
\[cardinality\] for every free Abelian group $G$ of rank $|X^{**}|$ there exists an $X$-valued invariant $\lambda$-mean on $G$.
\[bidualcase\] [there exists an $X$-valued invariant $\lambda$-mean on the additive group of $X^{**}$.]{}
The implication $\Rightarrow$ , has been already briefly explained and may be found, for example, in [@felix Theorem 1]. The implications $\Rightarrow$ and $\Rightarrow$ are clear. We shall prove the implications $\Rightarrow$ and $\Rightarrow$ in the subsequent section.
Every Abelian group is a quotient of a free Abelian group, so by Lemma \[normal\](ii), clause (iii) in Theorem A is not weaker than the assertion that for every Abelian group $G$ of cardinality $X^{**}$ there exists an $X$-valued invariant $\lambda$-mean on $G$.
The additive group of $X^{**}$ may seem large as already for separable Banach spaces that contain an isomorphic copy of $\ell_1$, it has the cardinality of the power set of the continuum; however, this is unavoidable. Indeed, it follows from [@kania Theorem 1.2] that the Banach space $X = \ell_\infty^c(\Gamma)$ of all countably-supported bounded functions on an uncountable set $\Gamma$ (endowed with the supremum norm) admits invariant means with respect to all countable amenable semigroups, yet $X$ is not complemented in $X^{**}$ ([@pelsud]).
We remark in passing that the mere existence of an $X$-valued invariant mean with respect to *some* commutative semigroup (that could be chosen as large as one wishes) is not sufficient for $X$ being complemented in $X^{**}$. To see this, fix an arbitrary Banach space $X$ (not necessarily complemented in $X^{**}$) and let $(S, \cdot)$ be a semigroup with an element $0\in S$ such that $0\cdot s = s\cdot 0 = 0$ for all $s\in S$. Define $M\colon \ell_\infty(S,X)\to X$ by $Mf = f(0)$. Certainly, $M$ is a norm-one linear operator and $M(x \mathds{1}_S)=x$ for any $x\in X$. Moreover, for any $s\in S$ one has $M f_s = f_s(0) = f(0\cdot s) = f(0) = Mf = M{}_s f$.
It is known that if a separable Banach space has a monotone basis (or more generally, the Metric Approximation Property), then $X$ is 1-complemented in $X^{**}$ if and only if there exists an $X$-valued 1-mean on $\mathbb N$ (an $X$-valued Banach limit); see [@spaniards Corollary 4.2.6]. It follows from Lemma \[groth\] that this equivalent to the existence of an $X$-valued invariant 1-mean on $\mathbb Z$. Thus, at least for very well-behaved separable Banach spaces complementability in the bidual can be indeed witnessed by the group of integers (the free abelian group of rank one).
The key result needed in order to establish Theorem A is related structures turning the family of all finite-dimensional subspaces of $X^{**}$ into a cancellative monoid.
Let $V$ be an infinite-dimensional vector space over an arbitrary field. Denote by ${\rm Fin}\, V$ the set of all finite-dimensional subspaces of $V$. Then there exists a binary operation $\ast$ on ${\rm Fin}\, V$ such that
$({\rm Fin}\, V, \ast)$ is a free commutative monoid;
for any $F,G \in {\rm Fin}\, V$ we have $F,G\subseteq F\ast G$.
Consequently, the Grothendieck group of $({\rm Fin}\, V, \ast)$ is a free Abelian group.
It is quite clear that there is no counterpart of Theorem B for a finite-dimensional vector space $V$ as seen by taking three different hyperplanes $W_1, W_2, W_3$ in $V$ and noticing that $V=W_1\ast W_3 = W_2\ast W_3$, which makes cancellativity impossible to satisfy.
It is noteworthy that the proof of Theorem B makes use of Zermelo’s well-ordering principle and does depend on the chosen well-ordering of ${\rm Fin}\, X^{**}\setminus \{0\}$, however the resulting group is always free Abelian of rank $|X^{**}|$.
Auxiliary facts
===============
We denote by $\mathbb N = \{0,1,2,3, \ldots\}$ the additive monoid of non-negative integers. As such, it is a submonoid of the group of integers $\mathbb Z$.
An Abelian group $G$ is *free*, when it is free as a module over $\mathbb Z$; as such $G$ is isomorphic to the direct sum $\mathbb Z^{(\alpha)}$ of $\alpha$-many copies of the group $\mathbb Z$. The number $\alpha$ is called the *rank* of $G$. In particular, two uncountable free Abelian groups of the same cardinality are isomorphic. Besides free Abelian groups, we will require the notion of a *free commutative monoid*, that is, a monoid isomorphic to the direct sum $\mathbb N^{(\alpha)}$ of $\alpha$-many copies of $\mathbb N$ for some cardinal $\alpha.$
Let $(S,+)$ be a commutative semigroup. Then $S$ embeds into a group if and only if $S$ is cancellative. In the latter case, $S$ embeds into its *Grothendieck group* $G(S)$, that is the group comprising equivalence classes of the relation $\sim$ on $S\times S$ given by $$(s_1, t_1)\sim (s_2, t_2) \iff s_1 + t_2 = t_1 + s_2\quad (s_1, s_2, t_1, t_2\in S).$$ Thus, formally $G(S) = \{s - t \colon s,t\in S\}$ and $0_S = 0_{G(S)}$ belongs to $S \cap (-S)$ if $S$ is already a monoid. Of course, $\mathbb Z$ is the Grothendieck group of $\mathbb N$ and more generally $\mathbb{Z}^{(\alpha)}$ is the Grothendieck group of the free commutative monoid $\mathbb{N}^{(\alpha)}$ for any cardinal number $\alpha$.
Even though all commutative semigroups are amenable, subsemigroups of amenable groups need not be amenable. In the vector-valued case, we need to justify separately the possibility of passing to a subsemigroup, even in the commutative case.
We shall require the following lemma concerning invariant means on groups that admit the infinite cyclic group as a quotient.
\[trivial\] Let $X$ be a Banach space and let $G$ be a group with a surjective homomorphism $\theta\colon G \rightarrow \mathbb{Z}$. Suppose that $M\colon \ell_\infty(G,X)\to X$ is an $X$-valued invariant mean. Then $Mf = 0$ for any function $f$ supported on the kernel of $\theta$.
Assume otherwise. Let $g \in G$ be an arbitrary element satisfying $\theta(g) = 1$. Then for each $n \in \mathbb{N}$ and $t \in G$ with $\theta(t) \neq -n$, we have that $\theta(t g^n) \neq 0$ and, therefore, $f_{g^n}(t) = f(t g^n) = 0$. As such, the functions $\{ f_{g^n}\colon n \in \mathbb{N} \}$ have pairwise disjoint support.
By the translation-invariance, $M(f_{g^n}) = M(f)$ for every $n \in \mathbb{N}$, so $$M(f_g + f_{g^2} + f_{g^3} + \cdots + f_{g^n}) = n M(f).$$ As the summands have pairwise disjoint supports, the sum has the same supremum norm as $f$ itself (and, in particular, is constant as a function of $n$). On the other hand, the norm of $n M(f)$ grows without bound as $n \rightarrow \infty$, contradicting the continuity of $M$.
\[groth\]Let $X$ be a Banach space and let $\lambda \geqslant 1$. If for some $\alpha$ there exists an $X$-valued invariant $\lambda$-mean on $\mathbb{Z}^{(\alpha)}$, then there exists such a mean on $\mathbb{N}^{(\alpha)}$ too.
Let $M\colon \ell_\infty(\mathbb{Z}^{(\alpha)}, X)\to X$ be an $X$-valued invariant $\lambda$-mean. For $t\in \mathbb{Z}^{(\alpha)}$ we define $|t|$ coordinate-wise, that is, $|t|(i) = |t(i)|$ for $i\in \alpha$. Let us consider the operator $M^\prime \colon \ell_\infty(\mathbb{N}^{(\alpha)}, X)\to X$ given by the formula $M^\prime(f) = M(f')$, where $f'(t) := f(|t|)$ for $t\in \mathbb{Z}^{(\alpha)}$ and $f\in \ell_\infty(\mathbb{Z}^{(\alpha)}, X)$. In particular, for the function $f = x \mathds{1}_{\mathbb{N}^{(\alpha)}}$ ($x\in X$), we have $$M^\prime f = M'(x\mathds{1}_{\mathbb{N}^{(\alpha)}})= M(x\mathds{1}_{\mathbb{Z}^{(\alpha)}}) = x.$$
Moreover, $f'$ and $f$ have the same supremum norm by definition; consequently, the operator norm of $M'$ is upper-bounded by the operator norm of $M$, which is in turn no greater than $\lambda$. To show that $M'$ is an $X$-valued invariant $\lambda$-mean, it remains to show that $M'$ is translation-invariant. Since every $g \in \mathbb{N}^{(\alpha)}$ can be written as a finite sum of ‘basis elements’ $e_i$ where $i \in \alpha$, it suffices to show that for every $i \in \alpha$ and every $f \in \ell_\infty(\mathbb{N}^{(\alpha)}, X)$, we have $M'(f_{e_i}) = M'(f)$.
Let $\pi_i\colon \mathbb{Z}^{(\alpha)} \rightarrow \mathbb{Z}$ be the ‘projection map’ homomorphism which satisfies $\pi_i(e_i) = 1$ and $\pi_i(e_j) = 0$ for all $j \neq i$. For a predicate $\phi\colon \mathbb{Z} \rightarrow \{ \textrm{true}, \textrm{false} \}$ and $h \in \ell_\infty(\mathbb{Z}^{(\alpha)}, X)$, we define $h[\phi(\pi_i)]$ to be the restriction of $h$ to the values where the predicate is true: $$h[\phi(\pi_i)](x) := \begin{cases}
h(x), & \textrm{ if } \phi(\pi_i(x)), \\
0, & \textrm{ otherwise.}
\end{cases}$$ Then we can write: $$f' = f'[\pi_i \leqslant -2] + f'[\pi_i = -1] + f'[\pi_i = 0] + f'[\pi_i \geqslant 1]$$ and similarly: $$(f_{e_i})' = (f_{e_i})'[\pi_i \leqslant -1] + (f_{e_i})'[\pi_i \geqslant 0].$$ Observe the following:
- $f'[\pi_i \leqslant -2]$ is a translate of $(f_{e_i})'[\pi_i \leqslant -1]$, so $$M(f'[\pi_i \leqslant -2]) = M((f_{e_i})'[\pi_i \leqslant -1]);$$
- $f'[\pi_i \geqslant 1]$ is a translate of $(f_{e_i})'[\pi_i \geqslant 0]$, so $$M(f'[\pi_i \geqslant 1]) = M((f_{e_i})'[\pi_i \geqslant 0]);$$
- $M(f'[\pi_i = 0])$ is zero by Lemma \[trivial\];
- $M(f'[\pi_i = 1])$ is also zero, by Lemma \[trivial\] combined with translation-invariance of $M$.
By linearity of $M$, it follows that $M(f') = M((f_{e_i})')$, and therefore (by definition) $M'(f) = M'(f_{e_i})$. The result follows.
It is a standard fact that subgroups and quotients of amenable discrete groups are amenable. Using exactly the same ideas one can prove that if a Banach space admits an invariant mean with respect to a group, then so does it with respect to subgroups and quotients of the said group.
\[normal\]Let $X$ be a Banach space and let $G$ be a group with a normal subgroup $H$. Suppose that there exists an $X$-valued invariant $\lambda$-mean $M\colon \ell_\infty(G,X)\to X$ on $G$. Then,
there exists an $X$-valued invariant $\lambda$-mean on $H$;
there exists an $X$-valued invariant $\lambda$-mean on $G/H$;
if, moreover, $S$ and $T$ are isomorphic semigroups and there exists an $X$-valued invariant mean on $S$, then there exists such a mean on $T$ too.
In order to prove (i), let $\iota\colon \ell_{\infty}(H, X)\to \ell_\infty(G,X) = \ell_{\infty}(\bigsqcup_{i\in I} Hg_i, X)$ be given by $\iota(f) = \bigsqcup_{i\in I} f_i$, where $f_i(hg_i) = f(h)$ ($i\in I$). Here, $(g_i)_{i\in I}$ are representative elements in $G$ chosen so that $G = \bigsqcup_{i\in I} Hg_i$. Then $\widehat{M} = M \circ \iota$ is the sought invariant mean on $H$.
For (ii), let $\pi\colon G\to G/H$ be the quotient map. Then the map $\widehat{M}(f) = M( f\circ \pi)$ ($f\in \ell_\infty(G/H,X)$) is the sought invariant mean on $G/H$.
Clause (iii) is quite trivial. Let $\theta\colon T\to S$ be an isomorphism and $M\colon \ell_\infty(S, X)\to X$ a mean on $S$. Then $Nf = M(f\circ \theta)$ ($f\in \ell_\infty(T,X)$) is an invariant mean on $T$
We shall require towards the proof the principle of local reflexivity due to Lindenstrauss and Rosenthal ([@lindros]), which we state here for the future reference.
\[LR\]Let $X$ be a Banach space and let $F\subset X^{**}$ be a finite-dimensional subspace. Then, for each $\varepsilon \in (0,1)$ there exists a linear map $P_F^\varepsilon \colon F\to \kappa_X(X)$ such that
$(1-\varepsilon)\|x\|\leqslant \|P_F^\varepsilon x\| \leqslant (1+\varepsilon)\|x\| \quad (x\in F)$;
$P_F^\varepsilon x = x$ for $x\in F\cap \kappa_X(X)$.
Proofs of the main results
==========================
We start with the proof of Theorem B on which Theorem A is reliant. The proof proceeds by partitioning $S := \textrm{Fin } V \setminus \{ 0 \} $ (the set of all non-zero finite-dimensional subspaces of an infinite-dimensional vector space $V$) into disjoint sets $\mathcal{P}$ and $\mathcal{Q}$. Then, we construct a bijection between $\textrm{Fin } V$ and the free commutative monoid $F_{\mathcal{P}}$, obtaining the operation $\ast$ by pulling back the monoid operation through this bijection.
We begin by using Zermelo’s well-ordering principle to construct a bijection $\theta\colon S \rightarrow \alpha$, where $\alpha$ is a cardinal number (that is, an initial ordinal). This induces a well-ordering $\phi$ on the collection $F_S$ of finite multisets of elements from $S$. Specifically, if $A \in F_S$ contains elements $U_1, U_2, \dots, U_k$ with respective multiplicities $c_1, c_2, \dots, c_k$, then we define: $$\phi(A) := \omega^{\theta(U_1)} \cdot c_1 + \dots + \omega^{\theta(U_k)} \cdot c_k,$$ where without loss of generality we have taken $\theta(U_1) > \dots > \theta(U_k)$. The well-ordering $\phi$ on $F_S$ is known as *colexicographical order*, and the order type is again $\alpha$.
$F_S$ is a graded monoid, where the grading $v(A) = c_1 + \dots + c_k$ is the total number of elements (counted with multiplicity) in the multiset. It is convenient to define, for each $n \in \mathbb{N}$, the set $$F_S^{(n)} := \{ A \in F_S\colon v(A) \geqslant n \}.$$
In particular, $F_S^{(0)} = F_S$, $F_S^{(1)}$ is the collection of non-empty multisets in $F_S$, and $F_S^{(2)}$ is the collection of non-empty non-singleton multisets in $F_S$.
For each ordinal $\beta \leqslant \alpha$, we define an injective partial function $g_{\beta}\colon F_S^{(2)} \rightarrow S$. Specifically:
- When $\beta = 0$ is the zero ordinal, $g_{\beta}$ is the empty partial function.
- When $\beta = \gamma^{+}$ is a successor ordinal, $g_{\beta}$ is obtained from $g_{\gamma}$ by extending it with another element $g_{\beta}(A) = W$. $A$ is defined to be the first (according to the colexicographical order $\phi$) element $A \in F_S^{(2)}$ such that $A$ is not in the domain of $g_{\gamma}$ and no element of $A$ is in the image of $g_{\gamma}$. $W$ is an arbitrary element of $S$ such that:
1. For all $B \subsetneq A$ with $v(B) \geqslant 2$, $g_{\gamma}(B)$ is a subspace of $W$ (note that the colexicographical ordering on $F_S^{(2)}$ ensures that we have already defined $g_{\gamma}$ on all such subsets $B$);
2. For all $U \in A$, $U$ is a subspace of $W$;
3. $W$ occurs later in the lexicographical ordering on $S$ than any element in $A$ or in the image of $g_{\gamma}$.
It is necessary to show that these conditions are consistent. Conditions (1) and (2) merely specify that $W$ contains a particular finite-dimensional subspace $W'$ (the Minkowski sum of all $U \in A$ and all $g_B$ where $B \subsetneq A$ with $v(B) \geqslant 2$); as $V$ is infinite-dimensional, the collection of spaces $\{ W \in \textrm{Fin } V\colon W' \subseteq W \}$ has the same cardinality $\textrm{Fin } V$ itself. Owing to our choice of an initial ordinal for the well-ordering, condition (3) only eliminates a strictly smaller set of candidates for $W$, so the set of admissible $W$ is non-empty.
- When $\beta$ is a limit ordinal, $g_{\beta}$ is the union of the partial functions $g_{\gamma}$ (for all $\gamma < \beta$).
At the end of this transfinite induction, we have a bijection $g_{\alpha} : F_{\mathcal{P}}^{(2)} \rightarrow \mathcal{Q}$, where $\mathcal{P}$ and $\mathcal{Q}$ are complementary subsets of $S$.
We extend $g_{\alpha}$ to a bijection $f\colon F_{\mathcal{P}} \rightarrow \textrm{Fin } V$ by setting:
- $f(\varnothing) := \{0\}$;
- $f(\{V\}) := V$ for all singleton sets $\{V\}$ where $V \in \mathcal{P}$;
- $f(A) := g_{\alpha}(A)$ if $v(A) \geqslant 2$.
At this point, one can define a binary operation $\ast\colon \textrm{Fin } V \times \textrm{Fin } V \rightarrow \textrm{Fin } V$ by $$U_1 \ast U_2 := f(f^{-1}(U_1) \sqcup f^{-1}(U_2)),$$ where $\sqcup$ is the usual ‘disjoint union’ operation that makes $F_{\mathcal{P}}$ into a free commutative monoid. This immediately establishes part (i) of the proposition, that $({\rm Fin}\, V, \ast)$ is a free commutative monoid.
What remains to be shown is part (ii) of Theorem B, that for any $F,G \in {\rm Fin}\, V$ we have $F,G \subseteq F \ast G$. We can assume that neither $F$ nor $G$ is the trivial space $\{0\}$, as this is the identity of the operation $\ast$ and the statement would trivially hold.
Consequently, $v(f^{-1}(F)) \geqslant 1$ and $v(f^{-1}(G)) \geqslant 1$, so $v(f^{-1}(F \ast G)) \geqslant 2$. This means that $A := f^{-1}(F \ast G)$ is in $F_{\mathcal{P}}^{(2)}$ and therefore appeared (together with the space $W := F \ast G$) in the transfinite induction. We shall show that $F$ is a subspace of $W := F \ast G$:
- If $v(f^{-1}(F)) = 1$, then $f^{-1}(F)$ is the singleton set $\{F\}$. As such, $F \in A$, and therefore $F$ is a subspace of $W$ by condition (2) in the transfinite induction.
- If $v(f^{-1}(F)) \geqslant 2$, then $f^{-1}(F) = B \subsetneq A$ and $F = f(B)$ is a subspace of $W$ by condition (1) in the transfinite induction.
By symmetry, it follows that $G$ is also a subspace of $F \ast G$, establishing the truth of Theorem B.
We are now ready to prove Theorem A.
We are to prove the implications $\Rightarrow$ and $\Rightarrow$ .
Without loss of generality we may assume that $X$ is infinite-dimensional. In order to prove the former implication, let us consider the free commutative monoid $$S = {\rm Fin}\, X^{**} \oplus \{2^{-k}\colon k\in \mathbb N\},$$ where ${\rm Fin}\, X^{**}$ is the monoid of finite-dimensional subspaces of $X^{**}$ considered in Theorem B and $\{2^{-k}\colon k\in \mathbb N\}$ is the monoid generated by $2$ in the multiplicative group of non-zero real numbers. By Theorem B, $S$ is a free commutative monoid and as such, it is isomorphic to $\mathbb N^{(|X^{**}|)}$.
Assume that there exists an $X$-valued invariant $\lambda$-mean on the Grothendieck group $G(S)$ of $S$, bearing in mind that $G(S)$ is isomorphic to $\mathbb Z^{(|X^{**}|)}$ and $S$ is isomorphic to $\mathbb N^{(|X^{**}|)}$. By Lemma \[groth\] (and Lemma \[normal\](iii)), we may fix an $X$-valued invariant $\lambda$-mean on $S$, $M\colon \ell_\infty(S, X)\to X$. Using the principle of local reflexivity (Theorem \[LR\]), for each $\varepsilon = 2^{-k}$ with $k\geqslant 1$ and $F\in {\rm Fin}\, X^{**}$, we fix $P^{\varepsilon}_{F}$ as in the statement. We also set $P^1_F = 0$, the zero operator on $F\in {\rm Fin}\, X^{**}$.
For an element $x\in X^{**}$, we define a function $f^x\colon S \to X$ by $$f^x(F, \delta) = P^{\min\{\delta, 1/2\}}_{F}x\quad \big((F,\delta)\in S\big),$$ where $[x]$ stands for the line spanned by $x$. Then $f^x \in \ell_\infty(G, X)$ with $\|f^x\| \leqslant 2\|x\|$.
We define $P\colon X^{**}\to X$ by $$Px = M(f^x)\quad (x\in X^{**})$$ and *claim* that it is the sought linear projection. Certainly, for $x\in \kappa_X(X)$, we have $$Px = M(f^x) = M (P^{\min\{\boldsymbol{\cdot}, 1/2\}}_{\boldsymbol{\cdot}}x)\! =M \big((P^{\min\{\boldsymbol{\cdot}, 1/2\}}_{\boldsymbol{\cdot}}x)_{([x],1)}\big) = M (P^{\min\{\boldsymbol{\cdot}, 1/2\}}_{\boldsymbol{\cdot}\ast[x]}x) = M (x \mathds{1}_{S}) = x.$$ Also, $P$ is linear. Indeed, homogeneity is clear. To see that $P$ is additive, we compute: $$\begin{array}{lcl}P(x+y)& =& M(f^{x+y})\\
&=& M \big(P^{\min\{\boldsymbol{\cdot}, 1/2\}}_{\boldsymbol{\cdot}}(x+y)\big)\\
&=& M \big((P^{\min\{\boldsymbol{\cdot}, 1/2\}}_{\boldsymbol{\cdot}}(x+y))_{([x,y], 1)}\big)\\
& = & M \big(P^{\min\{\boldsymbol{\cdot}, 1/2\}}_{\boldsymbol{\cdot}\ast [x,y]}x\big) + M \big(P^{\min\{\boldsymbol{\cdot}, 1/2\}}_{\boldsymbol{\cdot}\ast [x,y]}y\big)\\
& = & M \big(P^{\min\{\boldsymbol{\cdot}, 1/2\}}_{\boldsymbol{\cdot}}x\big) + M \big(P^{\min\{\boldsymbol{\cdot}, 1/2\}}_{\boldsymbol{\cdot}}y\big)\\
&=& Px + Py,\end{array}$$ where $[x,y]$ is the linear span of $x$ and $y$ ($x,y\in X^{**})$. Here, we used Theorem B from which it follows that for any $F\in {\rm Fin}\, X^{**}$, $[x,y]\subset F\ast [x,y]$. Finally, for any $x\in X^{**}$, by the translation-invariance of $M$, we have $$\|Px\| = \|M(f^x)\| \leqslant \lambda \cdot \inf_{k\geqslant 1}\|f^x_{([x], 1/2^k)}\| \leqslant \lambda\cdot \inf_{k\geqslant 1} \sup_{F\in {\rm Fin}\, X^{**}} \|P^{1/2^k}_F x\| \leqslant \lambda\|x\|,$$ which proves that indeed $X$ is $\lambda$-complemented in $X^{**}$.
In order to prove $\Rightarrow$ , we observe that $X^{**}$ is, in particular, a $\mathbb Q$-vector space, and being infinite-dimensional (as an $\mathbb R$-vector space), it is thus isomorphic to the direct sum $\mathbb Q^{(|X^{**}|)}$. As such, it contains a subgroup isomorphic to $\mathbb Z^{(|X^{**}|)}$. Thus, if there exists an $X$-valued invariant mean on $X^{**}$, by Lemma \[normal\], there is an $X$-valued invariant mean on a free Abelian group of rank $|X^{**}|$.
Countable amenable groups
-------------------------
As already explained, separable Banach spaces with the Metric Approximation Property are 1-complemented in their bidual if and only if they admit an invariant 1-mean on the group of integers. If one replaces the Metric Approximation Property with the $\gamma$-Bounded Approximation Property (with parameter $\gamma \geqslant 1$), then complementation in the bidual is still characterised by the existence of an invariant mean on the group of integers but the correspondence between the norm of the mean and the upper bound for the norm of a projection from the bidual is impeded by the parameter $\gamma$. Nonetheless, we do not know the answer to the following question.
> *Let $X$ be a separable Banach space. Suppose that for some countably infinite amenable group $G$ there exists an $X$-valued invariant mean on $G$. Is $X$ complemented in the bidual $X^{**}$?*
Acknowledgements {#acknowledgements .unnumbered}
----------------
The second-named author wishes to express his thanks to Rados[ł]{}aw [Ł]{}ukasik for numerous conversations concerning the problem.
[99]{} Y. Benyamini, Introduction to the uniform classification of Banach spaces, in: A. Aizpuru-Tomás and F. León-Saavedra, *Advanced courses of mathematical analysis I*, World Sci. Publ., Hackensack, New York, 2004, pp. 1–29. H. Bustos Domecq, Vector-valued invariant means revisited. *J. Math. Anal. Appl.* **275** (2002), no. 2, 512–520. F. J. García-Pacheco and F. J. Pérez-Fernández, *Vector-Valued Banach Limits and Vector-Valued Almost Convergence*, `arxiv.org/abs/1801.06235` (2018), 146 pp. T. Kania, Vector-valued invariant means revisited once again, *J. Math. Anal. App.* **445** (2017), no. 1, 797–802. J. Lindenstrauss and H.P. Rosenthal, The $\mathscr{L}_p$ spaces. *Israel J. Math.* **7** (1969), 325–349. R. [Ł]{}ukasik, Invariant means on Banach spaces, *Ann. Math. Sil.* **31** (2017), 127–140. A. Pe[ł]{}czyński, Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions, *Rozprawy Mat.* **58** (1968), 92 pp. A. Pe[ł]{}czyński and V.N. Sudakov, Remark on non-complemented subspaces of the space $m(S)$, *Colloq. Math.* **19** (1962), 85–-88.
[^1]: The second-named author acknowledges with thanks funding received from SONATA BIS no. 2017/26/E/ST1/00723.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We show that a law-invariant pricing functional defined on a general Orlicz space is typically incompatible with frictionless risky assets in the sense that one and only one of the following alternatives can hold: Either every risky payoff has a strictly-positive bid-ask spread or the pricing functional is given by an expectation and, hence, every payoff has zero bid-ask spread. In doing so we extend and unify a variety of “collapse to the mean” results from the literature and highlight the key role played by law invariance in causing the collapse. As a byproduct, we derive a number of applications to law-invariant acceptance sets and risk measures as well as Schur-convex functionals.'
author:
- '[^1]'
- '[^2], [^3]'
- '[^4]'
title: 'Law-invariant insurance pricing and its limitations'
---
0em
Introduction
============
In the actuarial literature it is customary to price insurance contracts by using law-invariant pricing functionals, i.e. pricing functionals whose outcomes are fully determined by the probability distribution of the contracts’ payoffs. Deprez and Gerber [@DeprezGerber1985] refer to such functionals as classical premium principles. Much of the original actuarial literature focused on pricing functionals applied to insurance claims, i.e. pricing functionals that are defined on the positive cone of a suitable space of random variables. More recently, efforts have been made to extend the theory to encompass pricing functionals defined on the whole model space; see e.g. Venter [@Venter1991], Gerber and Shiu [@GerberShiu1996], Wang et al. [@WangYoungPanjer1997], Wang [@Wang2000], Goovaerts et al. [@GoovaertsKaasDhaeneTang2004], Goovaerts and Laeven [@GoovaertsLaeven2008]. All of these papers study pricing functionals that, in addition to law invariance, satisfy a set of desirable properties such as convexity, sublinearity, comonotonicity, cash additivity, monotonicity with respect to stochastic orderings. We refer to Section \[sec: underlying model\] for the formal definitions of these and other properties.
In Castagnoli et al. [@CastagnoliMaccheroniMarinacci2004], it was shown that requiring pricing functionals to satisfy law invariance along with some of the desirable properties mentioned above can be highly restrictive. More specifically, the focus of Castagnoli et al. [@CastagnoliMaccheroniMarinacci2004] was essentially on law-invariant pricing functionals on the space of bounded random variables that are sublinear, nondecreasing, and comonotonic. The main result stated there is that, for such a pricing functional, one and only one of the following alternatives holds (see Theorem \[theo: castagnoli et al\] in the appendix):
- EITHER every payoff has zero bid-ask spread
- OR every risky payoff has a strictly-positive bid-ask spread.
In other words, either the underlying “market” is fully frictionless, or the only payoffs that can have zero bid-ask spread are risk-free payoffs. In the former case, the pricing functional collapses to a positive multiple of the expectation functional with respect to the reference probability measure. In the context of risk measures defined on the space of bounded random variables, the above result was extended by Frittelli and Rosazza Gianin [@FrittelliRosazza2005]. They kept monotonicity but replaced sublinearity by convexity and comonotonicity by cash additivity (see Theorem \[theo: frittelli and rosazza\] in the appendix). It is worth noting that comonotonicity implies cash additivity and, in the space of bounded random variables, cash additivity implies continuity. Hence, in the above results the functionals under scrutiny are always continuous.
In this paper we extend the results in Castagnoli et al. [@CastagnoliMaccheroniMarinacci2004] and Frittelli and Rosazza Gianin [@FrittelliRosazza2005] in two directions: (1) We establish the minimal properties that cause a pricing functional to “collapse” to the mean. In particular, neither monotonicity nor cash additivity are needed and continuity can be relaxed to (order) lower semicontinuity. (2) We go beyond the setting of bounded positions and show that the above results hold on general Orlicz spaces. We note that even though we drop monotonicity, we continue to use the language of pricing functionals to highlight the original motivation of our research.
As a byproduct, we obtain a variety of interesting applications. First, we show that a coherent law-invariant acceptance set is always pointed unless it is generated by the expectation under the reference probability measure. Second, we provide a characterization of law invariance for a risk measure based on acceptance sets and eligible assets. These risk measures go back to the original contribution of Artzner et al. [@ArtznerDelbaenEberHeath1999] and have been studied by Föllmer and Schied [@FoellmerSchied2002], Frittelli and Scandolo [@FrittelliScandolo2006], Artzner et al. [@ArtznerDelbaenKoch2009], and Farkas et al. [@FarkasKochMunari2015] among others. In the convex case, we show that a multi-asset risk measure is never law invariant unless it is a negative multiple of the expectation under the reference probability measure. As a final application, we extend the quantile representation for Schur-convex functionals obtained in Dana [@Dana2005] and Grechuk and Zabarankin [@GrechukZabarankin2012] from $L^p$ spaces to Orlicz spaces, and sharpen it by showing that one can always choose representing quantiles of bounded random variables.
The paper is structured as follows. Section 2 introduces terminology in the setting of $L^1$. In Section 3 we discuss some simple preliminary results on frictionless payoffs. These are needed in Section 4, which features our main results on sublinear and convex pricing functionals. In Section 5 we discuss applications to acceptance sets and risk measures. Finally, Section 6 is devoted to extending our results to general Orlicz spaces and highlighting applications to Schur-convex functionals. A brief review of the main results from the literature is provided in Section A and a general picture on some key results on (quantile) representations and extensions of law-invariant functionals is presented in Section B.
The underlying model {#sec: underlying model}
====================
We consider a one-period market with dates $t=0$ and $t=1$ in which future uncertainty is modeled by a nonatomic probability space $(\Omega,{{\mathcal{F}}},{\mathbb{P}})$. A random variable on $(\Omega,{{\mathcal{F}}},{\mathbb{P}})$ is interpreted as the state-contingent payoff of a financial contract at time $1$. We assume that payoffs at time $1$ belong to $$L^1:=L^1(\Omega,{{\mathcal{F}}},{\mathbb{P}}).$$ Each element of $L^1$ is an equivalence class with respect to almost-sure equality under ${\mathbb{P}}$ of random variables $X:\Omega\to{\mathbb{R}}$ such that $$\|X\|_1 := {\mathbb{E}}[|X|] < \infty,$$ where ${\mathbb{E}}$ denotes the expectation with respect to ${\mathbb{P}}$. As usual, we do not distinguish between an element of $L^1$ and any of its representatives. By abuse of notation, we will identify a real number with the random variable that is almost-sure identical to it. The space $L^1$ is partially ordered by the standard almost-sure ordering with respect to ${\mathbb{P}}$ and is equipped with the linear topology induced by the lattice norm $\|\cdot\|_1$. We denote by $L^\infty$ the subspace of $L^1$ consisting of bounded random variables.
We assume that each payoff demands a certain price at time $0$, which is represented by a functional $$\pi:L^1\to{\mathbb{R}}\cup\{\infty\}.$$ More precisely, for every payoff $X$ we interpret $\pi(X)$ as an [*ask price*]{}, i.e. the price from a seller’s perspective, expressed in a given numeraire. We adopt the standard convention according to which a positive value of $\pi(X)$ means that the seller receives $\pi(X)$ units of the numeraire when selling $X$ and a negative value of $\pi(X)$ means that the seller actually needs to pay $-\pi(X)$ units of the numeraire to “sell” $X$. As usual, the corresponding [*bid price*]{}, i.e. the price from a buyer’s perspective, is given by $-\pi(-X)$; see Jouini [@Jouini2000]. The difference between the ask and the bid price for $X$, the so-called [*bid-ask spread*]{} associated to $X$, is therefore given by the quantity $\pi(X)+\pi(-X)$.
We say that a payoff is frictionless when it has zero bid-ask spread and strongly frictionless when, in addition, its price per unit does not depend on the transacted volume. This is made precise by the following definition.
\[def: frictionless\] Let $\pi:L^1\to{\mathbb{R}}\cup\{\infty\}$. We say that a payoff $X\in L^1$ is:
(1) [*risk-free*]{} if it is constant.
(2) [*risky*]{} if it is not constant.
(3) [*frictionless (under $\pi$)*]{} if $\pi(-X)=-\pi(X)$.
(4) [*strongly frictionless (under $\pi$)*]{} if $\pi(mX)=m\pi(X)$ for every $m\in{\mathbb{R}}$.
When there is no ambiguity we omit the explicit reference to $\pi$ and speak simply of frictionless and strongly frictionless payoffs.
\(i) Clearly, a payoff $X$ is (strongly) frictionless if and only if $-X$ is (strongly) frictionless. Moreover, for $X$ to be frictionless we must have $\pi(X)\neq\infty$.
\(ii) It is clear that if the payoff $X$ is strongly frictionless, then $mX$ is automatically frictionless for every $m\in{\mathbb{R}}$. The converse implication is, however, not true in general unless $\pi$ is convex, see Proposition \[prop: additivity under convexity\]. To see this, take a nonzero $Z\in L^1$ and consider the functional $\pi:L^1\to{\mathbb{R}}$ defined by $$\pi(X)=
\begin{cases}
1 & \mbox{if $X=mZ$ for some $m\in(0,\infty)$},\\
-1 & \mbox{if $X=mZ$ for some $m\in(-\infty,0)$},\\
0 & \mbox{otherwise}.
\end{cases}$$ Then, $mZ$ is frictionless for every $m\in{\mathbb{R}}$ but $Z$ is not strongly frictionless under $\pi$.
For two payoffs $X$ and $Y$ we write $X\sim Y$ whenever $X$ and $Y$ have the same probability law under ${\mathbb{P}}$. We say that $X$ and $Y$ are comonotone if there exists a payoff $Z$ such that $X=f(Z)$ and $Y=g(Z)$ for suitable nondecreasing functions $f,g:{\mathbb{R}}\to{\mathbb{R}}$. By convention we set $0\cdot\infty:=0$. We recall the following terminology for functionals.
\[def: properties functionals\] A functional $\pi:L^1\to{\mathbb{R}}\cup\{\infty\}$ is:
(1) [*convex*]{} if $\pi(\lambda X+(1-\lambda)Y)\leq\lambda\pi(X)+(1-\lambda)\pi(Y)$ for all $\lambda\in[0,1]$ and $X,Y\in L^1$.
(2) [*positively homogeneous*]{} if $\pi(\lambda X)=\lambda\pi(X)$ for all $\lambda\in[0,\infty)$ and $X\in L^1$.
(3) [*sublinear*]{} if $\pi$ is convex and positively homogeneous.
(4) [*nondecreasing*]{} if $\pi(X)\geq\pi(Y)$ for all $X,Y\in L^1$ with $X\geq Y$.
(5) [*law-invariant*]{} if $\pi(X)=\pi(Y)$ for all $X,Y\in L^1$ with $X\sim Y$.
(6) [*comonotonic*]{} if $\pi(X+Y)=\pi(X)+\pi(Y)$ for all comonotone $X,Y\in L^1$.
(7) [*(norm) continuous*]{} if $\pi(X)=\lim\pi(X_n)$ for all $(X_n)\subset L^1$ and $X\in L^1$ such that $X_n\to X$ with respect to $\|\cdot\|_1$.
(8) [*(norm) lower semicontinuous*]{} if $\pi(X)\leq\liminf\pi(X_n)$ for all $(X_n)\subset L^1$ and $X\in L^1$ such that $X_n\to X$ with respect to $\|\cdot\|_\infty$.
(9) [*$Z$-additive*]{} (for $Z\in L^1$) if $\pi(X+mZ)=\pi(X)+m\pi(Z)$ for all $X\in L^1$ and $m\in{\mathbb{R}}$.
(10) [*cash-additive*]{} if it is $1$-additive and $\pi(1)=1$, i.e. if $\pi(X+m)=\pi(X)+m$ for all $X\in L^1$ and $m\in{\mathbb{R}}$.
\[rem: simple remark on convex and frictionless\] For a functional $\pi:L^1\to{\mathbb{R}}\cup\{\infty\}$ such that $\pi(0)=0$ the following statements hold:
(1) If $\pi$ is convex, then every payoff has a nonnegative bid-ask spread, i.e. $\pi(X)+\pi(-X)\ge 0$ for every $X\in L^1$.
(2) If $\pi$ is convex and $X\in L^1$ is frictionless, then $\pi(mX)=m\pi(X)$ for every $m\in[-1,1]$. In particular, $mX$ is frictionless for every $m\in[-1,1]$.
(3) Every risk-free payoff is automatically frictionless under any of the following conditions:
1. $\pi(X+Y)=\pi(X)+\pi(Y)$ for all independent $X,Y\in L^1$.
2. $\pi$ is comonotonic.
3. $\pi$ is cash-additive.
Preliminary results {#sec: preliminary results}
===================
In this brief section we establish some useful results about frictionless payoffs and their link to $Z$-additivity. We start with the following simple preliminary lemma.
\[lem: additivity\] Let $\pi:L^1\to{\mathbb{R}}\cup\{\infty\}$ and take $Z\in L^1$. If $\pi(X+mZ)\leq\pi(X)+m\pi(Z)$ for all $X\in L^1$ and $m\in{\mathbb{R}}$, then $\pi$ is $Z$-additive.
It suffices to observe that, for any arbitrary $X\in L^1$ and $m\in{\mathbb{R}}$, we have $$\pi(X) = \pi((X+mZ)-mZ) \leq \pi(X+mZ)-m\pi(Z)\leq \pi(X)+m\pi(Z)-m\pi(Z)=\pi(X)$$ where both inequalities follow by applying our assumption.
We show that, under a sublinear pricing functional, the property of $Z$-additivity is equivalent to $Z$ being frictionless or, equivalently, strongly frictionless.
\[prop: additivity under sublinearity\] Assume that $\pi:L^1\to{\mathbb{R}}\cup\{\infty\}$ is sublinear. Then, for every $Z\in L^1$ the following statements are equivalent:
(a) $Z$ is frictionless.
(b) $Z$ is strongly frictionless.
(c) $\pi$ is $Z$-additive.
Since $\pi(0)=0$, it is immediate to see that [*(c)*]{} implies [*(b)*]{}, which readily implies [*(a)*]{}. To conclude the proof, assume that [*(a)*]{} holds and take $X\in L^1$ and $m\in{\mathbb{R}}$. It follows from our assumption and from positive homogeneity that $\pi(mZ)=m\pi(Z)$ for every $m\in{\mathbb{R}}$. Hence, sublinearity implies that $$\pi(X+mZ) \leq \pi(X)+\pi(mZ) = \pi(X)+m\pi(Z).$$ That [*(c)*]{} holds now follows from Lemma \[lem: additivity\].
It is not difficult to show that the above equivalence does not hold if $\pi$ is only required to be convex. For instance, given a payoff $Z$, consider the convex functional $\pi:L^1\to{\mathbb{R}}\cup\{\infty\}$ defined by $$\pi(X) =
\begin{cases}
m & \mbox{if $X=mZ$ for some $m\in[-1,1]$},\\
\infty & \mbox{otherwise}.
\end{cases}$$ Then, $Z$ is easily seen to be frictionless, but not strongly frictionless. In addition, $\pi$ fails to be $Z$-additive. However, in the convex case, we can still characterize when a payoff $Z$ is strongly frictionless in terms of the $Z$-additivity of the recession functional associated with $\pi$.
Let $\pi:L^1\to{\mathbb{R}}\cup\{\infty\}$. The [*recession functional*]{} associated with $\pi$ is the map $\pi^\infty:L^1\to{\mathbb{R}}\cup\{\infty\}$ defined by $$\pi^\infty(X) := \sup_{\lambda\in(0,\infty)}\frac{\pi(\lambda X)}{\lambda}.$$
If $\pi(0)=0$, then $\pi^\infty$ is the smallest positively-homogeneous map dominating $\pi$, i.e. such that $\pi(X)\leq\pi^\infty(X)$ for every $X\in L^1$. If, in addition, $\pi$ is convex, then $\pi^\infty$ is the smallest sublinear map dominating $\pi$.
\[prop: additivity under convexity\] Assume that $\pi:L^1\to{\mathbb{R}}\cup\{\infty\}$ is convex and satisfies $\pi(0)=0$. Then, for every $Z\in L^1$ the following are equivalent:
(a) $Z$ is strongly frictionless under $\pi$.
(b) $mZ$ is frictionless under $\pi$ for every $m\in{\mathbb{R}}$.
(c) $Z$ is frictionless under $\pi^\infty$.
(d) $Z$ is strongly frictionless under $\pi^\infty$.
(e) $\pi^\infty$ is $Z$-additive.
Under any of the above conditions we have $\pi(mZ)=\pi^\infty(mZ)$ for every $m\in{\mathbb{R}}$. If, in addition, $\pi$ is finite valued, the preceding statements are also equivalent to:
(a) $\pi$ is $Z$-additive.
After Definition \[def: frictionless\], we already noted that [*(a)*]{} implies [*(b)*]{}. Conversely, if [*(b)*]{} holds we have, by Remark \[rem: simple remark on convex and frictionless\], that $\pi(\lambda mZ)=\lambda\pi(m Z)$ for every $m\in{\mathbb{R}}$ and $\lambda\in[-1,1]$. This implies that $$\pi(Z) = \pi\bigg(\frac{1}{m}mZ\bigg) = \frac{1}{m}\pi(mZ),$$ for every $m\in{\mathbb{R}}$ with $|m|\ge 1$. As a result, $\pi(mZ)=m\pi(Z)$ for every $m\in{\mathbb{R}}$, showing that $Z$ is strongly frictionless under $\pi$. Hence, [*(a)*]{} and [*(b)*]{} are equivalent. Moreover, [*(c)*]{} is equivalent to both [*(d)*]{} and [*(e)*]{} by Proposition \[prop: additivity under sublinearity\]. If [*(a)*]{} holds, then for every $m\in{\mathbb{R}}$ we have that $$\pi^\infty(mZ) = \sup_{\lambda\in(0,\infty)}\frac{\pi(\lambda m Z)}{\lambda} = \pi(mZ) = m\pi(Z).$$ This yields that [*(a)*]{} implies [*(c)*]{} and shows that $\pi^\infty(mZ)=\pi(mZ)$ for every $m\in{\mathbb{R}}$. Finally, assume that [*(e)*]{} holds. Note that, since every payoff has a nonnegative bid-ask spread when $\pi$ is convex and $\pi(0)=0$, we have $\pi(mZ)\geq-\pi(-mZ)$ for every $m\in{\mathbb{R}}$ and, hence, $$\pi^\infty(mZ) \geq \pi(mZ) \geq -\pi(-mZ) \geq -\pi^\infty(-mZ) = m\pi^\infty(Z)= \pi^\infty(m Z).$$ This yields $\pi(mZ)=m\pi^\infty(Z)$ for every $m\in{\mathbb{R}}$ and shows that [*(e)*]{} implies [*(a)*]{}. This concludes the proof of the equivalence of [*(a)*]{} to [*(e)*]{}
Since $\pi(0)=0$, it is clear that [*(f)*]{} always implies [*(a)*]{}. Now, assume that $\pi$ is finite valued and that [*(a)*]{} holds. For arbitrary $X\in L^1$ and $m\in{\mathbb{R}}$, convexity implies that $$\pi(X+mZ) \leq \lambda\pi\left(\frac{1}{\lambda}X\right)+(1-\lambda)\pi\left(\frac{m}{1-\lambda}Z\right) = \lambda\pi\left(\frac{1}{\lambda}X\right)+m\pi(Z)$$ for every $\lambda\in(0,1)$. Since the function $f:{\mathbb{R}}\to{\mathbb{R}}$ defined by $f(\alpha)=\pi(\alpha X)$ is convex and, thus, continuous, we infer that $\pi(X+mZ)\leq\pi(X)+m\pi(Z)$. Lemma \[lem: additivity\] implies that [*(e)*]{} holds.
Main results {#sec: main results}
============
We are now able to provide our first generalizations of the results in Castagnoli et al. [@CastagnoliMaccheroniMarinacci2004] and Frittelli and Rosazza Gianin [@FrittelliRosazza2005].
### Sublinear pricing functionals {#sublinear-pricing-functionals .unnumbered}
The result for sublinear pricing functionals is based on the quantile representation obtained by Dana [@Dana2005] in the setting of bounded random variables. In what follows, for every payoff $X$ we denote by $q_X$ a fixed but arbitrary quantile function for $X$, i.e. a function $q_X:(0,1)\to{\mathbb{R}}$ such that $$\inf\{m\in{\mathbb{R}}\,; \ {\mathbb{P}}(X\leq m)\geq\alpha\} \leq q_X(\alpha) \leq \inf\{m\in{\mathbb{R}}\,; \ {\mathbb{P}}(X\leq m)>\alpha\}$$ for every $\alpha\in(0,1)$. Note that, since the cumulative distribution function of $X$ has at most countably many discontinuity points, any two quantile functions for $X$ coincide almost surely with respect to the Lebesgue measure on $[0,1]$. We refer to Föllmer and Schied [@FoellmerSchied2011] for more on quantile functions.
\[lem: representation via quantile function\] Assume that $\pi:L^1\to{\mathbb{R}}\cup\{\infty\}$ is sublinear, lower semicontinuous, and law invariant. Then, there exists a set ${\mathcal{D}}\subset L^\infty$ such that for every $X\in L^1$ $$\pi(X) = \sup_{Y\in{\mathcal{D}}}\int_0^1q_X(\alpha)q_Y(\alpha)d\alpha.$$
Since $L^\infty$ is the topological dual of $L^1$, it follows from a classical result in convex analysis, see e.g. Corollary 5.99 in Aliprantis and Border [@AliprantisBorder2006], that $\pi$ is $\sigma(L^1,L^\infty)$ lower semicontinuous. The desired representation is now a direct consequence of Proposition \[prop: law invariance and schur convexity\].
\[theo: main collapse\] Assume that $\pi:L^1\to{\mathbb{R}}\cup\{\infty\}$ is sublinear, lower semicontinuous, and law invariant. Then, the following statements are equivalent:
(a) There exists a frictionless risky payoff $Z\in L^1$ with ${\mathbb{E}}[Z]\neq0$ .
(b) There exists $c\in{\mathbb{R}}$ such that $\pi(X)=c{\mathbb{E}}[X]$ for every $X\in L^1$.
(c) Every payoff is frictionless.
Clearly, we only need to prove that [*(a)*]{} implies [*(b)*]{}. To this effect, assume that [*(a)*]{} holds so that $\pi(Z)+\pi(-Z)=0$. Let ${\mathcal{D}}$ be the subset of $L^\infty$ from Lemma \[lem: representation via quantile function\]. Then, for every fixed $Y\in{\mathcal{D}}$ we must have $$\begin{aligned}
0
&\geq&
\int_0^1q_Z(\alpha)q_Y(\alpha)d\alpha+\int_0^1q_{-Z}(\alpha)q_Y(\alpha)d\alpha \\
&=&
\int_0^1q_Z(\alpha)q_Y(\alpha)d\alpha-\int_0^1q_Z(1-\alpha)q_Y(\alpha)d\alpha \\
&=&
\int_0^1q_Z(\alpha)[q_Y(\alpha)-q_Y(1-\alpha)]d\alpha \\
&=&
\int_0^{1/2}q_Z(\alpha)[q_Y(\alpha)-q_Y(1-\alpha)]d\alpha+
\int_{1/2}^1q_Z(\alpha)[q_Y(\alpha)-q_Y(1-\alpha)]d\alpha \\
&=&
\int_0^{1/2}[q_Z(\alpha)-q_Z(1-\alpha)][q_Y(\alpha)-q_Y(1-\alpha)]d\alpha.\end{aligned}$$ Since $q_Z$ is nondecresing, we have $q_Z(\alpha)-q_Z(1-\alpha)\leq0$ for almost every $\alpha\in(0,1/2]$. The same holds for $q_Y$. Together with the above inequality, this implies that $$\int_0^{1/2}[q_Z(\alpha)-q_Z(1-\alpha)][q_Y(\alpha)-q_Y(1-\alpha)]d\alpha = 0.$$ Since $Z$ is risky, we find $\beta\in(0,1/2)$ such that $q_Z(\alpha)-q_Z(1-\alpha)<0$ for almost every $\alpha\in(0,\beta]$. Hence, the above identity can only hold if $q_Y(\alpha)=q_Y(1-\alpha)$ for almost every $\alpha\in(0,\beta]$. Being nondecreasing, $q_Y$ must therefore be almost-surely constant. It follows that in the representation in Lemma \[lem: representation via quantile function\], the set ${\mathcal{D}}$ must consist of constant random variables, i.e. ${\mathcal{D}}\subset{\mathbb{R}}$, and for every $X\in L^1$ $$\pi(X)=
\begin{cases}
c_1{\mathbb{E}}[X] & \mbox{if} \ {\mathbb{E}}[X]<0\\
c_2{\mathbb{E}}[X] & \mbox{if} \ {\mathbb{E}}[X]\geq0
\end{cases}$$ where $c_1=\inf{\mathcal{D}}\le\sup{\mathcal{D}}=c_2$. Assuming without loss of generality that ${\mathbb{E}}[Z]>0$ (otherwise replace $Z$ by $-Z$), we use that $Z$ is frictionless to obtain $$c_2{\mathbb{E}}[Z] = \pi(Z) = -\pi(-Z) = c_1{\mathbb{E}}[Z].$$ Recall that the fact that $Z$ is frictionless also implies that $\pi(Z)$ is finite, so that $c_1$ and $c_2$ must coincide and be finite. This establishes [*(b)*]{} and concludes the proof.
The condition ${\mathbb{E}}[Z]\neq0$ in point [*(a)*]{} above is necessary for the equivalence to hold. To see this, consider the functional $\pi:L^1\to{\mathbb{R}}\cup\{\infty\}$ defined by $$\pi(X)=
\begin{cases}
{\mathbb{E}}[X] & \mbox{if} \ {\mathbb{E}}[X]\geq0,\\
0 & \mbox{otherwise}.
\end{cases}$$ It is easy to verify that $\pi$ is sublinear, law invariant, and lower semicontinuous. However, since a payoff $X$ is frictionless if, and only if, ${\mathbb{E}}[X]=0$, we see that condition [*(a)*]{} holds but conditions [*(b)*]{} and [*(c)*]{} are not satisfied.
The preceding theorem takes the following simpler form in the common situation where risk-free payoffs are assumed to be frictionless.
\[cor: main collapse\] Assume that $\pi:L^1\to{\mathbb{R}}\cup\{\infty\}$ is sublinear, lower semicontinuous, and law invariant. Moreover, assume that every risk-free payoff is frictionless (or equivalently, by sublinearity, some nonzero risk-free payoff is frictionless). Then, the following statements are equivalent:
(a) There exist a frictionless risky payoff.
(b) There exists $c\in{\mathbb{R}}$ such that $\pi(X)=c{\mathbb{E}}[X]$ for every $X\in L^1$.
(c) Every payoff is frictionless.
In view of Theorem \[theo: main collapse\], it suffices to prove that [*(a)*]{} implies the existence of a frictionless risky payoff $Z\in L^1$ such that ${\mathbb{E}}[Z]\neq0$. To this end, assume that [*(a)*]{} holds. Let $W\in L^1$ be a frictionless risky payoff and take any $m\in{\mathbb{R}}\setminus\{-{\mathbb{E}}[W]\}$. It is clear that $Z=W+m$ is a risky payoff with nonzero expectation. Moreover, $Z$ is frictionless. This follows from Proposition \[prop: additivity under sublinearity\] once we note that $$\pi(-Z) = \pi(0)-\pi(W)-\pi(m) = -\pi(Z+m) = -\pi(Z)$$ by $W$-additivity and $m$-additivity of $\pi$. This concludes the proof.
The above results extend Theorem 1 in Castagnoli et al. [@CastagnoliMaccheroniMarinacci2004] and Proposition 8 in Frittelli and Rosazza Gianin [@FrittelliRosazza2005] beyond the setting of bounded payoffs and by getting rid of the assumptions of comonotonicity, monotonicity, and cash-additivity as well as of the implicit assumption of continuity (see Remark \[rem: automatic continuity under cash additivity\]). In doing so, they show that the “collapse to the mean” is a general phenomenon caused by the interaction between law invariance and the existence of frictionless risky payoffs.
### Convex pricing functionals {#convex-pricing-functionals .unnumbered}
It is not difficult to verify that the preceding collapse to the mean does not generally hold if we consider a convex pricing functional instead of a sublinear one. This is illustrated by the following example.
\[ex: convex without collapse\] Consider the functional $\pi:L^1\to{\mathbb{R}}\cup\{\infty\}$ defined by $$\pi(X) = \inf\{m\in{\mathbb{R}}\,; \ X+m\geq-1\}.$$ It is clear that $\pi$ is convex, lower semicontinuous, and law invariant. Moreover, $\pi$ admits frictionless risky payoffs. For instance, taking $E\in{{\mathcal{F}}}$ such that ${\mathbb{P}}(E)=1/2$ and setting $X=1_E-1_{E^c}$, one easily sees that $\pi(X)=\pi(-X)=0$ so that $X$ is frictionless. However, $\pi$ does not coincide with the expectation under ${\mathbb{P}}$.
In this section we show how to characterize when a convex pricing functional collapse to the mean by exploiting the properties of the associated recession functional. As a preliminary step, it is worth observing the following simple fact.
Consider a functional $\pi:L^1\to{\mathbb{R}}\cup\{\infty\}$. If $\pi$ is convex, lower semicontinuous, and law invariant, then so is $\pi^\infty$.
It suffices to observe that, under the above assumptions, $\pi^\infty$ is the pointwise supremum of maps that are convex, lower semicontinuous, and law invariant and, as such, inherits all those properties.
\[theo: collapse under convexity\] Assume that $\pi:L^1\to{\mathbb{R}}\cup\{\infty\}$ is convex, lower semicontinuous, and law invariant. Then, the following statements are equivalent:
(a) There exists a strongly frictionless risky payoff $Z\in L^1$ with ${\mathbb{E}}[Z]\neq0$.
(b) There exists $c\in{\mathbb{R}}$ such that $\pi(X)=c{\mathbb{E}}[X]$ for every $X\in L^1$.
(c) Every payoff is frictionless.
In view of Proposition \[prop: additivity under convexity\], we only have to show that [*(a)*]{} implies [*(b)*]{}. To this effect, assume that [*(a)*]{} holds and note that $\pi(0)=\pi(0\cdot Z)=0\pi(Z)=0$. Since $Z$ is frictionless under $\pi^\infty$ by Proposition \[prop: additivity under convexity\], we infer from Theorem \[theo: main collapse\] that $\pi^\infty$ is a multiple of ${\mathbb{E}}$. Since $$\pi^\infty(X) \geq \pi(X) \geq -\pi(-X) \geq -\pi^\infty(-X) = \pi^\infty(X)$$ for every $X\in L^1$ by convexity of $\pi$ and linearity of $\pi^\infty$, we conclude that $\pi=\pi^\infty$ so that $\pi$ is also a multiple of ${\mathbb{E}}$. This establishes [*(b)*]{} and concludes the proof.
As above, the preceding theorem takes a simpler form in the case that risk-free payoffs are frictionless.
Assume that $\pi:L^1\to{\mathbb{R}}\cup\{\infty\}$ is convex, lower semicontinuous, and law invariant. Moreover, assume that every risk-free payoff is frictionless. Then, the following statements are equivalent:
(a) There exist a strongly frictionless risky payoff.
(b) There exists $c\in{\mathbb{R}}$ such that $\pi(X)=c{\mathbb{E}}[X]$ for every $X\in L^1$.
(c) Every payoff is frictionless.
In view of Theorem \[theo: collapse under convexity\], it suffices to prove that [*(a)*]{} implies the existence of a strongly frictionless risky payoff $Z\in L^1$ such that ${\mathbb{E}}[Z]\neq0$. To this effect, assume that [*(a)*]{} holds and let $W\in L^1$ be a strongly frictionless risky payoff. Note that $\pi(0)=\pi(0\cdot W)=0$. Take any $m\in{\mathbb{R}}\setminus\{-{\mathbb{E}}[W]\}$. It is clear that $Z=W+m$ is a risky payoff with nonzero expectation. Moreover, we have $$\pi^\infty(-Z) = \pi^\infty(0)-\pi^\infty(W)-\pi^\infty(m) = -\pi^\infty(W+m) = -\pi^\infty(Z)$$ by Proposition \[prop: additivity under convexity\]. Hence, $Z$ is frictionless under $\pi^\infty$. As a result, we may again apply Proposition \[prop: additivity under convexity\] to infer that $Z$ is strongly frictionless. This establishes the desired claim.
In view of Proposition \[prop: additivity under convexity\], the previous results extend Proposition 9 in Frittelli and Rosazza Gianin [@FrittelliRosazza2005] beyond the setting of bounded payoffs and monotonic and cash-additive functionals. Besides delivering a more general result, our proof is more direct and, by exploiting the properties of recession functionals, avoids the duality argument used there.
Applications {#sec: applications}
============
In this section we discuss applications of our results to acceptance sets and to risk measures with respect to multiple eligible assets. We start by recalling some standard notions for sets.
We say that a set ${\mathcal{A}}\subset L^1$ is:
(1) [*convex*]{} if $\lambda X+(1-\lambda)Y\in{\mathcal{A}}$ for all $\lambda\in[0,1]$ and $X,Y\in{\mathcal{A}}$.
(2) [*conic*]{} if $\lambda X\in{\mathcal{A}}$ for all $\lambda\in[0,\infty)$ and $X\in{\mathcal{A}}$.
(3) [*monotone*]{} if $X\in{\mathcal{A}}$ whenever $X\geq Y$ for some $Y\in{\mathcal{A}}$.
(4) [*law invariant*]{} if $X\in{\mathcal{A}}$ whenever $X\sim Y$ for some $Y\in{\mathcal{A}}$.
(5) [*(norm) closed*]{} if $X\in{\mathcal{A}}$ whenever $X_n\to X$ with respect to $\|\cdot\|_\infty$ for some sequence $(X_n)\subset{\mathcal{A}}$.
(6) an [*acceptance set*]{} if ${\mathcal{A}}$ is nonempty, monotone, and ${\mathcal{A}}\neq L^1$.
(7) a [*coherent acceptance set*]{} if ${\mathcal{A}}$ is a convex and conic acceptance set.
The basic construction principle for risk measures is captured in the next definition.
\[def: risk measure\] Consider a set ${\mathcal{A}}\subset L^1$, a vector space ${\mathcal{M}}\subset L^1$ containing a nonzero positive payoff $U\in L^1$, and a linear functional $\psi:{\mathcal{M}}\to{\mathbb{R}}$ such that $\psi(U)>0$. The [*risk measure*]{} associated to $({\mathcal{A}},{\mathcal{M}},\psi)$ is the functional $\rho:L^1\to{\mathbb{R}}\cup\{\pm\infty\}$ defined by $$\rho(X) := \inf\{\psi(Z) \,; \ Z\in{\mathcal{M}}, \ X+Z\in{\mathcal{A}}\}.$$ In particular, if $\dim({\mathcal{M}})=1$, then $$\rho(X) := \psi(U)\inf\{m\in{\mathbb{R}}\,; \ X+mU\in{\mathcal{A}}\}.$$ If $\dim({\mathcal{M}})=1$, we speak of a risk measure with respect to the single [*eligible payoff*]{} $U$. If $\dim({\mathcal{M}})>1$, we speak of a risk measure with respect to the multiple [*eligible payoffs*]{} in ${\mathcal{M}}$.
The acceptance set ${\mathcal{A}}$ represents the set of financial positions that are deemed to be acceptable, e.g. by a regulator. The subspace ${\mathcal{M}}$ and the functional $\psi$ represent, respectively, the marketed space and the pricing functional of a frictionless financial market where the Law of One Price holds. An element $X$ of the subspace ${\mathcal{M}}$ represents the payoff of a portfolio of traded securities and $\psi(X)$ the (unique) value of any portfolio with payoff $X$. For every payoff $X$, the quantity $\rho(X)$ is therefore interpreted as the minimal amount of capital that has to be raised and invested in a traded portfolio to ensure acceptability. Risk measures with respect to a single eligible payoff where introduced by Artzner et al. [@ArtznerDelbaenEberHeath1999]. The generalization to multiple eligible payoffs has been studied in Föllmer and Schied [@FoellmerSchied2002], Frittelli and Scandolo [@FrittelliScandolo2006], and Artzner et al. [@ArtznerDelbaenKoch2009]; see also Farkas et al. [@FarkasKochMunari2015] and Liebrich and Svindland [@LiebrichSvindland2017] for recent results in this direction.
Before applying our results to acceptance sets and risk measures with respect to multiple eligible payoffs we make a simple observation about risk measures with respect to a single riskless eligible payoff.
\[lem: cash additive rm and law invariance\] Assume that ${\mathcal{A}}\subset L^1$ is a closed acceptance set, $\dim({\mathcal{M}})=1$, and $U=1$. Then, the following statements are equivalent:
(a) $\rho$ is law invariant.
(b) ${\mathcal{A}}$ is law invariant.
Since ${\mathcal{A}}$ is closed, we clearly have $${\mathcal{A}}=\{X\in L^1 \,; \ \rho(X)\le 0\}.$$ This immediately shows that [*(a)*]{} implies [*(b)*]{}. To establish the converse implication, take an arbitrary $X\in L^1$ and note that $$\rho(X) = \psi(1)\inf\{m\in{\mathbb{R}}\,; \ X+m\in{\mathcal{A}}\}$$ by our assumption. Since $Y+m\sim X+m$ for every $Y\in L^1$ with $Y\sim X$ and every $m\in{\mathbb{R}}$, we see that [*(b)*]{} implies [*(a)*]{}.
### Acceptance sets {#acceptance-sets .unnumbered}
We show that every closed, coherent, law-invariant acceptance set is always pointed, unless it is the acceptance set generated by the expectation functional under ${\mathbb{P}}$.
Assume that ${\mathcal{A}}\subset L^1$ is a closed, coherent, law-invariant acceptance set. Then, one of the following two alternatives holds:
(i) ${\mathcal{A}}$ is pointed, i.e. ${\mathcal{A}}\cap(-{\mathcal{A}})=\{0\}$.
(ii) ${\mathcal{A}}=\{X\in L^1 \,; \ {\mathbb{E}}[X]\geq0\}$.
Consider the functional $\pi:L^1\to{\mathbb{R}}\cup\{\pm\infty\}$ defined by $$\pi(X) = \inf\{m\in{\mathbb{R}}\,; \ X+m\in{\mathcal{A}}\}.$$ Note that $\pi(0)=0$ for otherwise ${\mathcal{A}}$ would contain $L^\infty$ and, hence, $L^1$ by closedness. It follows from Lemma \[lem: cash additive rm and law invariance\] that $\pi$ is law invariant. Moreover, it is immediate to verify that $\pi$ is sublinear and lower semicontinuous. As in the proof of Lemma \[lem: cash additive rm and law invariance\], we have $$\label{eq: collapse acceptance set 2}
{\mathcal{A}}= \{X\in L^1 \,; \ \pi(X)\leq0\}.$$ Since $\pi(0)=0$ and $\pi$ is lower semicontinuous, it follows from Proposition 2.4 in Ekeland and Témam [@EkelandTemam1999] that $\pi(X)>-\infty$ for every $X\in L^1$. Note that for every $m\in{\mathbb{R}}$ we have $$\label{eq: collapse acceptance set}
-\pi(-m) = -\pi(0)-m = -m = \pi(0)-m = \pi(m),$$ showing that every risk-free payoff is frictionless. Now, assume that ${\mathcal{A}}$ is not pointed so that $Z\in{\mathcal{A}}\cap(-{\mathcal{A}})$ for some nonzero $Z\in L^1$. Then, $Z$ must satisfy $$0 = \pi(0) = \pi(Z-Z) \leq \pi(Z)+\pi(-Z) \leq 0$$ by sublinearity. In other words, $Z$ is frictionless. In addition, $Z$ must be risky because we would otherwise have $$0 \leq -\pi(-Z) = -Z = \pi(Z) \leq 0$$ by , implying $Z=0$. As a result, we infer from Corollary \[cor: main collapse\] that $\pi(X)=-{\mathbb{E}}[X]$ for every $X\in L^1$, where we have used the fact that $\pi(1)=\pi(0)-1=-1$. In view of , this yields alternative [*(ii)*]{} and concludes the proof.
The preceding result does not generally hold if ${\mathcal{A}}$ fails to be either monotone or conic. To see this, consider first the set $${\mathcal{A}}= \{X\in L^1 \,; \ \mbox{$X$ is risk-free}\}.$$ It is clear that ${\mathcal{A}}$ satisfies all the assumptions of the proposition apart from monotonicity but neither [*(i)*]{} nor [*(ii)*]{} holds. Now, consider the set $${\mathcal{A}}= \{X\in L^1 \,; \ {\mathbb{E}}[\min(X,0)]\geq-1\}.$$ The set ${\mathcal{A}}$ satisfies all the assumptions of the proposition apart from conicity but neither [*(i)*]{} nor [*(ii)*]{} holds.
Risk measures with respect to multiple eligible assets {#risk-measures-with-respect-to-multiple-eligible-assets .unnumbered}
------------------------------------------------------
We characterize when a risk measure of the above type is law invariant in the case that ${\mathcal{M}}$ contains a risky eligible payoff. In particular, we show that a risk measure with respect to multiple eligible assets is never law invariant unless it reduces to a multiple of the expectation under the reference probability measure. Here, we set $\ker(\psi):=\{X\in{\mathcal{M}}\,; \ \psi(X)=0\}$.
Assume that ${\mathcal{A}}\subset L^1$ is an acceptance set such that ${\mathcal{A}}+\ker(\psi)$ is convex and closed and that ${\mathcal{M}}$ contains a risky payoff. Moreover, assume that $\rho(0)=0$. Then, the following statements are equivalent:
(a) $\rho$ is law invariant.
(b) There exists $c\in(0,\infty)$ such that $\rho(X)=-c{\mathbb{E}}[X]$ for every $X\in L^1$.
In this case, there exists $c\in(0,\infty)$ such that $\psi(X)=c{\mathbb{E}}[X]$ for every $X\in{\mathcal{M}}$.
To establish the equivalence, we only need to show that [*(a)*]{} implies [*(b)*]{}. To this effect, assume that $\rho$ is law invariant and note that $\rho$ is convex and lower semicontinuous by Proposition 4 in Farkas et al. [@FarkasKochMunari2015]. Since $\rho(0)=0$, the lower semicontinuity of $\rho$ implies that $\rho(X)>-\infty$ for every $X\in L^1$ by Proposition 2.4 in Ekeland and Témam [@EkelandTemam1999]. Let $U$ be the payoff as in Definition \[def: risk measure\] and take a risky payoff $W\in{\mathcal{M}}$. Then, we find $\lambda\in{\mathbb{R}}$ such that $Z=U+\lambda W$ is risky and satisfies ${\mathbb{E}}[Z]\neq0$. In particular, if $U$ is risky, then it suffices to take $\lambda=0$ because $U$ is nonzero and positive by definition. Now, note that $$\rho(mZ) = \rho(0)-\psi(mZ) = -m\psi(Z) = m(\rho(0)-\psi(Z)) = m\rho(Z)$$ for every $m\in{\mathbb{R}}$, showing that $Z$ is strongly frictionless under $\rho$. Then, by Theorem \[theo: collapse under convexity\], there exists a constant $c\in{\mathbb{R}}$ such that $\rho(X)=c{\mathbb{E}}[X]$ for every $X\in L^1$. The desired statement follows by observing that $c=\rho(1)\leq\rho(0)=0$ and that $c=0$ is not possible because $\psi$ is nonzero.
To conclude the proof, assume that [*(b)*]{} holds and observe that $$\psi(X) = \rho(0)-\psi(-X) = \rho(-X) = -c{\mathbb{E}}[-X] = c{\mathbb{E}}[X]$$ for every $X\in{\mathcal{M}}$.
The augmented acceptance set ${\mathcal{A}}+\ker(\psi)$ is automatically convex whenever ${\mathcal{A}}$ is convex. However, in general, the closedness of ${\mathcal{A}}$ is not sufficient for ${\mathcal{A}}+\ker(\psi)$ to be closed. We refer to Baes et al. [@BaesKochMunari2018] for a variety of conditions ensuring that ${\mathcal{A}}+\ker(\psi)$ is closed when ${\mathcal{A}}$ is closed.
Extension to Orlicz spaces {#sec: Orlicz}
==========================
In this section we extend the preceding results to arbitrary Orlicz spaces. Here, payoffs at time $1$ are elements of the space $L^\Phi:=L^\Phi(\Omega,{{\mathcal{F}}},{\mathbb{P}})$ where $\Phi:[0,\infty)\to[0,\infty]$ is a nonconstant convex function with $\Phi(0)=0$. Each element of $L^\Phi$ is an equivalence class with respect to almost-sure equality under ${\mathbb{P}}$ of random variables $X:\Omega\to{\mathbb{R}}$ such that $$\|X\|_\Phi := \inf\{\lambda\in(0,\infty) \,; \ {\mathbb{E}}[\Phi(|X|/\lambda)]\leq1\} < \infty.$$ Note that $L^\infty\subset L^\Phi\subset L^1$. The space $L^\Phi$ is partially ordered by the standard almost-sure ordering with respect to ${\mathbb{P}}$ and is equipped with the linear topology induced by the lattice norm $\|\cdot\|_\Phi$. The standard space $L^p$, $p\in[1,\infty)$, corresponds to the choice $\Phi(t)=t^p$. In this case, the function $\Phi$ satisfies the [*$\Delta_2$ condition*]{}, i.e. there exist $s\in(0,\infty)$ and $k\in{\mathbb{R}}$ such that $\Phi(2t)<k\Phi(t)$ for all $t\in[s,\infty)$. Since our underlying probability space is nonatomic, it follows from Theorem 2.1.17 in Edgar and Sucheston [@EdgarSucheston1992] that $\Phi$ satisfies the $\Delta_2$ condition if, and only if, $L^\Phi$ coincides with its Orlicz heart $$H^\Phi := \{X\in L^\Phi \,; \ {\mathbb{E}}[\Phi(|X|/\lambda)]<\infty, \ \lambda\in(0,\infty)\}.$$ The space $L^\infty$ corresponds to the choice $\Phi(t)=0$ for $t\in[0,1]$ and $\Phi(t)=\infty$ otherwise. In this case we have $H^\Phi=\{0\}$. Recall that, if $\Phi$ is finite, then $H^\Phi$ coincides with the closure of $L^\infty$ with respect to the topology induced by $\|\cdot\|_\Phi$ by Theorem 2.1.14 in Edgar and Sucheston [@EdgarSucheston1992].
In this section we assume that prices at time $0$ are represented by a pricing functional $$\pi:L^\Phi\to{\mathbb{R}}\cup\{\infty\}.$$ All the notions recorded in Definitions \[def: frictionless\] and \[def: properties functionals\] will be freely adapted to the space $L^\Phi$. Pricing functionals and risk measures on Orlicz spaces are discussed, for instance, in Biagini and Frittelli [@BiaginiFrittelli2008] and Cheridito and Li [@CheriditoLi2009]. We also refer to the recent contributions by Gao et al. [@GaoLeungMunariXanthos2017] and Liebrich and Svindland [@LiebrichSvindland2017].
We start by showing that, in a general Orlicz space, the basic “collapse to the mean” established in Theorem \[theo: main collapse\] fails to hold.
Assume that $\Phi$ is finite and does not satisfy the $\Delta_2$ condition so that $L^\Phi\neq H^\Phi$ and set (see Theorem 1.2 in Gao et al. [@GaoLeungMunariXanthos2017]) $${\mathcal{A}}= \{X\in L^\Phi \,; \ {\mathbb{E}}[X]\geq0, \ \min(X,0)\in H^\Phi\}.$$ Clearly, the set ${\mathcal{A}}$ is convex, conic, closed, and law invariant. The map $\pi:L^\Phi\to{\mathbb{R}}\cup\{\pm\infty\}$ given by $$\pi(X) = \inf\{m\in{\mathbb{R}}\,; \ X+m\in{\mathcal{A}}\}$$ is therefore sublinear, lower semicontinuous, and law invariant. Note that, since $\pi(0)=0$, lower semicontinuity of $\pi$ implies that $\pi(X)>-\infty$ for every $X\in L^\Phi$ by Proposition 2.4 in Ekeland and Témam [@EkelandTemam1999]. Now, taking any risky $Z\in H^\Phi$ such that ${\mathbb{E}}[Z]\neq0$ it is easy to verify that $$0 = \pi(0) = \pi(Z+(-Z)) \leq \pi(Z)+\pi(-Z) = -{\mathbb{E}}[Z]+{\mathbb{E}}[Z] = 0,$$ where the inequality is due to sublinearity. This shows that $Z$ is frictionless. However, $\pi$ is not a multiple of ${\mathbb{E}}$ because, for instance, ${\mathbb{E}}[X]<\infty=\pi(X)$ for every negative $X\in L^\Phi\setminus H^\Phi$.
In order to obtain the same “collapse to the mean” in a general Orlicz spaces we need to replace lower semicontinuity with a stronger property, namely the so-called Fatou property.
We say that $\pi:L^\Phi\to{\mathbb{R}}\cup\{\infty\}$ has the [*Fatou property*]{}, or is [*order lower semicontinuous*]{}, if $\pi(X)\leq\liminf\pi(X_n)$ for every sequence $(X_n)\subset L^\Phi$ that converges almost surely to $X\in L^\Phi$ and admits $M\in L^\Phi$ such that $|X_n|\leq M$ for every $n\in{\mathbb{N}}$.
The following lemma highlights the link between the Fatou property and lower semicontinuity and shows that, for a convex and law-invariant functional on a general Orlicz space $L^\Phi$, the Fatou property is equivalent to $\sigma(L^\Phi,L^\infty)$ lower semicontinuity.
\[lem: fatou in orlicz spaces\] For a functional $\pi:L^\Phi\to{\mathbb{R}}\cup\{\infty\}$ the following statements hold:
(1) If $\pi$ has the Fatou property, then $\pi$ is lower semicontinuous.
(2) If $\pi$ is convex and law invariant, then the following statements are equivalent:
1. $\pi$ has the Fatou property.
2. $\pi$ is $\sigma(L^\Phi,L^\infty)$ lower semicontinuous.
If $L^\Phi=H^\Phi$ (e.g. if $\Phi$ satisfies the $\Delta_2$ condition) or $L^\Phi=L^\infty$ (or equivalently $\Phi$ takes nonfinite values), the above are also equivalent to:
1. $\pi$ is lower semicontinuous.
[*(1)*]{} Take a sequence $(X_n)\subset L^\Phi$ and $X\in L^\Phi$ such that $X_n\to X$ with respect to $\|\cdot\|_\Phi$. Since we also have $X_n\to X$ with respect to $\|\cdot\|_1$, we can extract a subsequence $(X_{n_k})$ such that $X_{n_k}\to X$ almost surely. Without loss of generality we can assume that $\|X_{n_k}\|_\Phi\leq2^{-k}$ for every $k\in{\mathbb{N}}$ (otherwise we extract a subsequence of $(X_{n_k})$ that does so). As a result, we have $M=\sum_{k\in{\mathbb{N}}}|X_{n_k}|\in L^\Phi$. Moreover, we clearly have that $|X_{n_k}|\leq M$ for every $k\in{\mathbb{N}}$. This implies that $\pi$ is lower semicontinuous whenever it satisfies the Fatou property.
[*(2)*]{} If $L^\Phi=L^\infty$, the equivalence follows from point [*(1)*]{} and by combining Theorem 3.2 in Delbaen [@Delbaen2002] and Proposition 1.1 in Svindland [@Svindland2010]. If $L^\Phi=L^1$, the equivalence between [*(a)*]{} and [*(c)*]{} follows from point [*(1)*]{} and from the dominated convergence theorem, and the equivalence between [*(b)*]{} and [*(c)*]{} follows from a classical result in convex analysis, see e.g. Corollary 5.99 in Aliprantis and Border [@AliprantisBorder2006]. In all the other cases, the equivalence follows from Theorems 1.1 and 1.2 in Gao et al. [@GaoLeungMunariXanthos2017].
Before we are able to extend our main results to a general Orlicz space we need the following additional preliminary lemma.
\[lem: convergence conditional expectations\] For every $X\in L^\Phi$ there exists an increasing sequence $({{\mathcal{F}}}_n(X))$ of finitely-generated $\sigma$-fields over $\Omega$ such that ${\mathbb{E}}[X\,|\,{{\mathcal{F}}}_n(X)]\to X$ with respect to $\sigma(L^\Phi,L^\infty)$.
Fix $X\in L^\Phi$ and take a sequence $({{\mathcal{F}}}_n(X))$ of finitely-generated $\sigma$-fields over $\Omega$ such that the $\sigma$-field generated by $\bigcup{{\mathcal{F}}}_n(X)$ coincides with the $\sigma$-field generated by $X$. Then, we have that ${\mathbb{E}}[X\,|\,{{\mathcal{F}}}_n(X)]\to X$ with respect to $\|\cdot\|_1$ by Lévy’s zero-one law. But this immediately implies that ${\mathbb{E}}[X\,|\,{{\mathcal{F}}}_n(X)]\to X$ also with respect to $\sigma(L^\Phi,L^\infty)$.
The following lemma establishes a quantile representation for law-invariant functionals on Orlicz spaces, which extends the representation recorded in Lemma \[lem: representation via quantile function\].
\[lem: quantile representation orlicz\] Let $\varphi:L^\Phi\to{\mathbb{R}}\cup\{\infty\}$ be a sublinear, law-invariant functional with the Fatou property. Then, there exists ${\mathcal{D}}\subset L^\infty$ such that for every $X\in L^\Phi$ $$\varphi(X) = \sup_{Y\in{\mathcal{D}}}\int_0^1q_X(\alpha)q_Y(\alpha)d\alpha.$$ If $L^\Phi=H^\Phi$ (e.g. if $\Phi$ satisfies the $\Delta_2$ condition) or $L^\Phi=L^\infty$ (or equivalently $\Phi$ takes nonfinite values), the Fatou property can be replaced by lower semicontinuity.
It follows from Lemma \[lem: fatou in orlicz spaces\] that $\pi$ is automatically $\sigma(L^\Phi,L^\infty)$ lower semicontinuous. The above representation in then an immediate consequence of the general result recorded in Proposition \[prop: law invariance and schur convexity\].
In view of the above lemmas, all the results from Sections \[sec: main results\] and \[sec: applications\] can be easily extended from $L^1$ to a general Orlicz space $L^\Phi$ provided that we replace lower semicontinuity by the Fatou property whenever $\Phi$ does not satisfy the $\Delta_2$ condition.
\[theo: collapse orlicz\] Assume that $\pi:L^\Phi\to{\mathbb{R}}\cup\{\infty\}$ is convex, law invariant, and has the Fatou property. Then, the following statements are equivalent:
(a) There exists a strongly frictionless risky payoff $Z\in L^\Phi$ with ${\mathbb{E}}[Z]\neq0$.
(b) There exists $c\in{\mathbb{R}}$ such that $\pi(X)=c{\mathbb{E}}[X]$ for every $X\in L^\Phi$.
(c) Every payoff is frictionless.
If $\pi$ is additionally sublinear, the above are equivalent to:
(a) There exists a frictionless risky payoff $Z\in L^\Phi$ with ${\mathbb{E}}[Z]\neq0$.
If $L^\Phi=H^\Phi$ (e.g. if $\Phi$ satisfies the $\Delta_2$ condition) or $L^\Phi=L^\infty$ (or equivalently $\Phi$ takes nonfinite values), the Fatou property can be replaced by lower semicontinuity.
\[cor: collapse orlicz\] Assume that $\pi:L^\Phi\to{\mathbb{R}}\cup\{\infty\}$ is convex, law invariant, and has the Fatou property. Moreover, assume that every risk-free payoff is frictionless. Then, the following statements are equivalent:
(a) There exist a strongly frictionless risky payoff.
(b) There exists $c\in{\mathbb{R}}$ such that $\pi(X)=c{\mathbb{E}}[X]$ for every $X\in L^\Phi$.
(c) Every payoff is frictionless.
If $\pi$ is additionally sublinear, the above are equivalent to:
(a) There exists a frictionless risky payoff.
If $L^\Phi=H^\Phi$ (e.g. if $\Phi$ satisfies the $\Delta_2$ condition) or $L^\Phi=L^\infty$ (or equivalently $\Phi$ takes nonfinite values), the Fatou property can be replaced by lower semicontinuity.
[*Proof of Theorem \[theo: collapse orlicz\] and Corollary \[cor: collapse orlicz\]*]{}. It follows from Lemma \[lem: fatou in orlicz spaces\] that $\pi$ is $\sigma(L^\Phi,L^\infty)$ lower semicontinuous. In view of Lemma \[lem: convergence conditional expectations\], we can then apply Proposition \[prop: restriction extension\] to uniquely extend $\pi$ to a convex, $\sigma(L^1,L^\infty)$ lower semicontinuous, and law-invariant functional $\overline{\pi}:L^1\to{\mathbb{R}}\cup\{\infty\}$. The desired assertions now follow by applying the results of Section \[sec: main results\] to the functional $\overline{\pi}$. Alternatively, one could start from the representation of $\pi$ in Lemma \[lem: quantile representation orlicz\] and follow the same arguments from Section \[sec: main results\] but replacing $L^1$ with $L^\Phi$.$\qed$
For all $X,Y\in L^\Phi$ we say that $X$ dominates $Y$ in the [*convex order*]{}, written $X\succeq_{cx}Y$, whenever ${\mathbb{E}}[f(X)]\geq{\mathbb{E}}[f(Y)]$ for every convex function $f:{\mathbb{R}}\to{\mathbb{R}}$. A functional $\pi:L^\Phi\to{\mathbb{R}}\cup\{\infty\}$ is said to be [*Schur-convex*]{}, or [*consistent with the convex order*]{}, whenever $\pi(X)\geq\pi(Y)$ for all $X,Y\in L^\Phi$ such that $X\succeq_{cx}Y$. For more information about Schur-convex functionals we refer to Dana [@Dana2005]. It follows from Lemma \[lem: fatou in orlicz spaces\] and Proposition \[prop: law invariance and schur convexity\] that every convex, Schur-convex functional $\pi:L^\Phi\to{\mathbb{R}}\cup\{\infty\}$ with the Fatou property can be represented as $$\varphi(X) = \sup_{Y\in L^\infty}\int_0^1q_X(\alpha)q_Y(\alpha)d\alpha-\pi^\ast(Y)$$ for every $X\in L^\Phi$, where $$\pi^\ast(Y) = \sup_{X\in L^\Phi}\int_0^1q_X(\alpha)q_Y(\alpha)d\alpha-\pi(X)$$ for every $Y\in L^\infty$. If $\pi$ is additionally sublinear, there exists ${\mathcal{D}}\subset L^\infty$ such that $$\varphi(X) = \sup_{Y\in{\mathcal{D}}}\int_0^1q_X(\alpha)q_Y(\alpha)d\alpha$$ for every $X\in L^\Phi$. If $L^\Phi=H^\Phi$ (e.g. if $\Phi$ satisfies the $\Delta_2$ condition) or $L^\Phi=L^\infty$ (or equivalently $\Phi$ takes nonfinite values), the Fatou property can be replaced by lower semicontinuity. The above quantile representation extends the representation in Dana [@Dana2005] and Grechuk and Zabarankin [@GrechukZabarankin2012] beyond the $L^p$ setting and sharpens it by showing that one can always use quantiles of bounded functions.
Results from the literature
===========================
For easy reference and comparability we reproduce here the key results from literature. In Castagnoli et al. [@CastagnoliMaccheroniMarinacci2004] the focus is on pricing functionals given by Choquet integrals on $L^\infty$. Their main result is the following “collapse to the mean”.
\[theo: castagnoli et al\] Consider a map $c:{{\mathcal{F}}}\to[0,1]$ satisfying the following properties:
1. $c(\Omega)=1$ and $c(E)=0$ for every $E\in{{\mathcal{F}}}$ such that ${\mathbb{P}}(E)=0$.
2. $c(E)\leq c(F)$ for all $E,F\in{{\mathcal{F}}}$ such that $E\subset F$.
3. $c(E\cup F)\leq c(E)+c(F)-c(E\cap F)$ for all $E,F\in{{\mathcal{F}}}$.
4. $c(E_n)\to 0$ for every decreasing sequence $(E_n)\subset{{\mathcal{F}}}$ such that $\bigcap E_n=\emptyset$.
Consider the functional $\pi:L^\infty\to{\mathbb{R}}$ defined by $$\pi(X) = \int_{-\infty}^0(c(X>x)-1)dx+\int_0^\infty c(X>x)dx.$$ If $\pi$ is law invariant, then the following statements are equivalent:
(a) There exists a frictionless risky payoff.
(b) $\pi(X)={\mathbb{E}}[X]$ for every $X\in L^\infty$.
The previous result was generalized by Frittelli and Rosazza Gianin [@FrittelliRosazza2005] in the context of risk measures. The reformulation in terms of pricing functionals is as follows.
\[theo: frittelli and rosazza\] Let $\pi:L^\infty\to{\mathbb{R}}$ be convex, nondecreasing, law invariant, and cash additive. Then, the following statements are equivalent:
(a) There exists a strongly frictionless risky payoff.
(b) $\pi(X)={\mathbb{E}}[X]$ for every $X\in L^\infty$.
If $\pi$ is additionally sublinear, the above are equivalent to:
(a) There exists a frictionless risky payoff.
\[rem: automatic continuity under cash additivity\] (i) Note that, by Schmeidler [@Schmeidler1986], the assumptions on $c$ in Theorem \[theo: castagnoli et al\] imply that $\pi$ is sublinear, nondecreasing, and cash additive. Hence, Theorem \[theo: frittelli and rosazza\] is a true generalization of Theorem \[theo: castagnoli et al\].
\(ii) The functional $\pi$ in Theorem \[theo: frittelli and rosazza\] is automatically continuous. This follows from Lemma 4.3 in Föllmer and Schied [@FoellmerSchied2011].
Law-invariant functionals
=========================
We consider a dual pairing $({{\mathcal{X}}},{{\mathcal{X}}}^\ast)$ where ${{\mathcal{X}}}$ and ${{\mathcal{X}}}^\ast$ are law-invariant subspaces of $L^1$ such that ${\mathbb{E}}[XY]$ is finite for all $X\in{{\mathcal{X}}}$ and $Y\in{{\mathcal{X}}}^\ast$. As usual, we denote by $\sigma({{\mathcal{X}}},{{\mathcal{X}}}^\ast)$ the coarsest topology on ${{\mathcal{X}}}$ that ensures the continuity of all the linear functionals $\varphi_Y:{{\mathcal{X}}}\to{\mathbb{R}}$ defined by $\varphi_Y(X)={\mathbb{E}}[XY]$.
Let $\pi:{{\mathcal{X}}}\to{\mathbb{R}}\cup\{\infty\}$. The conjugate of $\pi$ is the functional $\pi^\ast:{{\mathcal{X}}}^\ast\to{\mathbb{R}}\cup\{\pm\infty\}$ defined by $$\pi^\ast(Y) := \sup_{X\in{{\mathcal{X}}}}{\mathbb{E}}[XY]-\pi(X).$$ It follows from classical convex duality that, if $\pi$ is convex and $\sigma({{\mathcal{X}}},{{\mathcal{X}}}^\ast)$ lower semicontinuous, then $$\pi(X) = \sup_{Y\in{{\mathcal{X}}}^\ast}{\mathbb{E}}[XY]-\pi^\ast(Y)$$ for every $X\in{{\mathcal{X}}}$. If $\pi$ is also sublinear, the only finite value that $\pi^\ast$ can take is $0$ and, hence, there exists a set ${\mathcal{D}}\subset{{\mathcal{X}}}^\ast$ such that $$\pi(X) = \sup_{Y\in{\mathcal{D}}}{\mathbb{E}}[XY]$$ for every $X\in{{\mathcal{X}}}$. In what follows we adhere to the notation for quantile functions introduced at the beginning of Section \[sec: main results\]. The following characterization of the so-called convex order in terms of quantiles can be found e.g. in Dana [@Dana2005].
\[lem: characterization convex order\] For all $X,Y\in L^1$ the following statements are equivalent:
(a) $X\succeq_{cx}Y$, i.e. ${\mathbb{E}}[f(X)]\geq{\mathbb{E}}[f(Y)]$ for every convex function $f:{\mathbb{R}}\to{\mathbb{R}}$.
(b) For every nondecreasing function $g:(0,1)\to{\mathbb{R}}$ we have $$\int_0^1 q_X(\alpha)g(\alpha)d\alpha \geq \int_0^1 q_Y(\alpha)g(\alpha)d\alpha.$$
The following Hardy-Littlewood-type result can be found in Chong and Rice [@ChongRice1971]. For convenience, for every $X\in L^1$ we set ${\mathcal{C}}(X):=\{Y\in L^1 \,; \ Y\sim X\}$.
\[lem: chong rice\] For all $X,Y\in L^1$ such that ${\mathbb{E}}[XY]$ is finite we have $$\sup_{X'\in{\mathcal{C}}(X)}{\mathbb{E}}[X'Y] = \sup_{Y'\in{\mathcal{C}}(Y)}{\mathbb{E}}[XY'] = \int_0^1q_X(\alpha)q_Y(\alpha)d\alpha.$$
The next proposition extends to our general setting the quantile representation obtained by Dana [@Dana2005] in $L^\infty$ and by Grechuk and Zabarankin [@GrechukZabarankin2012] in the setting of $L^p$ spaces (for sublinear functionals). The notions of law invariance and Schur-convexity are defined as above.
\[prop: law invariance and schur convexity\] For a convex, $\sigma({{\mathcal{X}}},{{\mathcal{X}}}^\ast)$ lower semicontinuous map $\pi:{{\mathcal{X}}}\to{\mathbb{R}}\cup\{\infty\}$ the following statements are equivalent:
(a) $\pi$ is law invariant.
(b) $\pi$ is Schur-convex.
(c) $\pi^\ast$ is law invariant.
(d) $\pi^\ast$ is Schur-convex.
In any of the above cases, for every $X\in{{\mathcal{X}}}$ we have $$\pi(X) = \sup_{Y\in{{\mathcal{X}}}^\ast}\int_0^1 q_X(\alpha)q_Y(\alpha)d\alpha-\pi^\ast(Y)$$ and, similarly, for every $Y\in{{\mathcal{X}}}^\ast$ we have $$\pi^\ast(Y) = \sup_{X\in{{\mathcal{X}}}}\int_0^1 q_X(\alpha)q_Y(\alpha)d\alpha-\pi(X).$$ If $\pi$ is also sublinear, we find ${\mathcal{D}}\subset{{\mathcal{X}}}^\ast$ such that for every $X\in{{\mathcal{X}}}$ $$\pi(X) = \sup_{Y\in{\mathcal{D}}}\int_0^1 q_X(\alpha)q_Y(\alpha)d\alpha.$$
To establish the equivalence, we clearly only have to prove that [*(a)*]{} and [*(b)*]{} are equivalent and imply [*(c)*]{}. It is immediate to see that [*(b)*]{} implies [*(a)*]{}. To show the converse, assume that $\pi$ is law invariant. Then, it follows from classical convex duality that $$\pi(X)
=
\sup_{X'\in{\mathcal{C}}(X)}\pi(X')
=
\sup_{X'\in{\mathcal{C}}(X)}\sup_{Y\in{{\mathcal{X}}}^\ast}{\mathbb{E}}[X'Y]-\pi^\ast(Y)
=
\sup_{Y\in{{\mathcal{X}}}^\ast}\sup_{X'\in{\mathcal{C}}(X)}{\mathbb{E}}[X'Y]-\pi^\ast(Y)$$ for every $X\in{{\mathcal{X}}}$, where we have used law invariance in the first equality. As a result, we infer from Lemma \[lem: chong rice\] that $$\pi(X) = \sup_{Y\in{{\mathcal{X}}}^\ast}\int_0^1 q_X(\alpha)q_Y(\alpha)d\alpha-\pi^\ast(Y)$$ for every $X\in{{\mathcal{X}}}$. A direct application of Lemma \[lem: characterization convex order\] now shows that $\pi$ is Schur-convex. Hence, [*(a)*]{} implies [*(b)*]{}.
To prove that [*(a)*]{} implies [*(c)*]{}, assume that $\pi$ is law invariant and note that $$\pi^\ast(Y)
=
\sup_{X\in{{\mathcal{X}}}}\sup_{X'\in{\mathcal{C}}(X)}{\mathbb{E}}[X'Y]-\pi(X')
=
\sup_{X\in{{\mathcal{X}}}}\sup_{X'\in{\mathcal{C}}(X)}{\mathbb{E}}[X'Y]-\pi(X)$$ for every $Y\in{{\mathcal{X}}}^\ast$, where we have used law invariance in the second equality. Using Lemma \[lem: chong rice\] we conclude that $$\pi^\ast(Y) = \sup_{X\in{{\mathcal{X}}}}\int_0^1 q_X(\alpha)q_Y(\alpha)d\alpha-\pi(X)$$ for every $Y\in{{\mathcal{X}}}^\ast$. This shows that $\pi^\ast$ is law invariant and proves the desired implication.
We conclude this appendix by highlighting two key results about law-invariant functionals that are convex and $\sigma({{\mathcal{X}}},{{\mathcal{X}}}^\ast)$ lower semicontinuous. On the one side, they are uniquely determined by the values they take on $L^\infty$. On the other side, they can always be extended to convex, (norm) lower semicontinuous, and law-invariant functionals defined on the entire space $L^1$. For this to hold, every element of ${{\mathcal{X}}}$ has to be the limit of a suitable sequence of conditional expectations. Here, for every functional $\pi:{{\mathcal{X}}}\to{\mathbb{R}}\cup\{\infty\}$ and every subset ${{\mathcal{Y}}}\subset{{\mathcal{X}}}$ we denote by $\pi|_{{\mathcal{Y}}}$ the restriction of $\pi$ to ${{\mathcal{Y}}}$.
\[prop: restriction extension\] Assume that $L^\infty\subset{{\mathcal{X}}}$ and that for every $X\in{{\mathcal{X}}}$ there exists an increasing sequence $({{\mathcal{F}}}_n(X))$ of finitely-generated $\sigma$-fields over $\Omega$ such that ${\mathbb{E}}[X\,|\,{{\mathcal{F}}}_n(X)]\to X$ with respect to $\sigma({{\mathcal{X}}},{{\mathcal{X}}}^\ast)$. Then, the following statements hold:
(1) For every convex, $\sigma({{\mathcal{X}}},{{\mathcal{X}}}^\ast)$ lower semicontinuous, law-invariant map $\pi:{{\mathcal{X}}}\to{\mathbb{R}}\cup\{\infty\}$ we have $$\pi(X) = \lim\pi|_{L^\infty}({\mathbb{E}}[X\,|\,{{\mathcal{F}}}_n(X)])$$ for every $X\in{{\mathcal{X}}}$. In particular, $\pi$ is uniquely determined by its restriction to $L^\infty$.
(2) Assume that $L^\infty\subset{{\mathcal{X}}}^\ast$. Then, every convex, $\sigma({{\mathcal{X}}},{{\mathcal{X}}}^\ast)$ lower semicontinuous, law-invariant map $\pi:{{\mathcal{X}}}\to{\mathbb{R}}\cup\{\infty\}$ is $\sigma({{\mathcal{X}}},L^\infty)$ lower semicontinuous and admits a convex, lower semicontinuous, and law-invariant map $\overline{\pi}:L^1\to{\mathbb{R}}\cup\{\infty\}$ satisfying $\overline{\pi}|_{{\mathcal{X}}}=\pi$. In particular, for every $X\in L^1$ we have $$\overline{\pi}(X) = \sup_{Y\in L^\infty}\int_0^1 q_X(\alpha)q_Y(\alpha)d\alpha-\pi^\ast(Y).$$
[*(1)*]{} Take $X\in{{\mathcal{X}}}$. Note that for every $n\in{\mathbb{N}}$ and for every convex function $f:{\mathbb{R}}\to{\mathbb{R}}$ we have $${\mathbb{E}}[f(X)] = {\mathbb{E}}[{\mathbb{E}}[f(X)\,|\,{{\mathcal{F}}}_n(X)]] \geq {\mathbb{E}}[f({\mathbb{E}}[X\,|\,{{\mathcal{F}}}_n(X)])]$$ by Jensen’s inequality, so that $X\succeq_{cx}{\mathbb{E}}[X\,|\,{{\mathcal{F}}}_n(X)]$. Since $\pi$ is Schur-convex by Proposition \[prop: law invariance and schur convexity\], we infer that $\pi(X)\geq\pi({\mathbb{E}}[X\,|\,{{\mathcal{F}}}_n(X)])$ for every $n\in{\mathbb{N}}$ and thus $$\pi(X) \geq \limsup\pi({\mathbb{E}}[X\,|\,{{\mathcal{F}}}_n(X)]).$$ Moreover, $\sigma({{\mathcal{X}}},{{\mathcal{X}}}^\ast)$ lower semicontinuity implies that $$\pi(X) \leq \liminf\pi({\mathbb{E}}[X\,|\,{{\mathcal{F}}}_n(X)]).$$ This yields $\pi(X)=\liminf\pi({\mathbb{E}}[X\,|\,{{\mathcal{F}}}_n(X)])$, proving the desired claim.
[*(2)*]{} The restriction of $\pi$ to $L^\infty$ is clearly convex and law invariant. Moreover, it is lower semicontinuous with respect to $\|\cdot\|_\infty$. This is because for every sequence $(X_n)\subset L^\infty$ and every $X\in L^\infty$ such that $X_n\to X$ with respect to $\|\cdot\|_\infty$ we automatically have $X_n\to X$ with respect to $\sigma({{\mathcal{X}}},{{\mathcal{X}}}^\ast)$. Then, it follows from Proposition 1.1 in Svindland [@Svindland2010] that $\pi|_{L^\infty}$ is even $\sigma(L^\infty,L^\infty)$ lower semicontinuous. In particular, for every $X\in L^\infty$ we have $$\pi(X) = \sup_{Y\in L^\infty}\int_0^1 q_X(\alpha)q_Y(\alpha)d\alpha-\pi^\ast(Y)$$ by Proposition \[prop: law invariance and schur convexity\]. Now, define a map $\overline{\pi}:L^1\to{\mathbb{R}}\cup\{\infty\}$ by setting $$\overline{\pi}(X) = \sup_{Y\in L^\infty}\int_0^1 q_X(\alpha)q_Y(\alpha)d\alpha-\pi^\ast(Y).$$ It is clear that $\overline{\pi}$ is convex, lower semicontinuous, and law invariant. Note that the restriction of $\overline{\pi}$ to ${{\mathcal{X}}}$ is automatically $\sigma({{\mathcal{X}}},{{\mathcal{X}}}^\ast)$ lower semicontinuous because $L^\infty$ is contained in ${{\mathcal{X}}}^\ast$. Since $\overline{\pi}$ and $\pi$ have the same restrictions to $L^\infty$, it follows from point [*(1)*]{} that $\overline{\pi}$ is the unique desired extension of $\pi$. Since, being lower semicontinuous, $\overline{\pi}$ is automatically $\sigma(L^1,L^\infty)$ lower semicontinuous by Corollary 5.99 in Aliprantis and Border [@AliprantisBorder2006], we infer that $\pi$ is $\sigma({{\mathcal{X}}},L^\infty)$ lower semicontinuous.
We conclude this appendix by highlighting an interesting result about law-invariant sets. For every ${\mathcal{A}}\subset{{\mathcal{X}}}$ it is useful to consider the functional $\delta_{\mathcal{A}}:{{\mathcal{X}}}\to{\mathbb{R}}\cup\{\infty\}$ defined by $$\delta_{\mathcal{A}}(X) :=
\begin{cases}
0 & \mbox{if $X\in{\mathcal{A}}$},\\
\infty & \mbox{otherwise}.
\end{cases}$$
\[lem: conditional expectations and law invariance\] Assume that ${\mathcal{A}}\subset{{\mathcal{X}}}$ is convex, $\sigma({{\mathcal{X}}},{{\mathcal{X}}}^\ast)$ closed, and law invariant. Then, for every $X\in{\mathcal{A}}$ and every sub-$\sigma$-field ${{\mathcal{G}}}$ of ${{\mathcal{F}}}$ we have ${\mathbb{E}}[X\,|\,{{\mathcal{G}}}]\in{\mathcal{A}}$.
Note that $\delta_{\mathcal{A}}$ is convex, $\sigma({{\mathcal{X}}},{{\mathcal{X}}}^\ast)$ lower semicontinuous, and law invariant. Then, it follows from Proposition \[prop: law invariance and schur convexity\] that $\delta_{\mathcal{A}}$ is also Schur-convex. Let ${{\mathcal{G}}}$ be a sub-$\sigma$-field of ${{\mathcal{F}}}$ and $X\in{\mathcal{A}}$ and note that ${\mathbb{E}}[f(X)]\geq{\mathbb{E}}[f({\mathbb{E}}[X\,|\,{{\mathcal{G}}}])]$ for every convex function $f:{\mathbb{R}}\to{\mathbb{R}}$ by Jensen’s inequality, so that $X\succeq_{cx}{\mathbb{E}}[X\,|\,{{\mathcal{G}}}]$. It follows from the Schur-convexity of $\delta_{\mathcal{A}}$ that $0=\delta_{\mathcal{A}}(X)\geq\delta_{\mathcal{A}}({\mathbb{E}}[X\,|\,{{\mathcal{G}}}])$, which in turn implies that $\delta_{\mathcal{A}}({\mathbb{E}}[X\,|\,{{\mathcal{G}}}])=0$ and, thus, that ${\mathbb{E}}[X\,|\,{{\mathcal{G}}}]\in{\mathcal{A}}$.
Here, for every set ${\mathcal{A}}\subset{{\mathcal{X}}}$ we denote by ${\mathop{\rm cl}\nolimits}_{\sigma({{\mathcal{X}}},{{\mathcal{X}}}^\ast)}({\mathcal{A}})$ the closure of ${\mathcal{A}}$ with respect to $\sigma({{\mathcal{X}}},{{\mathcal{X}}}^\ast)$ and by ${\mathop{\rm cl}\nolimits}_{\|\cdot\|_1}({\mathcal{A}})$ the closure of ${\mathcal{A}}$ with respect to $\|\cdot\|_1$.
Assume that $L^\infty\subset{{\mathcal{X}}}$ and that for every $X\in{{\mathcal{X}}}$ there exists an increasing sequence $({{\mathcal{F}}}_n(X))$ of finitely-generated $\sigma$-fields over $\Omega$ such that ${\mathbb{E}}[X\,|\,{{\mathcal{F}}}_n(X)]\to X$ with respect to $\sigma({{\mathcal{X}}},{{\mathcal{X}}}^\ast)$. Then, for every convex, $\sigma({{\mathcal{X}}},{{\mathcal{X}}}^\ast)$ closed, and law-invariant set ${\mathcal{A}}\subset{{\mathcal{X}}}$ the following statements hold:
(1) ${\mathcal{A}}={\mathop{\rm cl}\nolimits}_{\sigma({{\mathcal{X}}},{{\mathcal{X}}}^\ast)}({\mathcal{A}}\cap L^\infty)$.
(2) If $L^\infty\subset{{\mathcal{X}}}^\ast$, then ${\mathcal{A}}={\mathop{\rm cl}\nolimits}_{\|\cdot\|_1}({\mathcal{A}})\cap{{\mathcal{X}}}$.
[*(1)*]{} Clearly, we only need to show that ${\mathcal{A}}\subset{\mathop{\rm cl}\nolimits}_{\sigma({{\mathcal{X}}},{{\mathcal{X}}}^\ast)}({\mathcal{A}}\cap L^\infty)$. To this effect, take $X\in{\mathcal{A}}$ and note that ${\mathbb{E}}[X\,|\,{{\mathcal{F}}}_n(X)]\in{\mathcal{A}}\cap L^\infty$ for every $n\in{\mathbb{N}}$ by Lemma \[lem: conditional expectations and law invariance\]. Since ${\mathbb{E}}[X\,|\,{{\mathcal{F}}}_n(X)]\to X$ with respect to $\sigma({{\mathcal{X}}},{{\mathcal{X}}}^\ast)$ by assumption, we conclude that $X\in{\mathop{\rm cl}\nolimits}_{\sigma({{\mathcal{X}}},{{\mathcal{X}}}^\ast)}({\mathcal{A}}\cap L^\infty)$.
[*(2)*]{} Clearly, we only need to show that ${\mathop{\rm cl}\nolimits}_{\|\cdot\|_1}({\mathcal{A}})\cap{{\mathcal{X}}}\subset{\mathcal{A}}$. To this end, take $X\in{{\mathcal{X}}}$ and assume that $X_n\to X$ with respect to $\|\cdot\|_1$ for a suitable sequence $(X_n)\subset{\mathcal{A}}$. Since $\delta_{\mathcal{A}}$ is $\sigma({{\mathcal{X}}},L^\infty)$ lower semicontinuous by Proposition \[prop: restriction extension\], we infer that ${\mathcal{A}}$ is $\sigma({{\mathcal{X}}},L^\infty)$ closed. Since $X_n\to X$ with respect to $\sigma({{\mathcal{X}}},L^\infty)$, we conclude that $X\in{\mathcal{A}}$.
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[^1]: Email: `[email protected]`
[^2]: Email: `[email protected]`
[^3]: Email: `[email protected]`
[^4]: Email: `[email protected]`
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